Low-Level Image Processing with the Structure Multivector
The present thesis deals with two-dimensional signal processing for computer vision. The main topic is the development of a sophisticated generalization of the one-dimensional analytic signal to two dimensions. Motivated by the fundamental property of the latter, the invariance - equivariance constraint, and by its relation to complex analysis and potential theory, a two-dimensional approach is derived. This method is called the monogenic signal and it is based on the Riesz transform instead of the Hilbert transform. By means of this linear approach it is possible to estimate the local orientation and the local phase of signals which are projections of one-dimensional functions to two dimensions. For general two-dimensional signals, however, the monogenic signal has to be further extended, yielding the structure multivector. The latter approach combines the ideas of the structure tensor and the quaternionic analytic signal. A rich feature set can be extracted from the structure multivector, which contains measures for local amplitudes, the local anisotropy, the local orientation, and two local phases. Both, the monogenic signal and the structure multivector are combined with an appropriate scale-space approach, resulting in generalized quadrature filters. Same as the monogenic signal, the applied scale-space approach is derived from the three-dimensional Laplace equation instead of the diffusion equation. Hence, the two-dimensional generalization of the analytic signal turns out to provide a whole new framework for low-level vision. Several applications are presented to show the efficiency and power of the theoretic considerations. Among these are methods for orientation estimation, edge and corner detection, stereo correspondence and disparity estimation, and adaptive smoothing.
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