A new complex basis for implicit polynomial curves and its simple exploitation for pose estimation and invariant recognition

New representations are developed for 2D IP (implicit polynomial) curves of arbitrary degree. These representations permit shape recognition and pose estimation with essentially single, rather than iterative, computation, and extract and use all the information in the polynomial coefficients. This is accomplished 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. These rotations in the two dimensional coefficient subspaces are related in simple ways to each other and to rotation in the x, y data space. By recasting this approach in terms of complex polynomials, i.e., z=x+iy and complex coefficients, further simplification occurs for rotations and some simplification occurs for translation.