Inverse formulas of length twelve parameterized orthogonal wavelets

Abstract

Inverse parameterizations of length 12 orthogonal wavelet filters are presented, which allow to determine parameter values from filter coefficients. Its applicability is shown in a case of study of image processing where the optimization of five parameters is required. The parameterization of length N filters involves $\frac{N}{2} - 1$ parameters, and it is easier to optimize shorter filters once they explore a subset of the search space. Under this approach, the optimization of length 12 filters is accelerated based on a nested optimization of length 4, 6, 8, and 10 filters by exporting the best solutions from shorter to larger filters via inverse parameterizations. Experimental results support the success of the nested optimization when exploring the search space. The conclusions are that the use of the inverse formulas accelerates the convergence and that parameterized filters provide better results as their length increases and achieve a better performance than standard filters.

Keywords

Wavelets filter parameterization image processing

1 Introduction

In wavelet analysis, a fundamental concept is that the wavelet transform develops the convolution between a signal f and wavelet basis [1]. In particular, the analysis of f can be implemented with the discrete wavelet transform by convolving f with low-pass and high-pass filters. For a perfect reconstruction scheme, the filters can be designed to be orthogonal under the quadrature mirror conditions [2, 3].

Orthogonal wavelets ψ (t) with compact support can be obtained from scaling functions φ (t) through the dyadic dilation equation [4 –6]: $φ (t) = \sum_{k = 0}^{N - 1} h_{k} φ (2 t - k)$ (1) where $N \in ℕ$ , $h_{k} \in ℝ$ , and h_k = 0 for k < 0 and k ≥ N. The sequences h_k and g_k = (-1) ^kh_N-1-k, for k = 0, 1, …, N - 1 represent the impulse response of low pass and high pass filters respectively, from which is possible to approximate the scaling and wavelet functions with the cascade algorithm [7].

In this work, and given the previous relationship, the low-pass filters will be referred as h_k sequences, low-pass coefficients, or scaling functions indistinctly, and the same for the high-pass filters that will be referred as g_k sequences, high-pass coefficients or wavelet functions.

There is an infinite number of sequences h_k and g_k that fulfill the quadrature mirror conditions and some of them have been widely used to be considered “standard” such as the Haar, Daubechies, Coiflets, and Symlets filters. One of the main reasons to choose “standard” filters is that they were designed to generate wavelets or scaling functions with vanishing moments, among other criterion, that does not focus on a specific signal. For this reason, they are also referred as general purpose wavelets.

General purpose wavelets does not always provide the best performance and personalized filters can be designed for specific applications or signals. But, one of the main drawbacks of designing personalized filters is the time required to calculate them, as well as the lack of specific criteria to guide the selection of optimal filters. But, it has been found that the filter parameterization provides a convenient method to find valid h_k and g_k sequences, at the same time it keeps the flexibility of the properties of the filters.

Note that, except for the Haar wavelet, neither of the previous wavelets (Daubechies, Coiflets and Symlets) have an explicit formulation, and this is a hurdle when trying to apply non-numerical gradient based optimization methods. Alternatively, the use of evolutionary algorithms has been proposed to optimize parameterized filters for specific applications. For example, [6] shows an application in image compression, [8] shows time series analysis with parameterized wavelets, and [9] presents an application of parameterized wavelet neural networks for data classification.

By considering that the parameterization of length N filters involves $\frac{N}{2} - 1$ parameters and that it is easier to optimize shorter filters (once they explore a subset of the search space) it is proposed to accelerate the optimization of length N filters based on the nested optimization of shorter filters, therein the best solutions are exported from shorter to larger filters via inverse parameterizations. In [10] it was shown the optimization of length 4, 6, 8 and 10 filters through inverse parameterizations.

