Fingerprint verification system based on curvelet transform and possibility theory

Fingerprint verification system based on curvelettransform and possibility theory

Hanene Guesmi & Hanene Trichili & Adel M. Alimi &Basel Solaiman

# Springer Science+Business Media New York 2013

Abstract A fingerprint feature extraction step represents the success key of the fingerprintverification process. In a matching step, the good processing of those features would generate ameasure that reflects more accurately the similarity degree between the input fingerprint andthe template. In our study, we propose a novel fingerprint feature extraction method based onthe Curvelet transform to reduce the dimensionality of the fingerprint image and to improvethe verification rate. Like all extractors, the features which are generated by the Curvelettransform are usually imprecise and reflect an uncertain representation. Therefore, we pro-posed to analyze these features by a possibility theory to deal with imprecise and uncertainaspect in our novel fingerprint matching method. Thus, this paper focused on presenting anovel fingerprint features extraction method and a novel matching method. The featuresextraction method consists of two main steps: decompose the fingerprint image into a set ofsub-bands by the Curvelet transform and extract the most discriminative statistical features ofthese sub-bands. A possibility based representation of those statistical features would beachieved by a possibility theory. So, the proposed fingerprint matching method is based onthe use of the possibility theory as a global framework, including knowledge representation (asa possibility measure); in order to build a possibility fingerprint knowledge basis to beexploited in order to make a fingerprint verification decision. An extensive experimentalevaluation shows that the proposed fingerprint verification approach is effective in terms offingerprint image representation and possibility verification reasoning.

Multimed Tools ApplDOI 10.1007/s11042-013-1785-1

H. Guesmi (*) : H. Trichili : A. M. AlimiREGIM: REsearch Groups on Intelligent Machines, National Engineering School of Sfax, Sfax, Tunisiae-mail: [email protected]

H. Trichilie-mail: [email protected]

A. M. Alimie-mail: [email protected]

H. Guesmi : H. Trichili : B. SolaimanDepartment of Image and Information Processing (ITI), Telecom-Bretagne, Brest, France

B. Solaimane-mail: [email protected]

Keywords Fingerprint verification . Possibility theory. Curvelet transform . Fingerprint featureextraction

1 Introduction

In 1880, Henry Fauld was the first who, scientifically, suggested the individuality and unique-ness of fingerprints. At the same time, Herschel asserted that he had practised fingerprintverification for about 20 years. This discovery established the foundation of modern fingerprintverification. In a fingerprint-based biometric system, a fingerprint image is represented by anextractor of relevant features. In literature, two main and most important approaches were usedto characterize a fingerprint image: image-based and minutiae-based approaches. The minutiaebased approach is more popular and is still used in most of the modern fingerprint recognitionsystems. The main steps for minutiae extraction are smoothing, local ridge orientation estima-tion, ridge extraction, thinning, and minutia detection. For the low quality images, it is difficultto generate a reliable minutiae set. However, the image-based approaches are more capable todeal with low quality images [13]. In addition, they are able to represent a fingerprint byextracting a fixed length feature vector in a multidimensionality space. The fixed lengthrepresentation makes the application of multidimensional indexing techniques easier (i.e. R-tree [10]). Fingerprint indexing deal with the need to increase the search speed for an unknownfingerprint in the verification problem, where the verification of a person may necessitate acomparison of his fingerprint with all the fingerprint templates stored in a database. Someresearchers such as [1, 2, 15] have shown that an indexing technique based on image-baseddescriptors (i.e. FingerCode [13]) outperforms an approach based on the minutiae-triplets. Also,the representation with a fixed length vector makes image-based approach suitable to becoupled with a learning process, thus approaching the fingerprint verification problem as atwo-class (genuine, imposter) pattern recognition problem [17]. In addition to all these advan-tages, it has been experimentally demonstrated that even if the performance of a stand-aloneimage-based matcher is lower than that obtained by a good minutiae-based one, the fusionbetween these two approaches outperforms the best-stand alone approach [16, 18]. Moreover,in some works [20] showed that using many fingerprint descriptors and several enhancementalgorithms can considerably improve the performance of image-based matchers to becomecomparable to that obtained by a minutiae-based matchers.

Having studied some works of the literature on a fingerprint classification, we chose torepresent the fingerprint by image-based approach and generate a statistical features vector.This type of representation does not take a large storage space. In addition, for the low qualityimages, it is difficult to generate a reliable minutiae set or reliable singular points. Whereas, theimage-based approaches are more capable in dealing with low quality images. Also, theimage-based approaches are able to represent a fingerprint by extracting a fixed length featurevector in a multidimensionality space. The fixed length representation makes the application ofmultidimensional indexing techniques easier. Thus, because of all of the advantages of theimage-based fingerprint, we were encouraged to propose a new image-based fingerprintverification system based on the curvelet transform. We note that in our prior works [9], wehave already integrated the curvelet transform in a fingerprint identification system and wehave proved its good performance to provide a good representation of the fingerprint image.

In the data analysis and pattern recognition field we manipulate some information, mostoften numerical, that is supposed to give an image which is as faithful as possible to reality.However, this information is often imperfect (imprecise, uncertain, vagueness, incomplete,…).It should be noted that imprecision and uncertainty are often abusively two disconcerted terms.

