Image Denoising Based on Genetic Algorithm - IEEE Xplore

2013 IEEE Congress on Evolutionary Computation June 20-23, Cancún, México

Image Denoising Based on Genetic Algorithm Claudio F. M. Toledo∗ , Lucas de Oliveira† , Ricardo Dutra da Silva† and Helio Pedrini† ∗ Institute

of Mathematics and Computer Science - University of São Paulo, Brazil, 13566-590 Email: [email protected] † Institute of Computing - University of Campinas, Brazil, 13083-852 Email: {lucas.oliveira, rdutra, helio}@ic.unicamp.br

Abstract—Digital images play an essential role in analysis tasks with applications in various knowledge domains, such as medicine, meteorology, geology, biology, among others. Such images can be degraded by noise during the process of acquisition, transmission, storage or compression. Although several image denoising methods have been proposed in the literature, noise suppression in images still remains a challenging problem for researchers since the process can cause the removal of relevant image features, such as edges and corners. This papers describes a novel image denoising method based on a genetic algorithm. A population of noisy images is evolved for several epochs applying tailor-made crossover and mutation operators. The population is reinitialized every time a convergence occurs, when only the best individual (image) is kept for the next epoch. Experimental results demonstrate that the proposed method is competitive in comparison with state-of-the-art approaches.

I.

I NTRODUCTION

Several image denoising approaches have been investigated in the literature, however, removing noise from images is still a challenging problem. Digital images can be acquired by different sensors, such as photographic cameras, laser scanners, weather satellites and medical scanners. Data sets captured by such image sensors are usually corrupted by noise, which can degrade the quality of the data under interest. Additionally, noise can be introduced by other processes, such as compression and transmission. Since images can be inherently contaminated by noise during acquisition, transmission, compression processes, it is fundamental to suppress the noise while preserving important features of the image, such as corners and edges. Noise suppression can introduce artifacts or cause image blurring, which makes image denoising a complex task. Several approaches have been proposed to remove noise in digital images, however, each one explores specific aspects of the problem. This paper proposes and evaluates a genetic algorithm for image denoising. The method evolves the images that are restorations of noisy images. Mutation is applied over such noisy images to create an initial population. The evolution happens for certain number of epochs aiming to find the best image. During this process, specific crossover and mutation operators are applied as well as population is reinitialized, except by the best individual (image) found so far. The text is organized as follows. Section II reviews some relevant concepts of image denoising. Section III describes the proposed image denoising method using a genetic algorithm. Section IV presents and discusses the experimental results

978-1-4799-0454-9/13/$31.00 ©2013 IEEE

1294

obtained by applying the proposed method to digital images. Section V concludes the paper with some final remarks. II.

BACKGROUND

The main purpose of image denoising methods is to recover a digital image that has been contaminated by additive or multiplicative noise. Several approaches have been developed to suppress noise from images, while preserving important features in the data. Some linear filtering [1], [2] have been proposed to remove Gaussian and uniform noise in images. A known linear filtering is the Wiener filter [3], which minimizes the mean squared error between the original and the recovered image. Nonlinear image filters [4], [5] have been developed to improve the effectiveness of linear filters. One class of such filters is the median filter [3]. Rational operators [6], [7] have been used to improve denoising techniques. Image denoising approaches based on partial differential equations (PDE) and computational fluid dynamics (CFD) have also been developed, such as total variation (TV) methods [8]–[11], level set methods [12], essentially nonoscillatory schemes [13], non-linear isotropic and anisotropic diffusion [14]–[17]. Some approaches have combined local adaptive filtering and impulse removal filters in the transform domain to suppress noise [18]. Non-local filtering has been demonstrated to be powerful for image denoising, such as the transform-based BM3D filter [19]. Singular value decomposition (SVD) has also been used in image noise filtering [20], [21]. Selection of thresholding parameters can be performed through cross-validation techniques [22], [23]. Bayesian procedures [24]–[26] have combined prior information and inference from data to estimate thresholding parameters. Other techniques have combined wavelet transforms, hidden Markov models and spatially adaptive methods [27]–[33]. Another category of denoising methods exploits the data decomposition into a wavelet basis and modifies the wavelet coefficients to suppress noise [34]–[41]. The assumption is that the coefficients affected by noise are replaced by zero or another proper value. The resulting denoised image is reconstructed from these modified wavelet coefficients. Several works using genetic algorithms (GA) have been proposed for image denoising. A distributed GA is proposed in [42] for the restoration of frames corrupted by vertical line scratches. All the scratches in the image are restored in parallel, where the GA evolves simultaneously sub-populations. A