In this work, the parameterization of length 12 filters given in [11] is reviewed and inverse formulas are presented to get the parameter values from filter coefficients. These inverse formulas can be used to optimize length 12 filters with 5 parameters based on a nested optimization of shorter filters that depend of 1, 2, 3, and 4 parameters. A first case of study shows the application of these inverse formulas to calculate parameters of “standard” filters, and a second case shows an application in image compression where the parameters are optimized to maximize the number of negligible wavelet coefficients. Some other applications are possible including watermarking, neural networks, denoising and signal processing where is relevant to increase the quality or precision as a priority. Indeed, this research aims to promote the use of longer filters optimized for specific purposes by providing acceleration convergence techniques.

The article is organized in sections as follows: In Section 2 the parameterization of length 12 filters is reviewed and their inverse formulas are obtained. In Section 3, several computational experiments are presented to validate the inverse formulas, and some properties about their domain are commented. In Section 4, some results of the validation are presented as well as an application in image processing. Finally, the conclusions are presented in Section 5.

2 Parameterization of filter coefficients

Given a length N filter, the sequence of coefficients h_k with k = 0, 1, …, N - 1 represents the impulse response of a low pass filter, whereas g_k = (-1) ^kh_N-1-k represents the mirrored high pass filter. With the h_k filter coefficients, is possible to build a trigonometric polynomial H (z) in the z domain given by [4, 11]: $H (z) = \sum_{k = 0}^{k = N - 1} h_{k} z^{k}, z = e^{i ω}, and ω \in ℝ .$ (2)

The orthonormality conditions (see [11 –14] and others) can be expressed as: $| H (z) |^{2} + | H (- z) |^{2} = 1$ (3)

$H (z = 1) = 1$ (4) and the zero moment condition: $H (z = - 1) = 0 .$ (5)

For example, for N = 2 the coefficients are h = {h₀, h₁}, and by applying Equation (2) it is simplified to: $H (z) = h_{0} + h_{1} z$ (6)

Therefore, Equation (3) yields to $h_{0}^{2} + h_{1}^{2} = \frac{1}{2}$ . Then, from Equation (33) h₀ - h₁ = 0, and consequently $h_{0} = h_{1} = \frac{1}{2}$ , that satisfies Equation (32). These coefficients correspond to the Haar filter, where the high-pass coefficients are $g_{0} = \frac{1}{2}$ and $g_{1} = - \frac{1}{2}$ .

The length two filter has unique solution and no parameters, but larger filters can be parameterized (see [3 , 15], and others). Consider that the filter parameterizations for N ≥ 4 are not unique, and that there are several methods of analysis including the spectral factorization [4] and the algebraic approach.

In this paper, the algebraic approach described in [11–13, 16] for length 4, 6, 8, 10 and 12 filters is revised as follows.

2.1 Parameterization of the length twelve filters

For the length twelve filters $H (z) = \sum_{k = 0}^{k = 11} h_{k} z^{k}$ the parameterizations of h_k given in [16] (Theorem 1) is applied as follows:

$h_{0} = \frac{p}{2} (cos β + cos (2 θ - β))$ (7)

$h_{1} = \frac{p}{2} (\sin β + \sin (2 θ - β))$ (8)

$h_{2} = \frac{\sqrt{2}}{4} sin α (- cos α + sin α) +$ (9) $\frac{q}{2} cos β - \frac{r}{2} sin β - \frac{s}{2} cos (2 θ - β) +$ (10) $\frac{t}{2} sin (2 θ - β) - h_{10}$ (11)

$h_{3} = \frac{\sqrt{2}}{4} cos α (cos α - sin α) +$ (13) $\frac{q}{2} sin β + \frac{r}{2} cos β -$ (14) $\frac{s}{2} sin (2 θ - β) - \frac{t}{2} cos (2 θ - β) - h_{11}$ (15)

$h_{4} = \frac{\sqrt{2}}{4} cos α (cos α + sin α) -$ (16) $\frac{q}{2} cos β + \frac{r}{2} sin β -$ (17) $\frac{s}{2} cos (2 θ - β) + \frac{t}{2} sin (2 θ - β)$ (18)

$h_{5} = \frac{\sqrt{2}}{4} sin α (cos α + sin α) -$ (19) $\frac{q}{2} sin β - \frac{r}{2} cos β -$ (20) $\frac{s}{2} sin (2 θ - β) - \frac{t}{2} cos (2 θ - β)$ (21)

