A Bayesian grouplet transform

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DOI:10.1007/s11760-019-01423-6

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A Bayesian grouplet transform

Lotfi Chaari1 _•2

49

Abstract

In the signal processing literature, wavelet transforms have been widely used for compression, restoration or texture processing. In this sense, grouplet transforms have been proposed to account for the geometrical image regularities. A grouplet transform (basis or frame) is based on an a priori fixed association field that groups image coefficients according to geometrical considerations. In this paper, we propose a method for estimating this association field in a Bayesian way. The resulting association field is therefore adaptive to the processed image content. A hierarchical Bayesian mode! is proposed and the inference is conducted using a Markov Chain Monte Carlo (MCMC) algorithm. The proposed method is tested on standard images in terms of association field quality and quantitative properties of the obtained wavelet coefficients. Specifically, the proposed method provides coefficients with low correlation level, and for which the highest level of energy is concentrated within the 20% most significant. These promising results confirm the potential of the proposed method for several image processing applications such as compression, denoising or restoration.

Keywords Grouplet • Wavelets • Bayesian models • MCMC

1 Introduction

Wavelet transforms are one of the most efficient tools in sig­ nal and image processing. Such transforms jointly provide space and spectral informations that could help processing the target signais. Several transforms have been proposed since the early literature such as the Haar, Daubechies or Symmlet transforms [l]. Redundant wavelet transforms have then been proposed to provide better directional analyses, such as the contourlets [2] or the dual tree [3] transforms.

Wavelets, either bases or frames, have been widely used in many applications in the recent signal and image process­ ing literature, such as remote sensing [4,5], medical imaging [6-8] or compressed sensing [9,10]. In the same research

avenue, adaptive wavelet transforms have also motivated many recent studies like in image compression [1 1-1 3], tex­ ture analysis [14-1 6], or image restoration [7,1 7]. The main idea is to provide transforms that better account for the sig­ nal/image content than standard transforms, which allows to finely explore the target signal/image properties.

181 Lotfi Chaari

Jotfi.chaari@enseeiht.fr

MIRACL laboratory, University of Sfax, Sfax, Tunisia 2 IRIT - INP-ENSEEIHT, University of Toulouse, Toulouse,

France

Recently, grouplet transforms have been proposed by Mal­ lat [18] using the geometry of the data to build the transform. As claimed, this helps better representing and exploring shapes and textures as the main information in an image. The grouplet motivation was partly inspired by recent stud­ ies about geometric perception through grouping processes [18,1 9].

As stated in [18], wavelet transforms can account for the local image regularity, in contrast to the geometrical direc­ tional regularity. Grouplets provide stable geometrical image representation using an orthogonal weighted Haar lifting, where a multi-scale association field is used. Both bases and tight frames have been proposed and tested on several image processing examples such as denoising and super-resolution [20,2 1].

However, finding an appropriate association field is a main issue to build a grouplet transform since the whole trans­ form is supposed to be adapted to the geometrical content of the target image. Finding the optimal field is crucial to build an efficient basis or frame with good sparsity proper­ ties as developed in a number of recent works of the literature [2 2,23]. To the best of our know ledge, all the existing method rely on a prior edge or line detection step, or even a block matching algorithm. As illustrated in our experiments, such methods provide association fields with low regularity

prop-erties, which may decrease the sparsity level of the result­ ing grouplet coefficients. Moreover, additional parameters adjustment is generally required. In this paper, we propose a Bayesian mode! with regularity constraints to directly esti­ mate this association field from the target image. Bayesian models are becoming more and more used in combination with wavelet transforms [24] and variational models [25]. Hierarchical Bayesian models have the advantage to be Jess user dependent by estimating the parameters directly from the data through putting suitable priors on them. We mode! the association field estimation as a denoising problem where the association information is estimated as a discrete variable. A fully automatic algorithm using an Markov Chain Monte Carlo (MCMC) scheme is proposed to build the inference.

The rest of this paper is organized as follows. Sect. 2 recalls the main principles of grouplet transforms, while Sect. 3 introduces the Bayesian mode! we propose to esti­ mate the association field. This mode! is validation in Sect. 4 through two main experiments. Finally, conclusions and future work are drawn in Sect. 5.

2 The grouplet transform

We first recall the principle of the Haar transform for a discrete signal I E

R

N

_.

