Non-Linear Shrinkage Estimation with Complex ... - CiteSeerX

Non-Linear Shrinkage Estimation with Complex Daubechies Wavelets J.-M. Lina

1;2

and B. MacGibbon

3;4

1:Centre de Recherches Mathematiques, Univ. de Montreal C.P. 6128 Succ. Centre-Ville, Montreal, Quebec H3C 3J7, Canada 2:Atlantic Nuclear Services Ltd., Fredericton, New Brunswick E3B 5C8, Canada 3:Departement de Mathematiques, UQAM C.P. 8888 Succ. Centre-Ville, Montreal, Quebec H3C 3P8, Canada 4:GERAD, Ecole des Hautes Etudes Commerciales 3000 Chemin de la C^ote-Sainte-Catherine, Montreal, Quebec H3T 2A7, Canada

GERAD G-97-44

and

Abstract

CRM-2478

One of the main advantages of the discrete wavelet representation is the near-optimal estimation of signals corrupted with noise. After the seminal work of De Vore and Lucier (1992) and Donoho and Johnstone (1995), new techniques for choosing appropriate threshold and/or shrinkage functions have recently been explored by Bayesian and likelihood methods. This work is motivated by a Bayesian approach and is based on the complex representation of signals by the Symmetric Daubechies Wavelets. Applications for two dimensional signals are discussed.

Resume

Une application importante des representations par ondelettes discretes est l'estimation de signaux alteres par un bruit. Apres les travaux de De Vore et Lucier (1992) et ceux de Donoho et Johnstone (1995), l'approche Bayesienne et les techniques de vraisemblance ont ete explorees pour determiner le seuil et la fonction de \retrecissement" de l'estimateur. Ce travail repose sur une approche Bayesienne, dans le contexte des representations complexes par ondelettes de Daubechies symetriques. Ce travail est illustre par des simulations en 2 dimensions.

1

1 Introduction A multiresolution analysis of signals or functions considers expansions of the form

f (x) =

X

k

ck '(x ? k) +

1X X j =0 k

dj;k j;k (x);

where the wavelets j;k (x) can be derived from a unique function translations: j

j;k (x) = 2 2 (2j x ? k ):

(1)

through dyadic dilations and discrete (2)

For a given scale labelled by j , this set of functions spans a wavelet space Wj . The set f'(: ? k); k 2 ZZg is a basis (possibly orthonormal) of a so-called approximation space V0 . The multiresolution framework consists in a hierarchy of such spaces built by scaling this basis. De ne Vj = Spanf'j;k (:); k 2 ZZg where j

'j;k (x) = 2 2 '(2j x ? k);

(3)

such that we have the orthonormal decomposition Vj+1 = Vj  Wj . In practice, a function (or signal) is known at some ne scale, i.e. in some approximation space Vjmax . The wavelet decomposition then consists in the computation of the coecients of the expansion (1) where only a nite number of scales \j" are present in the wavelet decomposition. Mostly advocated by Donoho and Johnstone [8], the wavelet regression estimators, based on orthogonal multiresolution bases and the shrinkage of the empirical wavelet coecients, provide near-optimal estimates of the true signal. Their wavelet regression technique relies on the de nition of the thresholding rule (hard or soft), the selection of the appropriate threshold (optimal or universal) and the choice of the basis. This regression estimator could be directly implemented on the complex coecients by considering only the amplitudes as advocated in previous work [13] and more recently by Gagnon and Smaili [10] and Downie and Silverman [9] for multiwavelets after an appropriate preprocessing. Here we have decided to use complex symmetric Daubechies wavelets where no preprocessing is required and to follow a Bayesian motivation in our choice of shrinkage rules in order to include the phase also in the choice of threshold. Abramovich, Sapatinas and Silverman [1] used a Bayesian approach to wavelet thresholding. In their work, a prior distribution consisting of a mixture of a mean zero normal distribution and the Dirac delta function at 0 (see comparable priors in Clyde et al. [5], Chipman et al. [4]) is speci ed on the unknown signal in order to capture the sparseness of wavelet expansions common to many applications. The prior model of Abramovich et al. (1997) can be adjusted so that the signal can be assumed to belong to a prespeci ed Besov space, since the hyperparameters of the model are related to the parameters of the Besov space, allowing for the incorporation of prior knowledge about the regularity conditions satis ed by the function of interest. Using a mixture of L1 losses they obtain a Bayesian thresholding rule which compares favorably to the soft and hard thresholding rules of Donoho and Johnstone (1994, 1995). However, we prefer to take advantage of the symmetry aorded by complex wavelets in order to include the possibility of also using the phase to determine the appropriate shrinkage. This is achieved by using both Bayesian and likelihood methods on the complex representation of the symmetric Daubechies wavelets and also second-order properties of complex signals (see, for example, Ref.[18]). Our method also retains the ability to incorporate prior knowledge about the regularity conditions, in terms of the Besov space parameters, that the function of interest satis es.

