Automated image segmentation by mathematical morphology and fractal geometry

Abstract

SUMMARY A new method for automated image segmentation is presented. It makes use of a combination of principles derived from mathematical morphology and fractal geometry. The method consists of searching for regions in which the grey‐tone function defining the black‐and‐white image can be represented, inside a certain range of resolutions, by a fractal surface with a given fractal dimension. The probability for a planar, circular structuring element, randomly positioned on a grey‐level plane inside a region of the digitized image, to intersect the surface which represents the grey‐tone function, is derived by two methods, fractal geometry and mathematical morphology. A fundamental equation is finally derived, relating, in a fractal region of the grey‐tone surface, the sum of discrete (pixel) grey‐level differences between the images obtained by ***multi‐grey‐level dilation and erosion of the original image, respectively, to the size of the structuring element and the fractal dimension. This equation represents the basis of the method. The automated image analysis program, written for the IBAS (Kontron), works on several memorized images, each produced by a subtraction between the images obtained by dilation and erosion of the original image, using several sizes for the structuring element. Although minimal computations are made, the algorithms are too slow for practical applications and will require massive parallel processing for future use. The method relies on the assumption that regions of an image having a particular structure will usually produce a fractal grey‐tone surface, with a particular value of the fractal dimension. The feasibility of such an approach is demonstrated with segmentations obtaining the isolation of various objects on different biological images. The natural structures we have studied indeed show a fractal grey‐tone surface, at least within a certain range of (low) resolutions. Our studies also allow us to consider whether human vision uses a combination of image transformations and principles of self‐similarity to segment images.