1

Anisotropic Diffusion Filtering with A-priori

Estimated Orientation Field for Enhancing

Low-quality Fingerprint ImagesCarsten Gottschlich and Carola-Bibiane Schonlieb

ABSTRACT

We present a novel combination of classic anisotropic diffusion filters, a-priori estimated orientation

fields and a locally adaptive contrast enhancement step for improving the quality of fingerprint images. By

applying our new approach to low-quality images of the FVC2004 fingerprint databases, we are able to

show its equivalency to state-of-the-art enhancement methods for fingerprints. Linear anisotropic diffusion

and three types of nonlinear anisotropic diffusion are examined: coherence-enhancing, incoherence-

enhancing and edge-enhancing diffusion. The performance of diffusion filtering is compared to existing

methods in verification tests according to the FVC protocol using the freely available NIST matcher

BOZORTH3. In addition, combining anisotropic diffusion filtering with curved Gabor filters leads to

considerable improvements in terms of lower equal error rates. The achieved performance improvements

and the computational efficiency of the method suggest to include anisotropic diffusion filtering as a

standard image enhancement add-on module for a future real-time fingerprint recognition system. In

order to facilitate the reproduction of these results by other researchers, a Matlab implementation of the

anisotropic diffusion filters is made available for download.

Carsten Gottschlich is with the University of Gottingen, Institute for Mathematical Stochastics, Goldschmidtstr. 7, 37077

Gottingen, Germany. Phone: +49-(0)551-39172100. Fax: +49-(0)551-3913505. Email: gottschlich@math.uni-goettingen.de

Carola-Bibiane Schonlieb is with the University of Cambridge, Department of Applied Mathematics and Theoretical Physics,

Wilberforce Road, CB3 0WA Cambridge, UK. Phone: +44-(0)1223-764251 Email: c.b.schoenlieb@damtp.cam.ac.uk

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INDEX TERMS

Anisotropic diffusion filtering, image enhancement, FVC2004, orientation field estimation, contrast

enhancement, structure tensor, Harris corner detector, image quality, BOZORTH3, verification tests,

fingerprint recognition, biometrics

I. INTRODUCTION

The matching performance of a fingerprint recognition system depends heavily on the image quality [1].

Image enhancement aims at improving the overall performance by preparing input images for later

processing stages. Most systems extract minutiae from fingerprints and use them as the main feature for

matching [28]. The presence of noise can interfere with the extraction. As a result, true minutiae may be

missed and false minutiae may be detected, both having a negative effect on the recognition rate. In order

to avoid these two types of errors, an enhancement step intends to improve image quality by removing

noise and increasing the clarity of the ridge and valley structure. Especially, if ridges are interrupted e.g.

due to creases, scars or dryness of the finger, an image enhancement method shall be able to reconnect

them. Ridges that falsely stick together, e.g. caused by wetness or smudges, shall be separated while true

ridge endings and bifurcations shall be preserved.

The enhancement of low quality images (occurring e.g. in all databases of FVC2004 [31]) and very low

quality prints like latents (e.g., NIST SD27 [14]) is in general a challenging task. Techniques based on

contextual filtering, particularly the Gabor filter (GF), are widely used for fingerprint image enhancement

[28]. Those methods heavily rely on a correct estimation of the local context, i.e., the local orientation

and ridge frequency taken as inputs for the GF. Errors done in estimating the local context may lead

to the creation of artifacts in the enhanced image which consequently tends to increase the number of

verification errors. In fact, for low quality images there is a substantial risk that an image enhancement

step may impair the recognition performance as shown in [13] (results are cited in Table I of Section III).

Hence, the choice of an adequate enhancement strategy is a crucial one in fingerprint matching.

