Multiscale Variational Approach to Simultaneous Image Regularization and Segmentation
The problem of finding a piecewise smooth approximation of an original image while preserving edges is closely related to the image segmentation problem. Indeed, if we know the cartoon version of the original image, it is easier to perform an image partition into homogeneous regions. From this point of view, we propose to simultaneously perform regularization and segmentation. By working in the space of functions of bounded total variation, we allow discontinuities in the result of the minimisation. Therefore we perform regularization by using a Total Variation (TV) minimisation based on the gradient descent method. Each gradient step involves solving a discrete approximation of the corresponding partial differential equation that results in a smooth estimate of the original image without blurring across edges. This non-uniform smoothing allows an explicit region growing segmentation scheme that minimises one form of the Mumford-Shah functional. Therefore, the major novelity of our approach consists in competing two processes, one for boundary detection and the other for intraregion smoothing.