A new fingerprint image compression based on wave atoms transform

Zehira HADDAD 1,2 , Azeddine BEGHDADI1, Amina SERIR 2, Anissa MOKRAOUI1

1L2TI, Institut Galilee, Universite Paris 1399, Avenue J. B. Clement, 93430 Villetaneuse, France

2LTIR, Faculte d'electronique et d'informatique, USTHBBP 32 EI alia, bab ezzouar, 16111 Alger, Algerie

zehira.haddad@univ-paris13.fr

Abstract- During the last decade the emergence of many trans­forms called geometric wavelets have attracted much attention of re­searchers working on image analysis. These new transforms proposea new representation richer than the classical wavelets multiscalerepresentation. This paper presents a comparative study of thesetransforms in order to determine what is the best transform dedicatedto a particular type of image that is the fingerprint image. Whilewe are not trying to compete with the current standard JPEG2000which is dedicated to all types of images, we are aware that for ~particul~r ~pe of images, we can do better by choosing for this typeof specific Image, a more appropriate tool than classical wavelets.The results show that for fingerprint images the wave atom offersbetter performance than the current transform based compressionstandard.

Keywords- image compression, wavelets, curvelets, wave atoms,wsq.

I. INTRODUCTION

The orthogonal transforms have been widely studied andused in image analysis, processing and coding. To overcomethe limitations of Fourier analysis many other orthogonaltransforms have been developed. The most important criteria tobe satisfied by the basis functions are localization in both spaceand spatial frequency and orthogonality. Over the past decades,various efficient and sophisticated wavelet-based schemes forimage analysis, processing and coding have been developed.In image compression, the use of orthogonal transform istwofold. First, it decorrelates the image components and allowsto identify the redundancy. Second it offers a high level ofcompactness of the energy in the spatial frequency domain.These two important properties allow to select the mostrelevant components of the signal in order to achieve efficientcompression. Many orthogonal transforms possess these char­acteristics and have been used for data compression. DiscreteFourier Transform (OFT) was the first orthogonal transformused in data compression. Another, Haar transform makesuse of rectangular basis functions (like Walsh Hadamardtransform). Slant transform (ST) is an attempt to match basisvectors to the areas constant luminance slope. It has betterdecorrelation efficiency. Discrete cosines transform (OCT) isone of the extensive families of sinusoidal transforms. Themost efficient transform for decorrelating input data is theKarhunen loeve transform (KLT) also known as HotellingTransform and Eigenvector Transform [1].

This work was supported by CMEP Tassili 06MDU688..

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Wavelet based compression has received considerable attentionin the 1990s and has been adopted by various importantstandards such as JPEG2000 [2] and MPEG4 [3]. The reasonsfor this interest are mainly due to competitive compressionratio achieved while maintaining good image quality levelwithout introducing annoying artifact such as blocking effectin JPEG [1], [4] . Although wavelets are very adapted toisotropic structure, they are not adapted for anisotropic struc­ture. This transform cannot effectively represent textures andfine details in images for lacking of directionality. In the lastdecade, many transforms have been developed to overcomethis limitation. Indeed, recent studies have shown that it ispossible to define new multiscale transforms more appropriateto the representation of geometric structures.The aim of this work is to show among the various newgeometric wavelets transform, which is the most appropriateto fingerprint images.This paper is organized as follows. After introducing WSQfingerprint compression standard and after proving the limitsof wavelets in section 2, we present in section 3 someadaptive geometric wavelets transforms. Section 4 discussesthe compression scheme followed by the results of differentcompression experiments. Section 5 is devoted to conclusionand perspectives.

II. WSQ STANDARD AND WAVELETS LIMITS

With the rapid development of biometric methods usingfingerprints and given the number and size of these images, theuse of lossy compression data method is inevitable. In responseto this need, the Federal Bureau of Investigations (FBI) ofUSA has developed a compression method specifically forfingerprints, called Wavelet Scalar Quantization (WSQ) [5].WSQ wavelets are bi-orthogonal wavelets (7/9). The structureof the tree decomposition can be determined by applyingdifferent tests on several reference images. From these tests,an optimal decomposition is derived and used for all images.This approach has been followed in WSQ standard. Thetests concluded that the best tree consists of 64 sub bands.WSQ quantization is a scalar quantization with adaptive sub­band. Quantization of each band is a based on a uniformquantization dead-zone scheme. It is adaptive in the sense thatthe quantization step is inversely proportional to the logarithmof the variance in the sub-band. In WSQ-based compression

89

(2)

(4)

a function x = j(x!, X2 )el, and position x~,l ) =

Rj(a,e,b) = JRj(r,e)a-1/2W(t_b) /adt

Where Rj is the Radon transform defined by:

Rj(t, e) = Jj(Xl , x2)8(- xlsine+ X2Cose-t)dxldx2 (3)