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Imprecision relates to the content of information and a quantitative defect of knowledge, on ameasure, while uncertainty is relative to the truth of the information, characterizing itsconformity to the reality [8]. The necessity to take into account, the imprecision and uncer-tainty of the information to be combined is obviously remarked during the experimental phase.So, we have chosen to combine the primitive issues from proposed feature extraction methodto, thereafter, generate a verification decision. This fusion is based on the possibility theory. Infact, the theories of the uncertain like the possibility theory and the theory of the belieffunctions appear to be of a good degree of success, as in graphic pattern recognition [14,21, 23, 29] even in handwriting recognition [19]. These classification methods seem promisingbecause the decisions don’t require any training stage. Thus, the proposed fingerprint decisionprocess is based on the use of the possibility theory (possibility measure) in order to build afingerprint knowledge basis, in a first step, which would be used in a second step, to perform afingerprint verification reasoning afterwards. The choice of this form of representation andreasoning is motivated by three main issues: the capacity of the possibility theory to deal withall the information types, its ability to cope with all types of information imperfections, and thedecision making based on the possibility measure. Promising results were obtained using theproposed system.

So, we proposed a fingerprint verification system based on the curvelet transform and thepossibility theory. In this system, we proceeded by decomposing the ROI of fingerprint imageinto sub-bands. From each sub-band, we calculated some statistical descriptors. Then, we builta fingerprint possibility knowledge basis from the extracted descriptors. Possibility reasoningwas applied in the matching module to be able to generate the verification decision of ourfingerprint-based biometric system. To prove the performance of our fingerprint verificationsystem based on curvelet transform and the possibility theory, we compared it with sevenfingerprint verification systems: the first, is based on 2D Gabor filter proposed by Jain et al.[13], the second is based on 2D Gabor filter and possibility theory( fingerprint verificationsystem of Jain et al.[13] integrating the possibility theory), the third is based on the invariantsmoment proposed by Yang et al. [26], the fourth is based on the invariant moments and thepossibility theory ( fingerprint verification system of Yang et al. [26] integrating the possibilitytheory), the fifth is the one proposed by Tang [24] which was based on the wavelet transform,the sixth is the system of Tang integrating the possibility theory, the seventh is based on thecurvelet transform without the use of the possibility theory in the fingerprint matching step.This comparison is based on an experimental comparative study. So, our approach was appliedto FVC2002 fingerprint image database and then compared with other approaches. Theremainder of this paper was organized as follows: Section 2 introduced the curvelet transformmethod. The possibility theory was detailed in section 3. In section 4, we described ourproposed fingerprint verification system based on curvelet transform and possibility theory.Section 5, was allocated to revealing and discussing the experimental results. Finally, con-cluding remarks were given in Section 5.

2 Curvelet transform

The Curvelet transform is a geometric transform developed by Emmanuel Candes et al. [5] toovercome the inherent limitations of wavelet like transforms. Curvelet transform is a multi-scale and multi-directional transform with needle shaped basis functions. Basis functions ofwavelet transform are isotropic and thus it requires large number of coefficients to representthe curve singularities. Curvelet transform basis functions are needle shaped and have highdirectional sensitivity and anisotropy. Curvelet obeys a parabolic scaling. Because of these

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properties, Curvelet transform allows almost optimal sparse representation of curve singular-ities [5]. The Curvelet transform at different scales and directions span the entire frequencyspace. So, the Curvelet transform was designed to represent edges and other singularities alongcurves much more efficiently than the traditional transforms, i.e., using fewer coefficients for agiven accuracy of reconstruction.

The implementation of the Curvelet transform can be summarized by the following steps:

1) Sub-band Decomposition: The image is decomposed into log2M (M is the size of theimage) wavelet sub-bands. Then, the Curvelet Sub-bands are formed by performingpartial reconstruction from these wavelet sub-bands at levels j∈{2s, 2s+1} . Thus theCurvelet Sub-band, s=1 corresponds to wavelet sub-bands j=0, 1, 2, 3, Curvelet Sub-band, s=2 corresponds to wavelet sub-bands j=4, 5 and so on.

2) Smooth Partitioning: Each sub-band is subdivided into an array of overlapping blocks.3) Renormalization: Each square resulting in the previous stage is renormalized to unit scale.4) Ridgelet Analysis: Ridgelet transform [3] is performed on each square resulting from the

previous stage and following these steps:

(a) Compute the 2-D Fast Fourrier Transform (FFT) of the image.(b) Perform cartesian to polar conversion. This is achieved by substituting the sampled

values of the Fourier transform obtained on the square lattice with the sampledvalues on a polar lattice.

(c) Compute the 1-D inverse FFT on each angular line.(d) Apply the wavelet transform on the resulting angular lines in order to obtain the

ridgelet coefficients.

So, the Curvelet transform is a ridge transform added with binary square window, whichmeans subdividing a curve into approximate straight enough windows to carry out ridgetransform. However, there exists a large data redundancy in the transform. Therefore, improv-ing the first generation curvelet transform led to the birth of a 2nd generation, which takes onfeatures of faster computation and less redundancy. Also, the Curvelet has a frequency supportin a parabolic-wedge area due to the anisotropic scaling law as width = length2 and the two fastalgorithms based on this theoretical basis are known as Discrete Curvelet Transform [4]. Thefirst algorithm is Fast Discrete Curvelet Transform (FDCT) via unequally spaced Fast FourierTransform (USFFT) in which the Curvelet coefficients are obtained by irregularly samplingthe Fourier coefficients of an image. The second algorithm is known as FDCT via wedge-wrapping based on the series of translation and wrapping of specially selected Fourier samples.Both algorithms have been used to give the same output [4]. In this work, we have applied analgorithm which used the unequally spaced Fast Fourier Transform (USFFT).

