2-Dimensional Wavelet Packet Spectrum for Texture Analysis Abdourrahmane Atto, Yannick Berthoumieu, Philippe Bolon

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Abdourrahmane Atto, Yannick Berthoumieu, Philippe Bolon. 2-Dimensional Wavelet Packet Spec- trum for Texture Analysis. IEEE Transactions on Image Processing, Institute of Electrical and Elec- tronics Engineers, 2013, pp.1-9. ￿10.1109/TIP.2013.2246524￿. ￿hal-00766713￿

HAL Id: hal-00766713 https://hal.archives-ouvertes.fr/hal-00766713 Submitted on 18 Dec 2012

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2-Dimensional Wavelet Packet Spectrum for Texture Analysis

Abdourrahmane M. ATTO1, Yannick BERTHOUMIEU2, Philippe BOLON3,

Abstract—The paper derives a 2-Dimensional spectrum es- that a large amount of frequencies contributes in the timator from some recent results on the statistical properties corresponding wavelet variances. of wavelet packet coefficients of random processes. It provides The present paper investigates the capability of 2-D an analysis of the bias of this estimator with respect to the wavelet order. The paper also discusses the performance of this wavelet packets to yield an accurate Power Spectral Density wavelet based estimator, in comparison with the conventional (PSD) estimator for characterizing second order statistical 2-D Fourier-based spectrum estimator on texture analysis and properties of random fields. In this paper, we consider content based image retrieval. It highlights the effectiveness of the standard Fourier spectrum as benchmark and seek for the wavelet based spectrum estimation. estimating this spectrum by using suitable wavelets. This positions the Shannon wavelet at the focus of the paper: Keywords – 2-D Wavelet packet transforms; Random as the decomposition level tends to infinity, the bias of the fields; Spectral analysis, Spectrum estimation, Similarity Shannon wavelet spectrum estimator tends to 0. The bias measurements. of an arbitrary wavelet spectrum estimator then relates to the closeness of the wavelet under consideration to the Shannon wavelet. This closeness is measured through a I.INTRODUCTION parameter called wavelet order. AVELET transforms have become appealing alterna- The results derived in the paper are in continuation of W tives to the Fourier transform for image analysis and [5] which introduces PSD estimation by using the statistical processing. Indeed, the availability of wavelet atoms with properties of the wavelet packet coefficients of random different shapes and different characteristics (vanishing processes. The paper focuses on presenting the specificities moments, compact support, symmetric or non-symmetric, that follow by dimensionality increasing from [5]: the analytic form of the wavelet packet based PSD, etc), as well as the wavelet multi-scale decomposition • the singular paths of fractional Brownian fields (Section scheme make wavelets enough flexible for concise/sparse • description of a wide class of images. In this respect, III), the 2-D wavelet packet PSD estimator (Section IV). wavelet transforms are, at present time, extensively used • for the description and the retrieval of image features, in Furthermore, for textured image analysis, the paper pro- many imaging systems such as those involved in machine vides experimental results for evaluating the relevance of vision, medical imaging, biometrics, radar and geoscience spectrum estimation by 1) comparing PSD estimated from applications, etc. Fourier and wavelet packet methods and 2) performing con- In contrast with this extensive use of wavelets as a repre- tent based image retrieval associated with spectral similarity sentation tool, spectral analysis by using wavelet transforms measurements in the Fourier and wavelet packet domains has not received much interest in the literature whereas (Section V). spectral analysis plays a crucial role for characterizing and understanding textures and natural images. II. 2-D WAVELET PACKETS Spectrum analysis is strongly connected to the distribu- This section provides a brief introduction to 2-D wavelet tion of variances in the frequency domain. In this respect, packet transforms. Advanced concepts and algorithms con- references [1], [2], [3], [4] have considered the wavelet cerning 1D and 2-D wavelet packet analysis can be found in framework in order to derive wavelet based spectra from [6], [7], [8]. The reader is also invited to refer to [9], [10], [11], subband variances computed at different scales. This yields [12], [13] (wavelets) and [5], [14] (wavelet packets) for more a wavelet spectrum definition that can differ from the con- details on the statistical properties of wavelet transforms, ventional definition of the Fourier based spectral density when the decomposition relates to a random process. function. Indeed, most wavelet functions have their Fourier frequency supports spreading over large size supports so A. Introduction on 2-D wavelet packets

1 Université de Savoie - Polytech Annecy-chambéry, LISTIC, We consider the 2-D separable wavelet packet decom- [email protected] position in a continuous time signal setting for presenting 2 Université de Bordeaux, IPB, IMS, CNRS UMR 5218, theoretical results (see [7] for the connection between the [email protected] 3 Université de Savoie - Polytech Annecy-Chambéry, LISTIC, continuous and the discrete wavelet transforms). In this [email protected] decomposition, the wavelet paraunitary filters H0 (low-pass, 2