The present work proposes to apply anisotropic diffusion filtering (1) combined with an a-priori

computed orientation field (OF) and a locally adaptive contrast enhancement for improving the quality

of noisy fingerprint images. As opposed to existing filtering methods, here the diffusion process is

directed by a tensor derived from a previously estimated OF. We examine linear anisotropic diffusion

and three types of nonlinear anisotropic diffusion: coherence-enhancing, incoherence-enhancing and edge-

enhancing diffusion. The performance of diffusion filtering for enhancing low-quality prints is compared

to state-of-the-art methods. Additional improvements in terms of lower equal error rates (EERs) are

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achieved by combining coherence-enhancing anisotropic diffusion filtering with curved Gabor filters. In

order to facilitate the reproduction of these results by other researchers, a Matlab implementation of the

diffusion filtering is made available for download1. While the focus of this paper is on fingerprint image

enhancement, the presented method is also applicable for other oriented, flow-like structures [24].

Related work

After the already successful application of nonlinear diffusion equations in imaging, e.g., [35], [34],

anisotropic diffusion filtering has been introduced into the image processing community by Weickert [42]

and Bigun [5]. In particular, coherence-enhancing anisotropic diffusion is proposed in [44]. There, the

author presents the effectiveness of the method in the context of fingerprint enhancement by applying it to

a medium quality print. Enhancement of low-quality fingerprint images by diffusion filtering is addressed

in [9]. Unfortunately, the authors give no details regarding used protocols, software and results, so that

this method could not be included in the comparison in Section III. The authors of [46] describe the

use of diffusion filtering for the binarization of fingerprints and show three samples from FVC 2002

[30]. Moreover, in [32] a method for sharpening edges in fingerprint images by anisotropic diffusion

is proposed. [10] suggests to replace the nonlinear anisotropic diffusion filter by an oriented isotropic

diffusion filter in order to improve the computational efficiency for its usage in real-time applications. A

similar approach is examined in [20]. The application of nonlinear anisotropic reverse diffusion equations

(NARDE) for enhancing prints is proposed in [18]. Furthermore, anisotropic diffusion is examined in

[39] for the classification of fingerprint images. In [36] and [8] diffusion filtering is used for smoothing

the orientation field of the fingerprint rather than the fingerprint itself. And in [47] the authors suggest a

singularity driven diffusion process for regularizing the orientation field.

Organization of the paper

Section II-A gives a concise introduction to anisotropic diffusion filtering, followed in Section II-B by

an explanation how a diffusion tensor is derived from an a-priori computed OF. Section II-B1 describes

an aggregation approach which combines two OF estimation methods in order to achieve a more reliable

estimation of the local context. After smoothing by diffusion filtering we compensate for differences in

gray-level intensities along ridges and valleys by applying a locally adaptive contrast enhancement as

detailed in Section II-C. For comparing different diffusion based techniques with existing enhancement

1www.stochastik.math.uni-goettingen.de/biometrics/

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methods, low quality images of FVC2004 [31] were enhanced and used in verification tests. Results

stated in Section III show the soundness of this approach and its equivalence to state-of-the-art image

enhancement methods in terms of EERs. Further improvements were achieved by combining anisotropic

diffusion filtering with curved Gabor filters. The paper concludes with a discussion of advantages and

drawbacks in Section IV.

II. FINGERPRINT IMAGE ENHANCEMENT

A. Anisotropic diffusion filtering - an introduction

Let Ω ⊂ R2 be a rectangular domain and f ∈ L∞(Ω) the given image of a fingerprint defined on Ω.

We consider a class of anisotropic diffusion filters given by the evolution ofut = div (D∇u) on Ω× (0,∞)

u(x, 0) = f(x) on Ω

〈D∇u, ~n〉 = 0 on ∂Ω× (0,∞),

(1)

where D : Ω 7→ R2×2 is the so-called diffusion tensor, ~n is the outward pointing unit normal vector

on ∂Ω and u is the enhanced fingerprint image. Depending on the construction of the diffusion tensor

D the diffusion favours different types of local structures. For its construction Weickert [42] defines the

structure tensor Jρ of an image u to be

Jρ(∇uσ) := Kρ ∗ (∇uσ ⊗∇uσ) , ρ > 0.