3 Possibility theory

After introducing the fuzzy-set theory [27], in 1978, Zadeh proposed the possibility theory[28]. This theory was then developed by many authors like Dubois and Prade [7], Yager [25]among others. It allows manipulating uncertainties of non probabilistic nature. Thus, in theframework of this theory, the imprecise knowledge and the uncertain knowledge can coexistand be treated jointly [8]. Therefore, it considers some knowledge (situation) more or lesspossible in relation to others. It does not model a degree of belief or truth, but it rather suggestsa way to say to what extent the occurrence of an event is possible and how it is certain. In thistheory, one formalizes these two subjective assessments through a measure of possibility and a

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measure of necessity. These two measures take their values in the interval [0, 1]. Therefore, anevent is quite possible if the measure of its possibility is equal to 1, and impossible if this one ishopeless.

A possibility measure provides information on the occurrence of an event A relative to areference set of finite W, but it is not enough to describe the uncertainty of the event. Forcomplete information on A, the Necessity measure (N(A)) indicates the degree to which theembodiment of A is certain. The necessity measure is a dual variable possibility measure; N (A)indicates the degree to which the occurrence of an event A is certain.

This theory allows to represent imprecision and uncertainty, through the intermediary ofpossibilities distributions π on a set of W (that are functions of adherence to fuzzy set) and oftwo functions characterizing the events: the possibility π and the necessity N, definite from thepossibility distribution of an event A ∈ W by:

π Að Þ ¼ Sup π wð Þ;w∈Af g; ð1Þ

N Að Þ ¼ Inf 1−π wð Þð Þ;w∉Af g ¼ 1−π Acð Þ ð2ÞWhere Ac: the complementary of A (the contrary event).Semantically, the logical of possibility is based on the notion of possibility distribution,

noted π, which is typically a function of the interpretations set W in the interval [0, 1]. π (w)represents the degree of compatibility of the interpretation w with the state of the beliefs on thereal world. By convention π(w)=1 means that it is completely possible that the world w is thereal world, whereas π(w)=0 means that it is certain that w is not the real world. A possibilitydistribution is said to be normalized if a world w exists as π(w)=1. Associated to a possibilitydistribution π, the degree of possibility of a formula Φ is the value π(Φ)=maxw|=Φπ(w) of thepossibility measure π at Φ. It values the degree of coherence of Φ with the available beliefs.

3.1 Possibility distributions

To avoid working on the set P(U) of parts of U, a function called possibility distribution isdefine on U. It associates a value in [0, 1] to each element of U. To normalize a distribution, itis sufficient that there is an element of U which is completely possible: ∃ xo ∈U, π(xo)=1.

The possibility distribution is directly related to the concept of the possibility measure by:

∀A∈P Uð Þ;π Að Þ ¼ sup π xð Þ; x∈Af g ð3Þ

π xð Þ ¼ π xf gð Þ ð4ÞThe expression (3) can be extended to the case where A is a fuzzy subset (characterized by

its membership function μA(x) ):

π Að Þ ¼ supx∈U μA xð Þ;π xð Þð Þ ð5Þ

It may be noted that (3) and (5) coincide in the case of ordinary sets. Just take in (5) themembership function corresponding to ordinary sets, called the characteristic function of the set.

The possibility distribution has an important role in the possibility theory just like themembership function in the fuzzy logic. Indeed, the possibility distribution and the member-ship function is the basis for modeling a more or less known event, that the degree ofknowledge is quantified in the interval [0, 1]. μA(x) denotes the degree to which element x

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belongs to the subset A. Similarly, πx(u) expresses the degree of possibility with which thevariable x takes the value u.

3.2 Possibility measure

Let’s consider a referential U. One defines a possibility measure П on the set of P(U) of theparts of U by the following three fundamental axioms:

1. One can define the possibility of an event by a coefficient (possibility degree) includedbetween 0 and 1: П : P(U) → [0, 1]

2. The set of reference U is completely possible or completely compatible with the availableknowledge; one assigns its the maximum possibility degree =1. For the inverse, the emptyset receives the degree zero: π (U)=1 and π (Ø)=0

3. The possibility of an event, formed by a collection of elements, is equal to the possibilitydegree of the preferred value among all elements, in the sense where the favorite value isthe one propertied biggest possibility degree: Ai ∈P(U), π (Ai) = sup π (Ai).

Since the verification problem is equivalent to verify the occurrence of an event and as thepossibility measure reflects the degree of occurrence we us rely on this measure to identify aninput fingerprint.

4 Our fingerprint verification system based on the curvelet transform and Includingthe possibility reasoning in the fingerprint matching step

In this section we present the different steps of our fingerprint verification system. We proceedwith the preprocessing system. Then, we present the proposed fingerprint feature extractionmethod based on the curvelet transform. Next, we describe how we build our possibilitydescription of our fingerprint database (templates). In the final sub-section, we present ourfingerprint matching method based on possibility theory. So, the general approach of ourverification process is presented by the following figure (Fig. 1)

In the training phase of our fingerprint verification system, we start with a preprocessingstep in which we enhance the quality of the image and localize the region of interest. Then, we

Fingerprint image

Preprocessing image Features extraction Possibility representation

Preprocessing image Features extraction Matching & decision

Training phase

Test phase

Data base

(1)

(2)

(1):fingerprint of training; (2): fingerprint of test

Fig. 1 Chart flow of our fingerprint verification system

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proceed to extract the fingerprint features based on the curvelet transform and the statisticalprimitives are calculated. These primitives are considered imprecise and the general fingerprintrepresentation can be uncertain. Therefore, after extracting the fingerprint features and beforethe matching phase, we proposed to append another module to give a possibility representationof the extracted fingerprint features. This representation allows a good way for an impreciseand uncertain aspect of the already extracted primitives.

