Member, IEEE Caroline Chaux

Senior Member, IEEE Jean-Christophe Pesquet

Member, IEEE Laurent Duval

Noise Covariance Properties in Dual-Tree Wavelet Decompositions

Keywords: Dual-tree, wavelets, frames, Hilbert transform, filter banks, cross-correlation, covariance, random processes, stationarity, noise, dependence, statistics

published or not. The documents may come

I. INTRODUCTION

The discrete wavelet transform (DWT) [START_REF] Mallat | A wavelet tour of signal processing[END_REF] is a powerful tool in signal processing, since it provides "efficient" basis representations of regular signals [START_REF] Cambanis | Wavelet approximation of deterministic and random signals: convergence properties and rates[END_REF]. It nevertheless suffers from a few limitations such as aliasing effects in the transform domain, coefficient oscillations around singularities and a lack of shift invariance. Frames (see [START_REF] Daubechies | The wavelet transform, time-frequency localization and signal analysis[END_REF], [START_REF] Gribonval | Sparse representations in unions of bases[END_REF] or [START_REF] Casazza | The art of frame theory[END_REF] for a tutorial), reckoned as more general signal representations, represent an outlet for these inherent constraints laid on basis functions.

C. Chaux and J.-C. Pesquet are with the Institut Gaspard Monge and CNRS-UMR 8049, Université de Paris-Est Marne-la-Vallée, 77454 Marne-la-Vallée Cedex 2, France. E-mail: {chaux,pesquet}@univ-mlv.fr.

L. Duval is with the Institut franc ¸ais du pétrole, IFP, Technology, Computer Science and Applied Mathematics Division, 1 et 4, avenue de Bois-Préau F-92852 Rueil-Malmaison, France. E-mail: laurent.duval@ifp.fr.

Redundant DWTs (RDWTs) are shift-invariant non-subsampled frame extensions to the DWT. They have proved more error or quantization resilient [START_REF] Coifman | Translation-invariant de-noising[END_REF]- [START_REF] Bölcskei | Frame-theoretic analysis of oversampled filter banks[END_REF], at the price of an increased computational cost, especially in higher dimensions. They do not however take on the lack of rotation invariance or poor directionality of classical separable schemes. These features are particularly sensitive to image and video processing. Recently, several other types of frames have been proposed to incorporate more geometric features, aiming at sparser representations and improved robustness. Early examples of such frames are shiftable multiscale transforms or steerable pyramids [START_REF] Simoncelli | Steerable wedge filters for local orientation analysis[END_REF]. To name a few others, there also exist contourlets [START_REF] Do | The contourlet transform: an efficient directional multiresolution image representation[END_REF], bandelets [START_REF] Pennec | Sparse geometric image representations with bandelets[END_REF], curvelets [START_REF] Candès | Fast discrete curvelet transforms[END_REF], phaselets [START_REF] Gopinath | The phaselet transform -an integral redundancy nearly shift-invariant wavelet transform[END_REF], directionlets [START_REF] Velisavljević | Directionlets: anisotropic multidirectional representation with separable filtering[END_REF] or other representations involving multiple dictionaries [START_REF] Figueras | Low rate and flexible image coding with redundant representations[END_REF].

Two important facets need to be addressed, when resorting to the inherent frame redundancy: 1) multiplicity: frame decompositions or reconstructions are not unique in general, 2) correlation: transformed coefficients (and especially those related to noise) are usually correlated, in contrast with the classical uncorrelatedness property of the components of a white noise after an orthogonal transform.

If the multiplicity aspect is usually recognized (and often addressed via averaging techniques [START_REF] Coifman | Translation-invariant de-noising[END_REF]), the correlation of the transformed coefficients have not received much consideration until recently. Most of the efforts have been devoted to the analysis of random processes by the DWT [START_REF] Pastor | Décomposition d'un processus stationnaire du second ordre. Propriétés statistiques d'ordre 2 des coefficients d'ondelettes et localisation fréquentielle des paquets d'ondelettes[END_REF]- [START_REF] Averkamp | A note on the discrete wavelet transform of second-order processes[END_REF]. It should be noted that early works by C. Houdré et al. [START_REF] Cambanis | On the continuous wavelet transform of second-order random processes[END_REF], [START_REF] Averkamp | Some distributional properties of the continuous wavelet transform of random processes[END_REF] consider the continuous wavelet transform of random processes, but only in a recent work by J. Fowler exact energetic relationships for an additive noise in the case of the non-tight RDWT have been provided [START_REF] Fowler | The redundant discrete wavelet transform and additive noise[END_REF]. It must be pointed out that the difficulty to characterize noise properties after a frame decomposition may limit the design of sophisticated estimation methods in denoising applications.

Fortunately, there exist redundant signal representations allowing finer noise behaviour assessment: in particular the dual-tree wavelet transform, based on the Hilbert transform, whose advantages in wavelet analysis have been recognized by several authors [START_REF] Abry | Multiresolution transient detection[END_REF], [START_REF] Olhede | Analytic wavelet thresholding[END_REF]. It consists of two classical wavelet trees developed in parallel. The second decomposition is refered to as the "dual" of the first one, which is sometimes called the "primal" decomposition.

The corresponding analyzing wavelets form Hilbert pairs [25, p.198 sq]. The dual-tree wavelet transform was initially proposed by N. Kingsbury [START_REF] Kingsbury | The dual-tree complex wavelet transform: a new technique for shift invariance and directional filters[END_REF] and further investigated by I. Selesnick [START_REF] Selesnick | Hilbert transform pairs of wavelet bases[END_REF] in the dyadic case. An excellent overview of the topic by I. Selesnick, R. Baraniuk and N. Kingsbury is provided in [START_REF] Selesnick | The dual-tree complex wavelet transform[END_REF] and an example of application is provided in [START_REF] Jalobeanu | Satellite image debbluring using complex wavelet packets[END_REF]. We recently have generalized this frame decomposition to the M -band case (M ≥ 2) (see [START_REF] Chaux | Hilbert pairs of M -band orthonormal wavelet bases[END_REF]- [START_REF]Image analysis using a dual-tree M -band wavelet transform[END_REF]). In the later works, we revamped the construction of the dual basis and the pre-processing stage, necessary in the case of digital signal analysis [START_REF] Rioul | Fast algorithms for discrete and continuous wavelet transforms[END_REF], [START_REF] Abry | On the initialization of the discrete wavelet transform algorithm[END_REF] and mandatory to accurate directional analysis of images, and we proposed an optimized reconstruction, thus addressing the first important facet of the resulting frame multiplicity. The M -band (M > 2) dual-tree wavelets prove more selective in the frequency domain than their dyadic counterparts, with improved directional selectivity as well. Furthermore, a larger choice of filters satisfying symmetry and orthogonality properties is available.

In this paper, we focus on the second facet, correlation, by studying the second-order statistical properties, in the transform domain, of a stationary random process undergoing a dual-tree M -band wavelet decomposition. In practice, such a random process typically models an additive noise. Preliminary comments on dual-tree coefficient correlation may be found in [START_REF] Wang | An investigation of 3D dual-tree wavelet transform for video coding[END_REF]. Dependencies between the coefficients already have been exploited for dual-tree wavelet denoising in [START_REF] Selesnick | Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency[END_REF], [START_REF] Rabbani | Image denoising based on a mixture of laplace distributions with local parameters in complex wavelet domain[END_REF]. A parametric nonlinear estimator based on Stein's principle, making explicit use of the correlation properties derived here, is proposed in [START_REF] Chaux | A nonlinear Stein based estimator for multichannel image denoising[END_REF]. At first, we briefly recall some properties of the dualtree wavelet decomposition in Section II, refering to [START_REF]Image analysis using a dual-tree M -band wavelet transform[END_REF] for more detail. In Section III, we express in a general form the second-order moments of the noise coefficients in each tree, both in the one and two-dimensional cases.

We also discuss the role of the post-transform -often performed on the dual-tree wavelet coefficients -with respect to (w.r.t.) decorrelation. In Section IV, we provide upper bounds for the decay of the correlations existing between pairs of primal/dual coefficients as well as an asymptotic result concerning coefficient whitening. The cross-correlations between primal and dual wavelets playing a key role in our analysis, their expressions are derived for several wavelet families in Section V. Simulation results are provided in Section VI in order to validate our theoretical results and better evaluate the importance of the correlations introduced by the dual-tree decomposition. Some final remarks are drawn in Section VII.

Throughout the paper, the following notations will be used: Z, Z * , N, N * , R, R * , R + and R * + are the set of integers, nonzero integers, nonnegative integers, positive integers, reals, nonzero reals, nonnegative reals and positive reals, respectively. Let M be an integer greater than or equal to 2, N M = {0, . . . , M -1} and N ⋆ M = {1, . . . , M -1}.

II. M -BAND DUAL-TREE WAVELET ANALYSIS

In this section, we recall the basic principles of an M -band [START_REF] Steffen | Theory of regular M -band wavelet bases[END_REF] dual-tree decomposition. Here, we will focus on 1D real signals belonging to the space L 2 (R) of square integrable functions. Let M be an integer greater than or equal to 2. An M -band multiresolution analysis of L 2 (R) is defined using one scaling function (or father wavelet)

ψ 0 ∈ L 2 (R) and (M -1) mother wavelets ψ m ∈ L 2 (R), m ∈ N ⋆ M .
In the frequency domain, the so-called scaling equations are expressed as:

∀m ∈ N M , √ M ψ m (M ω) = H m (ω) ψ 0 (ω), (1)
where a denotes the Fourier transform of a function a. 1. A pair of primal (top) and dual (bottom) analysis/synthesis M -band para-unitary filter banks.

G * 0 G * 1 G * M -1 ↓ M ↓ M ↓ M ↑ M ↑ M ↑ M G0 G1 GM-1 H * 0 H * 1 H * M -1 ↓ M ↓ M ↑ M ↑ M ↑ M H0 H1 HM-1 ↓ M Fig.
In order to generate an orthonormal M -band wavelet basis m∈N ⋆ M ,j∈Z {M -j/2 ψ m (M -j t-k), k ∈ Z} of L 2 (R), the following para-unitarity conditions must hold:

∀(m, m ′ ) ∈ N 2 M , M -1 p=0 H m (ω + p 2π M )H * m ′ (ω + p 2π M ) = M δ m-m ′ , (2)
where δ m = 1 if m = 0 and 0 otherwise. The filter with frequency response H 0 is low-pass whereas the filters with frequency response H m , m ∈ {1, . . . , M -2} (resp. m = M -1) are band-pass (resp. high-pass). In this case, cascading the M -band para-unitary analysis and synthesis filter banks, represented by the upper structures in Fig. 1, allows us to decompose and to perfectly reconstruct a given signal.

A "dual" M -band multiresolution analysis is built by defining another M -band wavelet orthonormal basis associated with a scaling function ψ H 0 and mother wavelets ψ H m , m ∈ N ⋆ M . More precisely, the mother wavelets are the Hilbert transforms of the "original" ones ψ m , m ∈ N ⋆ M . In the Fourier domain, the desired property reads:

∀m ∈ N ⋆ M , ψ H m (ω) = -ı sign(ω) ψ m (ω), (3)
where sign(•) is the signum function. Then, it can be proved [START_REF]2D dual-tree M -band wavelet decomposition[END_REF] that the dual scaling function can be chosen such

that ∀k ∈ Z, ∀ω ∈ [2kπ, 2(k + 1)π), ψ H 0 (ω) =      (-1) k e -ı(d+ 1 2 )ω ψ 0 (ω) if k ≥ 0 (-1) k+1 e -ı(d+ 1 2 )ω ψ 0 (ω) otherwise, (4)
where d is an arbitrary integer delay. The corresponding analysis/synthesis para-unitary Hilbert filter banks are illustrated by the lower structures in Fig. 1. Conditions for designing the involved frequency responses G m , m ∈ N M , have been recently provided in [START_REF]Image analysis using a dual-tree M -band wavelet transform[END_REF]. As the union of two orthonormal basis decomposition, the global dual-tree representation corresponds to a tight frame analysis of L 2 (R).

III. SECOND-ORDER MOMENTS OF THE NOISE WAVELET COEFFICIENTS

In this part, we first consider the analysis of a one-dimensional, real-valued, wide-sense stationary and zero-mean noise n, with autocovariance function

∀(τ, x) ∈ R 2 , Γ n (τ ) = E{n(x + τ )n(x)}. (5)
We then extend our results to the two-dimensional case.

A. Expression of the covariances in the 1D case

We denote by (n j,m [k]) k∈Z the coefficients resulting from a 1D M -band wavelet decomposition of the noise, in a given subband (j, m) where j ∈ Z and m ∈ N M . In the (j, m) subband, the wavelet coefficients generated by the dual decomposition are denoted by (n H j,m [k]) k∈Z . At resolution level j, the statistical second-order properties of the dual-tree wavelet decomposition of the noise are characterized as follows.

