Utilizing Moment Invariants and Gröbner Bases to Reason About Shapes

Shapes such as triangles or rectangles can be defined in terms of geometric properties invariant under a group of transformations. Complex shapes can be described by logic formulae w i th simpler shapes as the atoms. A standard technique for computing invariant properties of simple shapes is the method of moment invariants, known since the early sixties. We generalize this technique to shapes described by arb i t rary monotone formulae (formulae in proposit ional logic wi thout negation). Our technique produces a reduced Grobner basis for approximate shape descriptions. We show how to use this representation to solve decision problems related to shapes. Examples include determining if a figure has a particular shape, if one description of a shape is more general than another, and whether a specific geometric property is really necessary for characterizing a shape. Unlike geometry theorem proving, our approach does not require the shapes to be expl ici t ly defined. Instead, logic formulae combined w i th measurements performed on actual shape instances are used to compute well characterized least squares approximations to the shapes. Our results provide a proof that decision problems stated in terms of these approximations can be solved in a finite number of steps.