# Polynomial expansion for orientation and motion estimation

Abstract: This thesis introduces a new signal transform, called polynomial expansion, and based on this develops novel methods for estimation of orientation and motion. The methods are designed exclusively in the spatial domain and can be used for signals of any dimensionality.Two important concepts in the use of the spatial domain for signal processing is projections into subspaces, e.g. the subspace of second degree polynomials, and representations by frames, e.g. wavelets. It is shown how these concepts can be unified in a least squares framework for representation of nite dimensional vectors by bases, frames, subspace bases, and subspace frames.This framework is used to give a new derivation of normalized convolution, a method for signal analysis that takes uncertainty in signal values into account and also allows for spatial localization of the analysis functions.Polynomial expansion is a transformation which at each point transforms the signal into a set of expansion coefficients with respect to a polynomial local signal model. The expansion coefficients are computed using normalized convolution. As a consequence polynomial expansion inherits the mechanism for handling uncertain signals and the spatial localization feature allows good control of the properties of the transform. It is shown how polynomial expansion can be computed very efficiently.As an application of polynomial expansion, a novel method for estimation of orientation tensors is developed. A new concept for orientation representation, orientation functionals, is introduced and it is shown that orientation tensors can be considered a special case of this representation. By evaluation on a test sequence it is demonstrated that the method performs excellently.Considering an image sequence as a spatiotemporal volume, velocity can be estimated from the orientations present in the volume. Two novel methods for velocity estimation are presented, with the common idea to combine the orientation tensors over some region for estimation of the velocity field according to a parametric motion model, e.g. affine motion. The first method involves a simultaneous segmentation and velocity estimation algorithm to obtain appropriate regions. The second method is designed for computational efficiency and uses local neighborhoods instead of trying to obtain regions with coherent motion. By evaluation on the Yosemite sequence, it is shown that both methods give substantially more accurate results than previously published methods.Another application of polynomial expansion is a novel displacement estimation algorithm, i.e. an algorithm which estimates motion from only two consecutive frames rather than from a whole spatiotemporal volume. This approach is necessary when the motion is not temporally coherent, e.g. because the camera is affected by vibrations. It is shown how moving objects can robustly be detected in such image sequences by using the plane+parallax approach to separate out the background motion.To demonstrate the power of being able to handle uncertain signals it is shown how normalized convolution and polynomial expansion can be computed for interlacedvideo signals. Together with the displacement estimation algorithm this gives a method to estimate motion from a single interlaced frame.

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