Future Generation Computer Systems 28 (2012) 254–267

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Future Generation Computer Systems

journal homepage: www.elsevier.com/locate/fgcs

Fractional dual tree complex wavelet transform and its application to biometricsecurity during communication and transmissionGaurav Bhatnagar a,∗, Jonathan Wu a, Balasubramanian Raman b

a Department of Electrical and Computer Engineering, University of Windsor, Windsor, Ontario, ON, N9B 3P4, Canadab Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247 667, India

a r t i c l e i n f o

Article history:Received 31 May 2010Received in revised form7 November 2010Accepted 9 November 2010Available online 1 December 2010

Keywords:Fourier transformWavelet transformDual tree complex wavelet transformBiometricsSecurityEncryption

a b s t r a c t

In this paper, the dual tree complex wavelet transform, which is an important tool and recent advance-ment in signal and image processing, has been generalized by coalescing dual tree complexwavelet trans-form and fractional Fourier transform. The new transform, i.e. the fractional dual tree complex wavelettransform (FrDT-CWT) inherits the excellentmathematical properties of dual tree complexwavelet trans-form and fractional Fourier transform. Possible applications of the proposed transform are in biometrics,image compression, image transmission, transient signal processing etc. In this paper, biometric is cho-sen as the primary application and hence a new technique is proposed for securing biometrics duringcommunication and transmission over insecure channel.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Generally, all existing signals are represented in the time do-main and signal processing is applied on the signals in order toextract more information. For this purpose, the signals are trans-formed by different mathematical analysis functions. Among all,the Fourier transform (FT) is the most frequently used method toanalyze a time signal for its frequency content [1–3]. It usuallyrepresents a signal in a series of coefficients based on sinusoidsas analyzing functions and retrieves the global frequency compo-nents of the signal in terms of amplitudes and phases. Althoughthe phase contains the time information it is extremely difficult tojudge the point-to-point relationship between time and frequency.This is because the FT of a signal stretches in frequency the do-main as the time limits of the signal decrease, making it impossi-ble for the transform to concludewhich frequency components areresponsible for the generation of an arbitrarily small time-limitedpart of the signal. Therefore, the FT is only useful for stationary andpseudo-stationary signals and is not applicable to non-stationary,noisy and aperiodic signals. These types of signals can be analyzedusing local analysis methods which usually analyze signals in thelocal area of the signal throughout the signal domain. The first idea

∗ Corresponding author. Tel.: +1 519 563 7462.E-mail addresses: goravb@uwindsor.ca (G. Bhatnagar), jwu@uwindsor.ca

(J. Wu), balarfma@iitr.ernet.in (B. Raman).

0167-739X/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.future.2010.11.012

in this direction comes in the form of the Short Time Fourier Trans-form (STFT) [4,5]. The assumption is that a non-stationary, noisyor aperiodic signal can be divided into small stationary parts us-ing a kernel or window function. The product of the signal andkernel function localized at a specific instant of time can be sub-jected to a FT to obtain the STFT of the signal. The STFT is able to re-trieve both frequency and time information froma signal, but thereexists a quantum principle by which the conjoint time-frequencydomain cannot reach belowa certain value. Therefore, an appropri-ate trade-off between time and frequency information is needed.One of the best trade-offs was discovered by Gabor using Gaussianmodulated complex functions and the process coined as the Ga-bor transform (GT). Although the STFT and GT rectify all the short-comings of the FT, in some cases the STFT and GT are also still notapplicable, for example in the case of real signals having low fre-quencies of long duration and high frequencies of short duration.Such signals can be better described by a transform which has ahigh frequency and low time resolution at low frequencies anda low frequency and high time resolution at high frequencies. Inthese situations, the wavelet transform (WT) can provide a bet-ter description of the signal [6–8]. One advantage of the wavelettransform is its ability to perform local analysis, i.e. the WT isable to reveal signal aspects that other analysis techniques miss,such as trends, breakdown points, discontinuities etc. Alfred Haarwas the first person to mention wavelets and this led to the basictheory for the WT. The process of decomposing and reconstruct-ing a signal using the WT was derived by Morlet and Grossman.

G. Bhatnagar et al. / Future Generation Computer Systems 28 (2012) 254–267 255

The modern form of the WT is developed by Mallat and Mayerusing multiresolution analysis and a scaling function correspond-ing towavelets [9]. In recent advancements in the area of wavelets,a new concept, namely complex wavelets, was introduced byKingsbury [10,11]. Kingsbury shows the limitations of the WT andrectified these with the dual-tree complex wavelet transform (DT-CWT) [12].

Recently, researchers have come up with a new mathemat-ical transform that gives an enhanced directionality to the FT,namely the Fractional Fourier Transform (FrFT) [13–16]. FrFT canbe viewed as the rotation through an angle α of the FT. More pre-cisely, the FrFT can be defined by generalizing the concept of theFT, i.e. by rotating over an angle π/2. Like the classical FT corre-sponds to a rotation in the time frequency plane over an angleα = 1 × π/2 = π/2, the FrFT corresponds to a rotation over anarbitrary angle α = a × π/2 with a ∈ R. Based on the same idea,researchers tried to generalize the wavelet transform and came upwith a new transform, namely the Fractional wavelet transform(FrWT) [17,18]. The FrWT is derived from the essence of the FrFTand the wavelet transform. Due to the close relationship with thewavelet transform and the FrFT, the FrWT also serves as a generalmathematical and numerical tool in the field of signal and imageprocessing. But the issue with the FrWT is that the drawbacks ofthe WT still exist with this transform and in some way hinder itsuse in signal and image processing.

In this paper, a generalized version of the dual tree complexwavelet transform, i.e. the fractional dual tree complex wavelettransform (FrDT-CWT) is proposed in order to rectify thelimitations of the WT and the FrWT. For computer simulations, anefficient algorithm to compute the FrDT-CWT is also proposed. Inorder to explore its strength, biometric security is chosen as thepossible primary application of the proposed transform. For thispurpose, an encryption scheme is proposed to secure frequentlyused biometrics, viz. fingerprints during communication andtransmission over an insecure channel. The experimental resultsdemonstrate that FrDT-CWT is a very efficient tool in encryption,with a very high speed of computation. Some open questions arestill left concerning the physical analogies and further aspects ofFrDT-CWT; these are considered in the present work.

