This paper proposes a method to design multichannel cosine modulated filter bank for image compression using multiobjective optimization technique. The design problem is a combination of stopband residual energy, least square error of the overall transfer function of the filter bank, coding gain with dc leakage free condition as constraint. The proposed algorithm uses Non-dominated Sorting Genetic Algorithm (NSGA) to minimize the mutually contradictory objective function by minimizing filter tap weights of prototype filter. The algorithm solves this problem by searching solutions that achieve the best compromise between the different objectives criteria. The performance of this algorithm is evaluated in terms of coding gain and peak signal to noise ratio (PSNR). Simulation results on different images are included to illustrate the effectiveness of the proposed algorithm for image compression application.
Filter banks find application in various fields of signal, image and video processing for subband/transform coding [
In this paper, an approach based on multiobjective optimization i.e. non-dominated sorting genetic algorithm [
The organization of the paper is as follows: in Section 2, a relevant brief analysis of the pseudo cosine modulated filter bank is given. In Section 3 optimization problem is formulated. A brief overview of multiobjective optimization algorithm is explained in Section 4. In Section 5, design examples (cases) and results are presented with their application to image coding and conclusions are drawn in Section 6.
A typical M-band filter bank is shown in
where
tude distortion to input signal) and
In cosine modulated filter bank (CMFB) impulse response of all the analysis and synthesis filters generated from the prototype filter
For
Design criteria for effective filter bank are:
In M-band cosine modulated filter bank perfect reconstruction conditions [
where
The near perfect reconstruction condition measure is expressed as
A widely accepted measure of coding performance is the Coding Gain (CG) which measures the energy concentration capability of filter banks. By modeling a natural image as a one-dimensional Markovian source with a correlation factor
where:
As suggested in [
CMFB can be kept small by keeping the attenuation large for
where
In image coding, dc leakage free condition does not yield significant improvements in image quality [
For optimizing a number of conflicting objective functions simultaneously, multiobjective optimization techniques are used. These techniques provide Pareto-optimal solutions instead of unique optimal solution. These solutions are optimal in the sense that no other solutions in the parameter space are superior to them when all the objective functions are considered [
A multiobjective optimization can be stated mathematically as the minimization problem to minimize
where
Vectors x and y represent the independent variables and the corresponding values of the individual objective functions respectively; on the other hand, X and Y are called the solution space and objective space, respectively.
The notion of Pareto-optimality is an important concept for multiobjective and explained in terms of a “dominance relation” as a solution
If a solution is not dominated by any other solutions among the entire objective functions than that solution is said to be a nondominated solution. The solutions that are nondominated regarding the entire parameter space are called Pareto-optimal solutions.
In our design, we search prototype filter h that minimize the three individual objective functions, namely, the energy of the filter stopband, near perfect reconstruction measure and the coding gain. To confine the search process in a feasible solution space, we impose dc leakage free condition as constraint. We have to determine only half of the coefficients since the remaining coefficients are obtained, by applying symmetry.
Our multi-objective optimization problem is formulated as follows:
Subject to the constraint:
In our case, a set of coefficients of the analysis filter bank
Filter banks designed using non dominated sorting genetic algorithm here use some important parameters i.e. real valued filter coefficients for chromosome construction, crossover operator with distribution index of 20 and probability of 0.9, a polynomial mutation of distribution index 20 and probability of 0.01 are applied [
coefficients are set as
else are equal to zero, satisfying the unit energy constraint between [−1, 1]. NPR condition is set by keeping the distortion to 10−5. The maximum generation
Magnitude response of 8 Channel and 16 Channel NPR Pseudo QMF filter banks with 32 and 64 taps respectively, with dc leakage free constraint, designed using multiobjective optimization method, have been shown in
The performance of our designed Cosine Modulated Filter Bank’s is evaluated through a progressive block based EZW image coder [
The images chosen for the coding experiments are Barbara and Boat. Both of them are standard, well known 512 × 512 8-bit gray-scale test images. The objective distortion measure is the popular peak signal-to-noise ratio (PSNR)
where, MSE denotes the mean squared error. Filter banks 8 × 16 and 8 × 24 were included in the tests, and both were optimized for maximum coding gain. Reconstructed images “Boat” and “Barbara” using designed filter bank (8 × 16) and (8 × 24) are shown in
It is clear that longer filter banks with more no. of taps have the best characteristics with increase in PSNR. In this context, it is important to note that the filter length cannot be arbitrarily increased in order to increase PSNRs and achieve better frequency selectivity because long filters cause a ringing around edges artefacts when high frequency subband are coarsely quantized. Reconstructed images are better in visual quality as the frequency selectivity improves and overall distortion reduced, however, as the no. of taps increased artifacts become pronounced and PSNR decreases as the bit rates further reduced.
As shown in
Objective results for improvement in performance are shown in
Results shown in
Transform | Coding Gain | Stopband Attenuation (dB) |
---|---|---|
8 × 8 DCT | 8.83 | 9.96 |
8 × 16 LOT | 9.22 | 19.38 |
8 × 24 GenLOT | 9.35 | 23.20 |
Proposed (8 × 24) | 9.3749 | 26.026 |
M No. of channels | Length of prototype filter | With dc leakage | Without dc leakage | ||
---|---|---|---|---|---|
PSNR | CG | PSNR | CG | ||
8 | 16 | 28.02 | 8.89 | 29.99 | 8.85 |
8 | 24 | 30.11 | 9.37 | 31.18 | 9.21 |
8 | 32 | 32.02 | 9.374 | 33.63 | 9.257 |
The design of M-channel uniform cosine modulated filter banks using genetic algorithm is discussed in this paper. The design problem having multiple objectives is solved using non-dominated sorting algorithm with regularity constraint. Although dc leakage free condition is not an essential requirement for image processing in M- channel filter bank as PSNRs don’t improve considerably, its imposition identifies infeasible solutions thereby confines the search for the NPR filter banks simultaneously, and improve visual quality of images at low bit rates. From simulation result, it is shown that the algorithm results in lower tap filters with high stopband attenuation, the least overall distortion and higher coding gain for image compression. Use of progressive image coder based on block-based EZW improves PSNR via reducing blocking and ringing artefacts. The work can be further extended to design of biorthogonal cosine modulated filter banks for image coding application.