An Image Filter Based on Multiobjective Genetic Algorithm and Shearlet Transformation

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.

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. 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. Rician Noise

Noised MRI image 𝑣 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: 𝑝(𝑚)=𝑚𝜎²_𝑛𝑒^{−(𝑚2+𝐴2)/2𝜎2𝑛}𝐼₀𝐴𝑚𝜎²_𝑛. (1) 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: 𝑝(𝑚)=𝑚𝜎²_𝑛𝑒𝑚²𝜎²_𝑛. (2)

When SNR is high (i.e.,SNR>3), the Rician distribution is approximated as a Gaussian distribution. 𝑝(𝑚)=12𝜋𝜎²𝑒−𝑚²−𝐴²+𝜎²_𝑛²2𝜎²_𝑛. (3)

2.2. Shearlet Transform

Labate et al. [5–11] proposed Shearlet transform based on wavelet. With dimension 𝑛=2, consider the following affine system: Ψ_𝐴𝐵𝜓=𝜓_{𝑗,𝑙,𝑘}(𝑥)=|det𝐴|^𝑗/2𝜓𝐵^𝑙𝐴^𝑗𝑥−𝑘∶𝑗,𝑙∈ℤ,𝑘∈ℤ². (4) Here, 𝜓∈𝐿²(ℝ²), and 𝐴,𝐵 are 2*2 invertible matrices with |det𝐵|=1.

If Ψ_𝐴𝐵(𝜓) satisfied Parseval 𝐿²(ℝ²), then, those elements of Ψ_𝐴𝐵(𝜓) are called composite wavelets.

Shearlet is a special example of 𝐿²(ℝ²) only when 𝐴=𝐴₀=4002,𝐵=𝐵₀=0111. (5) Here, 𝐴=𝐴₀ is the anisotropic dilation matrix, and 𝐵=𝐵₀ is the shear matrix.

For 𝜉=(𝜉₁,𝜉₂)∈ℝ²,𝜉₁≠0, when 𝜓⁽⁰⁾,𝜓₁, and𝜓₂ satisfy 𝜓⁽⁰⁾(𝜉)=𝜓⁽⁰⁾𝜉₁,𝜉₂=𝜓₁𝜉₁𝜓₂𝜉₂𝜉₁,𝜓₁,𝜓₂∈𝐶^∞ℝ,supp𝜓₁⊂−121,−∪116,1162,supp𝜓₂⊂[−1,1_𝑗≥0],𝜓₁2^−2𝑗𝜔²|𝜔|≥1=1for8,𝑗≥0,_{2𝑗−1𝑙=−2𝑗}𝜓₂2^𝑗𝜔−𝑙²=1for|𝜔|≤1. (6)

Then, we get ²_𝑗≥0^𝑗−1_{𝑙=−2𝑗}𝜓⁽⁰⁾𝜉𝐴₀^−𝑗𝐵₀^−𝑙2=²_𝑗≥0^𝑗−1_{𝑙=−2𝑗}𝜓₁2^−2𝑗𝜉₁²𝜓₂2^𝑗𝜉₂𝜉₁−𝑙²=1. (7)

Then, {𝜓⁽⁰⁾(𝜉𝐴₀^−𝑗𝐵₀^−𝑙)} form a tiling of the set 𝐷₀=𝜉₁,𝜉₂∈ℝ²∶𝜉₁≥18,𝜉₂𝜉₁≤1. (8)

From the condition on the support of 𝜓₁and𝜓₂, it is easily deduced that 𝜓_{𝑗,𝑙,𝑘} have frequency support contained in the set as follows: supp𝜓⁽⁰⁾_{𝑗,𝑙,𝑘}⊂𝜉₁,𝜉₂∶𝜉₁∈−2^2𝑗−1,−2^2𝑗−4∪2^2𝑗−4,2^2𝑗−1,𝜉₂𝜉₁+𝑙2^−2𝑗≤2^−𝑗. (9)

Thus, every element in 𝜓_{𝑗,𝑙,𝑘} is supported on a pair of trapezoids of approximate size 2^2𝑗×2^𝑗, oriented along lines of slope𝑙2^−𝑗[].

