Multiscale difference equation signal models. I. Theory

The paper studies multiscale difference equation models for l-D and M-D signals. In this modeling technique, the signal of interest is viewed as a solution to a multiscale difference equation (MSDE). The model completely characterizes the signal as well as a number of its higher derivatives. It provides a recursive signal interpolation scheme as a function of scale. It also leads naturally to multigrid signal filtering, detection and estimation algorithms. An MSDE model must be uniquely decodable, i.e., it must correspond to a unique signal. Therefore, one must guarantee that the modeling MSDE has a unique solution. The authors investigate the existence and uniqueness of L₁ and L₂ solutions-to multiscale difference equations. Using Fourier domain techniques, they derive conditions for the existence of L₁ solutions to an MSDE. They provide conditions under which the L₁ solution is unique (up to a multiplicative constant) and has compact support. They also derive sufficient, but not necessary, conditions for the existence of a unique L₂ solution to a subclass of MSDEs. The results extend known facts about the solutions of two-scale difference equations. The paper concludes with several examples of MSDE signal models that highlight the modeling advantages of MSDEs over two-scale difference equation models