In the test phase, we go through the following steps: preprocessing, feature extraction withthe same principle as the learning phase. After these steps, we move to calculate the similarityscore between the input fingerprint and the template. The score is calculated in a matching steprelying on possibility reasoning. So in the first sub-section, we describe the preprocessingmodule used by our fingerprint verification system. Then, in the second sub-section, wepresent the proposed fingerprint feature extraction method. The third sub-section presentsour possibility fingerprint description method. Finally, and in the fifth sub-section, we describeour possibility reasoning in the matching step.

4.1 Preprocessing step

In preprocessing step, we proceed to enhance the fingerprint image, to improves its quality,using specific techniques (enhancement, ..). Then, we go on to detect a reference point (pointcore). The area within a certain radius around the detected reference point is then used as aregion of interest (ROI) for the feature extraction.

4.1.1 The fingerprint image enhancement

The fingerprint image has been enhanced using Fourier domain based block-wise contextualfilter approach described in Chikkerur et al. [6]. The image is first divided into smalloverlapping windows, in such a way that the signal can be assumed stationary and can bemodeled approximately as a surface wave. Then the short time Fourier transform (STFT)analysis is applied and the Fourier spectrum of each small region is analyzed to estimate theridge frequency and the ridge orientation. The resulting contextual information obtained fromSTFT analysis is used to filter each window in the Fourier domain.

4.1.2 Localization of the core point

In a fingerprint image, the core point presents a consistent point. We used the core pointdetection algorithm, based on the Poincaré index analysis [12], which is presented below.

1) Estimate the orientation field O using the least square orientation estimation algorithm.2) Smooth the orientation field in a local neighborhood. LetO' be the smoothed orientation field.3) Initialize A, a label image used to indicate the core point.4) For each pixel (i, j) in O', compute the Poincaré index and assign the corresponding pixels

in a value of one if the Poincaré index is (1/2). The Poincaré index at pixel (i, j) enclosedby a digital curve, which consists of a sequence of pixels that are on or within a distanceof one pixel apart from the corresponding curve.

5) Determine the connected components in A.

& If the area of a connected component is larger than 7, a core is detected at the centroidof the connected component.

& If the area of a connected component is larger than 20, two cores are detected at thecentroid of the connected component.

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6) If more than two cores are detected, go back to the second step.7) If two cores are detected, the center is assigned the coordinates of the core point. But, if

only one core is detected, the center is assigned the coordinates of the core point.8) If no core point is detected:

& For each point in the orientation field, compute the covariance matrix of the vectorfield in a local neighborhood.

& For each element in the orientation image, define a feature image I with the largesteigenvalue of the covariance matrix.

– A core is detected at the centroid of the largest connected component in thethresholded image and the center is assigned the coordinates of the core.

4.1.3 Localization of the fingerprint ROI

After the enhancement of the fingerprint image, we pass to the binarization step. Thisbinarization is done by a simple thresholding of the gray scale image. Then we pass to localizethe reference point. The used enhancement method, generates a good quality image, thus thereference point can be correctly located. So, we took the squared ROI (175×175) which isaround the reference point of the binary image. Finally, we pass to thin the ridge of the ROI.

4.2 The proposed Curvelet transform-based fingerprint feature extraction method

Previous research on multi-resolution texture analysis [22] suggested some statistical descrip-tors: energy, entropy, and standard deviation to be applied on the Curvelet sub-bands in orderto represent an image. Encouraging results were obtained in these research works which haveextracted these descriptors from texture decomposed by the Curvelet transform. Also, theentropy was refined and was applied by a lot of research for different fields of application like[11]. These results encouraged us to integrate these statistical parameters in the proposedfingerprint feature extraction method based on curvelet transform.

The construction of the curvelet basis obeys the anisotropic (parabolic) scaling relationbetween its length and width (length≈2−j/2,width≈2−j ) [4]. In addition, the curvelet basis isoscillatory in one direction (x1) and a low-pass filter in other direction (x2). At fine scale 2

−j, acurvelet is a little needle shaped basis whose envelope is a specified ridge of effective length2−j/2 and width 2−j displaying an oscillatory behavior across the irregular main ridge [4]. Onthe other hand, the fingerprint image contains intrinsic geometrical structures and the finger-print ridges have a shape of a curvature set. In fact, these fingerprint features will berepresented effectively because of the use of the curvelet transform and the parabolic scaling.The curvelet transform extracts the aforementioned geometrical fingerprint structures andprovides optimal sparse representation of the curve singularities [5] with very high directionalsensitivity and anisotropy for the fingerprint features.

Principally, the curvelet transform generates sub-band images from the decomposed image.So, extracting the features from these sub-band images and representing it in a compact form isa major problem. To overcome this problem, previous research on multi-resolution textureanalysis [22] suggests some statistical descriptors: energy, entropy, and standard deviation tobe applied on the curvelet sub-bands in order to represent an image. Encouraging results wereachieved in these research works which have extracted these descriptors from texture whichwas decomposed by curvelet transform. These results encouraged us to integrate this idea inour extractor of fingerprint features. In this framework, we have applied the curvelet transform

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on the ROI of the binarized and thinned fingerprint image. The curvelet decompositiongenerates several sub-band images. On these sub-bands, we calculated the statistical descrip-tors for generating the fingerprint features vector.