reduced execution time is reached when compared with the sequential version of the GA. A genetic algorithm has been applied by Korurek et al [43] to find parameter values in a model for a near field effect of Xray source. The objective is to eliminate the near field effect of X-ray source in bone images. An approach based on wavelet transform and multiobjective genetic algorithm (MOGA) is proposed by Thavavel et al [44]. The method is applied to tomographic reconstruction from noisy and sparse projections in the electron paramagnetic resonance imaging (EPRI). The MOGA optimizes values for wavelet denoising threshold. A GA is used to optimize a back propagation neural network in [45] applied to image restoration. The energy function for the reconstruction problem, formulated as a Bayesian framework, is minimized by a genetic algorithm in [46]. The method is evaluated in old films and artificially corrupted films, where competitive results are reached against film restoration methods. III.

M ETHODOLOGY

The genetic algorithm (GA) proposed in this paper evolves images that are restorations of noisy images. The noisy image I is received as input and the initial population is generated applying mutations over I. Next, the method runs by a fixed number of epochs (nEpochs) when population evolves until convergence. The population convergence is characterized by a generation without insertions of new individuals. A reinitialization process is applied in the population when convergence is achieved, such that only the best individuals are maintained and a new epoch begins. Each individual is represented as a matrix of pixels, whose entries are integer values ranging in the interval [0..255]. A pseudocode is shown in Algorithm 1, whose details of the proposed method are explained next. The objective of the proposed method is to find the uncorrupted image starting from the noisy one. This optimization search is guided based on the Markov Random Field model proposed in [47]. We adopt the related energy function used in this model as fitness function in the proposed method represented by Equation 1 X X 2 fitness(X) = µ |Xu − Xv | + I(v) − Xv (1) (u,v)∈N

v∈V

where X is the individual being evaluated, Xv is the v-th pixel of X and I(v) is the v-th pixel of the noisy image I. Two pixels (u, v) ∈ N if and only if u and v are neighboring pixels in the 4-neighborhood. The constant µ balances the smoothness of X and its similarity with I. Higher values of µ lead to smoother images, however, far from the noisy image, whereas lower values lead to unrepairable images (very close to I). The aim is to minimize Equation 1 which means that better individuals in the genetic algorithm will present smaller fitness values. As mentioned previously, the initial individuals are created from the noisy images. Each individual is a copy of I, where the mutation operator is applied. This generation process is repeated until the population size has reached a set limit.

1295

1: procedure G ENETIC R ESTORATION(Image Img) 2: initializePopulation(P op); 3: for epoch ← 1 to nEpochs do 4: converged ← false; 5: while not converged do 6: converged ← true; 7: for i ← 1 to crossoverRatio ∗ P op.size do 8: ind1, ind2 ← fathers(P op); 9: ind3 ←crossover(ind1, ind2); 10: if executesMutation? then 11: mutate(ind3); 12: end if 13: if ind3.f itness < ind1.f itness then 14: ind1 ← ind3; 15: converged ← false; 16: else 17: if ind3.f itness < ind2.f itness then 18: ind2 ← ind3; 19: converged ← false; 20: end if 21: end if 22: end for 23: end while 24: reinitialize(P op); 25: end for 26: return P op.best; 27: end procedure Figure 1.

Pseudocode of the proposed genetic denoising algorithm.

During the evolution process, a tournament selection [48] is applied to define two parents. The tournament selects two individuals randomly choosing the best of these as one parent. The crossover operator recombines two selected parents generating a new individual. The mutation operator can be applied over the new individual, if a uniformly random value in the real interval [0..1] is less than or equal to the mutation rate. The new individual replaces one of its parents if its fitness value is better. Otherwise, the new individual is not inserted into the population. The number of new individuals to be created is determined by crossoverRatio∗P op.size. After all new individuals have been created, one generation is counted and the convergence is evaluated. If any new individual was inserted into the population, there is no convergence and the same amount of new individuals is created again. If no individuals were inserted, convergence is assumed and the population is reinitialized as explained before. Two crossovers are proposed to be applied over the pixel matrix of the parents. For each new individual, each one of these crossovers has equal chance to be selected: •

one-point column: randomly defines a column of the pixel matrix of the individual. All pixels to the left of that column come from one parent and all the other pixels come from the other parent (Figure 2(c)).