scaling filter) and H1 (high-pass, wavelet filter) are used and Q is a polynomial, with the following equality holding to split the input functional space U W L2(R2) into almost everywhere (a.e.): = 0,0 ⊂ orthogonal subspaces (subbands): a.e. lim H r (ω) HS(ω), (6) r 0 = 0 W0,0 W1,[0,0] W1,[1,0] W1,[0,1] W1,[1,1], →∞ = ⊕ ⊕ ⊕ S filter H0 denoting the scaling filter associated with the Wj,[n1,n2] Wj 1,[2n1,2n2] Wj 1,[2n1 1,2n2] ... ³ ´ = + ⊕ + + ⊕ Shannon wavelet. Then the 1D multiscale filters Hr j,ni Wj 1,[2n1,2n2 1] Wj 1,[2n1 1,2n2 1] i 1,2 ⊕ + + ⊕ + + + have very tight supports when r is large (see [14, Figure= 1] for j > 1. The subbands involved in this quad-tree splitting for illustration). These multiscale filters tend to the Shannon S trick follow from a separable product Wj,[n1,n2] Wj,n1 ideal filters H when r increases (see Eq. (4) and Eq. = ⊗ j,[n1,n2] Wj,n2 of 1D wavelet packet subspaces Wj,n1 and Wj,n2 . We (6)). The consequence is that the spectrum of the input assume that U is the subspace spanned by an orthonormal random field is analyzed within a frequency window with © 2ª 2 2 S system τ[k1,k2]Φ :(k1,k2) Z with Φ L (R ) and where τ small size. Indeed, the restriction of the support of H ∈ ∈ j,[n1,n2] is the shift operator: τ f (t ,t ) f (t k ,t k ). j j [k1,k2] 1 2 = 1 − 1 2 − 2 to [ π,π] [ π,π] is with size 2− 2− . This makes possible Denote F f , the Fourier transform of f L1(R2) L2(R2), − × − × ∈ ∪ a local analysis of the spectrum, as performed in Section with IV below. Z i(ω1t1 ω2t2) F f (ω1,ω2) f (t1,t2)e− + dt1 dt2. = R2 B. Wavelet packet paths Since U does not necessarily depend on the wavelet filters, This section presents a specific wavelet packet path we assume in the following that U is the space of functions description derived from the binary sequence approach of having their Fourier transforms with supports in [ π,π] − × [16] for representing nested wavelet packet subspaces. This [ π,π]. In this respect, Φ is the sinc function: − description is suitable for establishing asymptotic proper- sin(πt1)sin(πt2) ties of 2-D wavelet packets with respect to the increase of Φ(t ,t ) sinc(t ,t ) . 1 2 1 2 2 the decomposition level. = = π t1t2 For the sake of simplifying notation, 2-D wavelet packets The wavelet packet subband Wj,[n1,n2] is, by construc- 2 2 are commonly presented through a quad-tree structure tion, the closure of the L (R ) subspace spanned by the 2 where a pair [n ,n ] ©0,1,...,2j 1ª of frequency indices orthornormal system of wavelet packet functions: 1 2 ∈ − is set in a one to one correspondence with an index © 2ª © j ª τ j j W :(k ,k ) Z . n 0,1,...,4 1 . This correspondence leads to a single [2 k1,2 k2] j,[n1,n2] 1 2 ∈ − ∈ frequency indexing of the wavelet packet subbands. Due to separability, the wavelet packet function Wj,[n1,n2] More precisely, this correspondence follows by associat- satisfies: ing a quaternary index µ {0,1,2,3} ∈  1 2 FWj,[n ,n ](ω1,ω2) FWj,n (ω1)FWj,n (ω2). (1) 0 if (² ,² ) (0,0) 1 2 = 1 2  =  1 if (²1,²2) (0,1) By decomposing the frequency indices n ,n in the form µ 2²1 ²2 = (7) 1 2 = + = 2 if (²1,²2) (1,0)  = j j  3 if (²1,²2) (1,1) X 1 j ` X 2 j ` = n1 ² 2 − , n2 ² 2 − , (2) ` ` 1 2 2 = ` 1 = ` 1 to the binary indices (² ,² ) {0,1} and by defining: = = ∈ with (²1,²2) {0,1}2 for every ` {1,2,..., j}, we have j ` ` X j ` ∈ ∈ n n(j) µ 4 − . (8) = = ` FW H F Φ, (3) ` 1 j,[n1,n2] = j,[n1,n2] = The above change of variable is a univoque correspon- where the multiscale wavelet packet filter H applied 2 j,[n1,n2] dence between (n ,n ) ©0,1,...,2j 1ª defined by Eq. (2) to obtain the subband (j,n) wavelet packet coefficients is 1 2 ∈ − and n ©0,1,...,4j 1ª defined by Eq. (8). This correspon- 2 ∈ − Y dence makes it possible to characterize the 4-band tree H (ω ,ω ) H (ω ) j,[n1,n2] 1 2 j,ni i structure of the 2-D wavelet packet decomposition with a = i 1 = single shift parameter n, instead of the more intricate tree with " j # description with double indexed frequency indices. j/2 Y ` 1 This characterization also makes it possible to use Hj,ni (ω) 2 H²i (2 − ω) (4) = ` 1 ` a straightforward path based description of the wavelet = i packet tree by associating an infinite quaternary sequence for ² {0,1}, i 1,2, where H0 and H1 are the scaling and ¡ ¢ ` ∈ = µ µ ,µ ,µ ,... to a wavelet packet path P P and the wavelet filter introduced above. = 1 2 3 ≡ µ vice versa or, equivalently, an infinite sequence of frequency Assume that the scaling filter is with order r : H H r , 0 ≡ 0 indices n (n(1),n(2),n(3),...), those associated with µ to where r is the largest non-negative integer such that [15] = P Pn from Eq. (8). µ iω ¶r ≡ 1 2 1 e− Let (² )k N and (² )k N be the binary sequences as- H r (ω) + Q(eiω) (5) k ∈ k ∈ 0 = 2 sociated with µ from the correspondence given by Eq. 3

