Design of Low-Complexity High-Performance Wavelet Filters for Image Analysis

1848 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 22, NO. 5, MAY 2013

Design of Low-Complexity High-PerformanceWavelet Filters for Image Analysis

Ameya K. Naik and Raghunath S. Holambe

Abstract— This paper addresses the construction of a familyof wavelets based on halfband polynomials. An algorithm is pro-posed that ensures maximum zeros at ω = π for a desired lengthof analysis and synthesis filters. We start with the coefficientsof the polynomial (x + 1)n and then use a generalized matrixformulation method to construct the filter halfband polynomial.The designed wavelets are efficient and give acceptable levelsof peak signal-to-noise ratio when used for image compression.Furthermore, these wavelets give satisfactory recognition rateswhen used for feature extraction. Simulation results show thatthe designed wavelets are effective and more efficient than theexisting standard wavelets.

Index Terms— Biometrics, computational complexity, filters,wavelet coefficients.

I. INTRODUCTION

OVER THE last few decades wavelets have gained sig-nificant importance and become a standard tool for

applications such as image coding [1], [2]. Wavelet basedimage coders are adopted in compression standards [3]–[6](JPEG 2000) and are found to outperform traditional codersbased on discrete cosine transform [7]. Recently waveletshave also been accepted as a standardized tool for imageanalysis and feature extraction [8]–[10]. It is well knownthat the performance of wavelet based systems is highlydependent on the choice of wavelet. Consequently waveletdesign becomes highly critical for such systems. Typicallyfor best performance the correlation between properties ofthe image, characteristics of filters and the nature of theapplication needs to be examined. One of the vital filtercharacteristics is the filter length which has direct impact onthe complexity and the system performance. Optimal filterdesigns have been proposed for image compression [11] andfeature extraction [8]–[10] independently. Existing literaturealso focuses on design of signal dependent wavelets [12] forcertain applications. However the computational complexityof such wavelets is usually high. Hence many issues are stillopen in the field of designing low-complexity filters for diverseapplications relating to different image sets.

Manuscript received January 24, 2012; revised May 27, 2012, August 31,2012, and December 20, 2012; accepted December 21, 2012. Date ofpublication January 9, 2013; date of current version March 14, 2013. Theassociate editor coordinating the review of this manuscript and approving itfor publication was Dr. Joan Serra-Sagrista.

The authors are with the S. G. G. S. Institute of Engineering andTechnology, Nanded 431606, India (e-mail: ameyaknaik@ yahoo.com;[email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIP.2013.2237917

This paper proposes a design algorithm for a class ofwavelets that can be used for image compression as well asfeature extraction. Our approach is to start with the design of ahalfband polynomial of the desired length. A method has beensuggested to design the halfband polynomial with maximumvanishing moments [1]. Low pass and high pass decompositionfilters of variable lengths can be obtained by factoring thehalfband polynomial and distribution of its factors. As aresult different wavelets can be obtained with different lengthsand properties. The performance of the designed waveletsis tested for applications such as image compression andfeature extraction. It is observed that in addition to the existingstandard wavelets, other wavelets do exist that perform wellfor compression and feature extraction.

The organization of the paper is as follows. Section IIgives a brief description about the existing wavelets and theircharacteristics. Section III gives the general conditions forconstruction of any wavelet. The construction of waveletsusing halfband polynomial (WCHP) is presented in Section IV.The relation of WCHP with the standard wavelets and theirfilter coefficients are given in Sections V and VI respectively.Section VII deals with the energy distributions of the designedwavelets in the various subbands. The performance of WCHPfor image compression and feature extraction is discussed inSection VIII. Section IX gives the comparison of WCHP withstate of art coders and standards followed by conclusions.

II. RELATED WORK

Although various wavelets exist, the most commonly usedbelong to the category of orthogonal and bi-orthogonal trans-forms. Daubechies [13] presented a method for construct-ing orthogonal wavelets with compact support. Orthogonalwavelets [14] ensure that the noise and error in subbands arenot amplified making the coding design comparatively simple.However such wavelets are not symmetric and do not providelinear phase. Phase linearity [15] is necessary for handlingboundary distortions of finite length signals such as images.Orthogonality and linear phase exists only in case of Haarfilters which are not continuous. Hence a new class of nearsymmetric wavelets [16] (symmlets) was designed. In orderto make the filter symmetric the phase is manipulated so thatit is almost linear. Symmlets find application in many signalprocessing areas such as feature extraction.

Linear phase symmetric filters can also be obtained byrelaxing the orthogonality constraint and using bi-orthogonalwavelets. Bi-orthogonal [17] wavelets are found to have

1057-7149/$31.00 © 2013 IEEE

NAIK AND HOLAMBE: DESIGN OF LOW-COMPLEXITY HIGH-PERFORMANCE WAVELET FILTERS 1849

good compression potential and low computational com-plexity. Various bi-orthogonal wavelets have been designedand proposed for different image processing standards suchas JPEG 2000. The most popular construction is the CDF9/7 [18] (Cohen–Daubechies–Feauveau) which have linearphase and give excellent compression ratios. However thesefilters are designed with coefficients that are irrational numbersand thus require floating point implementation. Subsequentlythe computational complexity increases making the hardwareimplementation difficult. Hence recently there has been muchemphasis on design of filters which yield hardware friendlydyadic coefficients. Furthermore additional filter propertiessuch as large vanishing moments, high frequency selectivity,flatness, etc. are also desired in certain applications. A class ofwavelets which give maximum vanishing moments and dyadiccoefficients are the bi-orthogonal splines. Godavarthy [19]showed how to algorithmically construct wavelets in thisfamily. However such wavelets give good performance onlyfor smaller lengths of reconstruction low pass filters.

