Binary Image Restoration by Signomial Programming - CiteSeerX

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Binary Image Restoration by Signomial Programming Yijiang Shen, Edmund Y. Lam and Ngai Wong Department of Electrical and Electronic Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong. phone: (852)28592696 fax: (852)25598738 {yjshen,elam,nwong}@eee.hku.hk

Abstract: We present a signomial programming optimization approach to restore binary images which are degraded by additive white Gaussian noise. Numerical experiments confirm the proposed approach is efficient with good accuracy. c 2007 Optical Society of America

OCIS codes: (100.2000) Digital image processing; (100.3020) Image reconstruction-restoration

1.

Introduction

Digital images are imperfect representations of scenes in the real world, because the data are subjected to degradations such as blur, noise and rounding error. One special case is when the true scenery is binary, such as for document, bar code, handwritten signatures, and vehicle license plates, but is then degraded by additive white Gaussian noise during imaging. Restoring the noisy binary images is therefore necessary. Numerous attempts have been made to deal with binary image restoration. The average mean square error (AMSE) method by Meloche and Zamar [1] is highly dependent on the content of the image, which means the AMSE criterion is inadequate in some cases. Hitchcock and Glasbey [2] tried to restore images of blob-like and filamentous objects. Neifeld et al. [3] included prior knowledge concerning local correlations among pixel values into the Viterbi-based restoration process. Gu et al. proposed the pulse coupled neural network to restore binary images degraded by white Guassian noise. Chan et al. [4] provided a convergent method of finding a minimizer of the total-variational functional to restore binary images. Generally speaking, noisy image formation can be modeled as: g(x1 , x2 ) = f (x1 , x2 ) + n(x1 , x2 ),

(1)

where f (x1 , x2 ) and g(x1 , x2 ) represent the true and the degraded image respectively, and n(x1 , x2 ) is the additive noise. The combinatorial nature of binary image denoising has been noted in [5, 6]: for each pixel position i of an image, the pixel value gi originates from either of two known prototype values u1 and u2 ; in practice, g is real-valued due to blurring and noise. To restore a discrete-valued image function represented by the vector x ∈ {−1, +1}n from the measurement g, we would like to minimize the functional: z(x) =

1X λ X ((u2 − u1 )xi + u2 + u1 − 2gi )2 + (xi − xj )2 , 4 i 2

(2)

in which λ is the smoothness term parameter, and the second term sums over all pairwise adjacent pixels on the regular image grid in both vertical and horizontal directions. The combinatorial problem is relaxed to a convex optimization problem, and can be solved using Positive Semidefinite Programming [7]. The computation time quickly grows with the number of variables, however. The optimization process is further explored in [8] to restore binary images degraded by blur and noise, where an overlapping method over image blocks is applied to decrease computation complexity to a more reasonable time. The development of Geometric Programming (GP) has the potential to make a much faster restoration of binary images. New solution methods can solve even large-scale GP problems efficiently and reliably with reasonable speed [9, 10]. In this paper, we aim to apply these developments in binary image restoration. 2.

Binary Image Restoration Using SP

GP is a type of mathematical optimization problem characterized by objective and constraint functions that have a special form [10]. The basic approach in GP modeling is to express a practical problem in the standard format. Let

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X1 , . . . , Xn denote n real positive variables, and X = (X1 , . . . , Xn ) a vector with component Xi , a Generalized Geometric Program (GGP) has the form: f0 (X) fi (X) ≤ 1, hi (x) = 1,

minimize subject to

i = 1, . . . , m, i = 1, . . . , p,

(3)

where h1 , . . . , hp are monomials and f0 , . . . , fm are generalized posynomials [10]. A signomial is a function with the same form as a posynomial, where the coefficients are allowed to be negative. A Signomial Program (SP) is a generalization of a geometric program, which has the form of a GGP in (4), but the objective and constraint functions can be signomials. Our task at hand is to minimize the expression in (2). If we consider only the first term and note that the variables in a GP have to be positive, without loss of generality we assign u1 = 1 and u2 = 3 and scale gi accordingly. Thus, the minimization of z(x) in (2) can be solved by the computation of the following optimization problem: X X X minimize x2i + gi2 − 2 xi gi i

subject to 3.

i

i

xi = 1 or 3.

