# An Image Filter Based on Multiobjective Genetic Algorithm and Shearlet Transformation

Authors: Zhi-yong Fan; Quan-sen Sun; Ze-xuan Ji; Kai Hu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Rician noise pollutes magnetic resonance imaging (MRI) data, making data’s postprocessing difficult. In order to remove this noise and avoid loss of details as much as possible, we proposed a filter algorithm using both multiobjective genetic algorithm (MOGA) and Shearlet transformation. Firstly, the multiscale wavelet decomposition is applied to the target image. Secondly, the MOGA target function is constructed by evaluation methods, such as signal-to-noise ratio (SNR) and mean square error (MSE). Thirdly, MOGA is used with optimal coefficients of Shearlet wavelet threshold value in a different scale and a different orientation. Finally, the noise-free image could be obtained through inverse wavelet transform. At the end of the paper, experimental results show that this proposed algorithm eliminates Rician noise more effectively and yields better peak signal-to-noise ratio (PSNR) gains compared with other traditional filters.


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## Body

## 1. Introduction

Magnetic resonance imaging (MRI) devices are important imaging equipment, and the image quality directly impacts the diagnosis accuracy. However, MRI images are frequently contaminated by Rician noise during image gaining or transmission [1]. This phenomenon makes noise reduction to be one of the most important problems in image processing. Preservation of image details and attenuation of noise are both critical, but they are contradictory in nature. Therefore, this research is focused on Rician noise elimination and data details preservation at the same time.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 processing. It provides better results and preserves more details compared with traditional algorithms. However, wavelet transform cannot achieve optimal sparse for images containing higher-dimension singularity. To overcome the limitation, multiscale geometric analysis theory is proposed, and, based on it, a series of methods sprang out, for example, ridgelet [2], curvelet [3], contourlet [4], and bandlet [4]. One of the most successful ideas is the curvelets of Candes and Donoho, which achieve an (almost) optimal approximation for 2D piecewise smooth functions with discontinuities along with C2 curves.Recently, Labate et al. described a new 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 [5–11].Simple threshold denoising method of classical Shearlet transform could yield good performance because of the method’s multiscale and multidirection characteristics and image sparse representation. However, there is still room for improvement because classical Shearlet algorithm does not take energy distribution of different scales and different directions into consideration; as a result, it kills the coefficient excessively; therefore, image details could be lost. In order to solve the problem, Sun and Zhao [12] proposed a particle swarm optimization; it uses adaptive algorithm to search for optimal threshold of the highest PSNR values.Based on these previous achievements, this paper proposed a new image-filtering algorithm. It has three characteristics: it uses soft threshold in Shearlet, it builds target function in MOGA by several evaluation methods, and it uses the MOGA to optimize coefficients of Shearlet wavelet threshold value in different scale and a different orientation.The rest of this paper is organized as follows. Section2 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. Rician Noise

Noised MRI imagev can be defined as v(i)=u(i)+n(i); here, u(i)represent original image pixels, and n(i)represent is noised pixels. When MR images are computed by using the magnitude of single-complex raw data, its distribution can be modeled as a Rician model [13–15]. Consider the following:
(1)p(m)=mσn2e-(m2+A2)/2σn2I0(Amσn2).
Here, σ2 is the standard deviation (STD) of Gaussian noise, Ais the amplitude of the signal without noise, x is the value of the magnitude image, and I0 is the 0th-order modified Bessel function. This model is used by the majority of the noise estimation methods.When SNR is small enough (i.e.,SNR=0), the Rician distribution is considered as a Rayleigh distribution. Consider the following:
(2)p(m)=mσn2e(m2σn2).When SNR is high (i.e.,SNR>3), the Rician distribution is approximated as a Gaussian distribution. 								(3)p(m)=12πσ2e(-(m2-A2+σn2)22σn2).