Cheng et al. [11] used lifting scheme [20], [21] to designwavelets with binary coefficients. However this constructionwas achieved at the cost of reduced vanishing moments.Similar constructions were proposed by Guangjun et al. [22]and Quan [23] in order to provide optimum filter banks.Patil et al. [24] used general halfband polynomial to designtwo channel bi-orthogonal wavelet filters. By imposing lessernumber of zeros at z = −1 some independent parameters wereobtained. These parameters were then manipulated to obtaindesired frequency response. Though it is explicit it is worthmentioning that for same degree of flatness a higher orderpolynomial is required than the standard ones. This increasesthe computational complexity making hardware implementa-tion difficult. Furthermore the performance of aforementionedwavelets was tested for standard photographic images [25].However many common class of images such as fingerprints,satellite images, etc. do not have the same statistical propertiesas photographic images. Hence it becomes necessary to designwavelets that match the characteristics of images of ourinterest. In addition even for the same class of images waveletsmay perform differently for distinct applications. In order toaddress this problem recently a new class of wavelets basedon THFB (Triplet Half-band Filter Bank) [26] were proposed.These wavelets give superior performances in applicationssuch as compression and feature extraction. However thesewavelets require a combined (analysis and synthesis) filterlength of 32. The higher filter lengths are generally not suitablefor hardware implementation. In order to overcome theseexisting limitations the design of a new class of waveletsis presented. The wavelets are optimal in terms of hardwarecomplexity, reconstruction quality, recognition accuracy andfilter length. Moreover these wavelets can prove useful indesigning a low-complexity framework [27] for diverse imageprocessing applications.

III. GENERAL BACKGROUND

In this article we consider only FIR filters for implementa-tion of a two channel filter bank (Fig. 1).

Fig. 1. Two channel filter bank with perfect reconstruction.

Let h and g denote low pass and high pass analysis filtersrespectively. The corresponding synthesis filters are given ash and g.

The perfect reconstruction (PR) conditions [1], [2] aregiven as

h(z) · h(z) + g(z) · g(z) = 2z−l (1)

where l is the amount of delay

h(z) · h(−z) + g(z) · g(−z) = 0. (2)

If g(z) = h(−z) and g(z) = −h(−z) then (2) is automati-cally satisfied and (1) reduces to an equation for the productfilter P(z) = h(z) · h(z)

P(z) − P(−z) = 2z−l where l is odd. (3)

The design of a PR filter bank can be given as below.

1) Choose a polynomial P(z) that satisfies (3).2) Factor P(z) = h(z) · h(z).3) Choose g(z) = h(−z) and g(z) = −h(−z).

However in general for an orthogonal filter set the relationbetween the filters can be given as

h(z) = h(−z) and g(z) = g(−z) (4)

h(z) and g(z) form a quadrature mirror filter set.

IV. WAVELETS CONSTRUCTED USING HALFBAND

POLYNOMIAL (WCHP)

In this section we present a method for designing waveletsusing halfband polynomials. The design starts with choosingthe best possible halfband polynomial P(z) of the desiredlength. The polynomial is selected so as to have maximumfactors of z = −1 for the chosen length.

A. Formation of Halfband Polynomial

Step 1: Let h be the length of the decomposition low passfilter and –λ be the length of the reconstruction low pass filter. hand –λ should satisfy the condition rem{(h + –λ), 4} = 0, whererem{.} is remainder after division.

Step 2: Calculate the length of the halfband polynomialP(z) = h(z) · h(z) as � = h + –λ − 1.

Step 3: Calculate the number of unknowns υ = {(�− 1)/2}− 1.

Step 4: The maximum number of vanishing moments forthe product filter P(z) can be given as ϑ = � − υ − 1. Thisimplies rem(P(z), (z + 1)ϑ) = 0.

Step 5: Find the polynomial ρ(z) = (z + 1)ϑ and thecorresponding vector ϕ1xυ containing first υ coefficients of ρ.That is ϕ = [1, ϕ2, ϕ3, . . . , ϕυ ].


TABLE I

FACTORS OF THE HALFBAND POLYNOMIAL FOR DIFFERENT LENGTHS

Step 6: Find the solution to the equation Aυ×υ · Xυ×1 =Bυ×1, where

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 0 0 0 0 0 0φ3 φ2 1 0 0 0 0 0 0 0φ5 φ4 φ3 φ2 1 0 0 0 0 0· · · · · · · · · ·

φυ−1 φυ−2 · · · φ4 φ3 φ2 1 01 φ2 φ3 φ4 · · · · φυ−1 φυ

0 0 1 φ2 φ3 φ4 · · · φυ−20 0 0 0 1 φ2 φ3 · · φυ−4· · · · · · · · · ·0 0 0 0 0 0 0 0 1 φ2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

X =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

χ1χ2χ3·····

χυ−1χυ

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−φ2−φ4−φ6

·−φυ

00000

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

Step 7: Construct the polynomial

Q(z) = 1 + χ1z + χ2z2 + χ3z3 + · · · + χυ−1zυ−1 + χυzυ .

Step 8: Find the roots of Q(z) which lie within the unitycircle and have phase between 0 to π/2. Tabulate the real rootsas Q� and complex roots as QI for different combinations ofh and –λ.