2 Symmetric Daubechies Wavelets: the role of the phase In the construction of a multiresolution wavelet analysis, the aim is to specify functions ' and satisfy the following equations

which

P

(4) '(x) = 2 k ak '(2x ? k); P k (x) = 2 k (?1) a1?k '(2x ? k); (5) P where the (complex-valued) coecients ak are only constrained by ak = 1. Using the Fourier transform of the scaling function, say '^(!), the re nement equation (4) can be written as X '^(2!) = m0 (!)'^(!); with m0 (!) = ak eik! (6) k

This Laurent series (also called a lter) has been widely studied in various contexts. By requiring constraints like orthogonality, compact support and vanishing moments for the wavelet basis functions, Daubechies investigated an important set of solutions of Eqs.(4) and (5). Details of her approach can be found in Ref.[6]. Already previously suspected by Lawton (see Ref.[6], p.253), the existence of symmetric Daubechies wavelets (SDW) has been investigated only recently ([11],[12]). The construction of the SDW bases is by now wellknown and we refer the reader to the Ref.[14] for details. For the sake of consistency and to motivate the use of this type of wavelet basis, we provide a short summary of its properties based on a comparison with other most common wavelet bases. An important prototype of a solution to Eq.(4) is given by splines. For instance, the lter associated with the linear spline (the hat function) is 



i! 2 m0 (!) = 1 +2e e?i! :

In 1981, Stromberg proposed

(7)

1 + ei! 2 e?i! e?i! + r ; with r = 2 + p3 (8) 2 e?i! + rei! in order to satisfy orthogonality of the integer translates of the scaling function. The resulting \Stromberg scaling function" displayed in Fig.1 is a linear combination of hat functions such that the resulting set of translates of ' is orthonormal; we lose however the original compactness of the support. Later, Battle [3] and Lemarie [16] modi ed this lter by adding a phase

m(0str) (!) =









m(0bl) (!) = m(0str) (!)ei((!)?(2!)) ;

(9)

p

p

1 + r cos ! ; sin  (!) = 3 +p 3 p r sin ! ; (10) cos  (!) = 3 +p 3 p 2 3 2 + cos ! 2 3 2 + cos ! such that the resulting multiresolution spaces are endowed with an orthonormal basis generated now by a symmetric scaling function (see Fig.1). The same procedure is possible with the Daubechies wavelets with an even number of vanishing moments. Let us consider for example the J=2 (two vanishing moments) Daubechies solution also called DAUB6 (see Fig.2). The associated lter is given by

m(0daub6) (!) =



    1 + ei! 3 e?i! ? r0 e?i! ? r0

2

1 ? r0

1 ? r0

); with r0 = 3 ? i 2

r

5 12 ?

q

10 3

p ? 2i 15 2

:

(11)

1.4

1

1.2 0.8 1 0.8

0.6

0.6 0.4 0.4 0.2

0.2

0 0 -0.2 -0.4

-0.2 -4

-2

0

2

4

6

-4

-3

-2

-1

0

1

2

3

4

Figure 1: The Stromberg (left) and Battle-Lemarie (right) scaling functions. 1.4

1

1.2

0.8

1

0.6

0.8

0.4

0.6 0.2 0.4 0

0.2

-0.2

0

-0.4

-0.2 -0.4

-0.6 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

-2

-1

0

1

2

3

Figure 2: The J=2 real Daubechies scaling function (left) and the SDW2 complex Daubechies scaling function (right). The imaginary part is in dashed line. But this solution is not unique and, similarly to the construction of Battle and Lemarie, it is possible to add a phase

m0(sdw2)(!) = m(0daub6) (!) Z (!) where

(12)

