On Rate-Distortion Models for Natural Images and Wavelet Coding Performance

Operational rate-distortion (RD) functions of most natural images, when compressed with state-of-the-art wavelet coders, exhibit a power-law behavior D alpha R(-gamma) at moderately high rates, with gamma being a constant depending on the input image, deviating from the well-known exponential form of the RD function D alpha 2(-xiR) for bandlimited stationary processes. This paper explains this intriguing observation by investigating theoretical and operational RD behavior of natural images. We take as our source model the fractional Brownian motion (fBm), which is often used to model nonstationary behaviors in natural images. We first establish that the theoretical RD function of the fBm process (both in 1-D and 2-D) indeed follows a power law. Then we derive operational RD function of the fBm process when wavelet encoded based on water-filling principle. Interestingly, both the operational and theoretical RD functions behave as D alpha R(-gamma). For natural images, the values of gamma are found to be distributed around 1. These results lend an information theoretical support to the merit of multiresolution wavelet compression of self-similar processes and, in particular, natural images that can be modelled by such processes. They may also prove useful in predicting performance of RD optimized image coders.