Q� = [Q�1, Q�2, . . . , Q�n

]

and

QI = [QI 1, QI 2, QI 3, . . . , QI m

]

where 2 · m + n = υ/2.Table I gives the factors of halfband polynomial other than

z = −1. The maximum vanishing moments possible fordifferent lengths of halfband polynomial are also given.

It can be noted that for h + –λ = 12 and 20 real roots do notexist other than at z = −1 whereas for h + –λ = 8 complexroots are absent.

Algorithm 1 Algorithm for Constructing Combinations forWCHP. (a) Biorthogonal Wavelets. (b) Orthogonal Wavelets

(a)

(b)

B. Algorithm for Construction of WCHP

We present the construction of wavelets for two halfbandpolynomial lengths namely � = 15 and 19. While designingthe wavelets we choose to have at least two zeros at z = −1for both decomposition and reconstruction low pass filters.

Moreover the filters are constructed with real coefficients.This implies for every complex root at zH B of the halfbandpolynomial there exists a root at z∗

H B . The constructionalgorithm is presented in Algorithm 1.

In the algorithm (Algorithm 1): �n = Number of real roots(other than z = −1) having magnitude less than unity. Im =Number of complex roots having magnitude less than unity.start = Minimum number of zeros at z = −1 for synthesislow pass filters. last = Maximum number of zeros at z = −1for synthesis low pass filters. combi = Various combinationsof I , I , �, � and Z possible.

The degree of the synthesis lowpass filter is given byDeg{h(z)} = 2 · I + 2 · I + � + � + 1 and the length is givenby Deg{h(z)}+ 1. The synthesis low pass filter is constructedby combining I complex roots (with conjugates), I complexinverses (with conjugates), � real roots, � real inverses and Zroots at z = −1. The remaining roots of halfband polynomialcan be used to construct analysis low pass filter. The procedureis repeated for all possible combinations available (combi).


TABLE II

VARIOUS COMBINATIONS FOR WCHP 16/i AND WCHP 20/i

Db: Daubechies Wavelets Rev. Db: Reverse Daubechies WaveletsSymm: Symmetric Wavelets Asymm: Asymmetric waveletsType A: Symmetric Biorthogonal Wavelets of even lengthType B: Symmetric Biorthogonal Wavelets of odd lengthType C: Asymmetric Biorthogonal Wavelets of even lengthType D: Asymmetric Biorthogonal Wavelets of odd length

The wavelets constructed by this method will be referred to asWCHP h+–λ/ i where i is the index number of the combination.It is found that 47 different combinations are possible for� = 15 and 35 for � = 19. Few of the combinations areshown in Table II.

It can be seen that many of the well known standardwavelets such as Daubechies, Symmlets, CDF and splinescan be constructed using this method. In addition some otheruseful wavelets can also be constructed for use in differentapplications.

V. RELATION OF WCHP WITH STANDARD WAVELETS

A. Daubechies Wavelets

Daubechies low pass reconstruction filters are minimumphase with all the roots within the unit circle. Hence the factorsof the reconstruction low pass polynomial can be given as

RDb(z) =n∏

t1=1

(z − Q�t1) ·m∏

t2=1

(z − QI t2)

·m∏

t3=1

(z − Q∗I t3) · (z + 1)z . (5)

Hence for combi(i) = {(I = Im) ∩ ( I = 0) ∩ (� = �n) ∩(� = 0)∩ (Z = ϑ/2)} the combination represents Daubechieswavelets.

B. Near Symmetrical Orthogonal Wavelets (Symmlets)

Symmlets are near symmetric wavelets constructed by tak-ing roots having magnitude less than unity and reciprocal ofthe remaining roots. A filter with coefficients an is called linearphase if the phase of the function a(ξ) = ∑

n ane−2nξ is alinear function of ξ . Symmlets are constructed by maintainingthe phase deviation from linear function as minimum aspossible. WCHP 16/7 and WCHP 20/4 belong to this category.

TABLE III

LOW PASS FILTER COEFFICIENTS FOR WCHP 16/i AND WCHP 20/i

C. Cohen–Daubechies–Feauveau Bi-Orthogonal Wavelets

CDF 9/7 low pass reconstruction filters are constructed byusing half the maximum zeros at z = −1 (4 zeros) and all thereal roots of Q(z) along with its reciprocals.

RCDF(z) = {R1 · R2}R1 = {

(z − QR1) · (z − QR2) · · · (z − QRR )}

R2 ={

1

(z − QR1)· 1

1(z − QR2)· 1

1(z − QRR )

· (z + 1)z}. (6)

Hence the combination combi(i) = {(I = 0) ∩ ( I = 0) ∩(� = �n) ∩ (� = �n) ∩ (Z = ϑ/2)} represents CDF 9/7wavelets.

D. Bi-Orthogonal Spline Wavelets

In case of splines the reconstruction low pass filter poly-nomial is of the form (z + 1)

–λ−1. Hence combi(i) = {(I =0) ∩ ( I = 0) ∩ (� = 0) ∩ (� = 0) ∩ (Z = h − 1)} representsthe combination for a spline wavelet of length –λ.

VI. CHARACTERISTICS OF THE PROPOSED WAVELETS

The performance of DWT (Discrete Wavelet Transform)based CODEC depends on the accuracy of the wavelet coef-ficients. A high precision representation ensures good perfor-mance, but at the cost of computational complexity [28]–[30].