•

one-point row: is similar to the one-point column, but a row is now randomly chosen (Figure 2(d)).

Three mutations are proposed to be applied over the new individual. Again, each one of them has the same chance to be randomly selected: •

blur: applies the effect of blur to the image of the

execution, the quality of the image restored, represented by the best individual, is evaluated using the peak signal to noise ratio (PSNR). This measure is defined in decibels for 8-bit gray-scale images, as presented by Equation 2. 2 LI PSNR = 10 log10 (2) MSE (a) Parent 1

(b) Parent 2

The mean-squared error (MSE) is defined for gray-scale images by Equation 3. MSE =

Figure 2.

(c) One-point column Crossover examples.

(d) One-point row

•

random: a small random perturbation can be applied with 5% of chance to each entry of the pixel matrix.

•

intensity: multiplies all the individual pixels of the image by a same random factor, which lightens or darkens the image as a whole. E XPERIMENTAL R ESULTS

This section reports and discusses the experimental results obtained with the proposed genetic algorithm (GA) for image denoising. The method was tested in several images, three of them were selected to illustrate the results (Figure 3), since they are commonly employed in image restoration: boat, glasses and lightning. These images present dimensions of 512×512, 512×256 and 512×256 pixels, respectively, and they were corrupted with additive Gaussian noise N (0, σ 2 ), where σ 2 is the estimated noise deviation with noise levels σ = 10, 15, 20, 30 and 35 applied.

A set of parameters was defined for GA based on empirical tests carried out, which are crossover rate of 2.0, mutation rate of 0.2 and population size of 15 individuals. It is worth mentioning that a crossover rate of 2.0 means that 30 new individuals are created at each iteration, as described in Algorithm 1. The constant µ in the fitness function was empirically √ chosen to be ν/4, where ν is the estimated variance of the noise in the image calculated by the method variance_noise provided by the CIMG library [49]. First, the GA performance is measured when the number of epochs increases. The parameter nEpochs determines how many times the method will try to improve the noisy image. As already explained, the GA evolves by minimizing the objective function presented by Equation 1 and the quality of the best individual found at the end of each execution is measured in terms of PSNR value. The average, minimum and maximum PSNR values achieved by GA are presented in Table I, which compares the results found by executing GA during 30, 35 and 40 epochs over boat image. Moreover, high PSNR means better GA performance. σ 10 15 20 25 30 35

(a) glasses

GA Avg 30.98 29.48 28.43 27.56 26.88 26.31

Table I.

- 30 Epochs Max Min 31.12 30.77 29.73 29.36 28.62 28.28 27.72 27.36 27.00 26.77 26.36 26.25

GA

GA Avg 30.94 29.50 28.43 27.59 26.86 26.32

- 35 Epochs Max Min 31.26 30.60 29.68 29.27 28.53 28.32 27.76 27.49 26.93 26.80 26.39 26.26

GA Avg 31.00 29.53 28.41 27.59 26.89 26.31

- 40 Epochs Max Min 31.28 30.57 29.73 29.32 28.57 28.28 27.69 27.55 26.98 26.81 26.38 26.21

PERFORMANCE IN TERMS OF PSNR EXECUTING FOR 35 AND 40 EPOCHS FOR BOAT IMAGE .

30,

As the other GA parameters, the initial value of 30 epochs was also determined based on previous tests. The results show a better performance running the GA for 35 and 40 epochs, where the values depicted in Table I remain very close to each other.

(c) boat Figure 3.

(3)

where M and N are the image dimensions, O is the original image, LO is the maximum possible pixel value of the image and K is the restored image. The average PSNR (Avg), as well as the minimum (Min) and maximum (Max) values found after 10 executions are considered for comparisons in the results reported next.

individual. Each pixel is assigned the value of a weighted average between it and its neighbors, making the image smoother. The procedure applied as blur effect is available in [49].

IV.

M −1 N −1 1 X X [O(i, j) − K(i, j)]2 M N i=0 j=0

(b) lightning Benchmark images.

