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The paper describes an algorithm to count and classify cells of different geometrical shapes on a given image. The algorithm assumes that it is known a priori the type of geometries to be recognized and it allows for many different geometrical shapes to appear in the same image with different sizes, locations and orientations. The algorithm combines classical tools, mainly the two dimensional Fourier transform, with newly developed tools for edge enhancements as well as the main technical contribution of the present paper, which consists in the definition of an over-complete set of spanning functions. These functions are constructed from geometrical templates of size comparable to the image cells; moreover, the resulting functions are scaled and rotated to assure the recognition of all image cells. We then describe an algorithm that decomposes the image in its most likely elements. The combination of ingredients used by the algorithm provides a cell recognition tool that is very robust, provides high resolution to discern among competing candidate cells and delivers practical computational efficiency.

Abstract. Given a finite collection of functions defined on a common domain, 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 which can be employed to perform lossy compression for the given set of input functions. The filtration associated to the martingale allows for a multiresolution 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.

Given a set of images we propose an algorithm that approximates all images simultaneously. The algorithm finds the best common partition of the images' domain at each step, this is accomplished by maximizing an appropriate inner product. The algorithm is a pursuit algorithm constrained to build a tree, the optimization is done over a large dictionary of wavelet-like functions. The approximations are given by vector valued discrete martingales that converge to the input set of images. Several computational and mathematical techniques are developed in order to encode the information needed for the reconstruction. Properties of the algorithm are illustrated through many examples, comparisons with JPEG2000 and MPEG4-3 are also provided.

Given a set of images we propose an algorithm that approximates all images simultaneously. The algorithm finds the best common partition of the images' domain at each step, this is accomplished by maximizing an appropriate inner product. The algorithm is a pursuit algorithm constrained to build a tree, the optimization is done over a large dictionary of wavelet-like functions. The approximations are given by vector valued discrete martingales that converge to the input set of images. Several computational and mathematical techniques are developed in order to encode the information needed for the reconstruction. Properties of the algorithm are illustrated through many examples, comparisons with JPEG2000 and MPEG4-3 are also provided.