Nonlinear binary wavelet transforms and their application to binary image compression

Most research on wavelet transforms of binary signals and images has been concentrated on linear transforms. This paper proposes a binary wavelet transform scheme that does not depend on the field structure of binary numbers and includes nonlinear transforms. As a result, the number of binary wavelet transforms increases drastically. It is demonstrated that the increased freedom in choosing a particular transform can be useful in binary image compression. Nonlinear transforms reduce the empirical entropy of the test images substantially better than linear transforms.

I. INTRODUCTION

In the last decade the discrete wavelet transform has been adapted to work on signals and images that are finite valued. In 1993 Caire, Grossman and Poor [1] proposed a linear wavelet transform that works on finite signals with values from a finite field. Some years later, Swanson and Tewfik [2] used binary filters to define binary wavelet transforms of binary images. More recently Klappenecker, May and Nückel have generalized this concept by introducing linear wavelet transforms of signals over finite commutative rings. Parallel to this development, Sweldens lifting scheme triggered [4] the research of nonlinear wavelets.