Pose estimation and object identification using complex algebraic representations

The comparison and alignment of two similar objects is a fundamental problem in pattern recognition and computer vision that has been considered using various approaches. In this work, we employ a complex representation for an algebraic curve, and illustrate how the algebraic transformation which relates two Euclidean equivalent curves can be determined using this representation. The idea is based on a complex representation of 2D points expressed in terms of the orthogonalx andy variables, with rotations of the complex numbers described using Euler's identity. We develop a simple formula for integer multiples of the rotation angle of the Euclidean transformation in terms of the real coefficients of implicit polynomial equations that are used to model 2D free-form objects. When there is a translation, it can be determined using some new results on the conic-line factors of implicit polynomial curves. Experimental results are presented for data sets characterised by both noisy and missing data points to illustrate and validate our procedures.