$h_{6} = \frac{\sqrt{2}}{4} sin α (- cos α + sin α) -$ (22) $\frac{q}{2} cos β + \frac{r}{2} sin β +$ (23) $\frac{s}{2} cos (2 θ - β) - \frac{t}{2} sin (2 θ - β)$ (24)

$h_{7} = \frac{\sqrt{2}}{4} cos α (cos α - sin α) -$ (25) $\frac{q}{2} sin β - \frac{r}{2} cos β +$ (26) $\frac{s}{2} sin (2 θ - β) + \frac{t}{2} cos (2 θ - β)$ (27)

$h_{8} = \frac{\sqrt{2}}{4} cos α (cos α + sin α) +$ (28) $\frac{q}{2} cos β - \frac{r}{2} sin β +$ (29) $\frac{s}{2} cos (2 θ - β) - \frac{t}{2} sin (2 θ - β) - h_{0}$ (30)

$h_{9} = \frac{\sqrt{2}}{4} sin α (cos α + sin α) +$ (31) $\frac{q}{2} sin β + \frac{r}{2} cos β +$ (32) $\frac{s}{2} sin (2 θ - β) + \frac{t}{2} cos (2 θ - β) - h_{1}$ (33)

$h_{10} = \frac{p}{2} (cos β - cos (2 θ - β))$ (34)

$h_{11} = \frac{p}{2} (\sin β - \sin (2 θ - β))$ (35)

$u = cos (β - γ) + cos (2 α + β - δ - 2 θ)$ (36)

$p = \frac{\sqrt{2}}{8} u (cos δ + sin δ + cos γ + sin γ)$ (37)

$q = \frac{\sqrt{2}}{4} (- cos (2 α + β - 2 θ) -$ (38) $sin (2 α + β - 2 θ) + u (cos δ + sin δ))$ (39)

$r = \frac{\sqrt{2}}{4} (- cos β + \sin β + u (cos γ - \sin γ))$ (40)

$s = \frac{\sqrt{2}}{4} (- cos β - \sin β + u (cos γ + \sin γ))$ (41)

$t = \frac{\sqrt{2}}{4} (cos (2 α + β - 2 θ) -$ (42) $sin (2 α + β - 2 θ) + u (- cos δ + sin δ))$ (43)

Now, with Equations (7) to (42) it will be calculated the inverse parameterizations, that consist in the determination of the parameters α, β, γ, δ, and θ in terms of the h_k coefficients, which is the main contribution of the paper.

2.2 Inverse parameterization of the length twelve filters

The inverse parameterization is divided in two stages. In the first stage the parameters α, β and θ are calculated from the h₀, h₁, h₂,..., h₁₁ filter coefficients. In the second stage temporary parameters and a rotation matrix M are used to get the δ and γ parameters in terms of known parameters, once they do not depend of the filter coefficients directly.

First stage. From (16) and (28) see that:

$h_{0} + h_{4} + h_{8} = \frac{1}{\sqrt{2}} cos α (cos α + \sin α)$ (44) and from (13) and (25) see that:

$h_{3} + h_{7} + h_{11} = \frac{1}{\sqrt{2}} cos α (cos α - \sin α)$ (45) then, by adding (44) to (45) it can be solved for α, hence: $α = arccos \sqrt{\frac{1}{\sqrt{2}} (h_{0} + h_{4} + h_{8} + h_{3} + h_{7} + h_{11})}$ (46)

The inverse parameterizations are not unique. For example, for α by combining (44) with (45) it yields to: $α = \frac{π}{8} + \frac{1}{2} arccos (h_{0} + h_{4} + h_{8} - h_{3} - h_{7} - h_{11})$ (47)

In the case of β, by adding (7) to (34), it takes to: $h_{0} + h_{10} = p cos β$ (48) and combining (8) and (35): $h_{1} + h_{11} = p sin β$ (49) then, by taking the quotient of (49) with (48) and solving for β it produces:

$β = arctan (\frac{h_{1} + h_{11}}{h_{0} + h_{10}})$ (50)