Proposition 1: For 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 have

E{n j,m [k + ℓ]n j,m ′ [k]} = Γ nj,m,n j,m ′ [ℓ] = ∞ -∞ Γ n (x) γ ψm,ψ m ′ x M j -ℓ dx (6)
E{n H j,m [k + ℓ]n H j,m ′ [k]} = Γ n H j,m ,n H j,m ′ [ℓ] = ∞ -∞ Γ n (x)γ ψ H m ,ψ H m ′ ( x M j -ℓ) dx (7) E{n j,m [k + ℓ]n H j,m ′ [k]} = Γ nj,m,n H j,m ′ [ℓ] = ∞ -∞ Γ n (x)γ ψm,ψ H m ′ x M j -ℓ dx, (8)
where the deterministic cross-correlation function of two real-valued functions f and g in L 2 (R) is expressed as

∀τ ∈ R, γ f,g (τ ) = ∞ -∞ f (x)g(x -τ ) dx. (9)
Proof: See Appendix I.

Proposition 2: For all (m, m ′ ) ∈ N M with m = m ′ = 0 or mm ′ = 0, we have γ ψ H m ,ψ H m ′ = γ ψm,ψ m ′ . As a consequence, Γ nj,m,n j,m ′ = Γ n H j,m ,n H j,m ′ . (10)
When mm ′ = 0, we have

∀τ ∈ R, γ ψm,ψ H m ′ (τ ) = -γ ψ m ′ ,ψ H m (-τ ) (11)
and, consequently,

∀ℓ ∈ Z, Γ nj,m,n H j,m ′ [ℓ] = -Γ n j,m ′ ,n H j,m [-ℓ]. (12)
Besides, the function γ ψ0,ψ H 0 is symmetric w.r. As a particular case of [START_REF] Do | The contourlet transform: an efficient directional multiresolution image representation[END_REF] when m = m ′ , it appears that the sequences (n j,m [k]) k∈Z and (n H j,m [k]) k∈Z have the same autocovariance sequence. We also deduce from Prop. 2 that, for all m = 0, γ ψm,ψ H m is an odd function, and the cross-covariance Γ nj,m,n H j,m is an odd sequence. This implies, in particular, that for all m = 0,

Γ nj,m,n H j,m [0] = 0. ( 13
)
The latter equality means that, for all m = 0 and k ∈ Z, the random vector

[n j,m [k] n H j,m [k]
] has uncorrelated components with equal variance.

The previous results are applicable to an arbitrary stationary noise but the resulting expressions may be intricate depending on the specific form of the autocovariance Γ n . Subsequently, we will be mainly interested in the study of the dual-tree decomposition of a white noise, for which tractable expressions of the second-order statistics of the coefficients can be obtained. The autocovariance of n is then given by Γ n (x) = σ 2 δ(x), where δ denotes the Dirac distribution. As the primal (resp. dual) wavelet basis is orthonormal, it can be deduced from ( 6)- [START_REF] Bölcskei | Frame-theoretic analysis of oversampled filter banks[END_REF]

(see Appendix III) that, for all (m, m ′ ) ∈ N 2 M and ℓ ∈ Z, Γ nj,m,n j,m ′ [ℓ] = Γ n H j,m ,n H j,m ′ [ℓ] = σ 2 δ m-m ′ δ ℓ (14)
Γ nj,m,n H j,m ′ [ℓ], = σ 2 γ ψm,ψ H m ′ (-ℓ) , (15)
where (δ k ) k∈Z is the Kronecker sequence (δ k = 1 if k = 0 and 0 otherwise). Therefore, (n j,m [k]) k∈Z and

(n H j,m [k]) k∈Z are cross-correlated zero-mean, white random sequences with variance σ 2 . The determination of the cross-covariance requires the computation of γ ψm,ψ H m ′ . We distinguish between the mother (m ′ = 0) and father (m ′ = 0) wavelet case.

• By using (3), for m ′ = 0, Parseval-Plancherel formula yields

γ ψm,ψ H m ′ (τ ) = 1 2π ∞ -∞ ψ m (ω) ψ m ′ (ω) H ) * exp(ıωτ ) dω = - 1 π Im ∞ 0 ψ m (ω) ψ m ′ (ω)) * exp(ıωτ ) dω , (16)
where Im{z} denotes the imaginary part of a complex z.

• According to (4), for m ′ = 0 we find, after some simple calculations:

γ ψm,ψ H 0 (τ ) = 1 π Re ∞ k=0 (-1) k 2(k+1)π 2kπ ψ m (ω) ψ 0 (ω) * exp ıω ( 1 2 + τ + d) dω , (17)
where Re{z} denotes the real part of a complex z.

In both cases, we have

|γ ψm,ψ H m ′ (τ )| ≤ 1 π ∞ 0 | ψ m (ω) ψ m ′ (ω)| dω. (18)
For M -band wavelet decompositions, selective filter banks are commonly used. Provided that this selectivity property is satisfied, the cross term | ψ m (ω) ψ m ′ (ω)| can be expected to be close to zero and the upper bound in [START_REF] Leporini | High-order wavelet packets and cumulant field analysis[END_REF] to take small values when m = m ′ . This fact will be discussed in Section VI-C based on numerical results. On the contrary, when m = m ′ , the cross-correlation functions always need to be evaluated more carefully. In Section V, we will therefore focus on the functions:

γ ψm,ψ H m (τ ) = - 1 π ∞ 0 | ψ m (ω)| 2 sin(ωτ ) dω, m = 0, (19)
γ ψ0,ψ H 0 (τ ) = 1 π ∞ k=0 (-1) k 2(k+1)π 2kπ | ψ 0 (ω)| 2 cos ω ( 1 2 + τ + d) dω. (20)
Note that, in this paper, we do not consider interscale correlations. Although expressions of the second-order statistics similar to the intrascale ones can be derived, sequences of wavelet coefficients defined at different resolution levels are generally not cross-stationary [START_REF] Leporini | High-order wavelet packets and cumulant field analysis[END_REF].

B. Extension to the 2D case

We now consider the analysis of a two-dimensional noise n, which is also assumed to be real, wide-sense stationary with zero-mean and autocovariance function

∀(τ , x) ∈ R 2 × R 2 , Γ n (τ ) = E{n(x + τ )n(x)}.
We can proceed similarly to the previous section. We denote by (n j,m [k]) k∈Z 2 the coefficients resulting from a 2D separable M -band wavelet decomposition [START_REF] Steffen | Theory of regular M -band wavelet bases[END_REF] of the noise, in a given subband (j, m) ∈ Z × N 2 M . The wavelet coefficients of the dual decomposition are denoted by (n H j,m [k]) k∈Z 2 . We 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 nj,m,n j,m ′ [ℓ] = E{n j,m [k + ℓ]n j,m ′ [k]} = ∞ -∞ ∞ -∞ Γ n (x 1 , x 2 )γ ψm 1 ,ψ m ′ 1 x 1 M j -ℓ 1 γ ψm 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]} = ∞ -∞ ∞ -∞ 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 nj,m,n H j,m ′ [ℓ] = E{n j,m [k + ℓ]n H j,m ′ [k]} = ∞ -∞ ∞ -∞ Γ n (x 1 , x 2 )γ ψm 1 ,ψ H m ′ 1 x 1 M j -ℓ 1 γ ψm 2 ,ψ H m ′ 2 x 2 M j -ℓ 2 dx 1 dx 2 . ( 23
)
From the properties of the correlation functions of the wavelets and the scaling function as given by Prop. 2, it can be deduced that, when (m

1 = m ′ 1 = 0 or m 1 m ′ 1 = 0) and (m 2 = m ′ 2 = 0 or m 2 m ′ 2 = 0), Γ nj,m,n j,m ′ = Γ n H j,m ,n H j,m ′ . (24)
Some additional symmetry properties are straightforwardly obtained from Prop. 2. In particular, for all m ∈ N ⋆2 M , the cross-covariance Γ nj,m,n H j,m is an even sequence. An important consequence of the latter properties concerns the 2 × 2 linear combination of the primal and dual wavelet coefficients which is often implemented in dual-tree decompositions. As explained in [START_REF]2D dual-tree M -band wavelet decomposition[END_REF], the main advantage of such a post-processing is to better capture the directional features in the analyzed image. More precisely, this amounts to performing the following unitary transform of the detail coefficients, for 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 when m 1 = 0 or m 2 = 0.) The covariances of the transformed fields of noise coefficients (w j,m [k]) k∈Z 2 and (w H j,m [k]) k∈Z 2 then take the following expressions:

Proposition 3: For all m ∈ N ⋆2 M and ℓ ∈ Z 2 , Γ wj,m,wj,m [ℓ] = Γ nj,m,nj,m [ℓ] + Γ nj,m,n H j,m [ℓ] (27)
Γ w H j,m ,w H j,m [ℓ] = Γ nj,m,nj,m [ℓ] -Γ nj,m,n H j,m [ℓ] (28)
Γ wj,m,w H j,m [ℓ] = 0. ( 29
)
Proof: See Appendix IV.

This shows that the post-transform not only provides a better directional analysis of the image of interest but also plays an important role w.r.t. the noise analysis. Indeed, it allows to completely cancel the correlations between the primal and dual noise coefficient fields obtained for a given value of (j, m). In turn, this operation introduces some spatial noise correlation in each subband.

For a two-dimensional white noise, Γ n (x) = σ 2 δ(x) and the coefficients

(n j,m [k]) k∈Z 2 and (n H j,m ′ [k]) k∈Z 2 are such that, for all ℓ = (ℓ 1 , ℓ 2 ) ∈ Z 2 , Γ nj,m,n j,m ′ [ℓ] = Γ n H j,m ,n H j,m ′ [ℓ] = σ 2 δ m1-m ′ 1 δ m2-m ′ 2 δ ℓ1 δ ℓ2 (30)
Γ nj,m,n H j,m ′ [ℓ] = σ 2 γ ψm 1 ,ψ H m ′ 1 (-ℓ 1 )γ ψm 2 ,ψ H m ′ 2 (-ℓ 2 ) . (31)
As a consequence of Prop. 2, in the case when ℓ = 0, we conclude that, for (m 1 = 0 or m 2 = 0) and k ∈ Z 2 , the vector

[n j,m [k] n H j,m [k]
] has uncorrelated components with equal variance. This property holds more generally for 2D noises with separable covariance functions.

IV. SOME ASYMPTOTIC PROPERTIES

In the previous section, we have shown that the correlations of the basis functions play a prominent role in the determination of the second-order statistical properties of the noise coefficients. To estimate the strength of the dependencies between the coefficients, it is useful to determine the decay of the correlation functions. The following result allows to evaluate their decay.

Proposition 4: Let (N 1 , . . . , N M -1 ) ∈ (N * ) M -1 and define N 0 = min m∈N ⋆ M N m . Assume that, for all m ∈ N M , the function | ψ m | 2 is 2N m + 1 times continuously differentiable on R and, for all q ∈ {0, . . . , 2N m + 1}, its q-th order derivatives (| ψ m | 2 ) (q) belong to L 1 (R). 1 Further assume that, for all m = 0, ψ m (ω) = O(ω Nm ) as ω → 0. Then, there exists C ∈ R + such that, for all m ∈ N M , ∀τ ∈ R * , |γ ψm,ψm (τ )| ≤ C |τ | 2Nm+1 (32)
and

∀τ ∈ R * , |γ ψm,ψ H m (τ )| ≤ C |τ | 2Nm+1 . ( 33
)
Proof: See Appendix V.

Note that, for all m ∈ N M , the assumptions concerning | ψ m | 2 are satisfied if ψ m is 2N m + 1 times continuously differentiable on R and, for all q ∈ {0, . . . , 2N m + 1}, its q-th order derivatives ψ

(q) m belong to L 2 (R). Indeed, if ψ m is 2N m + 1 times continuously differentiable on R, so is | ψ m | 2 .
Leibniz formula allows us to express its derivative of order q ∈ {0, . . . , 2N m + 1} as

(| ψ m | 2 ) (q) = q ℓ=0 q ℓ ( ψ m ) (ℓ) ( ψ * m ) (q-ℓ) .
Consequently, if for all ℓ ∈ {0, . . . , q}, ψ

(ℓ) m ∈ L 2 (R), then (| ψ m | 2 ) (q) ∈ L 1 (R).
Note also that, for integrable wavelets, the assumption

ψ m (ω) = O(ω Nm ) as ω → 0 means that the wavelet ψ m , m = 0, has N m vanishing moments.
Therefore, the decay rate of the wavelet correlation functions is all the more important as the Fourier transforms of the basis functions ψ m , m ∈ N M , are regular (i.e. the wavelets have fast decay themselves) and the number of vanishing moments is large. The latter condition is useful to ensure that Hilbert transformed functions ψ H m have regular spectra too. It must be emphasized that Prop. 4 guarantees that the asymptotic decay of the wavelet correlation functions is at most |τ | -2Nm-1 . A faster decay can be obtained in practice for some wavelet families.

