ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

ISSN 1651-6214ISBN 978-91-554-8923-6urn:nbn:se:uu:diva-2215682014

Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1137

Distance Functions and Their Use in Adaptive Mathematical Morphology

VLADIMIR ĆURIĆ

Dissertation presented at Uppsala University to be publicly examined in 2347,Lägerhyddsvägen 2, Hus 2, Uppsala, Friday, 23 May 2014 at 13:15 for the degree of Doctorof Philosophy. The examination will be conducted in English. Faculty examiner: HuguesTalbot (University Paris-Est - ESIEE).

AbstractĆurić, V. 2014. Distance Functions and Their Use in Adaptive Mathematical Morphology.Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science andTechnology 1137. 88 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-8923-6.

One of the main problems in image analysis is a comparison of different shapes in images. It isoften desirable to determine the extent to which one shape differs from another. This is usuallya difficult task because shapes vary in size, length, contrast, texture, orientation, etc. Shapes canbe described using sets of points, crisp of fuzzy. Hence, distance functions between sets havebeen used for comparing different shapes.

Mathematical morphology is a non-linear theory related to the shape or morphology offeatures in the image, and morphological operators are defined by the interaction between animage and a small set called a structuring element. Although morphological operators have beenextensively used to differentiate shapes by their size, it is not an easy task to differentiate shapeswith respect to other features such as contrast or orientation. One approach for differentiationon these type of features is to use data-dependent structuring elements.

In this thesis, we investigate the usefulness of various distance functions for: (i) shaperegistration and recognition; and (ii) construction of adaptive structuring elements and functions.

We examine existing distance functions between sets, and propose a new one, called theComplement weighted sum of minimal distances, where the contribution of each point to thedistance function is determined by the position of the point within the set. The usefulness of thenew distance function is shown for different image registration and shape recognition problems.Furthermore, we extend the new distance function to fuzzy sets and show its applicability toclassification of fuzzy objects.

We propose two different types of adaptive structuring elements from the salience map ofthe edge strength: (i) the shape of a structuring element is predefined, and its size is determinedfrom the salience map; (ii) the shape and size of a structuring element are dependent on thesalience map. Using this salience map, we also define adaptive structuring functions. We alsopresent the applicability of adaptive mathematical morphology to image regularization. Theconnection between adaptive mathematical morphology and Lasry-Lions regularization of non-smooth functions provides an elegant tool for image regularization.

Keywords: Image analysis, Distance functions, Mathematical morphology, Adaptivemathematical morphology, Image regularization

Vladimir Ćurić, Department of Information Technology, Division of Visual Information andInteraction, Box 337, Uppsala University, SE-751 05 Uppsala, Sweden.

© Vladimir Ćurić 2014

ISSN 1651-6214ISBN 978-91-554-8923-6urn:nbn:se:uu:diva-221568 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-221568)

To my mother Stana and my father ZarkoMami i tati

List of papers

This thesis is based on the following papers, which are referred to inthe text by their Roman numerals.

I Lindblad, J., Curic, V., Sladoje, N. (2009) On set distances andtheir application to image registration, In Proceedings of the 6thInternational Symposium on Image and Signal Processing andAnalysis (ISPA 2009), IEEE, pp. 449–454.

II Curic, V., Lindblad, J., Sladoje, N., Sarve, H., Borgefors, G.(2014) A new set distance and its application to shape registration,Pattern Analysis and Applications, Volume 17, Issue 1, pp.141–152.

III Curic, V., Lindblad, J., Sladoje, N. (2011) Distance Measuresbetween Digital Fuzzy Objects and Their Applicability in ImageProcessing, In Proceedings of the 14th International Workshop onCombinatorial Image Analysis (IWCIA 2011), LNCS–6636, pp.385–397.

IV Curic, V., Luengo Hendriks, C.L., Borgefors, G. (2012) SalienceAdaptive Structuring Elements, IEEE Journal of Selected Topics inSignal Processing, Special Issue on Filtering and Segmentation inMathematical Morphology, Volume 6, Issue 7, pp. 809–819.

V Curic, V., Luengo Hendriks, C.L., (2012) Adaptive StructuringElements Based on Salience Information, In Proceedings of theInternational Conference on Computer Vision and Graphics(ICCVG 2012), LNCS–7594, pp. 321–328.

VI Curic, V., Luengo Hendriks, C.L., (2013) Salience-Based ParabolicStructuring Functions, In Proceedings of the 11th InternationalSymposium on Mathematical Morphology (ISMM 2013),LNCS–7883, pp. 181–192.

VII Curic, V., Angulo, J., Morphological Image Regularization UsingAdaptive Structuring Functions, Manuscript for journalpublication.

VIII Curic, V., Landstrom, A., Thurley, M., Luengo Hendriks, C.L.Adaptive Mathematical Morphology – a Survey of the Field,Accepted to Pattern Recognition Letters

Reprints were made with permission from the publishers.

The author has contributed considerably to method development, im-plementations and writing of Papers II-VIII. The author also contributedto Paper I, but to a lesser extent. The work presented in Papers I-III weredeveloped under close discussions with Joakim Lindblad and NatasaSladoje, who also contributed in writing. The code for multi-modal im-age registration in Paper II was mostly developed by Hamid Sarve, andGunilla Borgefors contributed in writing. The work in Papers IV-VI wasdeveloped in close discussions with Cris L. Luengo Hendriks. The au-thor developed and implemented the methods, and wrote the papers,but with comments and advices from the coauthors. The method inPaper VII was developed in close discussions with Jesus Angulo. Theauthor implemented the method and wrote the paper. Paper VIII waswritten in a close collaboration with Anders Landstrom. The implemen-tations and writing was split between the two. The other coauthorscontributed in discussions and comments.

Related work

In addition to the papers included in this thesis, the author has alsowritten or contributed to the following publications.

1. Lindblad, J., Sladoje, N., Curic, V., Sarve, H., Johansson, C.B.,Borgefors, G. (2009) Improved quantification of bone remodellingby utilizing fuzzy based segmentation, In Proceedings of the 16thScandinavian Conference on Image Analysis (SCIA 2009), LNCS–5575, pp. 750–759.

2. Curic, V., Heilio, M., Krejic, N., Nedeljkov, M., (2010) Mathemati-cal Model for Efficient Water Flow Management, Nonlinear Analysis– Real World Applications, Volume 11, Issue 3, pp. 1600–1612.

3. Curic, V., Lindblad, J., Sladoje, N. (2010) The Sum of minimaldistances as a useful distance measure for image registration, InProceedings of the Swedish Symposium on Image Analysis (SSBA2010), pp. 55–58.

4. Allalou, A., Curic, V., Pardo Martin, C., Yanik, M.F., Wahlby, C.(2011) Approaches for increasing throughput and information con-tent of image-based zebrafish screens, In Proceedings of the SwedishSymposium on Image Analysis (SSBA 2011), pp. 5–8.

5. Curic, V., Luengo Hendriks, C.L., Borgefors G. (2012) Adaptivestructuring elements based on salience distance transform, In Pro-ceedings of the Swedish Symposium on Image Analysis (SSBA2012), pp. 127–130.

6. Gonzalez-Castro, V., Debayle, J., Curic, V. (2014) Pixel Classifica-tion using General Adaptive Neighbourhood-based Features, To ap-pear in Proceedings of the 22th International Conference on Pat-tern Recognition (ICPR 2014), IEEE.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Brief Introduction to Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Distance Functions Between Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Distance Functions Between Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Distance Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.1 Distance Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.2 Gray-Weighted Distance Transform . . . . . . . . . . . . . . . . . . . . . . . 242.4.3 Salience Distance Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Applications to Image Registration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Mathematical Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.1 Basic Morphological Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2 Adjunction Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 Opening and Closing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4 Other morphological operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.5 On the Selection of Structuring Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Adaptive Mathematical Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.1 Adaptivity in Mathematical Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Adjunction Property in Adaptive Mathematical

Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.3 Methods for Input-Adaptive Structuring Elements and

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.4 Salience-based Adaptive Mathematical Morphology . . . . . . . . . 504.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.3 Concluding Remarks and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6 Brief Summaries of the Included Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Summary in Swedish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

1. Introduction

“Essentially, all models are wrong, but some are useful.”George Edward Pelham Box

1.1 MotivationImage analysis deals with extracting relevant information from the datarepresented as digital images. This field started in the 1960’s with theextensive use of computers that evolved in previous few decades. To-day, different image analysis methods are used to solve some difficultproblems that arise in medicine, biology, astronomy, etc. Those meth-ods benefit from using appropriate mathematical models, and thereforeunderlying mathematical theories and tools are highly important.

Various mathematical models can be used in image analysis. For in-stance, images can be considered as matrices, and hence, tools devel-oped in linear algebra can be used. Digital images can be representedin a discrete grid, and methods derived in the field of discrete geom-etry can be a useful tool as well. Similarly, as a discrete structure, animage can be considered as a graph, which is sometimes important forreal applications due to a lower computational cost for a method. Fur-thermore, an image can be considered in a continuous framework byevolving a partial differential equation over time or using Fourier anal-ysis.

A comparison of shapes is a problem that often appears in imageanalysis. This is a difficult task because shapes vary in size, orientation,contrast and other features. Shapes in images can be represented assets, and distance functions between sets give a measure for how similarthe shapes are. These distance functions can be used for various imageanalysis related tasks, such as shape and image registration, and imageretrieval. Hence, it is of interest to study existing distance functionsbetween sets of points, and further develop new ones.

Possibly the first well-defined non-linear theory in image analysis ismathematical morphology, which can be used for image filtering, seg-mentation and measurement and characterization of objects in the im-age. Mathematical morphology began as a technique for studying ran-dom sets with applications in the mining industry. Mathematical mor-phology was first defined for binary images, and then extended to gray-level and color images. Furthermore, it is defined for lattices and graphs

11

as well as in the continuous domain. Mathematical morphology opera-tors (morphological operators) require a structuring element. Typicallythis is defined as a small rigid set. The construction of adaptive structur-ing elements that depend on the content in the image is now a popularand challenging task in mathematical morphology.

In this thesis, we study distance functions and their use for the con-struction of adaptive structuring elements. The main contributions ofthis thesis are:

• We develop new distance functions between set of points and eval-uate their applicability to the task of shape and image registration.

• We propose new methods for constructing adaptive structuring el-ements and adaptive structuring functions (adaptive morphologi-cal operators, in general) that are based on different distance func-tions.

• We present a unified framework for the proper computation ofadaptive morphological operators as well as the applicability ofadaptive mathematical morphology to image regularization.

1.2 Thesis outlineThis thesis consists of six chapters, including the introductory chapter.The next chapter, Distance functions, presents an overview of distancefunctions between sets of points and their applicability to image regis-tration. The third chapter, Mathematical Morphology, offers the readera brief overview on the field of mathematical morphology related tothis thesis. Methods for adaptive structuring elements as well as theo-retical advances and applications of adaptive morphological operatorsare presented in Adaptive Mathematical Morphology. Conclusions andFuture Work, discusses the contributions of the thesis and presents pos-sible extensions and improvements of this work. Finally, the last chapterpresents short summaries of the papers included in the thesis.

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2. Distance Functions

Distance functions have been used for many problems in image anal-ysis and are some of the older tools. They are also used for definingdifferent distance transforms, which is an efficient way to calculate dis-tances between pixels in the image. In this chapter, we consider distancefunctions between sets of points, which can be computed in linear timewith respect to the number of pixels in the image. Their extension todistance functions between fuzzy sets are also presented. We explorethe applicability of distance functions between sets of points to shaperegistration and object matching.

2.1 Brief Introduction to Distance FunctionsLet X be a non-empty set. A function d : X ×X → R is called a metricon X, if for all x, y, z ∈ X the following properties hold [38]:

(i) Nonnegativityd(x, y) ≥ 0,

(ii) Symmetryd(x, y) = d(y, x),

(iii) Reflexivityd(x, x) = 0,

(iv) Triangle inequalityd(x, y) ≤ d(x, z) + d(z, y).

The pair (X,d) is usually called a metric space.Some functions do not satisfy all properties of a metric, i.e., prop-

erties (i)–(iv). For instance, if a function d satisfies nonegativity (i),symmetry (ii) and triangle inequality (iv) it is called a pseudometric,while when satisfies properties (i)–(iii) it is called a semimetric. A func-tion d is a distance function if it satisfies the nonegativity property (i).Various distance functions are defined in the literature, and used fordifferent applications [38]. Statements in the rest of this section thatare not followed by a reference implicitly refer to Deza and Deza [38].

First, we review some well-known distance functions that will be usedlater in this thesis. Let x = (x1, ..., xn) ∈ Rn and y = (y1, ..., yn) ∈ Rn,n ∈ N. The Euclidean metric, and possibly the most used distancefunction, is defined as

d(x, y) =

(

n∑

i=1

(xi − yi)2

)1

2

. (2.1)

13

(a) (b) (c)

Figure 2.1. Open balls (white objects) of the same radius for different distancefunctions: (a) City block; (b) Euclidean; (c) Chessboard.

The Euclidean metric is a special case of Lp, p > 0 metrics defined by

dp(x, y) =

(

n∑

i=1

(xi − yi)p

)1

p

. (2.2)

For p = 1, this metric is called L1 distance or City block distance. Whenp→ +∞, then dp converges to the Chebyshev distance or L∞ distance,defined by

d∞(x, y) = max{|xi − yi| : i = 1, ..., n}. (2.3)

This distance is also known as the Chessboard distance, when one con-siders x, y ∈ Zn.

Given a metric space (X,d), the open ball with the centre x0 andradius r > 0 is defined by

B(x0, r) = {x : d(x0, x) < r}. (2.4)

The open balls for different Lp metrics are shown in Figure 2.1.Let P(a, b) = {a = x1, ..., xn = b} be a path between points a and b in

X, where xi and xi+1, i = 1, ..., n − 1, are adjacent points in the path.The cost of the path P(a, b) is defined by

c(P(a, b)) =n−1∑

i=1

c(xi, xi+1), (2.5)

where the distance function c is the cost to travel between two adjacentpoints in the path, i.e., the value c(xi, xi+1) corresponds to the cost totravel from a point xi to a point xi+1. The length of a shortest path froma point a to a point b is the minimal cost c of all paths between thesetwo points, i.e., the length is defined by

min{c(P(a, b)) : P(a, b) ∈ Π}, (2.6)

where Π denotes the set of all paths between points a and b. This typeof a distance function is usually refer to as path-based distance.

