A Note on Some Phase Differencing Algorithms for Disparity Estimation

Abstract

Disparity can be estimated from the local phase differences between a pair of stereo images after Gabor filterings. In this paper, some phase differencing algorithms are discussed from the point of view of numerical computation. They are all categorised as certain types of Newton iterations, hence are restricted by the preconditions of Newton iterations. Furthermore, a numerical sensitivity analysis is made for phase computation near its singularity points without referring to any specific filtering method. It makes clear that the numerical instability has its roots in the singularities of the phase functions themselves rather than in their derivatives, therefore severe errors cannot be properly reduced by Newton iterations. Hence, the singularity problem must be carefully treated for any disparity estimation using phase differencing techniques. In the case of foveal vision, such as target detection or vergence control, the central parts of scenes are of interest. A shift-trials algorithm is proposed to contain the effects of singularities instead of relying on detection and removal of singularity neighbourhoods through local computation. This algorithm directly tackles the singularity problem by posing the local disparity estimations as a minimisation of the phase differences of the left and right views in terms of a global matching residual norm. By a direct search for solution, no numerical derivative is needed, phase singularity can be tolerated and robust results are thus produced.