33 research outputs found
A General Geometric Fourier Transform Convolution Theorem
The large variety of Fourier transforms in geometric algebras inspired the
straight forward definition of ``A General Geometric Fourier Transform`` in
Bujack et al., Proc. of ICCA9, covering most versions in the literature. We
showed which constraints are additionally necessary to obtain certain features
like linearity, a scaling, or a shift theorem. In this paper we extend the
former results by a convolution theorem
Detection of Outer Rotations on 3D-Vector Fields with Iterative Geometric Correlation
Correlation is a common technique for the detection of shifts. Its
generalization to the multidimensional geometric correlation in Clifford
algebras has proven a useful tool for color image processing, because it
additionally contains information about rotational misalignment. In this paper
we prove that applying the geometric correlation iteratively can detect the
outer rotational misalignment for arbitrary three-dimensional vector fields.
Thus, it develops a foundation applicable for image registration and pattern
matching. Based on the theoretical work we have developed a new algorithm and
tested it on some principle examples
Detection of Total Rotations on 2D-Vector Fields with Geometric Correlation
Correlation is a common technique for the detection of shifts. Its
generalization to the multidimensional geometric correlation in Clifford
algebras additionally contains information with respect to rotational
misalignment. It has been proven a useful tool for the registration of vector
fields that differ by an outer rotation. In this paper we proof that applying
the geometric correlation iteratively has the potential to detect the total
rotational misalignment for linear two-dimensional vector fields. We further
analyze its effect on general analytic vector fields and show how the rotation
can be calculated from their power series expansions
Steigerung der Effizienz Hierarchischer Matrizen durch Verwendung gemeinsamer Basen
Viele physikalische Probleme führen zu Randwertproblemen. Dabei gilt es die Lösung einer Dfferentialgleichung zu finden, so dass auf dem Rand vorgegebene Funktionswerte, die so genannten Randbedingungen, angenommen werden. Differentialgleichungen können nur in wenigen Spezialfällen analytisch gelöst werden. Man muss also auf numerische Verfahren zurückgreifen. Ein Problem aus der Praxis ist in der Regel von zu hoher Komplexität. Wir können daher nicht davon ausgehen ein Black-Box-Verfahren zu finden, welches jede Dfferentialgleichung innerhalb akzeptabler Zeit löst. Deshalb brauchen wir auf die Problemklassen zugeschnittene Verfahren, welche ihre speziellen Eigenschaften ausnutzen. Wir beschränken uns hier auf elliptische Randwertprobleme. Sie werden zu Integralgleichungen umformuliert, mittels Randelementmethode diskretisiert und damit in ein lineares Gleichungssystem überführt. Zur Behandlung des Gleichungssystems bedienen wir uns Hierarchischer Matrizen. Obwohl diese bereits effektive Hilfsmittel darstellen, wollen wir versuchen ihre Effzienz durch Verwendung gemeinsamer Basen weiter zu steigern.
Convolution products for hypercomplex Fourier transforms
Hypercomplex Fourier transforms are increasingly used in signal processing
for the analysis of higher-dimensional signals such as color images. A main
stumbling block for further applications, in particular concerning filter
design in the Fourier domain, is the lack of a proper convolution theorem. The
present paper develops and studies two conceptually new ways to define
convolution products for such transforms. As a by-product, convolution theorems
are obtained that will enable the development and fast implementation of new
filters for quaternionic signals and systems, as well as for their higher
dimensional counterparts.Comment: 18 pages, two columns, accepted in J. Math. Imaging Visio
Reducing Occlusion in Cinema Databases through Feature-Centric Visualizations
In modern supercomputer architectures, the I/O capabilities do not keep up with the computational speed. Image-based techniques are one very promising approach to a scalable output format for visual analysis, in which a reduced output that corresponds to the visible state of the simulation is rendered in-situ and stored to disk. These techniques can support interactive exploration of the data through image compositing and other methods, but automatic methods of highlighting data and reducing clutter can make these methods more effective. In this paper, we suggest a method of assisted exploration through
the combination of feature-centric analysis with image space techniques and show how the reduction of the data to features of interest reduces occlusion in the output for a set of example applications
Topological Segmentation of 2D Vector Fields
Vector field topology has a long tradition as a visualization tool. The separatrices segment the domain visually into
canonical regions in which all streamlines behave qualitatively
the same. But application scientists often need more than just a
nice image for their data analysis, and, to best of our knowledge,
so far no workflow has been proposed to extract the critical
points, the associated separatrices, and then provide the induced
segmentation on the data level.
We present a workflow that computes the segmentation of the
domain of a 2D vector field based on its separatrices. We show
how it can be used for the extraction of quantitative information
about each segment in two applications: groundwater flow and
heat exchange
A Testing Environment for Continuous Colormaps
Many computer science disciplines (e.g., combinatorial optimization, natural
language processing, and information retrieval) use standard or established
test suites for evaluating algorithms. In visualization, similar approaches
have been adopted in some areas (e.g., volume visualization), while user
testimonies and empirical studies have been the dominant means of evaluation in
most other areas, such as designing colormaps. In this paper, we propose to
establish a test suite for evaluating the design of colormaps. With such a
suite, the users can observe the effects when different continuous colormaps
are applied to planar scalar fields that may exhibit various characteristic
features, such as jumps, local extrema, ridge or valley lines, different
distributions of scalar values, different gradients, different signal
frequencies, different levels of noise, and so on. The suite also includes an
expansible collection of real-world data sets including the most popular data
for colormap testing in the visualization literature. The test suite has been
integrated into a web-based application for creating continuous colormaps
(https://ccctool.com/), facilitating close inter-operation between design and
evaluation processes. This new facility complements traditional evaluation
methods such as user testimonies and empirical studies