Partial Differential Equations (PDEs) have dominated image processing research recently. The three main reasons for their success are: first, their ability to transform a segmentation modeling problem into a partial differential equation framework and their ability to embed and integrate different regularizers into these models; second, their ability to solve PDEs in the level set framework using finite difference methods; and third, their easy extension to a higher dimensional space. This paper is an attempt to survey and understand the power of PDEs to incorporate into geometric deformable models for segmentation of objects in 2D and 3D in still and motion imagery. The paper first presents PDEs and their solutions applied to image diffusion. The main concentration of this paper is to demonstrate the usage of regularizers in PDEs and level set framework to achieve the image segmentation in still and motion imagery. Lastly, we cover miscellaneous applications such as: mathematical morphology, computation of missing boundaries for shape recovery and low pass filtering, all under the PDE framework. The paper concludes with the merits and the demerits of PDEs and level set-based framework for segmentation modeling. The paper presents a variety of examples covering both synthetic and real world images.
Shape optimization of microstructural designs subject to local stress constraints within an XFEM-level set framework