SEMI-BLIND IMAGE RESTORATION USING A LOCAL NEURAL APPROACH

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1 SEMI-BLIND IMAGE RESTORATION USING A LOCAL NEURAL APPROACH Ignazio Gallo, Elisabetta Binaghi and Mario Raspanti Universitá degli Studi dell Insubria Varese, Italy ignazio.gallo@uninsubria.it ABSTRACT This work aims to define and experimentally evaluate an iterative strategy based on neural learning for semi-blind image restoration in the presence of blur and noise. A salient aspect of our solution is the local estimation of the restored image based on gradient descent strategies. This method can be viewed as a neural strategy where the pixels of the restored image are the synapse s weights that the neural network tries to modify during learning to minimize the output error measure; the learning strategy adopted is unsupervised. The method was evaluated experimentally using a test pattern generated by a checkerboard function in Matlab. To investigate whether the strategy can be considered an alternative to conventional restoration procedures, the results were compared with those obtained by a well known neural restoration approach. KEY WORDS Image restoration, deconvolution, neural network 1 Introduction Restoring an original image, when given the degraded image, with or without knowledge of the degrading point spread function (PSF) and degree and type of noise present is an ill posed problem [1] and can be approached in a number of ways [7]. Iterative image restoration techniques often attempt to restore an image linearly or non linearly by minimizing some measure of degradation such as maximum likelihood [1], or constrained least square error [3], by a wide variety of techniques. Generally the image degradation model suitable for most practical purposes is formed as a linear process with additive noise for which the matrix form is g = Hf + n (1) Where g and f are degraded and original vectors respectively, H is the degradation matrix, and n represents the noise. The aim of image restoration algorithms is to find an estimate that closely approximates the original image f, given g. In the presence of both blur and noise, the restoration process requires the specification of additional smoothness constraints on the solution. This is usually accomplished in the form of a regularization term in the associated cost function [4] Regularized image restoration methods aim to minimize the constrained least-squares error measure E = 1 2 g H λ D 2 (2) where is the restored image estimate, λ represents the regularization parameter and D is the regularization matrix. A small parameter value, which deemphasizes the regularization term, implies better feature preservation but less noise suppression for the restored image, whereas a large value leads to better noise suppression but blurred features. Imaging technology finds increasing applications motivated by the interest in acquiring images with which to observe complex phenomena and/or to estimate sophisticated parameters of the recorded scenes. The rapid evolution of imaging technology creates the premise for defining powerful and robust techniques for semi-blind image restoration, able to solve restoration problems in the presence of both blur and noise without requiring precise and then arbitrary estimation of degradation function. The aim of this work was to define and experimentally evaluate an iterative strategy based on neural learning for semi-blind image restoration in the presence of blur and noise. A salient aspect of our solutions is the local estimation of the restored image based on gradient descent strategy. This paper is organized as follows: Section 2 introduces the proposed neural-network algorithms, and Section 3 presents the neural-network algorithm used as comparison. Section 4 presents some experimental data from this investigation using synthetic images for which blur functions and original images are completely known, and Section 5 summarizes this paper. 2 Local Adaptive Neural Network for Image Restoration This work proposes an iterative method, called Local Adaptive Neural Network (LANN), which uses a gradient descent algorithm to minimize a local cost function derived from traditional global constrained least square measure (Equation 2).

