Image processing in catadioptric planes: spatiotemporal derivatives and optical flow computation

Images produced by catadioptric sensors contain a significant amount of radial distortion and variation in inherent scale. Blind application of conventional shift-invariant operators or optical flow estimators yields erroneous results. One could argue that given a calibration of such a sensor we would always be able to remove distortions and apply any operator in a local perspective plane. In addition to the inefficiency of such an approach, interpolation effects during warping have undesired results in filtering. In this paper, we propose to use the sphere as the underlying domain of image processing in central catadioptric systems. This does not mean that we will warp the catadioptric image into a spherical image. Instead, we will formulate all the operations on the sphere but use the samples from the original catadioptric plane. As an example, we study the convolution with the Gaussian and its derivatives and as well as the computation of optical flow in image sequences acquired with a parabolic catadioptric sensor.