Ridgelets analysis is a multiscale analysis performed alongeach radial direction. This mutiscale analysis could beachieved using normalized and transported ridgelets at dif­ferent scales . This is the idea of curvelets [8]. The image isassumed to contain locally rectilinear contours, based on thefact that a curve (a contour) can be represented by severalstraight segments.A curve let can be defined asat the scale z: ' , orientationR;}(k12-

j , k22-j / 2 ) by:

'Pj, l,k(X) = 'Pj (RO,1(x - x~,l ) ))

additional time during the synthesis phase in order to describethe configuration used in the analysis. In our study, we adoptthese approaches. Radon transform, ridgelets [7], curvelets[8], contourlets [17], complexe wavelets [18], cortex transform[19] and oriented pyramid [20] belong to these approaches.

A. Ridglelet and Curvelet transforms

Ridgelet transform [7] have been developed to analyzeobjects whose relevant information is concentrated aroundlinear discontinuities such as lines. Ridgelets coefficients areobtained by a ID wavelet transform of all projections of theimage derived from Radon transform. In summary Ridgelettransform is nothing that wavelet analysis on ID slices of theRadon transform, where the angle is fixed. The continuousRidgelet transform is defined by:

Rj(a,e,b) = JJj(Xl, X2)Wa,O,b(Xl, X2)dx ldx 2 (1)

Where Wa,O,b = a- 1/ 2w((Xl *Cos(e) + X2*sinCe) - b)/a) isa ID wavelet.Ridgelets are expressed through Radon transform by theequation:

"'" ,.1r

t ~

r: ~11'1i"" .

method, each quantized subband is first transformed intoa one-dimensional sequence, and then encoded using RLE(Run Length Encoding) and Huffman algorithm. The WSQcompression technique can compress fingerprint images withcompression ratio ranging from 10 to 1 and 20 to 1 [5].If the effectiveness of wavelets is not debatable for a goodnumber of applications, they are not suited to the represen­tation of anisotropic objects. Indeed, in classical wavelets de­composition the 2D-transform is decomposed into ID horizon­tal and vertical transforms. This creates a partial decorrelationof the image giving a number of high energy coefficientsalong the contours. Figure 1 illustrates the difficulties of thewavelet transform to represent the regularity of a contourcompared to new multi-scale transformed where geometricanisotropy and rotations are taken into account. The purpose ofimage compression is to find a good representation of imageswith few coefficients. The orthogonal wavelet bases makeit possible to obtain such representation. However, althoughthese transforms are very effective for regular areas , homoge­neous textures and point singularities, they do not exploit theregularity of geometric contours. For fingerprint compression,we suggest new transforms requiring less information to keepthe borders of the segments, such as geometric waveletstransforms.

Curvelet transform computation consists of the followingsteps :- Decomposition into subbands.- Partitioning.- Ridgelets analysis (Radon + wavelet transform ID) .The block size can change from a subband to another, accord­ing to the following algorithm:- Apply a wavelet transform (J subbands).- Initialize the block size: Bm in = B 1•

- For j = 1, ...., J do

Fig. 1. Comparison between wavelet transform and adapted transform.

III . GEOMETRIC WAVELETS TRANSFORMS

2D wavelet transforms generate many high-energy coeffi­cients along the contours. To overcome this limitation, somesolutions have been proposed [6]. A first solution consists inusing directional filter banks tuned at fixed scales, positionsand orientations. Another solution is to use an adaptive direc­tional filtering based on a geometric model.So, two main types of approaches, fixed and adaptive havebeen developed. Fixed approaches are based on directionalfilters banks, making them independent of the image toanalyze. Their main advantage is that they do not require

The Curvelet transform is defined by:

c(j , I, k) = (I, 'Pj,l,k) = r j(X)'Pi,l,k(X)dxJR2(5)

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We think that the description in terms of a and 13 willclarify the connections between various transforms of modernharmonic analysis. Wavelets (including Multi Resolution Anal­ysis [10], directional [11] and complex [12]) correspond toa = 13 = 1 , for ridgelets [13] a = 1,13 = 0 , Gabor transforma = 13 = 0 and curvelets [8] correspond to a = 1,13 = 1/2.Wave atoms are defined for a = 13 = 1/2 . Figure 3 illustratesthis classification [9].

- Partition the subbands W j into blocks.Bj .If Umodulo 2 = 1), then Bj +1 = 2Bj •

Otherwise, Bj+1 = Bj .

- Apply Ridgelet transform to each block.Figure 2 illustrates the curvelet tiling in space and frequencydomains.

Fig. 2. Curvelet tiling in space and frequency domains.