The rest of the paper is organized as follows: in Section 2, theintroduction to the dual tree complex wavelet transform is given,followed by its generalization to obtain the fractional dual treecomplex wavelet transform. The proposed efficient algorithm tocompute FrDT-CWT is given in Section 3. The proposed primaryapplication is briefly described in Section 4 and finally concludingremarks are given in Section 5.

2. Fractional dual-tree complex wavelet transform

The wavelet transform is one of the most important mathe-matical tools, having applications in several different fields suchas physical optics, linear system theory, signal processing and soon. In a nutshell, wavelets are the mathematical functions that cutup a signal into different frequency components and then studyeach component with a resolution matched to its scale. Gener-ally, wavelets are locally oscillating functions which form a basisfor the wavelet transformwhen the fundamental wavelet functionis stretched and shifted. If these stretched and shifted versions ofthe wavelet function are combined with the shift of another func-tion called the scaling function, they form an orthonormal basis ex-pansion of real-valued functions. In other words, any finite energysignal f (t) can be decomposed in terms of wavelets and scalingfunctions as

f (t) =

∞−n=−∞

Cj0,nφj0(t − n)+

∞−j=j0

∞−n=−∞

Dj,nψj(t − n) (1)

where j,φ andψ are the scale, scaling andwavelet function respec-tively. The function φ gives the scaling coefficients (C(n)), whereasψ gives the wavelet coefficients (D(j, n)), which are further ob-tained by the inner product with f (t) i.e.

Cj0,n = ⟨f , φj0⟩ =

∫∞

n=−∞

f (t)φj0(t − n)dt (2)

Dj,n = ⟨f , ψj⟩ =

∫∞

n=−∞

f (t)ψj(t − n)dt. (3)

Here, Eqs. (2) and (3) furnish a time-frequency analysis of signalf (t) by assessing the frequency content controlled by scale fac-tor j at different times controlled by time shift n. From a fine-scale representation of signal f (t), these coefficients Cj0,n and Dj,ncan be computed efficiently in linear time complexity using alow-pass filter (h0(n)), a high-pass filter (h1(n)), and upsamplingand downsampling operations. The wavelet transform provides anefficient representation of signals containing singularities (jumpsand spikes) which are separated by the piece-wise smooth func-tion consisting of lower order polynomials. Despite an efficientrepresentation of signals and linear time computation algorithm,the wavelet transform has some drawbacks which hinder its ap-plication. The main drawbacks of the wavelet transform are shiftvariance, aliasing and lack of directionality. The meaning of shiftvariance is that any small shift in the signal greatly affects thewavelet coefficient oscillation pattern around singularities. The useof iterated downsampling operations for wavelet coefficient com-putation results in substantial aliasing. However, aliasing can becanceled in the inverse transform, but only if there is no change(not even slight) in the wavelet and scaling coefficients. As it is al-ready known that the wavelet transform is a separable transform,which means a higher dimension wavelet transform can be ob-tained by applying a one dimensional wavelet transform along alldirections. But the separable extension of the wavelet transformcaptures very limited directional information. For instance, a 2-Dwavelet provides only three directional components namely hor-izontal, vertical and diagonal, which usually provide informationalong direction 0°, 90° and 45°. Further, the information along di-rection −45° is mixed with the information along 45° in the di-agonal component. To overcome these drawbacks of the wavelettransform, Kingsbury has suggested the use of complex waveletsand developed the dual tree complexwavelet transform (DT-CWT),which allows perfect reconstruction while providing shift invari-ance and directional selectivity. The core idea behind the DT-CWTis the use of a complex valued scaling and wavelet function in Eqs.(1)–(3) i.e.

φ(t) = φ1(t)+ iφ2(t), ψ(t) = ψ1(t)+ iψ2(t) (4)

where φ1 and ψ1 (real parts) are even functions whereas φ2 andψ2 (imaginary parts) and odd functions. The two real and imagi-nary parts are selected in such away that they form aHilbert trans-form pair i.e.

φ1(t) = H[φ2(t)], ψ1(t) = H[ψ2(t)] (5)

where H is the Hilbert transform operator. Using the concept of acomplex wavelet, the complex wavelet transform is given as (i.e.Eqs. (1)–(3) reduces to)

f (t) =

∞−n=−∞

C cj0,n(φ

1j0(t − n)+ iφ2

j0(t − n))

+

∞−j=j0

∞−n=−∞

Dcj,n(ψ

1j (t − n)+ iψ2

j (t − n)) (6)

256 G. Bhatnagar et al. / Future Generation Computer Systems 28 (2012) 254–267

Fig. 1. Decomposition and reconstruction process using the DT-CWT.

where C cj0,n

and Dcj,n are the scaling and wavelet coefficients asso-

ciated with the complex wavelet transform and are given as

C cj0,n = ⟨f , φj0⟩ =

∫∞

n=−∞

f (t)(φ1j0(t − n)+ iφ2

j0(t − n))dt

= C1j0,n + iC2

j0,n (7)

Dcj,n = ⟨f , ψj⟩ =

∫∞

n=−∞

f (t)(ψ1j (t − n)+ iψ2

j (t − n))dt

= D1j,n + iD2

j,n. (8)

Using Eqs. (7) and (8), Eq. (6) can be rewritten as

f (t) =

∞−n=−∞

(C1j0,n + iC2

j0,n)(φ1j0(t − n)+ iφ2

j0(t − n))

+

∞−j=j0

∞−n=−∞

(D1j,n + iD2

j,n)(ψ1j (t − n)+ iψ2

j (t − n)). (9)

The above equation can be rearranged as

f (t) =

∞−n=−∞

C1j0,n(φ

1j0(t − n)+ iφ2

j0(t − n))

+ i∞−

n=−∞

C2j0,n(φ

1j0(t − n)+ iφ2

j0(t − n))

+

∞−j=j0

∞−n=−∞

D1j,n(ψ

1j (t − n)+ iψ2

j (t − n))

+ i∞−j=j0

∞−n=−∞

D2j,n(ψ

1j (t − n)+ iψ2

j (t − n)) (10)

and separating the real and imaginary parts in Eq. (10), we get

f (t) =

∞−

n=−∞

C1j0,n(φ

1j0(t − n)+ iφ2

j0(t − n))

