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 image processing, i. e. , the manipulation of the value of an image function u(x) based upon values u(yk) elsewhere in the image. In fact there are situations in which it is useful to consider the greyscale value of an image u at a point x as a random variable that can assume a range of values Rg of R. One example is the characterization of the statistical properties of a class of images, e. g. , MRI brain scans, for a particular application, say image compression. Another example is statistical image processing as applied to the problem of image restoration (denoising or deblurring). Of course, it is not enough to know the greyscale values that may be assumed by an image u at a point x: one must also have an idea of the probabilities (or frequencies) of these values. As such, it may be more useful to represent images by measure-valued functions. 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: nonlocal-means denoising (same-scale) and multiparent block fractal image coding (cross-scale). In order to accomodate the latter, a new method of fractal transforms is formulated over the metric space (Y,dY). Nonlocal image processing has recently received a great deal of attention, fuelled in part by the exceptional success of the nonlocal means image denoising method. Fractal image coding is another example of a nonlocal image processing method. Both of these methods, which will be described briefly below, may be viewed under the umbrella of a more general model of affine image selfsimilarity, in which subblocks of an image are approximated by other sublocks of the image. Indeed, a number of other image processing methods that exploit self-similarity and the various example-based methods, also fit naturally under this nonlocal, self-similar framework.