4.1 Motivation
Up to now, each prediction filter has been separately optimized by minimizing the ℓ1-norm of the corresponding detail signal which seems appropriate to determine and . However, it can be noticed from Figure 1 that the diagonal detail signal is also used through the second and the third prediction steps to compute the vertical and the horizontal detail signals respectively. Therefore, the solution resulting from the previous optimization method may be suboptimal. As a result, we propose to optimize the prediction filter by minimizing the global prediction error, as described in detail in the next section.
4.2 Optimization of the prediction filter
More precisely, instead of minimizing the ℓ1-norm of , the filter will be optimized by minimizing the sum of the ℓ1-norm of the three detail subbands . To this respect, we will consider the minimization of the following weighted ℓ1 criterion:
(16)
where , o ∈ {HL, LH, HH}, are strictly positive weighting terms.
Before focusing on the method employed to minimize the proposed criterion, we should first express as a function of the filter to be optimized.
Let be the four outputs obtained from following the first prediction step (see Figure 1). Although for all i ∈ {0, 1, 2}, the use of the superscript will make the presentation below easier. Thus can be expressed as:
(17)
where is a filter which depends on the prediction coefficients of and .
Knowing that
(18)
where is the support of the predictor ), we thus obtain, after some simple calculations,
(19)
Where
(20)
(21)
Consequently, the proposed weighted ℓ1 criterion (Equation (16)) can be expressed as:
(22)
It is worth noting that in practice, the determination of and does not require to find the explicit expressions of and these signals can be determined numerically as follows:
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The first term (resp. the second one) in the expression of in Equation (20) can be found by computing from the components while setting (resp. while setting for i ∈ {0,1,2} and ).
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The vector in Equation (21) can be found as follows. For each i ∈ {0,1,2}, the computation of its component requires to compute by setting and for i' ∈ {0,1,2}. The result of this operation has to be considered for different shift values (r, s) (as can be seen in Equation (21)).
Once the different terms involved in the proposed weighted criterion in Equation (22) are defined (the constant values 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].
4.2.1 Douglas-Rachford algorithm in a product space
Consider the ℓ1 minimization problem:
(23)
where , o ∈ {HL,LH,HH}, are positive weights.
Since the Douglas-Rachford algorithm described hereabove is designed for the sum of two functions, we can reformulate (23) under this form in the 3-fold product space
(24)
If we define the vector subspace U as
(25)
the minimization problem (Equation 23) is equivalent to
(26)
where
(27)
We are thus back to a problem involving two functions in a larger space, which is the product space . So, the Douglas-Rachford algorithm can be applied to solve our minimization problem (see Appendix C). Finally, once the prediction filter is optimized and fixed, it can be noticed that the other prediction filters and can be separately optimized by minimizing and as explained in Section 3. This is justified by the fact that the inputs of the filter (resp. ) are independent of the output of the filter (resp. ).