 1 ? r0 (13) r0 ? 1 is of modulus one for all !. This particular solution leads to a symmetric scaling function with the usual properties that characterize the Daubechies multiresolution analyses. However, the main dierence is that, unlike the Battle-Lemarie solution, symmetry is only possible with a complex-valued scaling function as illustrated by Fig.2. Let us consider an expansion of the form (1) of a function f where the wavelet coecients are complex valued, i.e 

0 i! Z (!) = 1r??r0eei!



djk = h j;k jf i = rj;k eij;k with rj;k  0:

(14)

() We de ne a rotated wavelet basis j;k (x; j;k ) by ()

j;k (x; j;k ) = e

ij;k j;k (x)

(15)

2

1.5

1.5

1

1 0.5 0.5 0 0 -0.5 -0.5 -1

-1 -1.5

-1.5 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

-2

-1

0

1

2

3

Figure 3: The J=2 real Daubechies wavelet (left) and the SDW2 complex Daubechies wavelet (right). The imaginary part is in dashed line. and we observe that this set of functions is still an orthonormal set. In fact, we have a \second generation" () type of wavelet basis since j;k (x; j;k ) cannot be generated from genuine translations and dilations of a unique wavelet. We adopt this point of view here and consider the following expansion

f (x) =

X

k

ck 'k (x) +

jmax X?1 X j =0

k

() rj;k j;k (x; j;k )

(16)

() (x; j;k )jf (x)i. where the 's are given by Eq.(15) and rj;k = h j;k

A few remarks are in order. First, we notice that all the local rotations induced by the phases j;k somehow adapt the basis to the signal. Interestingly enough, there is approximately the same number of degree of freedom in those parameters as in a sampled real function. Second, there is an obvious link between the present scalar complex wavelets and the multiwavelet framework in which we consider vectors of real wavelets (see e.g. Downie and Silverman [9]). Leaving aside some distinctions between the notions of orthogonality, the essential dierence occurs in the interpretation of the interdependence between the two real wavelet coecients that describe the local behaviour of the function: modulus and phase truly exist with complex SDW analysis. The problem addressed here is the appropriate use of this phase in the computation of a threshold estimator of the (amplitude) wavelet coecients rj;k by introducing a Bayesian approach. Since this work is oriented towards \image processing", we conclude this section by recalling that a bidimensional multiresolution analysis can be built from the product of two multiresolution spaces Vi . We thus have one scaling function '(x)'(y) complemented with three wavelets (x)'(y); '(x) (y) and (x) (y). We will write these complex-valued basis functions as (x; y) = '(x)'(y) = (x; y) + i (x; y); 0 (x; y) = (x) (y) = 0 (x; y) + i 0 (x; y); 1 (x; y) = (x)'(y) = 1 (x; y) + i 1 (x; y); 2 (x; y) = '(x) (y) = 2 (x; y) + i 2 (x; y); and expansions like (1) and (7) now generalize in two dimensions to

I (x; y) =

X

k1 ;k2

ck1 ;k2 (x ? k1 ; y ? k2 ) +

2 jmax X?1 X X

=0 j =0 k1 ;k2

dj;k1 ;k2 j;k1 ;k2 (x; y)

(17) (18) (19) (20) (21)

where j;k1 ;k2 (x; y) = 2j (2j x ? k1 ; 2j y ? k2 ) span the wavelet spaces Wj respectively. In the sequel, we denote by W a wavelet transform: W fc ; d jmax g ?! fcm;n m;n 0;m;n ; d1;m;n ; : : : djmax ?1;m;n g

(22)

where d = (d0 ; d1 ; d2 ).

Figure 4: Complex Wavelet Analysis with SDW2. Right upper window: modulus of the complex wavelet coecients, decomposition with two levels. Right lower window: coecents d0 displayed in the complex plane. Left window: circular histogram for the phases (top), histograms for the real part (middle) and imaginary part (bottom) of the wavelet coecients d0 .