Table III gives the quantized low pass filter coefficients fora few WCHP types. It can be seen that the coefficients havedyadic values. Hence these filters can be easily implementedin hardware with only add and shift operations. Fig. 2 showsthe scaling and wavelet functions for a couple of WCHP types(WCHP 16/22 and WCHP 16/47). The corresponding impulseresponses and frequency responses are depicted in Figs. 3and 4 respectively. For the sake of comparisons, similar plotsare also provided for binary 9/7 filters [11].

The responses of WCHP 16/22 and binary 9/7 [11] filtersindicate good resemblance with each other and hence areexpected to give similar performances when used for com-pression and feature extraction applications.


(a)

(b)

Fig. 2. Scaling and wavelet functions of WCHP and binary wavelets.(a) Decomposition. (b) Reconstruction.

VII. ENERGY DISTRIBUTION OF THE

DESIGNED WAVELETS

The suitability of a wavelet for any application dependson the energy distribution in the subbands (LL, HL, LH, andHH) obtained after wavelet decomposition. Table IV givesthe percentage energy distribution for a few wavelets in theWCHP family. The values depicted are obtained after firstlevel wavelet decomposition of an image. The test imageconsidered is a general fingerprint image. It can be seen thatall the orthogonal wavelets retain nearly the same percentageof energy in the average subbands.

However for the bi-orthogonal case the same depends onthe length of the decomposition and reconstruction filters.In general the spline wavelets are found to retain maximumenergy in the average subband. As a result such wavelets cangive satisfactory performance for threshold based compressionapplications. The threshold is selected such that only thecoefficients in the desired subbands (usually LL) are retained.In the transformed image only the coefficients with magnitudeabove the threshold are retained and the rest are replaced withzeroes. The image can then be reconstructed from the modifiedtransform coefficients. Since maximum energy is retained inthe LL subband, the retention of coefficients in this subbandleads to good quality reconstruction. Subsequently the MeanSquare Error decreases leading to a higher PSNR. Howeverin case of feature extraction applications, vital informationpresent in detail subbands (HL, LH, and HH) is required.

(a)

(b)

Fig. 3. Impulse responses of low pass wavelet filters. (a) Decomposition.(b) Reconstruction.

Fig. 4. Frequency responses of WCHP and binary wavelets.

Hence it is desired to have at least some minimum energyin these regions.

VIII. EXPERIMENTAL RESULTS

The designed wavelets are tested for their possible appli-cation in image compression and feature extraction. For thispurpose we have chosen a few wavelets from the WCHPfamily that are found to retain sufficient energy in the averageand detail subbands. We have tested the performance of thesewavelets over seven standard images and well known finger-


TABLE IV

PERCENTAGE ENERGY DISTRIBUTION FOR WCHP 16/i AND WCHP 20/i

Ea : Percentage of Energy retained in LL subbandEh : Percentage of Energy retained in HL subbandEv : Percentage of Energy retained in LH subbandEd : Percentage of Energy retained in HH subband

print databases. For the purpose of fair comparisons we haveconsidered a few standard wavelet based image coders viz.Set Partitioning In Hierarchical tree (SPIHT) [31], EmbeddedZerotree Wavelet coder (EZW) [32] and Adaptive ScanningMethods for Wavelet Difference Reduction (ASWDR) [33].

In order to obtain an insightful analysis the objectivemeasure of peak signal to noise ratio [34], [35] (PSNR)for various bits per pixel (0.1, 0.25, 0.5, 1, and 2 bpp)are measured. Comparisons are also made by tabulating thenumber of additions and shift operations required to implementthe encoder in Canonical Signed Digit (CSD) form.

Table V(a) depicts the PSNR values obtained for sevencommon testing images [36], [37]: Lena (512×512), Barbara(512 × 512), Woman (2048 × 2560, Café (2560 × 2048),Bike (2560 × 2048), Tableware (600 spi, SCID N4A) andThree musicians (600 spi, SCID N7A). The first two images(Lena and Barbara) are the standard images used for testingclassic JPEG encoders. The next three images (Woman, Caféand Bike) are ISO/IEC high resolution images whereas thelast two (Tableware and Three musicians) are the test imagescommonly used in JPEG 2000 literature. It can be seen thatWCHP 16/22 gives the best performance as compared toother wavelets. The performance is comparable with the 9/7wavelets proposed in [11], [34].

Although WCHP 16/i wavelets perform better than WCHP20/i , there exist a few wavelets such as WCHP 20/34 and20/35 that give acceptable PSNR values.

PSNR values are known to be mathematically convenientand hence generally used for judging image quality. Howeverthey are not well matched to perceived visual quality. Hence,an index known as mean structural similarity (MSSIM)[38]–[40] is used for comparing the original image withthe reconstructed image. The similarity measure is given byS(x, y) = f (l(x, y), c(x, y), s(x, y))

where l(x, y) = luminance comparison function

c(x, y) = contrast comparison function

s(x, y) = structure comparison function.

Fig. 5. PSNR versus bit rate for WCHP and binary wavelets.

It can be noted that the MSSIM values (Table V(b)) are inaccordance with the respective PSNR values. However a fewexceptions do exist such as SI6 and SI7 for bpp = 0.25 wherehigher MSSIM value is obtained for the former but a higherPSNR value is obtained for the later.

Fig. 5 shows the performance comparison of WCHP fordifferent compression methods (SPIHT, EZW, and ASWDR).It can be seen that the performance of WCHP 16/22 is thebest when SPIHT encoding scheme is used.

Although comparable performances are obtained usingbinary wavelets [11] a marginal gain is still provided byWCHP. Additionally it may be noted that SPIHT encodinggives better results as compared to EZW and ASWDR.