+

∞−j=j0

∞−n=−∞

D1j,n(ψ

1j (t − n)+ iψ2

j (t − n))

+ i

∞−

n=−∞

C2j0,n(φ

1j0(t − n)+ iφ2

j0(t − n))

+

∞−j=j0

∞−n=−∞

D2j,n(ψ

1j (t − n)+ iψ2

j (t − n))

. (11)

Furthermore, Eq. (11) can be written asf (t) = TreeReal + iTreeImaginary. (12)

From the above equation, it is clear that twowavelet tree struc-tures are obtained using complex valued scaling andwavelet func-tions. Hence, this transform is usually called theDual Tree ComplexWavelet Transform (DT-CWT). Themost important property of theDT-CWT is that both the trees have the ability to reconstruct thesignal perfectly. Therefore, the inverse DT-CWT can be viewed asthe inverse wavelet transform of both the real and the imaginarytrees, which gives the two signals with the optimum signal beingcalculated by averaging these two obtained signals. These two realsignals are then averaged to obtain the final output. The implemen-tation of the DT-CWT can be derived from the implementation ofthe wavelet transform. Since two different wavelet trees are ob-tained, one needs two filter banks, one for real tree and another forimaginary tree. Let h0(n), h1(n) denote the low-pass/high-pass fil-ter pair for the real tree filter bank and let g0(n), g1(n) denote thelow-pass/high-pass filter pair for the imaginary tree filter bank. Theprocess for decomposition and reconstruction of the one dimen-sional function f (t) using the DT-CWT is depicted in Fig. 1. Sincethe DT-CWT gives double coefficients as compared to the wavelettransform, the DT-CWT is twice as expensive for one dimensionalsignals. This time complexity will increase as the dimension ofthe function increases. In general, for a N-d function f (tk) : k =

1, 2, . . . ,N the time complexity of the DT-CWT is given by

TDT−CWT = 2NTWT , (13)where TWT is the time complexity of the wavelet transform forN-d function. So theDT-CWT rectifies the drawbacks of thewavelettransform but for the cost of time complexity. A better way to re-duce time complexity is to select either the real or imaginary treeaccording to the requirements for desired purpose. If real tree ischosen then it is called the real DT-CWT whereas in case of imagi-nary tree it is called imaginary DT-CWT.

Now, if the used function in the definition of DT-CWT can bechosen as f (t) = Fα[g(x)](t) (say), i.e. the α-order fractionalFourier transform of the function g(x), then we come up with anew versatile transform, called the Fractional Dual Tree ComplexWavelet Transform (FrDT-CWT). Hence,

f (t) = Fα[g(x)](t) =

∫∞

−∞

Kα(t, x)g(x)dx (14)

where α is called the transform order and Kα(t, x) is the fractionalFourier transform kernel given as

Kα(t, x) =

√1 − i cotαei

t2+x22 cotα−ixt cscα α = nπ

δ(t − x), α = 2nπδ(t + x), α = 2nπ ± π

(15)

G. Bhatnagar et al. / Future Generation Computer Systems 28 (2012) 254–267 257

where n is a given integer. Further, rearranging the kernel Kα(t, x)as

Kα(t, x) = Cαkα(t, x)e−itx cscα (16)

where kα(t, x) = e(i/2)(t2+x2) cotα and Cα = eiα/2/

√2π i sinα, it is

clear that if α = απ/2 then Cα =

1−i cotα

2π . Similarly, if sinα =

0, then by a limiting process the kernel reduces to a Dirac delta(δ(x± t)). In general, the FrDT-CWT of any function corresponds tothe fractional Fourier transform followed by the DT-CWT to get theFrDT-CWT of the input function. The main properties of the FrDT-CWT are summarized as follows.

1. Dual Tree Complex Wavelet Transform Operator: The FrDT-CWTof order α = 0 is the dual tree complex wavelet transformoperator, i.e. using α = 0 one can get the DT-CWT of the inputsignal.

2. Dual Frequency Operator: The meaning of the dual frequencyoperator is that the input signal is transformed by two differenttransforms in succession. The FrDT-CWTwithα =

π2 is the dual

frequency operator, i.e. the FrDT-CWT of order α =π2 gives the

dual frequency (Fourier-complex wavelet) transformed signal.3. Successive applications of the FrDT-CWT : Successive applications

of the FrDT-CWT are equivalent to a single transform whoseorder is equal to the sum of the individual orders i.e.

FrDT − CWTα(FrDT − CWTβ) = FrDT − CWTα+β .

4. Higher Dimensional FrDT-CWT : Due to the separability of boththe transforms, the higher dimensional FrDT-CWT can beobtained by successively taking one dimensional FrDT-CWTsalong all the directions. For instance, the 2-D FrDT-CWT can beviewed as

FrDT − CWTαx,αy [f (tx, ty)](u, v)

= FrDT − CWTty→vαy {FrDT − CWTtx→u

αx{f (tx, ty)}}. (17)

3. Implementation of the fractional dual-tree complex wavelettransform

As far as the computer simulations are concerned, an efficientyet simple implementation of the FrDT-CWT is also proposed. Fromthe above discussion it is clear that the FrDT-CWT is a realization oftheDT-CWT in the fractional Fourier domain. The FrFT has a uniqueproperty of describing the information of the spatial and frequencydomain due to the rotation of time–frequency plane over anarbitrary angle. In contrast, the DT-CWT has a multiresolutionproperty. A combination of these twodomain results into the FrDT-CWT, which exhibits the multiresolution property, describing thespatial as well as the frequency domain information. Therefore,the forward FrDT-CWT can be obtained by first taking the FrFTwith the optimal fractional order α on the input signal followedby the DT-CWT, whereas the reconstruction can be done by takingthe inverse DT-CWT followed by the inverse FrFT to return backto the plane of the input signal. Fig. 2 shows the flow of theproposed implementation and can be summarized as described inthe following subsections.

3.1. Decomposition

1. Transform Order (α)Optimization: In this step, the optimal valueof the transform order (α) is calculated. The value of α isoptimized by the use of a trial-and-error algorithm.

2. Perform the FrFT of transform order α on the input signal.3. Perform the dual tree complex wavelet transform on the

obtained transformed signal from the previous step.

α

α

α α

Fig. 2. Decomposition and reconstruction process for FrDT-CWT.