_{Taking the resolution level j E} { 1, ... , J} and the scale 2j, the Haar transform computes approximation and detail coefficients for an index n E {O, ... , 2-j _{N} as follows:}

[ _{]_ 1[2n]+I[2n+l] dd·[ 1}_ I[2n+l]-1[2n] a n _

2 an 1 n - .,/2j-l • (1)

At the scale 21_{, the approximation coefficients are nor­}

malized as a1[n]

=

a[n]2112. The obtained coefficients

(a, { d _j

h

�h,J) are the projection of the signal I on the Haar orthonormal basis, where a

=

{a, [n]}o<_-n_-<₂-i N and

dj

=

{dj [n]}o�n�2-iN·

For the Haar transform, the approximation and detail coefficients are calculated based on horizontal neighbors (coefficient 2n and 2n + 1). Grouplet transforms relax this constraint and perform grouping between coefficients that are not necessarily neighbors. To transform images, this new grouping is based on an a priori defined association field that involves a group of embedded grids _{{Çj }i�j�J} with

9j+1 C Çj for every j [18] Fig. 1. Illustrates an example

of association field between odd and even columns where a pixel can be grouped with another one on a different image raw. It is worth noticing that a larger neighborhood can also be considered involving more neighboring columns [ 18]. How­ ever, without Joss of generality, we focus in this paper on the first order case as illustrated in Fig. 1 where each pixel of an even column is grouped with a pixel from the previous odd column, and this for the sake of presentation.

o /* 0/ o /* 0 * o/* o/* o/* 0�

:�

0 * o/* o/* o/* o/* o�* o+--* o+--* o..._,_* ..._,* 0 ..._,* ◊...__ 0 * ..._,* ◊...__ 0 * o /* o /* o /* o+--* ..._,* ◊...__ 0 * 0 ..._,* o+--* Fig. 1 Association field for a grouplet transform where each even pixel is associated with a neighboring odd one

y. (1, 1) ◄---� X y • (1,0) , ◄---◄---X (1,-1)

Fig. 2 Examples of association vectors with x = 1 and y E I = { -1, 0, 1}. Each vector groups a pixel from an even column to a pixel belonging to the previous odd column

By analyzing the association field in Fig. 1, we can notice that it could be represented by a vector field. Each vector has its origin on an even column. The first coordinate can be fixed to x

=

1, whereas the second coordinate y is an integer between -1 and + 1. Figure 2 illustrates some examples of association vectors where orientations of -45°_{, 0}° _{and +45}° can be identified. The vectors illustrated in Fig. 2 link a pixel from an even column with a pixel belonging to the intersec­ tion of the previous odd column and the current line with its top and bottom neighboring lines.

For an image I E JR.MxN, _{performing an orthogonal grou­}

plet transform consists of sequentially applying the transform over columns and then over lines. Each step requires the def­ inition of an association field (grid Ç j) that could be different from a step to another, and from a resolution level to another. The main issue in this paper is to estimate an appropriate association field that will allow us to built the grid Çj for

each resolution level j.

3 Bayesian association field 3.1 Problem formulation

Based on the adopted notations, and after setting the first coordinate of the vector field, each association field can be

modeled as a matrix A E IMxN 12, and this for any configu­

A denoising problem is therefore formulated where, for a coordinate x= (x, y), the image coefficient I (x) is assumed to be a Gaussian perturbation of another coefficient of the same image such that

I(x) = I(x + A(x)) + n (2) where I is the image, A is the association field, and n is an additive Gaussian noise with varianceσ_n2.

Estimating the best association field then consists of esti-mating a version of A based on the available data and the knowledge about the noise properties, in addition to some spatial regularity constraints. In this paper, we propose a hierarchical Bayesian model to perform this estimation in an automatic way. An appropriate likelihood is formulated in addition to suitable priors on the model parameters (associa-tion field and noise variance) in order to build the hierarchical model and derive estimators.

3.2 Hierarchical Bayesian model

3.2.1 Likelihood

Assuming that the image coefficients are i.i.d., and following the Gaussian noise assumption of (2) at each independent spatial position x, the likelihood can be expressed as:

f(I) = x f(I(x)) ∝ 1 σM_×N/2 n  x exp− 

(I(x) − I(x + A(x)))2

2σ2

n

 . (3) 3.2.2 Priors

In our model, the vector of unknown parameters is denoted byθ = {σ2

n, A}. Prior onσ_n2

Sinceσn2is real-positive, and to keep using a non-informative prior, a Jeffrey’s prior has been used which writes as

f(σ_n2) ∝ 1 σ2

n

1R+(σn2) (4)

where1_R+is the indicator function onR+. Using this prior is very common for positive parameters especially when a non-informative prior is needed since no additional hyperpa-rameters are required. The interested reader can refer to [26] for further details.