For example, when ψ m is compactly supported, γ ψm,ψm also has a compact support. In this case however, ψ H m cannot be compactly supported [START_REF]Image analysis using a dual-tree M -band wavelet transform[END_REF], so that the bound in (33) remains of interest. Examples will be discussed in more detail in Section V.

It is also worth noticing that the obtained upper bounds on the correlation functions allow us to evaluate the decay rate of the covariance sequences of the dual-tree wavelet coefficients of a stationary noise as expressed below.

Proposition 5: Let n be a 1D zero-mean wide-sense stationary random process. Assume that either n is a white noise or its autocovariance function is with exponential decay, that is there exist

A ∈ R + and α ∈ R * + , such that ∀τ ∈ R, |Γ n (τ )| ≤ Ae -α|τ | .
Consider also functions ψ m , m ∈ N M , satisfying the assumptions of Prop. 4. Then, there exists

C ∈ R + such that for all j ∈ Z, m ∈ N M and ℓ ∈ Z * , |Γ nj,m,nj,m [ℓ]| ≤ C 1 + |ℓ| 2Nm+1 (34)
|Γ nj,m,n H j,m [ℓ]| ≤ C 1 + |ℓ| 2Nm+1 . ( 35
)
Proof: See Appendix VI.

The decay property of the covariance sequences readily extends to the 2D case:

Proposition 6: Let n be a 2D zero-mean wide-sense stationary random field. Assume that either n is a white noise or its autocovariance function is with exponential decay, that is there 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 -α1|τ1|-α2|τ2| . ( 36
)
Consider also functions ψ m , m ∈ N M , satisfying the assumptions of Prop. 4. Then, there exists

C ∈ R + such that for all j ∈ Z, m ∈ N 2 M and ℓ = (ℓ 1 , ℓ 2 ) ∈ Z 2 , |Γ nj,m,nj,m [ℓ]| ≤ C (1 + |ℓ 1 | 2Nm+1 )(1 + |ℓ 2 | 2Nm+1 ) (37) |Γ nj,m,n H j,m [ℓ]| ≤ C (1 + |ℓ 1 | 2Nm+1 )(1 + |ℓ 2 | 2Nm+1 ) . (38)
Besides, for all

j ∈ Z, m ∈ N ⋆2 M and ℓ = (ℓ 1 , ℓ 2 ) ∈ Z 2 , |Γ wj,m,wj,m [ℓ]| ≤ 2 C (1 + |ℓ 1 | 2Nm+1 )(1 + |ℓ 2 | 2Nm+1 ) ( 39
)
|Γ w H j,m ,w H j,m [ℓ]| ≤ 2 C (1 + |ℓ 1 | 2Nm+1 )(1 + |ℓ 2 | 2Nm+1 ) . ( 40
)
Proof: Due to the separability of the 2D dual-tree wavelet analysis, ( 37) and ( 38) are obtained quite similarly to [START_REF] Abry | On the initialization of the discrete wavelet transform algorithm[END_REF] and [START_REF] Wang | An investigation of 3D dual-tree wavelet transform for video coding[END_REF]. The proof of ( 39) and ( 40) then follows from ( 27) and ( 28).

The two previous propositions provide upper bounds on the decay rate of the covariance sequences of the dualtree wavelet coefficients, when the norm of the lag variable (ℓ or ℓ) takes large values. We end this section by providing asympotic results at coarse resolution (as j → ∞).

Proposition 7: Let n be a 1D zero-mean wide-sense stationary process with covariance function

Γ n ∈ L 1 (R) ∩ L 2 (R). Then, for all (m, m ′ ) ∈ N 2 M , we have lim j→∞ Γ nj,m,n j,m ′ [ℓ] = Γ n (0) δ m-m ′ δ ℓ ( 41
)
lim j→∞ Γ nj,m,n H j,m ′ [ℓ] = Γ n (0) γ ψm,ψ H m ′ (-ℓ) . ( 42
)
Proof: See Appendix VII.

In other words, at coarse resolution in the transform domain, a stationary noise n with arbitrary covariance function Γ n behaves like a white noise with spectrum density Γ n (0). This fact further emphasizes the interest in studying more precisely the dual-tree wavelet decomposition of a white noise. Note also that, by calculating higher order cumulants of the dual-tree wavelet coefficients and using techniques as in [START_REF] Leporini | High-order wavelet packets and cumulant field analysis[END_REF], [START_REF] Touati | Some results on the wavelet packet decomposition of nonstationary processes[END_REF], 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 normal as j → ∞. Although Prop. 7 has been stated for 1D random processes, we finally point out that quite similar results are obtained in the 2D case.

V. WAVELET FAMILIES EXAMPLES

For a white noise (see ( 14), ( 15), ( 30) and ( 31)) or for arbitrary wide-sense stationary noises analyzed at coarse resolution (cf. Prop. 7), we have seen that the cross-correlation functions between the primal and dual wavelets taken at integer values are the main features. In order to better evaluate the impact of the wavelet choice, we will now specify the expressions of these cross-correlations for different wavelet families.

A. M -band Shannon wavelets

M -band Shannon wavelets (also called sinc wavelets in the literature) correspond to an ideally selective analysis in the frequency domain. These wavelets also appear as a limit case for many wavelet families, e.g. Daubechies's or spline wavelets. We have then, for all m ∈ N M ,

ψ m (ω) = 1 ]-(m+1)π,-mπ]∪[mπ,(m+1)π[ (ω),
where 1 S denotes the characteristic function of the set S ⊂ R:

1 S (ω) =      1 if ω ∈ S 0 otherwise.
In this case, (20) reads:

∀τ ∈ R, γ ψ0,ψ H 0 (τ ) = 1 π π 0 cos ( 1 2 + d + τ )ω dω =        (-1) d cos (πτ ) π( 1 2 + d + τ ) if τ = -d -1 2 1 otherwise. For m ∈ N ⋆ M , (19) leads to ∀τ ∈ R, γ ψm,ψ H m (τ ) = - 1 π (m+1)π mπ sin (ωτ )dω =      cos (m + 1)πτ -cos(mπτ ) πτ if τ = 0 0 otherwise.
We deduce from the two previous expressions that, for all ℓ ∈ Z,

γ ψ0,ψ H 0 (ℓ) = (-1) (d+ℓ) π(d + ℓ + 1 2 ) (43) ∀m = 0, γ ψm,ψ H m (ℓ) =      (-1) (m+1)ℓ 1 -(-1) ℓ πℓ if ℓ = 0 0 otherwise. (44)
We can remark that, for all (m, m

′ ) ∈ N ⋆2 M , γ ψm,ψ H m (ℓ) = (-1) (m ′ -m)ℓ γ ψ m ′ ,ψ H m ′ (ℓ) (45)
and γ ψm,ψ H m (ℓ) = 0, when ℓ is odd. Besides, the correlation sequences decay pretty slowly as ℓ -1 . We also note that, as the functions ψ m , m ∈ N M , have non-overlapping spectra, ( 6)-(8) (resp. ( 21)-( 23)) allow us to conclude that, dual-tree noise wavelet coefficients corresponding respectively to subbands (j, m) and

(j, m ′ ) with m = m ′ (resp. (j, m 1 , m 2 ) and (j, m ′ 1 , m ′ 2 ) with m 1 = m ′ 1 or m 2 = m ′ 2
) are perfectly uncorrelated.

B. Meyer wavelets

These wavelets [START_REF] Meyer | Ondelettes et opérateurs. I, ser. Actualités mathématiques[END_REF], [42, p. 116] are also band-limited but with smoother transitions than Shannon wavelets.

The scaling function is consequently defined as

ψ 0 (ω) =              1 if 0 ≤ |ω| ≤ π(1 -ǫ) W |ω| 2πǫ - 1 -ǫ 2ǫ 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) with ν : [0, 1] → [0, 1] such that ν(0) = 0 (47) ∀ θ ∈ [0, 1], ν(1 -θ) = 1 -ν(θ).
Then, it can be noticed that

∀ θ ∈ [0, 1], W 2 (1 -θ) = 1 -W 2 (θ). (48)
A common choice for the ν function is [42, p. 119]:

∀ θ ∈ [0, 1], ν(θ) = θ 4 (35 -84 θ + 70 θ 2 -20 θ 3 ). ( 49
)
For m ∈ {1, . . . , M -2}, the associated M -band wavelets are given by

ψ m (ω) =                    e ıηm(ω) W m + ǫ 2ǫ - |ω| 2πǫ if (m -ǫ)π ≤ |ω| ≤ (m + ǫ)π e ıηm(ω) if (m + ǫ)π ≤ |ω| ≤ (m + 1 -ǫ)π e ıηm(ω) W |ω| 2πǫ - m + 1 -ǫ 2ǫ if (m + 1 -ǫ)π ≤ |ω| ≤ (m + 1 + ǫ)π 0 otherwise (50)
while, for the last wavelet, we have

ψ M -1 (ω) =                    e ıηM-1(ω) W M -1 + ǫ 2ǫ - |ω| 2πǫ if (M -1 -ǫ)π ≤ |ω| ≤ (M -1 + ǫ)π e ıηM-1(ω) if (M -1 + ǫ)π ≤ |ω| ≤ M (1 -ǫ)π e ıηM-1(ω) W |ω| 2πǫM - 1 -ǫ 2ǫ if M (1 -ǫ)π < |ω| ≤ M (1 + ǫ)π 0 otherwise. (51)
Hereabove, the phase functions η m , m ∈ N ⋆ M , are odd functions and we have

∀ω ∈ (M π, M (1 + ǫ)π), η M -1 (ω) = -η M -1 (2M π -ω) mod 2π.
In addition, for the orthonormality condition to be satisfied, the following recursive equations must hold:

∀ω ∈ ((m -ǫ)π, (m + ǫ)π), η m (ω -2mπ) -η m-1 (ω -2mπ) = η m (ω) -η m-1 (ω) + π mod 2π.
by setting: ∀ω ∈ R, η 0 (ω) = 0. Generally, linear phase solutions to the previous equation are chosen [START_REF] Tennant | Solution to the orthogonal M -channel bandlimited wavelet construction proposition[END_REF].

Using the above expressions, the cross-correlations between the Meyer basis functions and their dual counterparts are derived in Appendix VIII. It can be deduced from these results that:

∀ ℓ ∈ Z, γ ψ0,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) = 2ǫ 1 0 W 2 1 + θ 2 sin (πǫxθ) dθ. ( 53
)
For the wavelets, we have when m ∈ {1, . . . , M -2}:

γ ψm,ψ H m (ℓ) =      (-1) (m+1)ℓ 1 -(-1) ℓ 1 πℓ -I ǫ (ℓ) if ℓ = 0 0 otherwise (54)
whereas

γ ψM-1,ψ H M -1 (ℓ) =      (-1) M ℓ 1 -(-1) ℓ πℓ + (-1) ℓ I ǫ (ℓ) -I M ǫ (ℓ) if ℓ = 0 0 otherwise. ( 55
)
Similarly to Shannon wavelets, for (m, m ′ ) ∈ {1, . . . , M -2} 2 , (45) holds and γ ψm,ψ H m (ℓ) = 0, when ℓ is odd. As expected, we observe that the previous cross-correlations converge point-wise to the expressions given for Shannon wavelets in ( 43) and ( 44), as we let ǫ → 0.

Besides, let us make the following assumption: W 2 is 2q + 2 times continuously differentiable on [0, 1] with q ∈ N * and, for all ℓ ∈ {0, . . . , 2q -1}, (W 2 ) (ℓ) (1) = 0. This assumption is typically satisfied by the window defined by ( 49) with q = 4. From [START_REF] Strömberg | A modified Franklin system and higher-order spline systems on R n as unconditional bases for Hardy spaces[END_REF], it can be further noticed that, for all ℓ ∈ {1, . . . , q + 1}, (W 2 ) (2ℓ) (1/2) = 0.

Then, when x = 0, it is readily checked by integrating by part that

1 0 W 2 1 + θ 2 sin (πǫxθ) dθ = 1 2πǫx + (-1) q-1 (W 2 ) (2q) (1) 2 2q (πǫx) 2q+1 cos(πǫx) + (-1) q (W 2 ) (2q+1) (1) 2 2q+1 (πǫx) 2q+2 sin(πǫx) + (-1) q+1 2 2q+2 (πǫx) 2q+2 1 0 (W 2 ) (2q+2) 1 + θ 2 sin (πǫxθ) dθ.
This shows that, as |x| → ∞,

I ǫ (x) = 1 πx + (-1) q-1 (W 2 ) (2q) (1) 2 2q-1 π 2q+1 ǫ 2q x 2q+1 cos(πǫx) + O(x -2q-2 ). (56)
For example, for the taper function defined by (49), we get

I ǫ (x) = 1 πx - 385875 4π 7 ǫ 8 x 9 cos(πǫx) + O(x -10 ).
Combining ( 56) with ( 52), ( 54) and ( 55) allows us to see that the cross-correlation sequences decay as ℓ -2q-1

when |ℓ| → ∞. Eq. ( 56) also indicates that the decay tends to be faster when ǫ is large, which is consistent with intuition since the basis functions are then better localized in time. Note that, as shown by ( 50) and ( 51), under the

considered differentiability assumptions, | ψ m | 2 is 2q -1 times continuously differentiable on R whereas ψ m (ω) = 0
for m ∈ N ⋆ M and |ω| < m -ǫ. Prop. 4 then guarantees a decay rate at least equal to |ℓ| -2q+1 (here, N m = q -1). In this case, we see that the decay rate derived from ( 56) is more acurate than the decay given by Prop. 4.