(7). Then, by taking into account Eq. (2), frequency index reduces to R [k ,k ,` ,` ] R [k ` ,k ` ] j,n 1 2 1 2 = j,n 1 − 1 2 − 2 = nP (j) is also associated with a pair of frequency indices R j,n[m1,m2]. Moreover, if we assume that PSD γ of X (n1[P ](j),n2[P ](j)). is bounded on the support of FWj,n, then function In the rest of the paper, we will use notation n,nP (j) in R j,n[m1,m2] has the form general whereas notation n ,n ,n [P ](j),n [P ](j) will be 1 2 1 2 1 Z used to specify the spatial location of the frequency support 2 R j,n[m1,m2] 2 γ(ω1,ω2) FWj,n(ω1,ω2) =4π R2 | | (12) of the wavelet packet subband under consideration. j i2 (m1ω1 m2ω2) To end this section, it is worth mentioning that some e + dω1 dω2. specific paths will present singular behavior, depending on the input random field: the wavelet coefficients of certain B. The case of non-stationary random fields: application to non-stationary random fields on the subbands associated separable and isotropic fractional Brownian fields with these singular paths will remain non-stationary. As a Associating a wavelet packet based PSD to a non-WSS matter of example, random field involves finding the WSS subbands (if any) The separable fractional Brownian field analyzed in • associated with the decomposition of this field. Specifically, Section III-B1 admits frequency indices n(j) such that we consider in this section, the wavelet packet decompo- n (j) 0 (resp. n (j) 0) for every j as singular 1 = 2 = sitions of some separable and isotropic constructions of frequency indices. The set of (singular) paths associ- random fields from the 1D fractional Brownian motion. ated with these frequency indices will be denoted by Such a random field is called a fractional Brownian field (resp. ). Pn/n1 0 Pn/n2 0 (fBf). The isotropic= fractional= Brownian field analyzed in • Section III-B2 admits a unique singular path: the ap- 1) Separable fractional Brownian field: This fBf, hereafter proximation path denoted by P0 and associated with s s denoted by Xα Xα(t1,t2), where α is the Hurst parameter, frequency indices nP0 (j) 0 for every j. = = follows from a separable 2-D extension (cartesian product) of a 1D fractional Brownian motion. Its autocorrelation III. 2-D WAVELETPACKETSOFSECONDORDERRANDOM function is FIELDS 2 s σ ¡ 2α 2α 2α¢ A. The case of stationary random fields R (t1,t2,s1,s2) t1 s1 t1 s1 = 4 | ¡ | + | | − | − | ¢ Let X X (t1,t2) be a second-order zero-mean real valued t 2α s 2α t s 2α = 2 2 2 2 WSS random field. We assume that X is continuous in × | | + | | − | − | quadratic mean. where σ is a constant that can be tuned so as to control The autocorrelation function of X is R(t ,t ,s ,s ) the variance of the fBf at a specific spatial coordinates. 1 2 1 2 = E Assume that the wavelet function Wj,n Wj,n(t1,t2) is [X (t1,t2)X (s1,s2)]. We assume that R satisfies = Z with compact support and has (at least) one vanishing- R(t ,t ,s ,s )τ j j W (t ,t ) 1 2 1 2 [2 k1,2 k2] j,n 1 2 moment in each variable t1, t2. This assumption is satisfied R4 for the frequency indices n such that n1 0 and n2 0. The τ j j W (s ,s )dt dt ds ds . [2 k1,2 k2] j,n 1 2 1 2 1 2 j j 1 6= 6= < ∞ corresponding subbands (4 2 + 1 subbands at decom- − + Under WSS assumption, this autocorrelation function position level j) are hereafter called detail wavelet packet reduces to a two variable function by using the convention: subbands since their generating functions have vanishing moments in each direction. R(t ,t ,s ,s ) R(t s ,t s ) R( , ) 1 2 1 2 1 1 2 2 ρ1 ρ2 j j 1 ≡ − − = The following Proposition 1 thus concerns the 4 2 + 1 − + where ρ t s for i 1,2. detail wavelet packet subbands. In these subbands, the i = i − i = We assume that γ F R exists, wavelet packet moments vanish and the autocorrelation = Z function Rs of cs reduces to i(ω1ρ1 ω2ρ2) j,n j,n γ(ω1,ω2) R(ρ1,ρ2)e− + dρ1 dρ2. (9) = R2 s R j,n[k1,k2,`1,`2] Function γ is the PSD of the WSS random field X . 2 Z = σ 2α 2α t s t s τ j j W (t ,t ) (13) Under assumptions mentioned above, the coefficients of 1 1 2 2 [2 k1,2 k2] j,n 1 2 4 R4 | − | | − | X on subband Wj,n define a discrete second order random τ[2j ` ,2j ` ]Wj,n(s1,s2)dt1 dt2 ds1 ds2. field 1 2 Z Assume furthermore that function Wj,n is with compact c [k ,k ] X (t ,t )τ j W (t ,t )dt dt . (10) j,n 1 2 1 2 2 [k1,k2] j,n 1 2 1 2 support or has sufficiently fast decay, and that the Fourier = R2 transform of W has a bounded derivative in the neigh- Random field c inherits the WSS property of X : the j,n j,n borhood of the origin, then we can state the following autocorrelation function R of c , j,n j,n Proposition 1. Z s R j,n[k1,k2,`1,`2] R(t1,t2,s1,s2)τ2j [k ,k ]Wj,n(t1,t2) Proposition 1: The discrete random field c represent- 4 1 2 j,n = R s τ j W (s ,s )dt dt ds ds ing the coefficients of X on Wj,n is WSS when the 2 [`1,`2] j,n 1 2 1 2 1 2 α (11) frequency index n is such that n 0 and n 0. Its 1 6= 2 6= 4

