Dynamically Quantized Pyramids

Introduction Dynamically Quantized (DO) spaces have been developed [O'Rourke] in response to the need for high-precision, high-dimensional Hough-like transforms. [Ballard; Sloan and Ballard) Their purpose is to cover a parameter space with a limited number of accumulators in such a way that fine precision is maintained where it is needed. O'Rourke's solution to this problem is to maintain a binary tree of cells. Each cell covers an n-dimensional rectangular region of the space. Under certain conditions, the cell may be split along a particular dimension, and two sons created. Under complementary conditions, sets of cells may merge. Cell splitting is relatively simple, but the process of cell merging is quite complicated, for reasons which are explored extensively in [O'Rourke]. The solution presented here is based on a pyramid data structure, in which the number and connectivity (between fathers and sons) of cells is fixed. This data structure has the advantage that its resource allocation is fixed, and the cells and their connections may be reduced to a hardware implementation (e.g., in VLSI . ) The customary difficulty with the pyramid is that the boundaries of the cells (and hence the spatial resolution) are fixed, also. In this Dynamically Quantized Pyramid (DQP), the boundaries of the cells are continually modified by means of a hierarchical warping process. Essentially, each cell tries to track the mean position of data points in its part of the space. This estimate of the local mean is used to define the boundaries of the cell's sons. An experimental implementation has been built and subjected to various distributions (spatial and temporal) of data. The resulting quantizations are shown and discussed. The tradeoff between resource allocation and precision of result is a familiar one. The task of developing a histogram is an example. For a given range of data values, and a given precision with which we need to locate features of the histogram (e.g., a peak), the usual procedure is to quantize the space uniformly, so that each cell covers a small part of the data space. Al l cells are of the same size, which is small enough to deliver an answer with the required precision. Usually, the required precision determines the amount of resources (histogram cells) which are allocated to this task. Sometimes, the precision we want cannot be achieved with the resources available. This is not often the case in oneor two-dimensional histograms. However, techniques which make use of histograms of high-dimensional data (four or more dimensions) are often resource limited.