000K utf8 1100 $c2012 1500 eng 2050 urn:nbn:de:101:1-201303157460 3000 Fleischmann, Oliver 4000 On the Analysis and Decomposition of Intrinsically One-Dimensional Signals and their Superpositions$hChristian-Albrechts-Universität zu Kiel [Fleischmann, Oliver] 4030 Kiel$nChristian-Albrechts-Universität zu Kiel 4209 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 low-level vision or low-level image processing. The field of low-level 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 low-level vision. Local signal structures are represented by (sinusoidal) intrinsically one-dimensional signals and their superpositions. Each intrinsically one-dimensional 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 one-dimensional 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 higher-order 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 one-dimensional 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 one-dimensional signals in the plane is proposed. The responses of the higher-order 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 one-dimensional 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 one-dimensional signals in the three-dimensional 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 non-Euclidean space, the two-sphere. The mathematical framework of Clifford analysis proposes a further generalization of the generalized Hilbert transform to the two-sphere 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. 4950 https://nbn-resolving.org/urn:nbn:de:101:1-201303157460$xR$3Volltext$534 4961 https://macau.uni-kiel.de/receive/macau_mods_00001778 5051 004 5550 analytic signal 5550 Clifford analysis 5550 Conformal embedding 5550 Curvature 5550 Directional correlation 5550 Equivariance 5550 Feature detection 5550 Hilbert transform 5550 Image Processing 5550 intrinsic dimension 5550 Invariance 5550 Local phase 5550 Monogenic signal 5550 Orientation decomposition 5550 Phase decomposition 5550 Riesz transform 5550 Scale-space theory 5550 Signal Processing 5550 Spherical harmonics 5550 spherical Hilbert transform 5550 Stereographic projection 5550 symmetric tensor 5550 Tensor decomposition