C. Wavelet families derived from wavelet packets 1) General form:

One can generate M -band orthonormal wavelet bases from dyadic orthonormal wavelet packet decompositions corresponding to an equal subband analysis. We are consequently limited to scaling factors M which are power of 2. More precisely, let (ψ m ) m∈N be the considered wavelet packets [START_REF] Coifman | Entropy-based algorithms for best-basis selection[END_REF], for all P ∈ N * an orthonormal M -band wavelet decomposition is obtained using the basis functions (ψ m ) 0≤m<M with M = 2 P . In this case, the basis functions satisfy the following two-scale relations: for all m ∈ N,

√ 2 ψ 2m (2ω) = A 0 (ω) ψ m (ω) (57) √ 2 ψ 2m+1 (2ω) = A 1 (ω) ψ m (ω), (58)
where A 0 and A 1 are the frequency responses of the low-pass and high-pass filters of the associated two-band para-unitary synthesis filter bank. We can infer the following result.

Proposition 8: For all τ ∈ R and m ∈ N * , we have

γ ψ2m,ψ H 2m (τ ) = γ a0 [0]γ ψm,ψ H m (2τ ) + ∞ k=1 γ a0 [k] γ ψm,ψ H m (2τ + k) + γ ψm,ψ H m (2τ -k) (59)
γ ψ2m+1,ψ H 2m+1 (τ ) = γ a1 [0]γ ψm,ψ H m (2τ ) + ∞ k=1 γ a1 [k] γ ψm,ψ H m (2τ + k) + γ ψm,ψ H m (2τ -k) , (60)
where, for all ǫ ∈ {0, 1}, (γ aǫ [k]) k∈Z is the autocorrelation of the impulse response (a ǫ [k]) k∈Z of the filter with frequency response A ǫ :

∀k ∈ Z, γ aǫ [k] = ∞ q=-∞ a ǫ [q] a ǫ [q -k].
Proof: See Appendix IX.

It is important to note that ( 59) and ( 60) are not valid for m = 0. These two relations define recursive equations for the calculation of the cross-correlations (γ ψm,ψ H m ) m>1 , provided that γ ψ1,ψ H 1 has been calculated first. For this specific class of M -band wavelet decompositions, it is possible to relate the decay properties of the cross-correlation functions to the number of vanishing moments of the underlying dyadic wavelet analysis.

Proposition 9: Assume that the filters with frequency response A 0 and A 1 are FIR and A 1 has a zero of order N ∈ N * at frequency 0 (or, equivalently, A 0 has a zero of order N at frequency 1/2). Then, there exists

C 0 ∈ R + such that ∀τ ∈ R * , |γ ψ0,ψ H 0 (τ )| ≤ C 0 |τ | -2N -1 . (61)
In addition, for all m ∈ N * , let (ǫ 1 , ǫ 2 , . . . , ǫ r ) ∈ {0, 1} r , r ∈ N * , be the digits in the binary representation of m,

that is m = r i=1 ǫ i 2 i-1 . (62)
Then, there exists

C m ∈ R + such that ∀τ ∈ R * , |γ ψm,ψ H m (τ )| ≤ C m |τ | -2N ( È r i=1 ǫi)-1 . ( 63
)
Proof: The filters of the underlying dyadic multiresolution being FIR (Finite Impulse Response), the wavelet packets are compactly supported. Consequently, their Fourier transforms are infinitely differentiable, their derivatives of any order belonging to L 2 (R). In addition, the binary representation of m ∈ N * being given by (62), Eqs. [START_REF] Rudin | Real and complex analysis[END_REF] and ( 58) yield

ψ m (ω) = ψ 0 ω 2 P r i=1 1 √ 2 A ǫi ω 2 i P i=r+1 1 √ 2 A 0 ω 2 i
that is H m (ω) = We see that the cross-correlation γ ψm,ψ H m decays all the more rapidly as the number of 1's in the binary representation of m is large. 22) The particular case of Walsh-Hadamard transform: The case M = 2 corresponds to Haar wavelets. In contrast with Shannon wavelets, these wavelets lay emphasis on time/spatial localization. We consequently have:

ψ 0 (ω) = sinc( ω 2 ) e -ı w 2 (64) ψ 1 (ω) = ı sinc( ω 4 ) sin( ω 4 ) e -ı w 2 , (65)
where

sinc(ω) =      sin(ω) ω if ω = 0 1 otherwise.
After some calculations which are provided in Appendix X, we obtain for all τ ∈ R,

πγ ψ0,ψ H 0 (τ ) = ∞ k=0 (-1) k 1 2 S k (3 + 2d + 2τ ) -S k (1 + 2d + 2τ ) + 1 2 S k (-1 + 2d + 2τ ) , (66)
where, for all k ∈ N and for all x ∈ R,

S k (x) = x (k+1)πx kπx sinc(u) du.
Furthermore, we have (adopting the convention: 0 ln(0) = 0):

πγ ψ1,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-correlations γ ψm,ψ H m , m ∈ {2, . . . , 2 P -1}, can be determined in a recursive manner thanks to Prop. 8. For Walsh-Hadamard wavelets, we have

∀ǫ ∈ {0, 1}, ∀k ∈ Z, γ aǫ [k] =              1 if k = 0 (-1) ǫ 2 if |k| = 1 0 otherwise (68)
and, consequently, for all m = 0 and τ ∈ R,

γ ψ2m,ψ H 2m (τ ) = γ ψm,ψ H m (2τ ) + 1 2 γ ψm,ψ H m (2τ + 1) + γ ψm,ψ H m (2τ -1) (69)
γ ψ2m+1,ψ H 2m+1 (τ ) = γ ψm,ψ H m (2τ ) - 1 2 γ ψm,ψ H m (2τ + 1) + γ ψm,ψ H m (2τ -1) . (70)
From (67), it can be noticed that γ ψ1,ψ H 1 (τ ) = 1/(8πτ

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

D. Franklin wavelets

Franklin wavelets [START_REF] Franklin | A set of continuous orthogonal functions[END_REF], [START_REF] Strömberg | A modified Franklin system and higher-order spline systems on R n as unconditional bases for Hardy spaces[END_REF] correspond to a dyadic orthonormal basis of spline wavelets of order 1 [42, p. 146 sq.]. With the Haar wavelet, they form a special case of Battle-Lemarié wavelets [START_REF] Battle | A block spin construction of ondelettes. Part I: Lemarié functions[END_REF], [START_REF] Lemarié | Une nouvelle base d'ondelettes de L 2 (R n )[END_REF]. The Fourier transforms of the scaling function and the mother wavelet are given by:

ψ 0 (ω) = 3 1 + 2 cos 2 (ω/2) 1/2 sinc 2 ω 2 (71)
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
)
The expression of the cross-correlation of the scaling functions readily follows from [START_REF] Cambanis | On the continuous wavelet transform of second-order random processes[END_REF]:

∀τ ∈ R, γ ψ0,ψ H 0 (τ ) = 6 π ∞ k=0 (-1) k T k (1 + 2d + 2τ ),
where, for all k ∈ N and x ∈ R,

T k (x) = (k+1)π kπ sinc 4 (u) 1 + 2 cos 2 (u) cos(ux) du.
The expression of the cross-correlation of the mother wavelet can be deduced from ( 19) and (72) and resorting to numerical methods for the computation of the resulting integral, but it is also possible to obtain a series expansion of the cross-correlation as shown next.

Taking the square modulus of (72), we find

2| ψ 1 (2ω)| 2 = | A 1 (ω)| 2 | χ(ω)| 2 , (73)
where

A 1 (ω) = 6 2 -cos(ω) 1 + 2 cos 2 (ω) 2 + cos(ω) 1/2 , χ(ω) = sin 2 (ω/2) ω/2 2 .
Let ( a 1 [k]) k∈Z (resp. χ) be the sequence (resp. function) whose Fourier transform is A 1 (resp. χ). Similarly to [START_REF] Schwartz | Analyse IV : applications à la théorie de la mesure[END_REF], (73) leads to the following relation

∀τ ∈ R, γ ψ1,ψ H 1 (τ ) = γ a1 [0] γ χ,χ H (2τ ) + ∞ k=1 γ a1 [k] γ χ,χ H (2τ + k) + γ χ,χ H (2τ -k) , (74)
where (γ a1 [k]) k∈Z denotes the autocorrelation of the sequence ( a 1 [k]) k∈Z .

We have then to determine γ χ,χ H and (γ ã1 [k]) k∈N . First, it can be shown (see Appendix XI for more detail) that

3πγ χ,χ H (τ ) = q 0 τ 3 ln |τ | + 4 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, the sequence (γ ã1 [k]) k∈N can be deduced from | A 1 (ω)| 2 by using z-transform inversion techniques (calculations are provided in Appendix XI). This leads to

∀k ∈ N,      γ ã1 [2k] = 2 √ 3 9 (2 - √ 3) k 7(-1) k + 4(2 - √ 3) k γ ã1 [2k + 1] = 8 √ 3 9 (2 - √ 3) k (-1) k (1 - √ 3) -(2 - √ 3) k+1 . (76)
Equations ( 74), ( 75) and (76) thus allow an accurate numerical evaluation of γ ψ1,ψ H 1 . Since

γ χ,χ H (τ ) ∼ -3/(2πτ 5 ) as |τ | → ∞ (77)
and

γ ã1 [k] = O((2 - √ 3) k/2 ) as k → ∞ (78)
the convergence of the series in (74) is indeed pretty fast.

From Prop. 4, we further deduce that γ ψ0,ψ H 0 (τ ) and γ ψ1,ψ H 1 (τ ) decay as |τ | -5 (here, we have N 0 = N 1 = 2). The decay rate of γ ψ1,ψ H 1 can be derived more precisely from (74). Indeed, we have

|τ | 5 ∞ k=-∞ |γ a1 [k]||γ χ,χ H (2τ -k)| ≤ 1 2 ∞ k=-∞ |γ a1 [k]| |2τ -k| 5 + |k| 5 |γ χ,χ H (2τ -k)| ≤ sup u∈R (|u| 5 |γ χ,χ H (u)|) + sup u∈R |γ χ,χ H (u)| ∞ k=-∞ (1 + |k| 5 )|γ a1 [k]| < ∞, (79)
where the convexity of |.| 5 has been used in the first inequality and the last inequality is a consequence of (77) and (78). It can be deduced from the dominated convergence theorem that

lim |τ |→∞ τ 5 γ ψ1,ψ H 1 (τ ) = ∞ k=-∞ γ a1 [k] lim |τ |→∞ τ 5 γ χ,χ H (2τ -k) = - 3 64π ∞ k=-∞ γ a1 [k] = - 3 64π | A 1 (0)| 2 = - 1 32π
.

Finally, we would like to note that similar expressions can be derived for higher order spline wavelets although the calculations become tedious.

VI. EXPERIMENTAL RESULTS

A. Results based on theoretical expressions

At first, we provide numerical evaluations of the expressions of the cross-correlation sequences obtained in the previous section when the lag variable (denoted by ℓ) varies in {0, 1, 2, 3}. The cross-correlations for lag values in {-3, -2, -1} can be deduced from the symmetry properties shown in Section III-A. We notice that cubic spline wavelets [START_REF] Unser | Splines, a perfect fit for signal processing[END_REF] have not been studied in Section V, so that the their cross-correlation values have to be computed directly from [START_REF] Averkamp | A note on the discrete wavelet transform of second-order processes[END_REF] and [START_REF] Cambanis | On the continuous wavelet transform of second-order random processes[END_REF]. The results concerning the dyadic case are given in Table II. They show that the cross-correlations between the noise coefficients at the output of a dual-tree analysis can take significant values (up to 0.64). We also observe that the wavelet choice has a clear influence on the magnitude of the correlations.

Indeed, while Meyer wavelet leads to results close to the Shannon wavelet, the correlations are weaker for the Haar wavelet. As expected, spline wavelets yield intermediate cross-correlation values between the Meyer and the Haar cases.

Our next results concern the M -band case with M ≥ 3. Due to the properties of the cross-correlations, the study can be simplified as explained below.