3.2. Reconstruction

1. Perform the inverse dual tree complexwavelet transformon thetransformed signal.

2. Perform the inverse FrFT of transform order α on the trans-formed signal which is obtained from the previous step to re-construct the original signal.

3.3. Trial-and-error algorithm

A trial-and-error algorithm is used to optimize the transformorder (α) for the FrDT-CWT. In this algorithm, the original signal istransformed via the FrDT-CWT for an arbitrary value of α and theoriginal signal is reconstructed via the inverse FrDT-CWT and thenthe error is calculated between the original and the reconstructedsignal using

ϵ =

∫∞

−∞

w(t)|f (t)−f (t)|2dt (18)

where f (t),f (t) and w(t) are the original signal, reconstructedsignal and weight function respectively. From Eq. (18), it is clearthat the value of α is determined in such a way that the weightedmean-square error between the original and the reconstructedsignal should be minimal. Hence, the value of α is chosen asthe optimized transform order, which gives the minimum errorbetween the original and the reconstructed signal. It is trivial thatthe weighted mean square error is the generalized form of themean square error and coincides with the mean square error ifw(x) = 1. The weight function must be chosen according tothe requirement of the problem. This trial-and-error algorithmmay be long and require many calculations. However, for a givensignal, this process should be done only once. Nevertheless, it isalso possible that the MSE values (ϵ) are equal for more thanone transform order. In this case, the user can choose any ofthe transform orders from the obtained values. The used weightfunctionw(x) reflects howmuch influencef (t) has on f (t), based,say, on ‘‘how far’’ or ‘‘how different’’f (t) is from f (t). For example,we could choose

w(t) =β

1 + |f (t)− f (t)|γ(19)

with

β = α

∫1

1 + |f (t)− f (t)|γdt−1

. (20)

This choice will make nearby points more ‘‘influential’’ on thedisplacement of every point on f (t) towards its new location. γis some positive constant, called the adjustability factor, which isusually set for better influence.

258 G. Bhatnagar et al. / Future Generation Computer Systems 28 (2012) 254–267

D1

D2

D3

D4

A4

Fig. 3. Visual comparison between the DT-CWT and the FrDT-CWT (with α = 0.5 π2 ) considering a sinc function.

In Fig. 3, the comparison between the DT-CWT and the FrDT-CWT is depicted considering normalized Sinc signal. Mathemati-cally, the normalized sinc function is expressed as

Sinc(x) =sinπxπx

. (21)

It is said to be normalized because its integral over all x is equal toone. First, we discuss theDT-CWT. In theDT-CWT, the approximatepart is the sub-sampled version of the signal and, hence, it followsthe shape of the signal. The detail part, on the other hand, actuallyprovides the changes in signal values in subsequent samples.This implies that the detail part describes the derivatives of thesignal itself. This can be explained using the figure. The peaks orextremas of the signal produce zero derivative. Thus, the detailpart is zero at these instances. Again, when the signal rapidlydecreases or increases, the changes in successive samples arelarger, producing maxima or minima in the derivative. Thus, thedetail part contain extrema at the sudden changes of the signal. Onthe other hand, the FrDT-CWT also follows the same phenomenabut will give randomness to the detail parts. Fig. 3 clearly showsthat the detail parts of the DT-CWT are smooth when comparedto the detail parts of the FrDT-CWT. This ability of the FrDT-CWT proves its superiority over the DT-CWT in the sense that forevery differentα it will produce different randomness, i.e. differentdetail coefficients. In Fig. 4, the variation in the randomness ofdetail parts with respect to varying α are depicted. Until now, thebehavior of approximate part of FrDT-CWT has not been discussed.Therefore, we first discuss the variation in the approximation partwith respect to α. In this experiment, the 1-level of decompositionis used in order to get better visualization. The FrDT-CWT willgive randomness to coefficients that lie on either local maxima orlocal minima whereas the randomness is much less (negligible) inthe case of global maxima or minima. Further, if the value of αis varied, the coefficients lying in local maxima or local minima

become smooth (constant) and only global maxima and minimaexist as α approaches 1. It is evident from the previous discussionthat the FrDT-CWT will give randomness to the detail parts andit is maximum when the value of α is close to 0. As it increases,the randomness decreases and is concentrated at the middlecoefficients of the signal. In the case of 2D functions, the FrDT-CWTproduces one complex-valued low-pass subband and six complex-valued high-pass subbands at each level of decomposition, whereeach high-pass subband corresponds to one unique directionθ ∈ {±15,±45,±75}. The impulse responses of the six complexwavelets of the 2-D FrDT-CWT are illustrated in Fig. 5(a). Thefrequency-domain partition resulting from a two-level 2-D FrDT-CWT decomposition is graphically depicted in Fig. 5(b).

4. Primary application: biometric security during communica-tion and transmission

Biometrics [19,20] is an automated method that uses measur-able, physical or physiological characteristics or behavioral traitsto recognize the identity or verify/authenticate the claimed iden-tity of an individual. Usually, biometrics can be divided into twocategories viz. physiological and behavioral biometrics. Physiolog-ical biometrics is related to the shape of the body and includesfingerprint, iris, face, DNA, hand and palm geometry, odour/scent,signature, keystroke dynamics etc. On the other hand, behavioralbiometrics is related to the behavior of a person and includes typ-ing rhythm, gait, voice etc. Biometrics is a unique characteristic ofan individual and it is believed that the chance of two persons, evenidentical twins, having the same biometrics is probably less thanone in a billion. This property makes biometrics not only a verypowerful tool for matching different pieces of information of in-dividuals across multiple databases, but also attempts to enhancesecurity. The main applications of biometrics include physical andlogical access controls, attendance recording, payment systems,security, crime/fraud prevention/detection and border security

G. Bhatnagar et al. / Future Generation Computer Systems 28 (2012) 254–267 259

D1

A1

D1

A1

Fig. 4. Variation in approximate and detail parts with varying transform orders considering the 1-level FrDT-CWT of sinc function.

controls. Among all applications, biometrics is gaining increasingsupport and interest from the research community for securitypurposes [21–26]. Almost all security systems based on biomet-rics use either biometric recognition or authentication. For thispurpose, the biometrics of an individual is usually compared to apreviously stored template and determines validity or authenticitybased on this comparison. However, biometrics is not a panaceafor security because it has some risks of being hacked, modifiedand reused during communication and transmission over insecurenetwork channels [27–32]. Hence, there is a strong need to protectbiometrics during communication and transmission.