Here, Kρ is a Gaussian kernel with variance ρ and uσ is the image u convolved with Kσ. The use of

∇uσ ⊗∇uσ := ∇uσ · ∇u⊥σ as a structure descriptor aims at making the tensor insensitive to noise and

sensitive to change in orientation only, i.e., the sign of the gradient should not be taken into account. As

pointed out in [5], the structure tensor also goes by the names “second order moment tensor”, “inertia

tensor”, “outer product tensor”, and “covariance matrix”. Bigun et al. introduced a generalized structure

tensor by analytically extending its standard form [4]. Recently, the structure tensor was also applied for

image synthesis and impainting in [37].

Jρ has two orthonormal eigenvectors v1 ‖ ∇uσ (points in the gradient direction) and v2 ‖ ∇u⊥σ (points

in the direction of the level lines) and corresponding eigenvalues µ1, µ2, which can be computed as

µ1 =1

2

(j11 + j22 +

√(j11 − j22)2 + 4j212

)µ2 =

1

2

(j11 + j22 −

√(j11 − j22)2 + 4j212

),

(2)

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where the jij’s are the components of Jρ

j11 = Kρ ∗(∂

∂xuσ

)2

j12 = Kρ ∗(∂

∂xuσ ·

∂

∂yuσ

)j22 = Kρ ∗

(∂

∂yuσ

)2

.

(3)

The eigenvalues of Jρ describe the ρ-averaged contrast in the eigendirections, e.g., if µ1 = µ2 = 0 it

means that the image is homogeneous in this area, if µ1 µ2 = 0 we are sitting on a straight line and

finally, if µ1 ≥ µ2 0 we are at a corner of an object. This property is utilized for detecting interest

points by the Harris corner detector [19]. The computation of the eigenvalues is avoided by instead

considering the determinant and the trace of the structure tensor which in that context is referred to as

autocorrelation matrix:det(Jρ) = (j11 · j22)− j212 = µ1 · µ2

trace(Jρ) = j11 + j22 = µ1 + µ2

And the corner response R is defined as

R = det(Jρ)− k · trace(Jρ)2,

where k is chosen empirically. Harris corner points are applied as features for fingerprint segmentation

in [45], and they are useful as interest points in many other situations because of their invariance under

affine transformations [33]. During the process of selecting keypoints for SIFT feautures, the Harris corner

response is applied for eliminating candidate points on edges (see Section 4.1 in [27]). Harris interest

points are proposed as a measure for image saliency in [26], linking them to models of preattentive

human visual perception.

Based on the eigenvalues, we can define the quantity

Coh = (µ1 − µ2)2 = (j11 − j22)2 + 4j212 (4)

as the local coherence of structures: this quantity is large for line-like structures whereas it is small for

constant areas in the image.

In [2], Bazen and Gerez showed that applying principal component analysis to the structure tensor

(which is called autocovariance matrix in that context) is equivalent to the averaging squared gradients

method for estimating the local orientation. They define coherence as follows:

Coh =µ1 − µ2µ1 + µ2

=

√(j11 − j22)2 + 4j212

j11 + j22.

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In our tests, both definitions of the coherence led to very similar results applying coherence-enhancing

diffusion filtering to the images of the FVC2004 databases. In the following, we shall concentrate on the

first definition (4) of the coherence.

With the derived structure tensor we consider the modified anisotropic diffusion (1) asut = div (D (Jρ(∇uσ))∇u) on Ω× (0,∞)

u(x, 0) = f(x) on Ω

〈D (Jρ(∇uσ))∇u, ~n〉 = 0 on ∂Ω× (0,∞),

(5)

where the eigenvectors of the new diffusion tensor D (Jρ(∇uσ)) : Ω 7→ R2×2 are parallel to the ones of

Jρ(∇uσ) and its eigenvalues λ1 and λ2 are chosen d epending on the desired enhancement method. In

the next section we shall discuss different choices for the eigenvalues of D, all in accordance with the

following assumptions which are necessary for the well-posedness of our equation, cf. [43].

