Evolution of Brain Tumor and Stability of Geometric Invariants

This paper presents a method to reconstruct and to calculate geometric invariants on brain tumors. The geometric invariants considered in the paper are the volume, the area, the discrete Gauss curvature, and the discrete mean curvature. The volume of a tumor is an important aspect that helps doctors to make a medical diagnosis. And as doctors seek a stable calculation, we propose to prove the stability of some invariants. Finally, we study the evolution of brain tumor as a function of time in two or three years depending on patients with MR images every three or six months.

DISCRETE CLOTHOID SPLINE SURFACES ON OPEN MESHES

By introducing a discrete Frenet frame, this paper first proposes 3D discrete clothoid splines to extend the planar discrete clothoid splines of Schneider and Kobbelt. On the basis of 3D discrete clothoid spline curves, discrete clothoid spline surfaces for arbitrary meshes are defined as a generalization of their closed discrete clothoid spline surfaces. Moreover, a discrete mean curvature normal operator instead of the conic surface fitting method is employed to compute curvature values. This induces a new iteration algorithm for generating the discrete clothoid spline surfaces. Since the curvature at a vertex is an area average over a region around this vertex and no linear system needs to be settled, the algorithm works more steadily and efficiently. Besides being directly used to produce fair mesh surfaces, the approach can also be employed to smooth control nets of spline surfaces.