In this paper, we concentrate our efforts on securing biometricsduring communication and transmission. For this purpose, themost popular and oldest biometric, i.e. fingerprints, are chosenand security is provided by the encryption technique. Theproposed primary application of the FrDT-CWT is concentrated onsecuring fingerprint images via an encryption technique. Althoughthe proposed encryption technique works efficiently for otherbiometric images, such as iris, palm, signatures etc, the visualresults are given for fingerprint images. The core idea behind the

proposed encryption technique is to first randomize the fingerprintimage using a Markov map and principal component analysis(PCA) followed by the confusion in the FrDT-CWT domain withthe help of a Hilbert space filling curve. The rest of the sectionis organized as follows: in Section 4.1, we briefly describe theproposed security solution for fingerprint images, followed by theexperimental results and discussions in Section 4.2 to show theperformance of the proposed security solution.

4.1. Proposed security solution: encryption technique for biometrics

In this subsection, some of the motivating factors in the designof our approach to biometric security are discussed. The proposedalgorithm uses biometric images and gives the encrypted biomet-ric images. Without loss of generality, assume that F representsthe original image of size M × N . First, the Markov map is iter-ated to get two chaotic sequences which are further stacked in thearrays of sizeM ×N and N ×M respectively, followed by the prin-cipal component computations. Using the principal components,

260 G. Bhatnagar et al. / Future Generation Computer Systems 28 (2012) 254–267

-15 -45 -75

+15+45+75

-15 -45 -75

+15+45+75

For

RealT

reeF

orIm

aginaryT

ree

I

+15

+45

+75

-15

-45

-75

R

a b

Fig. 5. (a) Typical wavelets associated with the 2D FrDT-CWT (b) 2-level FrDT-CWT decomposition of a 2D signal (where R and I are the real axis and the imaginary axis ofthe complex frequency domain, respectively).

( x, y)

( x, y)

α α

α α

Fig. 6. Block diagram of the proposed security solution for biometrics.

the original biometric image is first randomized. The randomiza-tion of the biometric image essentially changes the biometric im-age pixel values randomly. After obtaining a randomized biometricimage, the l-level FrDT-CWT is performed, followed by the confu-sion step using the Hilbert space filling curve. The confusion stepessentially makes the FrDT-CWT coefficients mix-up by changingthe position of coefficients in such a way that the image becomesunintelligible. Finally, the inverse FrDT-CWT is performed to getthe encrypted image. The block diagram of the proposed securitysolution is given in Fig. 6.

4.1.1. Hilbert space filling curveSpace-filling curves [33] can be viewed as methods of travers-

ing an image plane. Although thesemethods aremore complicated

than point movement by rows/columns they have the advantageof moving the point within an area before moving to anotherarea. In addition to their mathematical importance, space-fillingcurves have applications to dimension reduction, mathematicalprogramming, sparse multi-dimensional database indexing, elec-tronics and image compression. Generally, the space-filling curveorders points linearly to preserve the distance between two pointsin the 2D-space. This means that points which are close in spaceand represent similar data should be stored together in the linearsequence.

In 1891, the German mathematician David Hilbert presented awayof traversing two-dimensional space, namely theHilbert spacefilling curve (HSFC) orHilbert curve [34,35]. TheHilbert curve scansan image which has a size of 2m

× 2m pixels. The curve scans the

G. Bhatnagar et al. / Future Generation Computer Systems 28 (2012) 254–267 261

a b c

fedFig. 7. Hilbert space filling curve and their formation of order (a, d) 1 (b, e) 2 (c, f) 3.

Fig. 8. Non-linear Markov map trajectories with different values of µ.

array while never maintaining the same direction for more thanthree consecutive points. Once the curve has strayed three pointsin a straight line, it turns around and comes back. When the curve‘‘turns around’’ it turns right (with respect to the direction that it isfacing) andmoves to the next pixel and turns right again and startsheading back towards (see Fig. 7).

The basic pattern (Hilbert curve of order 1) is a curve whichstarts near the bottom left corner of a box and terminates near thebottom right corner. It has a kink in it, the kink takes it into thetop left and top right of the box. The next curve (Hilbert curve oforder 2) in the sequence is a refinement of this, we consider eachof the quarters to be a boxwith the appropriate orientation, so thatthe curve enters and leaves from the bottom left and leaves in thebottom right (see Fig. 7(d–f)). In a similar manner we can generateHilbert curves of higher orders.

4.1.2. Non-linear Markov mapA one dimensional map [36] M : U → U,U ⊂ R, U usually

taken to be [0,1] is defined by the difference relation

x(i + 1) = M(x(i)), i = 0, 1, 2, . . . (22)

whereM(·) is a continuous and differentiable function that definesthe map and x(0) is called the initial condition. Iterating this func-tion with a newly obtained value as the initial condition, one canget the sequence of desired lengths associated with the map, i.e.x(0), x(1) = M(x(0)), x(2) = M(x(1)), . . .. Further, the differentvalues of x(0) result in different sequences. The obtained sequenceis called the orbit of themap associatedwith x(0). In order to checkthe chaoticity of a map, the Lyapunov exponent(LE), which showsthe divergence rate between nearby orbits, is considered [37,38].Positive values of the LE are the sign of chaoticity. Mathematically,the LE is given as

λ = limL→∞

1L

L−1−l=0

ln |M′(x(l))|. (23)

The map M is said to be a non-linear Markov map [39,40] if it sat-isfies the following conditions1. The map is a piecewise, that is, there exist a set of points 0 =

µ1 < µ2 < · · · < µM = 1, coined the partition points.2. Every partition of map is represented by a non-linear function.3. The map satisfies the Markov property i.e. the partition points

are mapped to partition points:∀i ∈ [0,M], ∃j ∈ [0,M] : M(µi) = µj. (24)

4. The map is eventually expanding, i.e. there exist an integerr > 0 such that

infx∈[0,1]

ddxMr(x) > 1. (25)

For brevity, any map satisfying the above definition will be re-ferred to as non-linear Markov map. Any sequence obtained bythe Markov map has an exponential autocorrelation function anduniform distribution. Another key property, which makes Markovmaps better than others is that their spectral characteristics arecompletely controlled by the parameters of the map (µ). An ex-ample of a non-linear Markov map is illustrated in Fig. 8 and canbe expressed asM(x)

=

4µx1 −

xµ

, x ∈ [0, µ/2]

4x + 1 − µ

µ− 2

µ−

x + 1 − µ

2 − µ

, x ∈ (µ/2, 1].