Assumptions 2.1: For the well-posedness of (5) we make the following assumptions on D:

− D ∈ C∞(R2×2,R2×2) (6)

− D(J ) is symmetric (7)

− D is positive definite: for all w ∈ L∞(Ω;R2) with |w(x)| ≤ K on Ω,

there exists a positive lower bound ν(K) for all eigenvalues of D (Jρ(w)) , w 6= 0. (8)

In the following we present different choices for D in accordance with Assumptions 2.1.

B. Anisotropic diffusion filtering with given orientation field

The diffusion process (1) is steered by the diffusion tensor (or structure tensor), which is derived from

the image gradients smoothed by a Gaussian kernel. However, in areas of fingerprint images which are

affected by noise like scars, smudges, wetness or dryness of the finger, gradients are error-prone and

unreliable [17]. Depending on the level of noise, smoothing with a Gaussian kernel may be insufficient

for obtaining a feasible diffusion tensor (see Figure 1 for an example). The dilemma is that especially in

those regions which could profit the most from diffusion filtering, the image gradients are not correctly

estimated due to the influences of noise.

In contrast to previous works, we therefore decided to generate our diffusion tensor from a more

reliable orientation field (OF) estimation (see Figure 1). To be more specific, we propose an application

of anisotropic diffusions like (5) combined with an a-priori estimated orientation field, cf. Section II-B1,

for the enhancement of fingerprint images.

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Fig. 1. The original image (left, finger 84, impression 6, FVC2004 database 1) after enhancement by coherence-enhancing

diffusion filtering using the image gradients (center) and linear diffusion filtering with a-priori estimated OF as described in

Section II-B1 (right). The structure tensor derived from the aggregated OF clearly outperforms the classical gradient based tensor

in areas with a high noise level.

In fact, instead of (5) we considerut = div (D (Jρ(∇uσ),OF)∇u) on Ω× (0,∞)

u(x, 0) = f(x) on Ω

〈D (Jρ(∇uσ),OF)∇u, ~n〉 = 0 on ∂Ω× (0,∞),

(9)

where now the diffusion tensor D is not only dependent on the structure tensor Jρ, but also on the

orientation field OF. More precisely, D is chosen with eigenvectors parallel to the two orthogonal

directions

vOF1 = (cos (πγ/180), sin (πγ/180)) , vOF2 =(vOF1

)⊥given by the orientation field OF (γ measures angles in degrees), but with eigenvalues λ1 and λ2

determined by the structure tensor Jρ (∇uσ).

1) Orientation field estimation: In order to obtain robust OF estimations for low-quality prints, we

compute a combined OF by using the line sensor method [17] and the gradients based method [24], [2].

The two individual OFs are compared pixelwise, and if the angle between both estimations is smaller

than a threshold (we used t = 15 ), the orientation of the combined OF is set to the average of the

two. Otherwise, the pixel is marked as missing. In a final step, all inner gaps are reconstructed and the

orientation of the outer proximity is extrapolated up to a radius of 16 pixels, both as described in [17].

Results of verification tests on all available FVC databases [29], [30], [31] showed a better performance

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of the combined OF applied for contextual image enhancement than each of the two individual OF

estimations [16]. The same OF estimation was used in [15] for enhancing fingerprint images by curved

Gabor filters (for a comparison with this approach, see Table I). In general, an OF estimation based on

a global model like e.g. [22] could be an alternative to the afore described combined OF estimation,

however, in a recent comparison of OF estimation methods [7], the considered global model (FOMFE

[41]) yielded no advantages over the simple gradient based method. Providing an OF estimation which

is both robust to noise and precise enough for contextual filtering is certainly a not yet fully solved

challenge.

