Introduction

In a a number of applications in image processing, computer vision, and computer graphics, the data of interest is defined on non-flat manifolds and maps onto non-flat manifolds. A classical and important example is directional data, including gradient directions, optical flow directions, surface normals, principal directions, and chroma. Frequently, this data is available in a noisy fashion, and there is a need for noise removal. In addition, it is often desired to obtain a multiscale-type representation of the directional data, similar to those representations obtained for gray-level images, [2, 31, 36, 37, 55]. Addressing the processing of non-flat data is the goal of this chapter. We will illustrate the basic ideas with directional data and probability distributions. In the first case, the data maps onto an hypersphere, while on the second one it maps onto a semi-hyperplane.

Image data, as well as directions and other sources of information, are not always defined on the ℝ plane or ℝ3 space. They can be, for example, defined over a surface embedded in ℝ3. It is important then to define basic image processing operation for general data defined on general (not-necessarily flat) manifolds. In other words, we want to deal with maps between two general manifolds, and be able for example to isotropically and anisotropically diffuse them with the goal of noise removal. This will make it possible for example to denoise data defined on 3D surfaces.