Adaptive lifting scheme with sparse criteria for image coding

2.1 Principle of the considered 2D NSLS structure

In this article, we consider a 2D NSLS composed of three prediction lifting steps followed by an update lifting step. The interest of this structure is two-fold. First, it allows us to reduce the number of lifting steps and rounding operations. A theoretical analysis has been conducted in [13] showing that NSLS improves the coding performance due to the reduction of rounding effects. Furthermore, any separable prediction-update LS structure has its equivalent in this form [13, 14]. The corresponding analysis structure is depicted in Figure 1.

Let x denote the digital image to be coded. At each resolution level j and each pixel location (m, n), its approximation coefficient is denoted by x _j (m, n) and the associated four polyphase components by x_0,j(m, n) = x _j (2m,2n), x_1,j(m,n) = x _j (2m,2n+1), x_2,j(m,n) = x _j (2m+1,2n), and x_3,j(m,n) = x _j (2m + 1, 2n + 1). Furthermore, we denote by ${\mathbf{P}}_j^{\left(HH\right)}$, ${\mathbf{P}}_j^{\left(LH\right)}$, ${\mathbf{P}}_j^{\left(HL\right)}$, and U _j the three prediction and update filters employed to generate the detail coefficients $x_{j+1}^{\left(HH\right)}$ oriented diagonally, $x_{j+1}^{\left(LH\right)}$ oriented vertically, $x_{j+1}^{\left(HL\right)}$ oriented horizontally, and the approximation coefficients x_j+1. In accordance with Figure 1, let us introduce the following notation:

The set ${\mathcal{P}}_{i,j}^{\left(o\right)}$ (resp. ${\mathcal{U}}_j^{\left(o\right)}$) and the coefficients $p_{i,j}^{\left(o\right)}\left(k,l\right)$ (resp. $u_j^{\left(o\right)}\left(k,l\right)$) denote the support and the weights of the three prediction filters (resp. of the update filter). Note that in Equations (1)-(4), we have introduced the rounding operations ⌊.⌋ in order to allow lossy-to-lossless encoding of the coefficients [7]. Once the considered NSLS structure has been defined, we will focus now on the optimization of its lifting operators.

2.2 Optimization methods

where

The other optimal prediction filters ${\mathbf{p}}_j^{\left(HL\right)}$ and ${\mathbf{p}}_j^{\left(LH\right)}$ are obtained in a similar way.

Where

Now, we will introduce a novel twist in the optimization of the different filters: the use of an ℓ₁-based criterion in place of the usual ℓ₂-based measure.

3.1 Motivation

where $\Gamma \left(z\right)={\int }_0^{+\infty }{t}^{z-1}{e}^{-t}dt$ is the Gamma function, α > 0 is the scale parameter, and β > 0 is the shape parameter. We should note that in the particular case when β = 2 (resp. β = 1), the GGD corresponds to the Gaussian distribution (resp. the Laplace one). The parameters α and β can be easily estimated by using the maximum likelihood technique [42].

where (M _j ,N _j ) corresponds to the dimensions of the subband $x_{j+1}^{\left(o\right)}$.

This shows that there exists a close link between the minimization of the entropy of the detail wavelet coefficients and the minimization of their ${\ell }_{{{\beta }_{j+1}^{\left(o\right)}}_{}}$-norm. This suggests in particular that most of the existing studies minimizing the ℓ₂-norm of the detail signals aim at minimizing their entropy by assuming a Gaussian model.

Based on these results, we have analyzed the detail wavelet coefficients generated by the decomposition based on the lifting structure NSLS(2,2)-OPT-L2 described in Section 6. Figure 2 shows the distribution of each detail subband for the "einst" image when the prediction filters are optimized by minimizing the ℓ₂-norm of the detail coefficients. The maximum likelihood technique is used to estimate the β parameter.

It is important to note that the shape parameters of the resulting detail subbands are closer to β = 1 than to β = 2. Further experiments performed on a large dataset of images^b have shown that the average of β values are closer to 1 (typical values range from 0.5 to 1.5). These observations suggest that minimizing the ℓ₁-norm may be more appropriate than ℓ₂ minimization. In addition, the former approach has the advantage of producing sparse representations.

