Generalised Principal Component Analysis: exploiting inherent parameter constraints

Abstract

Generalised Principal Component Analysis (GPCA) is a recently devised technique for fitting a multi-component, piecewise-linear structure to data that has found strong utility in computer vision. Unlike other methods which intertwine the processes of estimating structure components and segmenting data points into clusters associated with putative components, GPCA estimates a multi-component structure with no recourse to data clustering. The standard GPCA algorithm searches for an estimate by minimising a simple algebraic misfit function. The underlying constraints on the model parameters are ignored. Here we promote a variant of GPCA that incorporates the parameter constraints and exploits constrained rather than unconstrained minimisation of a statistically motivated error function. The output of any GPCA algorithm hardly ever perfectly satisfies the parameter constraints. Our new version of GPCA greatly facilitates the final correction of the algorithm output to satisfy perfectly the constraints, making this step less prone to error in the presence of noise. The method is applied to the example problem of fitting a pair of lines to noisy image points, but has potential for use in more general multi-component structure fitting in computer vision.