Physically Based Adaptive Preconditioning for Early Vision

Several problems in early vision have been formulated in the past in a regularization framework. These problems, when discretized, lead to large sparse linear systems. In this paper, we present a novel physically based adaptive preconditioning technique which can be used in conjunction with a conjugate gradient algorithm to dramatically improve the speed of convergence for solving the aforementioned linear systems. A preconditioner, based on the membrane spline, or the thin plate spline, or a convex combination of the two, is termed a physically based preconditioner for obvious reasons. The adaptation of the preconditioner to an early vision problem is achieved via the explicit use of the spectral characteristics of the regularization filter in conjunction with the data. This spectral function is used to modulate the frequency characteristics of a chosen wavelet basis, and these modulated values are then used in the construction of our preconditioner. We present the preconditioner construction for three different early vision problems namely, the surface reconstruction, the shape from shading, and the optical flow computation problems. Performance of the preconditioning scheme is demonstrated via experiments on synthetic and real data sets. We note that our preconditioner outperforms other methods of preconditioning for these early vision problems, described in computer vision literature.