• Shannon wavelets: due to ( 45), the M -band cross-correlations are, up to a possible sign change, equal to the dyadic case cross-correlations (see Table II).

• Meyer wavelets: still due to [START_REF] Drmota | The sum of digits function of squares[END_REF], the first M -2 cross-correlations of the wavelets are easily deduced from the first one. So, we only need to specify γ ψ0,ψ H 0 , γ ψ1,ψ H 1 and γ ψM-1,ψ H M -1 . Tables III andIV give the related values when M ranges from 3 to 8, the ǫ parameter being set to its possible maximum value (M + 1) -1 .

• Walsh-Hadamard wavelets: when M = 2 P +1 , P ∈ N * , (ψ m ) 0≤m<M/2 is the set of basis functions of the (M/2)-band wavelet decomposition. In this way, the results in Table V allow us to evaluate the cross-correlation values for M ∈ {2, 4, 8}. III andIV, the cross-correlations in the Meyer case remain significant, their magnitudes being even slightly increased as the number of subbands becomes larger. Table V shows that the cross-correlation of Walsh-Hadamard wavelets are much smaller and that they are close to zero when the subband index m is large.

As shown in Tables

γ ψ 0 ,ψ H 0 γ ψ 1 ,ψ H 1 Wavelets \ ℓ 0 1 2 3 1 2

B. Monte Carlo simulations

A second approach for computing the cross-correlations consists in carrying out a Monte Carlo study. More precisely, a realization of a white standard Gaussian noise sequence of length L = M J ⌊ 2 14 M J ⌋ (with J = 3) is drawn and its 1D dual-tree decomposition over J resolution levels is performed. Then, the cross-covariances for each subband can be estimated by their classical sample estimates. In our experiments, average values of these cross-correlations are computed over 100 runs. This Monte Carlo study allows us to validate the theoretical expressions we have obtained for several wavelet families in Section V. In addition, this approach can be applied to wavelets whose Fourier transforms do not take a simple form. For instance, we are able to compute the cross-correlation values for symlets [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF][p.259] associated to filters of length 8 as well as for 4-band compactly supported wavelets (here designated as AC) associated to 16-tap filters [START_REF] Alkin | Design of efficient M -band coders with linear-phase and perfect-reconstruction properties[END_REF].

Table VI shows the estimations of the cross-correlations obtained in the dyadic case, while the results in the M -band case with M ≥ 3 are listed in Tables VII andVIII. By comparing these results with the ones in Tables V, III and IV, a good agreement is observed between the theoretical values and the estimated ones for Shannon, Meyer and cubic spline wavelets. For less regular wavelets such as Franklin or Haar wavelets, the agreement remains quite good at coarse resolution (j = 3) but, at fine resolution (j = 1), it appears that the correlations are stronger in practice than predicted by the theory. The fact that we use a discrete decomposition instead of the classical analog wavelet framework may account for these differences. Indeed, we use the implementation of the M -band dual-tree decomposition described in [START_REF]Image analysis using a dual-tree M -band wavelet transform[END_REF], which requires some digital prefilters. The selectivity of these filters is inherited from the frequency selectivity of the scaling function. As a side effect, the noise is colored by these prefilters. Some comments can also be made concerning symlets 8 and 4-band AC wavelets. We see that the symlets behave very similarly to Franklin wavelets whereas AC wavelets provide intermediate correlation magnitudes between the M -band Meyer and Hadamard cases.

γ ψ M -1 ,ψ H M -1 Wavelets \ ℓ 1 2 3
Meyer 3-band ǫ = 1/4 -0.58918 -6.0378×10

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

C. Inter-band cross-correlations

Although the cross-correlations between primal/dual basis functions corresponding to different subbands have not been much investigated in the previous sections, we provide in this part some numerical evaluations for them.

More precisely, we are interested in studying (γ ψm,ψ H m ′ (ℓ)) ℓ∈Z with m = m ′ , which represents the inter-band crosscorrelations. We are able to compute them thanks to ( 16) and [START_REF] Vanucci | Some findings on the covariance structure of wavelet coefficients: theory and models in a Bayesian perspective[END_REF]. Numerical results are given in Table IX. Some symmetry properties can be observed, which can be deduced from ( 16), [START_REF] Vanucci | Some findings on the covariance structure of wavelet coefficients: theory and models in a Bayesian perspective[END_REF] and the specific form of the considered wavelet functions. Most interestingly, it can be noticed that the inter-band cross-correlations often have a significantly smaller amplitude than the corresponding intra-band cross-correlations. As expected, the more frequency-selective the decomposition filters, the more negligible the values of the inter-band cross-correlations.

D. Two-dimensional experiment

We aim here at comparing the obtained theoretical expressions of the two-dimensional cross-covariances with Monte Carlo evaluations of these second-order statistics. We consider a two-dimensional 3-band Meyer dual-tree wavelet decomposition of a white standard Gaussian field of size 756 × 756. The Monte Carlo study is carried out over 10000 realizations. The decomposition is performed over J = 2 resolution levels and the results are provided at the coarsest resolution. The covariance fields are depicted in Fig. 2 as well as the ones derived from (31), ( 52)- [START_REF] Wornell | Wavelet-based representations for a class of self-similar signals with applications to fractal modulation[END_REF].

γ ψ 0 ,ψ H 0 γ ψ 1 ,ψ H 1 Wavelets j \ ℓ 0 1 2 3 0 1 2
For more readibility, a dashed separation line between the subbands has been added (for a 3-band decomposition,

9 covariance fields (Γ nj,m,n H j,m [ℓ]
) ℓ∈Z 2 have to be computed when m ∈ {0, 1, 2} 2 ). We compute these fields for ℓ ∈ {0, 1, 2, 3} 2 , thus resulting in 16 covariance values for each subband. Succinctly, each small gray-scaled square represents the intensity of the cross-covariance in a given subband m at spatial position ℓ. Comparing theoretical results with numerical ones (left and right sides of Fig. 2, respectively), it can be noticed that they are quite similar.

In addition, we observe that, due to the separability of the covariance fields and ( 13), for all m = (m 1 , m 2 ) and

ℓ = (ℓ 1 , ℓ 2 ), (Γ nj,m,n H j,m [ℓ]
) ℓ∈Z 2 vanishes when either (m 1 = 0 and ℓ 1 = 0) or (m 2 = 0 and ℓ 2 = 0).

VII. CONCLUSION

In this paper, we have investigated the covariance properties of the M -band dual-tree wavelet coefficients of widesense stationary 1D and 2D random processes. We have stated a number of results helping to better understand the structure of the correlations introduced by this frame decomposition. These results may be useful in the design of efficient denoising rules using dual-tree wavelet decompositions, when the noise is additive and stationary. In particular, if a pointwise estimator is applied to the pair of primal/dual coefficients at the same location and in the same subband, we have seen that the related components of the noise are uncorrelated. On the contrary, if a blockbased estimator is used to take advantage of some spatial neighborhood of the primal and dual coefficients around some given position in a subband, noise correlations generally must be taken into account. Recently, this fact has been exploited in the design of an efficient image denoising method using Stein's principle, yielding state-of-the-art performance for multichannel image denoising [START_REF] Chaux | A nonlinear Stein based estimator for multichannel image denoising[END_REF], [START_REF] Chaux | A new estimator for image denoising using a 2D dual-tree M -band wavelet decomposition[END_REF]. In future work, it would be interesting to extend our analysis to other classes of random processes. In particular, a similar study could be undertaken for self-similar processes [START_REF] Abry | Wavelets, spectrum analysis and 1/f processes[END_REF], [START_REF] Wornell | Wavelet-based representations for a class of self-similar signals with applications to fractal modulation[END_REF] and processes with stationary increments [START_REF] Averkamp | Some distributional properties of the continuous wavelet transform of random processes[END_REF], [START_REF] Krim | Multiresolution analysis of a class of nonstationary processes[END_REF].

γ ψ 0 ,ψ H 0 γ ψ 1 ,ψ H 1 Wavelets j \ ℓ 0 1 2 3 0 1 2
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
Finally, we would like to note that the expressions of the cross-correlations between the primal and dual wavelets which have been derived in this paper may be of interest for other problems. Indeed, let denote the dual-tree wavelet decomposition where D (resp. D H ) is the primal (resp. dual) wavelet decomposition.

T =   D D H   γ ψ M -1 ,ψ H M -1 Wavelet j \ ℓ 0 1 2
The studied cross-correlations then characterize the "off-diagonal" terms of the operator

T T * =   I D(D H ) * D H D * I   ,
where A * denotes the adjoint of a bounded linear operator A. The operator T T * is encountered in the solution of some inverse problems.

APPENDIX I PROOF OF PROPOSITION 1

The M -band wavelet coefficients of the noise are given by

∀m ∈ N M , ∀k ∈ Z, n j,m [k] = ∞ -∞ n(x) 1 M j/2 ψ m ( x M j -k) dx n H j,m [k] = ∞ -∞ n(x) 1 M j/2 ψ H m ( x M j -k) dx.
M -BAND MEYER WAVELETS γ ψm,ψ H m ′ IS ZERO WHEN |m -m ′ | > 1.
For all (m, m ′ ) ∈ N 2 M and (k, k ′ ) ∈ Z 2 , we have then

E{n j,m [k]n j,m ′ [k ′ ]} = ∞ -∞ ∞ -∞ E{n(x)n(x ′ )} 1 M j/2 ψ m ( x M j -k) 1 M j/2 ψ m ′ ( x ′ M j -k ′ )dx dx ′ .
After the variable change τ = x -x ′ , using the definition of the autocovariance of the noise in [START_REF] Casazza | The art of frame theory[END_REF], we find that

E{n j,m [k]n j,m ′ [k ′ ]} = ∞ -∞ Γ n (τ ) ∞ -∞ 1 M j/2 ψ m ( x M j -k) 1 M j/2 ψ m ′ ( x -τ M j -k ′ )dx dτ which readily yields E{n j,m [k]n j,m ′ [k ′ ]} = ∞ -∞ Γ n (τ )γ ψm,ψ m ′ ( τ M j + k ′ -k) dτ.
Note that, in the above derivations, permutations of the integral symbols/expectation have been performed. For these operations to be valid, some technical conditions are required. For example, Fubini's theorem [57, p. 164] can be invoked provided that

∞ -∞ Γ |n| (τ )γ |ψm|,|ψ m ′ | ( τ M j + k ′ -k) dτ < ∞,
where Γ |n| is the autocovariance of |n|.

Relations [START_REF] Pesquet | Time-invariant orthogonal wavelet representations[END_REF] and ( 8) follow from similar arguments. APPENDIX II PROOF OF PROPOSITION 2 For all (m, m

′ ) ∈ N 2 M , ∀τ ∈ R, 1 2π ∞ -∞ ψ H m (ω) ψ H m ′ (ω)) * e ıωτ dω = γ ψ H m ,ψ H m ′ (τ ).
Since the Fourier transform defines an isometry on L 2 (R), it can be deduced from (80 3 According to (3) and ( 4), when m = m ′ = 0 or mm ′ = 0, the latter function is equal to

) that γ ψ H m ,ψ H m ′ is in L 2 (R) and its Fourier transform is ω → ψ H m (ω) ψ H m ′ (ω)) * .
ω → ψ m (ω) ψ m ′ (ω)) * , thus showing that γ ψ H m ,ψ H m ′ = γ ψm,ψ m ′ .
The equality of the covariance sequences defined by ( 6) and ( 7) straightforwardly follows.

When mm ′ = 0, the Fourier transform of γ ψm,ψ

H m ′ is equal to ω → ı sign(ω) ψ m (ω) ψ * m ′ (ω) whose conjuguate is the Fourier transform of -γ ψ m ′ ,ψ H
m . This proves [START_REF] Pennec | Sparse geometric image representations with bandelets[END_REF], which combined with (8) leads to

∀ℓ ∈ Z, Γ nj,m,n H j,m ′ [ℓ] = - ∞ -∞ Γ n (x)γ ψ ′ m ,ψ H m - x M j + ℓ dx.
After a variable change and using the fact that Γ n is an even function, we obtain [START_REF] Candès | Fast discrete curvelet transforms[END_REF].

Consider now the Fourier transform ω → ψ 0 (ω)( ψ H 0 (ω)) * of γ ψ0,ψ H 0 . For all ω ≥ 0, there exists k ∈ N such as ω ∈ [2kπ, 2(k + 1)π) and, from (4), we get

ψ 0 (ω)( ψ H 0 (ω)) * = (-1) k e ı(d+ 1 2 )ω | ψ 0 (ω)| 2 = e ı(2d+1)ω ψ 0 (-ω)( ψ H 0 (-ω)) * .
For symmetry reasons, the equality between the first and last terms extends to all ω ∈ R. Coming back to the time domain, we find

∀τ ∈ R, γ ψ0,ψ H 0 (τ ) = γ ψ0,ψ H 0 (-τ -2d -1).
This shows the symmetry of γ ψ0,ψ H 0 w.r.t. -d -1/2. Eq. ( 8) then yields

∀ℓ ∈ Z, Γ nj,0,n H j,0 [ℓ] = ∞ -∞ Γ n (x)γ ψ0,ψ H 0 - x M j + ℓ -2d -1 dx = ∞ -∞ Γ n (x)γ ψ0,ψ H 0 x M j + ℓ -2d -1 dx = Γ nj,0,n H j,0 [-ℓ + 2d + 1].

