# Stopping criterion for linear anisotropic image diffusion: a fingerprint image enhancement case

- Tariq M. Khan†
^{1}Email author, - Mohammad A. U. Khan†
^{2}, - Yinan Kong
^{1}and - Omar Kittaneh
^{2}

**2016**:6

https://doi.org/10.1186/s13640-016-0105-x

© Khan et al. 2016

**Received: **14 May 2015

**Accepted: **17 January 2016

**Published: **8 February 2016

## Abstract

Images can be broadly classified into two types: isotropic and anisotropic. Isotropic images contain largely rounded objects while anisotropics are made of flow-like structures. Regardless of the types, the acquisition process introduces noise. A standard approach is to use diffusion for image smoothing. Based on the category, either isotropic or anisotropic diffusion can be used. Fundamentally, diffusion process is an iterated one, starting with a poor quality image, and converging to a completely blurred mean-value image, with no significant structure left. Though the process starts by doing a desirable job of cleaning noise and filling gaps, called under-smoothing, it quickly passes into an over-smoothing phase where it starts destroying the important structure. One relevant concern is to find the boundary between the under-smoothing and over-smoothing regions. The spatial entropy change is found to be one such measure that may be helpful in providing important clues to describe that boundary, and thus provides a reasonable stopping rule for isotropic as well as anisotropic diffusion. Numerical experiments with real fingerprint data confirm the role of entropy-change in identification of a reasonable stopping point where most of the noise is diminished and blurring is just started. The proposed criterion is directly related to the blurring phenomena that is an increasing function of diffusion process. The proposed scheme is evaluated with the help of synthetic as well as the real images and compared with other state-of-the-art schemes using a qualitative measure. Diffusions of some challenging low-quality images from FVC2004 are also analyzed to provide a reasonable stopping rule using the proposed stopping rule.

## Keywords

## 1 Introduction

In image processing problems, many times one comes across the task to enhance flow-like structures, for instance, the automatic assessment of wood surfaces or fabrics, fingerprint image analysis, scientific image processing in oceanography [1], seismic image analysis [2], or sonogram image interpolated for Fourier analysis [3]. All images as mentioned above have one thing common; they contain elongated structures [4–6]. Such images can be referred to as *anisotropic*. The isotropic, by contrast, is an image category having largely round objects. The isotropic as well as anisotropic images, once acquired from their respective sources are mostly noisy. The noise treatment is different based on the category they belong. The case of noise smoothing for anisotropic images is more interesting and is the focus of research presented here.

*L*(

*x*,

*y*) is smoothed with a Gaussian of small standard deviation. The result

*C*(

*x*,

*y*) is then differentiated in

*x*- and

*y*- direction to form

*C*

_{ x }(

*x*,

*y*) and

*C*

_{ y }(

*x*,

*y*), respectively. Next the covariance matrix components

*J*

_{1}(

*x*,

*y*)=2

*C*

_{ x }(

*x*,

*y*) and \(J_{2}(x,y) = {C_{x}^{2}}(x,y) -{C_{y}^{2}}(x,y)\), and \(J_{3}(x,y)= \sqrt {{C_{x}^{2}}(x,y) +{C_{y}^{2}}(x,y)}\) are computed. The components are smoothed again with a larger Gaussian. The local orientations and their anisotropy strength measure are computed as

The rest of this paper is organized as follows. In Section 3, a discrete image as a spatial distribution is discussed. The spatial entropy of linear isotropic diffusion process is described in Section 4. Section 5 talks about spatial entropy of a linear anisotropic diffusion process followed by results and discussion in Section 6. Finally, the paper is concluded in Section 7.

## 2 Related work

The research concerned here is to smooth noise present in fingerprint images (a representative of anisotropic class) without affecting their ridge/valley pattern. This aim can be conveniently served in a *scale-space* construction. A scale-space framework describes a noisy image as a stack of progressively evolving many smooth images, each one with their corresponding scale [8]. The stack is ordered in increasing smoothness scale, where the scale varies in fine-to-coarse. The fine-to-coarse transformation is implemented, in general, by a linear isotropic diffusion process, governed by a partial differential equation (PDE) as follows.