3 Shrinkage and Phase Let us consider an image corrupted with additive Gaussian noise (denoted by N ) projected into the approximation space of highest resolution X I~(x; y) = c~jmax ;m;njmax ;m;n(x; y) (23) m;n

We want to estimate a real signal

I0 (x; y) =

X

m;n

Cjmax ;m;njmax ;m;n(x; y)

(24)

from the observed image I~ = I0 + N . The wavelet shrinkage technique amounts to computing estimates from the wavelet representation of the observed signal. Let

I (x; y) =

X

k1 ;k2

c~k1 ;k2 (x ? k1 ; y ? k2 ) +

2 jmax X?1 X X

=0 j =0 k1 ;k2

dj;k1 ;k2 j;k1 ;k2 (x; y):

(25)

We have the following variational problem to solve: given a positive Lagrange parameter , nd an image

I  that minimizes the functional

L(I ) def = E (I~; I ) + S (I ):

(26)

Here E (I~; I ) is the root-mean square error between the observed and test images:

E (I~; I ) = jjI ? I~jj2 =

2 jmax X?1 X X

=0 j =0 k1 ;k2

jd~j;k1 ;k2 ? dj;k1 ;k2 j2 ;

(27)

whereas S (I ) expresses some constraints on the regularity of the optimal solution; in fact it regularizes the ill-posed problem of minimizing E alone. The choice of this constraint involves some a priori knowledge about the true signal we aim to restore. The parameter  controls the trade-o between goodness of t and a priori smoothness. This latter property, given by S (I ), can be quantiiedy by using a norm in some Besov space (see Ref.[7]). An important result in wavelet theory is the de nition of such a norm in the wavelet  (i.e. the space of functions \with  derivatives in Lp "), it representation. Considering the Besov space Bp;q can be shown that  pq  2 jmax X?1 X 0p X  p js j ; s0 =  + 1 ? 2 jd 2 S (I ) = (28) =0 j =0

k1 ;k2

j;k1 ;k2

p

de nes a norm in this space. For the sake of simplicity, we consider here p = q = 1 and  = 1; then

S (I ) =

2 jmax X?1 X X

jdj;k1 ;k2 j

(29)

F (dj;k1 ;k2 ; d~j;k1 ;k2 )

(30)

=0 j =0 k1 ;k2

and the functional de ned in Eq.(26) can be read as

L(I ) =

2 jmax X?1 X X

=0 j =0 k1 ;k2

with

F (d; d~) = jd~ ? dj2 +  jdj:

(31)

Equation (31) illustrates the eciency of the wavelet representation for solving the above optimization problem since they \diagonalize" the functional over each coecient of the decomposition. The solution d of our variational problem, d = Argmin F (d; d~) (32) is clearly, as illustrated on Fig.4, a point on the ray de ned by the phase of d~. In other words, the phase is preserved and rather plays the role of a parameter in this problem. Let us re-write F (d; d~) in terms of the amplitudes,

F (r; r~) = (~r ? r)2 +  r

(33)

The solution of Argmin F (r; r~) is obviously given by

r = (r ? 2 )+

(34)

where (x)+ is equal to x for x  0 and 0 elsewhere (see dashed line on Fig.6). We thus obtain a soft-shrinkage of the wavelet coecients amplitude with a threshold de ned by (j ). Denoting by T this shrinkage operator of the wavelet coecients, the resulting estimate is given by I  = (W ?1 T W ) I~ (35) Let us emphasize that this approximation is obtained by modifying only the amplitude of the wavelet coecients: the phases are preserved. This fact will be explicitely used in the Bayesian approach in the next Section.

