Wavelet-Based Statistical Modeling and Image Estimation

Abstract

Third, it has been noticed in image estimation practice that a translation invariant (TI) wavelet transform enhances estimation performance. We analyze the conventional complete wavelet transform and the TI wavelet transform from the viewpoints of approximation and estimation theory. First, we show that the TI expansion produces smaller approximation error when approximating smooth functions, and mitigates Gibbs artifacts when approximating discontinuous functions. Second, we study TI estimators and show that under mild conditions, replacing an estimator with its TI version will not worsen the estimation performance as measured by the minimax or Bayes risk.