APPENDIX III WHITE NOISE CASE

Recall that a white noise is not a process with finite variance, but a generalized random process [START_REF] Guelfand | Distributions Vols[END_REF], [START_REF] Yaglom | Correlation theory of stationary and related functions I. Basic results[END_REF]. As such, some caution must be taken in the application of ( 6)-( 8). More precisely, if n is a white noise, its autocovariance can be viewed as the limit as ǫ > 0 tends to 0 of

Γ n ǫ (τ ) = σ 2 √ 2πǫ exp(- τ 2 2ǫ 2 ), τ ∈ R.
Formula (8) can then be used, yielding for all (m, m ′ ) ∈ N 2 M and (j, ℓ) ∈ Z 2 ,

Γ n ǫ j,m ,n ǫH j,m ′ [ℓ] = σ 2 ∞ -∞ 1 √ 2π exp(- x 2 2 ) γ ψm,ψ H m ′ ǫx M j -ℓ dx.
Since ψ m and ψ H m ′ are in L 2 (R), γ ψm,ψ H m ′ is a bounded continuous function. By applying Lebesgue dominated convergence theorem, we deduce that

Γ nj,m,n H j,m ′ [ℓ] = lim ǫ→0 Γ n ǫ j,m ,n ǫH j,m ′ [ℓ] = σ 2 ∞ -∞ 1 √ 2π exp(- x 2 2 ) lim ǫ→0 γ ψm,ψ H m ′ ǫx M j -ℓ dx = σ 2 γ ψm,ψ H m ′ (-ℓ) ∞ -∞ 1 √ 2π exp(- x 2
2 )dx which leads to [START_REF] Figueras | Low rate and flexible image coding with redundant representations[END_REF]. Equations ( 14) are similarly obtained by further noticing that, due to the orthonormality

property, γ ψm,ψ m ′ (-ℓ) = γ ψ H m ,ψ H m ′ (-ℓ) = δ m-m ′ δ ℓ .
APPENDIX IV PROOF OF PROPOSITION 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 ′ ]} .
Using [START_REF] Olhede | Analytic wavelet thresholding[END_REF] and the evenness of Γ nj,m,n H j,m , one can easily deduce [START_REF] Selesnick | Hilbert transform pairs of wavelet bases[END_REF]. Concerning [START_REF] Selesnick | The dual-tree complex wavelet transform[END_REF], we proceed in the same way, taking into account 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, noting 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, invoking the same arguments, we see that w j,m [k] and w H j,m [k ′ ] are uncorrelated random variables.

APPENDIX V PROOF OF PROPOSITION 4

Since

ψ m ∈ L 2 (R), we have ∀τ ∈ R, γ ψm,ψm (τ ) = 1 2π ∞ -∞ | ψ m (ω)| 2 e ıωτ dω. Furthermore, | ψ m | 2 is 2N m + 1 times continuously differentiable and for all q ∈ {0, . . . , 2N m + 1}, (| ψ m | 2 ) (q) ∈ L 1 (R). It can be deduced [60][p. 158-159] that ∀τ ∈ R, (-ıτ ) 2Nm+1 γ ψm,ψm (τ ) = 1 2π ∞ -∞ (| ψ m | 2 ) (2Nm+1) (ω) e ıωτ dω which leads to ∀τ ∈ R, |τ | 2Nm+1 |γ ψm,ψm (τ )| ≤ 1 2π ∞ -∞ (| ψ m | 2 ) (2Nm+1) (ω) dω. (80)
Let us now consider the cross-correlation functions γ ψm,ψ H m with m = 0. Similarly, when m = 0, we have

∀τ ∈ R, γ ψm,ψ H m (τ ) = 1 2π ∞ -∞ α(ω)| ψ m (ω)| 2 e ıωτ dω, (81)
where

α(ω) = ı sign(ω). The function ω → α(ω)| ψ m (ω)| 2 is 2N m + 1 times continuously differentiable on R * ,
where its derivative of order q ∈ {0, . . . , 2N m + 1} is

(α| ψ m | 2 ) (q) = α (| ψ m | 2 ) (q) . (82)
Due to the fact that | ψ m (ω)| 2 = O(ω 2Nm ) as ω → 0, we have for all q ∈ {0, . . . , 2N m -1}, (| ψ m | 2 ) (q) (0) = 0.

From (82), we deduce that the function (α| ψ m | 2 ) (q) admits limits on the left side and on the right side of 0, which are both equal to 0. This allows to conclude that α| ψ m | 2 is 2N m -1 times continuously differentiable on R, its 2N m -1 first derivatives vanishing at 0. Besides, (α| ψ m | 2 ) (2Nm-1) is continuously differentiable on (-∞, 0] and on [0, ∞) ((α| ψ m | 2 ) (2Nm) may be discontinuous at 0). Using the same arguments as for γ ψm,ψm , this allows us to claim that

∀τ ∈ R, (-ıτ ) 2Nm γ ψm,ψ H m (τ ) = 1 2π ∞ -∞ α(ω)(| ψ m | 2 ) (2Nm) (ω)e ıωτ dω = - 1 π ∞ 0 (| ψ m | 2 ) (2Nm) (ω) sin(ωτ ) dω. (83)
We can note that

lim ω→∞ (| ψ m | 2 ) (2Nm) (ω) ∈ R as it is equal to (| ψ m | 2 ) (2Nm) + (0) + ∞ 0 (| ψ m | 2 ) (2Nm+1) (ν) dν where (| ψ m | 2 ) (2Nm) + (0) denotes the right-hand side derivative of order 2N m of | ψ m | 2 at 0. Since (| ψ m | 2 ) (2Nm) ∈ L 1 ([0, ∞))
, the previous limit is necessarily zero. Using this fact and integrating by part in (83), we find that, for

all τ ∈ R, τ ∞ 0 (| ψ m | 2 ) (2Nm) (ω) sin(ωτ ) dω = (| ψ m | 2 ) (2Nm) + (0) + ∞ 0 (| ψ m | 2 ) (2Nm+1) (ω) cos(ωτ ) dω.
Combining this expression with (83), we deduce that

∀τ ∈ R, |τ | 2Nm+1 |γ ψm,ψ H m (τ )| ≤ 1 π ∞ 0 |(| ψ m | 2 ) (2Nm+1) (ω)| dω + |(| ψ m | 2 ) (2Nm) + (0)| . (84)
Let us now study the case when m = 0. Eq. ( 81) still holds, but as shown by ( 4), α takes a more complicated form:

∀k ∈ Z, ∀ω ∈ [2kπ, 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.

So, the function α as well as its derivatives of any order now exhibit discontinuities at 2kπ where k ∈ Z * . However, from (1) and the low-pass condition ψ 0 (0) = 1, we have, for all m = 0,

H m (ω) = O(ω Nm ), as ω → 0.
As a consequence of the para-unitary condition (2), we get

M -1 m=0 |H m (ω)| 2 = M and M -1 p=0 |H 0 (ω + p 2π M )| 2 = M which allows to deduce that ∀p ∈ N ⋆ M , H 0 (ω + p 2π M ) = O(ω N0 ).
From (1), it can be concluded that

∀k ∈ Z * , ψ 0 (ω + 2kπ) = O(ω N0 ), as ω → 0. (85)
The derivatives of order q ∈ {0, . . . , 2N 0 + 1} of α| ψ 0 | 2 over R \ {2kπ, k ∈ Z * } are given by

(α| ψ 0 | 2 ) (q) = q ℓ=0 q ℓ (α) (ℓ) (| ψ 0 | 2 ) (q-ℓ) ,
As the left-hand side of (91) corresponds to an even function of ℓ, without loss of generality, it can be assumed that ℓ ≥ 0. We can decompose the above integral as

∞ -∞ e -α|x| 1 + |M -j x -ℓ| 2Nm+1 dx = ∞ 0 e -αx 1 + (M -j x + ℓ) 2Nm+1 dx + ∞ 0 e -αx 1 + |M -j x -ℓ| 2Nm+1 dx .
The first integral in the right-hand side can be upper bounded as follows

∞ 0 e -αx 1 + (M -j x + ℓ) 2Nm+1 dx ≤ (1 + ℓ 2Nm+1 ) -1 ∞ 0 e -αx dx = α -1 (1 + ℓ 2Nm+1 ) -1 .
Let ǫ ∈ (0, 1) be given. The second integral can be decomposed as

∞ 0 e -αx 1 + |M -j x -ℓ| 2Nm+1 dx = ǫM j ℓ 0 e -αx 1 + (ℓ -M -j x) 2Nm+1 dx + ∞ ǫM j ℓ e -αx 1 + |M -j x -ℓ| 2Nm+1 dx.
Furthermore, we have

ǫM j ℓ 0 e -αx 1 + (ℓ -M -j x) 2Nm+1 dx ≤ (1 + (1 -ǫ) 2Nm+1 ℓ 2Nm+1 ) -1 ǫM j ℓ 0 e -αx dx ≤ α -1 (1 -ǫ) -2Nm-1 (1 + ℓ 2Nm+1 ) -1 (92) ∞ ǫM j ℓ e -αx 1 + |M -j x -ℓ| 2Nm+1 dx ≤ ∞ ǫM j ℓ e -αx dx = α -1 e -αǫM j ℓ .
From the above inequalities, we obtain As lim ℓ→∞ (1 + ℓ 2Nm+1 )e -αǫM j ℓ = 0, it readily follows that there exists C ∈ R + such that (35) holds.

The left-hand side of (91) being also an upper bound for |Γ nj,m,nj,m [ℓ]|, ℓ = 0, (34) is proved at the same time.

APPENDIX VII PROOF OF PROPOSITION 7 Let us prove [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF], the proof of (41) being quite similar. We first note that ψ m ( ψ H m ′ ) * and therefore γ ψm,ψ H m ′ belong to L 2 (R) (see footnote 3). Applying Parseval's equality to [START_REF] Bölcskei | Frame-theoretic analysis of oversampled filter banks[END_REF], we obtain for all ℓ ∈ Z, When τ is an integer, this expression further simplifies in [START_REF] Abry | Wavelets, spectrum analysis and 1/f processes[END_REF].

Γ nj,m,n H j,m ′ [ℓ] = 1 2π ∞ -∞ Γ n (ω) M j ψ * m (M j ω) ψ H m ′ (M j ω)e ıM j ℓω dω = 1 2π ∞ -∞ Γ n ω M j ψ * m (ω) ψ H m ′ (ω)e ıℓω dω.
Finally, when m = M -1, we have, for all τ ∈ R * , Combining this result with (97) leads to (66).

γ ψM-1,ψ H M -1 (τ ) = - 1 π π(M -1+ǫ) π(M -1-ǫ) W 2 M -1 + ǫ 2ǫ - ω 2πǫ
On the other hand, according to (65) and ( 19), we have

∀τ ∈ R, γ ψ1,ψ H 1 (τ ) = - 1 π ∞ 0 sinc 2 ( ω 4 ) sin 2 ( ω 4
) sin(ωτ ) dω .

In [61, p.459], an expression of

P i=1 A ǫi ( 2 P

 2 -i ω). Moreover, by assumption A 1 (ω) = O(ω N ) as ω → 0, whereas A 0 (0) = √ 2 and | ψ 0 (0)| = 1. This shows that, when m = 0, ψ m (ω) = O(ω N ( È r i=1 ǫi) ) as ω → 0. From (33), we deduce the upper bound in (63). Furthermore, by applying Prop. 4 when M = 2, we have then N 0 = N 1 = N and (61) is obtained.

2 Fig. 2 .

 22 Fig. 2. 2D cross-correlations using 3-band Meyer wavelets. Theoretical results (left); experimental results (right).

  ∀ℓ ∈ N * , |Γ nj,m,n H j,m [ℓ]| ≤ A(1 + C)α -1 (1 + (1 -ǫ) -2Nm-1 )(1 + ℓ 2Nm+1 ) -1 + e -αǫM j ℓ .