*L*(

*x*,

*y*) denote a noisy grayscale input image and

*L*(

*x*,

*y*;

*t*) be an evolving image at scale

*t*, initialized with

*L*(

*x*,

*y*;0)=

*L*(

*x*,

*y*). Then, the linear isotropic diffusion process can be defined by the equation

*heat equation*. For image processing, the amount of heat is replaced with the intensity value at a certain location. The diffusivity parameter

*c*is constant across the image, making it a

*linear*isotropic equation. The linear isotropic equation has an elegant solution \(L(x,y;t)= G_{\sqrt {2ct}}(x,y) \ast L(x,y)\), where \(G_{\sigma }= \frac {1}{2 \pi \sigma ^{2}} \exp \left (- \frac {x^{2}+y^{2}}{2 \sigma ^{2}}\right)\). This solution provides the required interpretation in the form of low-pass filtering. Due to low-pass nature of this diffusion, as it progresses from fine scale images to coarser images, the blurring intensifies and may result in removing significant image structure, typically edges, lines, or other details, well before it had taken care of the noise. To protect the structure in a diffusion process, the diffusivity parameter should be made dependent on some characterization of image structure. This results in the famous nonlinear isotropic diffusion process, proposed by [11]. The diffusivity now becomes a function of gradients, so at the edge point the diffusion is completely inhibited and in smooth regions diffusion is allowed. However, computing gradients for a noisy image is an ill-posed problem. A remedy was pointed out by [12], that suggests the use of Gaussian smoothing before computing gradients. This modification lays the foundation for a well-behaved

*non-linear*isotropic diffusion process. Later on, instead of inhibiting diffusion at edge points, it was thought of to steer the diffusion in the direction parallel to the edge [13–16] rather than across it. This paved the way for the use of the diffusion matrix. This evolved the current form of

*non-linear*anisotropic diffusion. The diffusion matrix-based equation is defined as

*D*is the 2×2 diffusion matrix. The eigenvectors of the diffusion matrix provide the required steering while the eigenvalues as a function of gradients, add the non-linearity character. In our wish to keep connected with the Gaussian convolution interpretation that provides a mathematical tractability to the whole process, the research reported here is restricted to the linear anisotropic diffusion case. For that, the eigenvalues of the diffusion matrix are kept fixed. It is found that the Gaussian convolution connection is also useful for linking anisotropic diffusion with its earlier counterpart isotropic diffusion in a more natural way. The support for this modification came from the argument made in [14], that a non-uniform Gaussian can act as a solution of the Anisotropic Gaussian scale-space as long as the diffusion matrix is

*spatially constant*, i.e., it does not depend on (

*x*,

*y*) spatial location. Keeping in line with this argument, only spatially-invariant diffusion matrix is used; however, the steering was allowed. This leaves us with the so-called

*linear anisotropic diffusion*process. The constant eigenvalues are responsible for the linear part of the name, while the steering of the eigenvectors is what provided the word anisotropic in the nomenclature. The linear anisotropic diffusion equation has a convolution solution with a non-uniform Gaussian of the form:

where (*u*,*v*) are the rotated coordinates obtained using eigenvectors of the diffusion matrix. The eigenvalues *λ*
_{
u
}, *λ*
_{
v
} represent the standard deviations of the Gaussian in *u* and *v* direction, respectively. Normally, for noisy images, one of the eigenvalues is set to be much smaller than the other one, resulting in a non-uniform Gaussian function with more generalized elliptical support.

Searching for a suitable linear anisotropic diffusion strategy for noisy images in literature, we stumble upon considerable activity regarding the impact of a non-linear anisotropic diffusion equation on noisy images. The non-linear anisotropic literature is used as a stepping stone to reach a linear anisotropic diffusion strategy. The idea of non-linear anisotropic diffusion was pioneered by Nitzbeg et al. [17] and Cottet et al. [12]. Later on, Weickert [3] put forward a formal method for enhancing the elongated structure, referred to as coherence-enhanced diffusion (CED). The CED works by steering the diffusion process in a particular direction with the help of a spatially varying diffusion matrix. The design was further generalized by adopting a diffusion matrix to learn the local structure iteratively [18]. Since smoothing elongated structure is desired, the CED procedure comes in handy. The CED is adopted as it is, but with one major modification. That is, the eigenvalues are forced to be independent of spatial position without disturbing the eigenvectors. Thus, our proposed linear anisotropic diffusion process will steer the non-uniform Gaussian to lay along the structure, but its size will remain constant regardless of the position. Towards the end, we will desribe another variant of CED, where even the steering part of the diffusion matrix will also be precomputed and kept constant throughout the evolution process. This is referred to as the linear-oriented diffusion process.

