Variational Approach to BuildingAlgorithms of Signal and Image

Despite the dramatic variety of application problems of signal and image analysis , it is possible to set oo some single subclasses of problems which allow for treating them in the uniied terms of respective standard mathematical optimization problems for which there exist eeective methods of solving. It was even Leonard Euler who marked that everything what happens in the world bears the sense of some minimum or maximum. Concerning signal and image analysis, the verity of this observation, which procured the name of the variational principle, manifests itself in the fact that all the kinds of decision making rest on the intent to minimize an appropriate measure of discrepancy J(XjY) between the observed data set Y 2 Y and result of its processing within the permissible range X 2 X. The variational principle turns a practical problem into the formal one of minimizing the respective objective function and shifts oo the burden of heuristic constructing of the data analysis algorithm into the tilled eld of the optimization theory. We will consider a signal or image as a function y t of one t 2 T or two variables t = (t 1 ; t 2) 2 T that takes values from an appropriate set y 2 Y and is deened on the discrete axis of the variable forming the signal T = ftg = f1 Both in signals and images, each element of the data array t 2 T is naturally associated with a number of its neighbors. It is convenient to express the neighborhood relation in the form of an undirected graph G which is understood as set of pairs of neighboring elements G T T. The evident neighborhood graph for signals is chain (Fig.1,a), and the simplest neighborhood relation usually considered to be appropriate for images is rectangular lattice (Fig.1,b), but we will decompose it into a succession of trees (Fig.1,c).