Affine and Euclidean Geometric Heat Equation for Anisotropic Smoothing

Maxime Descoteaux$^\dag$

$^\dag$School of Computer Science
McGill University
ID no: 9932537
email: mdesco@cim.mcgill.ca

Abstract:

The study of geometric flows on 2D and 3D objects has received much attention in the past decade. The main applications of some of these evolution equations are for smoothing and for multi-scale representations. In this work, we study the geometric smoothing of curves and surfaces via the geometric heat equation known as the deformation by local curvature. We emphasize the paper on the affine analogue of this flow and its formulation in terms of Euclidean curvature and normal. We investigate the key mathematical properties of both the Euclidean and the affine geometric heat equation that make them sensible smoothing methods. We know that the Euclidean geometric heat equation corresponds to the shortening flow and we discuss the analogue in affine geometry. We question a statement made in the publications by Sapiro, Tannenbaum and others and try to suggest a more valid one. Our main concern is the application of the geometric flows on angiograms and MRI medical images. We present the implementation of the two different flows and nice results of both evolution method. We clearly see the advantages of the implemented flows. Both approaches have the property of smoothing along structure of images and not across them. In our medical imaging setting, this is excellent as we preserve blood vessels while removing artifacts and speckle noise.