Details:
Title  Utilizing Moment Invariants and Gröbner Bases to Reason About Shapes  Author(s)  Haim Schweitzer, Janell Straach  Type  Article in Journal  Abstract  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 formulas with 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 1960s. We generalize this technique to shapes described by arbitrary monotone formulas (formulas in propositional logic without negation). Our technique produces a reduced Gröbner basisfor 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 specifying a shape. Unlike geometry theorem proving, our approach does not require the shapes to be explicitly defined. Instead, logic formulas combined with measurements performed on actual shape instances are used to compute wellcharacterized 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.  Keywords  shaperecognition, geometric reasoning 
URL 
dx.doi.org/10.1111/08247935.00072 
Language  English  Journal  Computational Intelligence  Volume  14  Number  4  Pages  461  474  Publisher  Blackwell Publishing  Year  1998  Month  November  Translation 
No  Refereed 
No  Organization 
University of Texas 
