MATHEMATICAL MORPHOLOGY

I. INTRODUCTION

II. BINARY MORPHOLOGY

III. GREY-LEVEL MORPHOLOGY

INTRODUCTION

Mathematical morphology

• Self-sufficient framework for image processing and analysis, created at the École des Mines (Fontainebleau) in 70’s by Jean Serra, Georges Mathéron, from studies in science materials

• Conceptually simple operations combined to define others more and more complex and powerful

• Simple because operations often have geometrical meaning

• Powerful for image analysis

INTRODUCTION

• Binary and grey-level images seen as sets

X

Xc

f (x,y)

X = { (x, y, z) , z f (x,y) }

• Operations defined as interaction of images with a special set, the structuring element

INTRODUCTION

MATHEMATICAL MORPHOLOGY

I. INTRODUCTION

II. BINARY MORPHOLOGY

III. GREY-LEVEL MORPHOLOGY

BINARY MORPHOLOGY

1. Erosion and dilation

2. Common structuring elements

3. Opening, closing

4. Properties

5. Hit-or-miss

6. Thinning, thickenning

7. Other useful transforms :

i. Contour

ii. Convex-hull

iii. Skeleton

iv. Geodesic influence zones

X

B

BINARY MORPHOLOGY

No necessarily compactnor filled

A special set :the structuring element

-2 -1 0 1 2

-2 -1 0 1 2

Origin at center in this case, but not necessarily centered nor symmetric

Notation

x

y

Dilation : x = (x1,x2) such that if we center B on them, then the so translated B intersects X.

X

B

difference

BINARY MORPHOLOGY

BINARY MORPHOLOGY

Dilation : x = (x1,x2) such that if we center B on them, then the so translated B intersects X.

How to formulate this definition ?

1) Literal translation

2) Better : from Minkowski’s sum of sets

BINARY MORPHOLOGY

Minkowski’s sum of sets :

l

l

BINARY MORPHOLOGY

Dilation :

l

Dilation

l

BINARY MORPHOLOGY

Dilation is not the Minkowski’s sum

l

l

bbbb l

BINARY MORPHOLOGY

Dilation with other structuring elements

BINARY MORPHOLOGY

BINARY MORPHOLOGY

Dilation with other structuring elements

Erosion : x = (x1,x2) such that if we center B on them, then the so translated B is contained in X.

BINARY MORPHOLOGY

difference

BINARY MORPHOLOGY

2) Better : from Minkowski’s substraction of sets

Erosion : x = (x1,x2) such that if we center B on them, then the so translated B is contained in X.

How to formulate this definition ?

1) Literal translation

BINARY MORPHOLOGY

Erosion with other structuring elements

BINARY MORPHOLOGY

Did not belong to X

BINARY MORPHOLOGY

Erosion with other structuring elements

BINARY MORPHOLOGY

Common structuring elements shapes= origin

x

y

Note :

circledisk

segments 1 pixel wide

points

BINARY MORPHOLOGY

Problem :

BINARY MORPHOLOGY

Problem :

BINARY MORPHOLOGY

d/2

d

<d/2

BINARY MORPHOLOGY

Implementation : very low computational cost

0

1 (or >0)

Logical or

0

1

BINARY MORPHOLOGY

Implementation : very low computational cost

Logical and

Opening :

BINARY MORPHOLOGY

Supresses :• small islands• ithsmus (narrow unions)• narrow caps

difference

also

Opening with other structuring elements

BINARY MORPHOLOGY

Closing :

BINARY MORPHOLOGY

Supresses :• small lakes (holes)• channels (narrow separations)• narrow bays

also

BINARY MORPHOLOGY

Closing with other structuring elements

BINARY MORPHOLOGY

Application : shape smoothing and noise filtering

BINARY MORPHOLOGY

Application : segmentation of microstructures (Matlab Help)

original

negated

threshold

disk radius 6

closing

opening

and withthreshold

BINARY MORPHOLOGY

Properties

• all of them are increasing :

• opening and closing are idempotent :

• dilation and closing are extensive erosion and opening are anti-extensive :

BINARY MORPHOLOGY

BINARY MORPHOLOGY

• duality of erosion-dilation, opening-closing,...

• structuring elements decomposition

BINARY MORPHOLOGY

operations with big structuring elements can be doneby a succession of operations with small s.e’s

Hit-or-miss :

BINARY MORPHOLOGY

“Hit” part(white)

“Miss” part(black)

Bi-phase structuring element

Looks for pixel configurations :

BINARY MORPHOLOGY

doesn’t matter

background

foreground

BINARY MORPHOLOGY

isolated points at4 connectivity

Thinning :

BINARY MORPHOLOGY

Thickenning :

Depending on the structuring elements (actually, seriesof them), very different results can be achieved :

• Prunning• Skeletons• Zone of influence• Convex hull• ...

Prunning at 4 connectivity : remove end points by a sequence of thinnings

BINARY MORPHOLOGY

1 iteration =

BINARY MORPHOLOGY

1st iteration

2nd iteration 3rd iteration: idempotence

BINARY MORPHOLOGY

doesn’t matter

background

foreground

What does the following sequence ?

BINARY MORPHOLOGY

1. Erosion and dilation

2. Common structuring elements

3. Opening, closing

4. Properties

5. Hit-or-miss

6. Thinning, thickenning

7. Other useful transforms :

i. Contour

ii. Convex-hull

iii. Skeleton

iv. Geodesic influence zones

i. Contours of binary regions :

BINARY MORPHOLOGY

BINARY MORPHOLOGY

4-connectivity

8-connectivitycontour

8-connectivity

4-connectivitycontour

Important for perimeter computation.

ii. Convex hull : union of thickenings, each up to idempotence

BINARY MORPHOLOGY

BINARY MORPHOLOGY

iii. Skeleton :

BINARY MORPHOLOGY

Maximal disk : disk centered at x, Dx, such that Dx X and no other Dy contains it .

Skeleton : union of centers of maximal disks.

BINARY MORPHOLOGY

Problems :

• Instability : infinitessimal variations in the border of X cause large deviations of the skeleton

• not necessarily connex even though X connex

• good approximations provided by thinning with special series of structuring elements

BINARY MORPHOLOGY

1stiteration

BINARY MORPHOLOGY

result of1st iteration

2nd iteration reachesidempotence

BINARY MORPHOLOGY

20 iterationsthinning

40 iterationsthickening

BINARY MORPHOLOGY

Application : skeletonization for OCR by graph matching

Application : skeletonization for OCR by graph matching

BINARY MORPHOLOGY

and 3 rotations

Hit-or-Miss

iv. Geodesic zones of influence :

BINARY MORPHOLOGY

X set of n connex components {Xi}, i=1..n .The zone of influence of Xi , Z(Xi) , is the set ofpoints closer to some point of Xi than to a point ofany other component. Also, Voronoi partition.Dual to skeleton.

BINARY MORPHOLOGY

thr erosion7x7

GZI opening5x5

and

BINARY MORPHOLOGY

BINARY MORPHOLOGY

BINARY MORPHOLOGY

BINARY MORPHOLOGY

BINARY MORPHOLOGY

BINARY MORPHOLOGY

thr erosion7x7

dist