If a given ROI of fingerprint image were resized to 512×512 and decomposed by thecurvelet transform into 5 scales, the number of the directional sub-band images varies fromscale to scale. So, we would have 32 (8 directions×4 levels) sub-band images at the secondscale and 64 sub-band images in each of the two scales 3 and 4. Each sub-band image will becoded by three statistical features (Energy, entropy, standard deviation). So, in the totality, wewould have one sub-band which is coded by three feature values in the first scale, 32 sub-bandswhich are coded by 32×3 features values in the second scale. In the third scale, we would get 64sub-bands which are coded by 64×3 feature values. Even in the following scale, we would get64×3 feature values. Finally, in the fifth scale, we would get one sub-band which is coded by1×3 feature values. To end up the whole process, we obtain 164×3 feature values to encode theROI of the binarized fingerprint. Thus, our features vector has 3 rows. Each row has 164descriptors values (energy, entropy, or standard deviation) extracted from 164 sub-bands.

So, the structural activity extracted from the Curvelet transform of the image can beanalyzed statistically to generate a fingerprint feature vector. Thus, we have applied theCurvelet transform to the ROI of the binarized and thinned fingerprint. The Curvelet decom-position generates several sub-band images. Then, we calculate the statistical features from allthese sub-bands to generate the fingerprint features vector.

4.3 Possibility fingerprint description

The similar primitives between two fingerprint images descended from two different personproducts an uncertainty on the adherence of the primitives which values are very close. Therelocation of the spatial information, due to the regrouping the information in the same pixelcontained in a whole volume, is due to the sensor and its resolution. It constitutes animprecision on the localization of information on the fingerprint image (partial volume effect).Imprecision and uncertainties which rely on the analysis and fusion are then reinforced in theextracted primitives from the images. This antagonism often arises contradictions in dataanalysis and fusion, since there are several measurements on the same event: if the data isaccurate, then they are probably uncertain, and they may therefore be contradictory. It willtherefore be a decision fusion system that explicitly manages uncertainty and vagueness toavoid inconsistencies. What we’re trying to manage is our fusion features fingerprint to verifythe identification of a given person. For this, we relied on the possibility reasoning in order tomerge and analyze the features which are extracted from a fingerprint image.

4.3.1 Building the possibility knowledge basis

In order to build a possibility knowledge basis, the fingerprint description which is formulatedby the extractor of fingerprint features should be transformed into a possibility description.

Let Ω be the space of fingerprints database representation as Ω = {P1, P2,…, Pm,…, PM}Where Pm: a person number m (represented by a fingerprint image); m ∈1..M ; M : number ofpersons. For each person we have to consider the following steps:

1) On the ROI of fingerprint image, we apply an extractor to generate us a features vectorrepresented as follows: Vect [Fi1, Fi2,…,FiZ]

2) For each primitive Fiz, we estimate a possibility distribution πiz(Fiz) where i ∈ {1,2,…,M} ;z ∈ {1,2,…,Z} ; Ωiz = {definition space of Fiz}

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& This process generates us the RM × Zmatrix assembly the Z possibilities distributions of theprimitives of M persons and generate us a possibilities distributions vector of M persons.

4.3.2 Building the possibility distributions of the primitives

To estimate a possibilities distribution, it is vital to define a function that allows determiningthe possibility similarities degrees between the primitive value which is extracted from theinput fingerprint and all the values of definition space of this primitive.

Let us remind that every extractor generates a set of primitives. To estimate the possibilitiesdistribution of every primitive we should go through the following steps:

1) We determine the definition domain of every primitive while consulting the primitivevalues corresponding to the fingerprint training images:

& Calculate the deviation (D) of the triangular possibilities distribution of this primitive.

D ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

X

j¼1t v j −m

� �2r

ð6Þ

where t:number of templates; vj: value of the primitive Fiz of template j (j ∈ [1.. t]; t=5)and m=(∑ j=1

t (vj))/t

2) Constitute the triangular possibility distribution as follows:

& Provide the possibility degree 1 to the primitive which is extracted from the inputfingerprint image (template).

& From the deviation and the primitive having a possibility equal to 1, we determine thecoordinates of both upper and lower limit (A and B) of the triangular distributionhaving a hopeless possibility. In Fig. 4 we

x Aj ; yA

j� � ¼ v j −D; 0

� �

and x Bj ; y j

B

� � ¼ v j þ D; 0� �

– In the possibility theory, the triangular distribution is a continuous possibility distri-bution with lower limit A, upper limit B and mode X, where xA < xB and xA ≤ xC ≤ xB.The probability density function is given by

f xð Þ ¼a1xþ b1 if x < xCa2xþ b2 if x > xC0 otherwise

8

<

:

ð7Þ

Where a1=1/( xA- xC); b1=xA and a2=1/(xB- xC); b2=xBFigure 2 illustrates an example of a triangular possibilities distribution for one

primitive.

3) For each primitive, we store the parameters of f(x). Thus, in addition to the primitivevalue, we save a1, a2, b1 and b2 to be able, then, to calculate the possibility similarity withanother primitive of a test fingerprint.

4) Repeat 1,2 for each primitive Fiz (z∈ [1..Z] , i∈ [1..M]) )5) For each fingerprint of the training database, repeat 1,2 and 3 to generate the possibility

matrix RM × Z.

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4.4 Fingerprint matching step: possibility reasoning

The objective of the fingerprint verification decision is to compare a given fingerprintimage with one template of the fingerprint database, and thus, to “estimate” whether thisfingerprint is genuine or imposter. This comparison may be conducted by distanceestimation, similarity estimation, etc. In the proposed fingerprint matching method, thefingerprints similarities will be considered and estimated by the means of the possibilitymeasure П. The possibility measure of primitive-template similarity tells us the possibilitylevel that the considered primitive is similar to its corresponding stored template primitive.In, the Fig. 3. we present the chart flow of our matching method based on possibilitytheory.

