On a Variational Model for Selective Image Segmentation of Features with Infinite Perimeter

Variational models provide reliable formulation for segmentation of features and their boundaries in an image, following the seminal work of Mumford-Shah (1989, Commun. Pure Appl. Math.) on dividing a general surface into piecewise smooth sub-surfaces. A central idea of models based on this work is to minimize the length of feature’s boundaries (i.e., H1 Hausdorff measure). However there exist problems with irregular and oscillatory object boundaries, where minimizing such a length is not appropriate, as noted by Barchiesi et al. (2010, SIAM J. Multiscale Model. Simu.) who proposed to miminize L2 Lebesgue measure of the γ-neighborhood of the boundaries. This paper presents a dual level set selective segmentation model based on Barchiesi et al. (2010) to automatically select a local feature instead of all global features. Our model uses two level set functions: a global level set which segments all boundaries, and the local level set which evolves and finds the boundary of the object closest to the geometric constraints. Using real life images with oscillatory boundaries, we show qualitative results demonstrating the effectiveness of the proposed method.     

A variational model and its numerical solution for local,selective and automatic segmentation

Variational region-based segmentation models can serve as effective tools for identifying all features and their boundaries in an image. To adapt such models to identify a local feature defined by geometric constraints, re-initializing iterations towards the feature offers a solution in some simple cases but does not in general lead to a reliable solution. This paper presents a dual level set model that is capable of automatically capturing a local feature of some interested region in three dimensions. An additive operator spitting method is developed for accelerating the solution process. Numerical tests show that the proposed model is robust in locally segmenting complex image structures.     

Solitary wave solutions to the Tzitzeica type equations obtained by a new efficient approach

The properties of Tzitzeica equations in nonlinear optics have received a great attention of many recent studies. In this work, the so-called generalized exponential rational function method (GERFM) has been applied for finding the analytical solution of two nonlinear partial differential equations type of equations, namely Tzitzeica-Dodd-Bullough and Tzitzeica equation. The proposed method provides a wide range of closed-form travelling solutions leading to a very effective and simply-applied method by means of a symbolic computation system. The method not only provides a general form of solutions with some free parameters but also shows potential application to other types of nonlinear partial differential equations.