Local signal reconstruction via chromatic differentiation filter banks

Abstract

We investigate the theory of chromatic differentiation introduced by Ignjatovic (2001). The theory has a filter bank representation in which the analysis filters consist of high order chromatic differentiators, which are orthogonal linear combinations of standard differentiators allowing for practical FIR implementation. One distinctive aspect of chromatic differentiation is its local signal approximation: each reconstructed point depends only on the values of the chromatic derivatives at the nearest sample point. When all infinitely many chromatic derivatives are available at any point, the entire signal can be perfectly reconstructed. In this paper, we first consider the expected error of local reconstruction when only finitely many chromatic derivatives are available, and its impact on the design of the chromatic differentiation scheme. We next consider the problem of obtaining chromatic derivatives via FIR filtering and use the expected error term to justify a particular scheme proposed in Ignjatovic. Finally we consider the more general problem of obtaining higher order chromatic derivatives from available lower order chromatic derivatives. We conclude with an analysis of the ability of this technique to reduce the error of local signal reconstruction.