At this point, the filter coefficients, α and β are known, so it is possible to calculate θ by subtracting (34) from (7), as follows: $h_{0} - h_{10} = p cos (2 θ - β)$ (51) and, by subtracting (35) from (8): $h_{1} - h_{11} = p sin (2 θ - β)$ (52) hence, it can be solved for θ by dividing (52) with (51), so: $θ = \frac{1}{2} (arctan (\frac{h_{1} - h_{11}}{h_{0} - h_{10}}) + β) .$ (53)

Second stage. In this stage, the temporary parameters $\tilde{q}$ , $\tilde{r}$ , $\tilde{s}$ and $\tilde{t}$ are used. Their expressions only depend of filter coefficients (see [16]) as is described in equations (54) to (57):

$\tilde{q} = h_{0} - h_{4} + h_{8}$ (54)

$\tilde{r} = h_{1} - h_{5} + h_{9}$ (55)

$\tilde{s} = h_{2} - h_{6} + h_{10}$ (56)

$\tilde{t} = h_{3} - h_{7} + h_{11}$ (57) then, with the matrix M: $M = [\begin{matrix} cos β & - sin β & cos (2 θ - β) & - sin (2 θ - β) \\ sin β & cos β & sin (2 θ - β) & cos (2 θ - β) \\ cos β & - sin β & - cos (2 θ - β) & sin (2 θ - β) \\ sin β & cos β & - sin (2 θ - β) & - cos (2 θ - β) \end{matrix}]$ (58) that depends of the β and θ parameters, it is possible to calculate the p, q, r and t parameters since (see [16]): $M {[\begin{matrix} q & r & s & t \end{matrix}]}^{T} = {[\begin{matrix} \tilde{q} & \tilde{r} & \tilde{s} & \tilde{t} \end{matrix}]}^{T}$ (59) and consequently: ${[\begin{matrix} q & r & s & t \end{matrix}]}^{T} = M^{- 1} {[\begin{matrix} \tilde{q} & \tilde{r} & \tilde{s} & \tilde{t} \end{matrix}]}^{T}$ (60)

Additionally, the parameters q, r, s and t can be expressed in terms of other parameters $\bar{q}$ , $\bar{r}$ , $\bar{s}$ and $\bar{t}$ (see [16]) as is shown in equations (61) to (64): $q = \frac{\sqrt{2}}{4} (- cos (2 α + β - 2 θ) - sin (2 α + β - 2 θ) + \bar{q})$ (61)

$r = \frac{\sqrt{2}}{4} (- cos β + sin β + \bar{r})$ (62)

$s = \frac{\sqrt{2}}{4} (- cos β - sin β + \bar{s})$ (63)

$t = \frac{\sqrt{2}}{4} (cos (2 α + β - 2 θ) - sin (2 α + β - 2 θ) + \bar{t})$ (64) and then, it can be solved for $\bar{q}$ , $\bar{r}$ , $\bar{s}$ and $\bar{t}$ in terms of q, r, s, t, α, β and θ as follows: $\bar{q} = \frac{4}{\sqrt{2}} q + cos (2 α + β - 2 θ) + sin (2 α + β - 2 θ)$ (65)

$\bar{r} = \frac{4}{\sqrt{2}} r + cos β - sin β$ (66)

$\bar{s} = \frac{4}{\sqrt{2}} s + cos β + sin β$ (67)

$\bar{t} = \frac{4}{\sqrt{2}} t - cos (2 α + β - 2 θ) + sin (2 α + β - 2 θ)$ (68)

The last formulas are relevant since there is a relationship of the parameters $\bar{q}$ , $\bar{r}$ , $\bar{s}$ and $\bar{t}$ with the parameters u, γ and δ (see [16]) as is shown in (69) to (72): $\bar{q} = \sqrt{2} u cos (δ - \frac{π}{4})$ (69) $\bar{r} = \sqrt{2} u cos (γ - \frac{π}{4})$ (70)

$\bar{s} = \sqrt{2} u sin (γ - \frac{π}{4})$ (71)

$\bar{t} = \sqrt{2} u sin (δ - \frac{π}{4})$ (72)

Now, by dividing (72) with (69) and solving for δ: $δ = arctan \frac{\bar{t}}{\bar{q}} + \frac{π}{4}$ (73) and with equations (71) and (70), it can be solved for γ, hence: $γ = arctan \frac{\bar{s}}{\bar{r}} + \frac{π}{4}$ (74)

At this point, it has been shown how the inverse parameterizations of α, β and θ depend of filters coefficients directly, and how the δ and γ can be calculated using some temporary parameters and filter coefficients.

