Beschreibung: | 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 single intrinsically
one-dimensional signal, there exists no general method to decompose
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. |