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2.2 Distance Functions Between SetsA number of different distance functions are defined between sets ofpoints [38]. Here, we consider distance functions that are computablein linear time with respect to the number of points in the set, and weinvestigate their applicability to shape registration.

A distance between two sets is often based on the point-to-set dis-tance. Let A and B be two non-empty, closed and bounded subsets ofX. The distance between a point a ∈ X and a set B ⊂ X is defined as

d(a,B) =∧

b∈B

d(a, b), (2.7)

where∧

denotes the infimum. In the rest of the thesis, the symbol∨

denotes the supremum. Note that, the point-to-point distance d(a, b)can be any metric. An element b0 ∈ B is called an element of bestapproximation to a given point a if

d(a, b0) =∧

b∈B

d(a, b), (2.8)

i.e., if the infimum is attained.The earliest distance defined between two sets is possibly the Haus-

dorff metric, here denoted with dH, [49]

dH(A,B) = max

(

∨

a∈A

d(a,B),∨

b∈B

d(b,A)

)

. (2.9)

It is often considered that the Hausdorff metric is introduced in 1914,when this metric was presented in Hausdorff’s famous book [49]. How-ever, the first appearance of a similar distance function, that is nowa-days called the Hausforff metric, can be found in the PhD thesis ofPompeiu in 1905 [76]. Therefore, some authors suggest that this met-ric should be called the Hausdorff–Pompeiu metric or the Pompeiu–Hausdorff metric [9, 11].

The Hausdorff metric is based on an element with the largest distanceto the other set (see Figure 2.2 for 2D binary objects). This implies thatthe Hausdorff metric is highly sensitive to outliers, i.e., this metric de-pends on a few points that are far from the bulk. We show this propertyof the Hausdorff metric in Paper I and Paper II. Nonetheless, this metricis very widely used. An extensive list of disciplines where the Hausdorffmetric has been used was recently listed by Berinde and Pacurar [9].This is a really impressive list of disciplines ranging from mathematicsto biophysics and economics.

Different modifications of the Hausdorff metric have been proposedin the literature to deal with its aforementioned issues with outliers. An

15

Figure 2.2. Illustration of the Hausdorff distance for two sets represented as2D shapes. The Hausdorff distance is based on two maximal point-to-set dis-tances for sets A and B, i.e., the Hausdorff distance is the maximum of the twofollowing values: d(a, b0) =

∨

b∈B

d(a,B) and d(b, a0) =∨

a∈A

d(b, A).

empirical evaluation of different modifications of the Hausdorff metricwas presented by Dubuisson and Jain [41]. One modification of theHausdorff metric, that is the least sensitive to noise in images, is calledthe Modified Hausdorff distance, dMH, and defined as

dMH(A,B) = max

(

1

|A|

∑

a∈A

d(a,B),1

|B|

∑

b∈B

d(b,A)

)

, (2.10)

where |A| and |B| denote the cardinality of the sets A and B, respec-tively. dMH is not a metric, since the triangle inequality is not satis-fied [41]. This distance function is less sensitive to noise due to itsdefinition as the mean of point-to-set distances for all points from bothsets, rather than taking the supremum of those point-to-set distances,which is used in the definition of the Hausdorff metric (2.9).

A distance similar to the Modified Hausdorff distance is the Sum ofminimal distances, dSMD, defined by Eiter and Mannila [42],

dSMD(A,B) =1

2

(

∑

a∈A

d(a,B) +∑

b∈B

d(b,A)

)

. (2.11)

This distance is not a metric, since it does not satisfy the triangle in-equality [42]. As we show in Paper I, this distance function is less sen-sitive to outliers than the Hausdorff metric. This is especially importantfor image analysis tasks such as shape matching and image registration,which will be discussed in Section 2.5.

As can be seen from the definition of the Hausdorff metric (2.9), thisdistance function is dependent on the boundaries of objects (see Fig-ure 2.2). The Chamfer matching distance, dCH, was defined by Borge-

16

fors [16] as

dCH(A,B) =∑

a∈∂A

d(a, ∂B), (2.12)

where ∂A and ∂B are the boundaries of sets A and B, respectively. Thisdistance function is not a metric, since the symmetry property is notsatisfied. Furthermore, this distance function is computationally lessexpensive than previously introduced distance functions, since it usesonly points from the boundaries of the sets. On the other hand, it canbe very sensitive to noise that influences the object boundary, as weshow in Paper I and Paper II.

Apart from the distance functions defined for sets by the point-to-setdistance, they can be defined in some other ways too. Probably theeasiest way to define a distance between two sets is to take the infimumof all the distances between any two of their respective points, i.e.,

dINF(A,B) =∧

{d(x, y) : x ∈ A, y ∈ B}. (2.13)

This distance function is not suitable for comparing sets. For instance,the distance between two sets with non-empty intersection is zero, be-ing independent of the cardinality of the intersection between thesesets.

The symmetric difference between sets A and B, defined by A△B =(A \ B) ∪ (B \ A) provides information about these two sets. The met-ric that corresponds to this set operation, here denoted with dSD, wasdefined by Klette and Rosenfeld [58] as

dSD(A,B) = |A△B|. (2.14)

Its main disadvantage is that it does not take into account informationabout the spatial position of the points in the set, i.e., the position of theshape in the image, which is a specific application that we are lookingfor in this thesis. For instance, the distance between two disjoint binaryobjects is the same, independently how far the objects are, as it dependsonly on cardinality of the objects. Hence, the usefulness of this distancefunction for shape comparison is limited.

The above distance functions between sets (except the one defined by(2.13)) are compared in an empirical study on rigid body registrationof binary shapes in Paper I. In that paper, we introduce the notion of thecomplement distance function d, defined as d(A,B) = d(A,B), whereA and B are complements of sets A and B, respectively, with respect tothe universe X. The complement distance d adds new information to adistance d if and only if d 6= d. For instance, dSD = dSD and dSMD 6= dSMD.We explore different combinations of dSMD and dSMD, which are the twodistance functions with best properties for image registration (among

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(a) (b)

(c) (d)

Figure 2.3. (a) Binary set A; (b) Binary set B; (c) Contribution of every pointfrom sets A and B to dSMD(A,B); (d) Contribution of every point from sets Aand B to dCW(A,B).

the ones compared in Paper I). Our experiments show that it is betterto incorporate complements of the sets on a point level, rather than atthe whole set level.

Following this conclusion, in Paper II, we propose the Complementweighted sum of minimal distances as a weighted version of the Sumof minimal distances, where each point in the set contributes to thedistance function according to its spatial position within the set. Theunderlying idea is to define a distance function in which points that aredeeper inside of the object have higher importance and contribution tothe distance (see Figure 2.3). This assumption is suitable for problemsin image analysis, since points close to the boundary of an object areusually more affected by noise. Therefore, to each point a of a set Ais assigned a weight dependent on its position within the set A, i.e., aweight is calculated as d(a,A). The weighted contribution of all pointsof set A is normalized by the sum of all given weights. The contributionof all points from set B is computed in a similar way. The Comple-ment weighted sum of minimal distances, dCW, between sets A and B isdefined as (Paper II)

dCW(A,B) =1

2

∑

a∈A

d(a,B)d(a,A)

∑

a∈A

d(a,A)+

∑

b∈B

d(b,A)d(b,B)

∑

b∈B

d(b,B)

. (2.15)

18

Figure 2.4. Crisp (left) and fuzzy (right) representation of the same object.Crisp object is obtained from the fuzzy one taking the α−cut for α = 0.5.

This distance is a semimetric because the triangle inequality is not satis-fied. The denominators are non-zero, since for a non-empty set C ⊂ X,it holds that d(c,C) 6= 0, c ∈ C.

2.3 Distance Functions Between Fuzzy SetsBroadly speaking, a fuzzy set is a generalization of a crisp set. An el-ement either belongs to a crisp set, or not, while the belongingness ofan element to a fuzzy set can be partial, and it is described by a valuebetween zero and one. The theory of fuzzy sets is a good mathemati-cal model for gray-level images in general, as well as for more specificimage analysis problems as in fuzzy segmentation, which produces ob-jects with non-sharp boundaries [75]. For instance, a crisp and a fuzzyrepresentation of the same discrete object are shown in Figure 2.4.

Let us briefly introduce fuzzy set theory. A fuzzy set A on a referenceset X, is a set of ordered pairs A = {(x, µA(x)) : x ∈ X}, where µA :X → [0, 1] is the membership function of the fuzzy set A, [120]. Anα−cut of a fuzzy set A, is the set αA = {x ∈ X : µA(x) ≥ α}, α ∈ (0, 1].The height of a fuzzy set A is h(A) = max

x∈XµA(x), while the support of

A is defined as Supp(A) = {x ∈ X : µA(x) > 0}. The complement A ofa fuzzy set A, is A = {(x, 1 − µA(x)) : x ∈ X}.

There exist several ways to define a distance function between twofuzzy sets, and a good overview on this topic was presented by Bloch [12].Two different groups of distance functions between fuzzy sets were con-sidered in that paper. The first group is composed of methods thatare based solely on comparing membership functions of two fuzzy sets(A, µA) and (B, µB), and the best known are Lp, p > 0 distances, de-fined by

d(A,B) =(

∑

x∈X

|µA(x) − µB(x)|p)

1

p

. (2.16)

Several other distances of this type are listed in a more recent pa-per [119].

19

Distance functions that combine membership functions and spatialinformation of the support of fuzzy sets belong to the second group.Taking the spatial information into account, these distance functionsare more suitable for image analysis applications, and such distancefunctions are explored in Paper III. In that paper, we present extensionsof the distance functions between crisp sets of points that have the bestperformance for image registration (Paper I and Paper II), which are theSum of minimal distances, dSMD, and the Complement weighted sum ofminimal distances, dCW.

The most natural way to extend distance functions between crisp setsto distance functions between fuzzy sets is to consider integration overα−cuts [79]. Integration over α−cuts is a general principle for extend-ing properties and relations on crisp sets to corresponding ones on fuzzysets. Using the integration over α−cuts, a distance function betweentwo fuzzy sets A and B is defined as

dα(A,B) =

∫ 1

0

d(αA,αB) dα, (2.17)

where d is a distance function between crisp sets of points.Following the above discussion and using (2.17), the Sum of min-

imal distances, dSMD, and the Complement weighted sum of minimaldistances, dCW, are defined for fuzzy sets as (Paper III)

dαSMD(A,B) =

∫ 1

0

dSMD(αA,αB) dα, (2.18)

dαCW(A,B) =

∫ 1

0

dCW(αA,αB) dα. (2.19)

The main drawback with this approach is that dα(A,B) = ∞ if theheights of the two fuzzy sets are not the same, i.e., h(A) 6= h(B). Thisproblem also occurs when d is the Hausdorff metric in (2.17). Severalmodifications have been proposed to solve this issue [20, 30], but noneof them is widely accepted [21].

Despite that most distance functions between sets rely on the point-to-set distance (Section 2.2), the distances between two fuzzy sets arerarely defined using the point-to-set distance for fuzzy sets. Two def-initions of the point-to-set distance for fuzzy sets were proposed byBloch and Maıtre [13]. The first definition is based on integration overα−cuts, where the distance between a point a and a fuzzy set B is de-fined as

d(a,B) =

∫ 1

0

d(a,αB) dα =

∫ 1

0

∧

b∈αB

d(a, b) dα. (2.20)

20

The second definition is based on weighting the points from the support,Supp(B), of the fuzzy set B with their membership values,

d(a,B) =∧

b∈Supp(B)

d(a, b) · F (µB(b), (2.21)

where F (t) is a decreasing function of t. Note that the point-to-pointdistance d(a, b) is the spatial distance between two points and is inde-pendent on their membership values. A point-to-set distance can alsobe defined using morphology for fuzzy sets, where the value of such adistance is a fuzzy number [13].

When (2.20) is used, the Sum of minimal distances and the Comple-ment weighted sum of minimal distances for fuzzy sets are, respectively,defined as (Paper III)

dpsSMD(A,B) =

1

2

∑

a∈Supp(A)

d(a,B) +∑

b∈Supp(B)

d(b,A)

, (2.22)

dpsCW(A,B) =

1

2

∑

a∈Supp(A)

d(a,B) · d(a,A)

∑

a∈Supp(A)

d(a,A)+

∑

b∈Supp(B)

d(b,A) · d(b,B)

∑

b∈Supp(B)

d(b,B)

.

(2.23)In Paper III, we use only the definition (2.20) of the point-to-set dis-

tance. We tried different functions including F (t) = 1− t, F (t) = 1− t2

and F (t) = e−t for the point-to-set distance defined by (2.21), but wedid not observe any good performance. Furthermore, if a ∈ Supp(B),then for b = a

d(a, b) = 0 ⇒ d(a, b) · F (µ(b)) = 0 ⇒ d(a,B) = 0, (2.24)

which reduces the usefulness of this distance definition for the task ofobject matching. Therefore, we do not use the point-to-set distancedefined by (2.21).

When computing the point-to-set distance (2.20), we assumed thatµ(a) = 1 for all points a, and in this way the membership value of theobserved point in the point-to-set distance is neglected. This member-ship value was recently considered when defining a point-to-set distancefor fuzzy sets [64], by defining

d(a,B) =

∫ µ(a)

0

d(a,αB) dα. (2.25)

Another definition of a point-to-set distance is based on the length ofthe shortest path from a fuzzy point to a fuzzy set. This approach,

21

being based on the shortest paths, becomes more natural since it is con-sistent with the distance computations in images (Section 2.4). TheSum of minimal distances and the Complement weighted sum of min-imal distance were defined using the last two definitions for point-to-set distances for fuzzy sets, and their usefulness for the task of tem-plate matching and object classification was presented by Lindblad andSladoje [64].

2.4 Distance TransformsIn general, an image can be represented by a function f : D ⊂ Rn →Rm, n,m ∈ N. A binary image can be seen as a function f defined asfollows

f(x) =

{

1, x ∈ A,0, x /∈ A,

(2.26)

where A ⊂ D. A gray-level image can be considered as a function fwhen m = 1, while for color or multivalued images m > 1.

The distance transform is an image operator that computes distancesbetween points in the image. A distance transform is usually consideredto be defined for binary images, while for gray-level images one candefine a gray-weighted distance transform or a geodesic distance trans-form. Furthermore, a distance transform can be computed for colorimages, too. In these distance transforms, the underlying point-to-pointdistance determines the distance transform, as will be seen later in thissection.

In this section, we first review the distance transform for binary andgray-level images. Then, we present the salience distance transformthat is used in Chapter 4 and for the methods that are proposed in PaperIV, Paper V and Paper VI. Note that we present distance transforms for2D images, but they can be computed for arbitrary dimensions [14].