### 2.2. Shearlet Transform

Labate et al. [5–11] proposed Shearlet transform based on wavelet. With dimension n=2, consider the following affine system:
(4)ΨAB(ψ)={ψj,l,k(x)=|detA|j/2ψ(BlAjx-k):j,l∈ℤ,k∈ℤ2}.
Here, ψ∈L2(ℝ2), and A,B are 2*2 invertible matrices with |detB|=1.IfΨAB(ψ) satisfied Parseval L2(ℝ2), then, those elements of ΨAB(ψ) are called composite wavelets.Shearlet is a special example ofL2(ℝ2) only when
(5)A=A0=(4002),B=B0=(0111).
Here, A=A0 is the anisotropic dilation matrix, and B=B0 is the shear matrix.Forξ=(ξ1,ξ2)∈ℝ^2,ξ1≠0, when ψ(0),ψ^1, andψ^2 satisfy
(6)ψ^(0)(ξ)=ψ^(0)(ξ1,ξ2)=ψ^1(ξ1)ψ^2(ξ2ξ1),ψ^1,ψ^2∈C∞(ℝ^),suppψ^1⊂[-12,-116]∪[116,12],suppψ^2⊂[-1,1],∑j≥0|ψ^1(2-2jω)|2=1for|ω|≥18,j≥0,∑l=-2j2j-1|ψ^2(2jω-l)|2=1for|ω|≤1.Then, we get(7)∑j≥0∑l=-2j2j-1|ψ^(0)(ξA0-jB0-l)|2=∑j≥0∑l=-2j2j-1|ψ^1(2-2jξ1)|2|ψ^2(2jξ2ξ1-l)|2=1.Then,{ψ^(0)(ξA0-jB0-l)} form a tiling of the set
(8)D0={(ξ1,ξ2)∈ℝ^2:|ξ1|≥18,|ξ2ξ1|≤1}.From the condition on the support ofψ^1andψ^2, it is easily deduced that ψ^j,l,k have frequency support contained in the set as follows:
(9)suppψ^j,l,k(0)⊂{|ξ2ξ1+l2-2j|≤2-j(ξ1,ξ2):ξ1∈[-22j-1,-22j-4]∪[22j-4,22j-1],|ξ2ξ1+l2-2j|≤2-j}.Thus, every element inψj,l,k is supported on a pair of trapezoids of approximate size 22j×2j, oriented along lines of slopel2-j[].ForL2(D1)∨, here, D1 is the vertical cone when  the following formula was satisfied:
(10)D1={(ξ1,ξ2)∈ℝ^2:|ξ2|≥18,|ξ1ξ2|≤1},A1=(2004),B1=(1011),ψ^(1)(ξ)=ψ^(1)(ξ1,ξ2)=ψ^1(ξ2)ψ^2(ξ1ξ2).Then, collection{ψj,l,k(1)(x)=23j/2ψ(1)(B1lA1jx-k):j≥0,-2j≤l≤2j-1,k∈ℤ2} is a Parseval frame forL2(D1)∨.


### 2.3. Multiobjective Genetic Algorithm (MOGA)

Multiobjective genetic algorithm seeks feasible solutions to problems comprising multiple objectives which are often in conflict with each other. A general minimization problem ofMobjectives can be mathematically stated as x=[x1,x2,…,xd], wheredis the dimension of the decision variable space. Consider the following.Minimizef(x)=[fi(x),i=1,…,m] which satisfies
(11)gj(x)≤0,j=1,2,…,J,hk(x)=0,k=1,2,…,K,
where fi(x) is theith objective function, gj(x) is thejth inequality constraint, and hk(x) is thekth equality constraint. The multiobjective optimization problem then reduces to finding anx, such thatf(x)is optimized.