2) Choices for the eigenvalues: In the following we discuss different choices for the eigenvalues λ1, λ2

of D in (9). Choosing the eigenvalues λ1 and λ2 means to decide on the strength of the diffusion in each

eigendirection v1 and v2.

a) Linear anisotropic diffusion filtering: Our first approach is to choose them globally constant over

the image, i.e.,

λ1 = α, λ2 =

α if orientation could not be estimated

1− α wherever we have a trustable estimate for the orientation,(10)

with positive constant α 1. Then equation (9) results in a linear diffusion equation

ut = div (D (α,OF)∇u) ,

where the diffusion tensor D is constant in diffusion time. Similar approaches have been considered in

the literature, among them what is called directed diffusion [23].

b) Nonlinear anisotropic diffusion filtering: In order to make also the strength of the smoothing in

(5) dependent on the local image structure, the eigenvalues λ1 and λ2 of D are defined in dependency

of the image gradient. Such an approach results in a nonlinear diffusion equation.

Weickert [44], [43] proposes two choices for the eigenvalues of D, resulting in two diffusion filters

called edge-enhancing diffusion and coherence-enhancing diffusion. The edge-enhancing diffusion filter

possesses the two eigenvalues

λ1(µ1) = g(µ1), λ2 = 1,

where (for η > 0 and Cm > 0)

g(s) :=

1 s ≤ 0

1− e−Cm

(s/η)m s > 0,

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Fig. 2. The original images (left) are first smoothed by linear anisotropic diffusion filtering with a constant α = 0.01, 40

iterations and a stepsize of 0.25 (center) and as a final step, a locally adaptive contrast enhancement (right) is performed.

Impressions 1 (top row) and 3 of finger 2 in database 4 from FVC2004 [31] are displayed. Scores for matching both impressions

using BZ3 increase from 37 for the originals to 54 after diffusion and to 86 after contrast enhancement.

and µ1 is the first eigenvalue of Jρ. Having defined D with these eigenvalues the temporal evolution

of (9) enhances edges in f , caused by the rapidly decreasing diffusivity g. Here, the contrast of the

edge is decreased until the gradient reaches a value where no backward diffusion is possible anymore,

i.e, when ∇uσ < η. With this choice of the diffusion tensor equation (9) can be seen as an anisotropic

generalization of the Perona-Malik equation [35]. However, applying edge-enhancing diffusion filtering

to fingerprint images of FVC2004 resulted in the destruction of large parts of the image foreground. In

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our experience, edge-enhancing diffusion is inapplicable for fingerprint image enhancement.

Differently, coherence-enhancing diffusion wants to enhance the coherence of flow-like structures. With

µ1, µ2 being the eigenvalues of Jρ as before, we define (for Cm > 0)

λ1 = α, λ2 =

α if µ1 = µ2

α+ (1− α) · e−Cm

(µ1−µ2)2m else, (11)

where α ∈ (0, 1), α 1. Then, the smoothing of the coherence-enhancing diffusion is stronger in the

neighbourhood of coherent structures (where the radius of the neighbourhood is determined by ρ) while

stopping diffusion in homogeneous areas, at corners, and in general in incoherent (random) areas of the

image. Note, that this approach depends crucially on the correct choice of ρ. Choosing ρ too small the

filter might not be able to detect coherent structures anymore (depending on the level of noise contained

in f ). In contrast, assigning a value to ρ that is too large results in a smoothing which is too strong

and possibly merges single structures in the image. However, having found the correct ρ this approach

is even able to close gaps (smaller than ρ) in coherent structures in the image.

Additionally to these choices we propose another diffusion filter, which follows a philosophy opposite

to coherence-enhancing diffusion (11). We define the eigenvalues of the Incoherence-enhancing diffusion

as follows:

λ1 = α, λ2 =

α if |µ1 − µ2| = 1

α+ (1− α) · e− Cm

(1−(µ1−µ2)2)m else, (12)

where as before α 1 is a positive constant. Further, we choose C such that the exponential function

is very steep, and the structure tensor J has been normalised such that |µ1 − µ2| ≤ 1. Then, if the

coherence is large we have λ2 ≈ α and we smooth only a little bit, in both directions v1 and v2 with the

same strength α. The smoothing becomes stronger the smaller the coherence becomes. Such an approach

makes sense in its application to fingerprint images whenever they contain large areas of broken structures

and when choosing ρ small (such that gaps in coherent structures cannot be overseen and classified as

coherent).