See that, although it is not strictly necessary for the inverse parameterizations, the last parameter that can be calculated is u from Equation (36), because it requires the five parameters α, β, θ, γ, and δ.

3 Experiments

This section includes three experiments to validate the inverse formulas with a repair technique. Additionally, a case of study in image processing it is shown.

3.1 Experiment 1

In this experiment, a computer program was implemented to test the inverse parameterizations of Section 2.2. It is important to emphasize that, although all the real domain is valid for the parameters α, β, γ, δ, θ, they are set to [0, 2π) given the periodicity of the trigonometric functions.

In this way, the computer program generates random values for α, β, γ, δ, θ ∈ [0, 2π) and calculates the filter coefficients h_k with Equations (7) to (35). Then, the computer program applies the inverse parameterizations of Equation (46), Equation (50), Equation (53), Equation (73) and Equation (74), and calculates the approximation error (RMS) by comparing each parameter with the original value.

Since the inverse parameterizations uses the arcsin, arccos, and arctan functions and they do not match the [0, 2π) interval, the repair technique presented in [10] is applied to test alternatives of the values obtained with the inverse parameterizations. For example, for the α parameter, the next A alternatives are considered: $A = {MOD (\pm α \pm n * π / 4)} for n = {0, 1, \dots, 8}$ (75) where MOD (x) is a “modulo” function from x ∈ R → [0, 2π) described in [10].

In this computer experiment, MAX = 2700 cases were tested with the next results:

Using the inverse equations Equation (47), Equation (50), Equation (53), Equation (73) and Equation (74), the five parameters were recovered successfully in 75.8% of the cases, with an average error of 0.0011.

Replacing Equation (47) with Equation (46) for α, the five parameters were recovered successfully in 92.6% of the cases, with a lower average error of 0.00079.

Based on the previous results, the inverse parameterizations of Equation (46), Equation (50), Equation (53), Equation (73) and Equation (74) were chosen for the subsequent experiments.

3.2 Experiment 2

Experiment 2 takes into account that each parameter repair introduces |A|=34 alternatives, and not all of them could be useful. Then, MAX = 2700 test cases were executed to identify subsets of alternatives A_s = {A_α, A_β, A_γ, A_δ, A_θ} with the used options when applying Equation (75) to α, β, γ, δ and θ.

The subset identification was possible by counting the number of times that the options were effectively used when executing the experiments, and eliminating those with zero account.

The results of Experiment 2 are shown in Table 1. In the first column it is shown the parameter, in the second column the corresponding subset A_s for the parameter, in the third column the elements of these subsets, and in the last column their cardinality.

The size of the search space was reduced 97 : 1 since the exhaustive search space is 34⁵ and the reduced space is 10 × 13 × 14 × 16². The corresponding execution time was reduced from 4 minutes 44 seconds to 9.7 seconds, by preserving the success rate of 92% with an average error of 0.0001 in both cases. The experiments were executed on a Linux based system with 4 Intel Xeon CPU @2.53 GHz and 16 GBRAM.