The suggested linear anisotropic process for anisotropic images are confronted with one basic problem: when to stop the diffusion. For the case of a noisy image, the diffusion process initializes with an under-smooth situation that ultimately turns into an over-smooth one (the mean-value image at the end with no structure). Over-estimating stopping time will result in an over-smoothed blurry image while under-estimating may leave significant noise in the image. Therefore, it is crucial that an appropriate time is selected in an automatic way. The literature activity in this respect can be divided into two broad categories. One that deals with stopping criterion selection in additive noise model setting. These methods adopt the stopping time by treating the noisy image as the result of a noise addition, where the correlation between the diffused image and the initial noisy image minimized [3]. The authors in [19] introduced a multigrid algorithm using a normalized cumulative periodogram. A frequency approach to the problem was presented in [20]. Whereas, [21] uses the extent of noise smoothing in every iteration as a stopping parameter for diffusion. Later on, a spatially-varying stopping method was introduced that increased the computational cost significantly [22]. By identifying it as a Lyapunov functional of a large class of scalar-valued nonlinear diffusion filters, Weickert [23] introduced decreasing the variance of an evolving image as a stopping tool.

Since additive noise model may break down for some real-world images, where noise manifests itself in the form of gaps in regular ridge structures. Therefore, a second category of stopping rule was evolved. The category deals with examining entropy profile of the diffused image and proposed stopping criterion for the evolving image entropy distance from that of the entropy of the original noisy image [3]. The idea of local image entropy was introduced in [24], where the measure of local entropy defines the segmentation boundaries in multiple-object images. Local image entropy definition can be extended to define a global characteristic of the scale-space image, that is spatial entropy [25].

The research work reported here takes an investigative look at the stopping rule concerning the change in spatial entropy of an image as it goes through diffusion process. The connection, between last peak in spatial entropy curve and the size of the image structure, is found to be related to the start of significant information loss. This observation paves the way to the hypothesis that peak entropy change will happen at the time instant on diffusion time axis when dominant image structures just start blending with the background right at their boundaries. This finding, substantiated by extensive empirical evidence provided here, motivated us to put forward the idea that a maximum entropy change may well be posed as a good stopping time for the diffusion process.

## 3 A discrete image as spatial distribution

*x*is the row index and

*y*is the column index. This discrete image can be realized as spatially distribution light intensity [26]. Each spatial location that is (

*x*,

*y*) in the image registers the number of light quantum-hit. In this way, we may define

As stated in [26], the spatial entropy of the image increases monotonically towards an equilibrium state \(\log N\), where *N* is dimension *N*=*r*
*o*
*w*
*s*×*c*
*o*
*l*
*u*
*m*
*n*
*s*.

## 4 Spatial entropy of linear isotropic diffusion process

The linear diffusion process implemented by so-called heat equation is the oldest and well-investigated noise-smoothing process in the image processing domain. The linear diffusion process can be visualized as an evolution process with an artificial variable *t* denoting the *diffusion time*, where the noisy input image is repeatedly smoothed at a constant rate in all directions. No preference to any direction is what justifies the name *isotropic*. This evolution results in *scale space* representation of the noisy image. As we move up to coarser scales, the evolving images become more and more simplified since the diffusion process removes the image structures present at finer scales. In the process, noise also gets smoothed as it is considered a smaller size object while diffusion just reaches the point of touching the boundaries of the large dominating structure.