Remark 2.1: Note, that with the construction of the eigendirections of D from the OF in (9), if we

computed the coherence (4) according to these eigenfunctions, it always would equal one. To see this

we consider (2)-(4) and plug in the corresponding values for (3), e.g.,

j11 = cos2 (πγ/180).

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This gives

µ1 − µ2 =(cos2 (πγ/180)− sin2 (πγ/180)

)2+ 4 · (cos (πγ/180) · sin (πγ/180))2

= (cos2 (2πγ/180) + sin2 (2πγ/180))2 = 1.

Because of this, we decided to compute the coherence – used in the computation of the eigenvalues λ1

and λ2 of D and hence responsible for the diffusion strength – rather from the structure tensor Jρ(∇uσ),

and use the precomputed orientation only for the directional information.

C. Locally Adaptive Contrast Enhancement

These diffusion processes improve the image quality by smoothing along the local orientation. However,

the method also tends to reduce the overall image contrast and after the diffusion, there are also

considerable differences in gray-level intensities along ridges and valleys. In a final step, we compensate

for this by enhancing the contrast in a locally adaptive way. Our contrast enhancement is based on the

normalization formula from Section 2.3 in [21] which was proposed for a global image normalization.

Here, we first compute for each foreground pixel (i, j) the local mean mI and variance vI of the

grayvalue value I(i, j) by considering only neighboring pixels within a radius ≤ r (we used r = 6 for

all images). Next, a target gray-value T (i, j) is calculated for a given target mean mT and variance vT

(in our tests, we set mT = 127.5 and vT = 10, 000). Finally, the new gray value G(i, j) is obtained

by adjusting the current gray value I(i, j) a certain percentage pi,j towards the target value. Besides

the good performance of this algorithm in the inner areas of the fingerprint, it may create artifacts at

the border between image foreground and background. Overall, this may increase the detection of false

minutiae and hence result in higher EERs. To avoid this, the contrast enhancement is tuned to slowly

change in s steps from the foreground to the background area. Foreground pixels which have background

pixels as 4-connected neighbors start with pi,j = 1sp. Their 4-connected foreground neighbors are set

to pi,j = 2sp and so on for s steps, with pi,j = p for all other foreground pixels (see Figure 2 for an

example). We tested a number of parameter combinations on a few images from the four B databases

(prints of 10 fingers with 8 impressions per fingers; for training) and found that p = 0.5 and s = 60

produced a smooth transition from foreground to background. With this, the newly computed fingerprint

image G is computed pixel-wise as

G(i, j) =

I(i, j) + (pi,j ∗ (T (i, j)− I(i, j))) for I(i, j) ∈ foreground

mT otherwise

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with

T (i, j) = mT +

√vTvI

(I(i, j)−mI)

III. RESULTS

A. Test setup

The matcher referred to as “BZ3” is based on the freely available2 NIST biometric image software

package (NBIS) [40]. Minutiae were extracted using MINDTCT and templates were matched by BO-

ZORTH3. For the verification tests, we follow the FVC protocol in order to ensure comparability of

the results with [13] and other researchers. 2800 genuine and 4950 impostor recognition attempts were

conducted for each of the FVC databases. The FVC protocol and calculation of equal error rates (EERs)

are described in [29].

Our numerical implementation of (9) is based on the Nonlinear Diffusion Toolbox for Matlab by

D’Almeida [12] which starts with an initial condition u0 = f and computes each subsequent anisotropic

diffusion step as

uk+1 = uk + ∆tdiv(D∇uk

). (13)

The iterative approach (13) constitutes an explicit time stepping scheme for the time discretisation of (9).

The spatial derivatives div and ∇ are discretised with finite differences.

B. Combining Anisotropic Diffusion Filtering and Curved Gabor Filters

In order to improve the matching performance, we tested two information fusion strategies for combin-

ing anisotropic diffusion filtering and curved Gabor filters. For each fingerprint image of the FVC2004

databases, two enhanced versions are computed using both enhancement methods, and from each enhanced

image, a minutiae template is extracted using MINDTCT. First, let us consider fusion on the feature level.

