SELF-AVOIDING RANDOM WALK APPROACH TO IMAGE ENHANCEMENT ∗ Bogdan Smolka Andrzej Chydzinski Konrad Wojciechowski Artur Bal Andrzej Swierniak Silesian Technical University, Department of Computer Science Akademicka 16, 44-101 Gliwice, Poland Tel: +48322371912; fax: +48322371165
[email protected] ABSTRACT The paper presents a new technique of image enhancement. The described algorithm enables the suppression of noise and contrast enhancement. The interesting feature of this new algorithm is that its iterative use leads to the image segmentation. The algorithm is based on a concept of a virtual particle, which performs a special kind of a random walk - the so called self-avoiding random walk. The transition probabilities of the random walk are determined by the median distribution. The segmentation effect obtained using this method, together with the ability of the algorithm to eliminate impulse and Gaussian noise makes it an interesting preprocessing tool for image analysis. 1
SELF-AVOIDING RANDOM WALK ALGORITHM
The algorithm described here, enables the suppression of noise and contrast enhancement of images. This combination is quite novel since the commonly used algorithms are mostly not able to perform both of the tasks simultaneously, as the new procedure does [1,2,3]. The standard techniques of noise elimination, like low-pass filter, median filtering, Fourier transform based operations, cause the blurring of edges, which is a very undesirable effect. Unfortunately, the standard methods of contrast enhancement of noisy images makes the noise component even stronger, so that a compromise has to be found. The presented algorithm solves this dilemma, by performing at the same time contrast enhancement and noise smoothing. In this paper the concept of a walking particle performing a self-avoiding random walk is introduced. Selfavoiding random walk (SAW) is a special walk along the m-dimensional lattice, such that adjacent pairs of edges in the sequence share a common vertex of the lattice, but no vertex is visited more than once and in this way the trajectory never intersects itself. In other words SAW is a path on a lattice, that does not pass through the same point twice. On the two-dimensional lattice ∗ This work was supported by the Polish Committee for Science Research under grant 8T11E 03315.
m = 2 SAW is a finite sequence of distinct lattice points [4-8] (x0 , y0 ) , (x1 , y1 ) , . . . , (xn , yn ) which are in neighbourhood relation and (xi , yi ) = (xj , yj ) for all i = j. Let us introduce a virtual walking particle [9-11], which performs a SAW on a two-dimensional image lattice with eight-neighbourhood system and let the transition probabilities between points (i, j) and (k, l) in one step be described by exp {−β|F (i, j) − F (k, l)|} P {1, (i, j), (k, l)} = exp {−β|F (i, j) − F (m, n)|} (m,n)
(1) where the denominator is the statistical sum taken over all pixels (m, n) belongs to the eight-neighbourhood of the point (i, j) , β is the temperature parameter and F (i, j) is the image grey scale value of (i, j). Additionally let us define a smoothing operator J based on SAW J (i, j) = P [n, (i, j) , (k, l)] · F (k, l) (2) (k,l)
The assumption that the random walk is self avoiding enables the elimination of small image objects, which consist of fewer than n pixels, when n is the number of of steps of a SAW. This feature makes the new filtering technique similar to the mathematical morphology filters. It does not however the drawbacks of dilation or erosion, as the larger objects are not changed and the connectivity of objects is always preserved. Unlike in mathematical morphology no structuring element is needed. Instead, only a number of steps is given as input. This unique feature is clearly seen in the figures, where after the application of the new filter small objects are removed, the contrast is improved, the image is smoothed but the edges are well preserved. 2
EXAMPLES
We have used the new method for the enhancement of MRI and utrasound images (Fig. 1, 2, 3). Presented effects were obtained using β(1) = 25 in the formula β(k) = β(k − 1)δ, where k is the iteration step, δ was an experimentally chosen parameter. As can be clearly seen the proposed algorithm enables the noise reduction,
while preserving the edges. Using different staring values of β(1) enables smoothing of the image details like in the anisotropic diffusion [12]. 3
CONCLUSIONS
In this paper a new filtering technique has been proposed. Its performance resembles to some extent the classical anisotropic diffusion, but the introduction of the self avoiding random walk has opened new perspectives. First of all it is possible to tune the filter, so that only objects which are composed of a fewer than n points are smoothed or removed from the image (n is the number of steps of a SAW). Another advantage of this filter is its ability to preserve or even enhance the image edges, so that it can be used as a preprocessing tool for the detection of edges and image segmentation. References [1] Castleman K. R., Digital Image Processing, Prentice Hall, 1996. [2] Gonzalez R. C., R. E. Woods, Digital Image Processing, Addison-Wesley, 1992. [3] Pratt W. K., Digital Image Processing, New York, John Wiley & Sons, 1991. [4] Hayes B., How to avoid yourself, Amer. Sci., 86, 1998. [5] Lawler G.F., Intersections of random walks, Birkhauser, 1996. [6] Lee J. S., Digital Image Enhancement and Noise Filtering by Use of Local Statistics, PAMI 2, 165 -168, 1980. [7] Madras N., Slade G., The Self-Avoiding Walk, Birkhauser, 1996. [8] Roberts F. Applied Combinatorics, Prentice-Hall, Englewood Cliffs, 1984. [9] Smolka B., Wojciechowski K., Edge Preserving Probabilistic Smoothing Algorithm, Lecture Notes in Comp. Sci., 1689, 41-48, 1999. [10] Smolka B., Wojciechowski K., Contrast Enhancement of Grey Scale Images Based on the Random Walk, Lecture Notes in Comp. Sci., 1689, 411-418, 1999. [11] Spitzer F., Principles of Random Walk, D. Van Nostrand Company, 1975. [12] Perona P., Malik J., Scale space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell. 12, 629-639, 1990
Figure 1: From top to bottom : test ultrasound image and the results of the new algorithm after 3 and 15 iterations.
Figure 2: From left to right and top to bottom: test MRI image and the result of successive iterations (n = 4, β(0) = 25).
Figure 3: First column (from top to bottom): test MRI image and below images distorted by Gaussian noise with σ = 10, σ = 20 and σ = 30. Second column: the result of noise elimination for n = 2 and beside for n = 3.