## 2.1. Rician Noise

Noised MRI imagev can be defined as v(i)=u(i)+n(i); here, u(i)represent original image pixels, and n(i)represent is noised pixels. When MR images are computed by using the magnitude of single-complex raw data, its distribution can be modeled as a Rician model [13–15]. Consider the following:
(1)p(m)=mσn2e-(m2+A2)/2σn2I0(Amσn2).
Here, σ2 is the standard deviation (STD) of Gaussian noise, Ais the amplitude of the signal without noise, x is the value of the magnitude image, and I0 is the 0th-order modified Bessel function. This model is used by the majority of the noise estimation methods.When SNR is small enough (i.e.,SNR=0), the Rician distribution is considered as a Rayleigh distribution. Consider the following:
(2)p(m)=mσn2e(m2σn2).When SNR is high (i.e.,SNR>3), the Rician distribution is approximated as a Gaussian distribution. 								(3)p(m)=12πσ2e(-(m2-A2+σn2)22σn2).


## 2.2. Shearlet Transform

Labate et al. [5–11] proposed Shearlet transform based on wavelet. With dimension n=2, consider the following affine system:
(4)ΨAB(ψ)={ψj,l,k(x)=|detA|j/2ψ(BlAjx-k):j,l∈ℤ,k∈ℤ2}.
Here, ψ∈L2(ℝ2), and A,B are 2*2 invertible matrices with |detB|=1.IfΨAB(ψ) satisfied Parseval L2(ℝ2), then, those elements of ΨAB(ψ) are called composite wavelets.Shearlet is a special example ofL2(ℝ2) only when
(5)A=A0=(4002),B=B0=(0111).
Here, A=A0 is the anisotropic dilation matrix, and B=B0 is the shear matrix.Forξ=(ξ1,ξ2)∈ℝ^2,ξ1≠0, when ψ(0),ψ^1, andψ^2 satisfy
(6)ψ^(0)(ξ)=ψ^(0)(ξ1,ξ2)=ψ^1(ξ1)ψ^2(ξ2ξ1),ψ^1,ψ^2∈C∞(ℝ^),suppψ^1⊂[-12,-116]∪[116,12],suppψ^2⊂[-1,1],∑j≥0|ψ^1(2-2jω)|2=1for|ω|≥18,j≥0,∑l=-2j2j-1|ψ^2(2jω-l)|2=1for|ω|≤1.Then, we get(7)∑j≥0∑l=-2j2j-1|ψ^(0)(ξA0-jB0-l)|2=∑j≥0∑l=-2j2j-1|ψ^1(2-2jξ1)|2|ψ^2(2jξ2ξ1-l)|2=1.Then,{ψ^(0)(ξA0-jB0-l)} form a tiling of the set
(8)D0={(ξ1,ξ2)∈ℝ^2:|ξ1|≥18,|ξ2ξ1|≤1}.From the condition on the support ofψ^1andψ^2, it is easily deduced that ψ^j,l,k have frequency support contained in the set as follows:
(9)suppψ^j,l,k(0)⊂{|ξ2ξ1+l2-2j|≤2-j(ξ1,ξ2):ξ1∈[-22j-1,-22j-4]∪[22j-4,22j-1],|ξ2ξ1+l2-2j|≤2-j}.Thus, every element inψj,l,k is supported on a pair of trapezoids of approximate size 22j×2j, oriented along lines of slopel2-j[].ForL2(D1)∨, here, D1 is the vertical cone when  the following formula was satisfied:
(10)D1={(ξ1,ξ2)∈ℝ^2:|ξ2|≥18,|ξ1ξ2|≤1},A1=(2004),B1=(1011),ψ^(1)(ξ)=ψ^(1)(ξ1,ξ2)=ψ^1(ξ2)ψ^2(ξ1ξ2).Then, collection{ψj,l,k(1)(x)=23j/2ψ(1)(B1lA1jx-k):j≥0,-2j≤l≤2j-1,k∈ℤ2} is a Parseval frame forL2(D1)∨.