2.4.1 Distance Transform

To keep presentation simple, we will present distance transforms fortwo-dimensional digital images. Let (X,d), X ⊂ Z2 be a metric spaceand A ⊂ X. And let f be a binary image defined by (2.26). Then, thedistance transform of an image f is defined as

DT [f ](x) =∧

y∈A

d(x, y), x ∈ X. (2.27)

Note that equation (2.27) corresponds to the point-to-set distance d(x,A)defined by (2.7). A distance transform can be seen as an image, where

22

(a) Mask M1 (b) Mask M2

Figure 2.5. Masks for the computation of a distance transform for 3 × 3 neigh-bourhood.

each pixel in the object has a positive numerical value that correspondsto its distance to the background.

The basic idea, utilized for most distance transforms, is to approxi-mate the global distance by propagation of local steps, i.e., distances be-tween neighbouring points in the image. This approach was presentedby Rosenfeld and Pfaltz [86]. A distance transform is dependent on thepredefined weights (cost) to move from a pixel to the neighbouring one,and it can be computed by propagating local steps using the two-passchamfering algorithm, where the weights are given by two masks [86].An example of masks and weights for 3 × 3 neighbourhood is shownin Figure 2.5. Here, we present the case for 2D images, but a similarapproach can be used for images of higher dimensions [15]. Prior torunning the chamfering algorithm, the elements in the object are set tozero and the elements in the background are set to infinity. Next, theimage is scanned from the upper left corner, first right then down (for-ward scan) using the mask M1 (see Figure 2.5 (a)), followed by a scanfrom the lower right corner (backward scan) using the mask M2 (seeFigure 2.5 (b)).

The distance for a pixel a is computed by propagating distance valuesfrom the neighbouring pixels in the distance map DT. This process isperformed by adding the weight for the local step. The distance in apoint a is the minimum value of itself and the values in the neighbour-hood increased by respective local weights, i.e.,

DT(a) = minb∈M1

(DT(a+ b) + w(b)), (2.28)

where b is a vector from a to pixels in the mask M1, and w(b) is thecorresponding weight given in the mask. The final result is computedusing the second (backward) scan by

DT(a) = minb∈M2

(DT(a+ b) + w(b)). (2.29)

The size of the mask and the weights determine the distance trans-form. For instance, for a mask of size 3×3, the pair w = 〈w1, w2〉 denotes

23

the weights in the mask, where w1 is the weight for edge neighboursand w2 is the weight for vertex neighbours (see Figure 2.5). Althoughit seems natural to use the Euclidean distance for the weights, these arenot the best weights to use since this distance transform overestimateslong distances. Instead, Borgefors [15] recommended to use weights〈0.95509, 1.36930〉 or the integer weights 〈3, 4〉, which have better prop-erties. The use of the weights 〈1, 2〉 results in the City block distance,while weights 〈1, 1〉 yield to the Chessboard distance. It is possible toutilize the same concept by having larger neighbourhoods [106], as wellas for other grids different from Zn [107]. Note that the Euclidean dis-tance transform can be computed in linear time with the respect to thenumber of pixels in the image [70].

2.4.2 Gray-Weighted Distance Transform

The computational process of a distance transform for gray-level im-ages f : X ⊂ Z2 → R is a bit different than for binary images. A 2Dgray-level image can be represented as a surface, denoted here with S,embedded in 3D space, with two spatial coordinates and one coordinatethat represents the gray-level value in the image. A geodesic distancebetween two points (a, f(a)), (b, f(b)) ∈ S, a, b ∈ Z2 is the shortest pathbetween two points along the surface [57]. A distance transform thatcomputes the geodesic distance between points in the image is called atransform on gray-level surfaces [56].

Various different geodesic distances have been considered in the lit-erature [56, 63, 91, 100, 109]. Similarly to the distance transform forbinary images, the geodesic distances are based on the local step (cost)to move from one point to the neighbouring one along the surface S.The cost can be dependent on two incommensurate domains: (i) thespatial positions of pixels; (ii) gray-level values for pixels. For instance,in Section 4.4. we use the following cost c for the construction of adap-tive structuring elements [109]

c(xi, xi+1) = d(xi, xi+1) + σ|f(xi) − f(xi+1)|, (2.30)

where d is a spatial distance between two neighbouring points xi andxi+1 in a path that connects points a and b. The parameter σ > 0 makesthe two domains commensurate. Then, the geodesic distance betweenthe points a and b is computed with (2.6) by taking the minimal cost ofall possible paths between these points.

The main difference between the distance transform and the gray-weighted distance transform is how they can be computed. While thedistance transform can be computed by the two-pass chamfering al-gorithm, the gray-weighted distance transform requires performing a

24

number of iterations until the stability of the distance is obtained. Oneway to compute the shortest path between all points in an image is byusing Dijkstra’s algorithm [39]. Another method for distance compu-tations is the fast marching algorithm [97]. This method is based onpropagating a wave front from a set of pixels by using partial differ-ential equations, in particular the Eikonal equation. In addition, thecomputational cost to compute the gray-weighted distance transform isO(N logN), where N is the number of pixels in the image.

2.4.3 Salience Distance Transform

The classical distance transform uses as an input a binary image thatmight be obtained by thresholding. On the other hand, the saliencedistance transform eliminates the need for binarization, and it incorpo-rates edge attributes such as strength (gradient magnitude), length orcurvature. To keep the presentation simple, we focus on the approachwhere the salience distance transform is computed using the strengthof the edges and without other attributes of the edges.

The underlying assumption of the salience distance transform is thatthe edges are weighted by their importance, where stronger edges havehigher importance, i.e., salience. Several ways of incorporating salienceinto the distance transform have been studied by Rosin and West [88].Three algorithms were proposed in that study, and each one uses thetwo-pass chamfering algorithm.

The first algorithm ([88], Algorithm 1) is the simplest one. The dis-tance transform and the propagated magnitude of the edges are com-puted separately and stored in two images as follows:

b = arg minb∈M1

(DT(a+ b) + w(b)),

DT(a) = DT(a+ b) + w(b),

MP(a) = MP(a+ b) + w(b),

where DT(a) and MP(a) are the classical distance transform and thepropagated magnitude in a point a, respectively. The initialization forthe salience distance transform is similar to the classical distance trans-form, where the elements of the background (non-edge points) are setto infinity. The edge points are set to zero for the distance transform DT,and to the initial edge strength for the propagated magnitude MP. Afterthe forward scan, the backward scan should be performed in a similarmanner. Then, the salience distance transform is computed by simplydividing the classical distance transform with the propagated magni-tude, i.e., SDT(a) = DT(a)/MP(a), for each point in the image. Since

25

(a) (b) (c)

Figure 2.6. Salience distance transform computed with Algorithm 2. (a) Inputimage; (b) Edge strength image; (c) Salience distance transform.

the propagated edge magnitude map MP is discontinuous, the resultedsalience distance transform SDT is also discontinuous.

Similarly to the first algorithm, the authors proposed an algorithm([88], Algorithm 3) that propagates distance as well as magnitude ofthe edges in the same local step of the two-pass chamfering algorithm.The salience distance transform is computed during the propagationprocess by dividing the propagated distance with the propagated edgemagnitude, repeatedly as

b = arg minb∈M1

(DT(a+ b) + w(b)

MP(a+ b)

)

,

DT(a) = DT(a+ b) + w(b),

MP(a) = MP(a+ b) + w(b).

This initialization and the computation process is the same as for theabove described algorithm, i.e., the resulted salience distance transformis computed as SDT(a) = DT(a)/MP(a). This algorithm is the mostcomputationally expensive one, but it produces the salience distancetransform that is continuous.

Rosin and West also proposed an algorithm ([88], Algorithm 2) thattakes the edge strength image as an input (Figure 2.6 (b)), where theedge pixels are initialized with the negative values of their magnitudes.The non-edge pixels are set to infinity, similarly to as in the distancetransform. Then, the salience distance transform is computed with theclassical two-pass chamfering algorithm (Figure 2.6 (c)). The saliencemap produced with this algorithm is continuous.

An alternative algorithm to compute the salience distance transformis to use a threshold decomposition of the input edge image [87]. Forthis algorithm, the salience distance transform is computed as the sumof distance transforms over a threshold decomposition of the edge mag-nitude image; the result of this algorithm is dependent on the number of

26

(a) (b)

Figure 2.7. The salience distance transform with respect to edge strength usingAlgorithm 2, for 1D function. (a) Edges are far from each other; (b) Edges areclose to each other.

thresholds used. The resulting salience distance transform is zero at theedge pixels, and the magnitude becomes irrelevant at those points. Thesame conclusion is valid for Algorithms 1 and 3. However, the saliencedistance transform obtained by Algorithm 2 has non-zero values for theedge pixels, which makes a distinction between edges based on theirsalience.

Being based on the magnitude of edges in the image, the salience dis-tance transform is dependent on the spatial positions of the edge points.If a weak edge is far from a strong one, i.e., one with higher magnitude,then the resulting salience distance transform might be influenced bythe weaker edge (Figure 2.7 (a)). However, if two edges are close toeach other, the weaker one might be overshadowed by the stronger one(Figure 2.7 (b)).

The salience distance transform is a good alternative to gray-weighteddistance transform, taking advantage of easy computations of the clas-sical distance transform by incorporating salience in the chamferingalgorithm. The resulting salience map is continuous (for most of thepresented algorithms), which is a desirable property when constructingadaptive structuring elements (Section 4.4). For this purpose, we usethe second algorithm presented here ([88], Algorithm 2), where theedge strength is considered as salience.

2.5 Applications to Image RegistrationThe process of aligning two images of the same object is far from easy,since images, even those acquired with the same imaging technique, of-ten have large variations in illumination, pose, noise, etc. Moreover, ifobjects are imaged with different imaging modalities, they have differ-ent characteristics. Generally speaking, image registration is the processof finding the best geometric transformation that aligns two images, one

27

called an observed image O, and another a reference image R. The reg-istration is often considered as a process that seeks the transformationthat maximizes a similarity measure between two images [103]. In-stead, the geometric transformation that minimizes a distance functionbetween the images is good because it simplifies the search. Hence,the registration process can be identified by the following minimizationproblem

arg miny

d(O,T (R; y)), (2.31)

where d is a distance function, T is a geometric transformation and y be-longs to the set of all reasonable transformation parameters. The choiceof geometric transformation depends on the application. In the rigidbody case, i.e., when the object is not deformed between the differentimaging occasions, the transformation typically includes rotation andtranslation. On the other hand, if that is not the case, the registrationis considered to be non-rigid, i.e., the process where the objects mightbe deformed between imaging occasions. A rigid-body registration isoften required as the first step in a non-rigid registration problem. Ex-tensive studies on different image registration methods were presentedby Zitova and Flusser [121] and by Hajnal and Hill [47].

One of the key problems in image registration is how to find an ap-propriate distance function, which is the underlying requirement for agood registration algorithm, since even very sophisticated optimizationalgorithms do not have good performance with an inadequate distancefunction. A number of different distance functions have been used forimage registration. For instance, the most commonly used intensity-based distance functions are the sum of squared differences and crosscorrelation; they are useful for registration of images that are acquiredin the same modality [8, 90]. The distance function called mutual in-formation is used for multi-modal image registration [65, 118]. Sim-ilarly, the distance called normalized gradient fields was proposed formulti-modal image registration problems [46]. The Gromov–Hausdorffdistance was recently used for non-rigid shape comparison [24]. Ofthe distance functions presented in Section 2.2, the Hausdorff met-ric [54, 55] and the Chamfer matching distance [26, 93] have beenused for rigid-body registration.

In this thesis, we consider distance functions between sets that can becomputed in linear time (Section 2.2), and evaluate their usefulness forrigid-body shape registration. Desirable properties of a distance func-tion used for this task are:

(1) The distance between the object and a transformed version of theobject monotonically increases with increasing translation and ro-tation of the object.

(2) The distance function has low sensitivity to noise.

28

(a) (b) (c)

Figure 2.8. The Chamfer matching distance does not distinguish betweenpoints of the object and points of the background. (a): Reference image R,(b): Observed image O, (c): Observed image (blue color) and registered im-age (orange color) superimposed. The elements that belong to both shapes areshown in gray.

The first property enforces that an optimization process (2.31) has onlyone minimum, which is the global one. The second property is impor-tant for real world applications, where images are often corrupted bynoise.

The best way to evaluate the distance functions presented in Chapter2.2, especially considering property (1), is to use a greedy search in theregistration process. Therefore, we use this simple optimization algo-rithm since the objective of Paper II is to compare distance functions,and this method emphasizes properties of distances. An ideal situationwould be that the distance function has only one global minimum. Thisdistance behaviour is often impossible in practice, and the algorithmmight get stuck in a local minimum. Therefore, we also evaluate howfar the local minima are from the desirable global one. We performfour different experiments to show the usefulness of the Complementweighted sum of minimal distances to image registration (Paper II).

The first evaluation is conducted on a set of binary shapes with dif-ferent displacements and noise conditions. For most of the experiments,the Complement weighted sum of minimal distances has the best per-formance, followed by the Sum of Minimal distances (see Paper II, Table1). The Hausdorff distance has a low rate of perfect registrations, es-pecially when shapes are degraded by noise. The Chamfer matchingdistance, being based just on object boundaries, is highly influenced bythe boundary noise, and the greedy search usually got stuck in a localminimum not far from the global one (see Paper II, Figure 9). Also, theChamfer matching distance does not differentiate between points of theobject and points of the background, which may lead to a local mini-mum at a position where the registered objects have a large commonboundary while the interiors are not well matched (see Figure 2.8).

29

(a) (b) (c)

Figure 2.9. Multi-modal image registration. (a) Histological slice; (b) Corre-sponding slice in 3D SRµCT volume found by the Complement weighted sumof minimal distances; (c) Visualization of this 3D SRµCT volume.

We note an interesting property of the studied distance functions:the registration is occasionally better for shapes with noise than for thenoise free case, i.e., noise sometimes removes local minima. This prop-erty of distance functions shows their imperfections, and that the selec-tion and construction of a “perfect” distance function for a particulartask is not an easy problem.

In Paper II, we also present three examples of real registration prob-lems in which the Complement sum of minimal distances outperformsother distance functions presented in Chapter 2.2: (i) Recognition ofhandwritten digits by using the nearest neighbour classification algo-rithm. A shape is correctly classified if the reference shape with theminimal distance is the same digit as the observation shapes. (ii) Regis-tration of 2D images extracted from synchrotron radiation micro com-puted tomography (SRµCT) bone implants volume (Figure 2.9 (c)). (iii)Multi-modal registration of the exact location of a 2D histological sliceof a bone implant within a 3D SRµCT volume of the same implant (Fig-ure 2.9).

