1

Wavelets, Ridgelets and Curvelets for Poisson Noise Removal Bo Zhang∗ , Jalal M. Fadili and Jean-Luc Starck

Abstract In order to denoise Poisson count data, we have introduced a new Variance Stabilizing Transform (VST) to stabilize the variance of a low-pass filtered discrete Poisson process, yielding a near Gaussian process. This transform is fast for any dimensional data, and is more efficient than existing VSTs in low and very low-count situations. We then combine the VST with the wavelet, the ridgelet and the curvelet transforms, leading to multiscale VSTs (MS-VSTs) and nonlinear decomposition schemes. By doing so, the noise-contaminated coefficients of these MS-VST-modified transforms are asymptotically normally distributed with known variances. A classical hypothesis-testing framework is adopted to detect the significant coefficients, and a sparsity-driven iterative scheme reconstructs properly the final estimate. We show that this MS-VST approach provides an astounding denoiser capable of recovering important structures of various morphology in (very) low-count images. A range of examples illustrate the efficiency of this approach. Index Terms Poisson intensity estimation, filtered Poisson process, multiscale variance stabilizing transform, wavelets, ridgelets, curvelets.

EDICS Category: RST-DNOI Denoising B. Zhang is with the Quantitative Image Analysis Group URA CNRS 2582 of Institut Pasteur, 75015 Paris, France. Phone: +33(0)140613891. Fax: +33(0)140613330. E-mail: [email protected] J. M. Fadili is with the GREYC CNRS UMR 6072, Image Processing Group, 14050 Caen Cedex, France. Phone: +33(0)231452920. Fax: +33(0)231452698. E-mail: [email protected] J.-L. Starck is with the CEA-Saclay, DAPNIA/SEDI-SAP, Service d’Astrophysique, F-91191 Gif sur Yvette, France. Phone: +33(0)169085764. Fax: +33(0)169086577. E-mail: [email protected]

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I. I NTRODUCTION Images of Poisson counts arise from a variety of applications including astronomy and astrophysics [1], biomedical imaging [2], etc. Poisson noise removal is thus a vital task in these applications. This problem can be formulated as follows: we observe a discrete dataset of counts x = (xi )i∈Zq where xi is a realization of the Poisson random variable Xi of intensity λi , i.e., Xi ∼ P(λi ). Here we suppose that Xi ’s are mutually independent. The denoising aims at estimating the underlying intensity profile Λ = (λi )i∈Zq from x. Literature overview A host of estimation methods have been proposed in the literature. A classical solution is to preprocess the data by applying a variance stabilizing transform (VST), e.g., Anscombe transform [3][4] or Haar-Fisz transform [5][6]. Then the transformed data will be approximately stationary and Gaussian. Once we are brought to the Gaussian denoising problem, standard approaches such as wavelet thresholding [7][8] are used before we inverse the VST to get the final estimate. Other major estimation approaches consist of: 1) approximative oracle attenuation: Nowak and Baraniuk [9], and Antoniadis and Sapatinas [10] proposed a wavelet domain filter, which can be interpreted as an approximative oracle attenuation under a wavelet basis; 2) hypothesis testing: Kolaczyk first introduced a Haar domain threshold [11], which implements a hypothesis testing procedure based on a user-specified false positive rate (FPR). [12] extended the hypothesis tests (HTs) to the biorthogonal Haar domain, leading to a more regular reconstruction for smooth intensities. Although [13] and [14] attempted to generalize the HTs for wavelets other than Haar, [13] is more computationally complex than Haarbased methods, and [14] adopts an asymptotic approximation which may not allow reasonable solutions in low-count situations. 3) empirical Bayesian and penalized ML estimations: empirical Bayesian estimators are studied in [15][16][17][18]. The low-intensity case apart, Bayesian approaches generally outperform the direct wavelet filtering [9][10] (see also [19] for a comparative review). Poisson denoising has also been formulated as a penalized maximum likelihood (ML) estimation problem [20][21][22] within wavelet, wedgelet and platelet dictionaries. To our knowledge, no Poisson denoising method has been proposed for the ridgelet and curvelet transforms. This paper In this paper, we propose a VST to stabilize the variance of a low-pass filtered discrete Poisson process, yielding a near Gaussian process. This transform is fast for any dimensional data, and is shown to be more February 2, 2007

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efficient than existing VSTs in (very) low-count situations. The rationale behind the benefits of stabilizing a low-pass filtered version of the original process is as follows. It is well known that the performance of the Anscombe VST deteriorates as the intensity becomes low [1] (typically for λ < 10), i.e., as the SNR decreases. Hence, one can alleviate this limitation and enhance the performance of the VST if the SNR is increased before stabilization. This can be achieved by pre-filtering the original process provided that the filter acts as an “averaging” kernel, and more generally, a low-pass filter. A detailed asymptotic analysis will support these claims. As opposed to most existing methods, each being adapted to a particular analysis dictionary (e.g. wavelet), the proposed VST can be seamlessly combined with a large family of multiscale transforms, resulting in multiscale VSTs (MS-VSTs). Therefore, toward the goal of Poisson denoising, we are allowed to choose or design the most adaptive transform for the sources to be restored based on their morphology. Indeed, owing to recent advances in modern harmonic analysis, different multiscale transforms were shown to be very effective in sparsely representing different kinds of information. For example, to represent efficiently isotropic singularities and regular structures, a qualified candidate is the wavelet transform [23][1]. The ridgelet transform [24] has been shown to be very effective in representing global lines in an image. The curvelet system [25][26] is highly suitable for representing smooth (C 2 )

images away from C 2 contours. These transforms are also computationally attractive particularly in large-scale applications. We will show that our new VST can be easily combined with these different multiscale geometrical decompositions, yielding asymptotically normally distributed coefficients with known variances. A classical hypothesis testing framework is then adopted to detect the significant coefficients. Because of the non-linearity of the MS-VST and the over-completeness of the transforms, a sparsity-driven iterative scheme is proposed to reconstruct the final estimate. We show that the MS-VST approach provides an astounding denoiser capable of recovering important structures of various (isotropic, line-like and curvilinear) shapes in (very) low-count images. The paper is organized as follows. In Section II, a detailed analysis is provided to characterize the VST. Section III outlines the denoising setting using MS-VST with wavelets. Section IV and V show how the VST can be combined with the isotropic undecimated wavelet transform (i.e. one band per scale) and the standard undecimated wavelet transform (i.e. 2q − 1 directional bands per scale in q -dimensions (q D)), respectively. Denoising by MS-VST combined with ridgelets and curvelets are respectively presented in Section VI and VII. Section VIII provides a discussion on the numerical results obtained, followed by a brief conclusion and the prospects of our work. Given the space limitation, mathematical proofs are