(26)

262 G. Bhatnagar et al. / Future Generation Computer Systems 28 (2012) 254–267

-1 -0.5 0 0.5 1 1.5

0.2

0.4

0.6

0.8

-1.5 2µ

0

1

x(n)

Fig. 9. Bifurcation diagram of NLMM.

The non-linear Markov map (NLMM) mapped (0, 1) to (0, 1) withµ ∈ (0, 2). The dependence on µ of the proposed map is depictedin Fig. 8, in which the NLMM trajectories are given for µ = 0.25,0.5, 0.75, 1, 1.25, 1.5, 1.75. It is clear from the figure that theNLMM always follow a parabolic path. For µ = 1 the perfectparabolic path is obtained,whereas skewed parabolic paths are ob-tained for other values and all the paths have a center at µ/2. Inorder to judge the chaoticity of the NLMM, the LE is obtained usingEq. (23) and mathematically expressed as

λ = −1 +2 − µ

2ln

16(2 − µ)(µ− 2)

> 0

which is positive for all values of µ. Therefore, the nature of theNLMM is chaotic throughout the domain. Moreover, in order to seethe general behavior and uniform nature of the chaotic map a bi-furcation diagram is used. A bifurcation diagram is a plot of theorbits as a function of µ. Fig. 9 shows the bifurcation diagram ofthe NLMM. From the figure it is clear that the chaotic region startswhen µ = 0 and the generated sequence uniformly covers thewhole domain, i.e. [0, 1] for all values of µ.

4.2. Proposed security solution

4.2.1. Encryption processThe encryption process takes the biometric image as input and

gives an encrypted image as output. First, the biometric image israndomized followed by the confusion in the FrDT-CWT domain.The whole process can be summarized as follows.

1. By adopting k1, k2 andµ1, µ2 as the keys, iterate the non-linearMarkov map (Eq. (26)) to generate two sequences M1 and M2of lengthM × N .

2. Stack the obtained chaotic sequenceM1 andM2 into an array ofsizeM × N and N ×M row-wise respectively and denote themby M1 and M2.

3. Apply PCA [41] on both the arrays to obtain their principalcomponent matrix denoted by P M1

and P M2.

4. Randomization: Obtain the randomized biometric image (FR)using P M1

and P M2as

FR = P M1FP M2

. (27)

5. Perform the l-level (αx, αy)-FrDT-CWT on the randomizedbiometric image, which is denoted by f θR , where θ ∈ {A,H±15,V±45,D±75} for both the real and imaginary tree.

6. Confusion: Shuffle the FrDT-CWT coefficients of each sub-bandusing the Hilbert space filling curve.

7. Perform the l-level (αx, αy)-inverse FrDT-CWT to get the en-crypted biometric image (F e).

In the proposed scheme, the PCA is applied on the arrays of achaotic sequence in order to make those array uncorrelated [42,43]. Another benefit of using the principal components matrix isto ensure that thematrices used in randomization are invertible sothat the inverse randomization process will perfectly reconstructthe biometric image. It is evident that the principal componentma-trix is orthogonal and hence the inverse is just the transpose of thematrices.

4.3. Decryption process

The stressed motive of the decryption process is to obtain thebiometric image as perfectly as possible from the encrypted image.The decryption process can be summarized as follows.

1. Perform the l-level (αx, αy)-FrDT-CWT on the encrypted bio-metric image (F e) which is denoted by f e,θR , where θ ∈ {A,H±15,V±45,D±75} for both the real and imaginary tree.

2. Inverse Confusion: Deshuffle the FrDT-CWT coefficients of eachsub-band using the inverse Hilbert space filling curve.

3. Perform the l-level (αx, αy)-inverse FrDT-CWT to get thedecrypted randomized biometric image (F d

R ).4. By adopting k1, k2 andµ1, µ2 as the keys, step 1 to step 3 of the

encryption process are performed to get principal componentmatrices P M1

and P M2.

5. Inverse Randomization: Obtain the decrypted biometric imagefrom F d

R with the help of the principal component matrices as

Fd = inv(P M1)F d

R inv(P M2) = PTM1

F dRP

TM1. (28)

4.4. Results and discussions

The performance of the proposed security solution for biomet-rics, viz encryption technique, is demonstrated using the MATLABplatform. Among all existing biometrics, the proposed security so-lution is tested on fingerprint images. Fingerprint images are cho-sen because they are the cheapest, fastest, most convenient, mostpopular and reliable way to identify someone. The popularity offingerprints is due to their inherent ease in acquisition and the nu-merous sources (ten figures) available for collection, whereas thereliability can be viewed by a false-acceptance rate that is lowerthan other technologies. Therefore, the Fingerprint Database 3 partB (FVC-DB3-B) from the 2002 Fingerprint Verification Competition(FVC) is used for simulations. FVC-DB3-B consists of 10 differentusers each having enrolled 8 fingerprints from the same finger,a total of 80 fingerprints which are taken from capacitive sensor‘‘100 SC’’ by Precise Biometrics. Fig. 10(a) shows the randomly se-lected fingerprint image from FVC-DB3-B. In the proposed tech-nique, six parameters are used as the keys, these parameters arek1, k2, µ1, µ2 and transform orders of the FrWT (αx, αy). The firsttwo keys are used as the initial seed for NLMM and are taken ask1 = 0.0540 and k2 = 0.9572, whereas the corresponding val-ues of µ are µ1 = 1.5498 and µ2 = 0.7572 respectively. Accord-ing to the trial-and-error algorithm (see Section 3.3), the transformorders turn out to be αx = 0.2785 and αy = 0.5469. The ran-domly selected original fingerprint, and encrypted and decryptedimages using the above mentioned keys are shown in Fig. 10(a,b,c)respectively.