## 2.3. Multiobjective Genetic Algorithm (MOGA)

Multiobjective genetic algorithm seeks feasible solutions to problems comprising multiple objectives which are often in conflict with each other. A general minimization problem ofMobjectives can be mathematically stated as x=[x1,x2,…,xd], wheredis the dimension of the decision variable space. Consider the following.Minimizef(x)=[fi(x),i=1,…,m] which satisfies
(11)gj(x)≤0,j=1,2,…,J,hk(x)=0,k=1,2,…,K,
where fi(x) is theith objective function, gj(x) is thejth inequality constraint, and hk(x) is thekth equality constraint. The multiobjective optimization problem then reduces to finding anx, such thatf(x)is optimized.


## 3. Proposed Algorithm

### 3.1. Threshold Rule

Threshold rule is the most important problem in image denoising of transform domain, and the hard-threshold and the soft-threshold approaches are two options. Donoho and Johnstone [16] proposed the following threshold rule:
(12)δ=σ2ln(N).
Here,Nis the pixels number of image, and σ is the noise level.Research shows that Donoho threshold is the optimal threshold limit not the optimal threshold. With this considered, Donoho and Johnstone [16] proposed an improved threshold rule as follows:
(13)δk=σ2ln(N)*2(k-K)/2,k=0,1,…,K.As many researchers point out [12, 13], (12) did not consider energies of subwavelets in a different direction while being in the same scale, and this imperfection will make coefficients too much stifled.Considering the variability of image content and Shearlet transformation of multiscale and multidirection characteristics, a novel threshold selection rule is proposed based on Shearlet transform multiscale and multidirection; this rule is the following.Comprehensively considering complexity of image and the multiscale and multidirection characteristics of Shearlet transform,  this paper proposed the following adaptive threshold rule:(14)δk,j=Sigmoid(σ2ln(N)*2(k-K)/2),k=0,1,…,K,Sigmoid=11+e-v.Here, Sigmoid is adopted to build our rules. The Sigmoid curve is a mathematical concept which has been widely used to model the natural life cycle of many things, for its derivative is continuous and with higher accuracy.Kis the scale level, andjis thejth direction under thekth scale level.


### 3.2. Target Function

We build MOGA target function by the signal-to-noise ratio (SNR) and the mean square error (MSE).Signal-to-noise ratio (SNR) can be defined as(15)SNR=10lg∑i=1M∑j=1NA(i,j)2∑i=1M∑j=1N(A(i,j)-O(i,j))2.Here,Ois original image with size of M×N pixels,Ais filtered image of noised image, and(i,j)are coordinates of pixels.Mean square error (MSE) expressed the correlation between images, and it is defined as follows:(16)MSE=1M×N∑i=1M∑j=1N(A(i,j)-O(i,j))2.Here,Ois original image with size of M×N pixels,Ais filtered image of noised image, and(i,j)are coordinates of pixels.Our target function is defined as follows:(17)y=ω1·SNR+ω2·MSE,0≤ω1,ω2≤1,ω1+ω2=1.Here,ω1,ω2 are weight coefficients of SNR and MSE.


### 3.3. Proposed Model

The most critical problem which lies in our optimal filtering performance study is, under optimization criterion, how to decide coefficientsv, δk,j considering energy of subwavelets not only in different scale but also in different direction.Here, we proposed our algorithm which adopts MOGA algorithm to decide coefficientsv, δk,j of each subwavelet in different scale and direction of Shearlet transform, intending to get optimal filtering performance.Our algorithm works as follows [17, 18]; see Figure 1.Figure 1
Workflow of proposed algorithm.Step 1 (initialization).
Generate an initial population containingNpop strings, where Npop is the number of strings in each population. These strings contain weight coefficients of SNR, MSE, weight coefficients δk,j of Shearlet subwavelets,vofSfunction, and other parameters in MOGA; thus, we need the following.Step 2 (evaluation).
(1)
Use Shearlet transform to decompose target image.
(2)
Multiply subwavelets by weight coefficientsδk,j.
(3)
Filter subwavelets by threshold rule.
(4)
Reconstruct image by filtered subwavelets.
(5)
Calculate the values of the objective functions (16) for the generated strings.
(6)
Update a tentative setoff Pareto optimal solution.Step 3 (selection).
Calculate the fitness value of each string using the random weights in (3). Select a pair of strings from the current population according to the following selection probability.Step 4 (crossover).
For each selected pair, apply a crossover operation to generate two new strings.Npop new strings are generated by the crossover.Step 5 (mutation).
For each bit value of the strings generated by the crossover, apply a mutation with a prespecified mutation probability.Step 6 (elitist strategy).
Randomly removeNelite strings from the set of Npop strings generated by previous operations, and replace them with Nelite strings randomly selected from tentative set of Pareto optimal solutions.Step 7 (termination test).
If one stopping condition in the following is satisfied, go to Step8; if not,  return to Step 2.(i)
Maximum iterations are exceeded.(ii)
The optimal target value is achieved.Step 8 (algorithm termination).
Exit optimal algorithm.