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omitted here. They are made available for download to the interested reader.1 II. VST OF A FILTERED P OISSON PROCESS Given a discrete filter h, the filtered process is defined as Yj := original process. Let us define τk :=

P

i (h[i])

k

P

i h[i]Xj−i ,

where (Xi )i is our

for k = 1, 2, · · · . Instead of immediately considering

(Xi )i as a Poisson process, we begin with a more general analysis where (Xi )i is a Mixed-Poisson-

Gaussian (MPG) process. That is, each Xi is the sum of a Poisson variable Ui ∼ P(λi ) and a Gaussian

variable Vi ∼ N (µ, σ 2 ), all (Ui )i and (Vi )i being mutually independent. This kind of process typically models a CCD camera noise (i.e., shot + readout noise). In addition, we assume that ∀i, λj−i = λ in the support of h to simplify the analysis. We use Y to denote any one of Yj . A. VST-Heuristics It can be seen that the variance of Y (Var [Y ]) is an affine function of the intensity λ underlying the Poisson variable. To stabilize Var [Y ], we seek a transformation Z := T (Y ) such that Var [Z] is (asymptotically) constant, say 1, irrespective of the value of λ. Heuristically, the Taylor expansion gives us T (Y ) ≈ T (µY ) + T ′ (µY )(Y − µY ), where µY := E [Y ] =

(λ+µ)τ1 . We then have Var [Z] ≈ T ′ (µY )2 ·Var [Y ] = T ′ (µY )2 (λ+σ 2 )τ2 . Hence, by setting Var [Z] = 1, 

we obtain an ordinary differential equation (ODE): T ′ (µY ) = τ2 (σ 2 + µY /τ1 − µ) p

p

−1/2

. Solving this

ODE, we end up with T (Y ) = 2 τ1 /τ2 Y + (σ 2 − µ)τ1 . This implies that the square-root transform, √ i.e., T (Y ) = b Y + c, could serve as a VST. It is possible to use higher order Taylor expansions to find VST of different forms, but solving the associated ODEs is found difficult since they are highly non-linear.

B. VST-Rigor We define the square-root transform T as follows: T (Y ) := b · sgn(Y + c)|Y + c|1/2

(1)

where b is a normalizing factor. Lemma 1 rigorously confirms our heuristics that T is indeed a VST for a low-pass filtered MPG process (low-pass because of the condition τ1 6= 0 in Lemma 1) in that T (Y ) is asymptotically normally distributed with a stabilized variance as λ becomes large. This result holds 1

http://www.greyc.ensicaen.fr/∼jfadili/Pub/msvstPoisson proofs.pdf.

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true for any c ∈ R. However, the convergence rate in (2) varies with the value of c. The next section provides an analysis of the asymptotic rate and determines the optimal value of c for the Poisson case. Lemma 1 (Square root as VST) If τ1 6= 0, then we have: q

q



τ2 sgn(Y + c) |Y + c| − sgn(τ1 ) |τ1 |λ −→ N 0, λ→+∞ 4|τ1 | D



(2)

where sgn(·) is the sign function. C. Optimal Parameter of the VST For the sake of clarity, we consider in the rest of the paper low-pass filtered Poisson processes (i.e., µ = σ = 0 for the Gaussian component in MPG processes).

To facilitate the asymptotic analysis, we assume a non-negative filter h and a positive constant c (a non-positive h with a negative c can also be considered). Thus our VST is simplified to Z := T (Y ) = √ b Y + c. We can now derive the asymptotic expansions of E [Z] and Var [Z] as stated in Proposition 1. Note that the last point in the proposition results from Lemma 1 directly. Proposition 1 (Optimal parameter of the VST) √ (i) Define Z := b Y + c. Then we have: p

E [Z] = b λτ1 + b τ2 Var [Z] = b + b2 4τ1 2

4cτ1 − τ2 3/2 8τ1 7τ22 32τ13

λ−1/2 + Oλ→+∞ (λ−1 )

2τ2 c + τ3 − 8τ12

75τ23 17τ2 τ3 + 21cτ22 + − 32τ14 128τ15

!

!

λ−1 + b2

(3) 5τ4 + 16c2 τ2 + 16cτ3 64τ13

λ−2 + Oλ→+∞ (λ−5/2 )

(4)

(ii) For the VST to be second order accurate and Z to have asymptotic unit variance, b and c must satisfy: c=

7τ2 τ3 − 8τ1 2τ2

,

b = b1 := 2

√ D (iii) For b and c as above, Z − b1 τ1 λ −→ N (0, 1).

r

τ1 τ2

(5)

λ→+∞

Proposition 1 tells us that for the chosen value of c, the first order term in the expansion (4) disappears, and the variance is almost constant up to a second order residue. The value of b does not influence the convergence rate since it is just a normalizing factor. It can also be noted that if there is no filtering (h = δ ), the value of c in (5) equals 3/8, i.e., the value of the Anscombe VST.

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TABLE I CE

AND

CVar

OF DIFFERENT FILTERS

Filter1 h δ (Anscombe)

CE

6.25 × 10−2

2D Average = hA ⊗ hA

6.94 × 10−3

7.72 × 10−4

2D B3 -Spline = hB3 ⊗ hB3

−4.94 × 10−4

−3.45 × 10−4

1

6.25 × 10

CVar −2

hA = [1 1 1]/3; hB3 = [1 4 6 4 1]/16; ⊗ denotes the tensor product.

Once c is fixed by (5) and the asymptotic expectation is normalized to

√

λ, the coefficient of the

higher-order term O(λ−1/2 ) in (3) is given by (6). Similarly, the asymptotic variance being normalized to 1, the coefficient of the term O(λ−2 ) in (4) is given by (7). CE = CVar =

5τ22 − 4τ1 τ3 16τ12 τ2

(6)

5τ12 τ2 τ4 + 13τ24 − 4τ12 τ32 − 13τ1 τ22 τ3 16τ14 τ22

(7)

These higher-order coefficients (6) and (7) can be used to evaluate the stabilization efficiency for a given filter. The ideal filters will be those minimizing (6) and (7). Tab. I shows the values of CE and CVar for different filters, where h = δ corresponds to the Anscombe VST (no filtering). Note that the values for the Anscombe VST are 10 or even 100 times larger than for the other cases, indicating that the stabilization after filtering a Poisson process can be much more efficient than without filtering. This is also confirmed by the simulations depicted in Fig.1, where the estimates of E [Z] (resp. Var [Z]) obtained from 2 · 105 replications are plotted as a function of the intensity λ for Anscombe [3] (dashed√ dotted), Haar-Fisz [6] (dashed) and our VST (solid). The asymptotic bounds, i.e., λ for the expectation and 1 for the variance, are also plotted. Fig. 1(a) shows that the variable stabilized using the Anscombe VST can be considered unbiased for λ ' 1, while using our VST for λ ' 0.1 only. The Haar-Fisz VST is unbiased. However, it requires λ ' 1 to stabilize the variance as shown in Fig. 1(b), whereas our VST only requires λ ' 0.1. The Anscombe VST stabilizes the variance for λ ' 10. Consequently, Poisson variables transformed using the Anscombe VST can be reasonably considered to be stabilized and unbiased for λ ' 10, using Haar-Fisz for λ ' 1, and using our VST (after low-pass filtering with h) for λ ' 0.1.