During the process of diffusion from fine-scale image to the higher coarser scale images, the mean of the resulting image remains constant with a monotonic decrease in variance (a second-order statistic [13]). Later on, it was found that spatial entropy associated with linear isotropic diffusion process also rises smoothly in a monotonic fashion [25]. Motivated by the smoothness of the spatial entropy graph for the diffusion process, the first derivative of the entropy function on natural scale parameter \(\tau = \log (t)\) was investigated. It was shown that entropy change graph do show important peaks related to dominating structures present in the original fine scale image. However, their experiments did not involve smoothing noisy images, and the authors fell short of suggesting to use these peaks as stopping criterion. The empirical evidence is provided here to show that once a linear isotropic diffusion process is involved in smoothing noisy images, these peaks will come at a much later stage in diffusion time. Therefore, most of the noise being low size structure already wiped by the process, and thus the peaks could be regarded as a suitable stopping time. This proposition is tested by tracking experimental data.

## 5 Spatial entropy of a linear anisotropic diffusion process

In this section, spatial entropy analysis is carried out for the anisotropic diffusion process. What we are looking for is the finding whether we will get a smooth spatial entropy increasing function, and then will we get a distinct peak in the entropy change curve for the anisotropic diffusion process.

*L*(

*x*,

*y*) can be constructed by the diffusion equation

*D*is the 2×2 diffusion matrix, adapted to the local image structure, via a structural descriptor, called the second-moment matrix

*μ*, defined as

*L*

_{ x }

*L*

_{ y }, and \({L_{y}^{2}}\) represent the second order Gaussian-derivative filters, in the

*x*and

*y*directions. This symmetric 2×2 matrix has two eigenvalues

*λ*

_{1}and

*λ*

_{2}, given by:

*θ*represents the local orientations of the given image. What is observed here is that eigenvalues are dependent on the local structure. In order to transform CED process into a linear anisotropic process, fixed values are assigned to the eigenvalues. Specifically, the eigenvalue associated with eigenvector that goes parallel to the structure has given a larger value than that of the eigenvalue of an eigenvector that is perpendicular to the structure boundary. Our specific choice of

*λ*

_{1}and

*λ*

_{2}for this experiment are

with a step size of 0.01 to provide a stable diffusion process.

*D*can now be reconstructed with help of its structure-invariant eigenvalues and structure-dependent eigenvectors as

- 1.
Calculate the second-moment matrix for each pixel.

- 2.
Construct the diffusion matrix for each pixel.

- 3.
Calculate the change in intensity for each pixel as \( \nabla \left ({D\nabla L} \right)\).

- 4.Update the image using the diffusion equation as$$ L^{t+\bigtriangleup t}= L^{t}+ \bigtriangleup t \times \nabla\left({D\nabla L} \right). $$(16)

*D*is replaced by a scalar diffusivity, say

*c*. Spatial entropy change for linear isotropic diffusion process is given by

For both, anisotropic as well as isotropic cases, the spatial entropy change equation contains the same constant \(k=1-\log C\).

*linear anisotropic diffusion*process. The test anisotropic image for this purpose consists of three curves, as shown in Fig. 6. At the heart of the anisotropic process is the construction of diffusion matrix

*D*. The diffusion matrix handles steering the elliptical Gaussian to go around the structure. The geometric visualization in the form of ellipses corresponding to point-wise diffusion matrix is displayed in Fig. 6, where it can be seen that they align well with the local flow of the curve. The diffusion parallel to the edges is enabled due to the large eigenvalue while avoiding the cross-over edge problems due to small eigenvalues. The linear anisotropic diffusion character is made evident by having constant eccentricity for all the ellipses across the image. The term anisotropic used here is related to changing direction of the ellipse at each pixel due to the diffusion matrix eigenvector adaptability with the given local structure. Therefore, with each iteration, the ellipse does grow without changing the eccentricity ratio and for a given diffusion time, the size of the ellipse remains constant throughout the image. Since the major axis of the ellipse is parallel to the edge of the curve, so no harm in increasing it. The minor axis of the ellipse is aligned with the width of the curve. So increasing the ellipse minor axis will eventually make the ellipse protrude outside the boundary of the curve, and the disturbed structure is obtained, and that is precisely where the diffusion should stop eventually.

