Implementing the Context - Tree Weighting Method : Arithmetic Coding

The context-tree weighting algorithm 6] is an eecient universal source coding method for tree sources. Although a nite accuracy version of this algorithm has been analysed in 8], it is better to implement the algorithm as proposed in 7]. There it was suggested to store in each node s of the context tree, instead of an estimated probability P s e and a weighted probability P s w , the (logarithm of the) ratio P s e =P 0s w P 1s w. This leads to a considerable storage reduction. Here we present the arithmetic coding procedure that matches to this implementation. It is based on Tjalkens' Ph.D. thesis 4] in which tables are used to circumvent multiplications. We also present a very simple carry-blocking procedure and brieey analyse it. 1. The Rissanen-Langdon approach to arithmetic coding 1.1. Introduction. First we investigate the structure consisting of a Context-Tree Weight-ing (CTW) modeler followed by an arithmetic encoder or decoder. Suppose that the CTW modeler produces block probabilities as described in 6]. These block probabilities are delivered to the arithmetic encoder and decoder. This has the advantage that the arithmetic encoder and decoder do not have to multiply to subdivide the source interval in a 0-and a 1-subinterval. The disadvantage of this method is however that we have to be very careful when we deene the nite accuracy operations done in the context tree (see 8]). We will use the notation of 6] to study the arithmetic encoder and decoder that matches to such a modeler. 1.2. Floating point numbers. The CTW-method supplies block probabilities P c to the arithmetic encoder or decoder. These coding probabilities are assumed to satisfy 1 T 1) > 0, for all x T 1. (1) In addition to this, the coding probabilities P c (x t 1); t = 0; T are assumed to be represented by f-bit oating point numbers. An f-bit oating point number g can be written as g = m(g) 2 where the mantissa m(g) and the exponent e(g) are integer-valued. Note that storing a oating point number requires f binary positions for the mantissa 2 and an additional number of positions for the exponent. 1 Here, in contrast to 6], we do not require thesèprobabilities' to sum up to 1, and since all source sequences can occur, we want all probabilities to be positive. 2 Actually also f ? 1 positions …