On the uniqueness of nonlinear diffusion coefficients in the presence of lower order terms

We consider the identification of nonlinear diffusion coefficients of the form $a(t,u)$ or $a(u)$ in quasi-linear parabolic and elliptic equations. Uniqueness for this inverse problem is established under very general assumptions using partial knowledge of the Dirichlet-to-Neumann map. The proof of our main result relies on the construction of a series of appropriate Dirichlet data and test functions with a particular singular behavior at the boundary. This allows us to localize the analysis and to separate the principal part of the equation from the remaining terms. We therefore do not require specific knowledge of lower order terms or initial data which allows to apply our results to a variety of applications. This is illustrated by discussing some typical examples in detail

Template-Cut: A Pattern-Based Segmentation Paradigm

We present a scale-invariant, template-based segmentation paradigm that sets up a graph and performs a graph cut to separate an object from the background. Typically graph-based schemes distribute the nodes of the graph uniformly and equidistantly on the image, and use a regularizer to bias the cut towards a particular shape. The strategy of uniform and equidistant nodes does not allow the cut to prefer more complex structures, especially when areas of the object are indistinguishable from the background. We propose a solution by introducing the concept of a "template shape" of the target object in which the nodes are sampled non-uniformly and non-equidistantly on the image. We evaluate it on 2D-images where the object's textures and backgrounds are similar, and large areas of the object have the same gray level appearance as the background. We also evaluate it in 3D on 60 brain tumor datasets for neurosurgical planning purposes.Comment: 8 pages, 6 figures, 3 tables, 6 equations, 51 reference