Scale-Space Phase Stability
This Chapter shows that phase is typically stable with respect to small geometric deformations, and is quasi-linear as a function of spatial position. These issues are addressed in the restricted case of 1-d signals, where the relevant deformations are translations and dilations like those that occur between left and right views of stereo pairs of images; scale variations between left and right binocular views of a smooth surface are often as large as 20% [Ogle, 1956]. Using a scale-space framework, we simulate changes in the scale of the input by changing the tuning of a band-pass filter. In this context our concerns include the extent to which phase is stable under small perturbations of the filter tuning, and the extent to which phase is generally linear through space. These properties are shown to depend on the form of the filters and their frequency bandwidths; for a given type of filter, larger bandwidths provide better phase stability but poorer phase linearity through space. Therefore, when one designs filters for phase-based disparity measurement, it is important that the bandwidth be set according to the expected magnitude of deformation between left and right views. Situations in which phase becomes unstable or leads to inaccurate matching are discussed at length in the next chapter.
KeywordsPhase Behaviour Amplitude Spectrum Gabor Filter Image Velocity Scale Perturbation
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