## 3.1. Threshold Rule

Threshold rule is the most important problem in image denoising of transform domain, and the hard-threshold and the soft-threshold approaches are two options. Donoho and Johnstone [16] proposed the following threshold rule:
(12)δ=σ2ln(N).
Here,Nis the pixels number of image, and σ is the noise level.Research shows that Donoho threshold is the optimal threshold limit not the optimal threshold. With this considered, Donoho and Johnstone [16] proposed an improved threshold rule as follows:
(13)δk=σ2ln(N)*2(k-K)/2,k=0,1,…,K.As many researchers point out [12, 13], (12) did not consider energies of subwavelets in a different direction while being in the same scale, and this imperfection will make coefficients too much stifled.Considering the variability of image content and Shearlet transformation of multiscale and multidirection characteristics, a novel threshold selection rule is proposed based on Shearlet transform multiscale and multidirection; this rule is the following.Comprehensively considering complexity of image and the multiscale and multidirection characteristics of Shearlet transform,  this paper proposed the following adaptive threshold rule:(14)δk,j=Sigmoid(σ2ln(N)*2(k-K)/2),k=0,1,…,K,Sigmoid=11+e-v.Here, Sigmoid is adopted to build our rules. The Sigmoid curve is a mathematical concept which has been widely used to model the natural life cycle of many things, for its derivative is continuous and with higher accuracy.Kis the scale level, andjis thejth direction under thekth scale level.


## 3.2. Target Function

We build MOGA target function by the signal-to-noise ratio (SNR) and the mean square error (MSE).Signal-to-noise ratio (SNR) can be defined as(15)SNR=10lg∑i=1M∑j=1NA(i,j)2∑i=1M∑j=1N(A(i,j)-O(i,j))2.Here,Ois original image with size of M×N pixels,Ais filtered image of noised image, and(i,j)are coordinates of pixels.Mean square error (MSE) expressed the correlation between images, and it is defined as follows:(16)MSE=1M×N∑i=1M∑j=1N(A(i,j)-O(i,j))2.Here,Ois original image with size of M×N pixels,Ais filtered image of noised image, and(i,j)are coordinates of pixels.Our target function is defined as follows:(17)y=ω1·SNR+ω2·MSE,0≤ω1,ω2≤1,ω1+ω2=1.Here,ω1,ω2 are weight coefficients of SNR and MSE.