The designed wavelets are also tested over a fingerprintdataset. For fingerprint feature extraction two wavelet basedalgorithms are considered.

A. Fingerprint Feature Extraction Algorithm I [41]

The algorithm is based on the 2-D wavelet decompositionof J levels on a fingerprint image. The transformation resultsin 3 · J + 1 subbands viz. {aJ , [d1

j , d2j , d3

j ] j=1,...,J } where aJ

is a low resolution approximation of the original image anddk

j are the wavelet subands at different scales and orienta-tions. A feature vector of length 3 · J is formed by orderednormalized L2-norms of the wavelet subbands dk

j given by{[e1

j , e2j , e3

j ] j=1,...,J } where ekj is the L2-norn of kth subband

at the j th level. The similarity between two feature vectorsis measured based on the intersection operator proposed bySwain and Ballard [42]. For our simulations we have takenJ = 3 resulting in a feature vector of length nine for each ofthe fingerprint image.

B. Fingerprint Feature Extraction Algorithm II [43]

The fingerprint image undergoes single level waveletdecomposition. Each of the four sub-bands (LL, LH, HL,and HH) undergoes further decomposition. The originalimage is then represented by 64 sub-images (features)[a, d1, d2, . . . , d63] where a is the zero-th sub-image of the64 sub-images, which is a low resolution approximation of


TABLE V

(A) PSNR VALUES FOR WCHP AND BINARY WAVELETS (B) MSSIM VALUES FOR WCHP AND BINARY WAVELETS

SI1: Lena SI2: Woman (Barbara) SI3: Woman (Portrait) SI4: Café SI5: Bike SI6: Tableware SI7: Three musiciansWCHP1: Filter considering 2 factor during quantization WCHP2: Filter without considering 2 factor during quantization

the original image and d j , j = 1, 2, . . . , 63 are the waveletsub-images, containing the image details at different scalesand orientations.

The performance evaluation of WCHP is done on stan-dard databases (National Institute of Standards and Tech-nology NIST [44] and Biometric Systems Lab, Univer-sity of Bologna [45]) of the fingerprint research com-munity. All the images in the databases are gray scaleusing 8 bits per pixel. The NIST database includes2000 fingerprint images from 1000 different persons

whereas the Bologna dataset contains 168 images from21 individuals.

Each image is applied with some preprocessing techniquefor noise removal and quality enhancement. For compari-son [9] a False Acceptance Rate (FAR) and a False RejectionRate (FRR) test is performed. The FAR and FRR are definedas follows:

FAR = Number of accepted imposter claims

Total number of imposter accesses· 100% (7)


Fig. 6. ROC for comparison between WCHP and binary wavelets usingAlgorithm II.

FRR = Number of rejected genuine claims

Total number of genuine accesses· 100%. (8)

Finally the Equal Error Rate (EER) can be calculated asE E R = (F AR + F RR)/2.

For the FAR test the first impression of each finger ismatched against the first impression of all other fingerswhereas for FRR test, each impression of each finger ismatched against all other impressions of the same finger.Table VI gives the EER values for the designed wavelets.

It can be seen that the performance of orthogonal wavelets isusually better as compared to the bi-orthogonal construction.However it can be noted that the best performance can beobtained for the WCHP 16/i category for i = 22 (WCHP16/22). In general the WCHP 16/i wavelets perform betterthan WCHP 20/i . However, few exceptions do exist for theorthogonal case (WCHP 20/4). The spline wavelets (WCHP16/47 and 20/35) perform well only when the length of thereconstruction low pass filter is smaller.

For feature extraction applications it can be seen that theredo exist certain wavelets (WCHP 16/22 and 20/32) that giveexcellent EER performances for both the wavelet based featureextraction algorithms. Moreover it can be noted that cer-tain bi-orthogonal wavelets (WCHP 16/22, 16/32, and 20/35)perform satisfactorily for compression as well as featureextraction.

The performance of the designed wavelets is also comparedwith the existing binary [11], [21], [34] wavelets. Fig. 6 givesthe region operating characteristics (ROC) of WCHP andbinary wavelets as proposed by Cheng [11] and Arias [34].ROC is usually plotted as False Positive Rate (FPR) versusTrue Positive Rate (TPR). FPR and TPR values are calculatedbased on FAR and FRR values obtained from (7) and (8)respectively.

It can be observed that WCHP 16/22 outperforms binary9/7 wavelets [34] when feature extraction Algorithm II is used.Also the performance of WCHP 16/22 is comparable with thatof binary wavelets proposed in [11].

In order to validate the results obtained using a datasetof 2000 fingerprint images, similar plots are also shown forvarying database sizes (1800 and 2200). It can be observed(Fig. 7) that there is negligible deviation in the ROC curves.

TABLE VI

EER PERFORMANCE FOR WCHP AND BINARY WAVELETS

EER 1: Equal Error Rate obtained using fingerprint feature algorithm IEER 2: Equal Error Rate obtained using fingerprint feature algorithm II

Fig. 7. ROC obtained for WCHP 16/22 and binary wavelets for different.

We also compare the performance of DWT based featureextraction methods with other methods based on Fouriertransform [46] and Gabor filters [47]. It can be observed(Fig. 8) that the DWT based methods give superior perfor-mances as compared to non DWT based methods. We alsoinvestigate the performance (Fig. 9) of WCHP at differentdecomposition levels. The MSSIM values represent the qualityof reconstructed fingerprint images when SPIHT algorithm isused.

