Error Analysis and Condition Estimation of the Pyramidal Form of the Lucas-Kanade Method in Optical Flow

Abstract

Optical flow is the apparent motion of the brightness patterns in an image. The pyramidal form of the Lucas-Kanade (LK) method is frequently used for its computation but experiments have shown that the method has deficiencies. Problems arise because of numerical issues in the least squares (LS) problem minAx−b22, A∈Rm×2 and m≫2, which must be solved many times. Numerical properties of the solution x0=A†b = (ATA)−1ATb of the LS problem are considered and it is shown that the property m≫2 has implications for the error and stability of x0. In particular, it can be assumed that b has components that lie in the column space (range) R(A) of A, and the space that is orthogonal to R(A), from which it follows that the upper bound of the condition number of x0 is inversely proportional to cosθ, where θ is the angle between b and its component that lies in R(A). It is shown that the maximum values of this condition number, other condition numbers and the errors in the solutions of the LS problems increase as the pyramid is descended from the top level (coarsest image) to the base (finest image), such that the optical flow computed at the base of the pyramid may be computationally unreliable. The extension of these results to the problem of total least squares is addressed by considering the stability of the optical flow vectors when there are errors in A and b. Examples of the computation of the optical flow demonstrate the theoretical results, and the implications of these results for extended forms of the LK method are discussed.