Both templates are combined into one template by union. Duplicate entries are avoided by comparing

the x- and y-coordinates and the direction of the minutiae, and similar minutiae are only added once to

the combined template. Alignment is not required, since minutiae are extracted from enhanced versions

of the same original fingerprint image. The rationale behind template fusion is to reduce the number of

missed true minutiae, if both enhancement methods complement one another. However, in our experiments

matching the fused templates did not improve the performance and resulted in similar EERs as matching

templates of one enhancement method. MINDTCT is known for extracting false minutiae at the border

2http://fingerprint.nist.gov/

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Enhancement method DB1 DB2 DB3 DB4

Original images 14.5 9.5 6.2 7.3

Traditional Gabor filter

Hong, Wan, Jain [21](16.9) 14.4 7.1 9.8

Short time Fourier transform (STFT) analysis

Chikkerur, Cartwright, Govindaraju [11](19.1) 11.9 7.6 10.9

Pyramid-based filtering

Fronthaler, Kollreider, Bigun [13]12.0 8.2 5.0 7.0

Curved Gabor filters [15]

Curved region: 33× 65 pixels, σx = 4.0, σy = 4.09.7 6.3 5.1 6.5

Coherence-enhancing diffusion filtering ? 11.4 9.4 5.0 7.7

Weickert [44] with α = 0.001, C = 0.0001, ρ = 10 11.0 8.9 4.8 7.9

Anisotropic diffusion filtering & orientation:

Linear diffusion filtering ? 10.9 7.3 4.9 8.1

with α = 0.01 10.0 5.9 5.0 6.0

Linear diffusion filtering ? 11.6 7.4 4.7 7.3

with α = 0.001 9.9 5.9 4.9 5.9

Linear diffusion filtering ? 11.6 7.4 4.7 7.3

with α = 0.0001 10.0 5.7 5.0 5.8

Incoherence-enhancing diffusion filtering ? 11.0 7.2 5.6 8.0

with α = 0.001, C = 0.01, ρ = 10 10.0 5.6 5.2 6.1

Coherence-enhancing diffusion filtering ? 11.9 6.4 5.4 7.3

with α = 0.001, C = 0.001, ρ = 32 10.0 6.4 5.0 6.0

Combining curved Gabor filters and

coherence-enhancing anisotropic diffusion filtering:

Max rule 9.0 5.0 4.2 5.4

Sum rule 9.3 4.8 3.6 5.2

Template cross matching 8.9 4.3 3.4 4.9

TABLE I

EERS IN % FOR MATCHER BZ3 ON THE ORIGINAL AND ENHANCED IMAGES OF FVC2004 [31]. PARENTHESES INDICATE

THAT ONLY A SMALL FINGERPRINT AREA WAS USEFUL FOR RECOGNITION. RESULTS OF THE TOP FOUR ROWS ARE CITED

FROM [13]. ALL DIFFUSION PROCESSES STOPPED AFTER 40 ITERATIONS WITH STEP SIZE 0.25. ROWS MARKED WITH A

STAR (?) STATE RESULTS FOR MATCHING IMAGES DIRECTLY AFTER DIFFUSION FILTERING, ROWS WITH A DIAMOND ()

REPORT EERS FOR MATCHING IMAGES WITH ADDITIONAL CONTRAST ENHANCEMENT AS DESCRIBED IN SECTION II-C.

between foreground and background (see e.g. [45]) and the template fusion did not only reduce the

number of missed minutiae, but it also increased the number of falsely detected minutiae, and in doing

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so, the gains from template fusion were cancelled out.