It is worth noting that at higher decomposition levels theperformance of the wavelets is somewhat similar. Howeverat lower decomposition levels the performance of waveletswith total length (length of decomposition and reconstructionlowpass filters) equal to 20 is better as compared to otherwavelets.


Fig. 8. ROC for performance comparison between wavelet based methodsand other methods. The important region is indicated by the dotted ellipse,and its corresponding magnified version is shown in the dotted rectangle.

Fig. 9. MSSIM versus decomposition levels for WCHP and binary wavelets.The important region is indicated by the dotted ellipse, and its correspondingmagnified version is shown in the dotted rectangle.

TABLE VII

PERCENTAGE REDUCTION IN COMPUTATIONAL COMPLEXITY FOR WCHP

16/i , WCHP 20/i AND ADAPTIVE WAVELETS AS COMPARED TO CDF 9/7

One of the vital parameters for practical application of anyfilters is the hardware complexity. Existing literature focuseson multiplierless design for filter implementation. As seenfrom Table VII, for a particular length of analysis and synthesislowpass filters (e.g. h = 9 and –λ = 7) implementation ofWCHP requires much lesser hardware as compared to theexisting binary wavelets.

Fig. 10. PSNR versus bit rate for JPEG 2000, WCHP, and adaptive waveletsfor standard Lena image.

Fig. 11. MSSIM versus bit rate for JPEG 2000, WCHP, and adaptive waveletsfor standard Lena image.

Finally it can be concluded that certain wavelets such asWCHP 16/22 and 20/35 give satisfactory performances forboth image compression and feature extraction. Consequentlythey can be used for diverse image processing applications.

IX. COMPARISON WITH EXISTING STANDARDS

AND STATE-OF-THE-ART RESULTS

The performance of the designed wavelets is compared withstate of art encoder viz. JPEG 2000 and adaptive wavelet trans-forms [4]–[6]. For comparisons the most commonly employedmeasure viz. PSNR is used (Fig. 10). Also MSSIM values(Fig. 11) are also considered since they are more suitable forjudging perceived visual quality.

For objective performance evaluation, we replace thewavelet transform module of JPEG 2000 [48] by the othertransforms and use the same bit-plane coding (quantizationmodule) and EBCOT technique (entropy coding module) asin JPEG 2000. The same post compression rate distortion(PCRD) strategy of EBCOT (optimal rate control strategy)as recommended in JPEG2000 is used. This ensures a bestcode stream for a given target bit-rate. It can be seen thatthe designed wavelets give nearly same performance as that


of JPEG 2000. However, it may be noted that in WCHPcomputational complexity (Table VII) is reduced significantly.

In general for standard test images, the increase in codinggain offered by adaptive wavelets lies between 1% and 4%(0.71 dB for Lena, 1.03 dB for Bike and 0.49 dB for Caféusing Weighted Adaptive Lifting scheme [4]). Although theperformance of adaptive wavelets [4]–[6] (ADL, WAL, andDA-DWT) is better as compared to proposed wavelets itmay be argued that the implementation of these adaptiveschemes involve much higher computational costs. For DA-DWT with N direction candidates, the prediction step, alongwith the computation of the sum of absolute values, needs tobe performed N times. Hence in general the DA (DirectionalAdaptive)-DWT requires about 5 to 6 times more computationthan the conventional 2-D DWT. Similarly ADL (AdaptiveDirectional Lifting) based image encoder has considerablyhigher complexity since it has to compute the image dependentsegmentation tree, and then search for the optimal predictiondirection in each terminal node.

The complexity of WAL (Weighted Adaptive Lifting) basedtransform is comparable with that of ADL-based imageencoder since both ADL and WAL go through the directionestimation process to obtain the direction information. Inaddition, the sub-pixel interpolation process is required in bothADL and WAL, which further increases the computationalcomplexity. The Sinc 8-tap interpolation filter adopted byADL scheme is slightly more complex as compared to 6-tapdirectional interpolation filter used in WAL. Hence the fixedcomputations involved in WAL are less as compared to theADL scheme.

A brief comparison of computational costs involved inthe aforementioned schemes is presented in Table VII. Forcomparisons the CDF 9/7 wavelet is taken as a reference andthe percentage of reduction in add and shift operations arepresented for different (WCHP, binary and adaptive) wavelets.It can be seen that the adaptive wavelets require much highercomputations (negative percentage of reduction) as comparedto standard CDF 9/7 wavelets. In contrast to this, WCHPoffers a significant computational advantage over other theother wavelet types (binary and adaptive wavelets). Thus itcan be inferred that WCHP gives acceptable visual quality atmuch reduced computational costs. Subsequently they can beused in applications that require lower computations and lesserhardware.

X. CONCLUSION

In this paper design of wavelets based on halfband poly-nomial is discussed. These wavelets give acceptable PSNRvalues and are significantly less complex as compared tothe existing wavelets. Furthermore simulation results depictthat the designed wavelets can be used for fingerprint featureextraction. Thus using WCHP it is possible to design low-complexity systems for image analysis.

ACKNOWLEDGMENT

The authors would like to thank the associate editor andanonymous reviewers for valuable comments and helpfulsuggestions.

REFERENCES

[1] G. Strang and T. Nguyen, Wavelets and Filter Banks. Cambridge, MA:Wellesley-Cambridge, 1996.

[2] P. P. Vaidyanathan, Multirate Systems and Filter Banks. EnglewoodCliffs, NJ: Prentice-Hall, 1993.

[3] A. Skodras, C. Christopoulos, and T. Ebrahimi, “The JPEG 2000 stillimage compression standard,” IEEE Signal Process. Mag., vol. 18, no. 5,pp. 36–58, Sep. 2001.