## 3.3. Proposed Model

The most critical problem which lies in our optimal filtering performance study is, under optimization criterion, how to decide coefficientsv, δk,j considering energy of subwavelets not only in different scale but also in different direction.Here, we proposed our algorithm which adopts MOGA algorithm to decide coefficientsv, δk,j of each subwavelet in different scale and direction of Shearlet transform, intending to get optimal filtering performance.Our algorithm works as follows [17, 18]; see Figure 1.Figure 1
Workflow of proposed algorithm.Step 1 (initialization).
Generate an initial population containingNpop strings, where Npop is the number of strings in each population. These strings contain weight coefficients of SNR, MSE, weight coefficients δk,j of Shearlet subwavelets,vofSfunction, and other parameters in MOGA; thus, we need the following.Step 2 (evaluation).
(1)
Use Shearlet transform to decompose target image.
(2)
Multiply subwavelets by weight coefficientsδk,j.
(3)
Filter subwavelets by threshold rule.
(4)
Reconstruct image by filtered subwavelets.
(5)
Calculate the values of the objective functions (16) for the generated strings.
(6)
Update a tentative setoff Pareto optimal solution.Step 3 (selection).
Calculate the fitness value of each string using the random weights in (3). Select a pair of strings from the current population according to the following selection probability.Step 4 (crossover).
For each selected pair, apply a crossover operation to generate two new strings.Npop new strings are generated by the crossover.Step 5 (mutation).
For each bit value of the strings generated by the crossover, apply a mutation with a prespecified mutation probability.Step 6 (elitist strategy).
Randomly removeNelite strings from the set of Npop strings generated by previous operations, and replace them with Nelite strings randomly selected from tentative set of Pareto optimal solutions.Step 7 (termination test).
If one stopping condition in the following is satisfied, go to Step8; if not,  return to Step 2.(i)
Maximum iterations are exceeded.(ii)
The optimal target value is achieved.Step 8 (algorithm termination).
Exit optimal algorithm.


## 4. Experimental Results and Analysis

### 4.1. Evaluation Index

Peak signal-to-noise ratio (PSNR) is defined as(18)PSNR=10lg2552(1/M×N)∑i=1M∑j=1N(A(i,j)-O(i,j))2.Here,Ois original image with size of M×N pixels,Ais filtered image of noised image, and(i,j)are coordinates of pixels.