A total of 10 executions over each image was performed, running in a Probook with processor Intel i5, 2.30 GHz and 3.8 GB of RAM under Ubuntu 12.04. At the end of each

1296

Figure 4 compares the average execution time spent by GA in each epoch. The average computational time to restore the image increases following the number of epochs applied as expected. However, the GA spends less than 1.5 minutes

on average to restore an image executing for 30 and 35 epochs. This means that the proposed method is able to return 10 restored images (best individuals found) in less than 15 minutes. The same assessments using glasses and lightning are presented in Tables II and III, as well as in Figures 5 and 6, respectively.

epochs found maximum values superior than other approaches in the majority of results. Furthermore, the method with 35 epochs is faster than executing for 40 epochs, as shown in Figures 5 and 6, spending on average less than 37 seconds to restore the glasses and lightning images in all considered noise levels.

Figure 4. image.

Figure 6. Average CPU performance for 30, 35 and 40 epochs for lightning image.

σ 10 15 20 25 30 35

Average CPU performance for 30, 35 and 40 epochs for boat

GA Avg 38.23 36.51 35.28 34.42 33.48 32.80

Table II.

- 30 Epochs Max Min 38.57 37.85 36.81 36.21 35.55 35.08 34.76 34.13 33.75 33.37 33.07 32.60

GA Avg 38.39 36.50 35.31 34.42 33.54 32.84

- 35 Epochs Max Min 38.74 38.09 36.85 36.16 35.50 35.07 34.70 34.14 33.72 33.40 33.16 32.61

GA Avg 32.67 32.54 33.84 34.02 35.32 34.05

The previous results indicate a slightly better performance in terms of PSNR for GA running during 35 epochs. This number of epochs also represents a trade-off between solution quality and execution time for the three restored noisy images. Table IV compares the GA with 35 epochs, referred to only as GA, against other state-of-the-art approaches:

- 40 Epochs Max Min 38.34 38.66 36.48 36.61 35.42 35.75 34.62 34.86 33.50 33.88 32.89 33.10

GA PERFORMANCE IN TERMS OF PSNR EXECUTING 35 AND 40 EPOCHS FOR GLASSES IMAGE .

FOR

•

Wavelet-based approaches: Bayes [34], Bivariate [39], Wiener-chop [36], and Silva et al. [41].

•

Non-wavelet-based approaches: Wiener [3], BM3D [19], Anisotropic Diffusion (AD) [15], Median [3], and TV [9].

30,

Table IV shows the minimum (Min), maximum (Max) and average (Avg) PSNR attained by the GA. It is also presented the average confidence interval (CI) at 99% of significance level. The PSNR of state-of-the-art methods are available from [41].

Figure 5. image.

σ 10 15 20 25 30 35

Average CPU performance for 30, 35 and 40 epochs for glasses

GA Avg 36.24 34.53 33.57 32.74 31.94 31.25

Table III.

- 30 Epochs Max Min 36.42 35.75 34.71 34.39 33.99 33.28 32.89 32.61 32.20 31.74 31.43 31.00

GA Avg 36.44 34.67 33.75 32.81 32.11 31.27

- 35 Epochs Max Min 36.75 35.97 34.95 34.35 34.07 33.51 33.13 32.49 32.33 31.99 31.58 31.08

GA Avg 33.97 33.17 33.69 34.29 33.58 34.27

- 40 Epochs Max Min 36.36 36.75 34.74 34.94 33.63 33.81 32.79 33.02 32.11 32.31 31.37 31.67

GA PERFORMANCE IN TERMS OF PSNR EXECUTING 35 AND 40 EPOCHS FOR LIGHTNING IMAGE .