Table 1
Subset of alternatives to repair parameters

Parameter Subset A_s Alternatives |A_s|

α A _α $\begin{array}{l} α, α + \frac{π}{4}, \\ α + \frac{π}{2}, α + \frac{3 π}{4}, \\ α + π, - α + π, \\ α + \frac{5 π}{4}, \\ α + \frac{3 π}{2}, α + \frac{7 π}{4}, \\ - α + 2 π \end{array}$ 10

β A _β $\begin{array}{l} β, β + \frac{π}{4}, \\ - β + \frac{π}{4}, β + \frac{π}{2}, \\ - β + \frac{π}{2}, β + \frac{3 π}{4}, \\ - β + \frac{3 π}{4}, β + π, \\ - β + π, β + \frac{5 π}{4}, \\ - β + \frac{5 π}{4}, β + \frac{7 π}{4}, \\ - β + 2 π \end{array}$ 13

γ A _γ $\begin{array}{l} γ, γ + \frac{π}{4}, \\ γ + \frac{π}{2}, - γ + \frac{π}{2}, \\ - γ + \frac{3 π}{4}, γ + π, \\ - γ + π, γ + \frac{5 π}{4}, \\ - γ + \frac{5 π}{4}, γ + \frac{3 π}{2}, \\ - γ + \frac{3 π}{2}, γ + \frac{7 π}{4}, \\ - γ + \frac{7 π}{4}, - γ + 2 π \end{array}$ 14

δ A _δ $\begin{array}{l} δ, δ + \frac{π}{4}, \\ - δ + \frac{π}{4}, δ + \frac{π}{2}, \\ - δ + \frac{π}{2}, δ + \frac{3 π}{4} \\ - δ + \frac{3 π}{4}, δ + π, \\ - δ + π, δ + \frac{5 π}{4}, \\ - δ + \frac{5 π}{4}, δ + \frac{3 π}{2}, \\ - δ + \frac{3 π}{2}, δ + \frac{7 π}{4}, \\ - δ + \frac{7 π}{4}, - δ + 2 π \end{array}$ 16

θ A _θ $\begin{array}{l} θ, θ + \frac{π}{4}, - θ + \frac{π}{4}, \\ θ + \frac{π}{2}, - θ + \frac{π}{2}, \\ θ + \frac{3 π}{4}, - θ + \frac{3 π}{4}, \\ θ + π, - θ + π, \\ θ + \frac{5 π}{4}, - θ + \frac{5 π}{4}, \\ θ + \frac{3 π}{2}, - θ + \frac{3 π}{2} \\ θ + \frac{7 π}{4}, - θ + \frac{7 π}{4}, - θ + 2 π \end{array}$ 16

Parameter	Subset A_s	Alternatives	\|A_s\|
α	A _α	$\begin{array}{l} α, α + \frac{π}{4}, \\ α + \frac{π}{2}, α + \frac{3 π}{4}, \\ α + π, - α + π, \\ α + \frac{5 π}{4}, \\ α + \frac{3 π}{2}, α + \frac{7 π}{4}, \\ - α + 2 π \end{array}$	10
β	A _β	$\begin{array}{l} β, β + \frac{π}{4}, \\ - β + \frac{π}{4}, β + \frac{π}{2}, \\ - β + \frac{π}{2}, β + \frac{3 π}{4}, \\ - β + \frac{3 π}{4}, β + π, \\ - β + π, β + \frac{5 π}{4}, \\ - β + \frac{5 π}{4}, β + \frac{7 π}{4}, \\ - β + 2 π \end{array}$	13
γ	A _γ	$\begin{array}{l} γ, γ + \frac{π}{4}, \\ γ + \frac{π}{2}, - γ + \frac{π}{2}, \\ - γ + \frac{3 π}{4}, γ + π, \\ - γ + π, γ + \frac{5 π}{4}, \\ - γ + \frac{5 π}{4}, γ + \frac{3 π}{2}, \\ - γ + \frac{3 π}{2}, γ + \frac{7 π}{4}, \\ - γ + \frac{7 π}{4}, - γ + 2 π \end{array}$	14
δ	A _δ	$\begin{array}{l} δ, δ + \frac{π}{4}, \\ - δ + \frac{π}{4}, δ + \frac{π}{2}, \\ - δ + \frac{π}{2}, δ + \frac{3 π}{4} \\ - δ + \frac{3 π}{4}, δ + π, \\ - δ + π, δ + \frac{5 π}{4}, \\ - δ + \frac{5 π}{4}, δ + \frac{3 π}{2}, \\ - δ + \frac{3 π}{2}, δ + \frac{7 π}{4}, \\ - δ + \frac{7 π}{4}, - δ + 2 π \end{array}$	16
θ	A _θ	$\begin{array}{l} θ, θ + \frac{π}{4}, - θ + \frac{π}{4}, \\ θ + \frac{π}{2}, - θ + \frac{π}{2}, \\ θ + \frac{3 π}{4}, - θ + \frac{3 π}{4}, \\ θ + π, - θ + π, \\ θ + \frac{5 π}{4}, - θ + \frac{5 π}{4}, \\ θ + \frac{3 π}{2}, - θ + \frac{3 π}{2} \\ θ + \frac{7 π}{4}, - θ + \frac{7 π}{4}, - θ + 2 π \end{array}$	16

