Noise Covariance Properties in Dual-Tree Wavelet Decompositions

IEEE TRANSACTIONS ON INF ORMA TION THEOR Y , 2007 1 Noise Co v ariance Properti es in Dual-T ree W a ve let Decompositions Caroline Chaux, Member , IEEE , Jean-Christophe Pesquet, Senior Membe r , IEEE and Laurent Duval, Member , IEEE Abstract Dual-tree wavelet decompositions hav e recently gained much popularity , mainly due to their ability to provide an accurate directional analysis of images combined wit h a reduced redundancy . When the decomposition of a random process is performed – which occurs in particular when an additiv e noise is corrupting the si gnal to be analyzed – it is useful to characterize the statistical properties of t he dual-tree wav elet coefﬁcients of t his process. As dual-tree decompositions constitute o verco mplete frame expa nsions, correlation structures are introd uced among the coef ﬁci ents, e ven when a white noise is analyzed. In this paper , we show that it is possible to provid e an accurate description of the cova riance properties of the dual-tree coefﬁcients of a wide-sense stationary process. The expressions of the (cross-)cov ariance sequences of the coefﬁcients are derive d in the one and two -dimensional cas es. Asymptotic resu lts are also provided, al lo wing to predict the behaviour of the second-order moments for larg e lag value s or at coarse resolution. In addition, the cross-correlations between the primal and dual wav elets, which play a primary role in ou r theoretical analysis, are calculated for a number of classical wav elet families. Simulation results are ﬁnally provided to validate t hese r esults. Index T erms Dual-tree, wa velets, frames, Hilbert transform, ﬁlter banks, cross-correlation, cov ariance, random processes, stationarity , noise, dependence , st atistics. I . I N T R O D U C T I O N The discrete wav elet transform (DWT) [1] is a powerful tool in signal processing, since it p rovides “efﬁcient” basis repre sentations o f regular sign als [ 2]. It n ev ertheless suffers f rom a few limitation s such as aliasing effects in the transform domain, coefﬁcient o scillations around singularities and a lack of shift in variance. Frames (see [3], [4] or [5 ] for a tuto rial), reckoned as mor e general signal repr esentations, represent an outlet for these inh erent constraints laid on ba sis fun ctions. C. Chaux and J.-C. Pesquet are with the Instit ut Gaspard Monge and CNRS-UMR 8049, Uni versi t ´ e de Paris-Est Marne-la-V all ´ ee, 77454 Marne-la -V all ´ ee Cede x 2, France. E-mai l: { chaux,pesquet } @univ -mlv.fr . L. Duva l is with the Institut franc ¸ ais du p ´ etrole, IFP , T echnology , Computer Science and Applied Mathemat ics Divi sion, 1 et 4, ave nue de Bois-Pr ´ eau F-92852 Ruei l-Malmai son, France. E-mail: laurent.duval@ifp.fr . Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INF ORMA TION THEOR Y , 2007 2 Redundan t DW Ts (RDWTs) are shif t-in variant no n-subsamp led f rame exten sions to the DWT . They have proved more er ror o r q uantization r esilient [6 ]–[8] , at the p rice of an increased computation al cost, espe cially in hig her dimensions. They do no t ho wev er take on the lack of r otation in variance or poor directionality of classical sepa rable schemes. These f eatures are particula rly sensitive to image an d vid eo pr ocessing. Recen tly , several othe r types of frames h av e been pro posed to incorp orate mor e geometric feature s, aiming at spar ser rep resentations and impr oved robustness. Early examples of such f rames are shiftable mu ltiscale tran sforms or steerable pyramids [9]. T o name a few other s, there also exist con tourlets [10] , b andelets [1 1], curvelets [12], ph aselets [13], dir ectionlets [14] or other repr esentations inv olving mu ltiple dictionaries [15]. T wo impo rtant facets n eed to be ad dressed, when resortin g to the inh erent f rame redun dancy: 1) multiplicity : frame deco mposition s or reconstructio ns are not uniq ue in general, 2) corr elation : transform ed coefﬁcients ( and especially tho se r elated to no ise) are usu ally correlated , in con trast with the classical uncorrelated ness property of the componen ts of a wh ite noise after an orthogona l transform. If the multiplicity aspec t is usually reco gnized (and often add ressed via av eraging techniques [6]), the correlation of the transformed coefﬁcients have not receiv ed m uch conside ration u ntil recen tly . Most of the efforts have been dev oted to the ana lysis of rand om pro cesses by the DWT [ 16]–[ 19]. It shou ld be noted th at e arly works by C. Hou dr ´ e et al. [2 0], [21] co nsider the continu ous wa velet tra nsform of random pro cesses, but only in a recent work by J. Fo wler exact en ergetic relatio nships fo r an additive noise in the case o f the no n-tight RDWT h av e been provided [22]. It mu st be pointed out tha t the difﬁculty to char acterize noise pro perties after a fr ame decom position may limit the de sign of sop histicated estimatio n methods in den oising ap plications. Fortunately , ther e exist redu ndant signal repre sentations allowing ﬁne r n oise behaviour assessment: in pa rticular the dual-tree wa velet tr ansform, based on the Hilb ert transform , whose advantages in wa velet analy sis have b een recogn ized b y several au thors [23] , [24]. It con sists of two classical wa velet trees developed in parallel. The second decomp osition is refere d to as the “dual” of the ﬁr st one, which is so metimes called the “primal” decom position. The corre sponding an alyzing wavelets for m Hilber t pairs [2 5, p .198 sq]. The d ual-tree wa velet tran sform was initially proposed b y N. Kingsbury [26] and fu rther in vestigated by I. Selesnick [27 ] in the dyadic case. An excellent overview of the topic by I . Selesnick, R. Baraniuk an d N. King sbury is provided in [28] an d an example of application is pr ovided in [29]. W e recen tly have gen eralized this fram e deco mposition to the M -band case ( M ≥ 2 ) (see [3 0]–[3 2]). In the later works, we rev amped the constru ction of the dual basis and the p re-pro cessing stage, necessary in the case of digital signal an alysis [ 33], [34] and manda tory to accu rate dir ectional analysis of imag es, and we pr oposed an op timized recon struction, thus ad dressing the ﬁrst imp ortant facet of the resulting frame mu ltiplicity . The M -band ( M > 2 ) dual-tr ee wa velets prove more selective in the frequency domain than their dyadic counter parts, with im proved direc tional selectivity as well. Fur thermore , a larger c hoice o f ﬁlters satisfying symmetry and or thogon ality proper ties is av ailable. In this paper, we focu s on the second facet, correlation , by study ing the second-or der statistical pro perties, in the transfo rm d omain, of a stationary random process und ergoing a d ual-tree M -band wa velet deco mposition. In practice, such a ran dom pr ocess ty pically mod els an additive n oise. Prelimin ary comm ents on dual- tree coefﬁcient Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INF ORMA TION THEOR Y , 2007 3 correlation may b e f ound in [35] . Depende ncies between the coefﬁcients already have been exploited for dua l-tree wa velet denoising in [36 ], [37] . A param etric nonlin ear estimator based on Stein’ s principle , making explicit use o f the cor relation pro perties der iv ed h ere, is p roposed in [ 38]. At ﬁrst, we brieﬂy recall some prop erties of the dual- tree wa velet decom position in Section II, refering to [32] for more de tail. In Sec tion III, we expr ess in a gene ral form the second-ord er momen ts of the noise coefﬁcients in each tree, bo th in th e one and two-dimensiona l cases. W e also discu ss th e role of the post-transfor m — often per formed o n th e dual-tree wa velet coefﬁcients — with respect to (w .r .t.) de correlation . In Section I V, we provide u pper boun ds f or the de cay of th e corre lations existing between pairs of prim al/dual co efﬁcients as well as an asymptotic result concern ing coefﬁcient whitening. The cross-corre lations between prim al and dual w av elets playing a ke y role in ou r analysis, their expressions ar e deri ved for several wav elet families in Sectio n V. Simulation results are provided in Section VI in ord er to validate our theoretical results and better ev aluate the impor tance of the co rrelations intro duced b y the du al-tree deco mposition. Some ﬁnal rem arks are drawn in Sec tion VII. Throu ghout the paper, th e f ollowing notations will be used: Z , Z ∗ , N , N ∗ , R , R ∗ , R + and R ∗ + are the set of integers, nonzero integers, nonnegative integers, positiv e integers, reals, nonze ro reals, non negativ e reals and po siti ve reals, respectively . Let M be an integer greater tha n or equal to 2, N M = { 0 , . . . , M − 1 } and N ⋆ M = { 1 , . . . , M − 1 } . I I . M - B A N D D U A L - T R E E WA V E L E T A N A LY S I S In this section , we recall the basic principles of an M -band [39] du al-tree decompo sition. Here, we will fo cus on 1D real signals belonging to the space L 2 ( R ) of square integrable fun ctions. Let M be an integer greater than or equal to 2. An M -ban d multiresolutio n an alysis of L 2 ( R ) is d eﬁned using one scaling func tion (or father wa velet) ψ 0 ∈ L 2 ( R ) and ( M − 1) mother wavelets ψ m ∈ L 2 ( R ) , m ∈ N ⋆ M . I n the fr equency domain, th e so -called scalin g equations are expr essed as: ∀ m ∈ N M , √ M b ψ m ( M ω ) = H m ( ω ) b ψ 0 ( ω ) , (1) where b a deno tes the Fourier transfo rm o f a function a . In ord er to generate an orth onorm al M -b and wa velet basis S m ∈ N ⋆ M ,j ∈ Z { M − j / 2 ψ m ( M − j t − k ) , k ∈ Z } of L 2 ( R ) , the following para -unitarity co nditions must ho ld: ∀ ( m, m ′ ) ∈ N 2 M , M − 1 X p =0 H m ( ω + p 2 π M ) H ∗ m ′ ( ω + p 2 π M ) = M δ m − m ′ , (2) where δ m = 1 if m = 0 and 0 othe rwise. The ﬁlter with frequency response H 0 is low-pass whereas th e ﬁlters with f requen cy response H m , m ∈ { 1 , . . . , M − 2 } (resp. m = M − 1 ) are band -pass (resp. high -pass). In this case, cascadin g the M -band para-u nitary analysis and synth esis ﬁlter ban ks, rep resented by th e upp er structures in Fig. 1, allows us to deco mpose and to perfe ctly reco nstruct a gi ven signal. A “d ual” M -band multiresolution analy sis is built by d eﬁning anoth er M -band wa velet orthonor mal basis associated with a scaling fun ction ψ H 0 and m other wa velets ψ H m , m ∈ N ⋆ M . More p recisely , the mothe r wa velets are Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 4 the Hilbert tran sforms o f the “or iginal” o nes ψ m , m ∈ N ⋆ M . In the Fourier d omain, the desired pro perty rea ds: ∀ m ∈ N ⋆ M , b ψ H m ( ω ) = − ı sig n( ω ) b ψ m ( ω ) , (3) where sign( · ) is the sign um fun ction. Then, it can be proved [31] that the du al scaling f unction can be chosen su ch that ∀ k ∈ Z , ∀ ω ∈ [2 k π , 2( k + 1) π ) , b ψ H 0 ( ω ) =      ( − 1) k e − ı ( d + 1 2 ) ω b ψ 0 ( ω ) if k ≥ 0 ( − 1) k +1 e − ı ( d + 1 2 ) ω b ψ 0 ( ω ) o therwise, (4) where d is an arbitrar y integer delay . The correspon ding analysis/synthesis para-un itary Hilbert ﬁlter ban ks ar e illustrated by the lower structures in Fig. 1. Conditions for designing the in volved frequen cy respo nses G m , m ∈ N M , have b een recently p rovided in [32]. As the un ion of two or thonor mal basis dec omposition , the g lobal du al-tree representatio n correspo nds to a tight fr ame analysis of L 2 ( R ) . I I I . S E C O N D - O R D E R M O M E N T S O F T H E N O I S E WA V E L E T C O E FFI C I E N T S In this part, we ﬁrst consider the analysis of a one-d imensional, real-valued, wide-sense stationary and zero-me an noise n , with autocovariance func tion ∀ ( τ , x ) ∈ R 2 , Γ n ( τ ) = E { n ( x + τ ) n ( x ) } . (5) W e then extend our r esults to the two-dimen sional case. A. Expr e ssion o f the covariances in the 1 D ca se W e den ote by ( n j,m [ k ]) k ∈ Z the coefﬁcients resu lting fr om a 1 D M -band wavelet decom position of th e n oise, in a given subba nd ( j, m ) where j ∈ Z and m ∈ N M . In the ( j, m ) subb and, th e wa velet coefﬁcients generated by the dual decom position a re denoted by ( n H j,m [ k ]) k ∈ Z . At resolution level j , the statistical second-ord er prop erties of the d ual-tree wavelet decom position of the n oise are char acterized as follows. Pr op osition 1 : F or all ( m, m ′ ) ∈ N 2 M , ([ n j,m [ k ] n H j,m [ k ]]) k ∈ Z is a wide-sense stationary vector sequence. More precisely , for all ( ℓ , k ) ∈ Z 2 , we h av e E { n j,m [ k + ℓ ] n j,m ′ [ k ] } = Γ n j,m ,n j,m ′ [ ℓ ] (6) = Z ∞ −∞ Γ n ( x ) γ ψ m ,ψ m ′  x M j − ℓ  dx E { n H j,m [ k + ℓ ] n H j,m ′ [ k ] } = Γ n H j,m ,n H j,m ′ [ ℓ ] (7) = Z ∞ −∞ Γ n ( x ) γ ψ H m ,ψ H m ′ ( x M j − ℓ ) dx E { n j,m [ k + ℓ ] n H j,m ′ [ k ] } = Γ n j,m ,n H j,m ′ [ ℓ ] (8) = Z ∞ −∞ Γ n ( x ) γ ψ m ,ψ H m ′  x M j − ℓ  dx, Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 5 where the determ inistic cr oss-correlatio n function of two re al-valued f unctions f an d g in L 2 ( R ) is expre ssed as ∀ τ ∈ R , γ f ,g ( τ ) = Z ∞ −∞ f ( x ) g ( x − τ ) dx. (9) Pr oo f: See Appen dix I . The classical properties of covariance/corr elation f unctions are satisﬁed. I n particular, since for all m ∈ N M , ψ m and ψ H m are un it n orm f unctions, fo r all ( m, m ′ ) ∈ N 2 M , th e a bsolute values of γ ψ m ,ψ m , γ ψ H m ,ψ H m ′ and γ ψ m ,ψ H m ′ are upper boun ded by 1 . I n add ition, th e f ollowing symmetry pro perties are satisﬁed. Pr op osition 2 : F or all ( m, m ′ ) ∈ N M with m = m ′ = 0 o r mm ′ 6 = 0 , we have γ ψ H m ,ψ H m ′ = γ ψ m ,ψ m ′ . As a consequen ce, Γ n j,m ,n j,m ′ = Γ n H j,m ,n H j,m ′ . (10) When mm ′ 6 = 0 , we hav e ∀ τ ∈ R , γ ψ m ,ψ H m ′ ( τ ) = − γ ψ m ′ ,ψ H m ( − τ ) (11) and, con sequently , ∀ ℓ ∈ Z , Γ n j,m ,n H j,m ′ [ ℓ ] = − Γ n j,m ′ ,n H j,m [ − ℓ ] . (12) Besides, th e f unction γ ψ 0 ,ψ H 0 is symmetric w .r . t. − d − 1 / 2 , wh ich enta ils that Γ n j, 0 ,n H j, 0 is symm etric w . r .t. d + 1 / 2 . Pr oo f: See Appen dix I I. As a particu lar case of (10) when m = m ′ , it app ears that the sequen ces ( n j,m [ k ]) k ∈ Z and ( n H j,m [ k ]) k ∈ Z have the same auto covariance sequence. W e also ded uce from Prop. 2 that, for all m 6 = 0 , γ ψ m ,ψ H m is an od d fun ction, and the cross-covariance Γ n j,m ,n H j,m is an od d sequence. This implies, in par ticular, that for all m 6 = 0 , Γ n j,m ,n H j,m [0] = 0 . (13) The latter equ ality means that, f or all m 6 = 0 and k ∈ Z , the rand om vector [ n j,m [ k ] n H j,m [ k ]] has u ncorre lated compon ents with equ al variance. The previous results are ap plicable to an arb itrary stationary noise but the resulting expr essions may b e intricate depend ing on the speciﬁc fo rm of the au tocovariance Γ n . Subsequ ently , we will be m ainly interested in the study of the dual-tree decom position of a white noise, fo r which tractab le expressions of the secon d-ord er statistics o f the coefﬁcients can be obtained . The autocovariance of n is then given b y Γ n ( x ) = σ 2 δ ( x ) , where δ d enotes the Dirac distribution. As the prima l (r esp. dual) wavelet basis is or thonorm al, it can be dedu ced fr om ( 6)-(8) (see Append ix III) that, fo r a ll ( m, m ′ ) ∈ N 2 M and ℓ ∈ Z , Γ n j,m ,n j,m ′ [ ℓ ] = Γ n H j,m ,n H j,m ′ [ ℓ ] = σ 2 δ m − m ′ δ ℓ (14) Γ n j,m ,n H j,m ′ [ ℓ ] , = σ 2 γ ψ m ,ψ H m ′ ( − ℓ ) , (15) where ( δ k ) k ∈ Z is the Kro necker sequen ce ( δ k = 1 if k = 0 and 0 o therwise). Th erefore, ( n j,m [ k ]) k ∈ Z and ( n H j,m [ k ]) k ∈ Z are cross-corr elated zero-m ean, wh ite r andom sequen ces with variance σ 2 . The determ ination of the cross-covariance requ ires the computa tion of γ ψ m ,ψ H m ′ . W e d istinguish between the mother ( m ′ 6 = 0 ) an d father ( m ′ = 0 ) wavelet case. Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 6 • By using (3 ), fo r m ′ 6 = 0 , Parsev al-Plancherel fo rmula y ields γ ψ m ,ψ H m ′ ( τ ) = 1 2 π Z ∞ −∞ b ψ m ( ω )  b ψ m ′ ( ω ) H ) ∗ exp( ıω τ ) dω = − 1 π Im n Z ∞ 0 b ψ m ( ω )  b ψ m ′ ( ω )) ∗ exp( ıω τ ) dω o , (16) where Im { z } denotes the imag inary part of a comp lex z . • Accordin g to (4 ), for m ′ = 0 we ﬁnd, after some simple c alculations: γ ψ m ,ψ H 0 ( τ ) = 1 π Re n ∞ X k =0 ( − 1) k Z 2( k +1) π 2 kπ b ψ m ( ω ) ×  b ψ 0 ( ω )  ∗ exp  ıω ( 1 2 + τ + d )  dω o , (17) where Re { z } denote s the rea l part of a complex z . In both cases, we h av e | γ ψ m ,ψ H m ′ ( τ ) | ≤ 1 π Z ∞ 0 | b ψ m ( ω ) b ψ m ′ ( ω ) | dω . ( 18) For M -ban d wavelet decompo sitions, selectiv e ﬁlt er banks are commonly used. Provided that this selecti vity proper ty is satisﬁed, the cr oss term | b ψ m ( ω ) b ψ m ′ ( ω ) | can be expected to be c lose to zero and the up per bound in (18) to take small values when m 6 = m ′ . T his fact will be discussed in Section VI-C b ased on n umerical r esults. On the contrary , wh en m = m ′ , the cross-co rrelation fu nctions always need to be ev aluated m ore ca refully . In Section V, we will ther efore focus on the fu nctions: γ ψ m ,ψ H m ( τ ) = − 1 π Z ∞ 0 | b ψ m ( ω ) | 2 sin( ω τ ) dω, m 6 = 0 , (19) γ ψ 0 ,ψ H 0 ( τ ) = 1 π ∞ X k =0 ( − 1) k × Z 2( k +1) π 2 kπ | b ψ 0 ( ω ) | 2 cos  ω ( 1 2 + τ + d )  dω . (20) Note that, in this p aper, we do n ot consider interscale correla tions. Although expressions o f the secon d-ord er statistics similar to the intrascale ones can be derived, sequences of wa velet coefﬁcients deﬁned at different resolution lev els are generally not cross-stationar y [1 8]. B. Extension to the 2 D case W e now consider the analy sis of a two-d imensional n oise n , which is also assum ed to b e real, wide-sense stationary with zero -mean and autoc ov ariance function ∀ ( τ , x ) ∈ R 2 × R 2 , Γ n ( τ ) = E { n ( x + τ ) n ( x ) } . W e can p roceed similarly to the p revious sectio n. W e d enote by ( n j, m [ k ]) k ∈ Z 2 the coefﬁcients resulting from a 2 D separable M -b and wa velet deco mposition [39] of the noise, in a given subband ( j, m ) ∈ Z × N 2 M . Th e wavelet Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 7 coefﬁcients of the dual decompo sition are denoted by ( n H j, m [ k ]) k ∈ Z 2 . W e obtain expressions of the covariance ﬁelds similar to (6)-(8 ): for all j ∈ Z , m = ( m 1 , m 2 ) ∈ N 2 M , m ′ = ( m ′ 1 , m ′ 2 ) ∈ N 2 M , ℓ = ( ℓ 1 , ℓ 2 ) ∈ Z 2 and k ∈ Z 2 , Γ n j, m ,n j, m ′ [ ℓ ] = E { n j, m [ k + ℓ ] n j, m ′ [ k ] } = Z ∞ −∞ Z ∞ −∞ Γ n ( x 1 , x 2 ) γ ψ m 1 ,ψ m ′ 1  x 1 M j − ℓ 1  × γ ψ m 2 ,ψ m ′ 2  x 2 M j − ℓ 2  dx 1 dx 2 (21) Γ n H j, m ,n H j, m ′ [ ℓ ] = E { n H j, m [ k + ℓ ] n H j, m ′ [ k ] } = Z ∞ −∞ Z ∞ −∞ Γ n ( x 1 , x 2 ) γ ψ H m 1 ,ψ H m ′ 1  x 1 M j − ℓ 1  × γ ψ H m 2 ,ψ H m ′ 2  x 2 M j − ℓ 2  dx 1 dx 2 (22) Γ n j, m ,n H j, m ′ [ ℓ ] = E { n j, m [ k + ℓ ] n H j, m ′ [ k ] } = Z ∞ −∞ Z ∞ −∞ Γ n ( x 1 , x 2 ) γ ψ m 1 ,ψ H m ′ 1  x 1 M j − ℓ 1  × γ ψ m 2 ,ψ H m ′ 2  x 2 M j − ℓ 2  dx 1 dx 2 . (23) From th e proper ties of the co rrelation functio ns of the wa velets and the scaling function as g iv en by Pr op. 2 , it c an be deduce d that, whe n ( m 1 = m ′ 1 = 0 or m 1 m ′ 1 6 = 0 ) an d ( m 2 = m ′ 2 = 0 or m 2 m ′ 2 6 = 0 ), Γ n j, m ,n j, m ′ = Γ n H j, m ,n H j, m ′ . (24) Some