3.2 ℓ₁minimization technique

where x _i,j (m,n) is the (i + 1) ^th polyphase component to be predicted, ${\tilde{\mathbf{x}}}_j^{\left(o\right)}\left(m,n\right)$ is the reference vector containing the samples used in the prediction step, ${\mathbf{p}}_j^{\left(o\right)}$ is the prediction operator vector to be optimized (L will subsequently designate its length). Although the criterion in (11) is convex, a major difficulty that arises in solving this problem stems from the fact that the function to be minimized is not differentiable. Recently, several optimization algorithms have been proposed to solve nonsmooth minimization problems like (11). These problems have been traditionally addressed with linear programming [45]. Alternatively, a flexible class of proximal optimization algorithms has been developed and successfully employed in a number of applications. A survey on these proximal methods can be found in [46]. These methods are also closely related to augmented Lagrangian methods [47]. In our context, we have employed the Douglas-Rachford algorithm which is an efficient optimization tool for this problem [48].

3.2.1 The Douglas-Rachford algorithm

For minimizing the ℓ₁ criterion, we will resort to the concept of proximity operators [49], which has been recognized as a fundamental tool in the recent convex optimization literature [50, 51]. The necessary back-ground on convex analysis and proximity operators [52, 53] is given in Appendix A.

Now, we recall that our minimization problem (11) aims at optimizing the prediction filters by minimizing the ℓ₁-norm of the difference between the current pixel x _i,j and its predicted value. We note here that $x_{i,j}={\left(x_{i,j}(m,n)\right)}_{\underset{1\le n\le N_j}{1\le m\le M_j}}$ can be viewed as an element of the Euclidean space ${ℝ}^{K_j}$, where K _j = M _j × N _j . Thus, the minimization problem (11) can be rewritten as:

$\forall o\in \left\{HL,LH,HH\right\},\forall i\in \left\{1,2,3\right\},\underset{{\mathbf{z}}_j^{\left(o\right)}\in V}{\text{min}}\sum _{m=1}^{M_j}\sum _{n=1}^{N_j}\left|x_{i,j}\left(m,n\right)-z_j^{\left(o\right)}\left(m,n\right)\right|$

(12)

where V is the vector space defined as

$\begin{array}{l}V=\{z_j^{(o)}={\left(z_j^{(o)}(m,n)\right)}_{\underset{1\le n\le N_j}{1\le m\le M_j}}\in {ℝ}^{K_j}|\exists p_j^{(o)}\in {ℝ}^L,\\ \forall (m,n)\in \{1,\dots ,M_j\}\times \{1,\dots ,N_j\},z_j^{(o)}(m,n)={(p_j^{(o)})}^{\text{T}}{\tilde{x}}_j^{(o)}(m,n)\}.\end{array}$

Based on the definition of the indicator function ı _V (see Appendix A), Problem (12) is equivalent to the following minimization problem:

$\forall o\in \left\{HL,LH,HH\right\},\forall i\in \left\{1,2,3\right\},\underset{{\mathbf{z}}_j^{\left(o\right)}\in {ℝ}^{K_j}}{\text{min}}\sum _{m=1}^{M_j}\sum _{n=1}^{N_j}\left|x_{i,j}\left(m,n\right)-z_j^{\left(o\right)}\left(m,n\right)\right|+{\iota }_V\left({\mathbf{z}}_j^{\left(o\right)}\right).$

(13)

Therefore, Problem (13) can be viewed as a minimization of a sum of two functions f₁ and f₂ defined by:

$f_1\left({\mathbf{z}}_j^{\left(o\right)}\right)=\left|\right|{\mathbf{x}}_{i,j}-{\mathbf{z}}_j^{\left(o\right)}|{|}_{{\ell }_1}=\sum _{m=1}^{M_j}\sum _{n=1}^{N_j}\left|x_{i,j}\left(m,n\right)-z_j^{\left(o\right)}\left(m,n\right)\right|$

(14)

$f_2\left({\mathbf{z}}_j^{\left(o\right)}\right)={\iota }_V\left({\mathbf{z}}_j^{\left(o\right)}\right).$

(15)

In this case, the Douglas-Rachford algorithm can be applied to provide an appealing numerical solution to Problem (13) (see Appendix B).

Although it is an iterative algorithm, we have observed experimentally that the convergence of the Douglas-Rachford algorithm is generally ensured after a small number of iterations (often between 30 et 60 iterations). As an example, we plot in Figure 3a (resp. 3b) the evolution of the criterion ${\mathcal{J}}_{{\ell }_1}\left({\mathbf{p}}_0^{\left(HH\right)}\right)$ (resp. ${\mathcal{J}}_{{\ell }_1}\left({\mathbf{p}}_0^{\left(LH\right)}\right)$) w.r.t the iteration number for this algorithm.