For a test fingerprint (PFnew) features vector which is generated by our extractor: Vect =[F1x, F2x, …, Fzx… FZx] (Fzx: test fingerprint feature number z where z ∈ [1..Z]; Z is thenumber of the extracted features), and to be able to generate a decision (genuine or imposter),the proposed matching method consists in following these steps:

1) For each primitive (Fzx, vz,new) belonging to PFnew, we calculate the possibility measure that

this primitive value is equal to the value of the corresponding primitive of the template. Thispossibility measure is given as follows:PFnew ¼ Fzx; v z;new;Π v z;newð Þ ¼ f zi v

z;newð Þð Þf g

V z,new value of the primitive Fzx of the test fingerprintП(v z,new) possibility measure for that the primitive value v z,new of the test fingerprint is

similar to primitive of the template (Pi: fingerprint).f zi Membership function of triangular possibilities distribution of the primitive Fzi

(primitive of the template)

F1i … Fzi … FZi

Pi 1i … zi … Zi

.. .. .. .. .. ..1 z Z Mean

Matrix: RMxZ

Matching & decision module

Possibility decision

(Pi)

F1x … Fzx … FZx

1x … zx … Zx

Test fingerprint features

Global Possiblily similaritie

Ωi Ωi Ωi

π π π Π Π Π

Fig. 3 Chart flow of our matching method based on possibility theory

1

0xA xBxC

f(x)Fig. 2 Example of triangular pos-sibilities distribution

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2) The total similarity score between the considered test fingerprint and the stored fingerprintknowledge (possibility template) is then computed by means of possibilities combination,using a combination operator. In the literature, several combination operators werepresented. We can define the following combination operators as: average value, mini-mum, maximum, geometric mean, respectively given in the following equations,

Sim PFnew;PFtemp

� � ¼ 1

Z

X

i¼1Z Sim v i;new;PFtemp

� � ð8Þ

Sim PFnew;PFtemp

� � ¼Zmin Sim v i;new;PFtemp

� ��

i¼1ð9Þ

Sim PFnew;PFtemp

� � ¼Zmax Sim v i;new;PFtemp

� ��

i¼1ð10Þ

Sim PFnew;PFtemp

� � ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

∏i¼1

Z Sim v i;new;PFtemp

� �

Z

r

ð11Þ

where Sim can refer to the necessity or possibility measure. The combination operator selectionis generally related to the desired application purpose. In our matching method we have chosenthe possibility measure to reflect the similarity between two fingerprint images and we haveapplied the average operator which is presented by the Eq. (8)3) If s > Th then the test fingerprint is genuine; else, it is imposter. Th is the Threshold value

assigned; s is the similarity score =s ¼ 1Z ∑ i¼1

Z π v i;new� �

5 Experimental results

Our experiments were carried out on the FVC2002 DB1, DB2, DB3 and DB4 fingerprintdatabases (http://bias.csr.unibo.it/FVC2002/databases.asp). Each subset of FVC2002 databasecontains four distinctive databases: DB1, DB2, DB3, and DB4. The resolution of DB1, DB3,and DB4 is 500dpi, and that of DB2 is 569dpi. Each database consists of 800 fingerprintimages in 256 gray scale levels (100 persons, eight impressions per person). So, we havedivided each set of images into two sub-sets: a test set containing 3 * 100 images and a trainingsub-set containing 5*100 images.

The experimental results were reached on a Pentium running Windows OS with a 2.2 MHzclock speed and the implementations were carried out under MATLAB (R2009b) tools. Inorder to examine and prove the performance of the proposed method based on curvelettransform and the possibility theory, a number of experiments comparing performance withseven methods were carried out on the FVC2002 database. The first method is based on Gaborfilter proposed by Jain et al. [13]. The second is based on Gabor filter and possibility theory,the third is based on the invariants moment proposed by Yang et al. [26], the fourth is based onthe invariant moments and integrate the possibility theory, the fifth is based on the wavelet

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http://bias.csr.unibo.it/FVC2002/databases.asp

transform proposed by Tang [24], the six is based on the wavelet and integrate the possibilitytheory, the seven is our method based on curvelet transform without the possibility theory.Experiments under various conditions have been done to evaluate the performance of theproposed method in terms of accuracy, and to compare with other prominent methods usingpublic databases.

(a) Method of Yang et al. [26]:The fingerprint feature extraction method uses invariant moments to characterize a

fingerprint image. So, after the localisation of the ROI which is centered on the corepoint, the ROI was determined and tessellated into a predefined number of nonoverlap-ping square cells. A set of features consisting of seven invariant moments were extractedfrom each tessellated cell. Those features represent the fingerprint as information of thelocal structure. Thus, a features vector of fixed-length would be generated. Then, tomatch the two corresponding feature vectors of the input fingerprint image and templatefingerprint image, authors moved to calculate measures of similarity relying on theeigenvalue-weighted cosine distance.

(b) Method of Anil Jain et al. [13]:Authors proceeded to find a core point (center point) and define a spatial tessellation

of the image around the core point. Then, they moved to tessellate the ROI was centeredat the reference point. The ROI was divided into a series of concentric bands and eachband is sub-divided into sectors. To remove the effects of sensor noise and gray leveldeformation due to the finger pressure differences, the gray values in every sector werenormalized to a specified constant mean and variance. So, the ROI would be normalized.After that, the author passed to filter the ROI in eight different directions using a bank ofGabor filters, and produced a set of eight filtered images. To form the FingerCode, theycomputed the average absolute deviation from the mean (AAD) of gray values in each ofthe sectors for every filtered image. In the matching step, they rotated the features in theFingerCode cyclically to generate templates corresponding to the rotations of the originalfingerprint image. Finally, the FingerCode of the input fingerprint was matching witheach of the generated templates to obtain matching scores. The final matching score is theminimum of the generated matching scores, which corresponds to the dissimilarity scorebetween two fingerprints.