3.3 Experiment 3

In this experiment, the best parameters for a given image were calculated, under the criterion of concentrating the energy in the lowest number of coefficients when applying the discrete wavelet transform implemented with the corresponding filters. It also means to maximize the number of negligible coefficients and to minimize the entropy [6].

The energy of a gray-scale image with w × h pixels is: $E = \sum_{i = 0}^{w - 1} \sum_{j = 0}^{h - 1} p_{ij}^{2}, for 0 \leq p_{ij} \leq 255$ (76) where the p_ij is a pixel value.

After applying the discrete wavelet transform, the wavelet coefficients W_ij conserve the energy, so the energy in the time-frequency domain is: $E = \sum_{i = 0}^{w - 1} \sum_{j = 0}^{h - 1} W_{ij}^{2}$ (77)

By sorting the w × h wavelet coefficients W_ij in descendent order given by the index k, the cumulative energy of the first r coefficients of the signal is defined as: $E_{c} = \sum_{k = 0}^{r} W_{k}^{2}, 0 \leq r < w \times h$ (78) then, it is possible to use a genetic algorithm to optimize the filter parameters that minimize r and maximize the number of negligible wavelet coefficients NC which are less than a threshold value T_h. In fact, NC = w × h - r. The threshold T_h can be automatically calculated when the percentage of cumulative energy $PE = \frac{E_{c}}{E}$ reaches a predetermined value. So, the fitness function is: $MAX fitness (PE) = NC$ (79) Equation (79) means that the number of negligible wavelet coefficients depends on the desired percentage of the preserved cumulative energy, i.e. PE can be view as a quality factor. When PE is close to 1.0 the quality increases and viceversa. For a fixed PE provided by the user, it is calculated the best evolutionary wavelet filter parameters for a given image by appealing to the nested optimization of shorter filters.

Equation (79) aims to build good wavelet function basis based on the maximization of the number of negligible wavelet coefficients that reduces their entropy (maximize the zeros and produce a non-uniform distribution). It takes into account that there is a relationship betweeen the energy concentration and the entropy reduction [6].

The genetic algorithm has a population of even size P, with a temporal population of size 2P, pair wise for crossover between the i-th and the (P - 1 - i)-th sorted individuals, i ∈ [0, P - 1], deterministic selection, the best P individuals from the temporal population and annular crossover [17]. Each individual encodes the values of the parameters in the interval [0, 2π) and PE = 0.9999997. The parameters for the genetic algorithm were Pm = 0.02, Pc = 0.97, and P = 12 individuals, with 30 bits of precision.

For comparison purposes, the negligible wavelet coefficients for Haar, D4, D6, D8, D10, D12, LA12, B12 and SF12 filters [16] were calculated. The results are shown in Table 2, where the bold numbers remark the number of negligible coefficients (higher is better) of the best standard filter for each image.

The last column shows the best evolutionary filter identified as H12, whose performance is superior to the best standard filter in all the test cases.

It is very important to note that there is no a unique standard wavelet that outperforms the other standard filters in the experiments. So, in spite of the use of a reduced number of standard wavelets, it is necessary to compare their performance one by one to find the best standard solution. As an additional note, in the experiments it was preferred to use the parameters obtained with inverse equations to produce the standard filters instead of programming wired code for them, once the parameterization includes these standard filters as particular cases.