Once the different terms involved in the iterative algorithm (33) are defined, this one can be applied and further extended to optimize all the prediction filters.

4.1 Motivation

Up to now, each prediction filter ${\mathbf{p}}_j^{\left(o\right)}\left(o\in \left\{HL,LH,HH\right\}\right)$ has been separately optimized by minimizing the ℓ₁-norm of the corresponding detail signal $x_{j+1}^{\left(o\right)}$ which seems appropriate to determine ${\mathbf{p}}_j^{\left(LH\right)}$ and ${\mathbf{p}}_j^{\left(HL\right)}$. However, it can be noticed from Figure 1 that the diagonal detail signal $x_{j+1}^{\left(HH\right)}$ is also used through the second and the third prediction steps to compute the vertical and the horizontal detail signals respectively. Therefore, the solution ${\mathbf{p}}_j^{\left(HH\right)}$ resulting from the previous optimization method may be suboptimal. As a result, we propose to optimize the prediction filter ${\mathbf{p}}_j^{\left(HH\right)}$ by minimizing the global prediction error, as described in detail in the next section.

4.2 Optimization of the prediction filter ${\mathbf{p}}_j^{\left(HH\right)}$

where ${\kappa }_{j+1}^{\left(o\right)}$, o ∈ {HL, LH, HH}, are strictly positive weighting terms.

Before focusing on the method employed to minimize the proposed criterion, we should first express ${\mathcal{J}}_{w{\ell }_1}$ as a function of the filter ${\mathbf{p}}_j^{\left(HH\right)}$ to be optimized.

where $h_{i,j}^{\left(o,1\right)}$ is a filter which depends on the prediction coefficients of ${\mathbf{p}}_j^{\left(LH\right)}$ and ${\mathbf{p}}_j^{\left(HL\right)}$.

It is worth noting that in practice, the determination of $y_j^{\left(o,1\right)}\left(m,n\right)$ and ${\mathbf{x}}_j^{\left(o,1\right)}\left(m,n\right)$ does not require to find the explicit expressions of $h_{i,j}^{\left(o,1\right)}$ and these signals can be determined numerically as follows:

Once the different terms involved in the proposed weighted criterion in Equation (22) are defined (the constant values ${\kappa }_{j+1}^{\left(o\right)}$ are supposed to be known), we will focus now on its minimization. Indeed, unlike the previous criterion (Equation 11), which consists only of an ℓ₁ term, the proposed criterion is a sum of three ℓ₁ terms. To minimize such a criterion (22), one can still use the Douglas-Rachford algorithm through a formulation in a product space [46, 54].

5.1 Motivation

From Equations (20) and (21), it can be observed that $y_j^{\left(o,1\right)}$ and ${\mathbf{x}}_j^{\left(o,1\right)}$, which are used to optimize ${\mathbf{p}}_j^{\left(HH\right)}$, depend on the coefficients of the prediction filters ${\mathbf{p}}_j^{\left(HL\right)}$ and ${\mathbf{p}}_j^{\left(LH\right)}$. On the other hand, since ${\mathbf{p}}_j^{\left(HL\right)}$ and ${\mathbf{p}}_j^{\left(LH\right)}$ use $x_{j+1}^{\left(HH\right)}$ as reference signal in the second and the third prediction steps, their optimal values will depend on the optimal prediction filter ${\mathbf{p}}_j^{\left(HH\right)}$. Thus, we conclude that the optimization of the filters (${\mathbf{p}}_j^{\left(HL\right)}$, ${\mathbf{p}}_j^{\left(LH\right)}$) depends on the optimization of the filter ${\mathbf{p}}_j^{\left(HH\right)}$ and vice-versa.

A joint optimization method can therefore be proposed which iteratively optimizes the prediction filters ${\mathbf{p}}_j^{\left(HH\right)}$, ${\mathbf{p}}_j^{\left(HL\right)}$, and ${\mathbf{p}}_j^{\left(LH\right)}$.

5.2 Proposed algorithms

While the optimization of the prediction filters ${\mathbf{p}}_j^{\left(HL\right)}$ and ${\mathbf{p}}_j^{\left(LH\right)}$ is simple, the optimization of the prediction filter ${\mathbf{p}}_j^{\left(HH\right)}$ is less obvious. Indeed, if we examine the criterion ${\mathcal{J}}_{w{\ell }_1}$, the immediate question that arises is: which values of the weighting parameters will produce the sparsest decomposition?