[4] Y. Liu and K. N. Ngan, “Weighted adaptive lifting-based wavelettransform for image coding,” IEEE Trans. Image Process., vol. 17, no. 4,pp. 500–511, Apr. 2008.

[5] W. Ding, F. Wu, X. Wu, S. Li, and H. Li, “Adaptive directional lifting-based wavelet transform for image coding,” IEEE Trans. Image Process.,vol. 16, no. 2, pp. 416–427, Feb. 2007.

[6] C. L. Chang and B. Girod, “Direction-adaptive discrete wavelet trans-form for image compression,” IEEE Trans. Image Process., vol. 16,no. 5, pp. 1289–1302, May 2007.

[7] N. Ahmed, T. Natarajan, and K. R. Rao, “Discrete cosine transform,”IEEE Trans. Comput., vol. 23, no. 1, pp. 90–93, Jan. 1974.

[8] Z. Weina, H. Jun, Z. Xiaoyang, and Y. Weiwei, “Fingerprint verificationbased on wavelet and edge detection,” in Proc. 9th Int. Conf. Electron.Meas. Instrum., Aug. 2009, pp. 1001–1004.

[9] A. T. B. Jin, D. N. C. Ling, and T. S. Ong, “An efficient fingerprintverification system using integrated wavelet and Fourier-Mellin invarianttransform,” J. Image Vis. Comput., vol. 22, no. 6, pp. 503–513, 2004.

[10] D. Maltoni, D. Maio, A. K. Jain, and S. Prabhakar, Handbook ofFingerprint Recognition, 2nd ed. New York: Springer-Verlag, 2009.

[11] L. Cheng, D. L. Liang, and Z. H. Zhang, “Popular biorthogonal waveletfilters via a lifting scheme and its application in image compression,”IEE Proc.-Visual Image Signal Process., vol. 150, no. 4, pp. 227–232,Aug. 2003.

[12] H. Yoo and J. Jeong, “Signal-dependent wavelet transform and appli-cation to lossless image compression,” Electron. Lett., vol. 38, no. 4,pp. 170–172, Feb. 2002.

[13] I. Daubechies, “The wavelet transform, time-frequency localization andsignal analysis,” IEEE Trans. Inf. Theory, vol. 36, no. 5, pp. 961–1005,Sep. 1990.

[14] J. D. Villasenor, B. Belzer, and J. Liao, “Wavelet filter evaluation forimage compression,” IEEE Trans. Image Processing, vol. 4, no. 8,pp. 1053–1060, Aug. 1995.

[15] O. Ekiz and A. G. Constantinides, “Phase linearity in polyphase filters,”in Proc. Int. Conf. Acoust., Speech Signal Process., vol. 1. Istanbul,Turkey, Jun. 2000, pp. 321–323.

[16] I. Daubechies, “Orthonormal bases of compactly supported wavelets,”Commun. Pure Appl. Math. vol. 41, no. 7, pp. 909–996, 1988.

[17] I. Daubechies, Ten Lectures on Wavelets. Philadelphia, PA: SIAM, 1992.[18] C. I. Daubechies and J. C. Feauveau, “Biorthogonal bases of com-

pactly supported wavelets,” Commun. Pure Appl. Math., vol. 45, no. 5,pp. 485-560, 1992.

[19] S. Godavarthy, “Generating spline wavelets,” in Proc. 36th Annu.Southeast Regional Conf., Apr. 1998, pp. 8–14.

[20] W. Sweldens, “The lifting scheme: A custom-design construction ofbiorthogonal wavelets,” Dept. Math., Univ. South Carolina, Columbia,Tech. Rep. 7, 1994.

[21] G. Yang and S. Guo, “A new wavelet lifting scheme for image com-pression applications,” in Proc. Int. Workshop Intell. Comput. PatternAnal./Synth., Aug. 2006, pp. 465–474.

[22] Z. Guangjun, C. Lizhi, and C. Huowang, “A simple 9/7-tap waveletfilter based on lifting scheme,” in Proc. Int. Conf. Image Process. vol. 2.Thessaloniki, Greece, Oct. 2001, pp. 249–252.

[23] D. Quan and Y. S. Ho, “Image quality optimization based on waveletfilter design and wavelet decomposition in JPEG 2000,” in Proc. Int.Workshop Adv. Image Technol., Jan. 2009, pp. 1–6.

[24] B. D. Patil, P. G. Patwardhan, and V. M. Gadre, “On the design ofFIR wavelet filter banks using factorization of a halfband polyno-mial,” IEEE Signal Process. Lett., vol. 15, pp. 485–488, May 2008,DOI:10.1109/LSP.2008.922295

[25] Image Databases. (2008) [Online]. Available:http://www.imageprocessingplace.com/

[26] A. D. Rahulkar and R. S. Holambe, “Half-iris feature extraction andrecognition using a new class of biorthogonal triplet half-band filterbank and flexible k-out-of-n: A postclassifier,” IEEE Trans. Inf. ForensicsSecurity, vol. 7, no. 1, pp. 230–240, Feb. 2012

[27] O. Rioul and M. Vetterli, “Wavelets and Signal Processing,” IEEE SignalProcess. Mag., vol. 8, no. 4, pp. 14–38, Oct. 1991.


[28] Y. A. Reznik, A. T. Hinds, C. Zhang, L. Yu, and Z. Ni, “Efficientfixed-point approximations of 8×8 inverse discrete cosine transform,”in Proc. SPIE Appl. Dig. Image Process. XXX. vol. 6696, Aug. 2007,DOI: 10.1117/12.740228.

