Implicit algebraic curves and surfaces for shape modeling and recognition

Abstract

Implicit algebraic 2D curves and 3D surfaces are among the most powerful shape representations. With this approach, objects in 2D images are described by their silhouettes (em surfaces in 3D) and then represented by 2D implicit. polynomial curves; objects in 3D data are represented by the implicit polynomial surfaces. Implicit polynomial degrees of 2nd and. greater are used, typically up to 18th for 2D curves and 12th for 3D surfaces. In the thesis four contributions are presented whose common theme is using implicit polynomials for 2D and 3D shape modeling and object recognition purposes. Through the proposed concepts and algorithms, we argue that implicit polynomials provide a fast, and low computational cost model for low-error indexing into shape databases in 2D and 3D. The first contribution finds algebraic invariants of implicit polynomials under projective, affine, and Euclidean transformations by using and extending the symbolic method of classical invariant theory and shows how the invariants can be used for object recognition in invariant spaces. The second contribution is a novel model-based explicit algebraic expression for 2D-2D coordinate transformation estimation for Euclidean, similarity, and non-uniform scale transformations. The proposed explicit algebraic pose estimation expressions involve only the implicit polynomial coefficients for an object in two positions, and are drawn from an observation of how the implicit polynomial object model transforms under coordinate transformations. The third contribution is a 3D version of the 3L algorithm and an investigation of the specific problems in applying the technique in 3D as opposed to 2D. Finally, the fourth contribution explores the problem of getting affine invariant fits based on the 3L fitting as the 3L algorithm is inherently Euclidean invariant, but not affine invariant.