on analytical study of self-affine maps

self-affine maps were successfully used for edge detection, image segmentation, and contour extraction. they belong to the general category of patch-based methods. particularly, each self-affine map is defined by one pair of patches in the image domain. by minimizing the difference between these patches, the optimal translation vector of the self-affine map is obtained. almost all image processing methods, developed by using self-affine maps, take advantage of either the attracting or repelling behaviors which have been, only, experimentally investigated. in this paper, we analytically study the properties of self-affine maps and prove their attracting and repelling behaviors. furthermore, the new corner/edge pointing behavior is also proposed for contractive self-affine maps. we show that the conventional cost function of self-affine maps may cause critical uncertainty due to providing multiple equivalent optimal translation vectors. thus, a new cost function is suggested to effectively tackle this problem. for evaluation, it is used with the self-affine snake (sas) for contour extraction. experimental results demonstrated that the enhanced sas provides better performance compared to a number of different active contour methods in terms of both solution quality and cpu time.