Particle Swarm Optimization Based Support Vector Regression for Blind Image Restoration

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1 Dash R, Sa PK, Majhi B. Particle swarm optimization based support vector regression for blind image restoration. JOUR- NAL OF COMPUTER SCIENCE AND TECHNOLOGY 27(5): Sept DOI /s z Particle Swarm Optimization Based Support Vector Regression for Blind Image Restoration Ratnakar Dash, Pankaj Kumar Sa, Member, IEEE, and Banshidhar Majhi, Member, IEEE Computer Science and Engineering Department, National Institute of Technology Rourkela, Rourkela , India {pankajksa, Received October 14, 2011; revised February 22, Abstract This paper presents a swarm intelligence based parameter optimization of the support vector machine (SVM) for blind image restoration. In this work, SVM is used to solve a regression problem. Support vector regression () has been utilized to obtain a true mapping of images from the observed noisy blurred images. The parameters of are optimized through particle swarm optimization (PSO) technique. The restoration error function has been utilized as the fitness function for PSO. The suggested scheme tries to adapt the SVM parameters depending on the type of blur and noise strength and the experimental results validate its effectiveness. The results show that the parameter optimization of the model gives better performance than conventional model as well as other competent schemes for blind image restoration. Keywords image restoration, support vector regression, particle swarm optimization 1 Introduction The objective of image restoration is to obtain an estimate of the original image from a degraded observation. Image restoration finds its applications in many areas like remote sensing and medical imaging. Images can be degraded due to blur as well as noise. Blurring occurs in several situations like camera mis-focus, relative motion between object and camera, atmospheric turbulence and other similar conditions. In this paper, we treat the degraded images distorted with blur along with noise. Very often, the degraded image is expressed as 2-D convolution of the original image with the degradation function also known as point spread function (PSF) and is mathematically expressed as, g(x, y) = f(x, y) h(x, y) + η(x, y), (1) where, f(x, y) and g(x, y) represent true and blurred images respectively, h(x, y) is the PSF and η(x, y) is the additive noise. We assume the blur function to be linear shift invariant. In classical restoration, the knowledge of the PSF and statistics of additive noise are known a priori. So the restoration process is straightforward. However, in most practical situations, PSF and noise characteristics are not known a priori due to constraints in image acquisition systems and lead to a blind image restoration scenario. But pure blind systems are not possible and the problem is normally converted to a partial blind image restoration problem by assuming some of the characteristics of the degradation phenomenon [1]. In this paper, we propose an improved scheme for solving the blind image restoration problem. The scheme utilizes support vector regression () with its parameters optimized by particle swarm optimization (PSO). We exploit the existence of common characteristics between different images degraded by different blurs, i.e., one small block in an image looks similar to another block in the other image. The rest of the paper is organised as follows. Section 2 deals with the related work on the topic. Section 3 gives a brief introduction to. The role of the parameters in the performance of is discussed in Section 4. Section 5 presents the proposed parameter optimization scheme using PSO. Experimental results with discussions are given in Section 6. Finally Section 7 gives concluding remarks. 2 Related Work Various algorithms have been proposed in the past for blind image restoration [1-3]. Basically they are classified into two categories. In the first category, parametric form of PSF is assumed and blur parameters are estimated. Subsequently, the blur function is Short Paper 2012 Springer Science + Business Media, LLC & Science Press, China

2 990 J. Comput. Sci. & Technol., Sept. 2012, Vol.27, No.5 constructed and classical restoration techniques are used to restore the original image [4]. In the second category of algorithms, PSF and the image are estimated simultaneously. Our method falls in the second category in which we find the restored image directly from the blurred image. Blind deconvolution algorithms are ill posed [5-7], so some more assumptions about the image and the noise are required prior to restoration. Ayers and Dainty [8] suggested an iterative blind deconvolution (IBD) algorithm to determine the PSF and image simultaneously. Their algorithm alternates between the frequency and spatial domains after adding spatial and frequency domain constraints in each iteration. However, the output is not unique and the convergence is not guaranteed. The noise in the image also degrades the performance. To increase the noise robustness, Lucy-Richardson algorithm is used in conjunction with IBD. In another work, Kundur et al. [1] proposed a nonnegativity and support constraints recursive inverse filtering (NAS-RIF) algorithm for image restoration. The blurred image