Use of two-dimensional deformable mesh structures for video coding .I. The synthesis problem: mesh-based function approximation and mapping

This paper explores the use of a deformable mesh (also known as the control grid) structure for motion analysis and synthesis in an image sequence. We focus on the synthesis problem, i.e., how to interpolate an image function given nodal positions and values and how to predict a present image frame from a reference one given nodal displacements between the two images. For this purpose, we review the fundamental theory and numerical techniques that have been developed in the finite element method for function approximation and mapping using a mesh structure. Specifically, we focus on (i) the use of shape functions for node-based function interpolation and mapping; and (ii) the use of regular master elements to simplify numerical calculations involved in dealing with irregular mesh structures. In addition to a general introduction that is applicable to an arbitrary mesh structure, we also present specific results for triangular and quadrilateral mesh structures, which are the most useful two-dimensional (2-D) meshes. Finally, we describe how to apply the above results for motion compensated frame prediction and interpolation. It is shown that the concepts of shape functions and master elements are crucial for developing computationally efficient algorithms for both the analysis and synthesis problems.