(4)

Solving the Optimization Problem

The optimization problem above has the format of a SP. There are several ways to solve this. The first approach is a standard branch and bound technique, which does not utilize the special structure of SP and thus is not widely used. Another approach to solve it is to convert it into a complementary GP [9], which allows upper bound constraints on the ratio between two posynomials and then applies a monomial approximation iteratively. In this paper, we convert the SP into a reversed GP [9], and then apply a monomial approximation to solve the problem. Reversed GP refers to minimizing a posynomial subject to both upper and lower bound inequality constraints. Further details on the reversed GP are given P in [11].P P Let f01 (x) = i x2i + i gi2 , f02 (x) = 2 i xi gi , and the monomial terms with negative multiplicative coefficients f02 (x) are separated from those monomial terms with positive multiplicative coefficients f01 (x). We introduce an auxiliary variable t ≥ 0 and turn the objective to the minimization of t, with an additional constraint f01 (x) − f02 (x) ≤ t, which may be written as f01 (x) ≤ s ≤ f02 (x) + t, where s ≥ 0 is another auxiliary variable. If we can approximate the posynomial f02 (x) + t with a monomial, then a lower bound on f02 (x) + t becomes an upper bound on a monomial, which is allowed in standard form GP. We can use a simple approximation based on the geometric inequality that leads to the development of GP: that the arithmetic mean is greater than or equal to the geometric mean. Therefore,  α   X 2gi xi i t 0 f02 (x) + t = 2 xi gi + t ≥ Πi × = f02 (x), (5) α α i t i P where i αi + αt = 1. One possibility of computing αi and αt is to let αi = 2gi xi / (f02 (x) + t) , ∀i, and αt = t/ (f02 (x) + t) , where we choose gi to be xi . Thus the problem can now fit in a standard GGP format: minimize subject to

t 0 s−1 f02 (x)−1 ≥ 1, s−1 f01 (x) ≤ 1, xi ≤ 3, xi ≥ 1.

(6)

Several software packages are available to solve GPs, including MOSEK [12], TOMLAB [13] and YALMIP [14]. In our implementation, we use the GP software package GPCVX [15] to solve the optimization problem in (6), after the solution is obtained, xi ≤ 2 will be assigned value u1 , and xi > 2 will be assigned value u2 .

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4.

Experimental Results and Conclusions

We apply the image restoration algorithm on text images corrupted with additive white Gaussian noise. In Figure 1, (a) is the noisy image with Gaussian noise that has zero mean and variance of 0.01 (or standard deviation of 0.1). (b) is the restored image, which evidently is a lot cleaner. Figure 2 shows the same set of experiment with twice the variance, where (c) is the noisy image and (d) is the restored result. We can also see that the restored image is binary and has not many pixel errors.

(a) Noisy image

(b) Restored image

Fig. 1: Restoration of noisy binary image with noise variance of 0.01.

(a) Noisy image

(b) Restored image

Fig. 2: Restoration of noisy binary image with noise variance of 0.02. This paper has presented a novel approach to restore noisy binary images using Signomial Programming. Experimental results shows that the approach is fast and accurate. Further research will include the smoothness term in (2) for better restoration performance, and potentially leads to restore binary images degraded by noise and blur. References 1. J. Meloche and R. H. Zamar, “Binary image restoration,” The Canadian J. Statistics, vol. 22, no. 3, pp. 335–355, 1994. 2. D. Hitchcock and C. A. Glasbey, “Binary image restoration at subpixel resolution,” Biometrics, vol. 53, pp. 1010–1053, 1997. 3. M. Neifeld, R. Xuan, and M. Marcellin, “Communication theoretic image restoration for binary-valued imagery,” Applied Optics IP, vol. 39, no. 2, pp. 269–276, 2000. 4. T. F. Chan, S. Esedo¯glu, and M. Nikolova, “Finding the global minimum for binary image restoration,” ICIP 2005, pp. 121–124, 2005. 5. J. Keuchel, C. Schellewald, D. Cremers, and C. Schn˝orr, “Convex relaxations for binary image partitioning and perceptual grouping,” DAGM-Symposium 2001, pp. 353–360, 2001. 6. ——, “Binary partitioning, perceptual grouping, and restoration with semidefinite programming,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 25, no. 11, pp. 1364–1379, 2003. 7. Y. Shen, E. Y. Lam, and N. Wong, “Robust binary image deconvolution with positive semidefinite programming,” in Recent Advances in Engineering and Computer Science. Newswood, 2007, pp. 159–166. 8. ——, “Binary image restoration by positive semidefinite programming,” Optics Letters, vol. 32, pp. 121–123, Jan. 2007. 9. S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi, “A tutorial on geometric programming,” Optimization and Engineering, 2006, to be published. 10. S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. 11. M. Chiang, Geometric Programming for Communication Systems. Now Publishers Inc, 2005. 12. The MOSEK Optimization Tools Version 2.5. User’s Manual and Reference, MOSEK ApS, 2002. [Online]. Available: http://www.mosek.com 13. M. Edvall and F. Hellman, Uers’s Guide for TOMLAP/GP, 2005. [Online]. Available: http://tomlab.biz/docs/TOMLAB_GP.pdf 14. J. L˝ofberg, YALMIP. Yet Another LMI Parser. Version 2.4, 2003. [Online]. Available: http://control.ee.ethz.ch/˜joloef/yalmip.php 15. gpcvx, A MATLAB Solver for Geometric Programs in Convex Form, 2006. [Online]. Available: ˜ http://www.stanford.edu/boyd/ggplab/gpcvx.pdf