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1 0.9 Variance of stabilized variable

Mean of stabilized variable

0

10

−1

10

Anscombe Proposed VST Haar−Fisz sqrt(λ) bound −3

10

−2

10

−1

0

10 λ

10

0.8 0.7 0.6 0.5 0.4 0.3 0.2

Anscombe Proposed VST Haar−Fisz Unit bound

0.1 1

10

0 −3 10

−2

−1

10

(a) Fig. 1.

0

10 λ

10

1

10

(b)

Behavior of (a) E [Z] and (b) Var [Z] as a function of the underlying intensity, for the Anscombe VST, 2D Haar-Fisz

VST, and the proposed VST with a low-pass filter h = 2D B3 -Spline filter.

III. D ENOISING BY MS-VST+WAVELETS : G ENERAL SETTINGS A. The 1D Wavelet Transform In this section, the proposed VST will be incorporated within the multiscale framework offered by the (non-necessarily separable) undecimated wavelet transform (UWT), giving rise to the MS-VST. The undecimated transform is used because it provides translation-invariant denoising. Below, we will first discuss the 1D denoising case and the multidimensional extension is straightforward (Section IV-B and V-B). The UWT uses an analysis filter bank (h, g) to decompose a signal a0 into a set W = {d1 , . . . , dJ , aJ }, where dj is the wavelet (detail) coefficients at scale j and aJ is the approximation coefficients at the coarsest resolution. The passage from one resolution to the next one is obtained using the “`a trous” algorithm [27][28]: ¯ ↑j ⋆ aj )[l] = aj+1 [l] = (h

X

h[k]aj [l + 2j k],

dj+1 [l] = (¯ g ↑j ⋆ aj )[l] =

k

X

g[k]aj [l + 2j k]

(8)

k

¯ = h[−n], and “⋆” denotes convolution. The where h↑j [l] = h[l] if l/2j ∈ Z and 0 otherwise, h[n]

reconstruction is given by: aj [l] =

1 2

h

i

˜ ↑j ⋆ aj+1 )[l] + (˜ (h g ↑j ⋆ dj+1 )[l] . The filter bank (h, g, ˜h, g˜) needs

to satisfy the exact reconstruction condition.

¯ ↑j )j are low-pass filters, Now the VST can be combined with the UWT in the following way: since (h

we can first stabilize the approximation coefficients (aj )j using the VST, and then compute the detail coefficients from the stabilized aj ’s. Note that the VST now is scale-dependent (hence MS-VST). By February 2, 2007

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doing so, the asymptotic stabilized Gaussianity of the aj ’s will be transferred to the dj ’s, as will be shown later. Thus, the distribution of the dj ’s being known (Gaussian), we can detect the significant coefficients by classical HTs and zero the insignificant ones to achieve wavelet denoising. In summary, UWT denoising with the MS-VST involves the following three main steps: 1) Transformation (Sections IV and V): Compute the UWT and apply the MS-VST; 2) Detection (Section III-B): Detect significant detail coefficients by HTs2 ; 3) Estimation (Section V-C): Reconstruct the final estimate iteratively. The last step needs some explanation. The signal reconstruction requires inverting the MS-VST-combined UWT after the detection step. However, because of the nonlinearity of the MS-VST, direct inversion is not possible in the general case (see Section V-C). Even for the isotropic UWT (IUWT) which uses special filter banks yielding an invertible MS-VST, the direct inversion is far from optimal (see Sections IV-C and V-C). Hence, we propose to reformulate the reconstruction as a convex optimization problem and solve it by an iterative steepest descent algorithm (Section V-C). B. Detection by Wavelet-Domain Hypothesis Testing Let us first present the detection step, i.e., step 2) which is transform independent, by supposing that step 1) has been done so that the dj ’s have been Gaussianized. We want to test the null hypothesis H0 versus the alternative H1 for each wavelet coefficient d: H0 : d = 0 vs H1 : d 6= 0. HTs can be carried out individually in a coefficient-by-coefficient manner. First, the user pre-specifies a FPR in the wavelet domain, say α. Then the p-value of each coefficient pi := 2[1 − Φ(|d|/σ)] is calculated under H0 . Here Φ(x) is the standard normal cumulative distribution function, and σ is the asymptotic standard deviation of d after being stabilized by the MS-VST (see Sections IV-A and V-A). Finally, all the coefficients with pi > α will be zeroed. If we desire a control of global statistical error rates, multiple HTs should be used. For example, the Bonferroni over-conservative correction controls the probability of erroneously rejecting even one of the true null hypothesis, i.e., the Family-Wise Error Rate (FWER). Alternatively, one can carry out the Benjamini and Hochberg procedure [29] to control the false discovery rate (FDR), i.e., the average fraction of false detections over the total number of detections. The control of FDR has the following advantages over that of FWER: 1) it usually has a greater detection power; 2) it can easily handle correlated data [30]. The latter point allows the FDR control in non-orthogonal wavelet domains. Minimaxity of FDR 2

At this stage, one can apply his/her favorite thresholding or shrinkage rule on the stabilized detail coefficients.

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a0

T0

d1

+

¯ h

T1

T0

g¯

d1

¯ h

T1

g¯↑1

d2

¯ ↑1 h

T2

g¯↑2

d3

¯ ↑j h

Tj+1

a0







d2

+







¯ ↑1 h

d3

T2

T0−1

a0

dj+1

+







¯ ↑j

h

Tj+1 (aj+1 )

Tj+1

(a) Fig. 2.

Tj+1 (aj+1 )

(b)

Diagrams of the MS-VST in 1D for (a) the isotropic UWT and (b) the standard UWT.

has also been studied in various settings (see [31][32] for details). The interested reader may refer to [33][12] for further details of the denoising by wavelet-domain HTs. IV. MS-VST+I SOTROPIC UWT ˜ = δ, g˜ = δ) where h is a symmetric low-pass filter The IUWT [34] uses the filter bank (h, g = δ − h, h

such as the B3 -Spline filter. The particular structure of the analysis filters (h, g) leads to the iterative decomposition scheme shown in the left part of (9). The reconstruction is trivial, i.e., a0 = aJ +

PJ

j=1 dj .

