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, $\mu : X \to {\cal M}(\mathbb{R}_g)$, where $X$ is the base or pixel space and ${\cal M}(\mathbb{R}_g)$ is the set of probability measures supported on the greyscale range $\mathbb{R}_g$. 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 $u(y_k)$ 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 $M : Y \to Y$ is contractive, implying the existence of a unique fixed point measure-valued function $\bar \mu = M \bar \mu$. 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.