Automatic grey level thresholding through index of fuzziness and entropy

A utomatic grey level thresholding through index of fuzziness and entropy Abstract: Algorithms for automatic thresholding of grey levels (without reference to histogram) are described using the terms 'index of fuzziness' and 'entropy' of a fuzzy sel. Their values are seen to be minimum when the crossover point of an S-function corresponds to boundary levels among different regions in image space. The effectiveness of the algorithms is demonstrated for images having both bimodal and multimodal grey level distributions. The problem of grey level thresholding plays an important role in image processing. For example, in enhancing contrast in an image we need to select proper threshold levels from its histogram so that some suitable non-linear transformation can highlight a desirable set of pixel intensities com­ pared to others. Similarly, in image segmentation one needs proper histogram thresholding whose objective is to establish boundaries in order to par­ tition the image space into meaningful regions. The present work illustrates an application of theory of fuzzy sets to make this task automatic so that an optimum threshold (or set of thresholds) may be estimated without the need to refer directly to the histogram. These are explained by the terms 'index of fuzziness' (Kaufmann (1975» and 'en-tropy' (De Luca and Termini (1972» of a fuzzy set. Since these terms reflect the measures of closeness of a grey tone image to its two-tone version, they provide a quantitative measure (Pal (1982» of im­ age ambiguity when the cross-over point is set to a predetermined value. Modification of cross-over point will result in variation of these parameters and so a set of minima is obtained corresponding to the optimum threshold levels. the pattern corresponding to an M x N, L-level image array, where Iix(X mn) or Ilmn/xmn (0 :5, limn :5 1) denotes the grade of possessing some property limn (as defined in the next section) by the (m, n)th pixel intensity X II1f1 ' Let X = {IlX(x mn)} be similarly defined as the nearest ordinary plane to X, such that IiX(X mn) = 0 jf Iix(X mn) :50.5 and is equal to 1 for Iix(X mn) >0.5.