This algorithm is widely used in astronomical applications [1] and biomedical imaging [35] to detect isotropic objects. As stated in Section III-A, we apply the VST on the aj ’s resulting in the stabilization procedure shown in the right part of (9):    aj IUWT  

dj

¯ ↑j−1 ⋆ aj−1 = h = aj−1 − aj



 MS-VST  aj + =⇒  dj IUWT 

¯ ↑j−1 ⋆ aj−1 = h

(9)

= Tj−1 (aj−1 ) − Tj (aj )

In (9), the filtering on aj−1 can be rewritten as a filtering on a0 = x: aj = h(j) ⋆ a0 , where h(j) = ¯ ↑j−1 ⋆ · · · ⋆ h ¯ ↑1 ⋆ h ¯ for j ≥ 1 and h(0) = δ . Tj is the VST operator at scale j (see Lemma 1): h

(j)

Let us define τk := set to:

P  i

q

Tj (aj ) = b(j) sgn(aj + c(j) ) |aj + c(j) | k

h(j) [i]

. Then according to (5), the constant c(j) associated to h(j) should be (j)

c(j) :=

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(10)

7τ2

(j)

8τ1

(j)

−

τ3

(j)

2τ2

(11)

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10

This stabilization procedure is directly invertible as we have: 

a0 = T0−1 TJ (aJ ) +

J X

j=1



dj 

(12)

The decomposition scheme and the inverse of MSVST+IUWT are also illustrated in Fig. 2(a). A. Asymptotic Distribution of the Detail Coefficients Theorem 1 (dj under a high intensity assumption) Setting 

D dj −→ N 0, λ→+∞

(j−1) τ2 (j−1) 2 4τ1

+

(j) τ2 (j) 2 4τ1

b(j) −

:=

(j) sgn(τ1 )/

hh(j−1) , h(j) i (j−1) (j) τ1

2τ1



q

(j)

|τ1 |, then we have:



(13)

Here h., .i represents the scalar product. This is a very useful result showing that under H0 , the detail coefficients are normally distributed with an intensity-independent variance which only relies on the filter h and the current scale. Hence, the application of the MS-VST requires almost no additional computational (j)

complexity (except the square-root), since the constants b(j) , c(j) , τk

and the stabilized variance can all

be pre-computed once for any given h. B. Extension to the Multi-dimensional Case ˜ qD = δ, g˜qD = δ) where hqD = ⊗q h. The filter bank in q D (q > 1) becomes (hqD , gqD = δ − hqD , h i=1

Note that gqD is in general nonseparable. The MS-VST decomposition scheme remains the same as (9), (j)

and the asymptotic result above holds true. The complexity for pre-computing b(j) , c(j) , τk

and the

stabilized variance in (13) remains the same as in the 1D case. C. Denoising Algorithm HTs based on the asymptotic normality (Theorem 1) can be applied to detect the significant wavelet coefficients. After zeroing the insignificant coefficients, we obtain the estimate of Λ by directly inverting the MS-VST+IUWT [Eq. (12)]. This is summarized in step 1 – step 6 in Algorithm 1. In step 6, the term Var [T0 (a0 )] corrects the bias due to squaring an estimate. Indeed, if Z = 



λ = E [a0 ] = E Z 2 − c(0) = Var [Z] + E [Z]2 − c(0) .

p

a0 + c(0) , then

It is worth pointing point out that this estimate from the direct MS-VST inversion is far from optimal. Indeed, because of the non-linearity of the VST and the over-completeness of the IUWT (a non-injective frame), the significant coefficients are not reproducible when IUWT is applied once again on this direct inverse estimate, implying a loss of important structures in the estimate. A better way is to find a constrained sparsest solution to be described in Section V-C. This solution is obtained iteratively by February 2, 2007

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solving a convex optimization problem, where the direct inverse serves as an initialization (step 7). In step 7, W denotes the IUWT operator, and P+ represents the projection on the nonnegative orthant. Details are given in Section V-C. Algorithm 1 MS-VST + Isotropic UWT Require: a0 := x; a low-pass filter h, Detection 1: for j = 1 to J do 2: Compute aj and dj using (9). 3: HTs on dj assuming the normal statistics established by Theorem 1, and get the estimate dˆj . 4: end for Estimation PJ ˆ 5: Estimate E [T0 (a0 )] by: Td 0 a0 = j=1 dj + TJ (aJ ) 6: 7:

2

(0) Estimate E [a0 ] by: a ˆ0 = Var [T0 (a0 )] + Td 0 a0 − c (0) ˆ. Initialize d = WP+ a ˆ0 and apply the same iterative estimation steps as in Algorithm 2 to get Λ

D. Experimental Results Astronomical image restoration: We have simulated an image containing disk-like X-ray celestial sources on a constant background (see Fig. 3(a)). From the leftmost column to the rightmost one, source radii increase from 2 to 7 pixels. This image has been convolved with a Gaussian PSF with a standard deviation of 2 pixels. The source amplitudes range from 0.1 to 126. The observed photon counts are shown in Fig. 3(b). We present the restoration results given by Anscombe [4] (Fig. 3(c)), Haar-Fisz [6] (Fig. 3(d)), and MS-VST (Fig. 3(e)). In all these denoising methods, the IUWT is used with h = 2D B3 -Spline filter. Cycle spinning with 5 shifts for each axis (so 52 shifts in total) is applied in Haar-Fisz

denoising. In all the experiments, the FDR is controlled at the level 10−4 . As revealed by Fig. 3, all the estimators perform comparatively well at high intensity levels (right part of the images). For low-intensity sources MS-VST and Haar-Fisz approaches are more sensitive than Anscombe denoising (see the bottom-left image corners). By quantifying the ℓ1 loss, MS-VST is found to give the most accurate estimate:

1 ˆ 1 n kΛ − Λkℓ

is 0.31 for Anscombe, 0.12 for Haar-Fisz, and 0.06 for

MS-VST. Furthermore we can notice that despite the effort of cycle spinning, Haar-Fisz denoising still suffers from the staircase artifacts. This is due to the fact that this VST is based on the Haar transform. We would have fewer artifacts if more cyclic shifts were used. However, this comes at the price of a higher computational burden as the total number of shifts increases rapidly with the dimension of data. The execution time of the Haar-Fisz denoising in this example on a P4 2.66GHz PC is 467s, i.e., 6 times slower than MS-VST (79s) and 31 times slower than Anscombe (15s). Finally, because of the secondary February 2, 2007