Different values for PE can be chosen as a quality factor for the reconstruction error. This experiment shows a case of quasi-perfect reconstruction scheme with too high PE for images of 512 × 512 = 262144 pixels (wavelet coefficients after transforming) but it is applicable to larger images [10].

There was no preference in the selection of images, only the availability in gray-scale, a size of 512 × 512 pixels, and that they are already known in image processing.

The average difference between the best standard filter and the proposed best evolutionary filter H12 is 0.05%. Note that the best standard filters for the five test images have length 12 (see Table 2) and that there is no a single standard filter that outperforms the others in all the cases (Lena-Coiflet, Barbara-SF12, Boat-Symlet, Mandrill-LA12, Peppers-Symlet).

The worst case of standard filters corresponds to the Haar filter. In this case, the average difference between the Haar and the proposed best evolutionary filter is 1.63%. Since, there is a relationship between the number of negligible wavelet coefficients and their entropy [6] then is possible to improve the compression based on the optimization of the filter.

Although the margin of difference may be small, consider that the percentage of preserved energy is too high (PE = 99.99997%), so the most important is that no matter which standard filter it used, it is possible to calculate a better parameterized filter, and that the improvement will depend on the application.

Table 2

Filter comparison: maximization of negligible wavelet coefficients on gray-scale images, 512 × 512 pixels

	Lena	Barbara	Boat	Mandrill	Peppers
Haar	22569	18000	24193	10870	17726
D4	26929	23532	29442	12287	21454
D6	26687	24468	29516	12610	21543
D8	26884	25006	29812	12649	21567
D10	27031	25307	29804	12741	21499
e D12	27036	25326	29886	12648	21235
LA12	27037	25744	30129	12877	21672
B12	27035	25771	29817	12833	21414
SF12	27220	25944	29764	12816	21170
Coiflet	27475	25112	30068	12646	21634
Symlet	27340	25511	30145	12787	21675
H12	27537	26003	30383	13027	21886

4 Results and discussion

4.1 Experimental results

Experiment 1 validates the inverse equations obtained from the parameterizations of length twelve filters. There, a repair technique for the parameters is necessary to obtain the correct values of the parameters, given the periodicity properties of the trigonometric functions and the rotation matrices.

Experiment 2 makes possible to reduce the number of repair alternatives, with the consequent acceleration when searching for the correct parameters. In this experiment, it was possible to reach a percentage greater than 90% of efficiency.

Experiment 3 presents a quasi-perfect reconstruction scheme for image processing where 99.99997% of the energy is preserved and the RMS error is lower than 0.1 even a maximum number of wavelet coefficients are set to zero.

Experiment 3 also shows the feasibility of applying the inverse equations to optimize large filters based on a nested optimization of shorter filters, which in fact outperforms standard filters.

4.2 Discussion

The wavelet transform can be implemented with perfect reconstruction filters and general purpose filters are widely used, however for many applications it is preferable to design customized filters despite the time required to optimize them, such as offline image compression, watermarking and time series, among others where the quality or prediction are priorities.

A disadvantage of using large wavelet filters is that it is not easy to comply with the perfect reconstruction properties, so the parameterization offers a solution. But, it stills the task of optimizing the parameters of larger filters as quickly as possible. That is where the main contribution of the paper is focused on, and derived of this it is recommended the use of inverse parameterizations.

5 Conclusions and future work

The parameterizations of wavelet filters have already been studied and inverse equations for length twelve filters have been proposed to accelerate their optimization.

Experimental results motivate the study of personalized filters that offer better performance than general purpose filters. So, it is of interest to continue the study for larger filters. Since larger filters have more freedom degrees, it is possible to build filters with a variety of characteristics, and to get better results with respect to standard filters.

The application of the inverse equations opens the possibility of using larger parametric filters in applications where it is worth spending more time to optimizing them. Future work includes the study of parameterizations for larger filters, to explore other wavelet families in greater dimension, and to present applications in speech synthesis with wavelet neural networks, as well as time series forecasting.

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