As Γ n ∈ L 1 (ǫ 1 - 1 W 2 1 - 1 W 2

 1112112 R), the spectrum density Γ n is a bounded continuous function. According to Lebesgue dominatedconvergence theorem, * m (ω) ψ H m ′ (ω)e ıℓω dω = Γ n (0) 2π ∞ -∞ ψ * m (ω) ψ H m ′ (ω)e ıℓω dω = Γ n (0)γ ψm,ψ H m ′ (-ℓ).APPENDIX VIII CROSS-CORRELATIONS FOR MEYER WAVELETS Substituting (46) in (20), we obtain, for all τ ∈ R, ǫθ -1) d + 1 2 + τ dθ. (94) This allows us to rewrite (93) as γ ψ0,ψ H 0 (τ ) = sinc π(d (50), we have for all m ∈ {1, . . . , M -2} and τ ∈ R * , γ ψm,ψ H m (τ ) =m + 1 -ǫ)τ -cos π(m + ǫ)τ πτ + 1 + θ 2 sin π(ǫθ -m)τ dθ -ǫ 1 + θ 2 sin π(ǫθ + m + 1)τ dθ. By proceeding similarly to (93)-(94), we find γ ψm,ψ H m (τ ) = cos(π(m + 1)τ ) -cos(πmτ ) 1 πτ -I ǫ (τ ) .

ǫ 1 - 1 W 2 1 - 1 W 2 0 | 0 |A 0 0 | 0 |ν 2

 112112000002 πM (1 -ǫ)τ -cos π(M -1 + ǫ)τ πτ + 1 + θ 2 sin π(ǫθ -M + 1)τ dθ -ǫM 1 + θ 2 sin πM (ǫθ + 1)τ dθ = cos πM τ -cos π(M -1)τ πτ + cos π(M -1)τ I ǫ (τ ) -cos(πM τ )I M ǫ (τ ).This yields[START_REF] Wornell | Wavelet-based representations for a class of self-similar signals with applications to fractal modulation[END_REF].APPENDIX IXPROOF OFPROPOSITION 8 Let m ∈ N * . Given[START_REF] Averkamp | A note on the discrete wavelet transform of second-order processes[END_REF], (57) leads to-πγ ψ2m,ψ H 2m (τ ) = ∞ ψ 2m (ω)| 2 sin(ωτ ) dω = ∞ (ω)| 2 | ψ m (ω)| 2 sin(2ωτ ) dω(96)Furthermore, we have|A 0 (ω)| 2 = k γ a0 [k] exp(-ıkω) = γ a0 [0] + 2 ∞ k=1 γ a0 [k] cos(kω).Combining this equation with (96) and using classical trigonometric equalities, we obtain-πγ ψ2m,ψ H 2m (τ ) = γ a0 [0] ∞ ψ m (ω)| 2 sin(2ωτ )dω + ∞ k=1 γ a0 [k] ∞ | ψ m (ω)| 2 sin (2τ -k)ω dω + ∞ ψ m (ω)| 2 sin (2τ + k)ω dωwhich, again invoking Relation[START_REF] Averkamp | A note on the discrete wavelet transform of second-order processes[END_REF], yields[START_REF] Yaglom | Correlation theory of stationary and related functions I. Basic results[END_REF]. Eq. (60) can be proved similarly starting from[START_REF] Guelfand | Distributions Vols[END_REF].APPENDIX X CROSS-CORRELATIONS FOR HAAR WAVELETKnowing the expression of the Fourier transform of the Haar scaling function in (64) and using the crosscorrelation formula[START_REF] Cambanis | On the continuous wavelet transform of second-order random processes[END_REF], we obtain: cos ν (1 + 2τ + 2d) dν.(97)By integration by part, we find: for all (α, β, η) ∈ R 3 ,

∞ 0 sin 2 - 2 π ∞ 0 sin 8 4 x

 02084 (αx) sin 2 (βx) sin (2ηx)dx x 2 with (α, β, η) ∈ (R * + ) 3 is given. Using this relation yields (67) when τ > 0. The general expression for τ ∈ R follows from the oddness of γ ψ1,ψ H 1 .APPENDIX XI CROSS-CORRELATION FOR THE FRANKLIN WAVELETWe have, for all τ ∈ R,γ χ,χ H (τ ) = -1 π ∞ 0 | χ(ω)| 2 sin(ωτ )dω = (ω) ω 4 sin(2ωτ )dω.After two successive integrations by part, we obtainγ χ,χ H (τ ) = -(ω) sin 2(τ + 2)ω + sin 2(τ -2)ω -4 sin 2(τ + 1)ω -4 sin 2(τ -1)ω + 6 sin(2τ ω) sin 7 (ω) cos(ω) cos(2ωτ ) = 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 expressions in (98) yields3πγ 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. 459])∀x ∈ R, J(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 + 2τ + 1, Q 2 (τ ) = 1 8 (τ + 4) 2 .

  ), (69) and (70). The expressions given in TableIare in perfect agreement with the decay rates predicted by Prop. 9.

	m	1	2	3	4	5	6	7	8	9	10	11	12
	πγ ψm,ψ H m	(τ )

3

) + O(τ -5 ) when |τ | > 2, which corresponds to a faster asymptotic decay than with Shannon wavelets. The asymptotic behaviour of γ ψm,ψ H m (τ ), m > 2, can also be deduced from (67

TABLE I ASYMPTOTIC

 I FORM OF γ ψm,ψ H m (τ ) AS |τ | → ∞ FOR WALSH-HADAMARD WAVELETS.

TABLE II THEORETICAL

 II

				γ ψ 0 ,ψ H 0			γ ψ 1 ,ψ H 1
	Wavelets \ 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

CROSS-CORRELATION VALUES IN THE DYADIC CASE (d = 0).

TABLE III THEORETICAL

 III VALUES FOR THE FIRST TWO CROSS-CORRELATION SEQUENCES IN THE M -BAND MEYER CASE (d = 0).

TABLE V THEORETICAL CROSS

 VCROSS

-CORRELATION VALUES IN THE WALSH-HADAMARD CASE.

TABLE VI CROSS

 VI

-CORRELATION ESTIMATES IN THE DYADIC CASE (d = 0).

TABLE VII CROSS

 VII

-CORRELATION ESTIMATES IN THE M -BAND CASE (d = 0).

TABLE VIII ESTIMATION

 VIII OF THE LAST CROSS-CORRELATION SEQUENCE FOR M -BAND SHANNON AND MEYER WAVELETS.

TABLE IX INTER

 IX -BAND CROSS-CORRELATION VALUES FOR SOME WAVELET FAMILIES. WE RECALL THAT PROPERTY (12) HOLDS AND THAT, FOR

June 28, 2007 DRAFT

By convention, the derivative of order 0 of a function is the function itself.

The characterization of the sum of digits of integers remains an open problem in number theory[START_REF] Drmota | The sum of digits function of squares[END_REF],[START_REF] Allouche | Summation of series defined by counting blocks of digits[END_REF].June 28, 2007 DRAFT

As {ψ H m ′ (t -k), k ∈ Z} is an orthonormal family of L 2 (R), we have | ψ H m ′ (ω)| ≤ 1 and ψ H m ψ H m ′ ) * ∈ L 2 (R).

The series in (88) is convergent since all the other terms in (87) are finite.June 28, 2007 DRAFT

where ∀k ∈ Z, ∀ω ∈ (2kπ, 2(k + 1)π),

2 )ω otherwise.

We deduce that, for all q ∈ {0, . . . , 2N 0 + 1}, (α| ψ 0 | 2 ) (q) ∈ L 1 (R). Furthermore, combining (85) with (86) allows us to show that, for all q ∈ {0, . . . , 2N 0 -1}, the derivative of order q of α| ψ 0 | 2 at 2kπ, k ∈ Z * , is defined and equal to 0. Consequently,

Similarly to the case m = 0, this leads to

By integration by part, we deduce that

where 4 We conclude that

In summary, we have proved that [START_REF]Image analysis using a dual-tree M -band wavelet transform[END_REF] and [START_REF] Rioul | Fast algorithms for discrete and continuous wavelet transforms[END_REF] hold, the constant C being chosen equal to the maximum value of the left-hand side terms in the inequalities (80), ( 84) and (89).

APPENDIX VI PROOF OF PROPOSITION 5

Let m ∈ N M . Since ψ m is a unit norm function of L 2 (R), the function γ ψm,ψ H m is upper bounded by 1. As γ ψm,ψ H m further satisfies [START_REF] Rioul | Fast algorithms for discrete and continuous wavelet transforms[END_REF], it can be deduced that

The same upper bound holds for γ ψm,ψm .

For a white noise, the property then appears as a straightforward consequence of the latter inequality and Eqs. ( 14)

and [START_REF] Figueras | Low rate and flexible image coding with redundant representations[END_REF].

Let us next turn our attention to processes with exponentially decaying covariance sequences. From ( 8), ( 34) and

(90), we deduce that

Simple algebra allows us to prove that (99) is equivalent to (75).

On the other hand, | A 1 (ω)| 2 can be viewed as the frequency response of a non causal stable digital filter whose transfer function is

We next expand P A1 (z) in Laurent series on the holomorphy domain containing the unit circle, that is

We thus deduce from the partial fraction decomposition of P A1 (z) that

By identifiying the latter expression with ∞ k=-∞ γ ã1 [k]z -k , (76) is obtained.


Rahul Vanam
email: rahulv@ee.washington.edu

Eve A Riskin
email: riskin@ee.washington.edu

Richard E Ladner
email: ladner@cs.washington.edu

Francis M Ciaramello

Sheila S Hemami
email: hemami@ece.cornell.edu

JOINT RATE-INTELLIGIBILITY-COMPLEXITY OPTIMIZATION OF AN H.264 VIDEO ENCODER FOR AMERICAN SIGN LANGUAGE

This paper presents an H.264 standard-compliant video encoder optimized for region-of-interest (ROI) based coding tuned to American Sign Language (ASL) videos. Encoding modes are developed which allow the encoder to allocate both rate and computational resources differently across the ROIs. An objective measure of intelligibility is included in an encoder parameter optimization by modifying a fast offline distortion-complexity optimization algorithm, resulting in parameter selections that demonstrate excellent rateintelligibility-complexity performance. These parameters can be stored in a look-up table for use by an online algorithm which selects parameters based on available computational resources. The resulting parameter selections improve the encoder speed by up to 21.2% with a small decrease in intelligibility over the x264 default parameter settings.

INTRODUCTION

Cell phone technology has become ubiquitous due to its convenience and mobility. Current video cell phones are equipped with a camera and codecs, and have the potential for use in real-time mobile videoconferencing. However, the availability of high bandwidth 3G networks is limited to few cities in the United States, ultimately requiring such a system to operate at very low bandwidths. Furthermore, real-time capture, encoding, and transmission of digital video is difficult on devices with limited computational resources, such as mobile phones. This motivates the need for low complexity video compression algorithms which can provide video that is useful to the end user. In the past, the perceptual quality of videoconferencing has been improved by reducing distortions in the user's face [START_REF] Chai | Face segmentation using skin color map in videophone applications[END_REF][START_REF] Daly | Face-based visually-optimized image sequence coding[END_REF]. Region-of-interest (ROI) based video compression can be extended to American Sign Language (ASL) video. For ASL video, an observer is tracking the signer's face and hands and evaluating distortions only in This work is supported by the National Science Foundation under grant numbers CCF-0514353, CCF-0514357 and IIS-0811884.

those regions. This is supported by both the linguistic structure of sign language [START_REF] Ciaramello | Quantifying the effect of disruptions to temporal coherence on the intelligibility of compressed american sign language video[END_REF] (e.g. how information is conveyed) and by eye-tracking experiments [START_REF] Agrafiotis | A perceptually optimised video coding system for sign language communication at low bit rates[END_REF]. Because of this unique structure, several specialized algorithms have been proposed for encoding sign language video [START_REF] Agrafiotis | A perceptually optimised video coding system for sign language communication at low bit rates[END_REF][START_REF] Nakazono | Digital encoding applied to sign language video[END_REF][START_REF] Ciaramello | Complexity constrained rate-distortion optimization of sign language video using an objective intelligibility metric[END_REF].

In the author's previous work, an ASL optimized video encoder was developed using an objective measure of intelligibility incorporated into an H.264 rate-distortion (R-D) optimization algorithm [START_REF] Ciaramello | Complexity constrained rate-distortion optimization of sign language video using an objective intelligibility metric[END_REF]. A performance bound for the system is obtained using the Viterbi algorithm to search over all possible quantization parameters and encoding modes. For fixed levels of intelligibility, bitrate can be reduced by as much as 60% over an R-D optimization algorithm which measures distortion as MSE. The goal of this work is to achieve as much of this gain as possible while maintaining a computational complexity appropriate for low mobile devices with low processing power.

Traditionally, ROI-optimized encoders achieve bitrate savings by allocating rate only to the most relevant regions. In this work, we extend this concept and also allocate more computational resources to these important regions. Two additional encoding options are presented which allow variations in encoding complexity based on the relative importance of each macroblock. A fast offline algorithm is then used to search the space of possible encoding parameters available in H.264, including the proposed ROI-tuned options, to find parameters that give us improvement in encoding speed with only small decreases in intelligibility. The results demonstrate that appropriate parameter selections improve the encoder speed by up to 21.2% with a small decrease in intelligibility when compared to the x264 default parameter settings.