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(a)

(b)

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(c)

(d)

(e)

Fig. 3. Denoising a simulated astronomical X-ray image (image size: 1024×1024). IUWT is used with h = 2D B3 -Spline filter; the controlled FDR = 10−4 . (a) simulated sources (amplitudes ∈ [0.1, 126]; background = 0.05); (b) observed counts; (c) image ˆ − Λkℓ1 = 0.31; execution time = 15s); (d) image denoised by Haar-Fisz ( 1 kΛ ˆ − Λkℓ1 = 0.12; 5 denoised by Anscombe ( 1 kΛ n

n

ˆ − Λkℓ1 = 0.06; cyclic shifts for each axis, i.e., 52 shifts in total; execution time = 467s); (e) image denoised by MS-VST ( n1 kΛ Nmax = 10 iterations; execution time = 79s).

lobe of the B3 -Spline wavelet, all the denoising results exhibit some ringing artifacts around the restored sources. The ringing could be attenuated by redesigning the reconstruction filters of the IUWT [34]. Biomedical image restoration: Our method can be equally applied to biomedical image denoising. Fig. 4(a) and (c) show two optical slices of a 3D confocal image of a drosophila melanogaster ovary [36]. The ovary is in the development stage 7. The part of nurse cells consist of many nucleus surrounded by GFP (Green Fluorescent Protein)-marked Staufen genes. The slices of the denoised image are shown in Fig. 4(b) and (d). We can see clearly that the cytoplasm (homogeneous areas) is well smoothed and the Staufen genes are restored from the noise.

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(a)

(b)

(c)

(d)

Denoising of a 3D confocal image of a drosophila melanogaster ovary (image size: 512 × 512 × 24; max counts = 164; average counts I¯ = 19.6). 3D IUWT is used with h = 3D B3 -Spline filter; the controlled FDR = 10−10 ;

Fig. 4. Imax

Nmax = 10 iterations. (a) confocal slice at z = 18µm; (b) slice at z = 18µm of the MS-VST-denoised image; (c) confocal slice at z = 24µm; (d) slice at z = 24µm of the MS-VST-denoised image.

V. MS-VST+S TANDARD UWT In this section, we show how the MS-VST can be used to stabilize the wavelet coefficients of a standard separable UWT. Similarly to (9), we apply the VST on the approximation coefficients (aj )j , leading to the following scheme (see also the block-diagram of Fig. 2(b)):    aj UWT  

dj

¯ ↑j−1 ⋆ aj−1 = h = g¯↑j−1 ⋆ aj−1 q

MS-VST + =⇒ UWT

   aj  

dj

¯ ↑j−1 ⋆ aj−1 = h = g¯↑j−1 ⋆ Tj−1 (aj−1 )

(14)

where Tj (aj ) = b(j) sgn(aj + c(j) ) |aj + c(j) |, and c(j) is defined as in (11).

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A. Asymptotic Distribution of the Detail Coefficients Theorem 2 (dj under a high intensity assumption) Setting

b(j)

q

where σj2 =

1 (j−1)

τ2

X

g¯↑j−1 [m]¯ g ↑j−1 [n]

m,n

X k

(j)

(j)

D

:= 2 |τ1 |/τ2 , then dj −→ N (0, σj2 ), λ→+∞

h(j−1) [k]h(j−1) [k + m − n]

(15)

Parallel to Theorem 1, Theorem 2 shows the asymptotic normality of the detail coefficients with a zero mean under the null hypothesis and a variance depending only on the wavelet analysis filter bank and (j)

the current scale. Here the values of b(j) , c(j) , τk

and σj can all be pre-computed once (h, g) has been

chosen. B. Extension to the Multi-dimensional Case The scheme (14) can be extended straightforward to higher dimensional cases, and the asymptotic result above holds true. For example, in the 2D case, the UWT is given by the left part of (16) and the version combined with the MS-VST is given on the right:

UWT

   aj       1

dj

¯ ↑j−1 ⋆ aj−1 = g¯↑j−1 h

  d2j       d3

¯ ↑j−1 g¯↑j−1 ⋆ aj−1 = h

j

= g¯↑j−1 g¯↑j−1 ⋆ aj−1

¯ ↑j−1 h ¯ ↑j−1 ⋆ aj−1 = h =⇒

   aj     MS-VST   1

dj

¯ ↑j−1 ⋆ Tj−1 (aj−1 ) = g¯↑j−1 h

  d2j       d3

¯ ↑j−1 g¯↑j−1 ⋆ Tj−1 (aj−1 ) = h

j

= g¯↑j−1 g¯↑j−1 ⋆ Tj−1 (aj−1 )

+ UWT

¯ ↑j−1 h ¯ ↑j−1 ⋆ aj−1 = h

(16)

where hg ∗ a is the convolution of a by the separable filter hg , i.e., convolution first along the rows by (j)

h and then along the columns by g . The complexity for pre-computing the constants b(j) , c(j) , τk

and

σj remains the same as in the 1D case.

C. Denoising Algorithm HTs based on the asymptotic normality (Theorem 2) can serve to detect the significant Gaussianized wavelet coefficients. However, since the convolution (by g¯↑j−1 ) operator and the VST operator Tj−1 do not commute in (14), direct reconstruction scheme is unavailable. Thus, we propose below a sparsitydriven iterative approach for image reconstruction after wavelet coefficient detection. We first define the multiresolution support [37] M, which is determined by the set of the detected significant coefficients: M := {(j, l) | if dj [l] is significant}

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The estimation is then formulated as a constrained minimization problem in terms of the wavelet coefficients d: min J(d), d∈C

J(d) := kdk1

(18)

where C := S1 ∩ S2 , S1 := {d|d = Wx in M}, S2 := {d|Rd ≥ 0} where W represents the wavelet transform operator, and R its (weak-generalized) left inverse (synthesis

operator). It can be seen that we seek the sparsest solution by minimizing the ℓ1 objective [38][39] within the feasible set C := S1 ∩ S2 . The set S1 requires that the elements of d preserve the coefficients of interest; the set S2 ensures that the reconstructed estimate is positive-valued. (18) is a convex optimization problem which can be solved by Linear Programming (LP) using interiorpoint methods. However, the computational complexity of the LP solver increases dramatically with the size of the problem. Below we propose a much faster alternative based on the hybrid steepest descent (HSD) [40]. We first establish the following theorem. Theorem 3 Let d ∈ RL . Define the following optimization problem (ǫ ≥ 0): min Jǫ (d),

d∈CB

Jǫ (d) :=

PL

i=1

p

d[i]2 + ǫ

(19)

where CB := S1 ∩ S2 ∩ S3 , S3 := {d| kdk2 ≤ B, B ≥ kWxk1 } Define the HSD iteration scheme [40] (k ≥ 0):



