A novel hierarchical wavelet-based framework for pattern analysis and synthesis

We present a wavelet-based framework for modeling patterns in digital images. The wavelet coefficients of the underlying pattern template are modeled as independent Gaussian or Gaussian mixture random variables. Variations in pose and location of the pattern are accounted for by a finite collection of uniformly distributed transformations. The observation noise is assumed to be IID Gaussian. This hierarchical framework induces a statistical image model that can be used to synthesize instances of pattern observations. The underlying pattern, which is generally unknown, can be inferred from training data by means of an iterative alternating-maximization algorithm. This learning algorithm automatically infers a pattern template with a sparse wavelet representation. We can further promote an efficient representation by modeling the wavelet coefficients with a Gaussian mixture and placing a penalty on the number of "high" states.