FOR

30,

Tables II and III show again very similar average results found by GA with 35 and 40 epochs. However, the GA with 35

1297

The minimum value in the confidence interval of the average PSNR (Avg - CI) found by the proposed GA is greater than PSNR values of other approaches in 30 out of 54 results (55.5%) for boat image, 27 out of 54 (50%) for glasses image, and 35 out of 54 (64.8%) for lightning image. Thus, in such cases it is possible to claim with 99% of confidence that GA obtains better PSNR values on average. The best PSNR measure achieved by GA after 10 executions outperforms 31 results for boat, 29 for glasses, and 36 for lightning image. Even the worst PSNR value found in these executions is still better than 30 results reported for boat, 27 for glasses, and 34 for lightning image. The proposed method was not able to outperform the results found by Silva [41] and it did not always outperform BM3D, AD and Wiener-Chop. However, for images with higher noise level (σ = 35), the GA found better results than BM3D, AD and Wiener-Chop. Furthermore, it demonstrated to be superior than Median, Wiener filter and Bivariate, as well as it produced competitive results against the other methods. Although the GA requires relatively high computational time,

Methods Image

Noise

boat

glasses

lightning

10 15 20 25 30 35 10 15 20 25 30 35 10 15 20 25 30 35

BM3D

AD

Median

TV

Wiener

Bayes

Bivariate

Wiener-chop

Silva et al.

31.02 30.70 30.30 29.84 28.61 25.83 41.79 40.64 39.58 38.67 35.41 30.21 37.85 37.17 36.40 35.62 33.39 28.76

32.33 30.40 28.98 27.90 26.97 26.22 40.16 37.65 35.73 34.47 33.22 32.13 38.37 35.99 34.35 33.05 31.95 30.97

29.40 28.19 26.94 25.80 24.69 23.69 35.15 31.92 29.53 27.68 26.14 24.81 33.76 31.04 29.07 27.32 25.83 24.60

30.39 29.95 28.81 27.39 25.72 24.13 40.19 37.80 33.74 30.27 27.55 25.34 37.52 35.79 32.88 29.87 27.24 25.17

30.02 29.07 28.06 27.23 26.35 25.58 37.81 34.82 32.62 31.05 29.79 28.64 36.05 33.78 32.03 30.46 29.10 27.91

30.47 29.07 27.97 26.99 26.18 25.56 39.78 37.19 35.75 34.81 33.88 33.09 33.66 32.11 30.05 29.31 28.98 28.04

26.79 26.49 26.09 25.59 25.00 24.41 36.53 34.40 32.43 30.90 29.40 28.20 30.56 29.87 29.13 28.27 27.31 26.43

32.20 30.59 29.33 28.44 27.60 26.91 40.92 38.20 36.09 34.74 33.37 32.27 37.53 35.05 33.39 32.14 31.07 30.16

32.52 30.81 29.43 28.48 27.66 27.05 40.76 38.80 37.16 37.17 35.01 34.05 38.29 36.15 35.02 34.09 33.31 32.54

Table IV.

Avg 30.94 29.50 28.43 27.59 26.86 26.32 38.39 36.50 35.31 34.42 33.54 32.84 36.44 34.67 33.75 32.81 32.11 31.27

GA CI ±0.18 ±0.10 ±0.06 ±0.07 ±0.04 ±0.03 ±0.17 ±0.19 ±0.12 ±0.13 ±0.08 ±0.15 ±0.21 ±0.15 ±0.15 ±0.17 ±0.08 ±0.12

Max 31.26 29.68 28.53 27.76 26.93 26.39 38.74 36.85 35.50 34.70 33.72 33.16 36.75 34.95 34.07 33.13 32.33 31.58

Min 30.60 29.27 28.32 27.49 26.80 26.26 38.09 36.16 35.07 34.14 33.40 32.61 35.97 34.35 33.51 32.49 31.99 31.08

C OMPARING GA AGAINST OTHER STATE - OF - THE - ART APPROACHES .

such results justify the application of the proposed method to image denoising. Figures 7, 8 and 9 compare the original and corresponding noisy images with the restored images reached by the denoising methods for boat, glasses and lightning images, respectively. These images were corrupted with σ = 30 (noise level). The wavelet-based methods produces smoother results, while the GA tends to produce smooth results in homogeneous regions. The proposed denoising method has some drawback to remove noise in heterogeneous regions, however, it is able to improve such regions better than other wavelet and non-wavelet based approaches, such as Wiener, Bivariate and Median.

In fact, the method was able to outperform some of these methods in all results, taking into account the average and the best solutions found. As future work, an approach where the images are divided into several small frames is planned to be developed, such that one population is created and evolved from each small frame. Additionally, the combination of GA with denoising methods is another possibility under evaluation. ACKNOWLEDGEMENTS The authors are grateful to FAPESP, CNPq, and CAPES for the financial support. R EFERENCES

V.