(k) d(k+1) := TCB d(k) ǫ ǫ − βk+1 ∇Jǫ TCB dǫ



(20)

where ∇Jǫ is the gradient of Jǫ , and TCB := PS3 ◦ PS1 ◦ QS2 , PS3 d :=

d · min(kdk2 , B); kdk2

PS1 d :=

The step sequence satisfies: lim βk = 0,

k→∞

X

βk = +∞

   Wx   d

and

in M otherwise

X

k≥1

k≥1

;

QS2 d := WP+ Rd

|βk − βk+1 | < +∞

(21)

(22)

Suppose that in (ii)-(v) below W represents a tight frame decomposition and R its pseudo-inverse operator. Then we have: (i) The solution set of (18) is the same as that of (19) with ǫ = 0; (ii) TCB is nonexpansive, and its fix point set is F ix(TCB ) = CB 6= ∅; (0)

(k)

(iii) ∀ǫ > 0, with any dǫ ∈ RN , dǫ

−→ d∗ǫ , where d∗ǫ is the unique solution to (19);

k→+∞

(iv) As ǫ → 0+ , the sequence (d∗ǫ )ǫ>0 is bounded. Therefore, it has at least one limit point;

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(v) As ǫ → 0+ , every limit point of the sequence (d∗ǫ )ǫ>0 is a solution to (18). Theorem 3 implies that in practice instead of directly solving (18), one can solve its smoothed version (19) by applying (20) with a small ǫ. Moreover, Theorem 3 allows us to set a sufficiently large B so that the constraint S3 is always satisfied in real problems. Thus, TCB is simplified to TCB = TC := PS1 ◦ QS2 . We also point out that although Theorem 3 assumes a tight frame decomposition and pseudo-inverse reconstruction, in our experiments, it has been observed that the iterations (20) applied equally to general frame decompositions and inverses, and performed very well even with ǫ = 0 (see results in Sections IV-D and V-D). Below, (23) rewrites (20) with ǫ = 0. 

d(k+1) := TC d(k) − βk+1 ∇J TC d(k)



where

∇J(d) = sgn(d)

(23)

Assuming no changes of the signs of the wavelet coefficients when applying (23), (23) becomes indeed a soft thresholding with a threshold βk+1 . Finally, the MS-VST denoising using the standard UWT is summarized in Algorithm 2. Algorithm 2 MS-VST + Standard UWT Require: a0 := x; a wavelet filter bank (h, g, ˜h, g˜), Detection 1: for j = 1 to J do 2: Compute aj and dj using (14). 3: HTs on dj assuming the normal statistics established by Theorem 2 and update M. 4: end for Estimation 5: Initialize d(0) with the wavelet coefficients of x in M. 6: for k = 1 to Nmax do ˜ := PS ◦ QS d(k−1) 7: d 1 2 ˆ := d(k) := STβ [d] ˜. 8: d k 9: end for ˆ. ˆ = P+ Rd 10: Get the estimate Λ

In Algorithm 2, Nmax is the maximal number of iterations, and STβk is a coefficient-wise soft thresholding operator with threshold βk . A possible choice of the sequence (βk )k is a linearly decreasing one: βk =

Nmax −k Nmax −1 , k

= 1, 2, · · · , Nmax . It can be noted that for (βk )k chosen as above, the conditions in

(22) are all satisfied as Nmax → ∞. To initialize d(0) , although Theorem 3 suggests that any point can be used, different initial values do influence the number of iterations necessary for the convergence. A good choice of d(0) can be the wavelet coefficients of x with all insignificant coefficients set to zero (those not in M). The computational cost of the whole denoising is dominated by the iterative estimation step. February 2, 2007

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This step involves an analysis and a synthesis at each iteration and thus has a complexity of O(Nmax V ), where V = O(N log N ) is the complexity of UWT. Now if we reconsider the IUWT case (Section IV), it is clear that the iterative reconstruction will outperform the direct inverse. Indeed, every iteration of (23) involves a projection onto S1 that restores all the significant coefficients. This actually results in a better preservation of the important structures. D. Experimental Results Natural image restoration: We show the performance of the MS-VST with the standard UWT on Barbara image (Fig. 5(a)). To simulate low-count situations, the intensities of the original image are heavily scaled down, so that the maximal intensity in Fig. 5(a) is around 20 only. The noisy image is shown in Fig. 5(b). We compare the restorations given by Anscombe (Fig. 5(c)), Haar-Fisz (Fig. 5(d)) and MS-VST (Fig. 5(e)). In these denoisers, the UWT filter bank is the 7/9 biorthogonal filters [23]. Cycle spinning is adopted in Haar-Fisz where a total of 52 shifts are used. The FDR is controlled at the level 0.05. Visually the MS-VST-denoised image preserves more details and textures, so is sharper than the other results. Quantitatively, MS-VST gives again the smallest ℓ1 loss. VI. D ENOISING BY MS-VST+R IDGELETS A. The Ridgelet Transform The ridgelet transform [24] has been shown to be very effective for representing global lines in an image. Ridgelet analysis may be constructed as wavelet analysis in the Radon domain. Recall that the 2D Radon transform of an object f is the collection of line integrals indexed by (θ, t) ∈ [0, 2π) × R given by Rf (θ, t) =

Z

R2

f (x1 , x2 )δ(x1 cos θ + x2 sin θ − t) dx1 dx2

(24)

where δ is the Dirac distribution. Then the ridgelet transform is precisely the application of a 1D wavelet transform to the slices of the Radon transform where the angular variable θ is constant and t is varying. For each scale s > 0, position t ∈ R and angle θ ∈ [0, 2π), the 2D ridgelet function ψs,t,θ is defined from a 1D wavelet function ψ as: ψs,t,θ (x1 , x2 ) = s−1/2 · ψ((x1 cos θ + x2 sin θ − t)/s)

(25)

A ridgelet is constant along the lines x1 cos θ + x2 sin θ = const. Transverse to a ridge is a wavelet. Thus, the basic strategy for calculating the continuous ridgelet transform is first to compute the Radon transform Rf (t, θ) and second, to apply a 1D wavelet transform to the slices Rf (·, θ). Different digital February 2, 2007

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50

50

100

100

150

150

200

200

250

250

(a)

(b)

50

50

50

100

100

100

150

150

150

200

200

200

250

250

250

(c)

(d)

(e)

Poisson denoising of Barbara image (image size: 256 × 256). 7/9 biorthogonal filter bank is used in the UWT; the controlled FDR = 0.05. (a) original intensity image (max. intensity Imax = 19.7; average intensity I¯ = 9.9); (b) Poisson noisy Fig. 5.

