The complex representation of algebraic curves and its simple exploitation for pose estimation and invariant recognition

Representations are introduced for handling 2D algebraic curves (implicit polynomial curves) of arbitrary degree in the scope of computer vision applications. These representations permit fast, accurate pose-independent shape recognition under Euclidean transformations with a complete set of invariants, and fast accurate pose-estimation based on all the polynomial coefficients. The latter is accomplished by a centering of a polynomial based on its coefficients, followed by rotation estimation by decomposing polynomial coefficient space into a union of orthogonal subspaces for which rotations within two-dimensional subspaces or identity transformations within one-dimensional subspaces result from rotations in x, y measured-data space. Angles of these rotations in the two-dimensional coefficient subspaces are proportional to each other and are integer multiples of the rotation angle in the x, y data space. By recasting this approach in terms of a complex variable, i.e., x+iy=z, and complex polynomial-coefficients, further conceptual and computational simplification results. Application to shape-based indexing into databases is presented to illustrate the usefulness and the robustness of the complex representation of algebraic curves.