A Positional Representation for Noiseless Compression

The usual representation of a random sequence on a finite alphabet is obtained by recording the value occurring in each position as the positions are scanned in some standard order. Here, we propose a representation obtained by recording the positions occupied by each value as the values are scanned in some specified order. Entropy is preserved in converting to the positional representation. Also, the unknown positions can be arbitrarily rearranged as the occupied positions are revealed. Under control of a memory model, we propose a rearrangement acting to reduce the first-order entropy. This allows better compression using an adaptive method, such as the Lempel-Ziv algorithm. Memory effects over large sample distances, multiple dimensions, or large alphabets can be directly applied in predicting positions rather than slowly learned by the adaptive coder. Some empirical results for grey-scale television-quality images (480 rows by 512 columns by 256 intensities) are included.