In Paper III, we introduce the Complement weighted sum of mini-mal distances and Sum of minimal distances for fuzzy sets. We performdistance-based object classification (nearest neighbour rule) for crispand fuzzy discrete object representations of discrete disks and octagons.Note that a fuzzy discrete representation of objects (Figure 2.10 (c) and(f)) often appears visually more similar to the corresponding continu-ous objects (Figure 2.10 (a) and (d)) than a crisp discrete represen-tation at the same resolution (Figure 2.10 (b) and (e)). Fuzzy repre-sentations of disks and octagons were generated using pixel coveragedigitization [98] of continuous crisp disks and octagons, respectively.In the pixel coverage representation, the membership of a pixel is equalto the relative area of the pixel that is covered by the object [98]. Ithas been shown that this representation provides a good framework forhigher precision of estimation of different measures [99]. The corre-

30

(a) (b) (c)

(d) (e) (f)

Figure 2.10. Crisp and fuzzy discrete representations of continuous objects: (a)Continuous disk; (b) Crisp discrete disk; (c) Fuzzy discrete disk; (d) Continu-ous octagon; (e) Crisp discrete octagon; (f) Fuzzy discrete octagon.

sponding crisp object representations are obtained by taking the α−cutat α = 0.5, of the fuzzy object representations (Figure 2.10).

The recognition, i.e., correct classification rate, of the fuzzy discreteobjects using distances between fuzzy sets (Section 2.3) is better thanfor the discrete objects of the same spatial resolution, using the corre-sponding distances between crisp sets. We also show that the perfor-mance of distance functions between fuzzy sets depends on the numberof membership levels used for fuzzy set. Obviously, more membershiplevels provide more information to the distance function, but this willnot necessarily lead to the improvement of a distance function for aparticular task (Paper III).

31

3. Mathematical Morphology

This chapter presents the basics of mathematical morphology, and in-troduces the notation used in the following chapters. Mathematicalmorphology was introduced in mid 1960s by two French scientists,Georges Matheron and Jean Serra, who developed the main conceptsand tools [69, 95, 96]. Since then, mathematical morphology has beenwidely used in image analysis, and it is an integral part of almost allbasic image analysis courses. A number of excellent books exist on thistopic including ones by Soille [102], and Najman and Talbot [74].

Mathematical morphology is based on theories and tools from variousbranches of mathematics. The field began as a technique for analysingbinary images with applications in the mining industry. When math-ematical morphology was introduced, it was based on set-theoreticalnotions. Since then, the field developed to gray-level images using thenotion of umbra transform [105], and later using the notion of com-plete lattices [53, 83]. Nowadays, mathematical morphology is definedon various structures such as multivalued images [6], graphs [50, 115]and manifolds [80], and it can be defined using partial differential equa-tions [23].

In this thesis, we consider mathematical morphology for gray-levelimages, i.e., for functions f : D ⊂ Rn → R. We review basic mor-phological operators that use the set S ⊂ Rn to probe the image understudy. The set S is referred to as structuring element. Furthermore, suchstructuring element is often referred to as flat structuring elements be-cause it has two dimensions in the case of 2D images, i.e., when n = 2.The shape and size of a structuring element are chosen depending onthe application. Initially, structuring elements were fixed, i.e., one struc-turing element is used for every point in the image. In the rest of thethesis, we will refer to such structuring elements as rigid. Since theselection of structuring elements is not always an easy task, it recentlybecame an increased interest to define structuring elements that adaptto structures and orientations in the image. Chapter 4 is devoted to thistopic.

A structuring element is defined with respect to the origin. The ori-gin allows the positioning of the structuring element at a given point.Structuring element at point x ∈ D means that its origin coincides withthe point x. By translating a structuring element to a point x, its originwill coincide with x, i.e., Sx = {b + x : b ∈ S}. It should be stressed

33

(a) S1 (b) S2 (c) S3

Figure 3.1. Different structuring elements in Z2: (a) Square structuring ele-ment of side 5; (b) Discrete disk of radius 3; (c) Structuring element that doesnot contain its origin. The origin of these structuring elements is marked withX.

here that a structuring element does not necessarily contain its origin.In Figure 3.1, we show three different structuring elements in Z2.

Since an n dimensional gray-level image f corresponds to an n + 1dimensional surface, the structures in the image can be investigatedwith mathematical morphology using functions s : D ⊂ Rn → [−∞, 0].These functions can be seen as structuring elements with weights as-signed to each point. These functions are called non-flat structuringelements, or structuring functions. Similarly to structuring elements, itis important to define the origin of a structuring function.

Statements in the rest of this chapter that are not followed by a ref-erence implicitly refer to Soille [102].

3.1 Basic Morphological OperatorsLet L be a set of all gray-level images with the domain D ⊂ Rn, i.e.,L = {f |f : D → R}. Let (L,≤) be a complete lattice with a partialorder “≤” defined by

f ≤ g ⇔ (∀x ∈ D) f(x) ≤ g(x), (3.1)

where f, g ∈ L. A lattice is complete if all subsets have both an infimumand a supremum.

Each structuring element S has one corresponding structuring ele-ment S∗, called reflected or transposed structuring element, that is sim-ply the reflection of S through the origin. That means, S∗ = {−b : b ∈S}. For instance, S1 = S∗

1 , S2 = S∗2 , S3 = S∗

3 (see Figure 3.1) Thetwo basic morphological operators are the erosion εS : L → L and thedilation δS : L → L defined as [95]

[εS(f)](x) =∧

y∈Sx

f(y), x ∈ D, (3.2)

34

(a) Input f (b) Erosion εS2(f) (c) Erosion εS3

(f)

(d) Dilation δS1(f) (e) Dilation δS2

(f) (f) Dilation δS3(f).

Figure 3.2. Morphological operators with different structuring elements thatare shown in Figure 3.1.

[δS(f)](x) =∨

y∈S∗

x

f(y), x ∈ D. (3.3)

We now turn our focus to the non-flat case, i.e. structuring functions.Let sx : D → [−∞, 0] be an arbitrary structuring function with theorigin in x. Some typical examples of structuring functions are linear(cone) function sx(y) = −d(x, y) and quadratic (parabolic) functionsx(y) = −d(x, y)2 = −‖x− y‖2.

The corresponding reflected structuring function is defined as [19]

sx(y) = sy(x), x, y ∈ D. (3.4)

The erosion and the dilation by the structuring function sx are thendefined as [95]

[εs(f)](x) =∧

y∈D

(f(y) − sx(y)) , x ∈ D, (3.5)

[δs(f)](x) =∨

y∈D

(f(y) + sy(x)) , x ∈ D. (3.6)

It should be noted that the flat case is given by expressing a struc-turing element S using a corresponding structuring function s definedas

sx(y) =

{

0, y ∈ Sx,−∞, y /∈ Sx,

(3.7)

35

which, inserted into (3.5) and (3.6), gives (3.2) and (3.3), respectively.In Figure 3.2, we present the erosion and the dilation using structur-

ing elements depicted in Figure 3.1. The erosion shrinks light objects inthe image, and expands black objects (see Figure 3.2 (b) and (c)). Thedilation has the opposite effect (see Figure 3.2 (d), (e) and (f)). Thesetwo morphological operators satisfy several important properties. First,the erosion and the dilation are dual operators, i.e.,

εS(−f) = −δS(f), f ∈ L. (3.8)

Both operators preserve ordering of the functions, i.e., the propertycalled increasingness is satisfied

f ≤ g ⇒(

εS(f) ≤ εS(g) and δS(f) ≤ δS(g))

, f, g ∈ L. (3.9)

Extensivity of the dilation and antiextensivity of the erosion are satisfiedif the structuring element S contains the origin, i.e.,

εS(f) ≤ f ≤ δS(f), f ∈ L. (3.10)

If the origin is not included in the structuring element S, the morpho-logical operators can produce unintuitive result (see Figure 3.2 (c) and(f)).

3.2 Adjunction PropertyTwo operators ε : L → L and δ : L → L form an adjunction (ε, δ) when

δ(f) ≤ g ⇔ f ≤ ε(g), f, g ∈ L. (3.11)

The adjunction property (3.11) is one of the most fundamental notionsin mathematical morphology. This property can be considered as themorphological counterpart of Galois connection [37]. Also, the adjunc-tion property is closely related to the notion of complete lattices.

Erosion is any operator ε : L → L that preserves the infimum, i.e.,

ε

(

∧

i∈I

Xi

)

=∧

i∈I

ε(Xi), Xi ∈ L, (3.12)

where I is an index set. Dilation is any operator δ : L → L that pre-serves the supremum, i.e.,

δ

(

∨

i∈I

Xi

)

=∨

i∈I

δ(Xi), Xi ∈ L. (3.13)

36

(a) Input f (b) Opening γS2(f) (c) Closing φS2

(f)

Figure 3.3. Opening and closing with the discrete disk of radius three as struc-turing element (S2 from Figure 3.1 (b)).

It can be proven that any two operators ε and δ that form an adjunc-tion (ε, δ) are an erosion and a dilation, respectively [83]. Moreover,for every dilation δ there exists one and only one erosion ε that formsan adjunction (ε, δ). Similarly, for every erosion ε there exists a uniquedilation δ that forms an adjunction (ε, δ) [83].

Two different approaches exist to define an erosion and a dilation. Inone approach (presented in Section 3.1), for one operator a structur-ing element must be replaced with its reflected structuring element (see(3.2) and (3.3)), while for the other approach this is not required, i.e.,the same structuring elements is used to compute an erosion and a dila-tion. These approaches differ in the sense that the basic morphologicaloperators, erosion and dilation, satisfy the adjunction property (3.11)when they are defined by the first one, which is not generally true forthe later one.

3.3 Opening and ClosingCombinations of the two basic morphological operations, erosion anddilation, with the same structuring element, lead to two other importantmorphological operators: opening and closing. Opening can be definedby

γS = δS ◦ εS, (3.14)

and closing byφS = εS ◦ δS. (3.15)

Opening removes bright regions in which structuring element S doesnot fit, while closing removes dark regions in which S does not fit. Fig-ure 3.3 (b) and (c)) depicts an example of opening and closing.

Similarly to erosion and dilation, opening and closing are dual oper-ators, i.e.,

γS(−f) = −φS(f), f ∈ L. (3.16)

37

Furthermore, they are idempotent

γS ◦ γS = γS, φS ◦ φS = φS , (3.17)

increasing,

f ≤ g ⇒(

γS(f) ≤ γS(g) and φS(f) ≤ φS(g))

, f, g ∈ L, (3.18)

and opening is antiextensive, i.e.,

γS(f) ≤ f, f ∈ L, (3.19)

and closing is extensive, i.e.,

f ≤ φS(f), f ∈ L. (3.20)

If the origin belong to the structuring element S then the followingordering relation is valid

εS(f) ≤ γS(f) ≤ f ≤ φS(f) ≤ δS(f). (3.21)

Morphological operators defined so far are translation invariant, thatis, the two following operations give the same result: (1) first, a mor-phological operator is applied, and the image is then translated; (2)first, the input image is translated, and the morphological operator(same as in (1)) is applied. Mathematically,

ψS ◦ Tt = Tt ◦ ψS , (3.22)

where ψ is a morphological operator and Tt(C) = {x+ t : x ∈ C}, for aset C.

An operator that is increasing, idempotent, anti-extensive and can bedefined as an erosion followed by a dilation (3.14) is often referred toas morphological opening. If an operator has the same properties, butcannot be written as a unique erosion followed by dilation, then thisoperators is called an algebraic opening. An algebraic opening can bewritten as the supremum of a family of morphological openings as [69]

Γ(f) =∨

i∈I

γSi(f), (3.23)

where {Si : i ∈ I} is a family of structuring elements. Some of alge-braic openings are area opening [116], path opening [52], rank-maxopening [101]. For instance, area opening removes all connected com-ponents with an area that is smaller than a certain number of points.

38

(a) Input f (b) Gradient GS2(f)

Figure 3.4. Morphological gradient with the discrete disk of radius three asstructuring element S2.

3.4 Other morphological operatorsErosion and dilation, can be combined in various ways to form powerfulmorphological operators. Apart from basic morphological operators,other morphological tools were used for a number of image analysistasks. Here, we briefly summarize a few that will be used later in thisthesis.

One of the useful morphological operators, that will be used in Chap-ter 4.5, is the morphological gradient, defined by

GS(f) = δS(f) − εS(f), f ∈ L. (3.24)

Similarly to the gradient magnitudes, the morphological gradient is anoperator that indicates the strength of the edges in the image (see Fig-ure 3.4).

An operator ψ is a morphological filter if and only if ψ is increasingand idempotent. In other words, a filter preserves the order, and thestructures that are preserved by the filter will not be modified by furtheruse of the same filter. It is obvious that morphological closings andopenings are morphological filters.

The granulometry is an approach to compute a size distributions ofgrains, and it is obtained using a family of openings {γλ : λ ≥ 0} withincreasing size of λ. This family of openings is a granulometry if thesemi-group law (absorption property) is satisfied, i.e.,

γµ ◦ γλ = γmax{µ,λ}, (3.25)

or, equivalently, γλ ≤ γµ if 0 ≤ µ ≤ λ. For instance, if we consider theEuclidean space, the family of openings {γλS : λ > 0}, with convex setsas structuring elements {λS : λ > 0} satisfies the semi-group law (3.25)[69].

Alternating sequential filters are morphological filters that are con-structed by a series of openings and closings first with a small size ofstructuring elements and then with an increasing size until a certain

39

size is reached. The white alternating sequential filter that begins withan opening is defined by

(φSn◦ γSn

) ◦ (φSn−1◦ γSn−1

) ◦ ... ◦ (φS1◦ γS1

), (3.26)

On the other hand, the black alternating sequential filter, which startswith a closing, is defined as

(γSn◦ φSn

) ◦ (γSn−1◦ φSn−1

) ◦ ... ◦ (γS1◦ φS1

). (3.27)

Structuring elements should satisfy S1 ⊂ S2 ⊂ ... ⊂ Sn, for both alter-nating sequential filters.

