Phase relationships in stereoscopic computation

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Abstract

We apply the notion that phase differences can be used to interpret disparity between a pair of stereoscopic images. Indeed, phase relationships can also be used to obtain orientation and probabilistic measures from both edges and comers, as well as the directional instantaneous frequency of an image field. The method of phase differences is shown to be equivalent to a Newton-Raphson root finding iteration through the resolutions of band-pass filtering. The method does, however, suffer from stability problems, and in particular stationary phase.

The stability problems associated with this technique are implicitly derived from the mechanism used to interpet disparity, which in general requires an assumption of linear phase and the local instantaneous frequency. We present two techniques. Firstly, we use the centre frequency of the applied band-pass filter to interpret disparity. This interpretation, however, suffers heavily from phase error and requires considerable damping prior to convergence. Secondly, we use the derivative of phase to obtain the instantaneous frequency from an image, which is then used to improve the disparity estimate. The second measure is implicitly sensitive to regions that exhibit stationary phase. We prove that stationary phase is a form of aliasing. To maintain stability with this technique, it is essential to smooth the disparity signal at each resolution of filtering.

These ideas are extended into 2-D where it is possible to extract both vertical and horizontal disparities. Unfortunately, extension into 2-D also introduces a similar form of the motion aperture problem. The best image regions to disambiguate both horizontal and vertical disparities lie in the presence of comers. Fortunately, we introduce a measure for identifying orthogonal image signals based upon the same filters that we use to interpret disparity. We find that in the presence of dominant edge energy, there is an error in horizontal disparity interpretation that varies as a cosine function. This error can be reduced by iteration or resolving the horizontal component of the disparity signal.

These ideas are also applied towards the computation of deformation, which is related to the magnitude and direction of surface slant. This is a natural application to the ideas presented in this thesis.

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