Prior on A

Since the association field coefficients are assumed to be realizations of a discrete variable (A ∈ {−1, 0, 1}M×N/2

for example), a Markov random field prior with interaction parameterβ has been used:

f(A|β) = x f(A(x)|β) = x Z−1(β) exp ⎛ ⎝β y_∼x δ(A(x), A(y)) ⎞ ⎠ , (5)

whereδ(η, ζ ) = 1 if η = ζ and 0 otherwise, Z(β) is a nor-malization constant, and y∼ x denotes that y is a neighbor of y. This prior helps retrieving a spatially regular associ-ation field. The energy of the exponential term promotes homogeneity of neighboring coefficients according to the hyperparameterβ.

3.2.3 Hyperpriors

As regards the hyperparameterβ, it can be estimated through putting an appropriate hyperprior on it. Since this hyperpa-rameter is real-positive, an inverse Gamma (IG) distribution can be used: f(β|γ, η) = IG(β|γ, η) = _{(γ )}ηγ β−γ −1exp −η_β  1R+(β), (6)

where the scalarsγ and η can be manually set to 10−3to guar-antee a non-informative distribution. However, these values can be set using the expectation of theIG distribution η

γ − 1. This prior is widely used in the Bayesian literature since it is a conjugate distribution with respect to the Gaussian distri-bution.

3.3 The Bayesian inference scheme

In this paper, we adopt a maximum a posteriori (MAP) strat-egy in order to estimate the model parameter vectorθ. This is performed based on the combination of the model likelihood and priors formulated above in the hierarchical Bayesian model. The posterior distribution ofθ can be expressed as

f(θ|β) ∝ f (I|σn2) f (A|β) f (σn2) f (β|γ, η) ∝ 1 σM_×N/2 n  x exp− 

((I(x) − I(x + A(x)))2

2σ2 n  × 1 σ2 n 1R+(σn2) × β−γ −1exp −η β  1R+(β) × x Z−1(β) exp ⎛ ⎝β y∼x δ(A(x), A(y)) ⎞ ⎠ . (7)

Unfortunately, deriving closed-form estimators associated with the above posterior is a complicated task. For this

rea-son, we propose to numerically approximate this posterior by resorting to a Gibbs sampler (GS) [26] that allows generating samples asymptotically distributed according to (7).

The GS iteratively generates samples distributed accord-ing to the conditional distributions associated to (7). More precisely, it sequentially samples according to f(σ_n2|I, A),

f(A|I, β) and f (β|A). 3.3.1 The Gibbs sampler

The main steps of the designed GS are detailed in Algo-rithm1.

Algorithm 1: Gibbs sampler.

- Initialize with someθ(0)andβ(0)and set t= 0;

for r= 1 . . . T do

➀ Sample σ2

n(t)according to its posterior f(σn2|A, I);

➁ Sample A(t)_{according to its posterior f(A|I, β);}

➂ Sample β(t)_{according to its posterior f(β|A);}

➃ Set t ← t + 1 ;

end

Assuming that the convergence is reached after T/2 itera-tions (which corresponds to the burn-in period), the designed GS generates a chain of samples{A(t)}t=1,...,T that will be used to estimate A using the minimum means square error (MMSE) or the MAP estimators, where T is the total num-ber of iterations. The same procedure is used to estimate all the model parameters and hyperparameters ( σ2

n and β). It is worth noticing that the first half of any sampled chain has to be discarded before calculating the estimators since it con-tains samples corresponding to the burn-in period.

The obtained conditional distributions are detailed in Sect.3.3.2.

3.3.2 The conditional distributions

This section develops the conditional distributions of the model parameters and hyperparameters.

Samplingσn2

Straightforward calculations based on the joint posterior in (7) lead to the following conditional distribution ofσ_n2:

σ2 n|A, I ∼ IG  σ2 n| M× N 4 , 1 2  x ((I(x) − I(A(x)))2  . (8) f(A|I, β) ∝ x exp ⎛ ⎝−((I(x) − I(x+A(x)))2 2σ2 n +β y∼x δ(A(x), A(y)) ⎞ ⎠. (9) This conditional distribution is not easy to sample. For this reason, a Metropolis-Hastings (MH) move is requested to sample from. A uniform distribution overI is used as a pro-posal.

Samplingβ

Similar calculations based on the joint posterior in (7) lead to the following form of the conditional distribution ofβ:

f(β|A) ∝ β−γ −1exp −η β  1R+(β) × x Z−1(β) exp ⎛ ⎝β y∼x δ(A(x), A(y)) ⎞ ⎠ . (10) Since this conditional distribution is not easy to sample, an MH move is required to sample from. A truncated Gaussian distribution onR₊is used as a proposal.