C ONCLUSIONS AND F UTURE W ORK

[1]

This paper presented a novel image denoising method using a genetic algorithm to suppress noise from digital images. A representation of individuals is proposed based on the pixel matrix in such way that tailor-made crossover and mutation operators are designed. The initial population is created from the noised image through the application of the proposed operators. Thus, several images represented by individuals are evolved as restorations of noisy images until a convergence is achieved, according to a fitness function based on a Markov Random Field model. Also, a convergence criterion is adopted aiming to restart population when better individuals are not inserted after one epoch. Preliminary experiments conducted on a set of images indicate that the proposed method is promising, producing superior results compared to other state-of-the-art denoising methods. The genetic algorithm can be executed several times and, for practical purposes, this allows the method to offer several restored images at the end. The reported computational results indicated that several final solutions (images) can be returned by the proposed method within a short execution time. The solution quality measured using PSNR for these final solutions is similar to those found by the literature approaches.

1298

[2] [3] [4]

[5]

[6] [7]

[8]

[9] [10]

F. Russo, “A method for estimation and filtering of Gaussian noise in images,” IEEE Transactions on Instrumentation and Measurement, vol. 52, no. 4, pp. 1148–1154, Aug. 2003. ——, “Image filtering based on piecewise linear models,” in IEEE International Workshop IST, Stresa, Italy, May 2004, pp. 7–12. R. C. Gonzalez and R. E. Woods, Digital Image Processing. Upper Saddle River, NJ, USA: Prentice-Hall, Inc., 2006. C. Chaux, L. Duval, A. Benazza-Benyahia, and J.-C. Pesquet, “A nonlinear Stein-based estimator for multichannel image denoising,” IEEE Transactions on Signal Processing, vol. 56, no. 8, pp. 3855–3870, 2008. B. Chen, Y. Li, and J.-L. Cai, “Noisy image segmentation based on nonlinear diffusion equation model,” Applied Mathematical Modelling, vol. 36, no. 3, pp. 1197–1208, 2012. G. Ramponi, “The rational filter for image smoothing,” IEEE Signal Processing Letters, vol. 3, no. 3, pp. 63–65, Mar. 1996. F. Cocchia, S. Carrato, and G. Ramponi, “Design And Real-Time Implementation Of A 3-D Rational Filter For Edge Preserving Smoothing,” IEEE Transactions on Consumer Electronics, vol. 43, no. 4, pp. 1291– 1300, Nov. 1997. A. Chambolle, “An Algorithm for Total Variation Minimization and Applications,” Journal of Mathematical Imaging and Vision, vol. 20, no. 1–2, pp. 89–97, 2004. L. Rudin, S. Osher, and E. Fatemi, “Nonlinear Total Variation based Noise Removal Algorithms,” Physica D, vol. 60, pp. 259–268, 1992. C. Drapaca, “A nonlinear total variation-based denoising method with two regularization parameters,” IEEE Transactions on Biomedical Engineering, vol. 56, no. 3, pp. 582–586, 2009.

Figure 7.

(a) original

(b) noisy (σ = 30)

(c) Bayes [34]

(d) BM3D [19]

(e) AD [15]

(f) Median [3]

(g) Bivariate [39]

(h) TV [9]

(i) Wiener [3]

(k) Silva et al. [41]

(l) proposed genetic algorithm

(j) Wiener-chop [36] Denoising results for boat image.

1299

Figure 8.

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

(a) original

(b) noisy (σ = 30)

(c) Bayes [34]

(d) BM3D [19]

(e) AD [15]

(f) Median [3]

(g) Bivariate [39]

(h) TV [9]

(j) Wiener-chop [36] Denoising results for glasses image.