Moreover, score-level fusion is a popular approach for combining e.g. different fingerprint matching

algorithms or different biometric traits. Henceforth, ADi denotes the minutiae template extracted from

image i after enhancement by anisotropic diffusion filtering, CGi from image i enhanced by curved

Gabor filters. For each recognition attempt (RA), two fingerprints are matched denoted as a and b. For

a genuine RA, a and b are different impressions of the same finger, for an impostor RA, a and b are

impressions belonging to different fingers. Performance improvements were achieved by combining the

two scores obtained by matching ADa with ADb and matching CGa with CGb using the max rule or the

sum rule [25] (see Table I). Here, normalizing the scores [28] prior to fusion is not necessary, because

all scores were obtained using the same matching algorithm, BOZORTH3. Further improvements were

achieved by cross matching the templates, i.e. additionally matching templates ADa with CGb and CGa

with ADb, and considering the sum of the best two of the four scores as the combined score [16]. All

five tested combinations (the curved Gabor filter combined with each of the five variants of diffusion

filtering listed in Table I) led to significant reductions of the EER. The best combination is reported at

the bottom of Table I.

IV. DISCUSSION

In our opinion, the performance of anisotropic diffusion filtering for enhancing low-quality fingerprint

images is quite impressive in comparison to existing methods: e.g. linear anisotropic diffusion filtering

with α = 0.001 clearly outperforms the traditional Gabor filter [21] and enhancement based on the short

time Fourier transform [11] on all four databases, and pyramid-based filtering [13] on three of the four

databases. The EERs achieved by linear diffusion filtering are similar to those of the curved Gabor filters

which applied exactly the same OF estimation as the diffusion filtering for obtaining the results listed in

Table I. Advantages of diffusion filtering are that they do not require estimation of the ridge frequency

(RF) and they can be computed fast.

The performance of the curved Gabor filters heavily depends on the quality and reliability of the OF and

RF estimation. This fact is nicely illustrated when considering the use of Gabor filters for the generation

of synthetic fingerprints [6]: one black spike on a white image and the iterative application of the Gabor

filter is sufficient for creating a ridge pattern. In this sense, the pattern is completely defined by the OF

and RF image, and with respect to the input, the Gabor filter creates a perfect enhanced image. However,

when dealing with low-quality images, errors may occur during the estimation of the local context, and

if these erroneous estimations are passed on to the Gabor filter, it will create incorrect structures in the

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enhanced image. The diffusion filter in contrast proceeds more gently. Gray-value differences are evened

along the orientation, and in doing so, interrupted ridges are reconnected and conglutinated neighboring

ridges are separated by trend. The results in Section III show that the proposed anisotropic diffusion

filters followed by a contrast enhancing step are well-suited for enhancing low-quality images.

Comparing the different types of anisotropic diffusion filters, we conclude that coherence-enhancing,

incoherence-enhancing and linear anisotropic diffusion filtering diffusion achieve very similar EERs

matching images after the final contrast enhancing step. Of course, for real-life applications, the dif-

fusion with a constant α is the most favourable because it requires the least computational efforts. Its

computational efficiency suggest to include linear anisotropic diffusion filtering as a standard image

enhancement add-on module for a future real-time fingerprint recognition system.

Combining anisotropic diffusion filtering with curved Gabor filters led to additional advancements and,

to the best of our knowledge, the lowest EERs achieved so far using MINDTCT and BOZORTH3 on

the FVC2004 databases. Further performance improvements especially rest upon better OF estimations.

Automatic and reliable OF estimation methods for low-quality and very low-quality prints is a topic that

certainly deserves further research.

ACKNOWLEDGMENTS

The authors would like to thank Thomas Hotz, Stephan Huckemann and Axel Munk for their valuable

comments during the preparation of this manuscript. C. Gottschlich and C.-B. Schonlieb gratefully

acknowledge support by DFG RTG 1023 “Identification in Mathematical Models: Synergy of Stochastic

and Numerical Methods”. Moreover, C.-B. Schonlieb acknowledges the financial support provided by the

project WWTF Five senses-Call 2006, Mathematical Methods for Image Analysis and Processing in the

Visual Arts and the Cambridge Centre for Analysis (CCA). Further, this publication is based on work

supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology

(KAUST).

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