4 Experiments

In this section, we investigate the properties of the proposed Bayesian grouplet transform through two main experiments. The first experiment validates the Bayesian estimation of the association field, while the second one studies the energy properties of the obtained wavelet coefficients.

4.1 Association field

In this section, we investigate the convergence of the pro-posed algorithm to build Bayesian grouplet transforms. The proposed method is applied on a seismic image illustrated in Fig.3a. After running the proposed MCMC algorithm for 2000 iterations (91 sec with an R2014b Matlab implementa-tion on a 64 bits MacBook Pro with a 3.1 GHz Intel Core i7 processor) with a burn-in period of 1000 iterations, the obtained association field is illustrated in Fig.3b as a vec-tor field where an arrow links a pixel from an even column to a pixel from the previous odd column. Each coefficient of the association field has been obtained after calculating a hierarchical MAP estimator based on the sampled chains. For the sake of comparison, the association field obtained with the method in [23] is displayed in Fig. 3c. Through a visual inspection of the association field in Fig. 3b, one This IG distribution is easy to sample.

Sampling A

The conditional distribution of A can be obtained from (7) and writes

(a)

(b)

(c)

Fig. 3 Seismic image (a) and the computed association fields using the proposed method (b) and the method in [23). Arrows indicate pixels from add and even columns associated with the estimated field

can conclude that the proposed Bayesian scheme provides an association field that adapts to the local features of the image. This can be clearly noticed by analyzing the vector field close to the seismic wave on the bottom of the image. For homogeneous regions of the image, a horizontal association field is generally estimated, in contrast to the result obtained by the method in [23]. This shows that the proposed method is robust to the presence of noise in the processed image and provides regular association fields.

For some illustrative pixels of the image, the histogram

of the sampled chains are displayed in Fig. 4. Specifically,

this figure illustrates histograms of pixels with different esti­ mated association values (1, -1 and 0). The histograms show the good separation property of the sampled chains, which ensures accurate estimation with a hierarchical MAP estima­ tor.

A

1400 1200 1000 800 600 400 200 0 0 0.2 0.4 0.6 0.8 1.2 1.4 1.8 1.8 2

A

-_-

_-1

2000 1800 1600 1400 1200 1000 800 600 400 200 0 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

A

-_-

₀

1800 1600 1400 1200 1000 800 600 400 200 0 -2 -1.5 -1 -0.5 0 0.5 1.5 2

Fig. 4 Histograms of the sampled chains of the association parameter relative to three pixels of the seismic image

4.2 Coefficient properties

In this section, we investi gate some properties of the proposed

(a)

(b) Approximation coefficients

0.9 ==t::,::::Prop. Melh. -Grou lel:(23 OJI 0.7 �0.6 t{s (b) i-.oA (1) &Jo.3 0.2 0.1 0.95 � OJIS :>, 0.8 bO (c) � 01s � µ1 0.6

% of most significant coefficients

( c) Detail coefficients

...

-Symmlet -Grouplel: Prop. Melh. -Grou tet: 23

o.ss L-.__��---'-�-_._-� _ _._ _ _.__�____.

% of most significant coefficients

Fig. 5 Used Lena image (a) and energy percentage of approximation (b) and detail (c) wavelet coefficients with respect to the percentage of most significant coefficients, using the Haar, Symmlet and grouplet

transforms using the proposed method and the method in [23]

energy ratio of the wavelet coefficients generated using an orthogonal grouplet transform for which the association field has been estimated using our method.

For the 128 x 128 sized Lena image displayed in Fig. Sa, the energy contained in different ratios (from O to 1) of

the most significant coefficients is evaluated. The energy increase with respect to the ratio of the most significant coeffi­ cients is displayed separately for the approximation and detail coefficients in Fig. 5b,c, respectively. For the salœ of com­ parison, similar curves obtained with the Haar and Symmlet transforms are illustrated in the same figures. The same curve obtained with a grouplet transform whose association field is calculated using the method in [23] in displayed in the same figures. The reported curves indicate that for a given ratio of the most significant coefficients, the Bayesian grouplet trans­ form provides the highest energy level in comparison with the other transforms. This conclusion holds especially for detail coefficients. Moreover, the energy difference is more sig­ nificant for detail coefficients than for approximation ones, which confirms that the proposed Bayesian grouplet trans­ form has a good potential for applications such as image denoising or compression. For the detail coefficients, almost 95% of the energy is concentrated in 20% of the coefficients, which confirms the high sparsity level ensured by a grouplet transform obtained with an association field estimated by our method.

