Blind Image Restoration Combined Wavelet Transform and RBF Network

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1 Blind Image Restoration Combined Wavelet Transform and RBF Network Hongzhai Li and Ping Guo Department of Computer Science Beijing Normal University, Beijing, , P.R.China Abstract. It is well known that the blind image restoration is a quite difficult problem, especially for those images which degraded by Gaussian blur. To solve such kind problem, in this paper a novel technique is proposed, which combined wavelet transform and radial basis function (RBF) neural network. In the proposed technique, the wavelet transform decomposed the image into high frequency parts and low frequency part. Then the RBF neural network based technique is used to recover the given image. The inverse principal element method (IPEM) algorithm is applied to speed up the computation. Experimental results show that the proposed technique inherited the advantages of wavelet transform and IPEM, this new algorithm is efficient in computation and robust to the noise. 1 Introduction Image restoration represents one of the primary focuses of research in digital image processing. Blurring is hard to avoid in any image acquisition systems, and is caused not just by only one source. But by many, such as atmospheric turbulence, an out-of-focus optical system and aberrations in the imaging system,and so on. Obviously, all these causes cannot be simultaneously captured in a simple model. However, in general the result of these influences can be approximated by a Gaussian blur [1]. So the Gaussian blur is an important study subject in image restoration for the researchers. The existing linear image restoration algorithms assume that the Point Spread Function (PSF) is known a prior and attempt to invert it in cooperation to reduce noise by utilizing available information about the PSF, true image, and noise statistics [2]. Although many studies have been done to solve this problems, the more difficult one is the PSF unknown in many real situations. The process of restoring an unknown image using partial or no information about the imaging system is called blind image restoration. The traditional method for blind image restoration is to detect the parameters of the PSF firstly from the degraded image, and then recover the underlying image [3][4]. However, the restoration of the Gaussian blurred image is Corresponding Author 490

2 very difficult, especially blind image restoration problem. As we know, Fourier transformation of the Gaussian function is also a Gaussian type, and it has no zero crossing point, which is the most important feature for detecting the parameters, so it is hard to detect the parameters of the Gaussian function using the traditional method. In order to solve this kind of problem, we propose the RBF neural network based method, which can get better visual effects for some images [5] but is time-consuming in practice. Later, the IPEM is proposed in order to improve the computation efficiency [6]. This algorithm is relative faster than other neural network based algorithm, however, the shortage of the algorithm is that it is sensitive to the noise. There are many methods to reduce the noise, among them, the wavelet transform method should be considered firstly. Since there have been lots of algorithms based on wavelet techniques have successful applications in in the fields of image compression, image segmentation and image de-noising [7][10], etc. In this paper, we develop a new blind image restoration technique, which named wavelet inverse principal element method (WIPEM). The WIPEM contain a module of IPEM and a module of wavelet neural network. In the processing the noise is reduced by the module of wavelet neural network, so the algorithm is robust to the noise. 2 Background 2.1 The RBF neural network based method The image restoration can be considered as an inverse problem, that is, the true images (underlying images) degraded by various degraded systems in nature. In image restoration, we try to recover the underlying image from the observed image. The process of digital image degradation is generally modeled as Eq.1 [11]. g(x, y) = H[f(x, y)] + n(x, y). (1) where, f(x, y), g(x, y) are the original image and the degraded image, respectively. n(x, y) is the additive noise and supposed to have zero mean. (x, y) is the coordinate of the image pixel, 0 < x, y < N, here x, y, N are integers, the size of image is assumed as N N scale. H represents the degrade system, most of time H can be regard as a lineal system. The RBF neural network has an universal approximation capability [12] and has been successfully applied to many signal and image processing problems [13][14]. A model of RBF neural network is construct to solve the blind image restoration problem [5]. In the model of RBF neural network, the input is the coordinate and the output is the image pixel gray value, the Eq.1 can be rewritten as below based on the model of the RBF neural network. 491