Morphological operators can be connected with distance functions,presented in Chapter 2. The Hausdorff distance (2.9) between two non-empty sets A and B can be written as

dH(A,B) = min{r : A ⊂ δSr(B), B ⊂ δSr

(A)}, (3.28)

where Sr denotes a disk of radius r. Furthermore, it is shown by Stern-berg [104] that the distance transform of a set A can be computed asthe erosion of a function f defined as

f(x) =

{

0, x ∈ A,∞, x /∈ A,

(3.29)

with the cone structuring function sx(y) = −d(x, y). Moreover, theerosion of f with the parabolic structuring function sx(y) = −d(x, y)2

gives the squared distance transform [111].

3.5 On the Selection of Structuring ElementsGenerally speaking, any shape and size can be considered for a structur-ing element, and a structuring element contains its origin or not (Fig-ure 3.1 (c)). Morphological operators presented in this section are com-puted using one rigid structuring element for all points in the image.Those morphological operators and mathematical morphology we willrefer to as classical in the rest of the thesis.

The selection of structuring element depends on the application. Forinstance, circular structuring elements are well suited for images withcircular objects, as well as for any case where isotropic operation isneeded; and line segments are useful for images with a certain orienta-tion. Also, morphological operators with larger structuring elementspreserve only large features in the image, while those with smallerstructuring elements preserve also the finer details in the image.

Despite that morphological operators are a very useful tools in imageanalysis, morphological operators computed with one rigid structuring

40

element cannot differentiate shapes with respect to their contrast ororientation. For instance, classical morphological operators cannot dis-tinguish same-size shapes that have different contrast. Hence, it is ben-eficial to use structuring elements that adapt their size and shape to thelocal features in the image such as contrast, luminance, the magnitudeof the gradient, orientations in the image, or to the local curvature.Such structuring elements are called adaptive.

41

4. Adaptive Mathematical Morphology

4.1 Adaptivity in Mathematical MorphologyIn its original form, morphological operators are: (i) translation invari-ant (3.22); (ii) based on one rigid structuring element that is translatedto every point in the image and does not depend on its position. Hence,two possible ways to generalize mathematical morphology were dis-cussed by Roerdink [82]:

(i) Translation invariance can be replaced by other types of invari-ance; this approach leads to group morphology.

(ii) Rigid structuring elements can be replaced by adaptive structuringelements; this approach leads to adaptive mathematical morphol-ogy, also called adaptive morphology.

The basic morphological operators, defined in the classical way (Chap-ters 3.1 and 3.3), are translation invariant. If other types of invarianceare considered, then structuring elements change their size and shapewith respect to different geometric transformations that generate an al-gebraic structure called a group. Therefore, the name of this field ofmathematical morphology is called group morphology [81]. A compre-hensive overview on group morphology is given by Heijmans [51].

In this thesis, we study adaptive mathematical morphology, whichis a challenging topic that attracted a lot of attention in recent years[17, 19, 33, 60, 62, 72, 82]. Most likely, one of the earliest studiesof adaptive mathematical morphology was presented by Serra [96].Other early work was given by Charif-Chefchaouni and Schonfeld [29],who considered a general theory of adaptive morphology for binaryimages. More recently, comprehensive theoretical results of adaptivemathematical morphology were presented by Bouaynaya et al. [17],and Bouaynaya and Schonfeld [19], which were further explored byRoerdink [82]. Roerdink [82], among the other important aspects ofadaptive mathematical morphology, addressed theoretical issues impor-tant for the development of the field by noticing often overlooked is-sues when defining adaptive morphological operators. This paper byRoerdink [82] can be seen as a cornerstone paper for further develop-ment of adaptive mathematical morphology, since it presents a way toproperly define morphological operators with adaptive structuring ele-ments, i.e., under which conditions the erosion and the dilation com-puted with adaptive structuring elements will form an adjunction.

43

In the following text, we will refer to morphological operators com-puted with adaptive structuring elements as adaptive morphological op-erators. For instance, the erosion computed with adaptive structuringelements will refer to as adaptive erosion.

Adaptive structuring elements can be dependent on various image at-tributes such as spatial position in the image, image content, orientationin the image, etc. To distinguish different types of adaptive structur-ing elements, we follow the terminology introduced by Roerdink [82],who considered two categories of adaptiveness for structuring elements:location-adaptive structuring elements and input adaptive structuringelements.

Location-adaptive structuring elements

These structuring elements are dependent only on the position in theimage domain D, and not on the input image f . In other words, theseadaptive structuring elements are fixed a priori and do not depend onthe image content. These adaptive structuring elements are also calledextrinsic [33]. Most likely, one of the earliest examples of adaptivemathematical morphology was presented by Beucher et al. [10], whodefined adaptive structuring elements that are dependent on the posi-tion in the image. Vehicles are analysed, and the ones at the bottom ofthe image appear larger than the ones at the top, therefore structuringelements vary linearly with their vertical position in the image. Further-more, Verly and Delanoy [114] and Cuisenaire [32] designed adaptivestructuring elements using the similar approach.

If structuring elements are not computed from the input image, likethe case with location-adaptive structuring elements, but rather follow-ing some law regarding the imaging device (e.g. if structuring elementsare proportional to the distance to the imaging device), the adaptivemorphological operators might not be translation invariant. In otherwords, the equation (3.22) is not valid in the general case, i.e.,

ψ ◦ Tt 6= Tt ◦ ψ, (4.1)

where ψ is a morphological operator.

Input-adaptive structuring elements

These structuring elements are dependent on the image content andthereby the structuring elements are also dependent on the location inthe image. These adaptive structuring elements are also called intrin-sic [33], and they are usually computed from a smoothed version of

44

the input image f , called a pilot image [62]. Several interesting meth-ods for input-adaptive structuring elements have been proposed includ-ing morphological amoebas [62], region growing structuring elements[72], general adaptive neighbourhoods [33], elliptical adaptive struc-turing elements [60], salience adaptive structuring elements (Paper IV),etc.

As pointed by Roerdink [82], adaptive morphological operators aretranslation invariant if adaptive structuring elements are computed fromthe input (pilot) image. For instance, the results will be the same if wefirst translate both the input and the pilot image, and then compute theadaptive erosion or if we translate (using the same translation as above)the output of the adaptive erosion.

It should be mentioned here that adaptive morphological operators,i.e., operators that do not process all points in the image identically,can also be defined using other tools. For instance, for morphologicaloperators defined by partial differential equations, adaptivity can beincorporated directly into the partial differential equations underlyingthe basic morphological operators [22, 67].

More details about different approaches in adaptive mathematicalmorphology can be found in an overview paper presented by Maragosand Vachier [68]. In Paper VIII, we present the recent developments ofadaptive mathematical morphology, including latest methods for adap-tive structuring elements as well as proper definitions of adaptive mor-phological operators. We also present an application-oriented study ofdifferent methods for adaptive structuring elements, and briefly discusspossible future directions in which this field might further develop (Pa-per VIII).

4.2 Adjunction Property in Adaptive MathematicalMorphology

To obtain an adjunction (ε, δ) in the classical mathematical morphology,given by (3.11), for the erosion (3.2) and the dilation (3.3) a structur-ing element S used for the erosion is replaced by its reflected structuringelement S∗ for the dilation (Chapter 3.2). On the other hand, this con-struction can be problematic in the context of adaptive morphology andit is often overlooked in the literature, as noted by Roerdink [82].

Let f : D ⊂ Rn → R be an input image and f0 : D ⊂ Rn → R bea pilot image, which is obtained by smoothing the input image f . Let{Sf0 [x] : x ∈ D} be a set of adaptive structuring elements computedfrom a pilot image f0. To each point x ∈ D is associated a uniquestructuring element Sf0 [x]. The adaptive structuring elements derived

45

for different points may be independent one from another, i.e., they maycontain neighbouring points or not.

The adaptive morphological erosion ε : L → L and the correspondingdilation δ : L → L, are defined as [82]

[ε(f)](x) =∧

y∈Sf0 [x]

f(y), x ∈ D, (4.2)

[δ(f)](x) =∨

y∈S∗

f0[x]

f(y), x ∈ D, (4.3)

where S∗f0 [x] is the reflected structuring element of Sf0 [x] defined as

y ∈ Sf0 [x] ⇔ x ∈ S∗f0 [y]. (4.4)

First of all, note that if Sf0 [x] = S for all x ∈ D, i.e., if one considersthe same rigid structuring element for all points in the image, the ero-sion (4.2) and the dilation (4.3) lead to the classical erosion (3.2) andthe classical dilation (3.3). Hence, it is obvious that the adaptive math-ematical morphology is a generalization of the classical mathematicalmorphology.

Adaptive morphological operators ε and δ, defined by (4.2) and (4.3)will form an adjunction (ε, δ), if the structuring elements {Sf0[x] : x ∈D} are computed once from the pilot image f0, and then used to com-pute the adaptive erosion ε and the adaptive dilation δ. Here, we give aproof that ε and δ are adjunct morphological operators [82]:

δ(f) ≤ g ⇔ [δ(f)](x) ≤ g(x), x ∈ D

(equation (4.3)) ⇔∨

y∈S∗

f0[x]

f(y) ≤ g(x), x ∈ D

⇔ f(y) ≤ g(x), y ∈ S∗f0 [x], x ∈ D

(equation (4.4)) ⇔ f(y) ≤ g(x), x ∈ Sf0 [y], y ∈ D

⇔ f(y) ≤∧

x∈Sf0 [y]

g(x), y ∈ D

(equation (4.2)) ⇔ f(y) ≤ [ε(g)](y), y ∈ D

⇔ f ≤ ε(g).

Since the adaptive erosion ε and the adaptive dilation δ, given by(4.2) and (4.3), respectively, form an adjunction (ε, δ), the products ε◦δand δ ◦ ε will lead to operators that satisfy the algebraic properties ofthe closing and opening, respectively. Furthermore, it is now possible toproperly define alternative sequential filters, and other morphologicalfilters.

46

Figure 4.1. Adaptive structuring elements derived for points x, y and z. Sincex ∈ Sf0 [y] then y ∈ S∗

f0 [x]; and since x /∈ Sf0 [z] then z /∈ S∗f0 [x].

In other words, if we first compute the erosion with adaptive struc-turing elements derived from the pilot image, then derive new adaptivestructuring elements from the eroded image and use them to computethe dilation, the resulted operator will not necessarily be an opening,i.e., not necessarily satisfy the algebraic properties of opening (3.17)–(3.19).

Once adaptive structuring elements {Sf0 [x] : x ∈ D} are derivedfrom the pilot image f0, it is not necessary to compute their reflectedstructuring elements {S∗

f0 [x] : x ∈ D} in order to compute the adaptive

erosion and its adjoint dilation. For instance, the adaptive erosion εassigns to the centre x the infimum of f over the structuring elementSf0 [x]. This process may be called centripetal, as it brings to the centrex all gray-level values of their neighbours in Sf0 [x].

We observe that a point x is a part of the reflected structuring el-ement S∗

f0 [y] for any point y within the structuring element Sf0 [x], if

x ∈ Sf0 [y] (see Figure 4.1). Then, the adaptive dilation can be com-puted as follows: The value f(x) is distributed to all points y ∈ Sf0 [x]and contributes to their value of the dilation. In this process, the pointy receives the contribution of all centres x of all structuring elementsSf0 [x] which contain y. This process may be called centrifugal, as itdistributes the value of the central pixels to their neighbours. Note thatif x /∈ Sf0 [z] then z /∈ S∗

f0 [x], and f(z) do not contribute to the dilationin a point x (see Figure 4.1).

The adaptive erosion ε and adaptive dilation δ can be computed asfollows [62]:

Erosion:

[ε(f)](x) = +∞, ∀x ∈ Dfor each point x ∈ D do

compute Sf0 [x]for each point y ∈ Sf0 [x] do

[ε(f)](x) = min{f(y), [ε(f)](x)}

47

end forend for

Dilation:

[δ(f)](x) = −∞, ∀x ∈ Dfor each point x ∈ D do

for each y ∈ Sf0 [x] do[δ(f)](y) = max{f(x), [δ(f)](y)}

end forend for

Let us now consider the case when adaptive structuring functions(non-flat structuring elements) are derived from the pilot image f0,here denoted with {sx

f0 : x ∈ D}. In the theory provided by Bouaynaya

and Schofield [18, 19], non-adaptive morphological operators based onrigid structuring functions (Chapter 3) are directly generalized to theadaptive case. Hence, the erosion and dilation with adaptive structur-ing functions are defined, respectively, as

[ε(f)](x) =∧

y∈D

(f(y) − sxf0(y)), x ∈ D, (4.5)

[δ(f)](x) =∨

y∈D

(f(y) + sy

f0(x)), x ∈ D. (4.6)

Similarly to the case with adaptive structuring elements one has tofix adaptive structuring functions once when they are computed fromthe input (pilot) image. The erosion ε given by (4.5) and the dilationδ given by (4.6) form an adjunction (ε, δ). The proof for this claim isgiven by Bouaynaya and Schofield [19]:

δ(f) ≤ g ⇔ [δ(f)](x) ≤ g(x), x ∈ D

(equation (4.6)) ⇔∨

y∈D

(f(y) + sy

f0(x)) ≤ g(x), x ∈ D

⇔ f(y) + sy

f0(x) ≤ g(x), x, y ∈ D

⇔ f(y) ≤ g(x) − sy

f0(x), x, y ∈ D

(x = y) ⇔ f(x) ≤ g(y) − sxf0(y), y, x ∈ D

(equation (4.5)) ⇔ f(x) ≤∧

y∈D

(g(y) − sxf0(y)), x ∈ D

⇔ f(x) ≤ [ε(g)](x), x ∈ D

⇔ f ≤ ε(g).

48

4.3 Methods for Input-Adaptive StructuringElements and Functions

In this section, we briefly present a number of different methods foradaptive structuring elements and adaptive structuring functions thatexist in the literature. For more details, see provided reference and Pa-per VIII. In general, adaptive structuring elements are defined using dif-ferent features in the image, including gray-level image values, edges,gradients, the Hessian, orientation and connectivity between points.Hence, adaptive structuring elements often rely on a local similarity be-tween neighbouring points or are aligned to edges and contours in theimage. Apart from being defined using different attributes in the image,various constraints can be imposed to the shape and size of structuringelements, ranging from being adaptivity to the image content to beingconstrained by a set of predefined shapes. In this last case, the size andorientation of these structuring elements are adapted to the image.

In Paper VIII, we distinguish the two main groups of methods forthe construction of adaptive structuring elements by being based ondifferent features in the image: (i) local similarity; (ii) local structure.