s s autocorrelation function R j,n[k1,k2,`1,`2] R j,n[k1 `1,k2 2) Isotropic fractional Brownian field: This random field, s = − − i i ` ] R [m ,m ], is: denoted by X X (t1,t2) is such that any of its 1D section 2 = j,n 1 2 α = α by a line segment issued from the origin is a fractional s R j,n[m1,m2] Brownian motion. Its autocorrelation function is defined Z = 1 s 2 by 2 γ (ω1,ω2) FWj,n(ω1,ω2) (14) 4π R2 | | σ2 j i ¡ 2 2 α 2 2 α¢ i2 (m1ω1 m2ω2) R (t1,t2,s1,s2) (t t ) (s s ) e + dω1 dω2, = 2 1 + 2 + 1 + 2 2 σ £ 2 2¤α where (t1 s1) (t2 s2) . σ2Γ2(2α 1)sin2(πα) − 2 − + − s( , ) + (15) γ ω1 ω2 2α 1 2α 1 = ω1 + ω2 + By following a reasoning analog to that of Section III-B1 | | × | | above, we have and the Gamma function is defined by Γ(u) R u 1 x = i x − e− dx. R [k ,k , , ] R+ j,n 1 2 `1 `2 2 Z = σ £ 2 2¤α From Proposition 1, we derive that stationarity holds true (t s ) (t s ) τ j j W (t ,t ) 1 1 2 2 [2 k1,2 k2] j,n 1 2 j j 1 − 2 R4 − + − for the 4 2 + 1 detail wavelet subbands. Let us focus − + τ j j W (s ,s )dt dt ds ds on the frequency axes, that is when n 0 and n 0, or [2 `1,2 `2] j,n 1 2 1 2 1 2 1 = 2 6= (19) vice versa. For such an index n, we have under integrability condition for the term s s 2 2 α R [k ,k ,` ,` ] Λ [k ,k ,` ,` ] (t t ) Wj,n(t1,t2) (compact support or sufficiently j,n 1 2 1 2 = j,n 1 2 1 2 1 + 2 1 Z fast decay for W ) and provided that n 0 (the wavelet s 2 j,n 6= 2 γ (ω1,ω2) FWj,n(ω1,ω2) (16) +4π R2 | | function Wj,n then have at least one vanishing moment in j i2 (m1ω1 m2ω2) e + dω1 dω2 some direction). 2 2 3/4 Furthermore, if FWj,n(ω1,ω2)/(ω1 ω2) is bounded in where, as above, m k ` for i 1,2 and + i = i − i = the neighborhood of the origin, then we have the following ( 2α s result: M 2R [m2] if n1 0, n2 0 s j,0,k2 × j,n2 = 6= Λj,n[k1,k2,`1,`2] 2α s = M 1R [m1] if n2 0, n1 0 Proposition 2: The random field ci j,0,`1 × j,n1 = 6= j,n i = 2α (c j,n[k1,k2])[k1,k2] Z Z representing subband Wj,n coeffi- and Mj,n ,k is the 2α fractional moment of the 1D wavelet ∈ × i i cients of X i is WSS for n 0, with autocorrelation function function τ j W : α 2 ki j,ni i i 6= i R [k1,k2,`1,`2] R [k1 `1,k2 `2] R [m1,m2] given Z j,n ≡ j,n − − = j,n 2α 2α by: t τ j W (t)dt, (17) Mj,0,k 2 ki j,0 i = R | | Z i 1 i 2 s s R [m1,m2] γ (ω1,ω2) FWj,n(ω1,ω2) with 1R (resp. 2R ) denoting the autocorrelation func- j,n 2 2 j,n1 j,n2 = 4π R | | (20) j tion of a 1D section of an fBf (thus a fractional Brownian i2 (m1ω1 m2ω2) e + dω1 dω2, motion), where 1 Z j s 2 i2 m1ω1 1R [m1] γ∗(ω1) FWj,n (ω1) e dω1 (2α 1) 2 2 j,n1 1 2− + π σ 1 = 2π R | | γi(ω ,ω ) . (21) 1 2 2 ¡ ¢α 1 = sin(πα)Γ (1 α) ω2 ω2 + when n1 0 and + 1 2 6= + 1 Z j s 2 i2 m2ω2 2R j,n [m2] γ∗(ω2) FWj,n2 (ω2) e dω2, From Propositions 1 and 2, it follows that: 2 = 2π R | | Remark 1: Except for a few frequency indices given in when n2 0. In these equations, function γ∗ is given by 6= Propositions 1 and 2, the autocorrelation function of the 2 σ Γ(2α 1)sin(πα) wavelet packet coefficients of separable and isotropic fBfs γ∗(ω) + . (18) = ω 2α 1 can be written in the integral form involved in Eq. (12). | | + One can notice the presence of an additive nonstation- From now on, when no confusion is possible, the auto- s ary term Λj,n[k1,k2,`1,`2] in Eq. (16). This term is a 2α correlation close form given by Eq. (12) will also be used centered fractional moment of the scaling function Wj,0. to denote the autocorrelation function of separable fBf (see This fractional moment is centered at 2j k when n 0 − 2 1 = Eq. (14)) or isotropic fBf (see Eq. (20)). (resp. at 2j ` when n 0). The height of this fractional − 1 2 = More precisely, when the autocorrelation function of the moment depends on the scaling function associated with wavelet packet coefficients of an arbitrary, non-stationary the wavelet packet decomposition. random field can be written in the form given by Eq. (12) Note also that when both n n 0, the nonstationarity 1 = 2 = for a specific function γ, this function will be called the in wavelet coefficients is more intricate, mainly because the spectrum of the random field under consideration. analyzing function has no vanishing moments in neither of The following section addresses the estimation of spec- the variables t1,t2. trum γ. The method proposed relies on the frequency partitioning induced by the Shannon wavelet packets. 5