3 N g(x j ) = w i φ i ( x j µ i ) + w 0. (2) i=1 Where g(x j ), j = 1, 2,...,N, is the network s output vector, thus gray value of the degraded image. x = {x i i = 1, 2,...,N} is input vector. W = [w 1,w 2,...,w N ] T is the weight between the output layer and the hidden layer. µ i are the centers of the radial functions, µ i = x j here. w 0 stands for the bias neuron and for the noise in Eq.1. φ(x) stands for the basis function, here the Gaussian type function is used as basis function associated with each neuron: φ i ( x µ i ) = exp[ 1 2 (x µ i) T Σ 1 i (x µ i )]. (3) Where Σ i is a covariance matrix of the Gaussian type function. Then a RBF neural network model is designed to represent the observed image. In the training process, the RBF neural network generates the smooth function that minimizes the following cost function: E = Φw g 2 + λ Pg(x) 2. (4) where vector Φw are actual network output, and P is usually a differential operator with radial symmetry, λ in this equation is a positive constant called regularization parameter, and Φ now is a N N symmetrical matrix called radial basic function matrix (RBFM) φ( x 1 x 1 ) φ( x 2 x 1 ) φ( x N x 1 ) φ( x 1 x 2 ) φ( x 2 x 2 ) φ( x N x 2 ) Φ = (5) φ( x 1 x N ) φ( x 2 x N ) φ( x N x N ) After establishing the relationship between RBF neural network and image restoration, the Levenberg-Marquardt (L-M) algorithm then be applied to estimate parameters and recover the underlying image at the same time [5]. The L-M algorithm is an iterative algorithm with matrix operation and it is computational expensive. 2.2 The IPEM algorithm The IPEM simplifies the algorithm to reduce the computation. First of all, the signal-to-noise ratios (SNR) is supposed to be high, that is the noise can be ignored. Then the Eq.2 can be represented as matrix form as follow. G = ΦW. (6) Here G,W and Φ stand for the degrade image, the underlying image and the RBFM, respectively. Assume that the radio basic function be the simple form: 492

4 x µ 2 φ( x µ ) = exp[ 2σ 2 ]. (7) Here σ is the width of the radial function. Then the RBFM Φ is a block Toeplitz matrix for the problem discussed in this paper, and the diagonal elements of it are φ d = exp[ d2 ]. (8) 2σ2 Here d = x µ = 0, 1, 2,..., N 1, stands for Euclid distance of two point. Let some φ d be zero if they are small than a certain small value. Then only few φ d here are nonzero. When there are only 4 φ d which are nonzero, the underlying image W can be approximated by the follows [6]: W = G + (φ 2 φ 3 )F 1,3 G + (φ 3 φ 4 )F 1,5 G + φ 4 F 1,7 G φ 1 φ 2. (9) Here stands for the convolution operation, F 1,j, j = 3, 5, 7, stands for the size of 3 3, 5 5, 7 7 convolution kernel matrix whose elements are all ones. To recover the image, the IPEM does not detect the σ, but uses the iterative method based on Eq.6. G = Φ(σ 1 )Φ(σ 2 )W = Φ(σ 1 + σ 2 )W. (10) Eq.10 shows that the Gaussian blur is separable. The basic idea of IPEM is to deblur the image with a certain Gaussian basis function, which has small blur width. Even the function is not the real Gaussian basis function, the IPEM can deblur the image a little at each iterative step. So if we deblur the image with IPEM step by step, the W will be approaching to the underlying image at a proper step. After that point the image will be over-recovered, if we continue to deconvolute the image. Suppose that the underlying image W is smooth, and the blur width is relatively small, Eq.9 can be solved by iterative method. However, when the degraded image includes noise, the IPEM is not satisfied in recovering the image. 2.3 The Wavelet Neural Network The wavelet was reported well performing on image de-noising [15]. So the basic idea is to apply the wavelet transform technique. To combine the wavelet and the artificial neural network (ANN), the wavelet neural network (WNN) based method is used. The origin of WNN can be traced back to the work by Daugman, in which Gabor wavelets were used for image classification [16]. Wavelet networks have become popular after the work by Zhang and Benveniste [17]. They have shown that an arbitrary continuous function on a compact set can be approximated by a WNN within a precision small number. Kreinovich has proven that wavelet 493

