A generalized fractal transform for measure-valued images Original Research Article

Fractal image coding generally seeks to express an image as a union of spatially-contracted and greyscale-modified copies of subsets of itself. Generally, images are represented as functions u(x)u(x) and the fractal coding method is conducted in the framework of Ł2Ł2 or Ł∞Ł∞. In this paper we formulate a method of fractal image coding on measure-valued images: At each point xx, μ(x)μ(x) is a probability measure over the range of allowed greyscale values. We construct a complete metric space (Y,dY)(Y,dY) of measure-valued images, μ:X→M(Rg)μ:X→M(Rg), where XX is the base or pixel space and M(Rg)M(Rg) is the set of probability measures supported on the greyscale range RgRg. A generalized fractal transform MM is formulated over the metric space (Y,dY)(Y,dY). Under suitable conditions, M:Y→YM:Y→Y is contractive, implying the existence of a unique fixed point measure-valued function View the MathML sourceμ̄=Mμ̄.