is passed through an FIR filter which gives the estimate of the true image. Their algorithm gives good results for high signal to noise ratio (SNR) and is applicable for the images having uniform background. Lucy [9] and Richardson [10] used maximum likelihood estimation for image restoration when PSF is known and little information about the noise is available. Recently Seghouane [11] has proposed a scheme for blind image restoration using maximum likelihood algorithm. They have modelled the original image and the additive noise as multivariate Gaussian processes with unknown covariance matrices. Independent component analysis (ICA) has been used in [12] which assumes independence of pixels in the image. Li et al. [13] have suggested a principal component analysis (PCA) based algorithm for blind image deconvolution. They used the principle of variance maximization to obtain the restored image. However, proper selection of PSF support is needed, i.e., the size of the deblurring filter is essential to reduce the artifacts in the restored image. Multilayer multivalued neural networks have been used by Aizenberg et al. [14] to identify the blur type and blur parameters. They proposed a derivative free learning algorithm to train a feed-forward neural network. Machine learning techniques have also been used utilizing image super resolution techniques [15]. In another work, Li et al. [16] have used the tool for blind image deconvolution. In their method, the restoration problem is considered as a function approximation problem. An optimized mapping from a neighbourhood in a degraded image to the central pixel in the original image is accomplished. The image is deblurred pixel by pixel. However, a small variation in the blur or noise characteristics significantly affects the restored results. To alleviate such problems we optimize the parameters using PSO in our proposed scheme. 3 Support Vector Regression SVM is a theoretically well motivated algorithm and was originally developed at AT&T Bell laboratories by Vapnik [17]. SVM has been successfully applied in many applications like image recognition, text recognition, bioinformatics. The capability of SVM has been further improved [18]. In this work, we solve a regression problem using SVM. Let us consider a training set (X 1, y 1 ), (X 2, y 2 ),..., (X N, y N ) from a vector, X i R n with corresponding targets y i, i = 1, 2,..., N. ε- determines a linear function defined on X i as, f(x) = w, X + b, (2) where w is a high-dimensional weight vector and b R as the bias such that there is at most ε distance from the actual data and f(x) should be flat. No care is taken as long as errors are less than ε. But, any deviation from more than ε is not accepted. Flatness means the value of w should be as small as possible. In order to address non-linear regression problems, the input space is mapped to a high-dimensional feature space through a kernel function φ(x). Then the solves the following optimization function. subject to 1 min w,b,ξ,ξ 2 wt w + C N (ξ i + ξi ), (3) i=1 y i (w T φ(x i ) + b) ε + ξ i, (w T φ(x i ) + b) y i ε + ξ i, ξ i, ξ i 0; i = 1, 2,..., N, where ξ, ξi are the slack variables. Parameter C controls the trade-off between the complexity of the model and frequency of error. C is normally optimized using cross validation. However, for a large volume of data the cross validation technique increases the complexity of the problem. Typical kernel functions used in are given as, radial basis function (RBF) kernel: polynomial kernel: ( k(x i, x j ) = exp x i x j 2 ) 2σ 2, (4) k(x i, x j ) = (1 + x i x j ) d, (5)

3 Ratnakar Dash et al.: PSO-Based for Blind Image Restoration 991 linear kernel: k(x i, x j ) = x T i x j. (6) In the above equations x i and x j are input vectors. Parameter σ in (4) represents the spread of Gaussian kernel and parameter d in (5) is the degree of the polynomial. The operator ( ) between x i and x j represents the inner product. 4 Influence of Parameters on Performance of is characterised by a number of parameters and Vapnik [17] addressed that the two most relevant are the kernel parameter σ and the penalty parameter C. C determines the trade-off between the complexity of the model and approximated error. σ in plays a major role. If it is overestimated, the exponential will behave almost linearly and it would loose the power of transforming to a higher dimension plane. On the other hand, if it is underestimated, regularization will be affected and the decision boundary will be sensitive towards noise in training data. If is applied for image restoration without parameter optimization, the parameters are adjusted heuristically for good results. In real sense, for one set of parameters the model gives good results for a given blur and noise of different strengths. So if we can derive an adaptive scheme which adapts the model with different parameters for different types of blur and noise it would be useful for greater applications. In our scheme, we concentrate on different types of blurs along with Gaussian noise of different strengths found in practice. 