S IV. 2-D WAVELET PACKET BASED SPECTRUM ESTIMATION is such that every R , for j 1, has the form given by j,nP (j) > The spectrum estimation method presented in this sec- Eq. (25). tion follows from the asymptotic analysis of the autocorre- Let G(n1[P ](j))π lation functions of the 2-D wavelet packet coefficients. This ω1[P ] lim , (26) = j 2j asymptotic analysis is performed with respect to the wavelet →+∞ order r and the wavelet decomposition level j. When r in- G(n2[P ](j))π ω2[P ] lim . (27) creases, the asymptotic behavior of the sequence of wavelet = j 2j →+∞ functions is driven by the Shannon wavelet functions (see Assume that (ω1[P ],ω2[P ]) is a continuity point of the Eq. (6)). In this respect, we consider the Shannon wavelets spectrum γ. Then, the sequence of autocorrelation func- ³ ´ in Section IV-A below and derive asymptotic results with tions RS satisfies: respect to the wavelet decomposition level. The case of a j,nP (j) wavelet with order r will be considered as an approximation S lim R j,n[m1,m2] γ(ω1[P ],ω2[P ])δ[m1,m2] (28) j = of the Shannon limiting behavior when r is large enough. →+∞ uniformly in (m ,m ) Z2, where δ[ , ] is defined by 1 2 ∈ · · A. Autocorrelation functions of the Shannon wavelet packets ½ 1 if k k 0, and asymptotic analysis δ[k ,k ] 1 = 2 = 1 2 = 0 otherwise. When the Shannon ideal paraunitary filters are used, the Fourier transform of the Shannon wavelet packet function B. 2-D Wavelet packet based spectrum estimation W S satisfies: j,n The following provides a non-parametric method for ¯ ¯ ¯FW S (ω ,ω )¯ 2j 1l (ω ,ω ), (22) estimating spectrum γ of 2-D random fields on the basis ¯ j,n 1 2 ¯ ∆j,G(n1) ∆j,G(n2) 1 2 = × of the convergence criteria given by Eqs. (6) and (28). where 1l denotes the indicator function of the set K and K From Eq. (28), it follows that γ(ω1[P ],ω2[P ]) , with , S = ∆j,G(n) ∆−j,G(n) ∆+j,G(n) ∆−j,G(n) ∆+j,G(n) limj R [0,0] so that the continuity points of spec- = ∪ = − →+∞ j,n · ¸ trum γ can be estimated by subband variances (values G(n)π (G(n) 1)π n o ∆ , + . (23) S +j,G(n) j j R [0,0] ), provided that the Shannon wavelet is used = 2 2 j,n n and j is large enough. Furthermore, we can derive from The function G in Eq. (23) is a permutation (see [6], [7] for the convergence criteria given by Eq. (6), several spectrum details) defined by G(0) 0 and = estimators by considering wavelets with finite orders r ¹ º G(`) ² (Shannon wavelet corresponds to r S), the accuracy G(2` ²) 3G(`) ² 2 + , (24) = +∞ ≡ + = + − 2 of the spectrum estimation being dependent on the wavelet where z is the largest integer less than or equal to z and order as shown in Proposition 3 below. b c ² {0,1}. Assuming a uniform sampling (regularly spaced fre- ∈ From Eqs. (12), (15), (21), (22), the autocorrelation func- quency plane tiling), the method applies upon the following tions of the Shannon wavelet packet coefficients of the steps. WSS random field described above, the separable fBf (with n 0 and n 0 when separable fBf is considered) and the 2-D Wavelet Packet Spectrum 1 6= 2 6= isotropic fBf (assumes (n1,n2) 0 for the isotropic fBf) can 1) Define a frequency grid composed with frequency 6= p1π p2π © ª be written in the form points ( , ) for p ,p 0,1,...,2j 1 (natural 2j 2j 1 2 ∈ − 2j Z ordering). S 2 j j R j,n[m1,m2] 2 cos(2 m1ω1)cos(2 m2ω2) 2) Compute, from Eqs. (2), (7) and (8) the index n = π ∆+ ∆+ © j ª ∈ j,G(n1)× j,G(n2) 0,1,...,4 1 (corresponding to the wavelet packet − γ(ω1,ω2)dω1 dω2 ordering) associated with (p ,p ). (25) 1 2 3) Set, for any pair (p ,p ) given in step 1) and the where γ is the spectrum of the input random field: γ is 1 2 corresponding n obtained from step 2), given by Eq. (9) for a WSS random field and γ is given by µ ¶ Eq. (15) (resp. Eq. (21)) for the separable (resp. isotropic) p1π p2π γ , var[cr ]. (29) fBf. b 2j 2j = j,n The asymptotic analysis of these autocorrelations when j tends to infinity is performed by considering a path P From a practical point of view, the estimator of of the 2-D wavelet packet decomposition tree. We assume ³ p p ´ γ 1π , 2π given by Eq. (29) assumes i) approximating the that 2j 2j Shannon wavelet by a suitable wavelet and ii) estimating P P0 in the case of the isotropic fBf and • 6= © ª the variance of c j,n. P Pn/n1 0 Pn/n2 0 in the case of the separable • ∉ = ∪ = Approximating the Shannon wavelet can be performed by fBf. using Daubechies wavelets, depending to the Daubechies Thus, the sequence of autocorrelation functions ³ S ´ filter order r (see Eq. (5)), the larger the order, the more R j,n (j) of Shannon wavelet packets in path P relevant the approximation. In practice, Daubechies filters P j>1 6