ˆ − Λkℓ1 = 1.09; execution time = 6s); (d) image restored by Haar-Fisz with image; (c) image restored by Anscombe ( n1 kΛ ˆ − Λkℓ1 = 1.11; 5 cyclic shifts for each axis, so 52 shifts in total; execution time = 156s); (e) image cycle spinning ( n1 kΛ ˆ − Λkℓ1 = 1.06; Nmax = 10; execution time = 23s). restored by MS-VST ( 1 kΛ n

ridgelet transforms can be derived depending on the choice of both the Radon algorithm and the wavelet decomposition [41]. The Slant Stack Radon (SSR) transform [42][43] is certainly the one which leads to the best results because of its geometrical accuracy. The inverse SSR has however the drawback to be iterative. If computation time is an issue, the recto-polar Radon transform is a good alternative. More details on the implementation of these Radon transforms can be found in [26][42][43][41].

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B. MS-VST with Ridgelets As a Radon coefficient is obtained from an integration of the pixel values along a line, the noise in the Radon domain follows also a Poisson distribution. Thus, we can apply the 1D MS-VST denoising with wavelets described in Sections IV or V to the slices of the Radon transform. Hence, the Ridgelet Poisson denoising algorithm consists of the following three steps: Algorithm 3 MS-VST + Ridgelets 1: Apply the Slant Stack Radon transform. 2: For each Radon slice, apply the 1D UWT denoising with the MS-VST (Algorithm 1 or 2). 3: Apply the inverse Slant Stack Radon transform.

C. Experimental Results We have simulated an image with smooth ridges shown in Fig. 6(a). The maximal intensities and the average intensities of the 9 vertical ridges vary progressively from 0.03 to 0.1 and from 0.015 to 0.05, respectively; the inclined ridge has a maximal intensity of 0.06 and an average intensity of 0.02. The Poisson count image is shown in Fig. 6(b). For this extremely low-intensity case, the MS-VST estimate with IUWT cannot detect anything (Fig. 6(c)), whereas the MS-VST with the ridgelet transform correctly detects all the ridges (Fig. 6(d)). VII. D ENOISING BY MS-VST+C URVELETS A. The First Generation Curvelet Transform The ridgelet transform is efficient for finding only the lines of the size of the image. To detect line segments, a partitioning need to be introduced. The image is first decomposed into smoothly overlapping blocks of side-length B pixels, and the ridgelet transform is applied independently on each block. This is called the local ridgelet transform. The curvelet transform [44][45] opens the possibility to analyze an image with different block sizes, but with a single transform. The idea is to first decompose the image into a set of wavelet bands using the IUWT, and to analyze each band with a local ridgelet transform. The block size can be changed at each scale level. The coarsest description of the image (aJ ) is not processed. This method is very effective in detecting anisotropic structures of different lengths. More details can be found in [26].

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50

50

100

100

150

150

200

200

250

250

(a)

(b)

50

50

100

100

150

150

200

200

250

250

(c) Fig. 6.

(d)

Poisson denoising of smooth ridges (image size: 256 × 256). The controlled FDR = 10−7 ; Nmax = 10; positive

significant coefficients are detected. (a) image of smooth ridges (the maximal intensities and the average intensities of the 9 vertical ridges vary progressively from 0.03 to 0.1 and from 0.015 to 0.05, respectively; the inclined ridge has a maximal intensity of 0.06 and an average intensity of 0.02); (b) Poisson noisy image; (c) image restored by MS-VST+IUWT; (d) image restored by MS-VST+Ridgelets.

B. MS-VST with Curvelets As the first step of the algorithm is an IUWT, we can stabilize each resolution level in the same way as described in Section IV. We then apply the local ridgelet transform on each stabilized wavelet band. Insignificant Gaussianized curvelet coefficients will then be zeroed using the same hypothesis testing framework as in the wavelet case. Finally, a coarse-to-fine direct reconstruction can be performed by first inverting the local ridgelet transforms and then inverting the MS-VST+IUWT (see Section IV-C). We now present a sketch of the Poisson curvelet denoising algorithm:

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Algorithm 4 MS-VST + Curvelets 1: Apply the MS-VST+IUWT with J scales to get the stabilized wavelet subbands (dj )j . 2: set B1 = Bmin 3: for j = 1 to J do 4: Partition the subband dj with blocks of side-length Bj and apply the digital ridgelet transform to each block to obtain the stabilized curvelet coefficients. 5: if j modulo 2 = 1 then 6: Bj+1 = 2Bj 7: else 8: Bj+1 = Bj 9: end if 10: HTs on the stabilized curvelet coefficients. 11: end for 12: Inverse the ridgelet transform in each block before inverting the MS-VST+IUWT.

It is not as straightforward as with the wavelet and ridgelet transforms to derive the asymptotic noise variance in the stabilized curvelet domain. In our experiments, we derived them using simulated data with Poisson noise only. After having checked that the standard deviation in the curvelet bands becomes stabilized as the intensity level λ increases (which means that the stabilization is working properly), we stored this standard deviation σj,l for each wavelet scale j and each direction angle l. Then, once the stabilized curvelet transform is applied to our data, these values of (σj,l )j,l serve in the hypothesis testing framework described in Section III-B to test the significance of each stabilized curvelet coefficient at each scale j and direction angle l. In this algorithm, the estimation is given by the direct MS-VST inverse, but we can also introduce the same iterative estimation as described in Section V-C. That is, we derive first the multiresolution support M := {(j, l, k)}, where (j, l, k) means that the stabilized curvelet coefficient at scale j , direction l and position k is significant. Following the same strategy as described in Section V-C, once M is

determined the solution can be found by solving a constrained ℓ1 -minimization problem. For the same

reason explained in Sections IV-C and V-C, this second approach produces better results and is always used in our experiments. C. Experimental Results Fig. 7 shows a denoising example using MS-VST+Curvelet. The image (Fig. 7(a)) is obtained from a turbulence simulation code [46]. The Poisson noise is due to the fact that there is a limited number of particles. In this experiment, individual HTs are used with a 5σj,l detection level, which corresponds to a FPR α = 5.7 × 10−7 . Fig. 7(b) shows the intensity estimate obtained from ten iterations of the February 2, 2007

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(a)

(b) Fig. 7. Poisson denoising of a simulated turbulence image (image size: 512 × 256). The controlled FPR α = 5.7 × 10−7 (i.e., 5σ as detection level); Nmax = 10. (a) Poisson noisy data (max. counts Imax = 5; average counts I¯ = 0.2); (b) denoised image by MS-VST+Curvelet.

