Measure-valued images, associated fractal transforms, and the affine self-similarity of images

We construct a complete metric space (Y, d_Y) of measure-valued images, µ : X →M(R_g), where X is the base or pixel space and M(R_g) is the set of probability measures supported on the greyscale range R_g. Such a formalism is well suited to nonlocal (NL) image processing, i.e., the manipulation of the value of an image function u(x) based upon values u(yk) elsewhere in the image. We then show how the space (Y, d_Y) can be employed with a general model of affine self-similarity of images that includes both same-scale as well as cross-scale similarity. We focus on two particular applications: NL-means denoising (same-scale) and multiparent block fractal image coding (cross-scale). In order to accommodate the latter, a method of fractal transforms is formulated over the metric space (Y, d_Y). Under suitable conditions, a transform M : Y → Y is contractive, implying the existence of a unique fixed point measure-valued function µ = Mµ. We also show that the pointwise moments of this measure satisfy a set of recursion relations that are generalizations of those satisfied by moments of invariant measures of iterated function systems with probabilities.