1/2

o

Gabor

Wavelets

$f-----~ Curvelets

Ridgelet s

Wave atoms is noted as , with subscript . The indexes areinteger-valued associated to a point in the phase-space definedas follows:x p, = 2-jn ,wp, = 1r2jm,ca, ~ max i=1 ,2Imil ~ 022 j In[9], they suggest that two parameters are sufficient to index alot of known wave packet architectures. The index indicateswhether the decomposition is multi scale (a = 1) or not(a = 0) ; and 13 indicates whether basis elements are localizedand poorly directional (13 = 1) or, on the contrary, extendedand fully directional (13 = 0).

B. Wave atomsIn the classical wavelet transform, when we pass from one

stage to another, only the approximation is decomposed. Whilein the wavelets packets, the decomposition could be pursuedinto the other sets (details and approximation) , which is notoptimal. The optimality is related to the maximum energyof the decomposition. The idea is then to search for thepath yielding to the maximum energy through the differentsubbands.Wave atom is a new member in the family of oriented,multiscale transforms for image processing and numericalanalysis. For the sake of completeness, we recall here somefundamentals notions following [9].Let us define 2D Fourier transform as:

j(w) = Je- ixw f(x)dx

f( x) = (2~)2 Jeixw

j(w)dw

(6)

(7)

Fig. 3. (a , .B) diagram.

In order to introduce the wave atom , let us first consider theID case. In practice, wave atoms are constructed from tensorproducts of adequately chosen ID wavelet packets. An one­dimensional family of real-valued wave packets wtn,n(x) , j 2:0, m 2: 0, n E Z , centered in frequency around ±Wj,m =±1r2jm , with 012j ~ m ~ 022j ; and centered in spacearound Xj,n = 2-jn , is constructed. The one-dimensionalversion of the parabolic scaling inform that the support ofwtn,n(w) be of length 0(22j) , while Wj,m = 0(22j) [9] .The desired corresponding tiling of frequency is illustrated atthe botton of Figure 4. Filter bank-based wavelet packets isconsidered as a potential definition of an orthonormal basissatisfying these localization properties. The wavelet packettree, defining the partitioning of the frequency axis in ID ,can be chosen to have depth j when the frequency is 22j , asillustrated in Figure 4.

Figure 4 presents the wavelet packet tree corresponding towave atoms. More details on wavelet packet trees can be foundin [10]. The bottom graph depicts Villemoes wavelet packetson the positive frequency axis. The dot under the axis indicatesa frequency where a change of scale occurs. The labels "LH",respectively "RH" indicate a left-handed, respectively right­handed window [9].In 2D domain, the construction presented above can bemodified to suit certain applications in image processing ornumerical analysis: The orthobasis variant [9]. In practice, onemay want to work with the original orthonormal basis cpt (x )instead of a tight frame . Since cpt(x) = cp1(x) + cp~ (x) ,

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each basis function CPt (x) oscillates in two distinct directions,instead of one. This is called the orthobasis variant.

Fig. 4. The wavelet-packet tree corresponding to wave atoms.

W2

•WI

Fig. 5. The wave atom tiling of the frequency plane.

Figure 5 represents the wave atom tiling of the specialfrequency plane. The size of the squares doubles when thescale j increases by 1. At a given scale j, squares are indexedby mI, m2 starting from zero near the axes. The dot indicatesthe same change of scale as in Figure 4 and corresponds tothe basis function denoted 1lt~(Xl)Ilt~(X2) in the text [9].

IV. EXPERIMENT RESULTS

The current compression standard JPEG2oo0 was estab­lished for all types of images. This standard uses wavelets.We believe that each type of images can be characterized bya transform which highlights its main features. It is evidentthat wavelets give good results in image compression, butwe believe that for a particular type of image , we can find

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a transform that gives better results than wavelets transformif this appropriate transform characterize better this particulartype of images.

A. Representation and compactness evaluation - A compara­tive study

We adopt the classical method in compression to determinethe most appropriate transform. For each type of image ,we apply different transforms. We obtain the coefficientsrepresenting the image . Then , we apply a threshold tothe coefficients obtained in order to keep only the mostrepresentative coefficients of the image. We reconstruct theimage with the selected coefficients and we determine thequality of the reconstructed image by calculating the PSNRand the mean square error. We use different thresholds, alarge set of images and various transforms.The main objective of this work is to determine the mostappropriate transform for fingerprint image compression. Weuse the non adaptive geometric transforms. Since fingerprintimages contain mainly contours, we chose to use three typesof transforms: the curvelets transform, wave atom transformand wavelets transform. The curvelet transform is chosen forits ability to capture the geometric structure of contours. Thistransform is a generalization of ridgelets for non-rectilinearcontours. In [14], it has been shown that the curvelets yieldbetter compression results than ridgelets. We limit then thecomparison of this study to curvelets compression approach.We have not used contourlets transform because it resemblesto the curve lets transform [IS] . We have also chosen to usethe wave atom transform because it is based on anothertype of decomposition half multi-scale half multi-directional[9]. Finally, we chose to compare with the classical wavelettransform, which is the transform used in the current standardJEPG 2000 and fingerprint image compression standardWSQ.We used different image quality metrics such as SNRwav[16] and PSNR, these metrics yielded the same results, wechoose to present only the PSNR as it is well known.Table I contains the PSNR values with respect to the fractionof selected transform coefficient for different images.