(i) Wiener [3]


M. Kumar and S. Dass, “A total variation-based algorithm for pixellevel image fusion,” IEEE Transactions on Image Processing, vol. 18, no. 9, pp. 2137–2143, 2009. J. A. Sethian, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Sciences. Cambridge University Press, 1999. T. Chan and H. Zhou, “Adaptive ENO-wavelet Transforms for Discontinuous Functions,” Computational and Applied Mathematics Technical Report, Department of Mathematics, UCLA, Tech. Rep. 99-21, Jun. 1999. M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, “Robust Anisotropic Diffusion,” IEEE Transactions on Image Processing, vol. 7, no. 3, pp. 421–432, Mar. 1998. V. Katkovnik, K. Egiazarian, and J. Astola, Local Approximation Techniques in Signal and Image Processing. SPIE Press, Sep. 2006, vol. PM157. J. Monteil and A. Beghdadi, “A new adaptive nonlinear anisotropic diffusion for noise smoothing,” in IEEE International Conference on Image Processing, vol. 3, Chicago, IL, USA, Oct. 1998, pp. 254–258. J. S. Jin, Y. Wang, and J. Hiller, “An adaptive nonlinear diffusion algorithm for filtering medical images,” IEEE transactions on Information Technology in Biomedicine, vol. 4, no. 4, pp. 298–305, Dec. 2000. K. Egiazarian, J. Astola, M. Helsingius, and P. Kuosmanen, “Adaptive Denoising and Lossy Compression of Images in Transform Domain,” Journal of Electronic Imaging, vol. 8, no. 3, pp. 233–245, Jul. 1999. K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image Denoising with Block-Matching and 3D Filtering,” in SPIE Electronic Imaging: Algorithms and Systems, vol. 6064, Jan. 2006, pp. 606 414–1–606 414– 12. Y. Wongsawat, K. Rao, and S. Oraintara, “Multichannel SVD-based image De-noising,” in IEEE International Symposium on Circuits and Systems, vol. 6, May 2005, pp. 5990–5993.

1300


[21]

J. Orchard, M. Ebrahimi, and A. Wong, “Efficient Nonlocal-means Denoising using the SVD,” in IEEE International Conference on Image Processing, San Diego, CA, USA, Oct. 2008, pp. 1732–1735.

[22]

N. Weyrich and G. T. Warhola, “Wavelet Shrinkage and Generalized Cross Validation for Image Denoising,” IEEE Transactions on Image Processing, vol. 7, no. 1, pp. 82–90, Jan. 1998.

[23]

G. P. Nason, “Wavelet Shrinkage by Cross-Validation,” Journal of the Royal Statistical Society B, vol. 58, pp. 463–479, 1996.

[24]

H. Chipman, E. Kolaczyk, and R. McCulloch, “Adaptive Bayesian Wavelet Shrinkage,” Journal of the American Statistical Association, vol. 440, no. 92, pp. 1413–1421, 1997.

[25]

F. Ruggeri and B. Vidakovic, “A Bayesian Decision Theoretic Approach to Wavelet Thresholding,” Journal of The American Statistical Association, vol. 93, pp. 173–179, 1998.

[26]

B. Vidakovic, “Nonlinear Wavelet Shrinkage with Bayes Rules and Bayes Factors,” Journal of the American Statistical Association, vol. 93, no. 441, pp. 173–179, 1998.

[27]

S. Chang, B. Yu, and M. Vetterli, “Spatially Adaptive Wavelet Thresholding Based on Context Modeling for Image Denoising,” IEEE Transactions on Image Processing, vol. 9, no. 9, pp. 1522–1531, Sep. 2000.

[28]

M. Malfati and D. Roose, “Wavelet-based Image Denoising using a Markov Random Field a Priori Model,” IEEE Transactions on Image Processing, vol. 6, no. 4, pp. 549–565, Apr. 1997.

[29]

M. S. Crouse, R. D. Nowak, and R. G. Baraniuk, “Wavelet-based Statistical Signal Processing using Hidden Markov Models,” IEEE Transactions on Signal Processing, vol. 46, no. 4, pp. 886–902, Apr. 1998.

[30]

G. Fan and X. Xia, “Image Denoising using Local Contextual Hidden Markov Model in the Wavelet Domain,” IEEE Signal Processing Letters, vol. 8, no. 5, pp. 125–128, May 2001.

[31]

J. Liu and P. Moulin, “Complexity-Regularized Image Denoising,” IEEE

Figure 9.

[32]

[33]

[34]

[35] [36]

[37]

[38]

[39] [40]

(a) original

(b) noisy (σ = 30)

(c) Bayes [34]

(d) BM3D [19]

(e) AD [15]

(f) Median [3]

(g) Bivariate [39]

(h) TV [9]

(j) Wiener-chop [36] Denoising results for lightning image.

