Abstract: In this work we generalize the total variation restoration model, introduced by Rudin, Osher and Fatemi in 1992, to matrix valued data. In particular to Diffusion Tensor Images (DTI). Our model is a natural extension of the color total variation model proposed by Blomgren and Chan in 1996. We treat the diffusion matrix D implicitly as the product D = LL^T, and work with the elements of L as variables, instead of working directly on the elements of D. This ensures positive definiteness of the tensor during the regularization flow, which is essential when regularizing DTI. We perform numerical experiments on both synthetical data and 3D human brain DTI, and measure the quantitative behavior of the proposed model.
Abstract: In this paper we propose a variant of the level set formulation for identifying curves separating regions into different phases. In classical level set approaches, the sign of $n$ level set functions are utilized to identify up to $2^n$ phases. The novelty in our approach is to introduce a piecewise constant level set function and use each constant value to represent a unique phase. If $2^n$ phases should be identified, the level set function must approach $2^n$ predetermined constants. We just need one level set function to represent $2^n$ unique phases and this gains the storage capacity. Further, the reinitializing procedure requested in classical level set methods is superfluous using our approach. The minimization functional for our approach is convex and differentiable and thus avoid some of the problems with the non-differentiability of the Delta and Heaviside functions. Numerical examples will be given and we shall also compare our method with related approaches.
Abstract: In this work we propose a PDE based level set method. Traditionally interfaces are represented by the zero level set of continuous level set functions. We instead let the interfaces be represented by discontinuities of piecewise constant level set functions. Each level set function can at convergence only take two values, i.e.\ it can only be 1 or -1, thus our method is related to phase-field methods. Some of the properties of standard level set methods are preserved in the proposed method, while others are not. Using this new method for interface problems, we need to minimize a smooth convex functional under a quadratic constraint. The level set functions are discontinuous at convergence, but the minimization functional is smooth. We show numerical results using the method for segmentation of digital images.
Abstract: In this work we discuss variants of a PDE based level set method. Traditionally interfaces are represented by the zero level set of continuous level set functions. We instead use piecewise constant level set functions, and let interfaces be represented by discontinuities. Some of the properties of the standard level set function are preserved in the proposed method. Using the methods for interface problems, we minimize a smooth locally convex functional under a constraint. We show numerical results using the methods for image segmentation.
Abstract: We develop a two-phase incompressible
uid
ow simulator based on the Navier-Stokes equations.
The interface between the two
uids is modelled by a
level set approach. The Navier-Stokes equations are solved
by a standard splitting algorithm. We use a combined
continuous and discontinuous Galerkin approach for the
dierent subproblems of the splitting algorithm.We study
various phenomena related to the surface tension between
the two
uids. We then study how perturbations
in the surface tension can be used to induce
ow.