4 Likelihood and Bayesian analysis Let us consider the general case of a zero-mean complex Gaussain random variable  with a normal distribution (see for instance Ref.[17]). Following the recent work of Picinbono [18], we de ne the variance 2 and the \relation" C by = E (); C def = E (2 ) 2 def The complex Gaussian distribution is then described by the density function p(; ) = 12  p 1 2 e? 12 Q(;) 1? with ?1   2  Q(; ) = (; ) C C2 



and C = 2 ei :

(36) (37)

(38)

Straightforward computation yields the following expression for Q:  ? ) = 2 2 (39) Q(; ) = 2jj2 1 ? cos(2Arg 2 (1 ? 2 ) 2 ?() where  = ei and ? 2 ) (40) ?() = 1 ? (1cos(2  ? ) Following the discussion in section 3, we consider the following expression for the wavelet coecients computed from the observed image I~ (we omit to write the indices j; k): d~ = d0 + ; i:e: ei = rei ? ei (41) using the polar representation of the wavelet coecients, d~ = rei and d0 = ei . First we obtain the following likelihood function:

(r; j;  ; ; ; ; )  r e?

(r?)2 ?2r(1?cos(?)) 2 ?()

(42)

Second, since as shown in section 3 we are interested in an estimator that preserves the phase of the wavelet coecient, i.e. an estimator of the amplitude only, we de ne a likelihood on the amplitude rj;k by j;k (rj) = (r;  = j;k j;  = j;k ; ; ; ;  = j;k )  r e

)2 ? (2r?(?j;k )

(43)

Let us comment on the various \hidden" parameters (assumed to be known) in ?(j;k ). The angle 21 de nes the principal axis of the distribution (for the noise) in the complex plane; the parameter  is a real-valued taking values between 0 and 1 and it measures the \circularity" of the complex noise component[18]: if the data are circular (i.e. the phases are meaningless), then   0; however, real and imaginary parts are strongly correlated for   1 (the limiting case  = 1 is obtained with the Haar wavelet for which all the wavelet coecients lie on a line). The variance  intuitively measures the noise level. In a fully Bayesian analysis, it would be possible to assume prior distributions on each of these parameters in Eq.(43). Here, however, we perform a partial Bayesian analysis by rst using the wavelet coecients at the nest scale to estimate ?;  and and also using the fact that  =  = j;k . Thus, the following estimators of the parameters arise naturally:

2 = E[d~jmax ;k d~jjmax ;k ] ei = 12 E[d~2jmax ;k ]

(44) (45)

()   e? :

(46)

Now, following the discussion in the previous section, the prior distribution on , which can be considered as a measure of smoothness, should depend on the Lagrange parameter  in Eq.(33). Here, we choose to model this smoothness by a Gamma(2, 1 ) density; that is The resulting posterior density for  is thus found to be ? (r?)2 ?

2 ?(j;k ) j;k (jr) = R  e (r?r0 )2 0 1 0 ? 2 ?(j;k ) ?r dr0 0 r e

(47)

The Bayes estimates can now be speci ed as follows. Let us consider the signal contaminated with additive noise of section 2 expanded in some orthogonal discrete wavelet basis. The Bayes estimator with respect to the quadratic loss of the wavelet coecient amplitude is equal to the mean of the posterior distribution: that is, for any pair of indices (j; k), we have

j;k (r) =

1

Z

0

R1

? (r?r0 )2 ?r0

r02 e 2 ?(j;k ) dr0 0  j;k (jr) d = R 02 1 r0 e? (2r?(?rj;k) ) ?r0 dr0 0

(48)

Using the following probability integral[2], Z

0

1

p ?z ) xn e? 21 x2 ?zx dx = n! 2n?1  in erfc( p 2

(49)

where in erfc(x) denotes the repeated integrals of the Error function: in erfc(z ) =

1

Z

z

in?1 erfc(x) dx; i0 erfc(z ) = erfc(z )

(50)

S(x)

x

Figure 5: Shrinkage function S (x) we obtain the following expression for the estimator r

 s

?
j;k (r) =  ?(2j;k ) S 1 ?(2 ) r ? (2j;k ) j;k



(51)

with

() =  2 ?() and

erfc( ?px )

S (x) = x + p1 ierfc( ?p2x ) 2 2

(52) (53)