HSD reconstruction. We can see that the highly directional features contained in the data are very well restored. VIII. D ISCUSSION AND CONCLUSION In this paper, we have introduced a new variance stabilization method and shown that it can be easily combined with various multiscale transforms such as the undecimated wavelet (isotropic and standard), the ridgelet and the curvelet transforms. Based on our multiscale stabilization, we were able to propose a new strategy for removing Poisson noise and our approach presents the following advantages: February 2, 2007

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•

It is more efficient than existing methods in detecting faint features at a very low-count rate.

•

We have the choice to integrate the VST with the multiscale transform we believe to be the most suitable for restoring a given kind of morphological features (isotropic, line-like, curvilinear, etc).

•

The computation time is similar to that of a Gaussian denoising, which makes our denoising method capable of processing large data sets.

This work can be extended along several lines in the future. First, the curvelet denoising could be improved if the VST is applied after the Radon transform in the local ridgelet transform step, rather than on the wavelet coefficients as proposed here. This is however not trivial and requires further investigations. Second, new multiscale transforms have been recently proposed such as the fast curvelet transform [47] and the wave atom transform [48], and it would also be very interesting to investigate how our MSVST could be linked to them. Finally, here we have considered the denoising with a single multiscale transform only. If the data contains features with different morphologies, it could be better to introduce several multiscale transforms in the denoising algorithm. This could be done in a very similar way as in the Gaussian noise case [49]. ACKNOWLEDGMENT The authors would like to thank M. M. Mhlanga and J.-C. Olivo-Marin of the Institut Pasteur for kindly providing the image of drosophila melanogaster ovary (Fig. 4), and B. Afeyan for kindly providing the simulated turbulence image (Fig. 7). This work is partly supported by the Institut Pasteur, CNRS and CEA. R EFERENCES [1] J.-L. Starck, F. Murtagh, and A. Bijaoui, Image Processing and Data Analysis: The Multiscale Approach.

Cambridge

University Press, 1998. [2] J. Pawley, Handbook of Biological Confocal Microscopy, 3rd ed.

Springer, 2006.

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[8] ——, “De-noising by soft-thresholding,” IEEE Transactions on Information Theory, vol. 41, no. 3, pp. 613–627, 1995. [9] R. D. Nowak and R. G. Baraniuk, “Wavelet-Domain Filtering for Photon Imaging Systems,” IEEE Transactions on Image Processing, vol. 8, no. 5, pp. 666–678, May 1999. [10] A. Antoniadis and T. Sapatinas, “Wavelet shrinkage for natural exponential families with quadratic variance functions,” Biometrika, vol. 88, pp. 805–820, 2001. [11] E. D. Kolaczyk, “Nonparametric estimation of intensity maps using Haar wavelets and Poisson noise characteristics,” The Astrophysical Journal, vol. 534, pp. 490–505, 2000. [12] B. Zhang, M. J. Fadili, J.-L. Starck, and S. W. Digel, “Fast Poisson Noise Removal by Biorthogonal Haar Domain Hypothesis Testing,” Statsitcal Methodology, 2006, submitted. [13] A. Bijaoui and G. Jammal, “On the distribution of the wavelet coefficient for a Poisson noise,” Signal Processing, vol. 81, pp. 1789–1800, 2001. [14] E. D. Kolaczyk, “Wavelet shrinkage estimation of certain Poisson intensity signals using corrected thresholds,” Statist. Sinica, vol. 9, pp. 119–135, 1999. [15] ——, “Bayesian multiscale models for Poisson processes,” J. Amer. Statist. Ass., vol. 94, no. 447, pp. 920–933, Sep. 1999. [16] K. E. Timmermann and R. D. Nowak, “Multiscale Modeling and Estimation of Poisson Processes with Application to Photon-Limited Imaging,” IEEE Trans. Inf. Theo., vol. 45, no. 3, pp. 846–862, Apr. 1999. [17] R. D. Nowak and E. D. Kolaczyk, “A statistical multiscale framework for Poisson inverse problems,” IEEE Transactions on Information Theory, vol. 46, no. 5, pp. 1811–1825, Aug. 2000. [18] M. Jansen, “Multiscale Poisson data smoothing,” J. Roy. Statist. Soc. ser. B, vol. 68, no. 1, pp. 27–48, 2006. [19] P. Besbeas, I. D. Feis, and T. Sapatinas, “A Comparative Simulation Study of Wavelet Shrinkage Estimators for Poisson Counts,” Internat. Statist. Rev., vol. 72, no. 2, pp. 209–237, 2004. [20] S. Sardy, A. Antoniadis, and P. Tseng, “Automatic smoothing with wavelets for a wide class of distributions,” J. Comput. Graph. Stat., vol. 13, no. 2, pp. 399–421, Jun. 2004. [21] R. M. Willett and R. D. Nowak, “Fast, Near-Optimal, Multiresolution Estimation of Poisson Signals and Images,” in EUSIPCO, San Diego, CA, 2004. [22] ——, “Multiscale Poisson Intensity and Density Estimation,” Duke University, Tech. Rep., 2005. [23] S. G. Mallat, A Wavelet Tour of Signal Processing, 2nd ed.

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[24] E. Cand`es and D. Donoho, “Ridgelets: the key to high dimensional intermittency?” Philosophical Transactions of the Royal Society of London A, vol. 357, pp. 2495–2509, 1999. [25] E. J. Cand`es and D. L. Donoho, “Curvelets – A Surprisingly Effective Nonadaptive Representation For Objects with Edges,” in Curve and Surface Fitting: Saint-Malo 1999, A. Cohen, C. Rabut, and L. Schumaker, Eds.

Nashville, TN:

Vanderbilt University Press, 1999. [26] J.-L. Starck, E. Cand`es, and D. Donoho, “The Curvelet transform for image denoising,” IEEE Transactions on Image Processing, vol. 11, no. 6, pp. 131–141, 2002. [27] M. Holschneider, R. Kronland-Martinet, J. Morlet, and P. Tchamitchian, “A Real-Time Algorithm for Signal Analysis with the Help of the Wavelet Transform,” in Wavelets: Time-Frequency Methods and Phase-Space. Springer-Verlag, 1989, pp. 286–297. [28] M. J. Shensa, “Discrete wavelet transforms: Wedding the a` trous and Mallat algorithms,” IEEE Transactions on Signal Processing, vol. 40, pp. 2464–2482, 1992.

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