SIGN LANGUAGE INTELLIGIBILITY-OPTIMIZED VIDEO ENCODER

The ASL optimized encoder is implemented within x264 [7], an open-source H.264 encoder. The rate-distortion optimization uses an objective intelligibility measure, which is a function of the distortion in linguistically relevant regions and ac-Western NY Image Processing Workshop, Rochester, NY, September 2009 curately predicts an observer's subjective intelligibility rating [START_REF] Ciaramello | Quantifying the effect of disruptions to temporal coherence on the intelligibility of compressed american sign language video[END_REF]. Each frame of the input sequence is segmented into the signer's face, hands, torso, and background, using color-based skin detection and morphological processing. This segmentation operates in real-time on a mobile device [START_REF] Ciaramello | Real-time face and hand detection for videoconferencing on a mobile device[END_REF]. Given the region segmentation for a particular frame, the distortions affecting intelligibility are computed as the weighted combination of the mean squared error (MSE) in the face, hands, and torso of the signer:

D = W F M SE F + W H M SE H + W T M SE T , (1)
where W F = 1.6, W H = 0.5, and W H = 0.1. Because of the varying weights, a particular MSE in the signer's face will result in a higher total distortion than the same MSE in the signer's torso.

The ASL-tuned distortion measure in ( 1) is incorporated into a rate-distortion (R-D) optimization procedure similar to that of [START_REF] Ortega | Forward-adaptive quantization with optimal overhead cost for image and video coding with applications to mpeg video coders[END_REF] and applied to a collection of ASL videos. For a given Lagrangian λ, the parameter p that includes motion vector, mode and quantization step size (QP) is chosen such that it minimizes the joint R-D cost J(X, p) = D(X, p) + λR(X, p), where X is a particular macroblock. A consequence of using the distortion measure in 1 is that more rate is inherently allocated to the important regions (i.e., face, hands, and torso). The work presented in [START_REF] Ciaramello | Complexity constrained rate-distortion optimization of sign language video using an objective intelligibility metric[END_REF] identified a functional relationship between λ and the resulting optimal QPs. Ultimately, this allows for fast encoding by using a single parameter λ to quickly select a QP value for each of the region types. The motion vector and mode for each macroblock are still selected according to the minimum R-D cost. Rate control is performed at the frame-level by adjusting the Lagrangian parameter λ, according to λ(n + 1) = λ(n) -R target /R actual , where R target and R actual are the target bits and actual bits for frame n.

ROI-BASED COMPLEXITY ALLOCATION ENCODER OPTIONS

Four variable encoding parameters are varied to achieve different points in rate-distortion-complexity: sub-pixel motion estimation (subme); reference frames (ref); partition size (part); and entropy coding and quantization (trellis).

The subme has 7 options corresponding to the number of iterations for half-pel and quarter-pel motion estimation. A maximum of 16 reference frames can be specified using ref.

Ten different part options specify the partition size from 4 × 4 and above for intra (I), predictive (P) and bi-predictive (B) macroblocks [START_REF] Vanam | H.264/MPEG-4 AVC encoder parameter selection algorithms for complexity distortion tradeoff[END_REF]. The trellis parameter has four options that include uniform quantization with and without context adaptive arithmetic coding (CABAC) (options 1 and 0); and two schemes that use CABAC and Djikstra's algorithm for finding the quantization for a block of DCT coefficient such that the overall R-D cost is reduced (options 3 and 4).

Two additional encoding parameters are added to the x264 encoder that allow the encoding complexity to vary on a per-block basis, depending on the type of region being encoded (e.g. face, hand, or background). The first parameter (backgrd-part) restricts the partition search performed by the encoder in background blocks. In H.264, as many as 12-15 different modes need to be analyzed for a given macroblock. Since distortions in background macroblocks do not contribute to the overall distortion measure in Equation [START_REF] Chai | Face segmentation using skin color map in videophone applications[END_REF], background macroblocks can be encoded with very little rate (and consequently, very high distortion). Motivated by this, the encoder is modified to have two sets of available partition types, one for face and hand blocks and one for background blocks. When using both the backgrd-part and part parameters, the encoder uses backgrd-part option for background macroblocks and part option for the face and hands macroblocks. For ease of integration into the pre-existing encoder structures, the backgrd-part has the same 10 options as part. This allows the search for partitions in background macroblocks to be limited to only the coarsest partitions while still enabling the finer partitions for the relevant blocks.

In motion-compensated video coding, searching for optimal motion vectors comprises a significant portion of the total encoding time. To speed up the motion search, a parameter (ROI-ME) is included that specifies a potentially different motion search method for the face, hands, torso and background macroblocks. This approach was demonstrated to improve the encoding speed of the x264 encoder by up to 12% for ASL videos [START_REF] Vanam | Fast parameter setting selection algorithms for distortioncomplexity optimization of h.264 encoder[END_REF]. The space of possible x264 motion search methods are listed in increasing order of complexity: diamond (DIA), hexagon (HEX) and uneven multihexagon (UMH) search. The background macroblocks use only the DIA search, while the torso region uses equal or lower complexity search compared to the face and hand regions. The ROI-ME includes the following 8 options (1, . . . , 8) corresponding to the motion search in (face, torso, background) regions: (HEX, UMH, DIA), (UMH, HEX, DIA), (HEX, HEX, DIA), (UMH, DIA, DIA), (HEX, DIA, DIA), (DIA, DIA, DIA), (UMH, UMH, DIA) and (UMH, UMH, UMH).

For each of the encoding parameters, the options are indexed in order of increasing complexity. For example, a value of part = 10 is the most complex and enables the encoder to search over of all possible macroblock partitions. Conversely, a value of part = 1 restricts the search to only the coarsest partitions but offers the lowest complexity. The lower complexity options can increase the speed of the encoder but can reduce the overall rate-distortion performance.

JOINT RATE-INTELLIGIBILITY-COMPLEXITY OPTIMIZATION

The set of encoding options discussed in Section 3 made available to the encoder determine the achievable bitrate, distortion, and complexity. A vector of parameter options is referred to as parameter settings. The x264 default parameter setting is the vector (subme = 5, part = 8, ref = 1, trellis= 1, backgrd-part=0, ROI-ME=0). These values are also listed in Table 2. This parameter vector corresponds to high complexity sub-pixel motion estimation; all possible macroblock partitions; one reference frame; and the use of the context adaptive arithmetic coder (CABAC) with uniform quantization. The default settings do not use any of the region-based complexity optimization options. An ideal video encoder will select the parameter setting which results in a compressed video that meets the target rate and complexity constraints in an optimal way. An exhaustive search over all possible parameter settings requires 358400 encodings per video (7 × 16 × 10 × 4 × 10 × 8). Because it is impossible to perform an exhaustive search of this ratedistortion-complexity space in real-time on a mobile device, fast and accurate methods for choosing the appropriate set of encoding parameters must be employed. The dominant parameter setting pruning algorithm (DPSPA) [START_REF] Vanam | H.264/MPEG-4 AVC encoder parameter selection algorithms for complexity distortion tradeoff[END_REF] is applied to determine optimal parameter settings without performing a full search. DPSPA is a fast offline algorithm that uses significantly fewer encodings compared to an exhaustive search to estimate the distortion-complexity convex hull. For a fixed bitrate, DPSPA provides a collection of parameter settings which correspond to operating points lying approximately on the distortion-complexity (D-C) convex hull, as illustrated in Figure 1. These points are nearly optimal in terms of their D-C performance; for a fixed complexity constraint, the resulting distortion is minimized. Applying the algorithm over a range of target bitrates effectively provides a look-up table that specifies the parameter settings to use such that distortion is minimized given both a rate and complexity constraint.

EXPERIMENTAL RESULTS: APPLYING DPSPA TO ASL VIDEO SET

The DPSPA algorithm is applied to a set of nine training ASL videos and six test ASL videos each having 320 × 240 frame resolution, 200 frames and a frame rate of 15 fps. The results are reported for fixed values of λ. Fixing λ for a collection of sequences effectively fixes the QP value in each frame. This results in slight variations in rate and is an approximation of a constant rate scenario. Results are currently being generated at fixed values of bitrate using the rate control algorithm described in Section 2 and will be presented at the workshop. These experiments are conducted on a Windows XP PC having a 2.01 GHz AMD processor. The DPSPA algorithm is executed for values of λ = {0.5, 1, 5, 40, 150} corresponding to average bitrate of {215.5, 155.8, 74.2, 28.8, 16.1} kb/s for the default parameter setting on test videos. The optimal parameter settings computed by DPSPA are applied to the test set of ASL videos to obtain 1, the DPSPA parameter settings provide average speed improvements of approximately 15% with little decrease in intelligibility. A difference of approximately 1.5 dB corresponds to a statistical change in subjective intelligibility score [START_REF] Ciaramello | Complexity constrained rate-distortion optimization of sign language video using an objective intelligibility metric[END_REF]. Therefore, the average decreases in intelligibility shown in Table 1 will not significantly reduce the perceived intelligibility.

The reductions in complexity result from choosing an optimal combination of encoding parameters, including the use of the additional coding modes tuned for region-based coders. For example in Table 2, for λ DP SP A = 0.6, DPSPA picks three reference frames instead of one reference frame. The additional computation cost is mediated by the corresponding reduction in distortion.

CONCLUSION AND FUTURE WORK

This paper presents an ASL encoder based on the H.264 standard in which both rate and complexity can be allocated to the region-of-interest. The proposed encoder includes two new parameters that specify the partition size for the background macroblocks and ROI-based motion search complexity. DPSPA, a fast offline algorithm, is used to choose parameter settings that have excellent rate-intelligibility-complexity performance. These parameter settings can be stored in a look-up table that can be used by an online algorithm which chooses parameter settings based on the available computational resources and bandwidth. When compared to the default parameter settings, the DPSPA parameter settings gives up to 21.2% improvement in encoding speed with a small decrease in intelligibility.

DPSPA quickly provides a collection of parameter settings which are nearly optimal in terms of rate, distortion, and complexity. Real-time video encoding on a mobile device imposes constraints on both bitrate and encoding speed. Bitrate is a function of the available network bandwidth, which will vary depending on network load and geographical location. The encoding complexity depends on the maximum processing power of a given device and on the remaining battery life. Given these constraints on bitrate and encoding speed, the of-fline training results from DPSPA can be stored in a look-up table on the mobile device, allowing extremely fast selection of optimal encoding parameters based on the current network and processing resources. The ASL optimized encoder is currently being ported to an HTC TyTN II cell phone, being integrated into the MobileASL application in order to validate the results computed by DPSPA in the offline training. [START_REF] Riskin | The MobileASL Project[END_REF].

Fig. 1 .

 1 Fig. 1. Joint R-D I cost vs. encoding time for an ASL training video. DPSPA provides an approximation of the convex hull of the R-D-C space.

Table 1 .

 1 Relative performance of DPSPA parameter setting over the default parameter setting for ASL test videos. A negative bitrate gain implies lower bitrate for DPSPA. Tests were performed on a Windows XP PC with 2.01 GHz AMD processor.

	λ	Avg. rate Max speed Avg. speed	Avg.
		gain	gain	gain	Intell.
					loss (dB)
	0.6	-2.4%	15.7%	14.4%	0.08
	1.1	-0.47%	19.2%	16.8%	0
	5	-26.8%	21.2%	13.7%	0.16
	46	0.49%	20.1%	15%	0.67
	150	-0.66%	18%	13.7%	0.22
	the maximum and average encoding speed improvement,
	the relative gain in bitrate and the loss in intelligibility of
	DPSPA parameter setting over the default parameter setting.
	As demonstrated in Table

Table 2

 2 lists the encoder parameter settings which have been applied to the test videos. While the default parameter setting

Table 2 .

 2 The default parameter setting (fixed) and DPSPA parameter settings for different values of λ. I8 × 8, I4 × 4 for background macroblocks. Since DPSPA generates parameter settings that trade-off joint rateintelligibility cost with encoding time, it does not always pick parameters having lower complexity than the default setting.

	Parameter	Default		λ DP SP A
	name		0.6 1.1 5 46 150
	subme	5	3	3	3 2	2
	ref	1	3	1	1 1	1
	part	8	6	8	8 3	3
	trellis	1	0	0	2 2	2
	backgrd-part	-	8	7	3 6	7
	ROI-ME	-	6	5	6 6	6
	uses HEX search for the entire frame, the DPSPA parame-
	ter setting exploits the ROI-ME option, using either (HEX,
	DIA, DIA) or (DIA, DIA, DIA) for (face, torso, background),
	each of which have lower search complexity than the default
	setting. DPSPA often chooses lower complexity subme and
	backgrnd-part compared to the default subme=5 and
	P8 × 8,

Western NY Image Processing Workshop, Rochester, NY, September 2009