To further evaluate the properties of the generated coeffi­ cients, Fig. 6 displays the estimated autocorrelation fonction for the wavelet coefficients generated by the proposed Bayesian grouplet as well as a grouplet transform whose association field is calculated using the method in [23]. This figure also displays the same curves obtained with the Haar and Symmlet transforms, and this for the seismic image in Fig. 3 with J

=

1. This figure clearly shows that the proposed method generates wavelet coefficients with lower correlation levels in comparison with the other transforms. This confirms the conclusions made hereinabove. lndeed, a transform that generates coefficients with low autocorrelation level among which the most significant coefficients contain high energy level, has a great potential for applications where undesired content (noise, artifacts, .. ) generally lies in non-significant coefficients.

4.3 Comparison to other directional transforms

In this section, the performance of the proposed Bayesian grouplet transform is compared to other adaptive transforms. Specifically, the energy and correlation of the wavelet coef­ ficients obtained by the proposed transform are compared to those obtained with the Curvelet [27] and Dual Tree [3,28] transforms, in addition to the grouplet transform where the association field is calculated according to the method in [23]. These directional transforms have been used in several recent works in the image processing literature and have demon­ strated good performances in image denoising or deblurring [3,29,30], mainly due to their high directionality level which allows them to adapt to the perceptual content of the trans­ formed image.

0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 2 4

Sample Autocorrelation Functlon

6 8 10 12

Lag

14

Groupie!: Prop. Melh. ��plel:(�)

mmlel

16 18 20

Fig. 6 Autocorrelation function of the wavelet coefficients using the Haar, Symmlet and the grouplet transform wbere the association field is caJculated using the proposed method and the method in [23)

0.9 0.8 0.7 0.8 0.5 0.4 0.3 0.2 0.1 Approximation coefficients

--�ouplot: Prop. Mel.h. --Oroup_Cu�•I l•I: 12:JJ

--Ou-'Tl'eff 0 ,__..._ _ _._ _ _. __ ..._ _ _._ _ _. __ .._ _ _._ _ _. _ __, 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Detail coefficients •�s��-�-�--�-�-�-�--�-�-� --0.ouplel:Prop. Melh. --Grouplot:(23)

--

����

055'--0�.,-�o��-.�.3--0�.4--o�.s--o�.s--Q�7--o�.s-�Q-9 _ _. Fig. 7 Energy percentage of approximation and detail wavelet coef­ ficients using the Curvelet, Dual Tree, and grouplet transforms wbere the association field is caJculated using the proposed method and the method in [23) 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 ₀ 2

Sample Autoçorrelatlon Function

8 10 12 Lag

- Groupie!: Pro. Meth.

- Groupie!: [23) - Curvetel - DualTroo

14 16 18 20

Fig. 8 Autocorrelation function of the wavelet coefficients using the Curvelet, Dual Tree and the grouplet transform wbere the association field is caJculated using the proposed method and the method in [23)

However, it is worth noting that these transforms are overcomplete, which leads to a high number of coefficients in comparison with the image size. Moreover, using bases transforms can be preferred in some cases to design Jess com­ plicated resolution schemes such as in the inverse problem literature [31,32]. In this sense, Bayesian grouplets allow to perform a directional transform while being an orthonormal bases. This gives a high potential to use Bayesian grouplets for inverse problems resolution in the wavelet domain.

Figure 7 illustrates the evolution of the energy percentage contained in the most significant coefficients of the wavelet coefficients, and this for the same Lena image displayed in Fig. 5.

The same conclusions as for Sect. 4.2 hold. Indeed, the

Bayesian grouplet transform always provide detail coeffi­ cients for which the energy concentrates in a small ratio of the most significant coefficients: 20% of the coefficients con­ tain almost 95% of the energy.

As regards autocorrelation, comparisons are displayed in Fig. 8 where the estimated autocorrelation level is displayed for the proposed Bayesian grouplet transform in addition to the Curvelet and Dual Tree transforms. One can easily notice that the same conclusions drawn hereinabove hold since the proposed transform provides the lowest autocorrelation level in comparison with the used overcomplete transforms.

5 Conclusion

In this paper, a novel Bayesian model has been proposed to estimate the association field used for designing efficient and adaptive grouplet transforms. The proposed method relies on

a hierarchical Bayesian model, and enjoys a high flexibility and automating levels. The obtained results show the good properties of the resulting transform and demonstrates its ability to follow local features of the target image. Specif-ically, numerical results show that the proposed method allows designing a transform providing coefficients with low correlation level and whose 95% of the energy is concen-trated in 20% of the coefficients.

Future work will focus on the application of the proposed method to image restoration problems in remote sensing and medical imaging. Future work will also consider extending the proposed method to the multidimensional case for video processing or 3D object compression.

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