(i) Wiener [3]


Transactions on Image Processing, vol. 10, no. 6, pp. 841–851, Jun. 2001. M. K. Mihcak, I. Kozintsev, K. Ramchandran, and P. Moulin, “Lowcomplexity Image Denoising based on Statistical Modeling of Wavelet Coefficients,” IEEE Signal Processing Letters, vol. 6, no. 12, pp. 300– 303, Dec. 1999. J. K. Romberg, H. Choi, and R. G. Baraniuk, “Shift-invariant Denoising using Wavelet-domain Hidden Markov Trees,” in Conference Record of The Thirty-Third Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, USA, Oct. 1999, pp. 1277–1281. S. Chang, B. Yu, and M. Vetterli, “Adaptive Wavelet Thresholding for Image Denoising and Compression,” IEEE Transactions on Image Processing, vol. 9, no. 9, pp. 1532–1546, Sep. 2000. D. L. Donoho and I. M. Johnstone, “Ideal Spatial Adaptation via Wavelet Shrinkage,” Biometrika, vol. 81, pp. 425–455, 1994. S. Ghael, E. P. Ghael, A. M. Sayeed, and R. G. Baraniuk, “Improved Wavelet Denoising via Empirical Wiener Filtering,” in Proceedings of SPIE, vol. 3169, San Diego, CA, USA, Jul. 1997, pp. 389–399. M. Kazubek, “Wavelet Domain Image Denoising by Thresholding and Wiener Filtering,” IEEE Signal Processing Letters, vol. 10, no. 11, pp. 324–326, 2003. A. Pi˘zurica and W. Philips, “Estimating the Probability of the Presence of a Signal of Interest in Multiresolution Single- and Multiband Image Denoising,” IEEE Transactions on Image Processing, vol. 15, no. 3, pp. 654–665, Mar. 2006. L. Sendur and I. W. Selesnick, “Bivariate shrinkage with Local Variance Estimation,” IEEE Signal Processing Letters, vol. 9, no. 12, 2002. J. Weaver, X. Yansun, and D. H. L. Cromwell, “Filtering Noise from Images with Wavelet Transforms,” Magnetic Resonance in Medicine, vol. 21, no. 2, pp. 288–295, 1991.

1301

[41]

[42]

[43]

[44]

[45]

[46]

[47]

[48]

[49]


R. da Silva, R. Minetto, W. Schwartz, and H. Pedrini, “Adaptive Edge-Preserving Image Denoising Using Wavelet Transforms,” Pattern Analysis & Applications, pp. 1–14, 2013, DOI: 10.1007/s10044-0120266-x. F. Isgrò and D. Tegolo, “A distributed genetic algorithm for restoration of vertical line scratches,” Parallel Computing, vol. 34, no. 12, pp. 727– 734, 2008. M. Korürek, A. Yüksel, Z. Iscan, Z. Dokur, and T. Ölmez, “Retrospective correction of near field effect of X-ray source in radiographic images by using genetic algorithms,” Expert Systems with Applications, vol. 37, no. 3, pp. 1946–1954, 2010. V. Thavavel, J. J. Basha, M. Krishna, and R. Murugesan, “Heuristic wavelet approach for low-dose EPR tomographic reconstruction: An applicability analysis with phantom and in vivo imaging,” Expert Systems with Applications, vol. 39, no. 5, pp. 5717–5726, 2012. U. Farooq, S. Ting-Zhi, Z. San-Yuan, and M. Imran, “Image Restoration by Using New AGA Optimized BPNN,” Procedia Engineering, vol. 29, pp. 3028–3032, 2012. E. Y. Kim, K. tai Kim, and B. Kim, “Genetic algorithm-based reconstruction of old films corrupted by scratches and blotches,” Pattern Recognition Letters, vol. 34, no. 2, pp. 226–237, 2013. H. Ishikawa, “Global Optimization Using Embedded Graphs,” Ph.D. dissertation, Department of Computer Science, New York University, 2000. B. L. Miller, B. L. Miller, D. E. Goldberg, and D. E. Goldberg, “Genetic algorithms, tournament selection, and the effects of noise,” Complex Systems, vol. 9, pp. 193–212, 1995. The CImg Library, “C++ Template Image Processing Library,” 2013, http://cimg.sourceforge.net/reference/.