5 Fig.1. The incompact WNN neural networks are asymptotically optimal approximations for functions of one variable [18]. An interesting alternative to wavelet networks consists of using a dictionary of dyadic wavelets and to optimize only the weights. This approach is generally referred to as wave-net or wavenets. It was first proposed by Bakshi [19]. Originally, wavenets did refer to neural networks using dyadic wavelets. In wavenets, the position and dilation are fixed and the weights are optimized by the network algorithm. The theory of wavenets has been generalized by Marc Thuillard to biorthogonal wavelets [20]. This extension to biorthogonal wavelets has led to the development of fuzzy wavenets. Wavelet networks have been implemented successfully to continuous parameter WNN, frame function WNN and WNN on orthogonal basis function [21]. There are two main structures of WNN: incompact WNN and compact WNN [22]. The incompact WNN is showed in Fig.1. In this WNN, the input vector will be transformed by wavelet firstly, and then processed by the ANN to get the classified or approached result. 3 Wavelet Inverse Principal Element Method 3.1 The Architecture of WIPEM The IPEM is suitable to process the degraded image without noise. However, in the real world many images are noisy. To solve this problem, We develop the WIPEM under the framework of incompact WNN. In this method, incompact WNN framework is used to reduce the noise and the IPEM module is used to deblur the image. The WNN framework can reduce the noise well, for the wavelet provides an appropriate basis for separating noisy signal from the image signal. Because the wavelet transform is good at energy compaction, the small coefficient are more likely due to noise and large coefficient due to important signal features [23]. These small coefficients can be omitted without affecting the significant features of the image. That is so-called wavelet thresholding method [24], which is a traditional and effective image de-noise method. In the recent years there has 494

6 Fig.2. The architecture of the WIPEM been a fair amount of research on wavelet thresholding and threshold selection for signal de-noising. The architecture of the WIPEM is shown in fig.2. In the figure, G,G g,g h are degraded image, the high frequency part of the image and the low frequency part of the image, respectively. F,F g,f h is the restored image, the high frequency part of the image and the low frequency part of the image, respectively. 3.2 Algorithm Under the WNN framework, the steps of WIPEM are show below. The first step is to decompose the degraded image G using the wavelet transforms. G = G h1 + G gh1 + G gv1 + G gd1. (11) Where the coefficients matrices G h1 stands for the low frequency part, the G gh1, G gv1, G gd1 (horizontal, vertical and diagonal) stands for the high frequency part of the image. The coefficients matrices are all L L scale, L = N/2. Then we get some relatively small image, the high frequency part and low frequency parts of the degraded image. The second step, the noise is reduce in the high frequency parts using the wavelet thresholding method, and at the same time, the low frequency part is deblurred using the IPEM. Generally speaking, natural noises can be regarded as random signals, so the degraded image is more fluctuant than underlying image. In frequency domain, the degraded image has much more high frequency information than the underlying image. We can consider that the G g contain nearly all noise and the G h nearly noise free. Thus the WIPEM can denoise and deblur the degraded image at the same time. At the last step, the restored image is reconstructed by those decomposed high frequency parts and the low frequency part. F = F h1 + F gh1 + F gv1 + F gd1. (12) 495

7 (a) (b) (c) Fig. 3. The Example of blind image restoration by IPEM. (a) The underlying image: Lenna, blurred version of Lenna, (c) The restored image Where the coefficients matrices F h1 stands for the deblurred image of low frequency part, the F gh1, F gv1, F gd1 stands for the denoised results of high frequency part of the degraded image. 3.3 The advantages There are some advantages for the combination of the incompact WNN and the IPEM. First of all, the IPEM is a RBF neural network based method, when the size of input vector is N 2, the degree of the computing complexity is the order of N 4. After decomposing, the low frequency part is 1/4 times big of the degraded image. The input vector in WIPEM is one fourth big as original image. Experiments show that the WIPEM is more efficient than original IPEM in both computing time and storage space. The second, the WIPEM is much more robust to noise. If only the IPEM is applied, the noise will be magnified when deblurring the degraded image. In the WIPEM, most of the noise is reduced in the high frequency part by the WNN module. And the low frequency part is much smooth than the degraded image contained noise, so the IPEM module will not be infected by the noise. The third, the structure of the incompact WNN is much more simple than that of the compact WNN. Here, if the compact WNN is chosen, the computation will not be reduced. From the viewpoint of computation, the incompact WNN is better. 4 Experiments To verify the proposed WIPEM, two experiments results are represented below. In the experiments, we make the following assumptions about the image process, the form of PSF is assumed to be Gaussian function form, but the σ 496