2 The degradation measure we consider minimizing is a local cost function E x,y defined for each pixel (x, y) in an MxN image: E x,y = 1 2 [g x,y h ] λ x,y[d ] 2 (3) log λ a λ λ b 2 x where h denotes the convolution between a blur filter h centered in a point (x, y) and the restored image ; d denotes the convolution between a high-pass filter d centered in a point (x, y) and the restored image. In the present work the blur function h(x, y) is defined as a bivariate Gaussian function: h(x, y) = 1 2πσ x σ y e 1 2 (( x σx )2 +( y σy )2 ) where the parameters σ x and σ y are standard deviations. The restoration algorithm also works using a Gaussian function having σ x σ y or using blur functions of different analytical forms. In this work all the experiments were conducted with σ x = σ y. Moreover the dimension W H of the convolution mask is related to the parameters σ x and σ y by the relation W = 3σ x and H = 3σ y The high-pass filter d used in the present work corresponds to the Laplacian filter showed in Equation 5 d = (4) (5) Based on the observation that in low texture regions a high value of λ results in a visually pleasing result, while in high texture regions a low λ value works better [5], we investigate how the adaptive regularization strategy performs within the LANN algorithm. The regularization parameter is specified for each pixel as λ x,y = Y (S x,y ) where S x,y is the local variance of the g image varying from S min to S max, while Y corresponds to the log-linear function of Equation 9. To determine the function Y we need to fix two λ values (λ a and λ b ) corresponding to S min and S max. Considering that the degraded image is scaled in the range [0, 1] before computing the local variance S x,y, we compute each λ x,y through a function like that shown in Figure 1. The LANN restoration algorithm is described below (Algorithm 1). During each iteration the LANN algorithm tries to reduce the local error measure E x,y (Equation 3) changing weights xy (see Figure 2). An iteration involves all the pixels of the degraded image g; each neural weight representing a pixel value xy of the restored image is updated several times; the number of updates depends upon the dimension W H of the blur filter chosen. The training process stops when after an iteration the x,y x,y condition < ε holds for each pixel. In the present work we fixed ε = To facilitate the algorithm convergence the restored image estimate is initialized with random values in the range [0.1, 0.9] image variance t log(var ) MIN log log(var ) MAX Figure 1. An example of log-linear relationship relating the value of the constraint λ x,y to a pixel s local variance. Algorithm 1 LANN algorithm able to compute the restored image Require: Scale each pixel of the original gray level blurred image g using (1/2 b 1) as scale factor (b is the number of bits in each pixel of g image) Require: Set λ x,y values and Gaussian kernel h Require: Initialize with random values in the range [0.1, 0.9] repeat for x = 1 to M; y = 1 to N do select the pixel (x, y); compute each partial derivative and update as follows: s,r = end for t t + 1 until ( x,y = s,r η E x,y s,r s,r η(h g x,y )h i, j + +λ (d )h i, j (6) s = x + i, r = y + j s {x W/2,..., x + W/2} r {y H/2,..., y + H/2} x,y < ε, (x, y)) h fˆ h i, j ( x, y) d i, j d fˆ Figure 2. Graphic Representation of the proposed LANN Model. h and d denote the convolution computed in a point (x, y). fˆ

3 3 Hopfield neural network for image restoration This section briefly describes the restoration algorithm based on Hopfield neural model proposed by Perry and Guan [5] and used in this work for comparison. For an image where each pixel is able to take on any integer intensity between zero and S, the algorithm assigns each pixel in the image to a single neuron able to take any real value between zero and S. Equating the formula for the energy of a neural network with Equation 2 the bias inputs and interconnection strengths can be found such that as the neural network minimizes its energy function, the image will be restored [5]. From [8], setting L = MN, the interconnection strengths and bias inputs were shown to be w ij = h pi h pj λ d pi d pj (7) b i = g p h pi (8) where w ij is the interconnection strength between pixels i and j, and b i is the bias input to neuron (pixel) i. In addition, h ij is the (i, j) th element of matrixh from Equation 2 and d ij is the (i, j) th element of matrix D from Equation 2. The Hopfield neural restoration algorithm is described below (Algorithm 2). Perry and Guan propose an interesting method based on local regional statistics for adaptively varying the regularization parameter λ in Equation 2. They observed that in low texture regions, a high value of λ results in a visually pleasing result, while in high texture regions, a low λ value works best. The function Y (S i ) of the local statistic S i is proposed to locally determine the value λ i = Y (S i ). Due to the resulting complexity of the adaptive model, instead of assigning each pixel a different λ i, the authors assigned each statistically homogenous area a λ k. In this approach the largest and the smallest values of S i, S max and S min are identified and the corresponding λ a and λ b are fixed. In this way, assuming that there are K areas of the image with homogenous variance levels [S 1,..., S K ], it is possible to select the corresponding λ i value using the following log-linear function λ i = a log(s i ) + b (9) Rearranging the pixels of the degraded image g in such a way that the pixels in a homogenous area are consecutively indexed in to form a new vector T = [ 1 T,..., K T ], Algorithm 2 can be applied directly to re-computing Equation 7 for each k. 4 Experiments The LANN algorithm was experimentally evaluated and compared using a test image 64x64 generated by checkerboard function in Matlab [6] (Figure 3a). Algorithm 2 Hopfield based algorithm able to compute the restored image repeat for i = 1 to L do computes the input u i to neuron i: u i = b i + w ij j (10) j=1 changes the state i = 1, u i < 0 0, u i = 0 1, u i > 0 Let E ss = 0.5w ii ( i ) 2 u i i be the resulting energy change due to i if E ss < 0 then update pixel i: end if end for t t + 1 until ( i (t+1) i K(u) = = K( i + u i ) (11) w ii 0, u < 0 u, 0 u < S S, u S i = 0, i = 1,...,L) (12) In the experiments the test image was degraded by a Gaussian filter (Equation 4) having standard σ x = σ y = 1.33 and corrupted by Gaussian noise having standard deviation σ = 5 and σ = 15 (see Figure 3). Several configurations were considered for the LANN algorithm distinguished by an increasing number of Gaussian kernel dimensions which assumed values from 5 5 to and by different λ a and λ b assuming values ranging both from to 0.1 (Figure 5). For these experiments the neural learning parameter was fixed at η = 0.1. To evaluate the restoration performance of our approach quantitatively the improvement in signal-to-noise ratio measure (ISNR) [2] was adopted. This can be estimated as follows: x,y ISNR = 10 log 10 { (f x,y g x,y ) x,y ( x,y f x,y ) } (13) where g x,y is the given degraded image and x,y is the restored image. This performance measure can only be evaluated for controlled experiments in which the blur and noise have been synthetically introduced, because the original or undistorted image f x,y is required as well. The maximally achievable signal-to-noise ratio improvement depends strongly on the content of the image, the type of blur considered and the signal-to-noise ratio of the blurred image. Results obtained with the proposed LANN algorithm

