On the analysis and decomposition of intrinsically onedimensional signals and their superpositions
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 Zugl.: Kiel, Univ., Diss., 2012

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Computer and machine vision tasks can roughly be divided into a hierarchy of processing steps applied to input signals captured by a measuring device. In the case of image signals, the first stage in this hierarchy is also referred to as lowlevel vision or lowlevel image processing. The field of lowlevel image processing includes the mathematical description of signals in terms of certain local signal models. The choice of the signal model is often task dependent. A common task is the extraction of features from the signal. Since signals are subject to transformations, for example camera movements in the case of image signals, the features are supposed to fulfill the properties of invariance or equivariance with respect to these transformations. The chosen signal model should reflect these properties in terms of its parameters. This thesis contributes to the field of lowlevel vision. Local signal structures are represented by (sinusoidal) intrinsically onedimensional signals and their superpositions. Each intrinsically onedimensional signal consists of certain parameters such as orientation, amplitude, frequency and phase. If the affine group acts on these signals, the transformations induce a corresponding action in the parameter space of the signal model. Hence, it is reasonable, to estimate the model parameters in order to describe the invariant and equivariant features. The first and main contribution studies superpositions of intrinsically onedimensional signals in the plane. The parameters of the signal are supposed to be extracted from the responses of linear shift invariant operators: the generalized Hilbert transform (Riesz transform) and its higherorder versions and the partial derivative operators. While well known signal representations, such as the monogenic signal, allow to obtain the local features amplitude, phase and orientation for a \emph{single} intrinsically onedimensional signal, there exists no general method to decompose \emph{superpositions} of such signals into their corresponding features. A novel method for the decomposition of an arbitrary number of sinusoidal intrinsically onedimensional signals in the plane is proposed. The responses of the higherorder generalized Hilbert transforms in the plane are interpreted as symmetric tensors, which allow to restate the decomposition problem as a symmetric tensor decomposition. Algorithms, examples and applications for the novel decomposition are provided. The second contribution studies curved intrinsically onedimensional signals in the plane. This signal model introduces a new parameter, the curvature, and allows the representation of curved signal structures. Using the inverse stereographic projection to the sphere, these curved signals are locally identified with intrinsically onedimensional signals in the threedimensional Euclidean space and analyzed in terms of the generalized Hilbert transform and partial derivatives therein. The third contribution studies the generalized Hilbert transform in a nonEuclidean space, the twosphere. The mathematical framework of Clifford analysis proposes a further generalization of the generalized Hilbert transform to the twosphere in terms of the corresponding Cauchy kernel. Nonetheless, this transform lacks an intuitive interpretation in the frequency domain. A decomposition of the Cauchy kernel in terms of its spherical harmonics is provided. Its coefficients not only provide insights to the generalized Hilbert transform on the sphere, but also allow for fast implementations in terms of analogues of the convolution theorem on the sphere.
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