Adaptive vector-valued martingales: applications to image compression

Given a finite collection of functions defined on a common probability space, the paper describes an algorithm that constructs a vector-valued approximating martingale sequence. The orthonomal basis functions used to construct the martingale approximation are optimally selected, in each greedy step, from a large dictionary. The resulting approximations are characterized as generalized H-systems and provide scalar- and vector-valued orthonormal systems that can be employed to perform lossy compression for the given set of input functions. The filtration associated to the martingale allows for a multi-resolution analysis/synthesis algorithm to compute the approximating conditional expectation via a Fourier expansion. Convergence of the algorithm as well as several computational properties are established. Numerical examples are also provided for collection of images and video frames in order to study the approximating power of the constructed sequences.