Generalised Stochastic Automaton Image

It is well known that digital images can be generated by generalised stochastic automata (GSA). This fact has been taken as the basis for image compression by storing a GSA rather than the image that it generates. This paper investigates fundamental relationships between the parameters of GSA and their generated images. In particular, we are interested in expressing the magnitude of GSA weights, and the precision to which they must be speciied in image parameter terms. The signiicance of these relationships is connected to the compression ratio. An additional consequence of the use of GSA theory is a simpliication and extension of current work in automaton based image compression. 1 Basic Issues 1.1 Introduction Finite automaton based image compression was proposed and study in a series of papers by C ulik and Kari 7, 2, 3] and has been taken up by others, e.g., 5]. These investigators use the term weighted nite automaton (WFA). In fact, WFA are a particular instance of generalised stochastic automata (GSA), which have been studied for some time, largely in connection with formal languages and probabilistic computation. In this paper we apply notions and results from the theory of generalised stochastic automata (GSA) to issues related to digital image compression. We are chieey concerned with developing a principled account of the behaviour of basic GSA parameters in the context of image compression. To this end we adopt the formal language approach of GSA theory, and use some of its classical results, e.g., Theorem 2. Much of this theory can be found in 10, 9], and a more recent survey is presented in 8]. The three principal parameters of a GSA, so far as image compression is concerned are: (1) the number of states, (2) the maximum magnitude of its weights, and (3) the precision to which its weights must be speciied. The two principal parameters of a digital image are (1) its resolution and (2) its pixel quantisation level, which is just the number of discrete values a pixel may assume. We want to examine only the most primitive elements that can be associated to any image. The main items with which we will be concerned are: a) Can every image be generated by a GSA? b) For any image, is it always possible to nd a GSA with the minimum number of states such that its weights are bounded in some range, e.g., 0; …