The methods that belong to the first group are based on a local sim-ilarity between the origin of the structuring element and its neighbour-ing points. A structuring element includes points that are similar to theorigin according to some measure of similarity between points. For in-stance, a structuring element in a point x is defined as a connected com-ponent that contains points with similar gray-level values to f(x) [33].These adaptive structuring elements are called general adaptive neigh-bourhoods [35]. Structuring elements are also defined as balls in ageodesic metric space (often called geodesic balls), where a geodesicdistance can be defined using different image attributes. For example,morphological amoebas are computed using a geodesic distance thattakes spatial distance and gradient into account [62], while Grazziniand Soille [45] utilized the same attributes with different geodesic dis-tances. In the same line, we use a geodesic distance to define salienceadaptive structuring elements, which are derived from the salience mapbased on the edge strength (Paper IV). Furthermore, region growingstructuring elements include points with a region growing techniquethat is based on the difference between gray-level values between theadjacent points [72].

Structure-based methods belong to the second group, where adap-tive structuring elements are aligned to the structure in the image, suchas edges and contours. These methods are characterized by consider-ing orientation in the image, distances to the neighbouring edges andrate of anisotropy. For instance, Tankyevych et al. [108] proposed linestructuring elements where their orientations are obtained from a func-

49

tion that depends on the Hessian. Verdu-Monedero et al. [113] de-fined adaptive structuring elements (lines and rectangles) that are con-strained in width by the distance to the nearby edges, where orienta-tion of structuring elements is obtained using diffusive squared gradientfields. Furthermore, adaptive elliptical structuring elements are derivedusing the structure tensor, where their shape vary from disks to linesdepending on the rate of anisotropy [60].

Adaptive structuring elements can differ in shape and size. For in-stance, shapes of some adaptive structuring elements are completelyadaptive to the content of the image (still dependent on the input pa-rameter that determines the size), such as morphological amoebas [62]and general adaptive neighbourhoods [35], while certain constraintsare assigned for other methods [60, 113]. In addition, adaptive struc-turing elements can be defined to have one shape with adaptive sizes de-termined by the image content as proposed by Dokladal and Dokladalova[40], which used the distance transform to determine the size of rect-angles. Similarly, we use the salience distance transform to determinethe size of adaptive structuring elements, where any shape can be usedfor structuring elements (Paper V).

Adaptive structuring functions are not well explored as adaptive struc-turing elements, and there exists only a few methods for the construc-tion of adaptive structuring functions. Non-flat adaptive structuring el-ements, i.e., adaptive structuring functions are mostly inspired by thewell-known filtering methods, by considering similarity between pointsor between image patches. In this line, spatially variant bilateral struc-turing functions [2] are related to bilateral filtering [110], while non-local structuring functions [92, 112] are directly related to non-localmeans filtering method [25]. Recently, Angulo and Velasco-Forero [5],recently proposed adaptive structuring functions that are based on ran-dom walks, where the step from one point to the other depends on thedistance between points, and different distance functions were consid-ered for this task. In paper VI, we propose parabolic structuring func-tions based on the salience map of the edge strength.

4.4 Salience-based Adaptive MathematicalMorphology

Our methods for constructing adaptive structuring elements (Paper IVand Paper V) as well as for adaptive structuring functions (Paper VI) arebased on the salience map, denoted here with SM, which is a result ofthe salience distance transform (Algorithm 2, Chapter 2.4) applied tothe edge strength of the input image.

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(a) NMS(f) (b) SDT(f)

(c) SDT+(f) (d) SM(f)

Figure 4.2. Computation of the salience map SM(f) for a one dimensionalfunction f (see text).

To preserve most of the edges in the image, we use the gradientestimation and non-maximal suppression from the Canny edge detec-tor [27]. In this process, we use Gaussian derivatives to estimate thegradient in the input image, but exclude the hysteresis thresholdingfrom the Canny edge detector. This approach preserves even the edgeswith a small response in the gradient image. Formally, NMS(f) is theimage obtained by computing the gradient magnitude and non-maximalsuppression of the input image f . The edge pixels are initialized withthe negative values of their salience and the non-edge pixels are set toinfinity [88]. The salience distance transform SDT(f) is computed withthe classical two-pass chamfering algorithm [15] (Figure 4.2 (b)). Af-ter the salience distance transform is propagated from −NMS(f), thedistance image is offset to all positive values and the map SDT+(f) isobtained (Figure 4.2 (c)). By inverting the values of SDT+(f), we ob-tain the salience map SM(f) (Figure 4.2 (d)), which can be formallywritten as

[SM(f)](y) = Offset +∨

x∈D

(

NMS(f)(x)− d(x, y))

, y ∈ D, (4.7)

whereOffset =

∧

y∈D

∨

x∈D

(

NMS(f)(x)− d(x, y))

, (4.8)

and d(x, y) is a spatial distance. For instance, the example of the saliencemap SM(f) for “Le fifre” image is shown in Figure 4.3.

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(a) Input f (b) NMS(f)

(c) SDT+(f) (d) SM(f)

Figure 4.3. Steps for computing the salience map SM for the “Le fifre” image.

The salience map SM contains the information about the spatial dis-tance between points in the image and preserves the information aboutthe salience of the edges in the image, where the largest values in thesalience map SM correspond to the strongest edges in the input image.This salience map can be computed in linear time with respect to thenumber of pixels in the image N , i.e., O(N), since the salience distancetransform can be computed in linear time.

Our main inspiration to use the salience map SM for constructingadaptive structuring elements comes from the approach used for mor-phological amoebas [62]. Morphological amoebas are geodesic balls ina geodesic (gray-weighted) distance metric space, where the distancebetween two adjacent points xi and xi+1 is defined as

ca(xi, xi+1) = 1 + σ|f(xi) − f(xi+1)|, (4.9)

where σ > 0 is a constant used to scale the two incommensurate do-mains. For morphological amoebas, the following definition was alsoused [117]

ca(xi, xi+1) = d(xi, xi+1) + σ|f(xi) − f(xi+1)|, (4.10)

where d(xi, xi+1) is the Euclidean distance or a weighted distance, i.e.,〈3, 4〉 distance [15]. Then, a morphological amoeba Ar(x) centred in a

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(a) (b)

Figure 4.4. Salience adaptive structuring elements for one dimensional func-tion f . (a) Salience adaptive structuring elements for points x1 and x2, withthe same radius r; (b) Salience adaptive structuring elements for points x1 andx2, with with adaptive radii, where r1 < r2.

point x is defined as

Ar(x) = {y ∈ D : minP(x,y)

ca(P(x, y)) < r}, (4.11)

where the cost of the path P(x, y) is computed by using formulas (4.9)or (4.10), and r is the parameter that determines the size.

Morphological amoebas adapt well to the content in the image, i.e.,they are flexible in shape and strictly align to strong edges in the im-age (see Paper VIII, Figure 2). Furthermore, morphological amoebasstretch over the points with similar gray-level values rather then pointsin the neighbourhood, and structuring elements for neighbouring pointsmight have completely different shapes. Our intention is to constructadaptive structuring elements such that structuring elements for twoneighbouring points have similar shapes and not completely differentones. For this purpose, we use the salience map SM(f) that is a contin-uous map, and SM(f) gradually differs for neighbouring points.

Here, we briefly present our methods for adaptive structuring ele-ments and adaptive structuring functions that are based on the saliencemap described above. More details about the methods can be found inPaper IV, Paper V and Paper VI of this thesis.

In Paper IV, we define adaptive structuring elements, called the salienceadaptive structuring elements, that are dependent on the path-baseddistances computed on the salience map SM. We define the cost of thepath between two adjacent points xi and xi+1 as

cs(xi, xi+1) = [SM(f)](xi) + [SM(f)](xi+1), (4.12)

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(a) (b)

(c) (d)

Figure 4.5. Morphological operators with salience adaptive structuring ele-ments, (a) Input image f ; (b) Dilation for r(x) = 5 ·max(SM(f))− [SM(f)](x),x ∈ D; (c) Erosion for r(x) = 5 ·max(SM(f))− [SM(f)](x), x ∈ D; (d) Erosionfor r(x) = 9 · max(SM(f)) − [SM(f)](x), x ∈ D.

and the salience adaptive structuring element centered in a point x isdefined as

Sr(x) = {y ∈ D : minP(x,y)

cs(P(x, y)) < r}. (4.13)

Our main idea is that structuring elements that are close to salient edgesshould be smaller in size. Therefore, structuring elements located closeto edges in the input image are smaller in size, while structuring ele-ments in more homogeneous areas of the input image are larger (seeFigure 4.4 (a)).

We also define that the radii of the salience adaptive structuring ele-ments is adaptive and also dependent on the salience map SM. This isopposite to the other methods for adaptive structuring elements that usea fixed radius for each point in the image [45, 62] (or tolerance [33]).For example, we used r(x) = k · max(SM(f)) − [SM(f)](x), x ∈ D, todefine adaptive radii, where k > 1 is the input parameter. Two salienceadaptive structuring elements with adaptive radii are depicted in Fig-ure 4.4 (b).

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(a) (b) (c)

Figure 4.6. Morphological operators with salience-based rhomboidal (dia-mond) shapes as adaptive structuring elements. (a) Input image; (b) Erosionfor preserving weak edges; (c) Dilation for preserving weak edges.

We present a comparison between morphological amoebas and salien-ce adaptive structuring elements in Paper IV. An application-orientedstudy on different methods for adaptive structuring elements, includingmorphological amoebas [62], general adaptive neighbourhoods [35],adaptive elliptical structuring elements [60], and salience adaptive struc-turing elements (Paper IV), is presented in Paper VIII. In Paper VIII,among the other experiments, we examine shapes of adaptive structur-ing elements of the four aforementioned methods in some typical situa-tions in the image as well as their behaviour when noise is added to theimage (see Paper VIII, Figure 2). As opposed to the traditional morpho-logical operators and the ones compared to in Paper VIII, morphologi-cal operators with salience adaptive structuring elements process samesize objects differently depending on their contrast. For example, theadaptive erosion shrinks dark objects more than light ones, as shown inFigure 4.5 (c) and (d).

The computation cost of the salience adaptive structuring elements isrelatively high O(Nr2 log r2), whereN is the number of pixels in the im-age and r is the radius of salience adaptive structuring elements. This isdue to calculation of path-based distances for every point in the image.Therefore, in Paper V, we propose adaptive structuring elements witha predefined shape, and for which the size is adjusted by the saliencemap SM. These structuring elements can be computed in linear timewith respect to the number of pixels in the image. Our method is basedon the geometric relations between the spatial position of the pointsand their respective values in the salience map. More precisely, we uselocal minima and maxima in the salience map SM to propose two typesof adaptive structuring elements. One type of adaptive structuring el-ements preserves strong edges better than weak ones, while the othertype better preserves weak edges in the image. More details about thismethod can be found in Paper V, Section 3. For instance, the results of

55

(a) (b)

Figure 4.7. Salience-based parabolic structuring functions for 1D function. (a)Two structuring functions centered at edge points. (b) Two structuring func-tions where one is centered at the edge and one is close to the edge.

morphological operators with these structuring elements are depictedin Figure 4.6.

In the two methods for adaptive structuring elements that are brieflydescribed above (more details can be found in Paper IV and Paper V),the size of structuring elements decreases as the salience map SM in-creases and vice versa. In paper VI, we propose salience-based parabolicstructuring functions that are larger where the salience map SM is small-er, i.e., structuring functions are larger in points of weak edges than inpoints with strong edges (Figure 4.7 (a)). Formally, we define a fam-ily of salience-based parabolic structuring functions centered at pointx ∈ D as

sx(y) = −(

α([SM(f)](x), [SM(f)](y)) + β[SM(f)](x)‖x− y‖2)

, y ∈ D,(4.14)

where α([SM(f)](x), [SM(f)](y)) ≥ 0 is a monotonically non-decreasingfunction, β ≥ 0 is a constant. In addition, these structuring functionsare not symmetric and they decrease faster at the side closer to thehigher salience (Figure 4.7 (b)). In other words, the parabola sx3(y)centered in point x3 is skewed away from the edges in the input imagef .

4.5 ApplicationsClassical mathematical morphology has been used to solve various prob-lems in image analysis, such as image filtering and image segmentation.A detailed study of various real world applications of mathematical mor-phology can be found in the book edited by Najman and Talbot [74].

56

(a) (b)

(c) (d)

Figure 4.8. Isolation of the text in a historical document. (a) Color image ofa historical document; (b) Inverted gray-level image f ; (c) Adaptive erosionε(f); (d) Morphological gradient G(f), i.e., the difference between adaptivedilation δ(f) and adaptive erosion ε(f).

Adaptive mathematical morphology, similarly to the classical one,found its use in certain tasks, especially for detecting and enhancingelongated structures in the image. For instance, Morard et al. [72]used their region growing structuring elements to detect cracks. Forthe same application, adaptive elliptical structuring elements were usedby Landstrom and Thurley [60]. Similarly, adaptive dilation with struc-turing elements derived from distance functions based on the structuretensor were used for closing contours and curves [66]. Furthermore,Tankyevych et al. [108] used their adaptive morphological filters basedon the Hessian for enhancing vessels in medical images.

The usefulness of the general adaptive neighbourhoods was shownfor noise reduction and image segmentation [34]. Moreover, Gonzaleset al. [44] used adaptive morphological operators for pixel classifica-tion. The descriptor of each pixel is formed by a concatenation of theoriginal image and successive adaptive erosions and dilations of increas-ing size.

In Paper V, we also present the usefulness of salience-based linearadaptive structuring elements for problems of text recognition. Mor-phological operators with these structuring elements are used to detect

57

the text in a historical document, where the text on the back side of thepaper is highly visible on the front side (see Figure 4.8 (a) and (b)). Weconstruct adaptive structuring elements that preserve weak edges (seePaper V, Section 3 for the more details). These structuring elementsare larger where the text is, and smaller in the background. There-fore, adaptive morphological operators will affect mostly letters on thefront page, (see Figure 4.8 (c) for the adaptive erosion). Note that, forthis application, we used a scaled salience map SM(f) that is obtainedby taking NMS(f)/5 of the input image, which reduces the impact ofstrong edges.

Image Regularization

A mathematical problem is well-posed (in the sense of Hadamard) if thethree following properties are satisfied [94]:

(i) The solution exists.(ii) The solution is unique.

(iii) The solution is stable, in the sense that small perturbations in theequation only lead to small perturbations in the solution.

Otherwise, if one of the requirements (i)-(iii) is not satisfied the prob-lem is ill-posed.