with r 7,8 yield relevant approximations of the Shannon Since γ is continuous on ∆+ ∆+ and differentiable on = j,p1 × j,p2 filters. the interior of ∆+ ∆+ , we have, from the mean value j,p1 × j,p2 Estimating the variance Var[c j,n] of c j,n can be performed theorem: by using the standard unbiased variance estimator (also ¯ µ ¶¯ ¯ p1π p2π ¯ called unbiased sample variance). In the Section dedicated ¯γ(ω1,ω2) γ , ¯ 6 ¯ − 2j 2j ¯ to the experimental results, var[c j,n] will denote this unbi- s µ ¶2 µ ¶2 ased estimator when formula of Eq. (29) is under consid- ° ° p1π p2π sup ° γ(u1,u2)° ω1 ω2 (35) eration. ∇ × − 2j + − 2j (u1,u2) ∆+j,p ∆+j,p Assuming that an unbiased variance estimator is used, ∈ 1 × 2 the bias of the above sample spectrum estimator depends for any (ω1,ω2) ∆+ ∆+ , with ∈ j,p1 × j,p2 on j, r and the spectrum γ shape. More precisely, we have: s Z µ ¶2 µ ¶2 µ ¶3 p1π p2π π Proposition 3: Let X be a zero-mean random field hav- ω ω dω dω e . 1 j 2 j 1 2 j ∆+ ∆+ − 2 + − 2 = 2 ing Shannon wavelet packet autocorrelation function of j,p1 × j,p2 © j ª (36) the form given by Eq. (25). Let p1,p2 0,1,...,2 1 . ∈ − Proposition 3 then follows from Eqs. (32), (33), (34), (35) Let n be the frequency index associated with p1,p2 in and (36). the sense of the correspondence (j,n) (j,[n1,n2]) ¡ 1 1 ¢ ≡ = j,[G− (p1),G− (p2)] given in Section II. From Proposition 3, we derive that the bias of the estima- Assume that γ is continuous on ∆+ ∆+ and differ- tor given by Eq. (29) depends on the decomposition level j,p1 × j,p2 entiable on the interior of ∆+ ∆+ . Assume that the and wavelet order used. This bias tends to 0 when both j j,p1 × j,p2 standard unbiased estimator of variance is used in Eq. (29). r and r tends to infinity. Notice that εj,n tends to 0 as r tends We have: r to infinity, j,n, due to the convergence of the filter Hj,n ¯ · µ ¶¸¯ ∀ S ¯ p1π p2π ¯ e Cγ r to the Shannon filter H as r tends to infinity. ¯Bias γ , ¯ 6 πC γ ε , (30) j,n ¯ b 2j 2j ¯ ∇ 2j + 4π2 j,n ¡ ¢ V. EXPERIMENTAL RESULTS where constants e p2 ln(1 p2) /3 1, C γ ° ° = + + ¯ ¯ < ∇ = sup , C sup and ∆+ ∆+ ° γ° γ ∆+ ∆+ ¯γ¯ This section provides experimental results on spectral j,p1 × j,p2 ∇ = j,p1 × j,p2 analysis of textures. Section V-A addresses the quality of Z ¯ ¯ r ¯ r 2 S 2¯ the wavelet packet spectrum estimator to capture textural εj,n 6 ¯ FWj,n(ω1,ω2) FWj,n(ω1,ω2) ¯ dω1 dω2. R2 | | − | | information. Section V-B provides quantitative evaluation The proof sketch is the following. of the wavelet packet spectrum estimator in the framework of content based image retrieval. Proof: The bias of the estimator given by Eq. (29) is: · µ ¶¸ µ ¶ p1π p2π h i p1π p2π Bias γ , E var[cr ] γ , A. Spectral analysis and spectral texture contents b j j j,n j j 2 2 = − 2 2 1 µ ¶ Wavelet packet spectra of some texture images are pro- h i p1π p2π Bias var[cr ] Var[cr ] γ , . (31) vided in Figures 1 and 2. The wavelet packet spectra have j,n j,n j j = + − 2 2 been computed from the method given in Section IV-B, Thus, if we consider the standard unbiased estimator of h i where the decomposition level is 6 and the Daubechies variance (Bias var[cr ] 0), we obtain: wavelet with order r 7 is used. Spectra computed from j,n = = ¯ · µ ¶¸¯ the Fourier transform are also given in this figure, for ¯ p1π p2π ¯ ¯ h r i h S i¯ ¯Bias γ , ¯6¯Var c Var c ¯ comparison purpose. ¯ b 2j 2j ¯ ¯ j,n − j,n ¯ ¯ µ ¶¯ From a visual analysis of images given in Figures 1 and ¯ p1π p2π ¯ Var[cS ] γ , . (32) 2 (by focusing on image interpretation without watching ¯ j,n j j ¯ +¯ − 2 2 ¯ spectra given in the same figures), one can remark that most On one hand, we have of these textures exhibit non-overlapping textons replicat- ing repeatedly: thus, coarsely, we can distinguish several ¯ r S ¯ Cγ ¯Var[c ] Var[c ]¯ frequencies having significant variance contributions (from ¯ j,n − j,n ¯ 6 4π2 × Z ¯ ¯ a theoretical consideration), when the texture does not ¯ r 2 S 2¯ ¯ FWj,n(ω1,ω2) FWj,n(ω1,ω2) ¯ dω1 dω2.(33) reduce to the replications of a single texton. R2 | | − | | In addition, when these textons occupy approximately On the other hand, we have the same spatial area (see for instance “Fabric” textures in ¯ µ ¶¯ ¯ S p1π p2π ¯ Figure 2), the frequencies with high variance contributions ¯Var[c j,n] γ , ¯ ¯ − 2j 2j ¯ (peak in the spectrum) are close in terms of their spatial 2j ¯Z · µ ¶¸¯ 2 ¯ p1π p2π ¯ location (from a theoretical consideration). ¯ γ(ω ,ω ) γ , ¯ dω dω 2 ¯ 1 2 j j ¯ 1 2 = π ¯ ∆+ ∆+ − 2 2 ¯ 1 j,p1 × j,p2 Colors represented in Figures 1 and 2 are simulated from a light source 2j Z ¯ µ ¶¯ in order to ease 3-D visualization: red color [value 1] corresponds to fully 2 ¯ p1π p2π ¯ γ(ω ,ω ) γ , dω dω .(34) illuminated shapes whereas blue color [value 0] is associated to shaded 6 2 ¯ 1 2 j j ¯ 1 2 π ∆+ ∆+¯ − 2 2 ¯ areas, green color corresponds to value 0.5. j,p1 × j,p2 7