Security is a major issue of the encryption techniques. A goodencryption technique should be robust against all kinds of crypt-analytic, statistical and brute-force attacks. Here, a complete inves-tigation is made on the security of the proposed security solution,

G. Bhatnagar et al. / Future Generation Computer Systems 28 (2012) 254–267 263

a b c

d e f

g h i

j k l Fig. 10. (a) Original fingerprint image (b) Encrypted fingerprint image; Decrypted fingerprint image with (c) all correct keys (d) wrong k1 (e) wrong k2 (f) wrong µ1 (g)wrong µ2 (h) wrong k1, k2, µ1 and µ2 (i) wrong αx (j) wrong αy (k) wrong αx and αy (l) all wrong keys.

such as key-sensitivity analysis, key-space analysis, statistical anal-ysis and numerical analysis, to prove that the proposed encryptiontechnique is secure against most of the common attacks.

Key Sensitivity: According to the principle, a slight change inthe keys never gives perfect decryption for a good security. For thispurpose, the key sensitivity of the proposed technique is validated.In the proposed technique, six keys (k1, k2, µ1, µ2, αx and αy) areused. Among these, the first four are used for the NLMM and theremaining two are used for FrDT-CWT transform orders. In order

to check their sensitivity, we make a slight change in the values ofkeys and then try to decrypt the images. The new chosen values arek1 = 0.054111, k2 = 0.957311, µ1 = 1.549799, µ2 = 0.757111respectively. It is clear that the changes are made in a way suchthat the older values and newer values of keys k1, k2, µ1 and µ2are approximately same. Fig. 10 (d–h) shows the decrypted imagewhen k1, k2, µ1, µ2 and all (k1, k2, µ1, µ2) are wrong. Fig. 10(i–k)show the decrypted image when the transform orders are wrong.

264 G. Bhatnagar et al. / Future Generation Computer Systems 28 (2012) 254–267

a b

c

Fig. 11. The RE between the decrypted and the original fingerprint images as a function of the deviation of the transform order (a) along the x-axis (b) along the y-axis (c)along both axes.

From Fig. 10, it is clear that whenever the transform order alongany axis is wrong, perfect decryption does not occur along thatdirection. Finally, the decrypted imagewhen all six keys are wrongis depicted in Fig. 10(l). Hence, the proposed security solutionis highly sensitive to the keys and one cannot get the correctdecrypted image if even a single key among all is wrong.

Further, in order to quantitatively evaluate the effect causedby the deviation of different fractional orders, the relative error(RE) between the original and decrypted fingerprint image isconsidered. The RE can be defined as follows:

RE =

M∑i=1

M∑i=1

|F dij | − |Fij|

2M∑i=1

M∑i=1

|Fij|2(29)

where F and F d are original and decrypted fingerprint images. Thevalue of RE lies between [0, 1]. If the value of RE is equal to 1 thenthe decrypted image is in a degraded form, i.e. perfect decryption isnot achieved, whereas perfect decryption is achievedwhenever REis 0. For this purpose, the transform orders are deviated by factors△px and △py along x- and y-axis followed by the decryption. TheRE between the original and the decrypted fingerprint image as afunction of the deviation of transform order is shown in Fig. 11,where △px,△py ∈ {−0.05, 0.05}. It is clear from figure that theRE caused by Dp1 is relatively small and the change of RE is slowerwith a small increase, i.e. △px = △py = 0.01. In contrast, the REcaused by △px = △py = 0.05 is the biggest and rapid change.It proves that the decryption process is more sensitive to thefractional order (αx +0.05, αy +0.05) than (αx +0.01, αy +0.01).Hence, a slight change in transform orders gives a slow effect but asignificant change gives a rapid effect on the decryption process.

Key Space Analysis: According to the Kerckhoff’s principle [44],the information system should be secure even if everything aboutthe system, except the key, is publicly available. Hence, keys play avery vital role in the security of an information system. Accordingto the principle, the key space should be large enough for a goodencryption scheme. In the proposed technique, four keys (ki, µi|i =

1, 2) are used for the NLMM. Here, the key space is calculated foronly one key k1 as:

Generate two different sequence K1 and K1 by using k1 andk1 + d as initial values and a pre-defined length L1, i.e. K1 = {0 <k(g) < 1|1 ≤ g ≤ L1} and K1 = {0 < k(g) < 1|1 ≤ g ≤ L1}.Now, the key space is calculated by the mean absolute error be-tween the two generated sequences, i.e.

MAE(K1, K1) =1L1

L1−g=1

|k(g)−k(g)|. (30)

The key space for k1 is equal to 1d0, where d0 is the value of d for

which MAE = 0. After simulations, the value of d0 comes out to be7.2886 × 10−28

≈ 10−28. Similarly, the key spaces of k2, µ1 andµ2 can be computed. The total key space ofwhole technique comesout to be 10120, which is a sufficiently large key space and ensuresthe high security of the proposed system.

Statistical Analysis: While giving the communication theoryof a secrecy system [45], Shannon revealed that ‘‘It is possibleto evaluate the most of the encryption techniques by statisticalanalysis’’. Therefore he suggested two methods based on thehistogram and on the correlations of adjacent pixels in theencrypted image. The basic idea is to compare the histograms ofthe original and encryptedmedia. For a good encryption technique,the histogram of the encrypted media must be fairly uniform, andis significantly different from the histogram of the original one.The histogram of the original, encrypted and decrypted imagesare depicted in the Fig. 12(a–c), which clearly shows that afterencryption the histogram is uniform and decryption makes thehistogram of the decrypted images perfectly similar to the originalone and further shows the ability of perfect decryption of theproposed security solution. To test the correlation between twoadjacent pixels, there are three ways, either take two verticallyadjacent pixels, or take two horizontally adjacent pixels, or taketwo diagonally adjacent pixels, in the encrypted image. First,randomly select P pairs of adjacent pixels and then calculate theircorrelation coefficient as:

cov(x, y) = E(x − E(x))(y − E(y)) (31)

rxy =cov(x, y)

√var(x)

√var(y)

(32)

G. Bhatnagar et al. / Future Generation Computer Systems 28 (2012) 254–267 265

a b c

d e f

Fig. 12. Histogram and correlation plot of two horizontally adjacent pixels in (a, d) Original Fingerprint (b, e) Encrypted (c, f) Decrypted Fingerprint Image.

Table 1Correlation coefficients of two adjacent pixels.

