Image geometry through multiscale statistics

This study in the statistics of scale space begins with an analysis of noise propagation of multiscale differential operators for image analysis. It also presents methods for computing multiscale central moments that characterize the probability distribution of local intensities. Directional operators for sampling oriented local central moments are also computed and principal statistical directions extracted, reflecting local image geometry. These multiscale statistical models are generalized for use with multivalued data. The absolute error in normalized multiscale differential invariants due to spatially uncorrelated noise is shown to vary non-monotonically across order of differentiation. Instead the absolute error decreases between zeroth and first order measurements and increases thereafter with increasing order of differentiation, remaining less than the initial error until the third or fourth order derivatives are taken. Statistical invariants given by isotropic and directional sampling operators of varying scale are used to generate local central moments of intensity that capture information about the local probability distribution of intensities at a pixel location under an assumption of piecewise ergodicity. Through canonical analysis of a matrix of second moments, directional sampling provides principal statistical directions that reflect local image geometry, and this allows the removal of biases introduced by image structure. Multiscale image statistics can thus be made invariant to spatial rotation and translation as well as linear functions of intensity. These new methods provide a principled means for processing multivalued images based on normalization by local covariances. They also provide a basis for choosing control parameters in variable conductance diffusion.