A measure of edge ambiguity using fuzzy sets

AbSlracl: Algorilhms for providing a quanlitati\e measure of edge ambIguity are described through fuzzy measures in a set. An IOdex is derined whose value IS maximum fOJ a grey lone image and decreases as the fnzzmess In detecting edges decreases. The inherent fuzzifying properly of the S-funclion is found to enable one nOI 10 use the tNT operator for contrasl enhancement and 10 save the time of computation greatly as measured by (he index value. The index value IS also found to increase wllh standard deviation of injected random noise. The object of an ege detector is (0 detect the presence and location of changes in grey levels in an image. When an image is processed for ex­ tracting edges/contours of its various regions, it is ultimately up to the viewers to judge its quality for a specific application and how well a particular method works. The process of evaluation of edge­ enhancement quality (or edge ambiguity) therefore becomes a subjective one which makes the defini­ tion of a 'good edge-detected image' an elusive standard for comparison of algorithm perfor­ mance. The present work is an attempt to make this evaluation task somewhat objective by providing a quantitative measure of edge ambiguity in an im­ age. The fuzzy measures, namely the index of fuzziness (Kaufmann, 1975), the entropy (De Luca and Termini, 1972) and the index of nonfuzziness are used here in defining an index of edge ambi­ guity. The membership functions for implementing these measures are made here position dependent to incorporate the spatial relationship among the grey levels. The index value is seen to decrease as edge ambiguity decreases. Two contrast enhancement algorithms using the S-function and with/without the INT operator (Zadeh, 1975) are also presented here. The com­ parison of their performance is made on the basis of the index value when an X-ray image of a wrist is considered as input. The effect of noise on the index value is also studied in a part of the experi­ ment. be the fuzzy set representat ion of the pattern corresponding to an M x N, L-level im­ age array, where ,uX(x mn) or ,umn/xmn(O:5,uIllIl:5l) denotes the grade of possessing some property ,umll (as defi ned in the next section) by the (m, n)th pixel intensity X mn. Let ~= {,u5(x mn)} be similarly de­ fined as the nearest ordinary plane to X, such that ,u.y(xmJ=O jf ,ux(x …