FT-PSD “D3” WP-PSD “D3”

“D3” / Brodatz

FT-PSD “D10” WP-PSD “D10”

“D10” / Brodatz

FT-PSD “D17” WP-PSD “D17”

“D17” / Brodatz

Fig. 1. Textures images and their spectra γb computed by using discrete Fourier and wavelet packet transforms. Abscissa of the spectra images consist of a regular grid over [0,π/2] [0,π/2]. ×

The above heuristics, issued from visual image analysis, variance distribution. are confirmed by considering the wavelet packet spectra (see for instance spectra of “Fabric” textures in Figure Remark 2: In the above experimental results, we have considered a regular frequency grid over [0,π] [0,π]. Con- 2), whereas, in most cases, the two dimensional discrete × Fourier transform exhibits only one peak. sequently, we have performed a uniform sampling of the spectrum of the input texture. One can highlight that the poorness of the Fourier spectra Note that when we have some a priori on the spectrum is not due to a lack of resolution in the sampling step of shape, we can apply a non-uniform sampling scheme by the Fourier transform. This poorness can be explained by simply focusing on the particular tree structure associ- noting that Fourier transform is sensitive to global spatial ated with the subbands-of-interest. These subbands are regularity. In contrast wavelet packets can capture local those associated with wavelet packet functions having tight spatial regularity and lead to a more informative spectrum Fourier transform support across the frequencies where the estimator when several frequencies contribute in texture spectrum exhibits sharp components. 8

FT-PSD “D87” WP-PSD “D87”

“D87” / Brodatz

FT-PSD “Fabric.09” WP-PSD “Fabric.09”

“Fabric.09” / VisTeX

FT-PSD “Fabric.11” WP-PSD “Fabric.11”

“Fabric.11” / VisTeX

Fig. 2. Textures images and their spectra γb computed by using discrete Fourier and wavelet packet transforms. Abscissa of the spectra images consist of a regular grid over [0,π/2] [0,π/2]. ×

As a matter of example: by considering the spectrum of B. Content based image retrieval by using spectral texture texture “D87” (see Figure 2), we can estimate accurately features this spectrum by using a large amount of samples in [0,π/4] [0,π/4] and very few samples in [0,π] [0,π]\ Section V-A have shown qualitatively that wavelet packets × × [0,π/4] [0,π/4]. The corresponding wavelet packet de- are relevant in analyzing textures with rich spectral content. × composition concerns fewer subbands than a full wavelet In this section, we evaluate the contribution of wavelet packet decomposition and is thus with less computational packets in Content Based Image Retrieval (CBIR) from a complexity. database composed with such textures. The database under consideration is composed with M 94 images associated with the 6 texture classes given = in Figures 1 and 2. These M images are obtained from the splitting of every ‘large’ texture image into 16 non- overlapping images. An image from the database thus has 9

