Nonextensive Entropic Image Thresholding

In image processing, one of the most efficient techniques for image segmentation is entropy-based thresholding. In this work it was applied a generalized entropy formalism that represents a recent development in statistical mechanics. We propose, for the first time, an image thresholding method using a nonextensive entropy regarding the presence of nonadditive information content in some image classes. Preliminary results are shown. 1 Nonextensive Entropy The entropy of a discrete source is often obtained from the probability distribution p = {pi}, and the Shannon entropy may be described as S = − ∑k i=1 piln(pi), being k the total number of states. If we consider that a physical system can be decomposed in two statistical independent subsystems A and B, the Shannon entropy has the extensive property (additivity) S(A + B) = S(A) + S(B). This formalism has been shown to be restricted to the BoltzmannGibbs-Shannon (BGS) statistics. However, for nonextensive physical systems, some kind of extension appears to become necessary. Tsallis [1] has proposed a generalization of the BGS statistics which is useful for describing the thermostatistical properties of nonextensive systems. It is based on a generalized entropic form, Sq = 1− ∑k i=1(pi) q q − 1 (1) where the real number q is a entropic index that characterizes the degree of nonextensivity. This expression recovers to BGS entropy in the limit q → 1 . Tsallis entropy has a nonextensive property for statistical independent systems, defined by the following pseudo additivity entropic rule Sq(A+B) = Sq(A)+Sq(B)+(1−q)·Sq(A)·Sq(B) (2) 2 The Thesholding Technique Let pi = p1, p2, . . . , pk be the probability distribution for an image with k gray-levels. From this distribution, we derive two probability distributions, one for the object (class A) and the other for the background (class B), given by pA : p1 P A , p2 P A , . . . , pt P A and pB : p1 P B , p2 P B , . . . , pk P B where P = ∑t i=1 pi and P B = ∑k i=t+1 pi The Tsallis entropy of order q for each distribution is defined as S q (t) = 1− ∑t i=1(pA) q q − 1 S q (t) = 1− ∑k i=t+1(pB) q q − 1 (3) The Tsallis entropy Sq(t) is parametrically dependent upon the threshold value t for the foreground and background. It is formulated as the sum each entropy, allowing the pseudoadditive property, defined in equation (2). We try to maximize the information measure between the two classes (object and background). When Sq(t) is maximized, the luminance level t that maximizes the function is considered to be the optimum threshold value [2]. topt = argmax[S q (t) + S B q (t) + (1− q) · S q (t) · S q (t)] (4) Figure 1: (A) Grayscale 8 bit image, (B) Binary image using Shannon entropy (t = 121), (C) Binary image using Tsallis entropy, q = 5 (t = 171). 3 ConclusionsThe preliminary results obtained confirm the viability ofusing the nonextensive entropy formalism in image thresh-olding and other segmentation techniques. We believe thatTsallis entropy may trigger some pratical future applica-tions in such an area of image processing and recognition. References[1] C. Tsallis, J. Statistical Phys., 52, 480-487, (1988).[2] J. N. Kapur et al, C.V.G.I.P. 29, 273-285, (1985).[3] T. Yamano, Entropy 2001, 3, 280-292, (2001). Proceedings of the XV Brazilian Symposium on Computer Graphics and Image Processing (SIBGRAPI’02)1530-1834/02 $17.00 © 2002 IEEE