*τ*=4 in the entropy change graph, representing the characteristic width of the curves present in the image. By stopping the diffusion process by that peak location, the diffused image is shown in Fig. 7 d. The image is largely undisturbed with small diffusion effects at the boundaries and ends of the curves. The quantitative measures, of sensitivity and specificity, for the output image, are computed as 82 and 89 %. The peak in entropy change graph, thus, presents itself as a suitable stopping time for the linear anisotropic diffusion process.

## 6 Results and discussion for real fingerprint images

The acquired fingerprint images often show important illumination variations, poor contrast in some areas, and gaps in ridge/valley regions. To reduce the illumination imperfections and generate images more suitable for enhancement and minutia extraction, a preprocessing comprising the non-uniform illumination correction is applied. It occurs due to the very process of scanning a finger. The middle finger surface is thicker as compared to the surrounding region. This results in blocking the light in the middle while the outer surface is fairly highly illuminated. The fingerprint scanner registers this uneven illumination. Consequently, background variation will add bias for different regions of the same image to disturb the ridge/valley contrast. Since the ridge/valley pattern is identified and classified by its gray-level profile, this effect may worsen the performance of diffusion and disturb our spatial entropy analysis. With the purpose of removing this disturbing factor from our experimental analysis, a homomorphic filtering approach is adopted. The process is described below.

where *L*(*x*,*y*) is the fingerprint image, *i*(*x*,*y*) is the background illumination image, and *r*(*x*,*y*) is the reflectance image [29]. Reflectance *r* arises due to the object itself, but the illumination image *i* is independent of the object, is a pure representation of lighting conditions at the time of the image capture. To compensate for the non-uniform illumination, the illumination image part has to be made constant. Illumination is assumed to be slowly varying lending itself in the low-frequency region as compared to the reflectance image that contains abrupt changes, showing a considerable high-frequency attitude.

*i*(

*x*,

*y*) estimate. The difference

*d*(

*x*,

*y*) between original image

*L*(

*x*,

*y*) and background illumination

*i*(

*x*,

*y*) is calculated for every pixel,

*t*=2.13. The scale value

*t*in fingerprint images is linked to the width of the ridges as proposed in [14]. By stopping the process at

*τ*=1, a diffused image is obtained as shown in Fig. 12 c. If we let the diffusion process continue for long time (

*τ*=5), we get a mean image as shown in Fig. 12 d.

*n*is uncorrelated with the unknown signal

*u*(

*t*), it could be reasonable to minimize the covariance of the noise

*u*(0)−

*u*(

*t*) with the signal

*u*(

*t*). The covariance is represented by the correlation coefficient and is given by,

and choose the stopping time *T* so that the expression 31 is as small as possible.

A comparison. Total minutiae found by the detection algorithm enhanced by edge-width-based, correlation-based, and entropy-change-based. The sample images are used from FVC2004 DB2_B 101_1 to 101_6

Edge width-based | Correlation-based | Entropy-change-based | |
---|---|---|---|

Image1 | 220 | 367 | 155 |

Image2 | 200 | 333 | 135 |

Image3 | 222 | 370 | 150 |

Image4 | 208 | 330 | 140 |

Image5 | 224 | 380 | 160 |

Image6 | 206 | 320 | 130 |

Another set of experiments was conducted to assess the suitability of proposed stopping criterion for some extremely low-quality fingerprint images present in the FVC2004 database to assess the ultimate strength of the proposed stopping rule. One such challenging image is displayed in Fig. 17 c. The fingerprint shows broken ridges, salt and pepper noise, non-uniform illumination, and on top of it a dark square patch right at the center. The image was preprocessed first with small median filter of size 3×3 to tackle salt and pepper noise, and was then made to go through homomorphic filtering to eliminate to a larger extent the non-uniform background variations.