For 𝐿²(𝐷₁)^∨, here, 𝐷₁ is the vertical cone when the following formula was satisfied: 𝐷₁=𝜉₁,𝜉₂∈ℝ²∶𝜉₂≥18,𝜉₁𝜉₂,𝐴≤1₁=2004,𝐵₁=,𝜓1011₍₁₎(𝜉)=𝜓⁽¹⁾𝜉₁,𝜉₂=𝜓₁𝜉₂𝜓₂𝜉₁𝜉₂. (10)

Then, collection {𝜓⁽¹⁾_{𝑗,𝑙,𝑘}(𝑥)=2^3𝑗/2𝜓⁽¹⁾(𝐵^𝑙₁𝐴^𝑗₁𝑥−𝑘)∶𝑗≥0,−2^𝑗≤𝑙≤2^𝑗−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,…,𝑥_𝑑], where𝑑is the dimension of the decision variable space. Consider the following.

Minimize 𝑓(𝑥)=[𝑓_𝑖(𝑥),𝑖=1,…,𝑚] which satisfies 𝑔_𝑗(𝑥)ℎ≤0,𝑗=1,2,…,𝐽,_𝑘(𝑥)=0,𝑘=1,2,…,𝐾, (11) where 𝑓_𝑖(𝑥) is the𝑖th objective function, 𝑔_𝑗(𝑥) is the𝑗th inequality constraint, and ℎ_𝑘(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: 𝛿=𝜎2ln(𝑁). (12) 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: 𝛿_𝑘=𝜎2ln(𝑁)∗2^{(𝑘−𝐾)/2},𝑘=0,1,…,𝐾. (13)

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: 𝛿_𝑘,𝑗=Sigmoid𝜎2ln(𝑁)∗2^{(𝑘−𝐾)/2}1,𝑘=0,1,…,𝐾,Sigmoid=1+𝑒^−𝑣. (14)

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 ∑SNR=10𝑙𝑔^𝑀𝑖=1∑^𝑁_𝑗=1𝐴𝑖,𝑗²∑^𝑀_𝑖=1∑^𝑁_𝑗=1𝐴𝑖,𝑗−𝑂𝑖,𝑗². (15)

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: 1MSE=𝑀×𝑁^𝑀𝑁_𝑖=1𝑗=1𝐴𝑖,𝑗−𝑂𝑖,𝑗². (16)

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: 𝑦=𝜔₁⋅SNR+𝜔₂⋅MSE,0≤𝜔₁,𝜔₂𝜔≤1,₁+𝜔₂=1. (17)

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, 𝛿_𝑘,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, 𝛿_𝑘,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 containing 𝑁_pop strings, where 𝑁_pop is the number of strings in each population. These strings contain weight coefficients of SNR, MSE, weight coefficients 𝛿_𝑘,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 coefficients 𝛿_𝑘,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. 𝑁_pop 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 remove 𝑁_elite strings from the set of 𝑁_pop strings generated by previous operations, and replace them with 𝑁_elite 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 Step 8; 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 PSNR=10𝑙𝑔255²(1/𝑀×𝑁)∑^𝑀_𝑖=1∑^𝑁_𝑗=1𝐴𝑖,𝑗−𝑂𝑖,𝑗². (18)

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 Table 1. 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.
Figure 2 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 Table 2, 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.

Table 2 

MSE in different 𝜎 (%) and different algorithm to that in Figure 2.

Table 3 

PSNR in different 𝜎 (%) and different algorithm to that in Figure 2.

Figure 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 Figure 3.
Figure 3(a) is the original Lena. Adding 5% Rician noise level to Lena, we get Figure 3(b).
Filtering Figure 3(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 Figure 3(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. Figure 3(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. Figure 3(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.

(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=0.15

(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

Figure 3 

Experiment images in different noise levels and different algorithms.

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.

Acknowledgments

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 open-source code online. The authors would like to thank Professor Labate. The authors would also like to express their thanks to 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. 60773172) and a Grant from the Jiangsu Province Natural Science Foundation (BK2008411).