Typical examples for ill-posed problems are inverse problems. In gen-eral, inverse problems refer to problems used to reconstruct data thatare corrupted or processed. The inverse mapping is usually not acces-sible for real world problems, and inverse problems require regulariza-tion, which refers to a process of introducing additional information inorder to solve an ill-posed problem.

In image analysis, inverse problems can be formulated by

f = Ag + b, (4.15)

where A ∈ Rn×n is a linear operator, f ∈ Rn is the input image andg ∈ Rn is the desirable (reconstructed) image and b ∈ Rn is noise.

One way to find an optimal approximation of g is to consider thefollowing regularization problem

g∗ = arg ming∈Rn

(

1

2‖f −Ag‖2 + λJ(g)

)

, (4.16)

where J(g) is a regularization term and λ is a parameter used for bal-ancing between the data and the regularization term. The regulariza-tion term J(g) usually contains information about the edges in the im-age, and it depends on the image derivatives. For instance, the Tikhonovregularization term is defined by L2 norm, i.e., J(g) = ‖g‖, while the

58

total variation regularization is characterized by the gradient [89]

J(g) =n∑

i=1

|∇g(xi)|. (4.17)

Several regularization techniques utilize morphological operators inthe regularization term, and this regularization is referred to as morpho-logical regularization. For instance, Purkait and Chanda [77] proposedmulti-scaled morphological regularization defined as

J(g) =N∑

k=1

(φkB(g) − γkB(g)), (4.18)

where φkB and γkB are morphological closing and opening with disksof increasing size k as structuring elements. The same authors recentlypresented a morphological regularization as [78]

J(g) = φ(g) − γ(g), (4.19)

where φ and γ are adaptive morphological closing and opening, respec-tively.

In Paper VII, we propose a new framework for morphological imageregularization using adaptive structuring functions. Our approach isbased on the Lasry–Lions approximations that rely on infimal-supremalconvolution formulas, and provide Lipschitz regularization of non-smoothfunctions [61]. The approximations belong to the class of functionsC1,1, which means that the approximations are continuously differen-tiable with Lipschitz continuous gradient (see Appendix for more defi-nitions).

Let f : D ⊂ Rn → R ∪ {+∞} be a lower semi-continuous, convex

and bounded function from below. Lasry and Lions [61] defined thetwo following approximations of a function f as

fµ

λ(x) =

∨

z∈D

∧

y∈D

(

f(y) +1

2λd(z, y)2 −

1

2µd(z, x)2

)

, x ∈ D, (4.20)

and

fµ

λ(x) =∧

z∈D

∨

y∈D

(

f(y) −1

2λd(z, y)2 +

1

2µd(z, x)2

)

, x ∈ D. (4.21)

Following definitions (4.20) and (4.21), we have

fµ

λ≤ f ≤ f

µ

λ. (4.22)

Moreover, the approximations fµ

λ, f

µ

λ ∈ C1,1, and they converge point-

wise to f , when 0 < µ < λ and λ→ 0, µ→ 0.

59

Note that, if λ = µ then approximations (4.20) and (4.21) corre-spond to the morphological opening and closing of a function f with thequadratic structuring function sx(y) = −d(x, y)2, respectively. Since therequirement for the Lipschitz regularization is 0 < µ < λ, then we willrefer to the approximations (4.20) and (4.21) as pseudo-opening andpseudo-closing. Therefore, the above discussion implies that the Lasry–Lions regularization of non-smooth functions is a pseudo-morphologicaloperator of the initial function using quadratic structuring elements.

This type of regularization is further extended to non-convex func-tions, and this work is presented by Attouch and Aze [7]. They pre-sented an equivalence of Lasry–Lions approximations without imposingthe convexity constraint on a function f nor the boundedness. Instead,they assume that the function is quadratically minorized, i.e., there ex-ists c ≥ 0 such that

f(x) ≥ −c

2(1 + ‖x‖2), x ∈ D. (4.23)

Then, for all 0 < µ < λ < 1/c the function fµ

λ∈ C1,1 and fµ

λ→ f

converges to f , when µ → 0 and λ → 0. Similar statement is valid forthe approximation f

µ

λ [7].In the above discussion, all approximations are given for the fixed

quadratic structuring functions sx(y) = −d(x, y)2, x, y ∈ D. It is alsopossible to use structuring functions that are different from the quadraticones, and structuring functions that depend on the position in the do-main D [61]. Furthermore, Cepedello-Boiso [28] developed this theoryfor functions in Banach spaces and suggested that structuring functionswith the following properties can be used to obtain the Lipschitz regu-larization:

(i) sx(x) = 0,(ii) sx(y) = sy(x),

(iii) sx(y) is Lipschitz continuous on bounded sets.(iv) sx(y) → −∞, when y → ∞.

Note that there is an indefinite number of possible combinations howto define the structuring function and its finite support. In mathemati-cal morphology, parabolic structuring functions have been used for sev-eral reasons. For instance, morphology with parabolic structuring func-tions is the counterpart of the Gaussian convolution kernel, because itis the only structuring element that is rotationally invariant and separa-ble [111]. In Paper VII, we define adaptive structuring functions as

sx(y) = −d(x, y)p(x), x ∈ D, (4.24)

where the slope p(x) > 1 depends on the image content. These struc-turing functions satisfy the above properties (i)-(iv). We define p(x)

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(a) (b)

(c) (d)

Figure 4.9. Morphological regularization using a (pseudo openclose + pseudocloseopen)/2 filter using morphological amoeba as the finite support for struc-turing functions, and regularization parameters λ = 1, µ = 0.5. (a) Originalimage; (b) Image corrupted by 20% random impulse noise; (c) Regularizationusing quadratic structuring function for all points in the image. (d) Regular-ization using adaptive structuring functions.

as an increasing function of the salience map SM(f), i.e., the slope ofstructuring functions is higher in points with larger salience.

In Figure 4.9, we illustrate the described method using the followingfunction to determine the slope of adaptive structuring functions

p(x) = 1 + 3[SM(f)](x)

max(SM(f)). (4.25)

For the calculations, we use adaptive finite support for structuring func-tions, computed by morphological amoebas. The regularization withadaptive structuring functions produces more smoothing than the cor-responding regularization using quadratic structuring functions for ev-ery point in the image. Moreover, finite support of structuring functionshas significant influence on the morphological regularization presentedhere (Paper VII).

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5. Conclusions and Future Work

This chapter concludes the thesis and presents some directions for fu-ture work.

5.1 ContributionsIn this thesis, we have investigated distance functions and their useful-ness for several problems within image analysis. The major contribu-tions presented in this thesis are:

• We have proposed a new distance function between sets of points,called the Complement weighted sum of minimal distances, andinvestigated its properties in Zn. The applicability of this distancefunction has been presented to image registration. In additionto extensive evaluation with synthetic images, we have shown theusefulness of this distance function for several real world problems(Paper I and Paper II).

• We have proposed two different ways of extending the Comple-ment weighted sum of minimal distances to fuzzy sets. We haveshown that these distance functions can be a useful tool for classi-fying digital fuzzy objects (Paper III).

• We have proposed adaptive structuring elements based on thesalience map computed from the strength of the edges in the im-age. We have introduced two different types of structuring ele-ments:(i) The shape of a structuring element is predefined, and its size

is determined from the salience map (Paper V).(ii) Both the shape and size of a structuring element are depen-

dent on the salience map (Paper IV).• We have proposed parabolic adaptive structuring functions that

are based on the salience map (Paper VI). This method is a gen-eralization of the salience adaptive structuring elements proposedin Paper IV.

• We have proposed a new framework for morphological image reg-ularization using adaptive mathematical morphology (Paper VII).The methods proposed in Paper IV, Paper V and Paper VI can beused for this type of image regularization.

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• We presented an overview of adaptive mathematical morphology,which will be published in a journal special issue on mathematicalmorphology. There, we have presented an application-orientedstudy of various methods for adaptive structuring elements (PaperVIII).

• We have presented a unified framework for proper definition ofadaptive morphological operators, such that adaptive morpholog-ical erosion and dilation form an adjunction (Paper VIII).

In summary, we have proposed tools that can be used for differentproblems in image analysis. The distance functions, we have proposed,can be used for different tasks related to comparing various shapes inimages. In addition, we have proposed salience-based mathematicalmorphology that can be used to distinguish and modify shapes withrespect to their contrast and not just size.

5.2 Future WorkThe methods presented in this thesis can be further improved. Some ofthe possible future research topics related to the work presented in thisthesis are presented here.

Set distances

Like other distance functions between sets of points, the Complementweighted sum of minimal distances can be used for various applications.For instance, it can be used for evaluating different segmentation meth-ods, which is a problem that often appears in image analysis. Some ini-tial results clearly show this potential, and this distance function mightbe a powerful tool for other image analysis problems that include acomparison of shapes.

The Gromov–Hausdorff distance has been used for shape compari-son [24, 71], and this distance function measures how similar shapesare, where shapes are considered as metric spaces. Similarly to theHausdorff distance, the Complement weighted sum of minimal distancescan be extended to this framework.

We have shown that using more membership levels of fuzzy sets doesnot necessarily lead to improving the performance of distance functions.Hence, it would be valuable to determine how the number of member-ship levels of fuzzy sets influences a particular distance function.

Salience adaptive structuring elements

We have only used the magnitude of the edges in the image to com-pute the salience map SM thus far. It would be interesting to construct

64

similar salience maps using other attributes of the edges such as curva-ture or length. Moreover, apart from the salience map computed by thesalience distance transform, other salience maps could be consideredwhen constructing adaptive structuring elements.

The salience map SM can be modified by scaling attributes of theedges in the image. It would be valuable to further study this saliencemap, especially when considering a specific application.

An extension of the salience distance transform has been recentlypresented by Lagerstrom and Buckley [59]. Their method is refer to asattribute weighted distance transform, and it is dependent on differentimage attributes and not only edges. It might be valuable to computeadaptive structuring elements from the salience map that is computedusing an attribute weighted distance transform.

An extension of mathematical morphology to color images is a chal-lenging task, since it depends on ordering vectors [1, 6]. It wouldbe interesting to further extend the salience adaptive structuring ele-ments to color images. Some initial results (not yet published) showthat adaptive structuring elements computed from a salience map forcolor images might have better properties than other similar methodsof adaptive structuring elements for color images [36, 62].

There might be a connection between morphological operators withsalience adaptive structuring elements and the mean curvature equa-tion. This should be explored.

Adaptive mathematical morphology

Adaptive morphological operators are not well-known to the wider im-age analysis community. This might be due to a relatively high compu-tational cost required to compute adaptive structuring elements. Hence,efficient algorithms for computing such structuring elements are re-quired. Some of our initial results show that using GPU implementa-tions, at least for non-local structuring elements, lead to a significantspeed-up [48].

The theory of classical morphological operators is very rich. Possibly,most of the theory remains valid when using adaptive structuring ele-ments. For instance, it would be interesting to consider definitions ofscale-spaces and a granulometry with adaptive structuring elements aswell as other morphological operators.

Morphological regularization based on Lasry–Lions approximationsis a very elegant method for image regularization. Our initial studypresented in Paper VII gives a framework for morphological image reg-ularization using adaptive mathematical morphology. Further studieswill focus on finding appropriate adaptive structuring functions for par-ticular applications.

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Most of the methods for adaptive structuring elements can be con-sidered as special cases of the recently introduced mathematical mor-phology on Riemannian manifolds [4]. This approach seems to be aprominent future research direction when deriving adaptive structuringelements. In addition, it might be useful to define adaptive mathe-matical morphology for other image representations, such as orienta-tion scores [43] and Poincare upper-half plane [3], two frameworks forwhich classical morphological operators are already defined.

The current approach in adaptive mathematical morphology is to de-fine adaptive structuring elements for all points in the image. Eventhough this seems a natural way to define adaptive structuring ele-ments, not all points in the image are equally important and most of thepoints in the image could be processed with the same rigid structuringelement. Therefore, it seems reasonable to derive adaptive structuringelement only for some specific points in the image, possibly the pointsthat are more important according to a predefined criterion.

5.3 Concluding Remarks and PerspectivesBecause many various distance functions exist in the literature, it is notalways easy to select an appropriate distance function for a particularapplication. Furthermore, using existing distance functions might notalways be the best option, when considering a particular problem, andeven small modifications of the existing distance functions can signifi-cantly improve the performance of the distance. Therefore, it might bebetter to adapt existing distance functions to a particular problem orpropose new distance functions than use the old existing ones.

Mathematical morphology found its use in a number of different ap-plications including image filtering and segmentation. Although theneed for adaptive mathematical morphology is obvious, its future prob-ably depends on the applicability of adaptive morphological operatorsto real world problems.

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6. Brief Summaries of the Included Papers

This chapter presents a brief summaries of the papers included in thisthesis.

Paper I:

Distance functions between sets of points are analysed, and their ap-plicability to image registration problems is studied. The notion of thecomplement distance, as a distance function between complements oftwo sets, is introduced. An empirical evaluation of monotonicity of dis-tance functions with respect to translations and rotations of 2D binaryobjects for different noise conditions is given. It is concluded that dis-tance functions based on the contribution of all points from both setshave monotonic behaviour more frequent than Hausdorff-like distancefunctions, which are based on the maximum of point-to-set distancevalues.

Paper II

A novel distance function between sets of points, called the Comple-ment weighted sum of minimal distances, is proposed. For this distancefunction, the contribution of each point is weighted according to theposition within the set that it belongs to. An extensive study on theusefulness of the new distance function for the task of shape registra-tion is performed. Different distance functions between sets of pointsare compared, and it is concluded that the Complement weighted sumof minimal distances has the best performance overall. In addition, itsapplicability to two real problems is given: for the recognition of hand-written text characters and for the multi-modal 2D-3D image registra-tion of bone implants in the surrounding tissues.

Paper III

The Complement weighted sum of minimal distances is extended to thecase of fuzzy sets, using two different approaches. One extension isbased on the α−cut decomposition of fuzzy sets, and the second ap-proach is based on the point-to-set distance for fuzzy sets. The mono-tonicity property of the two novel distance functions with respect to

67

increasing translations and rotations of digital fuzzy objects is analysed,as well as the usefulness of those distance functions for the recogni-tion of different crisp and fuzzy digital objects. It is shown that the useof the proposed distance functions between fuzzy sets for classificationof fuzzy objects leads to improved performance, compared to a corre-sponding classification of binary objects at the same resolution usingcorresponding distance functions between binary objects.