### 4.2. Experimental Results

To verify the validity of the algorithm, this paper designed two kinds of experimental methods to verify its effectiveness. One is use of objective data such as PNSR and MSE to objectively analyze its performance; and the other is making us able to obverse filtering performance directly by naked eyes [19–21].Experiment 1.
We did filtering experiments on standard images Lena and Barbara in different noise level and listed results in Table1. As we have seen from Table 1, PSNR of proposed algorithm (Shearlet-MOGA) is higher than PSNR of classical Shearlet algorithm, and its performance will be better with noise level increased.
Figure2 is the original MR image we adopted to do experiments. Adding different noise level to Figure 2, we did filtering work by classical Shearlet and proposed algorithm in this paper and showed the statics data of MSE and PSNR as Tables 2 and 3.
In Table2, the excellent effect of our algorithm is dramatic, shown in and our proposed MSE is smaller than classical Shearlet algorithm. Similar good results were found when the same experiment was repeated on PSNR. In Table 3, the PSNR of proposed algorithm is greater than that of classical Shearlet algorithm.Table 1
Filtering results for Lena and Barbara.
Image
σ (%)
PSNR
Shearlet
Shearlet-MOGA
Lena
10
34.38
35.02
20
31.79
33.24
30
29.21
30.21
Barbara
10
33.07
33.07
20
29.40
29.40
30
26.31
27.51Table 2
MSE in differentσ (%) and different algorithm to that in Figure 2.
Noise level (%)
5
10
20
30
40
50
60
70
80
90
Classical
25.12
100.4
400.7
905.4
1584
2504
3579
4901
6398
8075
Proposed
10.8
26.52
73.96
147.6
240.9
381
528
705.9
910.7
1132Table 3
PSNR in differentσ (%) and different algorithm to that in Figure 2.
Noise level (%)
5
10
20
30
40
50
60
70
80
90
Classical
34.13
28.11
22.1
18.56
16.13
14.14
12.59
11.23
10.07
9.059
Proposed
40.38
35.93
31.21
28.03
25.77
23.66
22.16
20.82
19.65
18.65Figure 2
Original MR image.Experiment 2.
To evaluate the performance of proposed algorithm by naked eyes directly, several classical images such as Lena, Baboon, Barbara, and MRI are adopted to do filtering work, and all relative images are shown in Figure3.
Figure3(a) is the original Lena. Adding 5% Rician noise level to Lena, we get Figure 3(b).
Filtering Figure3(b) by classical Shearlet algorithm, we got Figure 3(c). Figure 3(d) is the output of the filtering work we did to Figure 3(b) by proposed algorithm.
We did similar experiment to the image of Baboon. Add 10% Rician noise level to Baboon, we get Figure3(e). Filtering Figure 3(f) by classical Shearlet algorithm, we got Figure 3(g). Figure 3(h) is the output of the filtering work we did to Figure 3(f) by proposed algorithm.
The image of Barbara is also adopted by us to test our algorithm. Figure3(i) is the original Barbara. Figure 3(j) is Barbara noised by 15% Rician noise level. Filtering Figure 3(j) by classical Shearlet algorithm, we got Figure 3(k). Figure 3(l) is the output of the filtering work we did to Figure 3(j) by proposed algorithm.
At last, we measured our algorithm performance on MRI image. Figure3(m) is the original MRI. Figure 3(n) is the MRI noised by 20% Rician noise level. Filtering Figure 3(n) by classical Shearlet algorithm, we got Figure 3(o). Figure 3(p) is the output of the filtering work we did to Figure 3(n) by proposed algorithm.
Through simple comparison, we can see that our proposed algorithm could effectively remove the noise from the degraded image of Rician noise with unknown intensity level and protect the image details better at the same time. To MRI image, experiments Paying particular attention data show that our algorithm has excellent performance in background. After strict analysis, we concluded that our algorithm retained the consistent component of low frequency in frequency domain by low-pass filtering, and background of MRI has this nature.Experiment images in different noise levels and different algorithms.
(a)
Lena
(b)
σ
=
0.05
(c)
Filter (b) by Shearlet
(d)
Filter (b) by proposed algorithm
(e)
Baboon
(f)
σ
=
0.1
(g)
Filter (f) by Shearlet
(h)
Filter (f) by proposed algorithm
(i)
Barbara
(j)
σ
=
0.1
5
(k)
Filter (j) by Shearlet
(l)
Filter (j) by proposed algorithm
(m)
MRI
(n)
σ
=
0.2
(o)
Filter (n) by Shearlet
(p)
Filter (n) by proposed algorithm


## 4.1. Evaluation Index

Peak signal-to-noise ratio (PSNR) is defined as(18)PSNR=10lg2552(1/M×N)∑i=1M∑j=1N(A(i,j)-O(i,j))2.Here,Ois original image with size of M×N pixels,Ais filtered image of noised image, and(i,j)are coordinates of pixels.


