1,796 research outputs found
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
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
Visualization and Analysis of Flow Fields based on Clifford Convolution
Vector fields from flow visualization often containmillions of data values. It is obvious that a direct inspection of the data by the user is tedious. Therefore, an automated approach for the preselection of features is essential for a complete analysis of nontrivial flow fields. This thesis deals with automated detection, analysis, and visualization of flow features in vector fields based on techniques transfered from image processing. This work is build on rotation invariant template matching with Clifford convolution as developed in the diploma thesis of the author. A detailed analysis of the possibilities of this approach is done, and further techniques and algorithms up to a complete segmentation of vector fields are developed in the process. One of the major contributions thereby is the definition of a Clifford Fourier
transform in 2D and 3D, and the proof of a corresponding convolution theorem for the Clifford convolution as well as other major theorems. This Clifford Fourier transform allows a frequency analysis of vector fields and the behavior of vectorvalued filters, as well as an acceleration of the convolution computation as a fast transform exists. The depth and precision of flow field analysis based on template matching and Clifford convolution is studied in detail for a specific application, which are flow fields measured in the wake of a helicopter rotor. Determining the features and their parameters in this data is an important step for a better understanding of the observed flow. Specific techniques dealing with subpixel accuracy and the parameters to be determined are developed on the way. To regard the flow as a superposition of simpler features is a necessity for this application as close vortices influence each other. Convolution is a linear system, so it is suited for this kind of analysis. The suitability of other flow analysis and visualization methods for this task is studied here as well. The knowledge and techniques developed for this work are brought together in the end to compute and visualize feature based segmentations of flow fields. The resulting visualizations display important structures of the flow and highlight the interesting features. Thus, a major step towards robust and automatic detection, analysis and visualization of flow fields is taken
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
Template Matching on Vector Fields using Clifford Algebra
Due to the amount of flow simulation and measurement data, automatic detection, classification and visualization of features is necessary for an inspection. Therefore, many automated feature detection methods have been developed in recent years. However, one feature class is visualized afterwards in most cases, and many algorithms have problems in the presence of noise or superposition effects. In contrast, image processing and computer vision have robust methods for feature extraction and computation of derivatives of scalar fields. Furthermore, interpolation and other filter can be analyzed in detail. An application of these methods to vector fields would provide a solid theoretical basis for feature extraction. The authors suggest Clifford algebra as a mathematical framework for this task. Clifford algebra provides a unified notation for scalars and vectors as well as a multiplication of all basis elements. The Clifford product of two vectors provides the complete geometric information of the relative positions of these vectors. Integration of this product results in Clifford correlation and convolution which can be used for template matching on vector fields. Furthermore, for frequency analysis of vector fields and the behavior of vector-valued filters, a Clifford Fourier transform has been derived for 2 and 3 dimensions. Convolution and other theorems have been proved, and fast algorithms for the computation of the Clifford Fourier transform exist. Therefore the computation of Clifford convolution can be accelerated by computing it in Clifford Fourier domain. Clifford convolution and Fourier transform can be used for a thorough analysis and subsequent visualization of vector field
Two-sided Clifford Fourier transform with two square roots of -1 in Cl(p,q)
We generalize quaternion and Clifford Fourier transforms to general two-sided
Clifford Fourier transforms (CFT), and study their properties (from linearity
to convolution). Two general \textit{multivector square roots} \in \cl{p,q}
\textit{of} -1 are used to split multivector signals, and to construct the left
and right CFT kernel factors.
Keywords: Clifford Fourier transform, Clifford algebra, signal processing,
square roots of -1 .Comment: 19 pages, 1 figur
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