Measure-Valued Images, Associated Fractal Transforms, and the Affine Self-Similarity of Images

We construct a complete metric space (Y,dY) of measure-valued images, μ : X → M(Rg), where X is the base or pixel space and M(Rg) is the set of probability measures supported on the greyscale range Rg. 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,dY) 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,dY). Under suitable conditions, a transform M : Y → Y is contractive, implying the existence of a unique fixed point measure-valued function μ = M μ. 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.