TOWARDS RANDOM FIELD MODELING OF WAVELET STATISTICS Z. Azimifar P. Fieguth E. Jernigan Department of Systems Design Engineering University of Waterloo Waterloo, Ontario, Canada, N2L-3G1 ABSTRACT This paper investigates the statistical characterization of sig- nals and images in the wavelet domain. In particular, in con- trast to common decorrelated-coefficient models, we find that the correlation between wavelet scales can be surpris- ingly substantial, even across several scales. In this pa- per we investigate possible choices of statistical-interaction models. One efficient and fast strategy which describes the wavelet-based statistical correlations is illustrated. Finally, the effectiveness of the proposed tool towards an efficient hierarchical MRF modeling of within-scale neighborhoods and across-scale dependencies will be demonstrated. 1. INTRODUCTION This paper presents a fast and efficient strategy for model- ing the statistical dependencies of 2-D wavelet coefficients. Specifically, we are interested in studying the Markovian nature of wavelet coefficient interactions, both within and across scales. We propose multiscale and Markov random field (MRF) models for the wavelet correlation structures. Our motivation is model-based statistical image pro- cessing, which requires some probabilistic description of the underlying image characteristics. Because of the com- plexity of spatial behaviour and pixel interactions, the raw statistics of pixels are extremely complicated and inconve- nient to specify. It is much more convenient to consider describing the statistics of a transformed image, where the transform is chosen to simplify or decorrelate, as much as possible, the starting statistics, analogous to the precondi- tioning of complicated linear system problems. The popu- larity of the wavelet transform (WT) stems from its effec- tiveness in this task: many operations, such as interpola- tion, estimation, compression, and denoising are simplified in the wavelet domain, because of its energy compaction and decorrelative properties [1, 2]. A conspicuously common assumption is that the WT is a perfect whitener, such that all of the wavelet coefficients The support of the Natural Science & Engineering Research Council of Canada and Communications & Information Technology Ontario are acknowledged. Coarse Scale Fine Scale Fig. 1. Illustration of a coefficient in the fine scale (shown by ), whose spatial neighbors come from different parents in the coarser scale. are independent, and ideally Gaussian. There is, however, a growing recognition that neither of these assumptions are accurate, nor even adequate for many image process- ing needs. There have been several recent efforts to study the wavelet statistics; most of these focus on the individual (marginal) statistics, only very little literature is present on the interrelationship (joint) statistics: 1. Marginal Models: (a) Non-Gaussian, i.e., heavy tail distribution [3], (b) Mixture of Gaussians [3], (c) Generalized Gaussian distribution [2], (d) Bessel functions [4]. 2. Joint Models: Hidden Markov tree models [1]. In virtually all marginal models, currently being used in wavelet shrinkage [2], the coefficients are treated individu- ally and as independent, i.e., only the diagonal elements of wavelet based covariance matrix are considered. This ap- proach, however, is not optimal in a sense that WT is not a perfect whitening process. The latter approach, however, examines the joint statis- tics of coefficients. Normally an assumption is present that the correlation between coefficients does not exceed the parent-child dependencies, e.g. given the state of its parent, a child is decoupled from the entire wavelet tree [1]. It is difficult to study both aspects simultaneously: that is, the development of non-Gaussian joint models with non-trivial neighborhood. The study of independent non- Gaussian models has been thorough; the complementary I - 361 0-7803-7622-6/02/$17.00 ©2002 IEEE IEEE ICIP 2002