8 (a) (b) (c) Fig. 4. The Example of blind image restoration by WIPEM. (a) The blurred version of Lenna, Gaussian type noise was added. (b) The restored image using WIPEM. (c) The restored image using deconvblind function in matlab image processing toolbox. parameter is unknown. Gaussian RBF is corresponding to Gaussian blur operator (it is easy to extend to other blur operator if we adopt other basis functions), and the degradation is described by the linear model of Eq.1. The blind restoration task is achieved by using partial information about the imaging process as a reference to recover the true image and PSF from the blurred image. Here we only assume the parametrical PSF model, other explicit knowledge of either original image or true PSF is not required. The first one is restoration without noise. The Lenna (Fig.3(a)) was blurred by Gaussian blur, shown as Fig.3(b). Fig.3(c) is the result of WIPEM, the blur is reduced. In the Fig.3(c), no ringing effect in the restored image is observed, that means this algorithm has good stability. Another experiment gives an example for the WIPEM to deal with the blur with noise. A little Gaussian noise was added to the blurred image, shown as Fig.4(a), now the image is hard to restore with IPEM algorithm only. The result shows as Fig.4(b) when WIPEM is applied. We did these experiments with matlab. There are already some classical methods for image restoration in the matlab image processing toolbox. Here we use the function deconvblind to restore the degraded image in the latter experiment with true PSF. The result is shown as Fig.4(c). Compared the Fig.4(b) to the Fig.4(c), the experiment shows that the WIPEM can reduce the noise well. The reason is that there are artifacts observed in the restored image by the function deconvblind, but no artifacts in the Fig.4(b). It seems that the WIPEM is more stable than the classical method for the sake of that the wonderful quality of the the wavelet multi-scale analysis capability. For some image, the effect of the deblurring is not better than the deconvblind method. One reason is that the true PSF is used in the deconvblind method. But, in the WIPEM, we iteratively approximate the true PSF. Another reason is the RBFM in IPEM is simplified, some information about the RBFM is omitted. 497

9 5 Conclusions In this paper, the WIPEM has been presented as a new blind image restoration method, which combines wavelet transform and RBF neural network based technique. Experimental results show that the WIPEM is a fast, robust algorithm compared to the IPEM. However, in the noiseless environment, the WIPEM could not get better result than that of the IPEM. Our further work will try to improve the proposed algorithm in order to obtain better visual effects. There may be possible three methods for improving the algorithm. Firstly, the IPEM itself should be improved. that is, let more φ d be nonzero in Eq.9, then the true PSF can be more approximated by the RBFM. Secondly, the compact WNN may be considered, most advantages of WNN can be used. Finally, the more suitable time to stop the iterative process of the IPEM should be found. In the IPEM, the stop time is detect by the error of G and ΦW, however, in WIPEM, the resolution of the image to the IPEM is actually half of which of the before one. So the algorithm should be changed according to the difference in the future. Acknowledgement The research work described in this paper was fully supported by a grant from the National Natural Science Foundation of China (Project No ). References 1. Vairy M. and Venkatesh Y. V., Debluring Gaussian Blur Using a Wavelet Arrray Transform. Pattern Recognition, 1995, 28. pp Katsaggelos A. K. Ed., Digital Image Restoration, Springer-Verlag, New York, Yitzhaky Y., Kopeika N. S., Identification of blur parameters from motion blurred images, CVGIP: graphical models and image processing. September 1997, 59(5). pp Michael M. Chang, Murat A. Tekalp, and Tanju A. Erdem. Blur identification using the bispectrum. IEEE Transactions on Signal Processing. Octomber (10). pp Ping Guo, Lei Xing. Blind Image Restoration Based on RBF neural networks. Proceedings of SPIE, Vol SPIE, Bellingham WA Li Hongzhai, Guo Ping. The Restoration of Gaussian Blurred Images Using Inverse Principal Element Method, Journal of Communication and Computer. 2004, 12(1). pp (In Chinese) 7. Xu Y, Weaver JB, Healy DM Jr., and Lu J. Wavelet Transform Domain Filters: a Spatially Selective Noise Filtration Technique. IEEE Transactions on Image Processing. 1994, 3(6). pp Ching P., So H.C., Wu S.Q. On Wavelet Denoising and Its Applications to Time Delay Estimation. IEEE Transactions on Image Processing. 1993, 2(3). pp

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