The Six-Line Problem and Polynomial Resultants*
The "Six-Line Problem" arises in the automated analysis of images. A three-dimensional object is photographed and from the photographs (or other means) a set of geometric invariants are obtained by techniques of algebraic geometry. These three-dimensional invariants represent six lines abstracted from the object. Suppose that later an object is encountered and a photograph taken. The problem is to decide if this is the same object as before. From the flat photograph, two-dimensional invariants may be obtained.
The relationship between the three- and two-dimensional invariants may be expressed as a set of polynomial equations. The photographed object is the right one if the set of equations has a solution. One well known method to solve such sets of equations is with resultants. The number of variables and the size of the matrix seemed overwhelming until recently. This talk will describe how the solution was obtained using the Dixon-Kapur-Saxena-Yang resultant.
Robert H. Lewis
Department of Mathematics
Fordham University
rlewis@murray.fordham.edu
*Research supported by the Office of Naval Research.