Image Original Encrypted Decrypted

Horizontal 0.9402 −0.0317 0.9378Vertical 0.9134 0.0587 0.9043Diagonal 0.9067 0.0677 0.9023

where x and y are the gray levels of two adjacent pixels in theimage. This phase essentially shows that after encryption thecorrelation among the image pixels is broken, whereas decryptionwill bind the pixelswith the original correlation. Fig. 12(d–f) showsthe correlation distribution of two horizontally adjacent pixelsin the original, encrypted and decrypted fingerprint images. Thecorrelation coefficients values for all images in all directions areshown in Table 1, which are far apart. Hence, the proposed schemeis able to break the correlation among image pixels.

Numerical Analysis: Numerical analysis includes the valuesof the objective metrics. A metric which provides more efficienttest methods and is suitable for computer simulations is calledobjective metrics. PSNR, Universal Image Quality Index (UIQ) [46]and Structural Similarity (SSIM) [47] index are used as the objectivemetrics to evaluate the proposed multiple encryption scheme.Fig. 13 shows the distribution of objective metrics betweenoriginal–encrypted and original–decrypted images with correctkeys for all 80 images of FVC-DB3-B. From the figure, it is clear thatthe proposed security solution perfectly encrypts and decrypts thefingerprint image. The mathematical definitions of the objectivemetrics are as follows:

1. Peak Signal to Noise Ratio (PSNR): The PSNR indicates thesimilarity between two images. The higher the value of PSNR,the greater similarity in the images. Mathematically, the PSNRis defined as

PSNR(f , g) = 10log102552

1MN

M∑i=1

N∑j=1

[fi,j − gi,j]2(33)

where MN is the total number of pixels in the image, fi,j andgi,j are the values of the ijth pixel in the original and encryptedimage.

2. Universal Image Quality Index (UIQ): The UIQ indicates thestructural similarity between two images. The UIQ lies between[−1, 1], and the closer the value to 1, the greater similarity in theimages. Mathematically, UIQ is defined as

UQI(f , g) =σfg

σf σg.2µfµg

µ2f + µ2

g.

2σf σgσ 2f + σ 2

g(34)

where µf , µg , σf , σg and σfg are the mean of f and g, variance fand g and the covariance of f and g respectively.

3. Structural Similarity Index Measure (SSIM): The SSIM is theextended version of the UIQ index. The SSIM lies between [−1,1] and the closer the value to 1, the greater similarity in theimages. Mathematically, SSIM is defined as

SSIM(f , g) =(2µfµg + C1)(2σfg + C2)

(µ2f + µ2

g + C1)(σ2f + σ 2

g + C2)(35)

where C1, C2 are two constants used to stabilize the divisionwith a weak denominator.

5. Conclusions

In this paper, a novel fractional transformationwas defined, thefractional dual tree complex wavelet transform (FrDT-CWT), alongwith its fast implementation for computer simulations. The newtransform possesses all the properties of the wavelet transformand fractional Fourier transform. The FrDT-CWT may be used fordifferent applications in signal and image processing. In orderto explore its effectiveness, biometric security is chosen as theprimary application and hence a encryption technique is proposedto secure biometric data over transmission and communication. Toachieve the desired goal, the FrDT-CWT, non-linear Markov map,PCA and space filling curve are used. The non-linear Markov mapand PCA are used to randomize the biometric image, which furtherundergoes confusion via the Hilbert space filling curve in the FrDT-CWT domain to get encrypted biometric data. The experimentalresults have been carried out with detailed key-space, keysensitivity, statistical, numerical analysis which demonstrate theefficiency and robustness of the newly proposed biometric securitysolution. The proposed technique is particularly suitable forsecuring biometrics during communication and transmission overan insecure channel.

266 G. Bhatnagar et al. / Future Generation Computer Systems 28 (2012) 254–267

Fig. 13. Numerical analysis for FVC database FVC-DB3-B.

Acknowledgements

The work is supported by the Canada Research Chair program,the NSERC Discovery Grant. Further, the authors thank the anony-mous Referees and the Editor for the valuable suggestions andmany constructive comments that resulted in the improvementand readability of this paper.

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Gaurav Bhatnagar has been a member of the ComputerVision and Sensing Systems Laboratory in the Departmentof Electrical and Computer Engineering at University ofWindsor since 2009. He received his B.Sc from C.C.SUniversity in 2003 and M.Sc in Applied Mathematicsfrom the Indian Institute of Technology, Roorkee in2005. He submitted his Ph.D. from Indian Institute ofTechnology, Roorkee, India in 2009. So far he has publishedin 1 book chapter, 8 international journals and 14conference proceedings. His areas of research includeDigital Watermarking, Image Analysis, Image Fusion,

Biometrics, Wavelet Analysis and Cryptography.

Jonathan Wu received the Ph.D. degree in electricalengineering from the University of Wales, Cardiff, U.K., in1990. In 1995, he joined the National Research Councilof Canada, Ottawa, ON, Canada, where he was a SeniorResearch Officer and Group Leader. He is currently a fullProfessor with the Department of Electrical and ComputerEngineering, University of Windsor, Windsor, ON. He isa holder of the Canada Research Chair in automotivesensors and sensing systems. He is the author of over 100published scientific papers in the areas of computer vision,neural networks, fuzzy systems, robotics, micro sensors

and actuators, and integrated Microsystems. Dr. Wu is an Associate Editor forthe IEEE Transactions on Systems Man and Cybernetics–PART A: Systems andHumans. His current research interests include 3-D image analysis, active videoobject extraction, vision-guided robotics, sensor analysis and fusion,wireless sensornetworks, multimedia security and integrated Microsystems.

Balasubramanian Raman has been an Assistant Professorin the Department of Mathematics at the Indian Instituteof Technology, Roorkee since February 2006. He receivedhis Ph.D. in Mathematics from the Indian Institute ofTechnology, Madras, India in 2001. He received hisB.Sc and M.Sc in Mathematics from the University ofMadras in 1994 and 1996 respectively. So far he haspublished in 26 international journals, 41 conferenceproceedings, 4 book chapters and a technical report.His areas of research include Computer Vision, Graphics,Satellite Image Analysis, Scientific Visualization, Imaging

Geometry, Reconstruction problems, Biometrics and Watermarking.