Extreme compression of grayscale images

Given an grayscale digital image, and a positive integer $n$, how well can we store the image at a compression ratio of $n:1$? In this paper we address the above question in extreme cases when $n>>50$ using "$\mathbf{V}$-variable image compression".

V-variable image compression

V-variable fractals, where $V$ is a positive integer, are intuitively fractals with at most $V$ different "forms" or "shapes" at all levels of magnification. In this paper we describe how V-variable fractals can be used for the purpose of image compression.