As seen on the graph of the Fig.5, S (x) gives a \supersmooth" version of a \soft-shrinkage" estimator. This function and the phase-dependent estimator given by Eq.(51) are our main results. In Fig.6, we show the phase dependency of the threshold for various values of the \circularity". As expected, we observe that the limiting case   1 does correspond to the usual real case for which a single phase (e.g 0 or ) is present in the decomposition. Conversely, the threshold does not depend on the phase when the data are circular, i.e when   0. Note that approximating S (x) by (x)+ (the dashed line on Fig.5) simpli es the result in (51) and leads to the estimator j;k (r) = (r ? (2j;k ) )+ : (54) Images displayed in Fig.7 illustrate the technique presented here. In the upper and lower images both columns contain the noisy observed images and the corresponding estimates respectively. It should be mention that

-2

Threshold

phase

1 0

2

0.8

15 0.6 10 Shrink 5

0.4

0 10 0.2

8 6 4

-3

-1

-2

1

2

3

phase

2

r

0

Figure 6: Left: Plots of 0:5?(), for various values of . (The threshold is at for  = 0 and peaky for   1). Right: j;k (r).

p

we have used  = 2 ln n?1 in Eq.(51) (n being the size of the image) in order to recover the universal threshold of Ref.[8] when  = 1.

5 Conclusion This work described a preliminary study of the Bayesian approach to the wavelet regression problem with complex valued Daubechies wavelets. To some extent, we exploit the redundancy in the representation of real signals by the complex wavelet coecients. The technique presented here is based on the following observation: modulus and phase of the wavelet coecients encompass very dierent information about the underlying signal. As an application, we have computed a Bayesian estimator of the modulus of the wavelet coecients that depends on the phase. Let us emphasize at this point that previous studies have shown the \importance of the phase" of the complex wavelet representation of signals [15]. Current research is focused on the following extensions of this present work:  A sensitivity analysis for the choice of prior density (46): for instance, if more smoothness were required,  2 the choice of a prior such as ()   e? 2  leads to the estimator

S j;k (r) = 2

p

2 ?(j;k )?1  p+?(j;k )?1 r p  + ?(j;k )?1



1  1 + ?( j;k ) r

(55)

 A fully Bayesian approach is mandated and careful estimates of the various parameters is under investigation. In particular, it is anticipated that approriate prior distributions on the phase (circular distributions) could render the estimator more robust.  Empirical and hierarchical Bayesian methods are also being studied. This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada.

Figure 7: Noisy and estimated images

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[8] D. Donoho and I. Johnstone, Wavelet shrinkage: Asymptopia?, J. R. Statist. Soc. B, p. 301, 1995 (and reference therein). [9] T.R. Downie and B.W. Silverman, The discreste multiple wavelet transform and the thresholding methods, Tech. Report, School of Mathematics, Univ. of Bristol (UK), 1996. [10] L. Gagnon and F. Drissi Smaili, Speckle noise reduction of airbone SAR images with symmetric Daubechies wavelets, SPIE-2759, 1996. [11] W. Lawton, Applications of Complex Valued Wavelet Transforms to Subband Decomposition, IEEE Trans. on Signal Proc., vol.41, p.3566, 1993. [12] J.M. Lina and M. Mayrand, Complex Daubechies Wavelets, App. Comp. Harmonic Anal., vol.2, p.219, 1995. [13] J.M. Lina, Image processing with Complex Daubechies Wavelets, CRM Report 2335 (1995), to appear in Int. J. of Mathematical Imaging and Vision (1997). [14] , J.M. Lina, Complex Daubechies Wavelets: lters design and applications, CRM Report 2449 (1996). [15] , J.M. Lina and P. Drouilly, The importance of the phase of the symmetric Daubechies Wavelets representation of signals, Proc. IWISP'96, Manchester (UK), p.69, 1996. [16] P.G. Lemarie, Une nouvelle base d'ondelettes de L2(IRn ), J. de Math. Pures et Appl., 67, p.227, 1988. [17] K.S. Miller, Complex Stochastic Processes, Addison-Wesley Pub., 1974. [18] B. Picinbono, Second-Order Complex Random Vectors and Normal Distributions IEEE Trans. Sig. Proc. 44, p.2637, 1996. [19] J. Stromberg, A modi ed Franklin system and higher order spline systems on IRn as unconditional bases for Hardy spaces, Conference in honor of A. Zygmund, Vol.II, Wadsworth math. series, p.475, 1982.