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 ((Formula presented.)) 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 ((Formula presented.)) problem (Formula presented.), (Formula presented.) and (Formula presented.), which must be solved many times. Numerical properties of the solution (Formula presented.) = (Formula presented.) of the (Formula presented.) problem are considered and it is shown that the property (Formula presented.) has implications for the error and stability of (Formula presented.). In particular, it can be assumed that b has components that lie in the column space (range) (Formula presented.) of A, and the space that is orthogonal to (Formula presented.), from which it follows that the upper bound of the condition number of (Formula presented.) is inversely proportional to (Formula presented.), where (Formula presented.) is the angle between b and its component that lies in (Formula presented.). It is shown that the maximum values of this condition number, other condition numbers and the errors in the solutions of the (Formula presented.) 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 (Formula presented.) method are discussed.