## 4.2. Experimental Results

To verify the validity of the algorithm, this paper designed two kinds of experimental methods to verify its effectiveness. One is use of objective data such as PNSR and MSE to objectively analyze its performance; and the other is making us able to obverse filtering performance directly by naked eyes [19–21].Experiment 1.
We did filtering experiments on standard images Lena and Barbara in different noise level and listed results in Table1. As we have seen from Table 1, PSNR of proposed algorithm (Shearlet-MOGA) is higher than PSNR of classical Shearlet algorithm, and its performance will be better with noise level increased.
Figure2 is the original MR image we adopted to do experiments. Adding different noise level to Figure 2, we did filtering work by classical Shearlet and proposed algorithm in this paper and showed the statics data of MSE and PSNR as Tables 2 and 3.
In Table2, the excellent effect of our algorithm is dramatic, shown in and our proposed MSE is smaller than classical Shearlet algorithm. Similar good results were found when the same experiment was repeated on PSNR. In Table 3, the PSNR of proposed algorithm is greater than that of classical Shearlet algorithm.Table 1
Filtering results for Lena and Barbara.
Image
σ (%)
PSNR
Shearlet
Shearlet-MOGA
Lena
10
34.38
35.02
20
31.79
33.24
30
29.21
30.21
Barbara
10
33.07
33.07
20
29.40
29.40
30
26.31
27.51Table 2
MSE in differentσ (%) and different algorithm to that in Figure 2.
Noise level (%)
5
10
20
30
40
50
60
70
80
90
Classical
25.12
100.4
400.7
905.4
1584
2504
3579
4901
6398
8075
Proposed
10.8
26.52
73.96
147.6
240.9
381
528
705.9
910.7
1132Table 3
PSNR in differentσ (%) and different algorithm to that in Figure 2.
Noise level (%)
5
10
20
30
40
50
60
70
80
90
Classical
34.13
28.11
22.1
18.56
16.13
14.14
12.59
11.23
10.07
9.059
Proposed
40.38
35.93
31.21
28.03
25.77
23.66
22.16
20.82
19.65
18.65Figure 2
Original MR image.Experiment 2.
To evaluate the performance of proposed algorithm by naked eyes directly, several classical images such as Lena, Baboon, Barbara, and MRI are adopted to do filtering work, and all relative images are shown in Figure3.
Figure3(a) is the original Lena. Adding 5% Rician noise level to Lena, we get Figure 3(b).
Filtering Figure3(b) by classical Shearlet algorithm, we got Figure 3(c). Figure 3(d) is the output of the filtering work we did to Figure 3(b) by proposed algorithm.
We did similar experiment to the image of Baboon. Add 10% Rician noise level to Baboon, we get Figure3(e). Filtering Figure 3(f) by classical Shearlet algorithm, we got Figure 3(g). Figure 3(h) is the output of the filtering work we did to Figure 3(f) by proposed algorithm.
The image of Barbara is also adopted by us to test our algorithm. Figure3(i) is the original Barbara. Figure 3(j) is Barbara noised by 15% Rician noise level. Filtering Figure 3(j) by classical Shearlet algorithm, we got Figure 3(k). Figure 3(l) is the output of the filtering work we did to Figure 3(j) by proposed algorithm.
At last, we measured our algorithm performance on MRI image. Figure3(m) is the original MRI. Figure 3(n) is the MRI noised by 20% Rician noise level. Filtering Figure 3(n) by classical Shearlet algorithm, we got Figure 3(o). Figure 3(p) is the output of the filtering work we did to Figure 3(n) by proposed algorithm.
Through simple comparison, we can see that our proposed algorithm could effectively remove the noise from the degraded image of Rician noise with unknown intensity level and protect the image details better at the same time. To MRI image, experiments Paying particular attention data show that our algorithm has excellent performance in background. After strict analysis, we concluded that our algorithm retained the consistent component of low frequency in frequency domain by low-pass filtering, and background of MRI has this nature.Experiment images in different noise levels and different algorithms.
(a)
Lena
(b)
σ
=
0.05
(c)
Filter (b) by Shearlet
(d)
Filter (b) by proposed algorithm
(e)
Baboon
(f)
σ
=
0.1
(g)
Filter (f) by Shearlet
(h)
Filter (f) by proposed algorithm
(i)
Barbara
(j)
σ
=
0.1
5
(k)
Filter (j) by Shearlet
(l)
Filter (j) by proposed algorithm
(m)
MRI
(n)
σ
=
0.2
(o)
Filter (n) by Shearlet
(p)
Filter (n) by proposed algorithm


## 5. Conclusions

In order to eliminate Rician noise and preserve image details as much as possible, this paper proposed a new image-filtering algorithm based on MOGA and classical Shearlet transform. It builds target functions in MOGA by several evaluation methods such as SNR and MSE. It also uses MOGA to find optimal Shearlet wavelet threshold coefficients in a different scale and different orientation. Computer simulations results are given to verify the effectiveness of this algorithm. At last, experiments data show that our algorithm has excellent performance in MRI imaging.



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*Source: 463760-2013-09-30.xml*