Mathematical Problems in Engineering

Volume 2015, Article ID 414561, 9 pages

http://dx.doi.org/10.1155/2015/414561

## An Image Filter Based on Shearlet Transformation and Particle Swarm Optimization Algorithm

^{1}Nanjing University of Information Science and Technology, Nanjing 210044, China^{2}Southeast University, Nanjing 210000, China^{3}PLA University of Science and Technology, Nanjing 210000, China

Received 26 August 2014; Revised 18 October 2014; Accepted 19 October 2014

Academic Editor: Erik Cuevas

Copyright © 2015 Kai Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Digital image is always polluted by noise and made data postprocessing difficult. To remove noise and preserve detail of image as much as possible, this paper proposed image filter algorithm which combined the merits of Shearlet transformation and particle swarm optimization (PSO) algorithm. Firstly, we use classical Shearlet transform to decompose noised image into many subwavelets under multiscale and multiorientation. Secondly, we gave weighted factor to those subwavelets obtained. Then, using classical Shearlet inverse transform, we obtained a composite image which is composed of those weighted subwavelets. After that, we designed fast and rough evaluation method to evaluate noise level of the new image; by using this method as fitness, we adopted PSO to find the optimal weighted factor we added; after lots of iterations, by the optimal factors and Shearlet inverse transform, we got the best denoised image. Experimental results have shown that proposed algorithm eliminates noise effectively and yields good peak signal noise ratio (PSNR).

#### 1. Introduction

Images are frequently contaminated by noise on the processes of formation, transmission, and reception and make following processes such as segmentation, recognition difficult. This phenomenon makes noise reduction one of the most important problems in image processing.

Basically, there are three common methods to solve this problem, such as transform domain method [1, 2], spatial domain method [3], and partial differential equations (PDE) [4] method; here, our work is one branch of transform domain method. In transform domain method, because of its good performance in both time domain and frequency domain, wavelet transform has become one of the most active research fields in image enhancing. In this field, multiscale geometric analysis theory is the hottest and the most advanced research area. Under this theory, many famous achievements are proposed, for example, ridgelet [5], curvelet [6], contourlet [7], and bandlet [8]. However, with these achievements, there are also many problems to overcome.

Recently, Labate et al. proposed a novel class of multidimensional representation systems, which is called Shearlet. One advantage of this approach is that these systems can be constructed using generalized multiresolution analysis and implemented efficiently using a classical cascade algorithm [1, 9–15]. Due to the fact that Shearlet transforms have multiscale and multidirection feature, simple hard threshold denoising method could not yield bad performance in its practical application of image denoising [16]. Fan and Zhao [17–19] researched the problem of how to calculate the best threshold in this denoising method; they proposed a good idea that used optimal algorithm to get best threshold, but, in their papers, they did not research the fitness function. We hold this opinion that this last step is worth studying and the finding has theoretical value and practical directive meaning.

Among many optimal algorithms [20–33], PSO algorithm has been the most famous global optimization algorithm which is based on swarm intelligence. And it is easy to be realized and already applied to many fields. In this paper, we adopted PSO algorithm to optimize the problem we mentioned above. In this process, as an important part of PSO algorithm, an objective fitness function should be built to evaluate quality of reconstructed image, so we adopted a fast and rough method to evaluate noise level of image, and this method is just asked to be able to reflect the change trend of noise level in image. Many experiments data have shown that proposed algorithm can achieve better performance than classical Shearlet transform, but like all optimization algorithms, it also has drawback of slow computation speeds.

The remaining paper is organized as follows. Section 2 introduces related theories. Section 3 explains our algorithm, including workflow, Section 4 presents the experiment results of proposed algorithm, and Section 5 concludes this paper.

#### 2. Related Theories

##### 2.1. Shearlet Transform

Labate et al. [1, 2] proposed Shearlet transform based on wavelet. In dimension , affine systemhere, and , are invertible matrices with .

If satisfied Parseval , then those elements of are called composite wavelets.

Shearlet is a special example of , for only when here is the anisotropic dilation matrix and is the shear matrix.

For , when , , satisfy

Then, we get

Then, form a tiling of the set

From the condition on the support of , , it is easily deduced that have frequency support contained in the setThus, every element in is supported on a pair of trapezoids of approximate size , oriented along lines of slope .

For , here is the vertical cone, when formula (7) was satisfied:

Then, collection is a Parseval frame for .

##### 2.2. Particle Swarm Optimization Algorithm

The classical PSO algorithm is described as follows [34].

*Step 1*. Initialize a population of particles with random positions and velocities in a -dimension problem space.

*Step 2*. For each particle, evaluate its fitness value.

*Step 3*. Compare each particle’s fitness evaluation with the current particle’s* pbest*. If current value is better than* pbest*, set its* pbest* value to the current value and the* pbest* location to the current location in -dimensional space.

*Step 4*. Compare fitness evaluation with the population’s overall previous best. If current value is better than* gbest*, then reset* gbest* to the current particle’s array index and value.

*Step 5*. Change the velocity and position of particle according to Step and Step , respectively:

*Step 6*. Loop to Step until a stopping criterion is met.

*Step 7*. Over.

The vector is the position of th particles, is the velocity of the th particles, and is the best previous position (the position giving the best fitness value) of the th particles. The index is the index of the best particle among all the particles in the swarm. Variable is the inertia weight, and are positive constants, and rand is random number in range generated according to a uniform probability distribution. Particles’ velocities along each dimension are clamped to a maximum velocity . If the sum of acceleration causes the velocity on that dimension to exceed , which is a parameter specified by the user, then the velocity on that dimension is limited to .

The inertia weight represents the degree of the momentum of the particles. The second part is the “cognition” part, which represents the independent behavior of the particle itself. The third part is the “social” part, which represents the collaboration among the particles. The constants and represent the weighting of “cognition” and “social” parts that pull each particle towards* pbest* and* gbest* positions.

#### 3. Proposed Algorithm

This chapter is divided into 3 parts. Section 3.1 introduced the new threshold rule we designed which is based on classical Shearlet theory. And the fitness function we quoted is shown in Section 3.2. Then, our all algorithm workflow (Figure 1) is detailed and introduced in Section 3.3.