Analysis of Fractals, Image Compression, Entropy Encoding, Karhunen-Loève Transforms

The purpose of this paper is to show that algorithms in a diverse set of applications may be cast in the context of relations on a finite set of operators in Hilbert space. The Cuntz relations for a finite set of isometries form a prototype of these relations. Such applications as entropy encoding, analysis of correlation matrices (Karhunen-Loève), fractional Brownian motion, and fractals more generally, admit multi-scales. In signal/image processing, this may be implemented with recursive algorithms using subdivisions of frequency-bands; and in fractals with scale similarity. In Karhunen-Loève analysis, we introduce a diagionalization procedure; and we show how the Hilbert space formulation offers a unifying approach; as well as suggesting new results.