*λ*

_{1}=0.01 and

*λ*

_{2}=1−0.01. The diffusion matrix was constructed as before:

*θ*is now precomputed orientation field from the use of directional filter bank framework for the image [40]. The orientation field

*θ*was kept constant in the whole evolution process. The diffusion process was evolved starting from scale \(\tau _{i}= \log (t=\exp (-3))\) and reaching final scale\(\tau _{f}=\log (t=\exp (5.5))\) (providing mean value image) with a step size of \(t=\exp (-3)\). The spatial entropy was computed along the way and reported to be growing entity with steady value at the end, as depicted in Fig. 16 d. The entropy graph contains a multitude of discontinuities corresponding to a small leftover noise particles in the fingerprint after preprocessing. The curve can be smoothed by fitting a piecewise spline while caring for some real big discontinuities. To do so, a smoothing spline function was fitted to the noisy entropy curve with a coarser soothing parameter of value 0.95 on a scale of [0,1]. The entropy change curve is constructed from fitted spline curve and is depicted in Fig. 16 e. It shows a number of peaks representing different structures dominating at different scales. There may well be some small broken parts of otherwise long ridges. The last peak at the farthest end represents the largest dominating structure that may be linked tom average ridge width of the fingerprint. By stopping the linear diffusion process at that peak

*τ*=3.2, the diffused image is displayed in Fig. 16 f. The uneven image contrast can be straightforwardly improved using well-known block-based contrast enhancement scheme such as contrast limited adaptive histogram equalization (CLAHE) [41], to provide evenly-contrasted image, as in Fig. 16 g. The contrast-adjusted image was then binarized with a block-by-block process to result in Fig. 16 h. The binarized result shows a clear fingerprint with ridge/valley structure largely intact (minimum mixing of nearby ridges) with greatly diminishing the intensity of noise. Most of the genuine minutia points (ridge ending and bifurcation points) are still valid and can be easily detected by the subsequent extraction process.

*p*represents the paired minutiae (between the manually extracted and machine extracted),

*a*represents the missing minutiae,

*b*represents the spurious minutiae and

*t*represents the true minutiae. The measure is suppose to give a number between 0 and 1. This goodness index is applied on Fig. 17 c. The GI without enhancement is found to be 0.34, with enhancing using CED [18] is 0.45 and after applying the proposed method is 0.52. A larger test is performed on the 40 images of FVC2000 DB4_B (101 to 105). The averaged GI without enhancement comes out to be 0.26, with enhancing using CED [18] is 0.37 and after applying the proposed method is 0.43.

The proposed stopping rule being an iterated process can be analyzed with its computation complexity profile. The stopping rule involves three nested loops. First one is the do-while loop that let the process runs till it reaches the farthest peak in the entropy change graph, and the remaining two are FOR loops that span the dimensionality of the fingerprint. Therefore, an estimate of the computational complexity associated with the proposed stopping rule can be described as a product *N*×*M*×*I*
*T*
*E*
*R*
*A*
*T*
*I*
*O*
*N*
*S*, where *N* and *M* represents the rows and columns of the fingerprint and ITERATIONS are the count of repetitions to reach the required peak. Since the peaks represent the dominating structure, which is this case is the width of the ridges, an experiment was conducted to see that linkage more explicitly. A sequence of same dimension fingerprint images was created by increasing zoom values and center cropping the resultant image. For each of these images, an identical linear diffusion scheme with precomputed orientation filed was run to locate the desired peak in their respective entropy-change graphs. A plot in Fig. 17 is shown connecting logarithmic scale at which the process stopped and the average width of the ridges in the respective zoomed images. The graph in fig shows the dots, obtained from this experiment, and were fitted with a linear curve having 95 *%* confidence interval. The logarithmic scale, at which the diffusion process stopped, in turn, can provide the number of iterations knowing the step size involved in the diffusion process. Thus, given dimension of the input fingerprint and an estimate of the average ridge width, a reasonable guess at the computation complexity of the proposed stopping rule can be reached.

## 7 Conclusions

In this paper, the entropy-change for an anisotropic diffusion of a fingerprint image is investigated. a unique peak is found, associated with blurring of the dominant structure. This provides a reasonable stopping rule for the anisotropic diffusion process, whose goal is to smooth the image without disturbing the structural information. The numerical results validated the existence of the boundary between under-smooth and over-smooth regions of anisotropic diffusion.

## Declarations

### Acknowledgements

The authors would like to acknowledge the support of the Department of Engineering, Macquarie University, Sydney, Australia for the work presented in this paper.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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