Direct and least square fitting of coupled geometric objects for metric vision

A new approach to fitting coupled geometric objects, such as concentric circles, is presented. The objects can be coupled via common Grassmannian coefficients and through a correlation constraint on their coefficients. The implicit partitioning and partial block diagonal structure of the design matrix enables an efficient orthogonal residualisation based on a generalised Eckart-Young-Mirsky matrix approximation. The residualisation prior to eigen- or singular-value decomposition improves the numerical efficiency and makes the result invariant to the residuals of the independent portions. Analysis is performed for the generalised case of coupled implicit equations and examples of parallel lines, orthogonal lines, concentric circles, concentric ellipses and coupled conics are given. Furthermore, numerical tests and applications in image processing are presented.