The obtained results show that wavelets give the best PSNRexcept for fingerprint images for which wave atom gives thebest PSNR. We note that for barbabra image, there is rivalitybetween wavelets transform and wave atom. This is explainedby the fact that barbabra image contains many geometricstrucutres of various directions.We conclude that the wave atom transform is the best trans­form for fingerprint images compression. We noticed also thatthe curve let transform does not give good results even if theybetter represent the contours that the classical wavelets. This isdue to the fact that the curvelets are very redundant transformand therefore are not very suitable for compression purpose[IS].

92

45r-~-~-~-~-~-~--,--------,

REFERENCES

Fig. 6. Rate-distortion curves for fingerprint image using wave atoms,curvelets and wavelets.

produces also smoothing curves and other degradations inthe backgroud. These results are in good agreements withthe objective evaluation shown in table 1 and figure 6. Theobtained results confirm the efficiency of wave atom basedcompression compared to wavelets and curvelets compressionfor fingerprint image.

- - - .wavelets

--wave atom-- curvelets

10 15 20 25 30 35 40 45compression rate

25

20

\

35 II\ \

\ \

\\\

"- "

- ~:-:.~ -----

40

'"~ 30a.

V. CONCLUSIONS

We have shown that for a particular type of images, we canfind a transform that is more appropriate than the wavelets.In the first part of our work, we tested for several typesof images, different transforms and we have observed thatwavelets transform give good results for several types ofimages except for fingerprint images for which wave atomtransform is better appropriate. We have also demonstratedthrough the whole chain of compression the superiority ofwave atom over the others transforms.

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Image Transform Percentage of selected coefficients

10"10 20"10 30"10 40"10 50"10

Lena Wa 33.60 36.94 38.67 40.33 42.24

W 33.36 38.08 41.12 43.78 46.52

C 28.29 34.46 38.65 43.11 45.54

Peppers Wa 14.14 15.17 17.39 19.89 22.61

W 29.16 36.32 40.03 43.37 46.43

C 23.15 28.26 33.54 36.39 40.36

Boat Wa 30.35 33.15 34.16 35.84 36.54

W 31.05 35.91 38.26 40.93 44.59

C 28.04 32.91 36.48 39.07 42.74

Flinstones Wa 22.21 24.92 26.66 28.88 30.22

W 21.43 26.33 31.82 35.83 39.17

C 20.48 23.14 26.00 29.17 32.11

House Wa 31.18 34.99 37.52 40.18 40.60

W 34.97 39.79 43.11 46.05 48.61

C 30.14 34.55 37.15 41.3 1 45.05

Barbara Wa 30.94 33.30 37.44 39.60 41.54

W 27.93 32.28 36.84 40.73 44.25

C 29.08 34.17 37.27 40.19 44.12

Fingerprint w. 27.71 29.49 32.63 36.77 37.56

W 23.98 27.07 30.40 33.33 37.22

C 22.18 26.83 30.60 31.70 35 .65

Fingerprint w. 23.50 25.94 28.64 30.51 31.98

W 19.33 22.04 25 08 27.62 33.14

C 20.63 24.24 26.60 28.84 31.31

Table 1. PSNR v.s the fraction of selected coefficients for various images.

B. Compression performance evaluation

In the second part of this work, we developed the wholecompression chain in order to evaluate the performance ofcompression in terms of image quality. Here, we consideronly the main three steps of any transform based compression:transform, quantization and coding. We chose to use a uniformscalar quantization with dead zone since this quantization isused in WSQ and JPEG2000. This quantization is followed byRun length coding (RLE) and Huffman coding used in WSQ.

Figure 6 represents the distortion rate curves for fingerprintimage using the wave atom transform, curvelet transform andwavelet transform. The results show that the most appropriatetransform for fingerprint image compression is wave atomtransform.

Figure 7 represents the original image (a) and three im­ages resulting from: wave atom compression (b), waveletscompression (c) and curvelets compression (d), at the samecompression rate (around 15).At the same bitrate, we obtain a compressed image withoutany visible degradation when using wave atom transform.Whereas wavelets based compression introduces some artifactssuch as visible blocking effect. Curvelet based compression,

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Fig. 7. (a) original image, the results of compression using (b) wave atoms,(c) wavelets and (d) curvelets.

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[19) A. B. Watson, "The cortex transform: rapid computation of simulated

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