Representing arbitrarily-shaped regions: a case study of overcomplete representations

Efficiently representing the interior of an arbitrarily-shaped region is an important problem in many applications, including object- or region-based image/video compression. This paper focuses on representing the interior as a linear combination of vectors defined on a superset basis. Two seemingly different, though highly related, problem formulations are given and a number of approaches that result are discussed and analyzed. A geometric interpretation of the problem is given to provide insight into the approaches and examine their properties. The incorporation of quantization is also briefly discussed. In addition, the chosen set of vectors corresponds to a special/structured class of overcomplete representations with O(N log ₂ N)-type processing, low memory requirements, and other important properties