additional symmetr y prope rties are straightforwardly obtain ed from Prop. 2. In particular, for all m ∈ N ⋆ 2 M , the cross-covariance Γ n j, m ,n H j, m is an e ven sequence. An importan t co nsequen ce o f the latter pro perties concer ns the 2 × 2 linear combination o f th e primal and dual wavelet coefﬁcients which is o ften imp lemented in dual-tree decomp ositions. As e xplained in [31], the main advantage o f such a post-proce ssing is to better captu re the directional features in th e an alyzed imag e. M ore precisely , this amou nts to p erform ing the f ollowing unitary transfo rm of the detail coefﬁcients, fo r m ∈ N ⋆ 2 M : ∀ k ∈ Z 2 , w j, m [ k ] = 1 √ 2 ( n j, m [ k ] + n H j, m [ k ]) (25) w H j, m [ k ] = 1 √ 2 ( n j, m [ k ] − n H j, m [ k ]) . (26) (The transform is usually not applied whe n m 1 = 0 or m 2 = 0 .) The c ovariances of the tr ansformed ﬁelds of noise coefﬁcients ( w j, m [ k ]) k ∈ Z 2 and ( w H j, m [ k ]) k ∈ Z 2 then take the following exp ressions: Pr op osition 3 : F or all m ∈ N ⋆ 2 M and ℓ ∈ Z 2 , Γ w j, m ,w j, m [ ℓ ] = Γ n j, m ,n j, m [ ℓ ] + Γ n j, m ,n H j, m [ ℓ ] ( 27) Γ w H j, m ,w H j, m [ ℓ ] = Γ n j, m ,n j, m [ ℓ ] − Γ n j, m ,n H j, m [ ℓ ] ( 28) Γ w j, m ,w H j, m [ ℓ ] = 0 . (29) Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 8 Pr oo f: See Appen dix I V. This shows that the post-transfor m no t on ly provides a b etter directiona l analysis of the image of interest but also plays an importan t role w .r .t. th e noise analysis. I ndeed, it allo ws to co mpletely can cel th e co rrelations between the prim al an d dual n oise coefﬁcient ﬁelds obtained f or a gi ven value of ( j, m ) . In tu rn, this op eration introdu ces some spatial no ise cor relation in each sub band. For a two-dimension al white noise, Γ n ( x ) = σ 2 δ ( x ) an d the coefﬁcients ( n j, m [ k ]) k ∈ Z 2 and ( n H j, m ′ [ k ]) k ∈ Z 2 are such that, f or all ℓ = ( ℓ 1 , ℓ 2 ) ∈ Z 2 , Γ n j, m ,n j, m ′ [ ℓ ] = Γ n H j, m ,n H j, m ′ [ ℓ ] = σ 2 δ m 1 − m ′ 1 δ m 2 − m ′ 2 δ ℓ 1 δ ℓ 2 (30) Γ n j, m ,n H j, m ′ [ ℓ ] = σ 2 γ ψ m 1 ,ψ H m ′ 1 ( − ℓ 1 ) γ ψ m 2 ,ψ H m ′ 2 ( − ℓ 2 ) . (31) As a consequence of Prop . 2 , in th e case wh en ℓ = 0 , we c onclude that, f or ( m 1 6 = 0 or m 2 6 = 0) and k ∈ Z 2 , the vector [ n j, m [ k ] n H j, m [ k ]] has uncorrelated comp onents with equal v ariance. Th is p roperty ho lds mo re g enerally for 2D noises w ith separab le covariance function s. I V . S O M E A S Y M P T OT I C P RO P E RT I E S In the previous section, we hav e sh own that the correlation s of the basis function s play a promine nt role in the deter mination o f the second-o rder statistical prop erties of the n oise coefﬁcients. T o estimate the streng th of th e depend encies between the coefﬁcients, it is useful to determine the decay of the corre lation function s. The following result allows to e v aluate their decay . Pr op osition 4 : Let ( N 1 , . . . , N M − 1 ) ∈ ( N ∗ ) M − 1 and deﬁne N 0 = min m ∈ N ⋆ M N m . Assume that, f or a ll m ∈ N M , the fun ction | b ψ m | 2 is 2 N m + 1 times continuou sly d ifferentiable on R and, for all q ∈ { 0 , . . . , 2 N m + 1 } , its q -th order der iv atives ( | b ψ m | 2 ) ( q ) belong to L 1 ( R ) . 1 Further assume that, for all m 6 = 0 , b ψ m ( ω ) = O ( ω N m ) as ω → 0 . Then, there exists C ∈ R + such that, f or all m ∈ N M , ∀ τ ∈ R ∗ , | γ ψ m ,ψ m ( τ ) | ≤ C | τ | 2 N m +1 (32) and ∀ τ ∈ R ∗ , | γ ψ m ,ψ H m ( τ ) | ≤ C | τ | 2 N m +1 . (33) Pr oo f: See Appen dix V. Note that, for all m ∈ N M , th e assum ptions concer ning | b ψ m | 2 are satisﬁed if b ψ m is 2 N m + 1 times continuously differentiable on R a nd, for all q ∈ { 0 , . . . , 2 N m + 1 } , its q -th orde r derivati ves b ψ ( q ) m belong to L 2 ( R ) . Indeed, if b ψ m is 2 N m + 1 times con tinuously differentiable on R , so is | b ψ m | 2 . Le ibniz formu la allows us to express its deriv ati ve o f order q ∈ { 0 , . . . , 2 N m + 1 } as ( | b ψ m | 2 ) ( q ) = q X ℓ =0  q ℓ  ( b ψ m ) ( ℓ ) ( b ψ ∗ m ) ( q − ℓ ) . 1 By con ventio n, the deri vati ve of order 0 of a function is the function itself. Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 9 Consequently , if for all ℓ ∈ { 0 , . . . , q } , b ψ ( ℓ ) m ∈ L 2 ( R ) , then ( | b ψ m | 2 ) ( q ) ∈ L 1 ( R ) . Note also that, f or in tegrable wa velets, the assumption b ψ m ( ω ) = O ( ω N m ) as ω → 0 mea ns that the wa velet ψ m , m 6 = 0 , has N m vanishing mom ents. Therefo re, the decay rate of th e wavelet corre lation fu nctions is all the more imp ortant as th e Fourier tran sforms of the basis function s ψ m , m ∈ N M , are regular ( i.e. the wa velets have fast decay themselves) and the n umber of vanishing m oments is large. The latter condition is useful to ensure that Hilbert transfo rmed functio ns ψ H m have r egular spectra too. It must be emphasized th at Prop. 4 g uarantees that the asymp totic decay of the wavelet correlation fun ctions is at mo st | τ | − 2 N m − 1 . A faster d ecay can b e obtain ed in practice fo r so me wav elet families. For example, wh en ψ m is compa ctly sup ported , γ ψ m ,ψ m also has a compact support. In this case howe ver , ψ H m cannot be compa ctly su pported [ 32], so that the boun d in ( 33) rema ins of in terest. Examples will be discussed in more detail in Section V. It is also worth noticin g th at the obtained up per bound s on the cor relation f unctions allow us to evaluate the decay rate o f th e cov ariance sequen ces of the dual-tree wa velet coefﬁcients of a station ary noise as expressed below . Pr op osition 5 : Let n be a 1D zero -mean wide -sense stationary ran dom process. Assume tha t either n is a white noise or its a utocovariance fun ction is with exponential dec ay , th at is there exist A ∈ R + and α ∈ R ∗ + , such that ∀ τ ∈ R , | Γ n ( τ ) | ≤ Ae − α | τ | . Consider also fun ctions ψ m , m ∈ N M , satisfy ing the assum ptions o f Prop . 4. Th en, there exists e C ∈ R + such that for all j ∈ Z , m ∈ N M and ℓ ∈ Z ∗ , | Γ n j,m ,n j,m [ ℓ ] | ≤ e C 1 + | ℓ | 2 N m +1 (34) | Γ n j,m ,n H j,m [ ℓ ] | ≤ e C 1 + | ℓ | 2 N m +1 . (35) Pr oo f: See Appen dix VI . The decay pr operty o f the covariance sequences read ily extends to the 2D case: Pr op osition 6 : Let n be a 2 D zer o-mean wide-sense stationary random ﬁeld . Assume that either n is a white noise or its au tocovariance fun ction is with exponen tial decay , th at is th ere exist A ∈ R + and ( α 1 , α 2 ) ∈ ( R ∗ + ) 2 , such that ∀ ( τ 1 , τ 2 ) ∈ R 2 , | Γ n ( τ 1 , τ 2 ) | ≤ Ae − α 1 | τ 1 |− α 2 | τ 2 | . (36) Consider also fun ctions ψ m , m ∈ N M , satisfy ing the assum ptions o f Prop . 4. Th en, there exists e C ∈ R + such that for all j ∈ Z , m ∈ N 2 M and ℓ = ( ℓ 1 , ℓ 2 ) ∈ Z 2 , | Γ n j, m ,n j, m [ ℓ ] | ≤ e C (1 + | ℓ 1 | 2 N m +1 )(1 + | ℓ 2 | 2 N m +1 ) (37) | Γ n j, m ,n H j, m [ ℓ ] | ≤ e C (1 + | ℓ 1 | 2 N m +1 )(1 + | ℓ 2 | 2 N m +1 ) . (38) Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 10 Besides, for all j ∈ Z , m ∈ N ⋆ 2 M and ℓ = ( ℓ 1 , ℓ 2 ) ∈ Z 2 , | Γ w j, m ,w j, m [ ℓ ] | ≤ 2 e C (1 + | ℓ 1 | 2 N m +1 )(1 + | ℓ 2 | 2 N m +1 ) (39) | Γ w H j, m ,w H j, m [ ℓ ] | ≤ 2 e C (1 + | ℓ 1 | 2 N m +1 )(1 + | ℓ 2 | 2 N m +1 ) . (40) Pr oo f: Due to th e sep arability of th e 2 D d ual-tree wavelet analysis, (37) and (38 ) are obtain ed quite similarly to (34 ) and ( 35). The pro of of ( 39) and (4 0) th en follows fro m ( 27) and (28 ). The two previous pr opositions provide u pper bound s o n the dec ay rate of the covariance sequ ences of th e dual- tree wa velet co efﬁcients, wh en th e no rm of the lag variable ( ℓ or ℓ ) takes large values. W e end this sectio n by providing asy mpotic r esults at coa rse r esolution (as j → ∞ ). Pr op osition 7 : Let n be a 1D zero- mean wide-sen se station ary proce ss with cov ariance function Γ n ∈ L 1 ( R ) ∩ L 2 ( R ) . Then, f or all ( m, m ′ ) ∈ N 2 M , we hav e lim j →∞ Γ n j,m ,n j,m ′ [ ℓ ] = b Γ n (0) δ m − m ′ δ ℓ (41) lim j →∞ Γ n j,m ,n H j,m ′ [ ℓ ] = b Γ n (0) γ ψ m ,ψ H m ′ ( − ℓ ) . (42) Pr oo f: See Appen dix VI I. In other words, at coar se resolution in the transfo rm domain, a stationary noise n with arbitrary covariance fun ction Γ n behaves like a white noise with spectru m den sity b Γ n (0) . This fact fur ther emp hasizes th e in terest in stud ying more p recisely the dual-tree wavelet d ecompo sition o f a white n oise. Note also that, by calcu lating hig her orde r cumulants of the dual-tr ee wa velet coefﬁcients and using techniq ues as in [18 ], [40] , it could be proved that, for all ( m, m ′ ) ∈ N 2 M and ( k , k ′ ) ∈ Z 2 , [ n j,m ( k ) n H j,m ′ ( k ′ )] is asymptotically no rmal as j → ∞ . A lthough Pro p. 7 h as been stated f or 1D random proce sses, we ﬁnally point ou t that quite similar results are obtained in th e 2 D case. V . W A V E L E T FA M I L I E S E X A M P L E S For a white noise (see ( 14), (15), ( 30) and (3 1)) or for arbitrar y wide-sense statio nary n oises a nalyzed at coar se resolution (cf. Prop. 7), we h av e seen th at the cross-corr elation fu nctions b etween the primal and dua l wavelets taken at integer values are the ma in f eatures. In or der to better ev aluate the impact of the wavelet choice, we will now specify the expr essions of these cro ss-correlation s for different wav elet families. A. M - band S hanno n wavelets M -band Shann on wav elets (also ca lled sinc wavelets in th e liter ature) correspo nd to an ideally selectiv e an alysis in the frequency d omain. These wavelets also a ppear as a limit ca se fo r m any wa velet families, e .g. Daubech ies’ s or spline wavelets. W e ha ve then , for all m ∈ N M , b ψ m ( ω ) = 1 ] − ( m +1) π , − mπ ] ∪ [ mπ , ( m +1) π [ ( ω ) , where 1 S denotes the char acteristic f unction of the set S ⊂ R : 1 S ( ω ) =      1 if ω ∈ S 0 otherwise. Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 11 In this case, (20) read s: ∀ τ ∈ R , γ ψ 0 ,ψ H 0 ( τ ) = 1 π Z π 0 cos  ( 1 2 + d + τ ) ω  dω =        ( − 1) d cos ( πτ ) π ( 1 2 + d + τ ) if τ 6 = − d − 1 2 1 otherwise. For m ∈ N ⋆ M , (19 ) leads to ∀ τ ∈ R , γ ψ m ,ψ H m ( τ ) = − 1 π Z ( m +1) π mπ sin ( ω τ ) dω =      cos  ( m + 1 ) π τ  − cos( mπ τ ) π τ if τ 6 = 0 0 other wise. W e deduce fr om th e two previous expressions that, for all ℓ ∈ Z , γ ψ 0 ,ψ H 0 ( ℓ ) = ( − 1) ( d + ℓ ) π ( d + ℓ + 1 2 ) (43) ∀ m 6 = 0 , γ ψ m ,ψ H m ( ℓ ) =      ( − 1) ( m +1) ℓ 1 − ( − 1) ℓ π ℓ if ℓ 6 = 0 0 otherwise . (44) W e can rem ark th at, for all ( m, m ′ ) ∈ N ⋆ 2 M , γ ψ m ,ψ H m ( ℓ ) = ( − 1) ( m ′ − m ) ℓ γ ψ m ′ ,ψ H m ′ ( ℓ ) (45) and γ ψ m ,ψ H m ( ℓ ) = 0 , wh en ℓ is odd. Besides, the correlatio n sequen ces decay pretty slo wly as ℓ − 1 . W e also no te that, as the fu nctions ψ m , m ∈ N M , have n on-overlappin g spectra, (6)-(8 ) (resp. (21)-(23)) allow us to conclude that, dual-tree noise wavelet co efﬁcients correspo nding respectively to subband s ( j, m ) and ( j, m ′ ) with m 6 = m ′ (resp. ( j, m 1 , m 2 ) and ( j, m ′ 1 , m ′ 2 ) with m 1 6 = m ′ 1 or m 2 6 = m ′ 2 ) are per fectly u ncorrelated . B. Me yer wav e le ts These wa velets [4 1], [42, p. 11 6] are also band-lim ited but with smoother transition s th an Shannon wa velets. The scaling fun ction is consequen tly deﬁned as b ψ 0 ( ω ) =              1 if 0 ≤ | ω | ≤ π (1 − ǫ ) W  | ω | 2 π ǫ − 1 − ǫ 2 ǫ  if π (1 − ǫ ) ≤ | ω | ≤ π (1 + ǫ ) 0 otherwise, (46) where 0 < ǫ ≤ 1 / ( M + 1) and ∀ θ ∈ [0 , 1] , W ( θ ) = cos  π 2 ν ( θ )  Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 12 b ψ m ( ω ) =                    e ıη m ( ω ) W  m + ǫ 2 ǫ − | ω | 2 π ǫ  if ( m − ǫ ) π ≤ | ω | ≤ ( m + ǫ ) π e ıη m ( ω ) if ( m + ǫ ) π ≤ | ω | ≤ ( m + 1 − ǫ ) π e ıη m ( ω ) W  | ω | 2 π ǫ − m + 1 − ǫ 2 ǫ  if ( m + 1 − ǫ ) π ≤ | ω | ≤ ( m + 1 + ǫ ) π 0 otherwise (50) b ψ M − 1 ( ω ) =                    e ıη M − 1 ( ω ) W  M − 1 + ǫ 2 ǫ − | ω | 2 π ǫ  if ( M − 1 − ǫ ) π ≤ | ω | ≤ ( M − 1 + ǫ ) π e ıη M − 1 ( ω ) if ( M − 1 + ǫ ) π ≤ | ω | ≤ M (1 − ǫ ) π e ıη M − 1 ( ω ) W  | ω | 2 π ǫM − 1 − ǫ 2 ǫ  if M (1 − ǫ ) π < | ω | ≤ M (1 + ǫ ) π 0 otherwise. (51) with ν : [0 , 1 ] → [0 , 1] su ch that ν (0) = 0 (47) ∀ θ ∈ [0 , 1] , ν (1 − θ ) = 1 − ν ( θ ) . Then, it can b e noticed that ∀ θ ∈ [0 , 1] , W 2 (1 − θ ) = 1 − W 2 ( θ ) . ( 48) A comm on ch oice fo r the ν fun ction is [ 42, p. 11 9]: ∀ θ ∈ [0 , 1] , ν ( θ ) = θ 4 (35 − 8 4 θ + 70 θ 2 − 20 θ 3 ) . (49) For m ∈ { 1 , . . . , M − 2 } , the associated M -band wavelets ar e g i ven b y (50 ) while, for the last wa velet, we ha ve (51). Hereabove, the phase f unctions η m , m ∈ N ⋆ M , are odd functions an d we have ∀ ω ∈ ( M π , M (1 + ǫ ) π ) , η M − 1 ( ω ) = − η M − 1 (2 M π − ω ) mo d 2 π . In addition , for th e ortho norma lity con dition to b e satisﬁed, the f ollowing recursive e quations must ho ld: ∀ ω ∈ (( m − ǫ ) π , ( m + ǫ ) π ) , η m ( ω − 2 mπ ) − η m − 1 ( ω − 2 mπ ) = η m ( ω ) − η m − 1 ( ω ) + π mo d 2 π . by setting: ∀ ω ∈ R , η 0 ( ω ) = 0 . Gene rally , linear phase solution s to the p revious equatio n are chosen [4 3]. Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 13 Using the above expressions, the cross-correlation s between the Meyer basis function s and their dual counterparts are derived in Appe ndix VIII . I t can be ded uced from these results th at: ∀ ℓ ∈ Z , γ ψ 0 ,ψ H 0 ( ℓ ) = ( − 1) d + ℓ π ( d + ℓ + 1 2 ) − ( − 1) d + ℓ I ǫ  d + ℓ + 1 2  , (52) where ∀ x ∈ R , I ǫ ( x ) = 2 ǫ Z 1 0 W 2  1 + θ 2  sin ( π ǫxθ ) dθ . ( 53) For the wa velets, we h av e wh en m ∈ { 1 , . . . , M − 2 } : γ ψ m ,ψ H m ( ℓ ) =      ( − 1) ( m +1) ℓ  1 − ( − 1) ℓ   1 π ℓ − I ǫ ( ℓ )  if ℓ 6 = 0 0 oth erwise (54) whereas γ ψ M − 1 ,ψ H M − 1 ( ℓ ) =      ( − 1) M ℓ  1 − ( − 1) ℓ π ℓ + ( − 1) ℓ I ǫ ( ℓ ) − I M ǫ ( ℓ )  if ℓ 6 = 0 0 otherwise. (55) Similarly to Sh annon wav elets, for ( m, m ′ ) ∈ { 1 , . . . , M − 2 } 2 , (45 ) holds and γ ψ m ,ψ H m ( ℓ ) = 0 , when ℓ is o dd. As expected, we observe that th e previous cross-correlations conver ge point-wise to th e exp ressions given f or Shann on wa velets in (4 3) and (44 ), as w e let ǫ → 0 . Besides, let us make the fo llowing assumption: W 2 is 2 q + 2 times c ontinuo usly differentiable on [0 , 1] with q ∈ N ∗ and, for all ℓ ∈ { 0 , . . . , 2 q − 1 } , ( W 2 ) ( ℓ ) (1) = 0 . This assump tion is typically satisﬁed by the window deﬁned by (4 9) with q = 4 . Fro m (48), it can be furth er n oticed that, for all ℓ ∈ { 1 , . . . , q + 1 } , ( W 2 ) (2 ℓ ) (1 / 2) = 0 . Then, when x 6 = 0 , it is readily ch ecked by in tegrating b y part that Z 1 0 W 2  1 + θ 2  sin ( π ǫxθ ) dθ = 1 2 π ǫx + ( − 1) q − 1 ( W 2 ) (2 q ) (1) 2 2 q ( π ǫx ) 2 q +1 cos( π ǫx ) + ( − 1) q ( W 2 ) (2 q +1) (1) 2 2 q +1 ( π ǫx ) 2 q +2 sin( π ǫx ) + ( − 1) q +1 2 2 q +2 ( π ǫx ) 2 q +2 Z 1 0 ( W 2 ) (2 q +2)  1 + θ 2  sin ( πǫxθ ) dθ . This shows that, as | x | → ∞ , I ǫ ( x ) = 1 π x + ( − 1) q − 1 ( W 2 ) (2 q ) (1) 2 2 q − 1 π 2 q +1 ǫ 2 q x 2 q +1 cos( π ǫx ) + O ( x − 2 q − 2 ) . (56) For examp le, for th e tap er function deﬁne d by (49), we g et I ǫ ( x ) = 1 π x − 38587 5 4 π 7 ǫ 8 x 9 cos( π ǫx ) + O ( x − 10 ) . Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 14 Combining (56 ) with (5 2), (54) an d (55) allows us to see that the cross-co rrelation sequences decay as ℓ − 2 q − 1 when | ℓ | → ∞ . Eq. ( 56) also indicates that the decay tends to be faster when ǫ is large, which is co nsistent with intuition since the b asis fu nctions a re then better lo calized in time. Note that, as shown by (50) and (51), under the considered differentiability ass umption s, | b ψ m | 2 is 2 q − 1 times co ntinuou sly differentiable on R whereas b ψ m ( ω ) = 0 for m ∈ N ⋆ M and | ω | < m − ǫ . Pro p. 4 th en guaran tees a decay r ate at least equ al to | ℓ | − 2 q +1 (here, N m = q − 1 ). In this case, we see that the decay rate derived from (56 ) is more acurate than th e d ecay given b y Prop. 4. C. W av e let fa milies derived fr om wav e le t pa ck ets 1) General form: One c an generate M -band orthonor mal wa velet b ases from dyadic ortho normal wav elet packet decomp ositions correspo nding to an equ al sub band analysis. W e are conseq uently limited to scaling factors M which are p ower of 2 . Mo re precisely , let ( ψ m ) m ∈ N be the co nsidered wav elet packets [4 4], for all P ∈ N ∗ an orthon ormal M -band wav elet decomp osition is obtained u sing the basis function s ( ψ m ) 0 ≤ m 1 , provid ed that γ ψ 1 ,ψ H 1 has b een calculated ﬁrst. For this speciﬁc class of M -band wa velet decompo sitions, it is po ssible to relate the decay properties of the cross-corre lation functio ns to the nu mber of vanishing mo ments of the underly ing dyad ic wav elet analysis. Pr op osition 9 : Assume that the ﬁlters with frequency respo nse A 0 and A 1 are FIR and A 1 has a zer o of order N ∈ N ∗ at f requency 0 (or, equivalently , A 0 has a zero o f ord er N at fr equency 1 / 2 ). Then , th ere exists C 0 ∈ R + Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 15 such that ∀ τ ∈ R ∗ , | γ ψ 0 ,ψ H 0 ( τ ) | ≤ C 0 | τ | − 2 N − 1 . (61) In ad dition, for all m ∈ N ∗ , let ( ǫ 1 , ǫ 2 , . . . , ǫ r ) ∈ { 0 , 1 } r , r ∈ N ∗ , b e the d igits in the binar y representatio n of m , that is m = r X i =1 ǫ i 2 i − 1 . (62) Then, there exists C m ∈ R + such that ∀ τ ∈ R ∗ , | γ ψ m ,ψ H m ( τ ) | ≤ C m | τ | − 2 N ( P r i =1 ǫ i ) − 1 . (63) Pr oo f: The ﬁlters of the u nderly ing d yadic multireso lution being FIR (Fin ite Impu lse Response), th e wavelet packets are compactly sup ported. Consequen tly , their Fourier transforms are inﬁnitely differentiable, their deriv ati ves of any order b elonging to L 2 ( R ) . In a ddition, the bin ary r epresentation of m ∈ N ∗ being given by ( 62), Eqs. (5 7) and (58) yield b ψ m ( ω ) = b ψ 0  ω 2 P  r Y i =1  1 √ 2 A ǫ i  ω 2 i   P Y i = r +1  1 √ 2 A 0  ω 2 i   that is H m ( ω ) = Q P i =1 A ǫ i (2 P − i ω ) . Moreover , by assumption A 1 ( ω ) = O ( ω N ) as ω → 0 , whereas A 0 (0) = √ 2 and | b ψ 0 (0) | = 1 . This shows that, when m 6 = 0 , b ψ m ( ω ) = O ( ω N ( P r i =1 ǫ i ) ) as ω → 0 . From (33 ), we ded uce the upper bo und in (63 ). Furtherm ore, by applyin g Prop. 4 wh en M = 2 , we have th en N 0 = N 1 = N and (61) is obtained. W e see that the cross-correlation γ ψ m ,ψ H m decays all the more rapidly as the number of 1’ s in the binary representation of m is large. 2 2) The particula r case of W alsh-Had amar d transform: Th e case M = 2 co rrespond s to Haar wa velets. In contrast with Shann on wav elets, these wavelets lay emphasis on time/spatial localization. W e consequen tly have: b ψ 0 ( ω ) = sinc( ω 2 ) e − ı w 2 (64) b ψ 1 ( ω ) = ı s inc( ω 4 ) sin( ω 4 ) e − ı w 2 , (65) where sinc( ω ) =      sin( ω ) ω if ω 6 = 0 1 otherwise. After some calc ulations wh ich are pr ovided in Appen dix X, we obtain for all τ ∈ R , π γ ψ 0 ,ψ H 0 ( τ ) = ∞ X k =0 ( − 1) k  1 2 S k (3 + 2 d + 2 τ ) − S k (1 + 2 d + 2 τ ) + 1 2 S k ( − 1 + 2 d + 2 τ )  , (66) 2 The chara cteri zatio n of the sum of digits of int ege rs remains an open proble m in number theory [45], [46]. Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 16 where, for all k ∈ N and for all x ∈ R , S k ( x ) = x Z ( k +1) π x kπ x sinc( u ) du. Furthermo re, we have (ad opting th e co n vention: 0 ln(0) = 0 ): π γ ψ 1 ,ψ H 1 ( τ ) = 6 τ ln | τ | + ( τ + 1) ln | τ + 1 | + ( τ − 1 ) ln | τ − 1 | − 4  τ + 1 2  ln    τ + 1 2    − 4  τ − 1 2  ln    τ − 1 2    . (67) For M = 2 P with P > 1 , the cross-co rrelations γ ψ m ,ψ H m , m ∈ { 2 , . . . , 2 P − 1 } , can be deter mined in a recursive manner thank s to Prop. 8. For W alsh -Hadamard wav elets, we hav e ∀ ǫ ∈ { 0 , 1 } , ∀ k ∈ Z , γ a ǫ [ k ] =              1 if k = 0 ( − 1) ǫ 2 if | k | = 1 0 otherwise (68) and, con sequently , for all m 6 = 0 an d τ ∈ R , γ ψ 2 m ,ψ H 2 m ( τ ) = γ ψ m ,ψ H m (2 τ ) + 1 2  γ ψ m ,ψ H m (2 τ + 1) + γ ψ m ,ψ H m (2 τ − 1)  (69) γ ψ 2 m +1 ,ψ H 2 m +1 ( τ ) = γ ψ m ,ψ H m (2 τ ) − 1 2  γ ψ m ,ψ H m (2 τ + 1) + γ ψ m ,ψ H m (2 τ − 1)  . (70) From (67), it can b e noticed that γ ψ 1 ,ψ H 1 ( τ ) = 1 / (8 π τ 3 ) + O ( τ − 5 ) when | τ | > 2 , which c orrespon ds to a faster asymptotic decay than with Shanno n w av elets. The asymptotic behaviour of γ ψ m ,ψ H m ( τ ) , m > 2 , can also be deduced from (67), ( 69) and ( 70). The expressions given in T able I are in p erfect ag reement with the d ecay ra tes pr edicted by Prop. 9 . D. F ranklin wavelets Franklin wavelets [4 7], [48 ] cor respond to a dyadic o rthono rmal b asis o f spline wavelets of or der 1 [42, p . 1 46 sq.]. With the Haar wavelet, they form a special case o f Battle-Lem ari ´ e wa velets [49], [5 0]. The Fourier transf orms of the scaling f unction and the mo ther wavelet are g i ven b y: b ψ 0 ( ω ) =  3 1 + 2 cos 2 ( ω / 2)  1 / 2 sinc 2  ω 2  (71) b ψ 1 ( ω ) = − 3(1 + 2 sin 2 ( ω / 4))  1 + 2 cos 2 ( ω / 2)  1 + 2 cos 2 ( ω / 4)  ! 1 / 2 × sin 2  ω 4  sinc 2  ω 4  exp ( − ı ω 2 ) . (72) Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 17 The expression of the cr oss-correlation of the scaling functions r eadily f ollows from (2 0): ∀ τ ∈ R , γ ψ 0 ,ψ H 0 ( τ ) = 6 π ∞ X k =0 ( − 1) k T k (1 + 2 d + 2 τ ) , where, for all k ∈ N and x ∈ R , T k ( x ) = Z ( k +1) π kπ sinc 4 ( u ) 1 + 2 cos 2 ( u ) cos( ux ) du. The expression of the cross-cor relation of th e m other wav elet can b e deduced fro m (19 ) and ( 72) and resortin g to numerical metho ds f or the c omputatio n of th e resulting in tegral, but it is also possible to ob tain a series expansion of the cr oss-correlation as shown next. T aking the squa re modulu s of (72 ), we ﬁnd 2 | b ψ 1 (2 ω ) | 2 = | e A 1 ( ω ) | 2 | b χ ( ω ) | 2 , (73) where e A 1 ( ω ) = 6  2 − cos( ω )   1 + 2 cos 2 ( ω )  2 + cos( ω )  ! 1 / 2 , b χ ( ω ) =  sin 2 ( ω / 2) ω / 2  2 . Let ( e a 1 [ k ]) k ∈ Z (resp. χ ) be the sequence (resp. fun ction) whose Fourier transfor m is e A 1 (resp. b χ ). Similarly to (60), (73) leads to the f ollowing relation ∀ τ ∈ R , γ ψ 1 ,ψ H 1 ( τ ) = γ e a 1 [0] γ χ,χ H (2 τ ) + ∞ X k =1 γ e a 1 [ k ]  γ χ,χ H (2 τ + k ) + γ χ,χ H (2 τ − k )  , (74) where ( γ e a 1 [ k ]) k ∈ Z denotes the autocorrelation of the sequence ( e a 1 [ k ]) k ∈ Z . W e have then to determin e γ χ,χ H and ( γ ˜ a 1 [ k ]) k ∈ N . First, it can be shown (see Append ix XI for more detail) that 3 π γ χ,χ H ( τ ) = q 0 τ 3 ln | τ | + 4 X p =1 q p  ( τ + p ) 3 ln | τ + p | + ( τ − p ) 3 ln | τ − p |  , (75) where q 0 = − 35 16 , q 1 = 7 4 , q 2 = − 7 8 , q 3 = 1 4 , q 4 = − 1 32 . Secondly , th e seq uence ( γ ˜ a 1 [ k ]) k ∈ N can be d educed fr om | e A 1 ( ω ) | 2 by using z -tra nsform inversion tec hniques (calculations are p rovided in Appe ndix X I). This lead s to ∀ k ∈ N ,      γ ˜ a 1 [2 k ] = 2 √ 3 9 (2 − √ 3) k  7( − 1) k + 4(2 − √ 3) k  γ ˜ a 1 [2 k + 1] = 8 √ 3 9 (2 − √ 3) k  ( − 1) k (1 − √ 3) − (2 − √ 3) k +1  . (76) Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 18 Equation s (74), (75 ) and (76) thus allow an accurate n umerical ev a luation o f γ ψ 1 ,ψ H 1 . Since γ χ,χ H ( τ ) ∼ − 3 / (2 π τ 5 ) as | τ | → ∞ (77) and γ ˜ a 1 [ k ] = O ((2 − √ 3) k/ 2 ) as k → ∞ (78) the convergence of the series in (7 4) is indee d pretty fast. From Prop. 4 , we furthe r dedu ce that γ ψ 0 ,ψ H 0 ( τ ) and γ ψ 1 ,ψ H 1 ( τ ) decay as | τ | − 5 (here, we h ave N 0 = N 1 = 2 ) . The decay rate o f γ ψ 1 ,ψ H 1 can be derived more pr ecisely from (74). Ind eed, we hav e | τ | 5 ∞ X k = −∞ | γ e a 1 [ k ] || γ χ,χ H (2 τ − k ) | ≤ 1 2 ∞ X k = −∞ | γ e a 1 [ k ] |  | 2 τ − k | 5 + | k | 5  | γ χ,χ H (2 τ − k ) | ≤  sup u ∈ R ( | u | 5 | γ χ,χ H ( u ) | ) + sup u ∈ R | γ χ,χ H ( u ) |  ∞ X k = −∞ (1 + | k | 5 ) | γ e a 1 [ k ] | < ∞ , (79) where th e conve xity o f | . | 5 has b een used in the ﬁrst ine quality and the last in equality is a con sequence of (77) and (78). It can be deduced f rom the dom inated c on vergence theorem that lim | τ |→∞ τ 5 γ ψ 1 ,ψ H 1 ( τ ) = ∞ X k = −∞ γ e a 1 [ k ] lim | τ |→∞ τ 5 γ χ,χ H (2 τ − k ) = − 3 64 π ∞ X k = −∞ γ e a 1 [ k ] = − 3 64 π | e A 1 (0) | 2 = − 1 32 π . Finally , we would like to note that similar expressions can b e derived for h igher order sp line wa velets althoug h the calculations beco me tedio us. V I . E X P E R I M E N TA L R E S U LT S A. Results b ased on theo r e tical expr ession s At ﬁrst, we provide num erical ev aluations o f the expressions of the c ross-correlatio n sequ ences ob tained in the previous section when the lag variable (denoted by ℓ ) varies in { 0 , 1 , 2 , 3 } . The cro ss-correlation s for lag values in {− 3 , − 2 , − 1 } can be deduced from the symm etry pr operties sh own in Section I II-A. W e notice that cu bic spline wa velets [51] have no t been studied in Sectio n V, so that the their cro ss-correlation values have to be com puted directly from (19 ) and (20). The re sults concernin g the dya dic case are given in T able II. Th ey show th at the cross-corre lations between the noise coe fﬁcients at the output of a d ual-tree analysis can take sign iﬁcant values (up to 0.64 ). W e also observe that the wa velet cho ice has a clear inﬂuen ce o n the magnitud e of the cor relations. Indeed , while Meyer wa velet lead s to r esults close to the Shanno n wa velet, th e corr elations are weaker for the Haar Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 19 wa velet. As expected, spline wavelets yield interm ediate cr oss-correlation values between th e Meyer an d the Haar cases. Our next results conc ern the M -band case with M ≥ 3 . Due to the properties of the cro ss-correlation s, the study can be simp liﬁed a s explained below . • Shanno n wav elets: due to ( 45), the M -band cro ss-correlation s are, up to a possible sign ch ange, equal to the dyadic case cross-co rrelations (see T able II ). • Meyer wav elets: still du e to (45), the ﬁrst M − 2 cross-co rrelations of the wa velets a re easily d educed from the ﬁrst o ne. So, w e o nly need to specify γ ψ 0 ,ψ H 0 , γ ψ 1 ,ψ H 1 and γ ψ M − 1 ,ψ H M − 1 . T a bles III and IV give the re lated values when M r anges from 3 to 8, the ǫ par ameter b eing set to its po ssible maximu m value ( M + 1) − 1 . • W alsh-Hadam ard wa velets: wh en M = 2 P +1 , P ∈ N ∗ , ( ψ m ) 0 ≤ m 0 ten ds to 0 of Γ n ǫ ( τ ) = σ 2 √ 2 π ǫ exp( − τ 2 2 ǫ 2 ) , τ ∈ R . Formula (8 ) can then be used, y ielding for all ( m, m ′ ) ∈ N 2 M and ( j, ℓ ) ∈ Z 2 , Γ n ǫ j,m ,n ǫ H j,m ′ [ ℓ ] = σ 2 Z ∞ −∞ 1 √ 2 π exp( − x 2 2 ) γ ψ m ,ψ H m ′  ǫx M j − ℓ  dx. Since ψ m and ψ H m ′ are in L 2 ( R ) , γ ψ m ,ψ H m ′ is a bo unded continuo us fun ction. By ap plying Lebesgu e dom inated conv ergence theo rem, we d educe that Γ n j,m ,n H j,m ′ [ ℓ ] = lim ǫ → 0 Γ n ǫ j,m ,n ǫ H j,m ′ [ ℓ ] = σ 2 Z ∞ −∞ 1 √ 2 π exp( − x 2 2 ) lim ǫ → 0 γ ψ m ,ψ H m ′  ǫx M j − ℓ  dx = σ 2 γ ψ m ,ψ H m ′ ( − ℓ ) Z ∞ −∞ 1 √ 2 π exp( − x 2 2 ) dx which leads to (15). E quations (14) are similarly o btained by fu rther noticing that, du e to the o rthono rmality proper ty , γ ψ m ,ψ m ′ ( − ℓ ) = γ ψ H m ,ψ H m ′ ( − ℓ ) = δ m − m ′ δ ℓ . A P P E N D I X I V P R O O F O F P R O P O S I T I O N 3 From (25) and (26) deﬁning the unitary transform applied to the detail noise coefﬁcients ( n j, m [ k ]) k ∈ Z 2 and ( n H j, m [ k ]) k ∈ Z 2 : E { w j, m [ k ] w j, m [ k ′ ] } = 1 2  E { n j, m [ k ] n j, m [ k ′ ] } + E { n j, m [ k ] n H j, m [ k ′ ] } + E { n H j, m [ k ] n j, m [ k ′ ] } + E { n H j, m [ k ] n H j, m [ k ′ ] }  . Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 24 Using (2 4) and th e evenness of Γ n j, m ,n H j, m , o ne can easily deduce (27 ). Concernin g (2 8), we pr oceed in the same way , ta king into acco unt the relation: E { w H j, m [ k ] w H j, m [ k ′ ] } = 1 2  E { n j, m [ k ] n j, m [ k ′ ] } − E { n j, m [ k ] n H j, m [ k ′ ] } − E { n H j, m [ k ] n j, m [ k ′ ] } + E { n H j, m [ k ] n H j, m [ k ′ ] }  . Finally , notin g that E { w j, m [ k ] w H j, m [ k ′ ] } = 1 2  E { n j, m [ k ] n j, m [ k ′ ] } − E { n j, m [ k ] n H j, m [ k ′ ] } + E { n H j, m [ k ] n j, m [ k ′ ] } − E { n H j, m [ k ] n H j, m [ k ′ ] }  and, inv oking the same argu ments, we see th at w j, m [ k ] and w H j, m [ k ′ ] are u ncorrela ted rando m variables. A P P E N D I X V P R O O F O F P R O P O S I T I O N 4 Since ψ m ∈ L 2 ( R ) , we ha ve ∀ τ ∈ R , γ ψ m ,ψ m ( τ ) = 1 2 π Z ∞ −∞ | b ψ m ( ω ) | 2 e ıω τ dω . Furthermo re, | b ψ m | 2 is 2 N m + 1 times c ontinuo usly differentiable an d for all q ∈ { 0 , . . . , 2 N m + 1 } , ( | b ψ m | 2 ) ( q ) ∈ L 1 ( R ) . It can be d educed [60 ][p. 158 –159 ] that ∀ τ ∈ R , ( − ıτ ) 2 N m +1 γ ψ m ,ψ m ( τ ) = 1 2 π Z ∞ −∞ ( | b ψ m | 2 ) (2 N m +1) ( ω ) e ıω τ dω which leads to ∀ τ ∈ R , | τ | 2 N m +1 | γ ψ m ,ψ m ( τ ) | ≤ 1 2 π Z ∞ −∞   ( | b ψ m | 2 ) (2 N m +1) ( ω )   dω . (80 ) Let us n ow consider the cross-co rrelation fu nctions γ ψ m ,ψ H m with m 6 = 0 . Similar ly , when m 6 = 0 , we h ave ∀ τ ∈ R , γ ψ m ,ψ H m ( τ ) = 1 2 π Z ∞ −∞ α ( ω ) | b ψ m ( ω ) | 2 e ıω τ dω , (81 ) Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 25 where α ( ω ) = ı sign( ω ) . The function ω 7→ α ( ω ) | b ψ m ( ω ) | 2 is 2 N m + 1 times c ontinuo usly dif ferentiable on R ∗ , where its derivati ve o f order q ∈ { 0 , . . . , 2 N m + 1 } is ( α | b ψ m | 2 ) ( q ) = α ( | b ψ m | 2 ) ( q ) . (82) Due to the fact that | b ψ m ( ω ) | 2 = O ( ω 2 N m ) as ω → 0 , we hav e for all q ∈ { 0 , . . . , 2 N m − 1 } , ( | b ψ m | 2 ) ( q ) (0) = 0 . From ( 82), we ded uce th at th e fun ction ( α | b ψ m | 2 ) ( q ) admits limits o n the lef t side an d on the rig ht side of 0 , which are both equal to 0. This allows to conclude that α | b ψ m | 2 is 2 N m − 1 times con tinuously differentiable on R , its 2 N m − 1 ﬁrst d eriv ati ves vanishing at 0 . Besides, ( α | b ψ m | 2 ) (2 N m − 1) is continu ously d ifferentiable on ( − ∞ , 0 ] and on [0 , ∞ ) ( ( α | b ψ m | 2 ) (2 N m ) may be discon tinuous at 0). Using the same argumen ts as for γ ψ m ,ψ m , this allows us to claim that ∀ τ ∈ R , ( − ıτ ) 2 N m γ ψ m ,ψ H m ( τ ) = 1 2 π Z ∞ −∞ α ( ω )( | b ψ m | 2 ) (2 N m ) ( ω ) e ıω τ dω = − 1 π Z ∞ 0 ( | b ψ m | 2 ) (2 N m ) ( ω ) sin( ω τ ) dω . (83) W e can note that lim ω →∞ ( | b ψ m | 2 ) (2 N m ) ( ω ) ∈ R as it is equal to ( | b ψ m | 2 ) (2 N m ) + (0) + R ∞ 0 ( | b ψ m | 2 ) (2 N m +1) ( ν ) dν where ( | b ψ m | 2 ) (2 N m ) + (0) deno tes the right-han d side der i vati ve of order 2 N m of | b ψ m | 2 at 0. Sin ce ( | b ψ m | 2 ) (2 N m ) ∈ L 1 ([0 , ∞ )) , the pr evious limit is n ecessarily zero. Using th is fact and integrating by pa rt in (83), we ﬁnd that, f or all τ ∈ R , τ Z ∞ 0 ( | b ψ m | 2 ) (2 N m ) ( ω ) sin( ω τ ) dω = ( | b ψ m | 2 ) (2 N m ) + (0) + Z ∞ 0 ( | b ψ m | 2 ) (2 N m +1) ( ω ) cos( ω τ ) dω . Combining this expre ssion with (83 ), we deduce th at ∀ τ ∈ R , | τ | 2 N m +1 | γ ψ m ,ψ H m ( τ ) | ≤ 1 π  Z ∞ 0 | ( | b ψ m | 2 ) (2 N m +1) ( ω ) | dω + | ( | b ψ m | 2 ) (2 N m ) + (0) |  . (84) Let u s now study the ca se whe n m = 0 . Eq. ( 81) still holds, but as shown by (4), α takes a more complicated form: ∀ k ∈ Z , ∀ ω ∈ [2 k π , 2( k + 1) π ) , α ( ω ) =      ( − 1) k e ı ( d + 1 2 ) ω if k ≥ 0 ( − 1) k +1 e ı ( d + 1 2 ) ω otherwise. Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 26 So, the function α as well as its deri vati ves of any order n ow exhibit discontinuities at 2 k π whe re k ∈ Z ∗ . Howe ver , from (1) and th e low-pass con dition b ψ 0 (0) = 1 , we have, f or all m 6 = 0 , H m ( ω ) = O ( ω N m ) , as ω → 0 . As a con sequence of the para-un itary cond ition (2 ), we get M − 1 X m =0 | H m ( ω ) | 2 = M and M − 1 X p =0 | H 0 ( ω + p 2 π M ) | 2 = M which allows to d educe that ∀ p ∈ N ⋆ M , H 0 ( ω + p 2 π M ) = O ( ω N 0 ) . From (1), it can b e conclu ded that ∀ k ∈ Z ∗ , b ψ 0 ( ω + 2 k π ) = O ( ω N 0 ) , as ω → 0 . (85) The deriv ativ es of orde r q ∈ { 0 , . . . , 2 N 0 + 1 } of α | b ψ 0 | 2 over R \ { 2 k π , k ∈ Z ∗ } are given by ( α | b ψ 0 | 2 ) ( q ) = q X ℓ =0  q ℓ  ( α ) ( ℓ ) ( | b ψ 0 | 2 ) ( q − ℓ ) , where ∀ k ∈ Z , ∀ ω ∈ (2 k π , 2( k + 1) π ) , α ( ℓ ) ( ω ) =      ( − 1) k ı ℓ ( d + 1 2 ) ℓ e ı ( d + 1 2 ) ω if k ≥ 0 ( − 1) k +1 ı ℓ ( d + 1 2 ) ℓ e ı ( d + 1 2 ) ω otherwise. W e deduce th at, for all q ∈ { 0 , . . . , 2 N 0 + 1 } , ( α | b ψ 0 | 2 ) ( q ) ∈ L 1 ( R ) . Fur thermor e, combin ing (8 5) with (86 ) allows us to show that, for all q ∈ { 0 , . . . , 2 N 0 − 1 } , the derivati ve of order q o f α | b ψ 0 | 2 at 2 k π , k ∈ Z ∗ , is d eﬁned and equal to 0. Consequ ently , α | b ψ 0 | 2 is 2 N 0 − 1 times c ontinuo usly differentiable o n R wh ile ( α | b ψ 0 | 2 ) (2 N 0 − 1) is continuo usly differentiable on ∪ k ∈ Z (2 k π , 2( k + 1) π ) . Similar ly to the case m 6 = 0 , this leads to ∀ τ ∈ R , ( − ıτ ) 2 N 0 γ ψ 0 ,ψ H 0 ( τ ) = 1 2 π Z ∞ −∞ ( α | b ψ 0 | 2 ) (2 N 0 ) ( ω ) e ıω τ dω = 1 2 π ∞ X k = −∞ Z 2( k +1) π 2 kπ ( α | b ψ 0 | 2 ) (2 N 0 ) ( ω ) e ıω τ dω . (86) Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 27 By integration b y part, we d educe that ∀ τ ∈ R , ( − ıτ ) 2 N 0 +1 γ ψ 0 ,ψ H 0 ( τ ) = 1 2 π ×  Z ∞ −∞ ( α | b ψ 0 | 2 ) (2 N 0 +1) ( ω ) e ıω τ dω + β  (87) β = X k ∈ Z ∗  ( α | b ψ 0 | 2 ) (2 N 0 ) + (2 k π ) − ( α | b ψ 0 | 2 ) (2 N 0 ) − (2 k π )  e ı 2 πk τ , (88) where ( α | b ψ 0 | 2 ) (2 N 0 ) + ( ω 0 ) (r esp. ( α | b ψ 0 | 2 ) (2 N 0 ) − ( ω 0 ) ) denotes the right- side (resp. left-side) deriv ati ve of order 2 N 0 of α | b ψ 0 | 2 at ω 0 ∈ R . 4 W e con clude th at ∀ τ ∈ R , | τ | 2 N 0 +1 | γ ψ 0 ,ψ H 0 ( τ ) | = 1 2 π  Z ∞ −∞   ( α | b ψ 0 | 2 ) (2 N 0 +1) ( ω )   dω + | β |  . (89) In sum mary , we h av e pr oved that ( 32) and ( 33) hold , the co nstant C being cho sen equal to the max imum value of the left-h and side ter ms in the in equalities (80), (84 ) and ( 89). A P P E N D I X V I P R O O F O F P R O P O S I T I O N 5 Let m ∈ N M . Since ψ m is a unit norm fun ction of L 2 ( R ) , th e function γ ψ m ,ψ H m is upper bo unded by 1. As γ ψ m ,ψ H m further satisﬁes (3 3), it can be dedu ced th at ∀ τ ∈ R , | γ ψ m ,ψ H m ( τ ) | ≤ 1 + C 1 + | τ | 2 N m +1 . (90) The same up per b ound hold s for γ ψ m ,ψ m . For a white noise, the property then appe ars as a straightf orward consequence of the latter inequality and Eqs. ( 14) and (15). Let us next turn ou r attention to proce sses with expon entially decaying covariance sequ ences. From ( 8), (34) an d (90), we ded uce th at ∀ ℓ ∈ Z , | Γ n j,m ,n H j,m [ ℓ ] | ≤ A (1 + C ) Z ∞ −∞ e − α | x | 1 + | M − j x − ℓ | 2 N m +1 dx. (91) As the left-h and side of (91 ) correspond s to an even function of ℓ , without loss o f gener ality , it c an be assumed 4 The series i n (88) is c on vergent since al l the othe r terms in (87) a re ﬁnite . Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 28 that ℓ ≥ 0 . W e can decom pose the ab ove integral as Z ∞ −∞ e − α | x | 1 + | M − j x − ℓ | 2 N m +1 dx = Z ∞ 0 e − αx 1 + ( M − j x + ℓ ) 2 N m +1 dx + Z ∞ 0 e − αx 1 + | M − j x − ℓ | 2 N m +1 dx . The ﬁrst integral in the right-hand side can be upper b ounde d as follows Z ∞ 0 e − αx 1 + ( M − j x + ℓ ) 2 N m +1 dx ≤ (1 + ℓ 2 N m +1 ) − 1 Z ∞ 0 e − αx dx = α − 1 (1 + ℓ 2 N m +1 ) − 1 . Let ǫ ∈ (0 , 1) be g iv en. Th e seco nd integral can b e deco mposed as Z ∞ 0 e − αx 1 + | M − j x − ℓ | 2 N m +1 dx = Z ǫM j ℓ 0 e − αx 1 + ( ℓ − M − j x ) 2 N m +1 dx + Z ∞ ǫM j ℓ e − αx 1 + | M − j x − ℓ | 2 N m +1 dx. Furthermo re, we have Z ǫM j ℓ 0 e − αx 1 + ( ℓ − M − j x ) 2 N m +1 dx ≤ (1 + (1 − ǫ ) 2 N m +1 ℓ 2 N m +1 ) − 1 Z ǫM j ℓ 0 e − αx dx ≤ α − 1 (1 − ǫ ) − 2 N m − 1 (1 + ℓ 2 N m +1 ) − 1 (92) Z ∞ ǫM j ℓ e − αx 1 + | M − j x − ℓ | 2 N m +1 dx ≤ Z ∞ ǫM j ℓ e − αx dx = α − 1 e − αǫM j ℓ . From the above ineq ualities, we obtain ∀ ℓ ∈ N ∗ , | Γ n j,m ,n H j,m [ ℓ ] | ≤ A (1 + C ) α − 1  (1 + (1 − ǫ ) − 2 N m − 1 ) × (1 + ℓ 2 N m +1 ) − 1 + e − αǫM j ℓ  . As lim ℓ →∞ (1 + ℓ 2 N m +1 ) e − αǫM j ℓ = 0 , it readily follows that there exists e C ∈ R + such that (3 5) h olds. The left- hand side of (91 ) being also an upper bound for | Γ n j,m ,n j,m [ ℓ ] | , ℓ 6 = 0 , (3 4) is p roved at th e same time. Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 29 A P P E N D I X V I I P R O O F O F P R O P O S I T I O N 7 Let us prove (42), the pro of of (41) being quite similar . W e ﬁrst no te that b ψ m ( b ψ H m ′ ) ∗ and theref ore γ ψ m ,ψ H m ′ belong to L 2 ( R ) (see foo tnote 3 ). Applyin g Parsev al’ s equ ality to (8), we obta in for all ℓ ∈ Z , Γ n j,m ,n H j,m ′ [ ℓ ] = 1 2 π Z ∞ −∞ b Γ n ( ω ) M j b ψ ∗ m ( M j ω ) b ψ H m ′ ( M j ω ) e ıM j ℓω dω = 1 2 π Z ∞ −∞ b Γ n  ω M j  b ψ ∗ m ( ω ) b ψ H m ′ ( ω ) e ıℓω dω . As Γ n ∈ L 1 ( R ) , the spectru m d ensity b Γ n is a bo unded continuo us fun ction. Acco rding to Lebe sgue do minated conv ergence theo rem, lim j →∞ Γ n j,m ,n H j,m ′ [ ℓ ] = 1 2 π Z ∞ −∞ lim j →∞ b Γ n  ω M j  b ψ ∗ m ( ω ) b ψ H m ′ ( ω ) e ıℓω dω = b Γ n (0) 2 π Z ∞ −∞ b ψ ∗ m ( ω ) b ψ H m ′ ( ω ) e ıℓω dω = b Γ n (0) γ ψ m ,ψ H m ′ ( − ℓ ) . A P P E N D I X V I I I C RO S S - C O R R E L AT I O N S F O R M E Y E R W A V E L E T S Substituting (46) in (20), we o btain, for all τ ∈ R , γ ψ 0 ,ψ H 0 ( τ ) = 1 π  Z π (1 − ǫ ) 0 cos  ω ( d + 1 2 + τ )  dω + Z π (1+ ǫ ) π (1 − ǫ ) W 2  ω 2 π ǫ − 1 − ǫ 2 ǫ  cos  ω ( d + 1 2 + τ )  dω  =(1 − ǫ )sinc  π (1 − ǫ )( d + 1 2 + τ )  + ǫ Z 1 − 1 W 2  1 + θ 2  cos  π ( ǫθ + 1)  d + 1 2 + τ   dθ. (93) Using (48), we g et Z 0 − 1 W 2  1 + θ 2  cos  π ( ǫθ + 1)  d + 1 2 + τ   dθ = Z 1 0 cos  π ( ǫθ − 1)  d + 1 2 + τ   dθ − Z 1 0 W 2  1 + θ 2  cos  π ( ǫθ − 1)  d + 1 2 + τ   dθ. (94) This allows us to rewrite (93) as γ ψ 0 ,ψ H 0 ( τ ) = sinc  π ( d + 1 2 + τ )  − sin  π  d + 1 2 + τ   × I ǫ  d + 1 2 + τ  . (95) Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 30 After simpliﬁcation, (5 2) f ollows. According to (19 ) and (50), we have fo r a ll m ∈ { 1 , . . . , M − 2 } and τ ∈ R ∗ , γ ψ m ,ψ H m ( τ ) = − 1 π  Z π ( m + ǫ ) π ( m − ǫ ) W 2  m + ǫ 2 ǫ − ω 2 π ǫ  sin( ω τ ) dω + Z π ( m +1 − ǫ ) π ( m + ǫ ) sin( ω τ ) dω + Z π ( m +1+ ǫ ) π ( m +1 − ǫ ) W 2  ω 2 π ǫ − m + 1 − ǫ 2 ǫ  sin( ω τ ) dω  = cos  π ( m + 1 − ǫ ) τ  − cos  π ( m + ǫ ) τ  π τ + ǫ Z 1 − 1 W 2  1 + θ 2  sin  π ( ǫθ − m ) τ  dθ − ǫ Z 1 − 1 W 2  1 + θ 2  sin  π ( ǫθ + m + 1 ) τ  dθ. By proc eeding similarly to (93)-(94), we ﬁnd γ ψ m ,ψ H m ( τ ) =  cos( π ( m + 1) τ ) − co s( π mτ )   1 π τ − I ǫ ( τ )  . When τ is an integer, th is expression f urther simpliﬁes in (54 ). Finally , when m = M − 1 , we h av e, fo r all τ ∈ R ∗ , γ ψ M − 1 ,ψ H M − 1 ( τ ) = − 1 π  Z π ( M − 1+ ǫ ) π ( M − 1 − ǫ ) W 2  M − 1 + ǫ 2 ǫ − ω 2 π ǫ  sin( ω τ ) dω + Z π M (1 − ǫ ) π ( M − 1+ ǫ ) sin( ω τ ) dω + Z π M (1+ ǫ ) π M (1 − ǫ ) W 2  ω 2 π ǫM − 1 − ǫ 2 ǫ  sin( ω τ ) dω  = cos  π M (1 − ǫ ) τ  − cos  π ( M − 1 + ǫ ) τ  π τ + ǫ Z 1 − 1 W 2  1 + θ 2  sin  π ( ǫθ − M + 1) τ  dθ − ǫM Z 1 − 1 W 2  1 + θ 2  sin  π M ( ǫθ + 1 ) τ  dθ = cos  π M τ  − cos  π ( M − 1) τ  π τ + cos  π ( M − 1) τ  I ǫ ( τ ) − cos ( π M τ ) I M ǫ ( τ ) . This yields (5 5). Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 31 A P P E N D I X I X P R O O F O F P R O P O S I T I O N 8 Let m ∈ N ∗ . Given (19) , (57 ) leads to − π γ ψ 2 m ,ψ H 2 m ( τ ) = Z ∞ 0 | b ψ 2 m ( ω ) | 2 sin( ω τ ) dω = Z ∞ 0 | A 0 ( ω ) | 2 | b ψ m ( ω ) | 2 sin(2 ω τ ) dω (96) Furthermo re, we have | A 0 ( ω ) | 2 = X k γ a 0 [ k ] exp( − ık ω ) = γ a 0 [0] + 2 ∞ X k =1 γ a 0 [ k ] cos( k ω ) . Combining this eq uation with (96) and u sing classical trigon ometric equ alities, we obtain − π γ ψ 2 m ,ψ H 2 m ( τ ) = γ a 0 [0] Z ∞ 0 | b ψ m ( ω ) | 2 sin(2 ω τ ) dω + ∞ X k =1 γ a 0 [ k ]  Z ∞ 0 | b ψ m ( ω ) | 2 sin  (2 τ − k ) ω  dω + Z ∞ 0 | b ψ m ( ω ) | 2 sin  (2 τ + k ) ω  dω  which, again inv o king Relation (19), yields (59). E q. (60) can be proved similarly startin g from (5 8). A P P E N D I X X C RO S S - C O R R E L AT I O N S F O R H A A R WA V E L E T Knowing the expression of the Fourier tr ansform o f the Haar scaling f unction in (64) an d using the cr oss- correlation for mula (2 0), w e obtain: ∀ τ ∈ R , γ ψ 0 ,ψ H 0 ( τ ) = 1 π ∞ X k =0 ( − 1) k Z 2( k +1) π 2 kπ sinc 2 ( ω 2 ) cos  ω ( 1 2 + τ + d )  dω = 2 π ∞ X k =0 ( − 1) k Z ( k +1) π kπ sin 2 ( ν ) ν 2 cos  ν (1 + 2 τ + 2 d )  dν. (97) Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 32 By integration b y part, we ﬁnd : for all ( α, β , η ) ∈ R 3 , Z β α sin 2 ( ω ) ω 2 cos( η ω ) dω = sin 2 ( α ) cos( η α ) α − sin 2 ( β ) cos( η β ) β + 1 4 (2 + η ) Z β α sin  (2 + η ) ω  ω dω − η 2 Z β α sin( η ω ) ω dω + 1 4 (2 − η ) Z β α sin  (2 − η ) ω  ω dω = sin 2 ( α ) cos( η α ) α − sin 2 ( β ) cos( η β ) β + 1 4 ( η + 2) Z β ( η +2) α ( η +2) sin( ν ) ν dν − η 2 Z β η αη sin( ν ) ν dν + 1 4 ( η − 2) Z β ( η − 2) α ( η − 2) sin( ν ) ν dν. Combining this resu lt with (97) lea ds to ( 66). On the oth er hand, accord ing to (6 5) an d (19), we h av e ∀ τ ∈ R , γ ψ 1 ,ψ H 1 ( τ ) = − 1 π Z ∞ 0 sinc 2 ( ω 4 ) sin 2 ( ω 4 ) sin( ω τ ) dω . In [6 1, p.45 9], a n expression of R ∞ 0 sin 2 ( αx ) sin 2 ( β x ) sin (2 ηx ) dx x 2 with ( α, β , η ) ∈ ( R ∗ + ) 3 is given. Usin g this r elation yields (67) wh en τ > 0 . The ge neral expr ession for τ ∈ R f ollows from the o ddness o f γ ψ 1 ,ψ H 1 . A P P E N D I X X I C RO S S - C O R R E L AT I O N F O R T H E F R A N K L I N W A V E L E T W e ha ve, for all τ ∈ R , γ χ,χ H ( τ ) = − 1 π Z ∞ 0 | b χ ( ω ) | 2 sin( ω τ ) dω = − 2 π Z ∞ 0 sin 8 ( ω ) ω 4 sin(2 ω τ ) dω . After two successive in tegrations by p art, we ob tain γ χ,χ H ( τ ) = − 4 3 π  4 Z ∞ 0 sin 7 ( ω ) cos( ω ) sin(2 ω τ ) ω 3 dω + τ Z ∞ 0 sin 8 ( ω ) cos(2 ω τ ) ω 3 dω  = − 2 3 π  28 Z ∞ 0 sin 6 ( ω ) cos 2 ( ω ) sin(2 ω τ ) ω 2 dω − 2(2 + τ 2 ) Z ∞ 0 sin 8 ( ω ) sin(2 ω τ ) ω 2 dω + 16 τ Z ∞ 0 sin 7 ( ω ) cos( ω ) cos(2 ω τ ) ω 2 dω  . (98) Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 33 Standard trigon ometric manip ulations a llow us to wr ite: sin 6 ( ω ) cos 2 ( ω ) sin(2 ω τ ) = 1 8 sin 4 ( ω )  sin(2 τ ω ) − 1 2 sin  2( τ + 2 ) ω  − 1 2 sin  2( τ − 2) ω   sin 8 ( ω ) sin(2 ω τ ) = 1 16 sin 4 ( ω )  sin  2( τ + 2) ω  + sin  2( τ − 2) ω  − sin  2( τ + 1) ω  − 4 sin  2( τ − 1) ω  + 6 sin(2 τ ω )  sin 7 ( ω ) cos( ω ) cos(2 ω τ ) = 1 16 sin 4 ( ω )  sin  2( τ − 2) ω  − sin  2( τ + 2) ω  + 2 sin  2( τ + 1) ω  − 2 sin  2( τ − 1) ω   . Inserting these expre ssions in (9 8) y ields 3 π γ χ,χ H ( τ ) = Q 0 ( τ ) J ( τ ) − Q 1 ( τ ) J ( τ + 1 ) − Q 1 ( − τ ) J ( τ − 1) + Q 2 ( τ ) J ( τ + 2) + Q 2 ( − τ ) J ( τ − 2) , (99) where (see [ 61, p . 4 59]) ∀ x ∈ R , J ( x ) = 2 Z ∞ 0 sin 4 ( ω ) ω 2 sin(2 ω x ) dω = − 3 2 x ln | x | + (1 + x ) ln | 1 + x | − (1 − x ) ln | 1 − x | − 2 + x 4 ln | 2 + x | + 2 − x 4 ln | 2 − x | and Q 0 ( τ ) = 3 4 τ 2 − 2 , Q 1 ( τ ) = τ 2 2 + 2 τ + 1 , Q 2 ( τ ) = 1 8 ( τ + 4) 2 . Simple algebra allows us to prove that (99 ) is equivalent to (7 5). On the other h and, | e A 1 ( ω ) | 2 can be v iewed as the fr equency respo nse of a non causal stab le digital ﬁlter whose transfer fun ction is P e A 1 ( z ) = 6(2 − z + z − 1 2 )  1 + 2  z + z − 1 2  2 )(2 + z + z − 1 2 ) = 2 √ 3 9  4(2 + √ 3) z + 2 + √ 3 − 4(2 − √ 3) z + 2 − √ 3 + 7(2 + √ 3) − 4(1 + √ 3) z z 2 + 2 + √ 3 − 7(2 − √ 3) − 4(1 − √ 3) z z 2 + 2 − √ 3  . W e next expa nd P e A 1 ( z ) in Lau rent series on th e h olomor phy domain contain ing th e u nit c ircle, that is D P e A 1 = n z ∈ C | √ 3 − 1 √ 2 < | z | < √ 3 + 1 √ 2 o . Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 34 W e thus ded uce fr om the partial fra ction decom position of P e A 1 ( z ) that P e A 1 ( z ) = 2 √ 3 9  4 ∞ X k = −∞ ( − 1) k (2 − √ 3) | k | z − k + 7 ∞ X k = −∞ ( − 1) k (2 − √ 3) | k | z − 2 k + 4(1 − √ 3) ∞ X k =0 ( − 1) k (2 − √ 3) k  z 2 k +1 + z − 2 k − 1   . By identiﬁyin g the latter expression with P ∞ k = −∞ γ ˜ a 1 [ k ] z − k , ( 76) is ob tained. R E F E R E N C E S [1] S. Ma llat, A wav elet tour of s ignal pr ocessing . Sa n Die go, CA, USA: Ac ademic Press, 1998. [2] S. Ca mbanis and E. Masry , “W av elet app roximati on of determin istic and random signa ls: con verge nce propertie s and rates, ” IEEE T rans. on Inform. Theory , vol. 40, no . 4, pp . 1013–1029, Jul. 19 94. [3] I. 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Theoretical results (lef t); experimenta l results (right). 38 L I S T O F T A B L E S I Asymptotic for m of γ ψ m ,ψ H m ( τ ) as | τ | → ∞ fo r W alsh-Hadam ard wavelets. . . . . . . . . . . . . . . 37 II Theoretical cross-cor relation values in the d yadic c ase ( d = 0 ). . . . . . . . . . . . . . . . . . . . . . 37 III Theoretical values for th e ﬁrst two cro ss-correlation seq uences in the M -band M eyer case ( d = 0 ). . 38 IV Theoretical values for th e last cro ss-correlation seq uence in the M -b and Meyer case ( d = 0 ). . . . . . 39 V Theoretical c ross-correlatio n values in th e W alsh-Ha damard case. . . . . . . . . . . . . . . . . . . . . 39 VI Cross-correlatio n estimates in th e dyad ic case ( d = 0 ). . . . . . . . . . . . . . . . . . . . . . . . . . . 40 VII Cross-correlatio n estimates in th e M -band case ( d = 0 ). . . . . . . . . . . . . . . . . . . . . . . . . . 41 VIII Estimation o f th e last cro ss-correlation seq uence for M -band Shanno n and Meyer wa velets. . . . . . 42 IX Inter-band cross-co rrelation values f or som e wa velet families. W e rec all that Proper ty ( 12) hold s and that, fo r M -b and Meyer wa velets γ ψ m ,ψ H m ′ is zero when | m − m ′ | > 1 . . . . . . . . . . . . . . . . . 4 3 IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 37 fb_M_ortho_d ual.pstex G ∗ 0 G ∗ 1 G ∗ M − 1 ↓ M ↓ M ↓ M ↑ M ↑ M ↑ M G 0 G 1 G M − 1 H ∗ 0 H ∗ 1 H ∗ M − 1 ↓ M ↓ M ↑ M ↑ M ↑ M H 0 H 1 H M − 1 ↓ M Fig. 1. A pair of primal (top) and dual (bot tom) analysi s/synthesi s M -band para-unit ary ﬁlter banks. m 1 2 3 4 5 6 7 8 9 10 11 12 π γ ψ m ,ψ H m ( τ ) 1 2 3 τ 3 1 2 5 τ 3 − 3 2 7 τ 5 1 2 7 τ 3 − 3 2 9 τ 5 − 3 2 11 τ 5 45 2 14 τ 7 1 2 9 τ 3 − 3 2 11 τ 5 − 3 2 13 τ 5 45 2 16 τ 7 − 3 2 15 τ 5 T ABLE I A S Y M P T O T I C F O R M O F γ ψ m ,ψ H m ( τ ) A S | τ | → ∞ F O R W A L S H - H A D A M A R D WA V E L E T S . γ ψ 0 ,ψ H 0 γ ψ 1 ,ψ H 1 W ave lets \ ℓ 0 1 2 3 1 2 3 Shannon 0.63662 -0.2122 1 0.12732 − 9 . 094 6 × 10 − 2 0.63662 0 0.21221 Meye r ǫ = 1 / 3 0.63216 -0.1991 6 0.10668 − 6 . 416 6 × 10 − 2 0.59378 − 4 . 1412 × 10 − 2 0.11930 Splines order 3 0.62696 -0.18538 8.8582 × 10 − 2 -4.6179 × 10 − 2 0.55078 -5.8322 × 1 0 − 2 8.2875 × 10 − 2 Splines order 1 0.60142 -0.12891 3 . 4815 × 10 − 2 − 9 . 2967 × 10 − 3 0.38844 − 5 . 7528 × 10 − 2 1 . 8659 × 10 − 2 Haar 0.5128 8 − 1 . 1338 × 10 − 2 − 1 . 0855 × 10 − 3 − 2 . 6379 × 10 − 4 0.10816 5 . 6994 × 10 − 3 1 . 5610 × 10 − 3 T ABLE II T H E O R E T I C A L C R O S S - C O R R E L ATI O N VAL U E S I N T H E DYAD I C C A S E ( d = 0 ) . Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 38 inter2D.eps Fig. 2. 2 D cross-correlati ons using 3 -band Mey er wa velets. Theoretical results (left ); expe rimenta l results (right) . γ ψ 0 ,ψ H 0 γ ψ 1 ,ψ H 1 W avelets \ ℓ 0 1 2 3 1 2 3 Meyer 3 -band ǫ = 1 / 4 0.63411 -0.20478 0.11530 -7.4822 × 10 − 2 0.62662 0 0.18391 Meyer 4 -band ǫ = 1 / 5 0.63501 -0.20742 0.11950 -8.0293 × 10 − 2 0.63020 0 0.19367 Meyer 5 -band ǫ = 1 / 6 0.63550 -0.20887 0.12184 -8.3419 × 10 − 2 0.63216 0 0.19917 Meyer 6 -band ǫ = 1 / 7 0.63580 -0.20975 0.12327 -8.5357 × 10 − 2 0.63334 0 0.20255 Meyer 7 -band ǫ = 1 / 8 0.63599 -0.21033 0.12421 -8.6637 × 10 − 2 0.63411 0 0.20478 Meyer 8 -band ǫ = 1 / 9 0.63612 -0.21072 0.12486 -8.7525 × 10 − 2 0.63463 0 0.20632 T ABLE III T H E O R E T I C A L VAL U E S F O R T H E FI R S T T W O C R O S S - C O R R E L ATI O N S E Q U E N C E S I N T H E M - B A N D M E Y E R C A S E ( d = 0 ) . Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 39 γ ψ M − 1 ,ψ H M − 1 W ave lets \ ℓ 1 2 3 Meye r 3 -band ǫ = 1 / 4 -0.58918 -6.0378 × 10 − 2 -0.11965 Meye r 4 -band ǫ = 1 / 5 0.58555 -7.0840 × 10 − 2 0.11961 Meye r 5 -band ǫ = 1 / 6 -0.58278 -7.7359 × 10 − 2 -0.11940 Meye r 6 -band ǫ = 1 / 7 0.58063 -8.1773 × 10 − 2 0.11914 Meye r 7 -band ǫ = 1 / 8 -0.57893 -8.4944 × 10 − 2 -0.11888 Meye r 8 -band ǫ = 1 / 9 0.57755 -8.7324 × 10 − 2 0.11863 T ABLE IV T H E O R E T I C A L VAL U E S F O R T H E L A S T C RO S S - C O R R E L A T I O N S E Q U E N C E I N T H E M - B A N D M E Y E R C A S E ( d = 0 ) . ℓ 1 2 3 γ ψ 2 ,ψ H 2 6 . 