Distribution of run lengths over scanned rectangles

Abstract

Two of the most common operations performed on pictorial data are thresholding to delineate areas of interest, and run-length encoding of similar regions along a scan line in a television-like scan of this binary data. This run-length encoding of binary images requires little auxiliary storage and can often be performed almost simultaneously with the scanning operation. In this paper, we show that the information-compressive run-length histogram, commonly used to set up the code tables for Huffman encoding, carries significant shape information for a population of scanned rectangles. Our method is to decompose the problem by introducing auxiliary variables and finding conditional probability densities of run lengths using these variables. Probability distributions are found for run lengths derived from a line-by-line scan of a population of uniform intensity rectangles of one size but arbitrary orientation on a background of a different uniform intensity. For this idealized situation, the probability densities have two singularities. New, more robust estimates of rectangle shape derived from the run-length histogram are given. Sources of these singularities are examined and used to extend the results to more complex and less idealized situations. We conclude that the probability density of run lengths and similar random variables over similar populations will still have peaks or singularities. Thus sampled representations such as histograms may be very sensitive to sampling methods and parameters.