5 Proposed PSO-Based for Blind Image Restoration In the present work, we improve the approach towards blind image deconvolution by optimizing the parameters of. A genetic algorithm (GA) based approach has been proposed [19] to optimize the parameters. Authors in [20] have also suggested a GA-based approach for optimizing the SVM parameters. We have utilized the particle swarm optimization (PSO) for faster convergence and easier implementation. PSO also has fewer adjusting parameters than GA. We discuss PSO in a nutshell followed by detailed procedure of our proposed method in sequel. 5.1 Particle Swarm Optimization PSO is a stochastic optimization technique initially developed by Eberhart and Kennedy [21] and subsequently modified to a more generalised form [22-23]. It is an evolutionary computation technique based on intelligent behaviour of swarms. A swarm consists of particles which represent the solution. They fly in a multidimensional searching space. Each particle changes its position according to its own experience and experience collected from the neighbouring particles. In this way, the particles move towards the best solution. The performance of each particle is measured using a fitness function which is application dependent. In this paper, we utilize root mean square error function (RMSE) as the fitness function and defined as RMSE = { 1 MN MN ( ˆf f) 2} 1/2, (7) i=1 where f, ˆf are true and restored images respectively and both of size M N. The present particle position (presentx) and velocity (V ) of each particle are updated using the following two update equations. and V =V + c1 rand() (pbest presentx)+ c2 rand() (gbest presentx) (8) presentx = presentx + V, (9) where, rand() is a random number between 0 and 1 and c1, c2 are two weighting constants or accelerating constants. The local best solution of the particle is defined as pbest and gbest is the global best solution of all the particles achieved so far. 5.2 Parameter Optimization of The proposed scheme is similar to the approach [16] in which a pair of blurred and true images are used for training purpose. A mapping from the blurred image to the true image is done. To create a training pattern of pixels, a 7 7 window from the blurred image is taken and stacked in a column format. The corresponding center pixel from the true image is used as target value. Thus the size of one pattern becomes The window is shifted from top to bottom and the grey values of the true image are recorded. The is trained with a random initial set of parameters. The parameters of the i.e., C, σ are updated till the RMSE between the restored image and true image is below a threshold level (T ). We have used a threshold value of 0.01 in our experiments. The trained with the optimized set of parameters is used to restore images pixel by pixel. The flow chart of the proposed optimization scheme is shown in Fig. 1.

4 992 J. Comput. Sci. & Technol., Sept. 2012, Vol.27, No.5 The images used in training and testing are normalised in the range [0, 1]. In all experiments, images of size are considered. The PSO parameters are kept fixed in each experiment. Population size of 30 has been used in the simulation. The values of c1 and c2 were chosen to be 1.4 through experiments. The number of iterations varies in each experiment for a new test image. Blurred images are created using the standard blur functions including motion blur, out-of-focus blur and Gaussian blur. Then Gaussian noise is added to the blurred images. To compare the results, the images are also restored with PCA[13], [16] and with maximum likelihood estimation technique for blind image deconvolution[24]. The overall simulation study is divided into three experiments and are discussed in the sequel. Experiment 1. Restoration of Lena Image with Motion Blur. Standard Lena image is degraded with motion blur parameters L = 10 and θ = 45. Subsequently, Gaussian noise of strength (SNR = 40 db) is added to the resultant image. Training patterns are accumulated from the 7 7 window of blurred noisy image having its target pixel as the corresponding center pixel in the original image. In Lena model, the optimized parameters are found to be C = 2.3, σ = 0.8. The Lena model is subjected to restoration of degraded Lena image, with different motion blur parameters and Gaussian noise (L = 8 and θ = 30, SNR = 40dB). Restoration results are shown in Fig. 2. Further, Lena images degraded with defocus aberration and noise (R = 3 and SNR = 30 db) and Gaussian blur Fig.1. Flow chart of the proposed scheme for parameters optimization. 6 Experimental Results Experiments are carried out in the MATLAB environment to validate the efficacy of the proposed scheme in different blur and noise conditions. We have tested our approach on several standard images including Lena, Cameraman, Pepper with the model. The model trained with a particular image is used to restore a different blurred image, e.g., the model created with Lena image is used to restore Pepper image. We denote the model trained with Lena image as Lena. The peak signal to noise ratio (PSNR) is used as the performance measure to compare our result with other standard schemes and is defined as M P N P PSNR = 10 log10 Fig.2. Restoration of Lena Image. (a) Original Lena image. (b) 2552 i=1 j=1 M P N P i=1 j=1 (f (i, j) Motion blurred Lena image (SNR = 40 db). (c) Restored with. fˆ(i, j))2 (10) PCA. (d) Restored with maximum likelihood. (e) Restored with without parameter optimization (C = 1). (f) Restored with PSO-based.

