The Complex Representation of Algebraic Curves and Its Simple Exploitation for Pose Estimation and Invariant Recognition

• Authors: Jean-Philippe Tarel;David B. Cooper
• Venue: IEEE Transactions on Pattern Analysis and Machine Intelligence
• Year: 2000

New 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 new 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.