Simultaneous fitting of several planes to point sets using neural networks

It is a simple problem to fit one line to a collection of points in the plane. But when the problem is generalized to two or more lines then the problem complexity becomes exponential in the number of points because we must decide on a partitioning of the points among the lines they are to fit. The same is true for fitting lines to points in three-dimensional space or hyperplanes to data points of high dimensions. We show that this problem despite its exponential complexity can be formulated as an optimization problem for which very good, but not necessarily optimal, solutions can be found by using an artificial neural network. Furthermore, we show that given a tolerance one can determine the number of lines (or planes) that should be fitted to a given point configuration. This problem is prototypical of a class of problems in computer vision, pattern recognition, and data fitting. For example, the method we propose can be used in reconstructing a planar world from range data or in recognizing point patterns in an image.