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).
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 [
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 [
Among many optimal algorithms [
The remaining paper is organized as follows. Section
Labate et al. [
If
Shearlet is a special example of
For
Then, we get
Then,
From the condition on the support of
For
Then, collection
The classical PSO algorithm is described as follows [
The vector
The inertia weight
This chapter is divided into 3 parts. Section
Workflow of proposed algorithm.
Threshold rule is the most important problem in image denoising of transform domain, and the hard threshold and the softthreshold approach are two options. Fan et al. [
Research shows that Fan et al. threshold is not the optimal threshold. Considering this, Fan et al. [
All formula (
However, in optimization algorithm, the key problem is how to design an objective and good fitness.
Fan and Zhao [
Wang [
Define
Here, let us suppose the classical Gaussian filter is perfect; its output is noiseless, so we can suppose the median data of signal to be the estimate of signal intensity in image.
Then, we suppose signal in image is at low frequencies, and
Based on this definition of fitness, we proposed our algorithm.
The most critical problem is how to find the best weighted factor
Our algorithm main routine is as follows.
Our PSO subroutine is shown as follows.
Maximum iterations are exceeded.
The optimal target value is achieved.
For the sake of evaluating the performance of fitness function in proposed algorithm, we added Gaussian noise in several classical images with different variance and then calculated their fitness function data and plotted in Figure
Performance of our fitness function.
Boat
Our fitness function of image (a) in different noise level
Barbara
Our fitness function of image (c) in different noise level
In Figure
These two images are
Based on the above, we can draw a conclusion that this fitness function (formula (
In order to validate the performance of this algorithm, 4 images were filtered by proposed algorithm and classical Shearlet algorithm that we downloaded code from website [
Comparison chart of two algorithms on some images.
Head MRI
Baboon
Barbara
Brain MRI
Comparison chart of two algorithms on (a)
Improved PSNR of (e)
Comparison chart of two algorithms on (b)
Improved PSNR of (g)
Comparison chart of two algorithms on (c)
Improved PSNR of (i)
Comparison chart of two algorithms on (d)
Improved PSNR of (k)
Figures
So the result of these experiments data had shown that our proposed algorithm has better denoising performance than classical Shearlet algorithm, and this advantage will decrease when noise level increased.
To show the “improved PSNR” data in Figure
Improved PSNR of Figure
Average of Figure 
Average of Figure 
Average of Figure 
Average of Figure 

0.057  0.072  0.127  0.119 


Best of Figure 
Best of Figure 
Best of Figure 
Best of Figure 


0.132  0.243  0.281  0.542 


Worst of Figure 
Worst of Figure 
Worst of Figure 
Worst of Figure 


−0.043  0.005  −0.076  −0.081 
In all above experiments, 9 main PSO algorithm’s parameters adopted are shown in Table
Some PSO parameters.
Population size  24 
Local best influence  2 
Initial inertia weight  0.9 
Minimum global error gradient 

VarRange 

Maximum number of iterations  100 
Global best influence  2 
Final inertia weight  0.4 
Max particle velocity  0.1 
In order to eliminate Gaussian noise and preserve image details as much as possible, this paper proposed a novel image filter algorithm which combined merit of PSO algorithm and classical Shearlet transform. It decomposes noised image into many subwavelets under multiscale and multiorientation by classical Shearlet transform; design a fitness functions in PSO and give weighted factors to each subwavelet; use PSO to find optimal weighted factors; after that, reconstruct new and denoised image. It has two contributions: one is proposing PSO to optimal Shearlet transform and another is quoting a rough, objective, fast fitness function to measure images quality. Computer simulations results are given to verify the effectiveness of this algorithm.
The authors declare that they have no conflict of interests.
All authors drafted the paper and read and approved the final paper.
Whatever implementation the authors come up with needs to be based on classical Shearlet algorithm, and they cannot execute their work without Professor Labate’s earlier work and his opensource code online. Thanks go to Professor Labate. The authors would like to thank the anonymous reviewers whose comments greatly improved the paper. This paper was funded under a Grant from the National Natural Science Foundation of China (no. 61105115), “Six Talent Peaks Program” of Jiangsu Province of China (2014XXRJ007), a Grant from the Jiangsu Province Natural Science Foundation (nos. BK20131000, S7013028001), National Training Program for Undergraduate on Innovation and Entrepreneurship (no. 201310300039Z), Industry Academia Research Cooperative Innovation Fund of Jiangsu Province (BY201300701), and Open Program based on Laboratory of Nanjing University of Information Science and Technology (nos. 2014KF001, 2014KF006, and 2014KF0019).