Implicit Polynomial Based Geometric Shape Modeling and Recognition

This paper presents a brief overview and focuses on two key aspects of a technology for representing and recognizing complicated 2D and 3D shapes subject to partial occlusion and missing data, based on implicit polynomials. The two key aspects are new concepts and results for fast, robust, repeatable tting of implicit poly-nomials to data, and new approaches to representing and recognizing complicated shapes based on these polynomials. This representation is built around signature curves of vector valued algebraic invariants which permit a recognition machine to start anywhere in a shape data set, choose a data patch of appropriate length, t an implicit polynomial to the patch, and recognize the shape or index into a database by comparing algebraic invariants of the polynomial with those stored in the signature curves. Fitting and recognition or indexing can be multiscale. This technology can also be applied to curves obtained from shape data, such as Kimia's shocks and time to shock-formation 1 , and our set-of-curves representation for generalized cylinders 2. Shape modeling for what purpose? Shape model development should depend on the intended use. In this paper, we are interested in shape models for the purposes of machine recognition of shapes and shape-based indexing into databases for 2D curve and 3D surface shapes, and shape models that have good representation power and are easy to manipulate for animation and deformation in computer graphics or virtual reality. Desirable characteristics for the rst two uses are low computational cost and low error rate. Of interest are both the case of speciic rigid or articulated objects, and the more diicult case of speciic deformable objects or determining the class association of a shape in the data. This paper presents a brief overview and focuses on two key aspects of a technology for representing and recognizing or indexing complicated 2D and 3D shapes subject to partial occlusion and missing data, based on implicit polynomials. The two key aspects are new concepts and results for fast, robust, repeatable tting of implicit polynomials to data, and new approaches to invariantly representing and recognizing complicated shapes based on these polynomials. Implicit polynomial planar curves and 3D surfaces can be thought of as extensions of quadric curves and surfaces to polynomials of higher de