Paper IV

A method for the construction of salience adaptive structuring elementsthat locally adjust their shape and size according to the salience of theedges in the image is proposed. Salience adaptive structuring elementsadapt differently to different contrasts of same-size objects, and adap-tive morphological operators process these objects differently. The com-parison with morphological amoebas is given, and it is concluded thatthe salience adaptive structuring elements have a more compact shape,and are less sensitive to noise than the morphological amoebas.

Paper V

The computational cost for calculating adaptive structuring elements isrelatively high, and often higher than linear with respect to the numberof pixels in the image. Therefore, structuring elements with predefinedshape and adaptive size are proposed. The size of adaptive structuringelements is determined by the salience of the edges in the image, andthey can be computed in linear time. A visual comparison betweenthe new adaptive structuring elements and morphological amoebas isshown. The applicability of the new adaptive morphological operatorsis demonstrated for isolating text in a historical document.

Paper VI

Parabolic structuring functions based on the salience of the edges inthe image are introduced. They are not necessarily symmetric withrespect to the origin, and these adaptive structuring functions can beconsidered as a generalization of the salience adaptive structuring el-ements introduced in Paper IV. Furthermore, morphological operatorswith adaptive flat structuring elements obtained by thresholding thesalience-based parabolic structuring functions are also presented. In ad-dition, the proper way of computing adjunct morphological operators isdiscussed.

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Paper VII

The applicability of adaptive mathematical morphology to the task ofimage regularization is studied. The underlying theory is based onLasry–Lions regularization of non-smooth functions. The Lasry–Lionsregularization is Lipschitz continuous, and it corresponds to morpholog-ical or pseudo-morphological operators with adaptive structuring func-tions. An empirical evaluation on using different regularization param-eters is presented and different adaptive structuring functions are con-sidered. It is concluded that the finite support of structuring functionshas significant impact on this regularization process.

Paper VIII

An overview of adaptive mathematical morphology is presented. Differ-ent aspects of adaptivity in mathematical morphology are considered,and the focus is given to different methods for constructing adaptivestructuring elements. An application-oriented study of four methodsfor adaptive structuring elements is presented, where advantages anddisadvantages of each method are discussed. The proper definitionsof adaptive morphological operators for both structuring elements andstructuring functions are given. Furthermore, various possible direc-tions how the field of adaptive mathematical morphology might evolvein future are discussed.

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Appendix

Norm

Given a vector space V , a norm on V is a function ‖ · ‖ : V → R thatsatisfies the following properties, for all x, y ∈ V and α ∈ C:

1. ‖x‖ ≥ 0,2. ‖x‖ = 0 if and only if x = 0,3. ‖αx‖ = |α|‖x‖,4. ‖x+ y‖ ≤ ‖x‖ + ‖y‖.

Lipschitz continuity

Let (X,dX) and (Y, dY ) be two metric spaces. A function f : X → Y isLipschitz continuous if there exists a constant λ > 0 such that

dY (f(x1), f(x2)) ≤ λdX(x1, x2),

for all x1, x2 ∈ X. A constant λ is referred to as Lipschitz constant.

Banach space

A metric space (X,d) is a Banach space if every Cauchy sequence haslimit in X.

Hilbert space

An inner product 〈·, ·〉 : H ×H → C satisfies the following properties:1. 〈x, y〉 ≥ 0, x ∈ H,2. 〈x, x〉 = 0 if an only if x = 0, x ∈ H,

3. 〈y, x〉 = 〈x, y〉, x, y ∈ H,4. 〈ax1 + bx2, y〉 = a〈x1, y〉 + b〈x2, y〉, x1, x2 ∈ H, a, b ∈ C,5. 〈x, ay1 + by2〉 = a〈x, y1〉 + b〈x, y2〉, y1, y2 ∈ H, a, b ∈ C.A Hilbert space is a vector space H with an inner product 〈x, y〉 such

that the norm defined by

‖x− y‖ =√

〈x− y, x− y〉.

A distance d between two points x, y ∈ H can be defined in terms ofthe norm by

d(x, y) = ‖x− y‖ =√

〈x− y, x− y〉.

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Acknowledgements

First of all, I am very grateful for these five years as a graduate studentat the Uppsala University. My PhD position was funded by the GraduateSchool in Mathematics and Computing and I am grateful for this finan-cial support. I would like to thank all the people at our centre/division.Special thanks to the following people who contributed to the thesis inone way or the other:

• Gunilla Borgefors, my main supervisor, for giving me the opportu-nity to do research in image analysis, for the support during theseyears, for having confidence in me and for encouraging me to workon adaptive mathematical morphology.

• Cris Luengo, my assistant supervisor, for encouraging me to de-velop my own ideas and for great discussions that we had on re-search, mathematical morphology and life in general. It was agreat pleasure learning from you!

• Joakim Lindblad and Natasa Sladoje, my assistant supervisors, forintroducing me to image analysis and for encouraging me to applyfor this PhD position, and for their valuable support in the first twoyears of my PhD studies.

• Jesus Angulo, for being such a good host during my visit at theCentre for Mathematical Morphology, Fontainebleau, France. Forsharing a lot of interesting ideas on mathematical morphology andresearch in general, and for our close collaboration on Paper VII.

• Anders Landstrom and Matthew Thurley, for our excellent collab-oration during these years. I had the great please working withAnders on Paper VIII.

• Researchers in the mathematical morphology community: SebastienLefevre, Johan Debayle, Santiago Velasco-Forero and Victor Gonzales-Castro, for our collaborations and for your influence on my work.

• Ewert Bengtsson, Ingela Nystrom, Lena Nordstrom, and Olle Eriks-son for taking a good care of our division and helping me with alladministrative issues.

• The more permanent staff: Ingrid Carlbom, Ida-Maria Sintorn,Carolina Walhby, Robin Strand, Anders Brun and Filip Malmberg,for all ideas and discussions about research and life in general.

• My officemates during all these years, Erik Wernersson, Cather-ine Ostlund, Jimmy Azar, Fredrik Wahlberg, Kristina Lidayova andOlle Eriksson for all nice discussions and fun we had.

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• Hamid Sarve and Amin Allalou, also known as Braintrust or BT,for such a great fun we had all these years, for all discussions wehad and for everything.

• Milan Gavrilovic and Khalid Niazi for late night working hours andfor fruitful discussions on image analysis and life in general.

• Patrik Malm for being a really good friend and my teacher into theSwedish culture, from surstromming to innebandy.

• Gustaf Kylberg, for helping me whenever I had a problem withplots and graphs.

• Azadeh Fakhrzadeh, for being a great friend, for all fun we hadand for our dreams about the future.

• Bettina Selig, for friendship, fun, support, and for always beingaround.

• Lennart Svenson for being a friend and a great lab partner. Alsofor inviting me to his beautiful home in Vasteras.

• Pontus Olsson, for interesting discussions about the curve of lifeand its derivatives.

• Elisabeth Linner, for knitting during the seminars and making mehappier.

• Alexandra Pacureanu and Christophe Avenel, for bringing Franceto our division.

• The Nysjo brothers, Johan and Fredrik, for being similar and dif-ferent at the same time.

• Andreas Karsnas and Fei Liu, for being good friends despite spend-ing only a few days in Uppsala.

• More recent PhD students at our division: Omer Ishaq, KaylanRam, Sajith Sadanandan Kecheril, for the nice discussions we had.

• My closest Uppsala friends: Katja, Markus, Abhi, Majid, Else, Juan,Camille, Vera, Camij, Pavol, Jonathan, Per, Anders, Fredrik andothers, for social activities that we had all these years.

• My friends from volleyball: Andrea, Eliza, Joanna, Heidi, Gurdip,Marco (el capitano), Justin, Frank, Alex and Dimitris, for havingsuch a great time on the volleyball court. And for being officiallythe best team in Campus 1477!

• Many thanks to my closest friends: Ceca, Sloba, Mica, Mira, Jens,Vuk, Lidia, Djudja, Ceba, Dejan (VC), Kosta-kumic, and others, foryour friendship.

• My dear colleagues from the Faculty of Technical Sciences, Univer-sity of Novi Sad, Serbia, for your support.

• Gunilla Borgefors, Cris Luengo, Natasa Sladoje, Robin Strand, HamidSarve and Johan Nysjo, for proofreading and commenting on draftsof this thesis.

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• My extended family for all support I had during these years. Forall SMS messages, Skype calls and for awaiting me at the NikolaTesla International airport in Belgrade and making my life easier.

• The end of PhD studies can be seen as the end of an educationaljourney that in my case started in 1988 in Klek, a small villagein Serbia. My mother Stana and my father Zarko have been myconstant support during all these years. Mama i tata, hvala vam!Najbolji ste! Volim vas!

• My finance, Iva Lucic, for bringing love into my life. Words cannotdescribe how lucky I am to have you in my life. Thank you foryour unconditional love, for your time and support, and all themoments we shared. I look forward to our lifelong journey. I loveyou! Volim te najvise na svetu, ljubavi moja! ♥

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Summary in Swedish

Denna avhandling behandlar avstandstransformer och dessas anvand-ning inom datoriserad bildanalys. Speciellt behandlas tva delomraden:avstandstransformer mellan punktmangder och deras tillampning pabildregistreringsproblem; och anvandning av avstandstransformer foratt definiera adaptiva strukturelement och strukturfunktioner som i sintur anvands for att konstruera adaptiva morfologiska operatorer.

Ett antal olika avstandsfunktioner har anvants for ett antal olika prob-lemlosningar. Trots att den vetenskapliga litteraturen beskriver mangaolika avstandsfunktioner ar det inte latt att valja den lampligaste av demfor att losa ett specifikt problem. Darfor ar det intressant att studeraolika funktioner och beskriva deras egenskaper, liksom att introduseranya avstandsfunktioner.

Har studerar vi avstandsfunktioner mellan mangder och vi utvarderarderas teoretiska egenskaper och deras anvandbarhet for bildregistre-ringsproblem. Speciellt undersoker vi monotonitet hos avstandsfunktio-nerna under translation och rotation samt bruskanslighet. Forutomexisterande avstandsfunktioner mellan mangder, dar tidsatgangen forberakningen ar en linjar funktion av antalet pixlar i bilden, foreslarvi en ny avstandsfunktion, som vi kallar “komplementet till den vik-tade summan av minimala avstand”. Denna avstandsfunktion mellanmangder lagger hogre vikt vid punkter som ligger djupare inne i ob-jektet och lagre vikt vid punkter som ar narmare objektets kant. Dettaoverensstammer val med situationen i olika tillampningar, dar det oftastar punkter nara objektens kanter som mest paverkas av brus.

Vi visar att komplementet till den viktade summan av minimala av-stand ar overlagen vid bildregistrering och matchning av 2D-objekt narvi jamfor den med andra liknande avstandsfunktioner. Vi visar dessanvandbarhet for igenkanning av handskrivna symboler och for multi-modal registrering fran 2D till 3D: det tva dimensionella histologiskasnittet genom ett benimplantat med kringliggande vavnad registrerastill motsvarande lage i en tredimensionell SRµCT-volym av samma im-plantat. Dessutom utvidgar vi komplementet till den viktade summanav minimala avstand till ett avstand mellan oskarpa mangder och vivisar hur anvandbart detta ar vid avstandsbaserad klassifikation.

Den andra delen av avhandlingen fokuserar pa avstandstransformerfor adaptiv matematisk morfologi. Matematisk morfologi utgor ett kraft-fullt koncept for icke-linjar bildbehandling, och skapar darmed varde-

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fulla verktyg for manga uppgifter, sasom bildfiltrering, bildsegmente-ring, formjamforelse, mm. Det vanligaste sattet att definiera morfolo-giska operatorer ar att anvanda begreppet strukturelement. Strukturele-ment ar vanligtvis sma punktmangder som anvands for att undersokabilder. Resultatet av den morfologiska operationen beror pa interaktio-nen mellan bilden och strukturelementet.

Adaptiv matematisk morfologi har nyligen blivit ett populart forsk-ningsomrade inom matematisk morfologi. Den underliggande iden aratt definiera morfologiska operatorer som inte behandlar alla punk-ter i bilden pa samma satt, utan tar hansyn till hur viktig varje punktar. Darfor definieras adaptiva morfologiska operatorer genom adap-tiva strukturelement som anpassas till lokala forhallanden i bilden. Forvarje punkt i bilden kan det adaptiva strukturelementet beraknas medhjalp av olika bildegenskaper, till exempel likhet mellan granivaerna forpunktgrannar, gradienter och orientering.

Vi skapar adaptiva strukturelement som baseras pa en signifikans-karta som i sin tur baseras pa styrkan hos kanterna i bilden, det villsaga pa hur viktig varje kantpunkt ar. En sadan signifikanskarta ska-pas genom att berakna en avstandstransform som tar hansyn till badeavstand mellan punkter och styrkan hos bildens kanter. Egenskapernasprids over bilden med hyvlingsalgoritmen.

Tva olika typer av adaptiva strukturelement har definierats. I detforsta fallet ar strukturelements form forbestamd och endast dess stor-lek bestams av signifikanskartan. I det andra fallet beror bade struk-turelementets form och storlek pa signifikanskartan. Dessutom definie-rar vi adaptiva strukturfunktioner, som ar strukturelement dar varjepunkt har en vikt mellan minus oandligheten och noll. Dessa adap-tiva strukturfunktioner baseras ocksa pa signifikanskartan och utgor engeneralisering av de adaptiva strukturfunktioner vi forst introducerade.De nya adaptiva strukturfunktionerna ar icke-symmetriska paraboliskafunktioner.

Den har avhandlingen visar aven pa anvandbarheten av adaptiv ma-tematisk morfologi for bildregularisering. Samspelet mellan adaptivmatematisk morfologi och Lipschitz-regularisering av Lasry–Lions-typfor icke-slata funktioner skapar ett elegant verktyg for morfologisk bil-dregularisering. Med andra ord anvander denna typ av bildregulariser-ing adaptiva morfologiska operatorer fr att skapa Lipschitz-regulariseringav bilder som innehaller brus. Avhandlingen innehaller aven en over-sikt av adaptiv matematisk morfologi, tillsammans med en jamforelse avolika metoder for att skapa adaptiva strukturelement och avslutas medolika mojliga framtida forskningsinriktningar inom adaptiv matematiskmorfologi.

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Acta Universitatis Upsaliensis Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1137

Editor: The Dean of the Faculty of Science and Technology

A doctoral dissertation from the Faculty of Science and Technology, Uppsala University, is usually a summary of a number of papers. A few copies of the complete dissertation are kept at major Swedish research libraries, while the summary alone is distributed internationally through the series Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology. (Prior to January, 2005, the series was published under the title “Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology”.)

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