160 160 pixels when it pertains to one of the classes the sample spectra γ (ω ,ω ),γ (ω ,ω ) under compari- × 1 q ` 2 q ` “D3”, “D10”, “D17” or “D87” and 128 128 pixels for classes son. Indeed, for large values of γ (ω ,ω )/γ (ω ,ω ), the × 1 q ` 2 q ` “Fabric.09” and “Fabric.11” (the larger images have 640 640 log function imposes a higher penalty in Kullback-Leibler × pixels in the Brodatz album and 512 512 pixels in the measurements than in Log-Spectral measurements. × VisTeX album). TABLE I In order to compare a query image Iq with an arbitrary RETRIEVAL RESULTS PER TEXTURE AND AVERAGE RETRIEVAL RESULTS FOR THE image I`, 1 6 q,` 6 M, we use the Log-Spectral distance and 6 TEXTURECLASSESGIVENIN FIGURES 1 AND 2. EXPERIMENTAL RESULTS ARE the symmetric version of the Kullback-Leibler divergence. PERFORMEDWITHTHELOG-SPECTRALSPECTRALDISTANCEANDTHE KULLBACK-LEIBLERDIVERGENCE.SPECIFICALLY, FT, WT AND WPT For images I1, I2 with sample spectra γ1,γ2, their Log- DESIGNATE THE DISCRETE FOURIER,WAVELET AND WAVELET PACKET Spectral distance is given by TRANSFORMS, RESPECTIVELY. Ã Ã 2 !!1/2 X γ1(ωq ,ω`) LS(γ ,γ ) log (37) Measure: Log-Spectral Kullback-Leibler 1 2 2 Texture FT WT WPT FT WT WPT ∝ γ (ωq ,ω`) q,` 2 D3 44.5 88.7 99.2 39.5 58.2 91.0 D10 71.5 82.8 90.2 90.6 58.2 85.6 and their symmetric Kullback-Leibler divergence is defined D17 93.0 100 99.6 100 99.6 100 by D87 77.3 100 96.1 87.9 87.5 99.2 KL(γ ,γ ) KL(γ γ ) KL(γ γ ) (38) Fabr.09 84.4 100 83.6 82.8 80.5 98.1 1 2 = 1|| 2 + 2|| 1 Fabr.11 69.5 95.7 77.0 74.6 73.4 97.3 where µ ¶ X γ1(ωq ,ω`) KL(γ1 γ2) γ1(ωq ,ω )log ` VI.CONCLUSION || ∝ q,` γ2(ωq ,ω`) Table I provides the CBIR retrieval rates per texture and Two issues are addresses in the paper: (i) associating a the average CBIR retrieval rate over the 6 texture classes. PSD to the wavelet packet coefficients of random fields and In this table, the above spectral similarity measurements (ii) estimating this PSD from the statistical properties of the are computed upon discrete Fourier, wavelet and wavelet wavelet packet coefficients. packet spectra. The Daubechies wavelet of order r 7 and Issue (i) has been addressed for stationary random fields = j 4 have been used to compute wavelet and wavelet and some non-stationary fractional Brownian fields. Issue = packet spectra from Eq. (29). (ii) has been tackled from asymptotic properties of the This table emphasizes Shannon wavelet packets and the spectrum estimation method proposed is more effective for wavelet filters with the suitability of wavelet based approaches, in com- • large order. parison with the Fourier approach, The PSD estimation method derived in the paper has the sensitivity of the wavelet transform to the similar- • shown relevancy in estimating the PSD of texture images. ity measures considered, which may follow from the This method outperforms Fourier based spectral estimation coarse-irregular spectrum sampling induced by using when the textures under consideration have substantial wavelet method. spectral content (when spectrum do not reduces to a peak the good performance of the wavelet packet spectral • at single frequency). analysis for the Log-Spectral and Kullback-Leibler sim- Prospects regarding this work may concern: ilarity measurements. (p ) The extension of the method to adaptive sampling One can note that the wavelet transform is a particular 1 of the PSD: a best basis can be derived in order case of the wavelet packet transform so that the wavelet to achieve an efficient strategy. Indeed, the experi- spectrum can be seen through the wavelet packet spectrum mental results have shown that for many textures, by zooming on the neighborhood of the zero frequency. a large frequency domain is unoccupied whereas This wavelet spectrum thus follows from a non-uniform estimating several spectrum sample points in this sub-sampling of the wavelet packet spectrum (non-regular domain could increase estimation errors due to frequency grid). From similarity evaluations, this sampling the decimation step involved by the wavelet packet scheme can be interpreted as a binary sparse weighting, decomposition. where the non-zero values are concentrated in low fre- (p ) The extension of the method with respect to non- quencies. This sampling scheme consequently penalizes 2 orthogonal wavelet packet transforms: the redun- highly medium and high frequencies. This strategy is suit- dancy of these transforms will probably benefit to able for textures with spectral supports localized in the the spectrum estimation. Note that the spectrum neighborhood of zero (up to 90 % of textures from natural needs to be defined with respect to the transform images). However, this strategy can justify the contrasted and its estimation cannot reduce to variances per results obtained by using the wavelet transform for the subbands unless any given subband has been tests performed upon the database of Table I: the spectral processed accordingly. content of these textures is not restricted to low frequencies. In addition to the above binary weighting regarding Some applications of the method are: spectra sample points, the log function involved in Log- (a1) Hurst parameter estimation for self-similar medi- Spectral and Kullback-Leibler measures impacts differently cal images, see for instance [2]. 10

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