Table 2 EER(%) comparison of the Gabor-based methods (before and after integrating the possibility theory)

% DB1 DB2 DB3 DB4 Average

Jain et al. [13] 3.51 5.25 6.12 6.86 5.44

Gabor + possibility theory (modified method of Jain et al.) 3.05 3.82 3.86 4.13 3.72

Table 1 EER(%) comparison of the four image-based methods (before and after integrating the possibilitytheory)


Jain et al. [13] 3.51 5.25 6.12 6.86 5.44

Yang et al. [26] 1.63 3.78 4.20 4.68 3.57

Tang [24] 1.49 2.89 4.08 4.17 3.16

Proposed system 1.79 3.03 3.04 4.51 3.09

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(c) Method of Tang [24]The authors were used the wavelet transform to extract fingerprint features. They

proceeded to divide the 128×128 ROI into 16×16 blocks. Then, two-level decomposi-tion is performed using wavelet from each block of ROI. To characterize the ROI,statistical measures (mean energy and standard deviation) of wavelet domain, were used.In the matching module, they calculated a normalized Euclidean distance between theinput feature vector and the template feature vectors.

In the following table, we present the ERR of four fingerprint verification systems Jain et al.[13] system, Yang et al. [26] system, Tang [24] system and the proposed fingerprint verifica-tion system based on curvelet transform and the possibility theory.

Relying on the results of the four fingerprint verification image-based systems, we canconclude that the curvelet transform is more efficient than one based on the Gabor filter [13], theone based the invariant moments [26] and the one based on wavelet transform [24]. Thisapproach gave an average error rate lower than those given by the other three approaches. Thus,from the point of view performance and comparing the proposed method with Jain’s, as shownin Table 1, our proposed method clearly gave encouraging results. When evaluated on theFVC2002 database, our method achieves a better ERR (−2.35 %). It also gave excellent resultswhen evaluated on FVC2002 (DB1) database, the Equal Error Rate (ERR) is only 1.79 %.These results are quite promising and indicate the high performance of the proposed method. Inaddition, our method achieves a better ERR rate (−0.48%) when compared to the method basedon invariant moments proposed by Yang et al. [26]. Also, our method achieves an ERR rate(−0.08%) lightly better, when compared to themethod based on wavelet transform proposed byTang [24]. Thus, we can conclude that our fingerprint verification method based on the curvelettransform might be considered as one of the most successful fingerprint verification methods.

In the table below, we present the ERR of the fingerprint verification system of Jain et al.[13] before and after the integration of the possibility theory in its matching module:

After integrating the possibility theory in the matching module, the verification systemproposed by Jain et al. [13] who used the Gabor filter to extract the fingerprint features, theerror rate has decreased distinctly. For each sub base of the fingerprint database FVC 2002, wedetermined the error rate of the system of Jain et al. before and after incorporating thepossibility theory (Table 2). Therefore, the average error rate of the system of Jain et al. [13]is 5.44 % while after the integration of the possibility theory; the rate decreased to just 3.72 %.

In Table 3, we present the ERR of the fingerprint verification system of Yang et al. [26] andthe same one but in which we integrated the possibility theory in its matching module:

Table 4 EER(%) comparison of the wavelet-based methods (before and after integrating the possibility theory)


Tang et al. [24] 1.49 2.89 4.08 4.17 3.16

Wavelet + possibility theory (modified method of Tang) 1.11 2.31 3.12 2.65 2.30

Table 3 EER(%) comparison of the Invariant moment-based methods (before and after integrating the possi-bility theory)


Yang et al. [26] 1.63 3.78 4.20 4.68 3.57

Invariant moments + possibility theory (modified method of Yang et al.) 1.02 2.64 2.88 3.01 2.39

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According to Table 3, the possibility theory has reduced the error rate of fingerprintverification system based on the invariant moments relative to a system based on momentinvariants [26] without incorporating possibility theory. The average error rate was reducedby 1.18 %, which proves the advantage of the integration of the possibility theory at thematching phase.

In Table 4, we present the ERR of the fingerprint recognition system of Tang [24] and thesame one but in which we integrated the possibility theory in its matching module.

The error rate, of the Tang [24] system, has decreased distinctly, after integrating thepossibility theory in its matching module. Therefore, the average error rate of the system ofTang [24] is 3.16 % while after the integration of the possibility theory; the rate decreased tojust 2.30 %.

We note that our verification process based on curvelet transform (without integrating thepossibility theory), in the matching step, calculate the Euclidian distances between the rows offeatures vectors of the two query fingerprints. Then, the norm of these distances will be takenas a dissimilarity score which will be compared to a threshold to take a final verificationdecision (genuine or imposter).

As the other three systems of [13, 26] and [24], the integration of the possibility theory atthe decision step and the use of the possibility representation of the curvelet primitives tocalculate the dissimilarity possibility score, increased the performance of our fingerprintverification system by decreasing the average ERR rate from 3.09 % to 2.09 % (Table 5).Also, the figure (Fig. 4) confirms the results presented in Table 5.

Table 5 EER(%) comparison of the Curvelet-based methods (before and after integrating the possibility theory)


Curvelet transform 1.79 3.03 3.04 4.51 3.09

Curvelet transform + possibility theory 1.02 1.94 2.63 2.77 2.09

Fig. 4 ROC curve of our fingerprint verification methods (before and after integrating the possibility theory) onFVC2002 DB4 database

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Finally, we present a summarizing table (Table 6) to give a general idea on the performanceof the four systems in which the possibility theory was integrated in their matching methods.So, the four fingerprint verification systems are: the three first systems are those of Jain et al.[13], Yang et al. [26] and Tang [24] integrating the possibility theory in their matching modulesand our system based on the curvelet transform and the possibility theory.