5 Ratnakar Dash et al.: PSO-Based for Blind Image Restoration 993 and noise (σ = 0.9, SNR = 40 db) respectively are also subjected to restoration using the aforesaid Lena model. The objective parameter PSNR (db) is computed and compared with other schemes in Table 1. Table 1. PSNR (in db) Comparison of Restored Lena Image Using Lena Model and Other Schemes for Different Degradations Blur & Noise PCA Maximum Likelihood Motion blur (L = 8, θ = 30, SNR = 40 db) Out-of-Focus (R = 3, SNR = 30 db) Gaussian (σ = 0.9, 5 5, SNR = 40 db) PSOBased Fig.3. Restoration of Pepper Image. (a) Original Pepper image. (b) Out-of-focus blurred Pepper image (SNR = 40 db). (c) Re- It may be observed that the proposed optimized shows superior performance. It may also be noted that, optimized on one degradation phenomenon works well for restoring images degraded with similar type of blur as well as for other types of blurs in the same image. Experiment 2. Restoration of Images Not Considered During Training. To validate the efficacy of the trained Lena model (Experiment 1), Pepper images with various degradations are used for testing. The restored images for out-of-focus blur (R = 3, SNR = 30 db) are shown in Fig. 3 and comparative analysis of PSNR (db) is shown in Table 2. Further, an optimized Pepper model is generated in the similar direction and Cameraman image is used for testing with different degradations. The performance analysis is shown in Table 3. It is inferred that the optimized trained with one image on a particular degradation is capable of restoring different images with various degradations. stored with PCA. (d) Restored with maximum likelihood. (e) Restored with without parameter optimization (C = 1). (f) Restored with PSO-based. Experiment 3. Restoration of Photographic Blurred Image. The proposed scheme is subjected to two naturally blurred images captured on a camera (Canon EOS 400D). The canon image has the out-of-focus blur and the key image has the motion blur. The noises on both images are assumed to be Gaussian. The restoration performances on the images using the Lena model along with other schemes are shown in Figs. 4 and 5 respectively. It may be observed that the proposed PSO-based scheme has superior performance as compared with the existing schemes. 7 Conclusions In this paper, we proposed a PSO-based optimization scheme for which in turn utilized for blind image restoration. The parameters of play a vital Table 2. PSNR (in db) Comparison of Restored Pepper Image Using Lena Model and Other Schemes for Different Degradations Blur & Noise Motion blur (L = 8, θ = 30, SNR = 40 db) Out-of-Focus (R = 3, SNR = 40 db) Gaussian (σ = 1, 3 3, SNR = 40 db) PCA Maximum Likelihood PSO-Based Table 3. PSNR (in db) Comparison of Restored Cameraman Image Using Pepper Model and Other Schemes for Different Degradations Blur & Noise Motion blur (L = 12, θ = 25, SNR = 40 db) Out-of-Focus (R = 3, SNR = 30 db) Gaussian (σ = 1.5, 3 3, SNR = 40 db) PCA Maximum Likelihood PSO-Based

6 994 Fig.4. Restoration of canon Image. (a) Blurred and noisy canon image. (b) Restored with PCA. (c) Restored with maximum likelihood. (d) Restored with without parameter optimization (C = 1). (e) Restored with PSO-based. Fig.5. Restoration of key Image. (a) Blurred and noisy key image. (b) Restored with PCA. (c) Restored with maximum likelihood. (d) Restored with without parameter optimization (C = 1). (e) Restored with PSO-based. role in restoration performance and become the basis of the proposition. parameters were optimized on known training patterns from standard images. The optimized was then subjected to the same as well as different images not used for training. Both subjective as well as objective (PSNR in db) restoration performances were studied and compared with some competent schemes. It was observed that, the proposed scheme outperforms others. References [1] Kundur D, Hatzinakos D. Blind image deconvolution. IEEE Signal Processing Magazine, 1996, 13(3): J. Comput. Sci. & Technol., Sept. 2012, Vol.27, No.5 [2] Chen L, Yap K H. A soft double regularization approach to parametric blind image deconvolution. IEEE Transactions on Image Processing, 2005, 14(5): [3] Bronstein M M, Bronstein A M, Zibulevsky M, Zeevi Y Y. Blind deconvolution of images using optimal sparse representations. IEEE Transactions on Image Processing, 2005, 14(6): [4] Gonzalez R, Woods R. Digital Image Processing (3rd edition). Addison-Wesley Longman, [5] Lagendijk R L, Biemond J, Boekee D E. Regularized iterative image restoration with ring reduction. IEEE Transactions on Acoustics, Speech, and Signal Processing, 1988, 36(12): [6] Tikhonov A N, Arsenin V Y. Solutions of Ill-Posed Problems, Winston, [7] Hansen P C. Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. SIAM, [8] Ayers G R, Dainty J C. Iterative blind deconvolution method and its applications. Optics letters, 1988, 13(7): [9] Lucy L B. An iterative technique for the rectification of observed distributions. Astronomical Journal, 1974, 79(6): [10] Richardson W H. Bayesian-based iterative method of image restoration. Journal of Optical Society of America, 1972, 62(1): [11] Seghouane A K. Maximum likelihood blind image restoration via alternating minimization. In Proc. the 17th IEEE International Conference on Image Processing, September 2010, pp [12] Bell A J, Sejnowski T J. An information-maximization approach to blind separation and blind deconvolution. Neural Computing, 1995, 7(6): [13] Li D, Mersereau R M, Simske S. Atmospheric turbulencedegraded image restoration using principal components analysis. IEEE Geoscience and Remote Sensing Letters, 2007, 4(3): [14] Aizenberg I, Paliy D V, Zurada J M, Astola J T. Blur identification by multilayer neural network based on multivalued neurons. IEEE Transactions on Neural Networks, 2008, 19(5): [15] Freeman W T, Jones T R, Pasztor E C. Example-based superresolution. IEEE Computer Graphics Applications, 2002, 22(2): [16] Li D, Mersereau R M, Simske S. Blind image deconvolution through support vector regression. IEEE Transactions on Neural Networks, 2007, 18(3): [17] Vapnik V N. The Nature of Statistical Learning Theory. Springer-Verlag, [18] Joachims T. Making large-scale support vector machine learning practical. In Advances in Kernel Methods: Support Vector Learning, Scho lkopf B, Burges C J C, Smola A J (eds.), MIT Press, 1999, pp [19] Wu C H, Tzeng G H, Lin R H. A novel hybrid genetic algorithm for kernel function and parameter optimization in support vector regression. Expert Systems with Applications, 2009, 36(3): [20] Huang C L, Wang C J. A GA-based feature selection and parameters optimization for support vector machines. Expert Systems with Applications, 2006, 31(2): [21] Kennedy J, Eberhart R C. Particle swarm optimization. In Proc. International Conference on Neural Networks, November 27-December 1, 1995, pp [22] Kennedy J, Eberhart R C. A discrete binary version of the particle swarm algorithm. In Proc. IEEE International Conference on Systems, Man and Cybernetics, October 1997, pp

7 Ratnakar Dash et al.: PSO-Based for Blind Image Restoration 995 [23] Fan H. A modification to particle swarm optimization algorithm. Engineering Computations, 2002, 19(8): [24] Katsaggelos A K, Lay K T. Maximum likelihood blur identification and image restoration using the EM algorithm. IEEE Transactions on Signal Processing, 1991, 39(3): Ratnakar Dash earned his BTech degree from National Institute of Science and Technology, India, in 2002 and MTech degree from University College of Engineering, India in He is currently pursuing his Ph.D. degree in image processing at National Institute of Technology Rourkela, India. His research interest also includes digital signal processing and communication. Pankaj Kumar Sa earned his Bachelors degree from Bharathidasan University, India. He has also completed his Masters and Ph.D. degrees from National Institute of Technology Rourkela, India, in image processing. His research interest also includes computer vision and computer graphics. Banshidhar Majhi earned his Masters and Ph.D. degrees in computer science and engineering in 1998 and 2003 respectively from Sambalpur University, India. He is with the Department of Computer Science and Engineering, National Institute of Technology Rourkela, India since 1991 and currently serving as a professor. His research interest includes image processing, soft computing and network security.