[29] H. Kiya, M. Yae, and M. Iwahashi, “A linear phase two channel filterbank allowing perfect reconstruction,” in Proc. IEEE Int. Symp. CircuitsSyst., San Diego, CA, May 1992, pp 951–954.

[30] M. Martina and G. Masera, “Low-complexity efficient 9/7 wavelet filtersVLSI implementation,” IEEE Trans. Circuits Syst., vol. 53, no. 11,pp. 1289–1293, Nov. 2006.

[31] A. Said and W. A. Pearlman, “A new, fast, and efficient image codecbased on set partitioning in hierarchical trees,” IEEE Trans. CircuitsSyst. Video Tech., vol. 6, no. 3, pp. 243–250, Jun. 1996.

[32] J. M. Shapiro, “Embedded image coding using zerotrees ofwavelet coefficients,” IEEE Trans. Signal Process., vol. 41, no. 12,pp. 3445–3462. Dec. 1993.

[33] J. S. Walker and T. Q. Nguyen, “Adaptive scanning methods for waveletdifference reduction in lossy image compression,” in Proc. IEEE Int.Conf. Image Process., Sep. 2000, pp. 182–185.

[34] G. Strang, Lecture Notes in Control and Information Sciences, NewYork: Springer-Verlag, pp. 143–153, 1996.

[35] D. Salomon, Data Compression: The Complete Reference, 4th ed. NewYork: Springer-Verlag, 2007.

[36] Standard Images. (2010) [Online]. Available: http://www.hlevkin.com/[37] Graphic Technology—Prepress Digital Data Exchange—Part 1: CMYK

Standard Colour Image Data (CMYK/SCID), ISO 12640-1, 2005.[38] Z. Wang, A. C. Borik, H. R. Sheikh, and E. P. Simoncelli, “Image quality

assessment: From error visibility to structural similarity,” IEEE Trans.Image Process., vol. 13, no. 4, pp. 600–612, Apr. 2004.

[39] A. M. Eskicioglu and P. S. Fisher, “Image quality measures and theirperformance,” IEEE Trans. Commun., vol. 43, no. 12, pp. 2959–2965,Dec. 1995.

[40] A. Horé and Z. Ziou, “Image quality metrics: PSNR vs. SSIM,” in Proc.20th Int. Conf. Pattern Recognit., Aug. 2010, pp. 2366–2369.

[41] M. Tico, E. Immonen, P. Ramo, P. Kuosmanen, and J. Saarinen,“Fingerprint recognition using wavelet features,” in Proc. IEEE Int.Symp.Circuits Syst., May 2001, pp. 21–24.

[42] M. J. Swain and D. H. Ballard, “Color indexing,” Int. J. Comput. Vis.,vol. 7, no. 1, pp. 11–32, 1991.

[43] L. Shen and A. Kot, “A new wavelet domain feature for finger-print recognition,” Biomed. Soft Comput. Human Sci., vol. 14, no. 1,pp. 55–59, 2009.

[44] National Institute of Standards and Technology, Gaithersburg, MD[Online]. Available: http://www.nist.gov/srd/nistsd4.cfm

[45] Fingerprint Database. (2000). Biometric Systems Lab, Univ.Bologna, Cesena, Italy [Online]. Available: http:// biolab.csr.unibo.it/databasesoftware.asp

[46] R. Zewail, A. Seif, M. Saeb, and N. Hamdy, “Fingerprint recognitionbased on spectral feature extraction,” in Proc. IEEE 46th Midwest Symp.Circuits. Syst., vol. 1. Dec. 2003, pp. 328–332.

[47] C. J. Lee and S. D. Wang, “Fingerprint feature extraction using gaborfilters,” IEE Electron. Lett., vol. 35, no. 4, pp. 288–290, Feb. 1999.

[48] JPEG 2000 Image Coding System: Reference Software. (2009) [Online].Available: http://www.jpeg.org

Ameya K. Naik received the B.E. degree in elec-tronics from Shah and Anchor Kutchhi Engineer-ing College, University of Mumbai, Mumbai, India,and the M.E. degree from the Vivekanand Educa-tion Society’s Institute of Technology, University ofMumbai, in 2002 and 2004, respectively.

He is currently a Research Scholar with the ShriGuru Gobind Singhji Institute Institute of Engi-neering and Technology, Swami Ramanand TeerthMarathwada University, Nanded, India, and anAssistant Professor with the K. J. Somaiya College

of Engineering, Mumbai. His current research interests include image process-ing, biometrics, and applications of wavelet transform.

Raghunath S. Holambe received the B.E. andM.E. degrees in instrumentation from the Shri GuruGobind Singhji (SGGS) Institute of Engineeringand Technology, Vishnupuri, Nanded, India, in 1985and 1990, respectively, and the Ph.D. degree fromthe Indian Institute of Engineering and Technology,Kharagpur, India, in 1997.

He was a Consultant with Siemens InformationSystems Ltd., Bangalore, India, for two years. Heis currently a Professor with the Department ofInstrumentation Engineering, the SGGS Institute of

Engineering and Technology. His current research interests include digitalsignal processing, image processing, filter banks, applications of wavelettransform, biometrics, and real-time processing of signals using digital signalprocessors.

Design of Low-Complexity High-Performance Wavelet Filters for Image Analysis

Documents

Transcript of Design of Low-Complexity High-Performance Wavelet Filters for Image Analysis