By observing the above table, we can prove that the fingerprint verification system based onthe curvelet transform is still more efficient than the other three systems even after theintegration of possibility theory at the matching module. Also, the possibility theory hasimproved the performance of the four fingerprint verification systems: the one based on theGabor filter [13], the other based on the moment invariants [26], the one based on wavelettransform and our system based on the curvelet transform. Therefore, the error rate of 2.09 %of our fingerprint verification system based on the possibility theory and the curvelet transformis the smallest rate of all the error rates given by the seven other systems based on: the Gaborfilter [13], the invariant moments [26], the wavelet transform [24], the curvelet transform, theGabor filter and the possibility theory, the invariant moments and possibility theory, thewavelet and the possibility theory.

Figure 5 shows the ROC curves of the modified methods of Jain, Yang and Tang along withour proposed method on FVC2002 DB2 database. It can be seen that the proposed algorithmcompares favorably with the other methods.

After having proved that the methods based on the possibility theory are better than thosewhich do not use the possibility theory, we give in the table below an overview of the

Table 6 EER(%) comparison of the four image-based methods (after integrating the possibility theory)


System of Jain et al. [13] + possibility theory 3.05 3.82 3.86 4.13 3.72

System of Yang et al. [26] + possibility theory 1.02 2.64 2.88 3.01 2.39

System of Tang [24] + possibility theory 1.11 2.31 3.12 2.65 2.30

Proposed system + possibility theory 1.02 1.94 2.63 2.77 2.09

Fig. 5 ROC curves of different methods on the FVC2002-DB2 database

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complexity in terms of the time spent by the methods which were used the possibility theory intheir matching modules.

According to Tables 6 and 7, we can conclude that our method has better performances interms of matching accuracy and computational speed comparing with other methods. Formatching accuracy, the proposed algorithm can achieve the average EER of 2.09 %, on theFVC2002 databases. This ERR is the smallest one of all the others ERR of other methods. Thecomputation speed of the proposed algorithm is also much faster than others by showingnearly 0.43 s, for each fingerprint enrollment and matching.

6 Conclusion

In this paper, we introduced a new fingerprint feature extraction method based on the curvelettransform. This method yields competitive results to the three methods which are based onGabor filter, invariant moment and wavelet transform. Also, at the matching stage, we used anew reasoning based on the possibility theory. We also tested this reasoning through itsintegration in the four fingerprint verification systems: a system based on invariant moments,another system based on Gabor filter, a system based on wavelet transform and our systembased on the curvelet transform. This evaluation has demonstrated the good performance ofthis reasoning since this theory has reduced the error rate of these systems.

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Table 7 Average time (s) spent for enrollment and matching using the methods of Jain, Yang, Tang and theproposed method (After integrating possibility theory) on the databases of FVC2002

The method ofJain et al. [13]

The method ofYang et al. [26]

The method ofTang [24]

The proposed method

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DB2 1.09 0.70 0.69 0.41

DB3 2.20 0.93 0.85 0.50

DB4 2.15 0.79 0.94 0.38

Average 1.64 0.77 0.83 0.43

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Automatique des Images Satellite, Thèse de 3ème cycle présentée devant l’Université Paul Sabatier. Toulouse

Hanene Guesmi graduated with a Master’s Thesis in Computer Science from the University of Sfax in Tunisia.She is a researcher in the REGIM laboratory (Research Group on Intelligent Machines). Currently, she ispreparing her Ph.D at the University of Sfax in Tunisia (ENIS). Her research interests include biometrics, patternrecognition and image processing.

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Dr. Hanène Trichili associated researcher at the Department Image and Information Processing (ITI) ofthe Ecole Nationale Superieure des Telecommunications de Bretagne (ENST-Br) and in the REGIMlaboratory (Research Group on Intelligent Machines, Sfax-Tunisia). She holds a PhD in “SignalProcessing and Telecommunications” Ecole Nationale Superieure des Telecommunications de Bretagne(ENST-Br, France), a Ph.D. in Electronics from the National School of Engineers of Sfax (ENIS,Tunisia, within the framework of a joint thesis), a postgraduate degree in Industrial Computing ENIS,an engineering degree in electrical engineering (option: Automatic Control and Computer Engineering)from the ENIS, and a master’s degree in English from the Bourguiba Institute of Modern Languages(University of Tunis El Manar).

Pr. Adel M. Alimi is an IEEE senior member of IAPR, INNS. He graduated in Electrical Engineering in 1990,obtained a Ph.D. and then an HDR both in Electrical Engineering in 1995 and 2000, respectively. He is now anassociate professor in Electrical and Computer Engineering at the University of Sfax. His research interestincludes of intelligent methods to pattern recognition, robotic systems, vision systems, intelligent patternrecognition and industrial processes. He is the associate editor of the international journal Pattern RecognitionLetters.

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Pr. Basel Solaiman received the telecommunication engineering degree from Ecole Nationale Supérieure destélécommunications de Bretagne (Télécom Bretagne). Brest, France, in 1983 and the PhD degree from theUniversity de Rennes I, Brest, France, in 1989. From 1984 to 1985, he was a Research Assistant in theCommunication Group at the Centre d’Etudes et de recherche Scientifique, Damascus, Syria. He joined theImage and Information Processing Department at Télécom Bretagne, Brest, France, in 1992. His current researchinterests include the fields of remote sensing, medical image processing, pattern recognition, neural network, andartificial intelligence.

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Fingerprint verification system based on curvelet transform and possibility theory

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