4 (a) (b) (c) (d) Figure 3. Original (a), blurred with Gaussian filter 9 9 (b); blurred image plus Gaussian noise (σ = 5) (c); blurred image plus Gaussian noise (σ = 15) (d). are summarized in Figure 5. Globally results obtained were good, showing a stable behavior of the restoration algorithm as noise level increase. Better performances were obtained when blur function was overestimated than underestimated. Best values were obtained for all three images considered when using Gaussian blur of dimension and λ a = and λ b = 0.001; in these conditions the overestimation of the blurring function optimally compensate the de-noise performed by the regularization terms. 4.1 Comparison analysis The LANN algorithm was compared with the Hopfield restoration algorithm described above. Performances of the Hopfield restoration algorithm are shown in Figure 4. The LANN algorithm prevailed reaching higher ISNR values. The behavior of the Hopfield algorithm appeared more stable: its performances were less influenced by the variation of the two parameters λ a and λ b. Results obtained in nonnoisy and noisy conditions using the LANN algorithm are reported in Figure 5. Comparing results in Figure 4 with those in Figure 5, our method prevailed under the three conditions of noise considered. It is noteworthy that, unlike the LANN algorithm, the optimal setting of blur kernel dimensions depends on the level of noise in the image to be restored. Visual inspection of images restored by the two methods in Figure 6 highlights perceptual differences. The image restored by the LANN method lacks the ringing effect that is evident near the boundary of the image restored by the Hopfield algorithm. 5 Conclusion The objective of this work was a preliminary experimental investigation of the potentialities of a new local restoration method based on neural adaptive learning. Results obtained demonstrate the feasibility of the approach. The main limitation of the method lies in the high sensitivity to the variation of regularization range. Considering the high flexibility of the LANN algorithm derived from the local character of the restoration procedure, future plans Figure 4. ISNR comparison between different configuration of Perry s algorithm applied to the same blurred image but with three different conditions of noise: in (a) the degraded image corresponds to that showed in Figure 3b, in (b) the degraded image corresponds to that showed in Figure 3c, in (c) the degraded image corresponds to that shown in Figure 3d The PSF size varies from 5 5 to while the two parameters λ a and λ b assume all the combination of six different considered values. contemplate the insertion of an adaptive error driven setting of the regularization range. References [1] H. C. Andrews and B. R. Hunt. Digital Image Restoration. Prentice Hall Professional Technical Reference, [2] M. R. Banhom and A. k. Katsaggelos. Digital image restoration. IEEE Signal Processing Mag., [3] R. C. Gonzalez and R. E. Woods. Digital Image Processing. Addison-Wesley Longman Publishing Co., Inc., Boston, MA, USA, [4] A. K. Katsaggelos. Digital Image Restoration. Springer-Verlag New York, Inc., Secaucus, NJ, USA, 1991.

5 LANN Figure 5. ISNR comparison between different configuration of the LANN algorithm applied to the same blurred image but with three different conditions of noise: in (a) the degraded image corresponds to that showed in Figure 3b, in (b) the degraded image corresponds to that showed in Figure 3c, in (c) the degraded image corresponds to that shown in Figure 3d The PSF size varies from 5 5 to while the two parameters λ a and λ b assume all the combination of six different considered values. [5] S. W. Perry and L. Guan. Weight assignment for adaptive image restoration by neural networks. IEEE Trans. on Neural Networks, 11: , Perry (a) (b) (c) (d) Figure 6. In the first row from (a) to (d): Original, blurred with Gaussian filter 9 9, blurred image plus Gaussian noise (σ = 5), blurred image plus Gaussian noise (σ = 15). In the second row correspondent best restored image obtained by LANN algorithm. In the third row correspondent best restored image obtained by Perry s algorithm. [6] R. E. Woods R. C. Gonzalez and S. L. Eddins. Digital Image Processing Using MATLAB. Prentice-Hall, Inc., Upper Saddle River, NJ, USA, [7] M. I. Sezan and A. M. Tekalp. Survey of recent developments in digital image restoration. Optical Engineering, 29: , [8] B.K. Jenkins Y.T. Zhou, R. Chellappa. Image restoration using a neural network. IEEE Trans. Acoust,. Speech, Sign. Proc., 36:38 54, 1988.