A linear solution to 1-dimensional subspace fitting under incomplete data

Abstract

Computing a 1-dimensional linear subspace is an important problem in many computer vision algorithms. Its importance stems from the fact that maximizing a linear homogeneous equation system can be interpreted as subspace fitting problem. It is trivial to compute the solution if all coefficients of the equation system are known, yet for the case of incomplete data, only approximation methods based on variations of gradient descent have been developed. In this work, an algorithm is presented in which the data is embedded in projective spaces. We prove that the intersection of these projective spaces is identical to the desired subspace. Whereas other algorithms approximate this subspace iteratively, computing the intersection of projective spaces defines a linear problem. This solution is therefore not an approximation but exact in the absence of noise. We derive an upper boundary on the number of missing entries the algorithm can handle. Experiments with synthetic data confirm that the proposed algorithm successfully fits subspaces to data even if more than 90% of the data is missing. We demonstrate an example application with real image sequences. © 2011 Springer-Verlag Berlin Heidelberg.