0560 × 10 − 2 1 . 5848 × 10 − 3 4 . 0782 × 10 − 4 γ ψ 3 ,ψ H 3 − 4 . 9162 × 10 − 2 − 3 . 0109 × 10 − 4 − 3 . 4205 × 10 − 5 γ ψ 4 ,ψ H 4 3.2069 × 10 − 2 4.0952 × 10 − 4 1.0319 × 10 − 4 γ ψ 5 ,ψ H 5 -2.8899 × 10 − 2 -8.0753 × 10 − 5 -8.7950 × 10 − 6 γ ψ 6 ,ψ H 6 -2.4899 × 10 − 2 -2.6077 × 10 − 5 -2.4511 × 10 − 6 γ ψ 7 ,ψ H 7 2.4297 × 10 − 2 1.0608 × 10 − 5 4.8118 × 10 − 7 T ABLE V T H E O R E T I C A L C R O S S - C O R R E L A T I O N V A L U E S I N T H E W A L S H - H A DA M A R D C A S E . Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 40 γ ψ 0 ,ψ H 0 γ ψ 1 ,ψ H 1 W avelets j \ ℓ 0 1 2 3 0 1 2 3 1 0.63538 -0.21134 0.12586 -9.1515 × 10 − 2 9.97 × 10 − 6 0.63680 -1.7137 × 10 − 4 0.21165 Shannon 2 0.63558 -0 .21347 0.12970 -8.7908 × 10 − 2 2.6426 × 10 − 6 0.63404 7.0561 × 10 − 4 0.21210 3 0.63467 -0.20732 0.13168 -9.0116 × 10 − 2 -1.0078 × 10 − 4 0.63846 -1.2410 × 10 − 3 0.20975 1 0.63091 -0.19828 0.10517 -6.4650 × 10 − 2 1.8257 × 10 − 5 0.61092 -1.2433 × 10 − 2 0.15307 Meyer 2 0.63112 -0.20043 0.10903 -6.1060 × 10 − 2 -7.5431 × 10 − 6 0.59115 -4.0881 × 10 − 2 0.11888 ǫ = 1 / 3 3 0.62971 -0.19391 0.11084 -6.3378 × 10 − 2 4.0868 × 10 − 4 0.59522 -4.2624 × 10 − 2 0.11651 1 0.62587 -0.18459 8.7088 × 10 − 2 -4.6635 × 10 − 2 -1.4511 × 10 − 4 0.58458 -1.2651 × 10 − 2 0.12557 Splines 2 0.62606 -0.18679 9.1068 × 10 − 2 -4.3124 × 10 − 2 1.9483 × 10 − 4 0,54841 -5.8083 × 10 − 2 8.2386 × 10 − 2 order 3 3 0.62398 -0. 17984 9.2793 × 10 − 2 -4.5682 × 10 − 2 1.2400 × 10 − 3 0.55204 -5.9368 × 10 − 2 8.0105 × 10 − 2 1 0.60016 -0.12749 3.2975 × 10 − 2 -9.7419 × 10 − 3 -4.5287 × 10 − 4 0.47691 1.6224 × 10 − 2 6.9681 × 10 − 2 Splines 2 0.60059 -0.13045 3.7613 × 10 − 2 -6.5441 × 10 − 3 6.6358 × 10 − 4 0.38507 -5.7502 × 10 − 2 1.8042 × 10 − 2 order 1 3 0.59771 -0. 12303 3.9388 × 10 − 2 -9.3208 × 10 − 3 2.2725 × 10 − 3 0.38958 -5.8143 × 10 − 2 1.6160 × 10 − 2 1 0.50297 -3.3557 × 10 − 3 -1.1706 × 10 − 3 2.7788 × 10 − 4 3.8368 × 10 − 4 0.22455 7.2451 × 10 − 2 4.6418 × 10 − 2 Haar 2 0.50966 -1.0083 × 10 − 2 7,2357 × 10 − 6 1.5087 × 10 − 3 -1.2135 × 10 − 3 9.9745 × 10 − 2 5.1371 × 10 − 3 1.0847 × 10 − 3 3 0.51023 -8.3267 × 10 − 3 2.7936 × 10 − 3 7.0343 × 10 − 5 1.2329 × 10 − 3 0.10703 6.7651 × 10 − 3 2.2422 × 10 − 3 1 0.59822 -0.12059 2.3566 × 10 − 2 -3.3325 × 10 − 3 -5.0189 × 10 − 4 0.46392 2.1155 × 10 − 2 6.1137 × 10 − 2 Symlets 8 2 0.59899 -0.12432 2.8865 × 10 − 2 -2.8960 × 10 − 4 6.7795 × 10 − 4 0.36368 -5.7692 × 10 − 2 9.7533 × 10 − 3 3 0.59654 -0.11703 3.0357 × 10 − 2 -2.8071 × 10 − 3 1.8568 × 10 − 3 0.37012 -5.8376 × 10 − 2 6.9416 × 10 − 3 T ABLE VI C R O S S - C O R R E L AT I O N E S T I M AT E S I N T H E DYAD I C C A S E ( d = 0 ) . Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 41 γ ψ 0 ,ψ H 0 γ ψ 1 ,ψ H 1 W avelets j \ ℓ 0 1 2 3 0 1 2 3 Meyer 1 0.63337 -0.20549 0.11431 -7.1877 × 10 − 2 -6.8977 × 10 − 4 0.62533 -1.3630 × 10 − 4 0.18236 3 -band 2 0.63284 -0.19932 0.11938 -7.5331 × 10 − 2 -1.7781 × 10 − 4 0.63013 1. 2830 × 10 − 3 0.18409 ǫ = 1 / 4 3 0.63886 -0.19987 0.11763 -6.6380 × 10 − 2 -3.9622 × 10 − 4 0.61503 8. 4042 × 10 − 4 0.17519 Meyer 1 0.63383 -0.20856 0.12176 -7.7150 × 10 − 2 2.1961 × 10 − 5 0.62739 7. 6636 × 10 − 4 0.19339 4 -band 2 0.63648 -0.19903 0.11757 -7.7337 × 10 − 2 4.8821 × 10 − 4 0.62676 3. 8876 × 10 − 3 0.18683 ǫ = 1 / 5 3 0.64642 -0.19651 0.12202 -6.9984 × 10 − 2 2.3054 × 10 − 3 0.63384 -1.6254 × 10 − 3 0.19233 Meyer 1 0.63338 -0.20818 0.12534 -8.0594 × 10 − 2 8.6373 × 10 − 4 0.62902 8. 3871 × 10 − 4 0.1981 5 -band 2 0.64020 -0.20288 0.12135 -7.3844 × 10 − 2 5.3607 × 10 − 4 0.62230 4. 6651 × 10 − 4 0.19093 ǫ = 1 / 6 3 0.6566 -0.19609 0.12891 -7.6061 × 10 − 2 -2.8654 × 10 − 3 0.62281 -4.7324 × 10 − 3 0.19364 Meyer 1 0.63403 -0.20818 0.12711 -8.2124 × 10 − 2 4.5293 × 10 − 4 0.63229 -1.9919 × 10 − 3 0.20228 6 -band 2 0.64471 -0.20716 0.13141 -8.4914 × 10 − 2 3.7150 × 10 − 4 0.62450 6. 5942 × 10 − 4 0.20313 ǫ = 1 / 7 3 0.66409 -0.19532 0.14401 -9.3486 × 10 − 2 2.0490 × 10 − 3 0.63619 1. 5614 × 10 − 2 0.17595 Meyer 1 0.63323 -0.20781 0.12663 -8.3335 × 10 − 2 1.5731 × 10 − 3 0.63528 -8.6821 × 10 − 4 0.20509 7 -band 2 0.64286 -0.20057 0.12881 -8.1995 × 10 − 2 -1.6505 × 10 − 4 0.62782 -7.9119 × 10 − 3 0.20007 ǫ = 1 / 8 3 0.68445 -0.1845 0.12065 -9.0295 × 10 − 2 -5.9955 × 10 − 3 0.62572 -5.3033 × 10 − 2 0.17409 Meyer 1 0.63426 -0.20592 0.12928 -8.6766 × 10 − 2 -2.1756 × 10 − 4 0,63658 -1.3977 × 10 − 3 0.20385 8 -band 2 0.64743 -0.19970 0.12725 -7.7096 × 10 − 2 1.4856 × 10 − 3 0.63725 -2.4313 × 10 − 3 0.20396 ǫ = 1 / 9 3 0.69342 -0.20505 0.11257 -6.0075 × 10 − 2 -3.6363 × 10 − 3 0.61590 1. 3830 × 10 − 2 0.22112 1 0.59148 -0.11001 1.9635 × 10 − 2 2.4318 × 10 − 3 -6.6559 × 10 − 6 0.36856 -6.0858 × 10 − 2 8.4608 × 10 − 5 AC 2 0.59855 -0.10412 1.6012 × 10 − 2 1.8921 × 10 − 4 -7.1462 × 10 − 3 0.37379 -5.8026 × 10 − 2 -4.4309 × 10 − 3 4 -band 3 0.60057 -9.5335 × 10 − 2 2.0094 × 10 − 2 7.6430 × 10 − 3 2.5313 × 10 − 3 0.37514 -5.6207 × 10 − 2 6.8164 × 10 − 3 γ ψ 2 ,ψ H 2 γ ψ 3 ,ψ H 3 1 -1.9012 × 10 − 4 -0.34054 5.5692 × 10 − 2 4.6899 × 10 − 5 -5.5011 × 10 − 5 0.36755 4. 1274 × 10 − 2 5.6594 × 10 − 2 AC 2 1.0139 × 10 − 3 -0.32275 5.4137 × 10 − 2 -6.7903 × 10 − 3 3.6460 × 10 − 3 0.18371 -4.1645 × 10 − 2 6.8637 × 10 − 3 4 -band 3 6.8587 × 10 − 3 -0.32199 4.5083 × 10 − 2 -9.7023 × 10 − 3 8.3037 × 10 − 3 0.19070 -3.7675 × 10 − 2 -4.5919 × 10 − 4 1 2.4712 × 10 − 4 0.20479 6. 9476 × 10 − 2 4.4200 × 10 − 2 -1.8669 × 10 − 4 -6.1810 × 10 − 2 -1.2677 × 10 − 3 2.4199 × 10 − 5 Hadamard 2 3. 5680 × 10 − 3 5.9530 × 10 − 2 -5.3171 × 10 − 3 4.3827 × 10 − 3 6.2437 × 10 − 4 -5.0635 × 10 − 2 4.6773 × 10 − 3 -8.7358 × 10 − 3 3 1.1391 × 10 − 2 5.9541 × 10 − 2 8.3376 × 10 − 4 -1.4604 × 10 − 3 1.9009 × 10 − 3 -5.5798 × 10 − 2 -5.7086 × 10 − 3 -1.1253 × 10 − 2 T ABLE VII C R O S S - C O R R E L ATI O N E S T I M AT E S I N T H E M - B A N D C A S E ( d = 0 ) . Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 42 γ ψ M − 1 ,ψ H M − 1 W avelet j \ ℓ 0 1 2 3 1 5.2467 × 10 − 6 0.63606 2.0952 × 10 − 3 0.21261 Shannon 2 5.9145 × 10 − 6 0.63592 -4.1893 × 10 − 3 0.21083 4 -band 3 -1.2667 × 10 − 4 0.62746 -5.7616 × 10 − 3 0.2020 Meyer 1 4.1334 × 10 − 4 -0.60986 -2.5395 × 10 − 2 -0.16095 3 -band 2 3.9059 × 10 − 4 -0.58694 -6.1089 × 10 − 2 -0.11754 ǫ = 1 / 4 3 3.5372 × 10 − 3 -0.5879 -5.1057 × 10 − 2 -0.11499 Meyer 1 3.9730 × 10 − 4 0.60845 -2.9936 × 10 − 2 0.16111 4 -band 2 -1.3788 × 10 − 3 0.58530 -7.5797 × 10 − 2 0.11985 ǫ = 1 / 5 3 1.0644 × 10 − 3 0.57418 -7.6790 × 10 − 2 0.10690 Meyer 1 -7.2077 × 10 − 6 -0.60862 -3.4588 × 10 − 2 -0.16162 5 -band 2 -3.2301 × 10 − 3 -0.58482 -8.6826 × 10 − 2 -0.11844 ǫ = 1 / 6 3 -8.8877 × 10 − 3 -0.56937 -9.3811 × 10 − 2 -0.11512 Meyer 1 8.2632 × 10 − 4 0.60806 -3.7209 × 10 − 2 0.16215 6 -band 2 -1.2448 × 10 − 3 0.58023 -8.3257 × 10 − 2 0.11022 ǫ = 1 / 7 3 5.5425 × 10 − 3 0.58196 -8.4671 × 10 − 2 0.12368 Meyer 1 2.7863 × 10 − 4 -0.60863 -3.9804 × 10 − 2 -0.16443 7 -band 2 -5.9703 × 10 − 3 -0.57749 -9.9056 × 10 − 2 -0.11228 ǫ = 1 / 8 3 1.8490 × 10 − 3 -0.58901 -6.4289 × 10 − 2 -0.13516 Meyer 1 -2.5084 × 10 − 4 0.60811 -4.1611 × 10 − 2 0.16612 8 -band 2 1.0345 × 10 − 3 0.57216 -9.4172 × 10 − 2 0.12014 ǫ = 1 / 9 3 -1.0777 × 10 − 2 0.56259 -0.12183 0.10776 T ABLE VIII E S T I M ATI O N O F T H E L A S T C R O S S - C O R R E L ATI O N S E Q U E N C E F O R M - B A N D S H A N N O N A N D M E Y E R W A V E L E T S . Novem ber 27, 2024 DRAFT IEEE TRANSACTIONS ON INFORMA TION THEOR Y , 2007 43 W ave lets \ ℓ -3 -2 -1 0 1 2 3 Meye r 2-band γ ψ 0 ,ψ H 1 ( ℓ ) 9.1502 × 10 − 2 -0.10848 0.11800 -0.11800 0.10848 -9.1502 × 10 − 2 7.0491 × 10 − 2 ǫ = 1 / 3 γ ψ 1 ,ψ H 0 ( ℓ ) -8.1258 × 10 − 2 0.10073 -0.11434 0.11924 -0.11434 0.100 73 -8.1258 × 10 − 2 Splines γ ψ 0 ,ψ H 1 ( ℓ ) -8.2660 × 10 − 2 0.13666 -0.18237 0.18237 -0.13666 8.2660 × 10 − 2 -4.5433 × 10 − 2 order 3 γ ψ 1 ,ψ H 0 ( ℓ ) 6.1604 × 10 − 2 -0.10838 0.16319 -0.18941 0.16319 -0 .10838 6.1604 × 10 − 2 Haar γ ψ 0 ,ψ H 1 ( ℓ ) -9.2323 × 10 − 3 -2.2034 × 10 − 2 -0.16656 0.44127 -0.16656 -2.203 4 × 10 − 2 -9.2323 × 10 − 3 γ ψ 1 ,ψ H 0 ( ℓ ) -3.1567 × 10 − 3 -1.9621 × 10 − 2 0.35401 -0.35401 1.9621 × 10 − 2 3.1567 × 10 − 3 1.0758 × 10 − 3 Meye r γ ψ 0 ,ψ H 1 ( ℓ ) -8.4807 × 10 − 2 8.8904 × 10 − 2 -8.8904 × 10 − 2 8.4807 × 10 − 2 -7.7120 × 10 − 2 6.6763 × 10 − 2 -5.4904 × 10 − 2 3 -band γ ψ 1 ,ψ H 0 ( ℓ ) 6.0944 × 10 − 2 -7.2206 × 10 − 2 8.1363 × 10 − 2 -8.7347 × 10 − 2 8.9428 × 10 − 2 -8.7347 × 10 − 2 8.1363 × 10 − 2 ǫ = 1 / 4 γ ψ 1 ,ψ H 2 ( ℓ ) -6.3891 × 10 − 2 -7.4738 × 10 − 2 -8.3192 × 10 − 2 -8.8252 × 10 − 2 -8.9297 × 10 − 2 -8.6196 × 10 − 2 -7.9333 × 10 − 2 Meye r γ ψ 0 ,ψ H 1 ( ℓ ) 6.5090 × 10 − 2 -6.9156 × 10 − 2 7.1274 × 10 − 2 -7.1274 × 10 − 2 6.9156 × 10 − 2 -6.5090 × 10 − 2 5.9394 × 10 − 2 4 -band γ ψ 1 ,ψ H 0 ( ℓ ) -6.2421 × 10 − 2 6.7350 × 10 − 2 -7.0473 × 10 − 2 7.1543 × 10 − 2 -7.0473 × 10 − 2 6.7350 × 10 − 2 -6.2421 × 10 − 2 ǫ = 1 / 5 γ ψ 1 ,ψ H 2 ( ℓ ) 6.0949 × 10 − 2 6.6274 × 10 − 2 6.9878 × 10 − 2 7.1475 × 10 − 2 7.0939 × 10 − 2 6.8312 × 10 − 2 6.3804 × 10 − 2 γ ψ 2 ,ψ H 3 ( ℓ ) -6.5090 × 10 − 2 6.9156 × 10 − 2 -7.1274 × 10 − 2 7.1274 × 10 − 2 -6.9156 × 10 − 2 6.5090 × 10 − 2 -5.9394 × 10 − 2 T ABLE IX I N T E R - BA N D C R O S S - C O R R E L ATI O N V A L U E S F O R S O M E WA V E L E T F A M I L I E S . W E R E C A L L T H AT P R O P E R T Y (12) H O L D S A N D T H AT , F O R M - B A N D M E Y E R W A V E L E T S γ ψ m ,ψ H m ′ I S Z E R O W H E N | m − m ′ | > 1 . Novem ber 27, 2024 DRAFT

Noise Covariance Properties in Dual-Tree Wavelet Decompositions

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