668 research outputs found
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
Connecting spatial and frequency domains for the quaternion Fourier transform
The quaternion Fourier transform (qFT) is an important tool in multi-dimensional data analysis, in particular for the study of color images. An important problem when applying the qFT is the mismatch between the spatial and frequency domains: the convolution of two quaternion signals does not map to the pointwise product of their qFT images. The recently defined ‘Mustard’ convolution behaves nicely in the frequency domain, but complicates the corresponding spatial domain analysis.
The present paper analyses in detail the correspondence between classical convolution and the new Mustard convolution. In particular, an expression is derived that allows one to write classical convolution as a finite linear combination of suitable Mustard convolutions. This result is expected to play a major role in the further development of quaternion image processing, as it yields a formula for the qFT spectrum of the classical convolution
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
Clifford wavelets for fetal ECG extraction
Analysis of the fetal heart rate during pregnancy is essential for monitoring
the proper development of the fetus. Current fetal heart monitoring techniques
lack the accuracy in fetal heart rate monitoring and features acquisition,
resulting in diagnostic medical issues. The challenge lies in the extraction of
the fetal ECG from the mother's ECG during pregnancy. This approach has the
advantage of being a reliable and non-invasive technique. For this aim, we
propose in this paper a wavelet/multi-wavelet method allowing to extract
perfectly the feta ECG parameters from the abdominal mother ECG. The method is
essentially due to the exploitation of Clifford wavelets as recent variants in
the field. We prove that these wavelets are more efficient and performing
against classical ones. The experimental results are therefore due to two basic
classes of wavelets and multi-wavelets. A first-class is the classical Haar
Schauder, and a second one is due to Clifford valued wavelets and
multi-wavelets. These results showed that wavelets/multiwavelets are already
good bases for the FECG processing, provided that Clifford ones are the best.Comment: 21 pages, 8 figures, 1 tabl
An Optimized Architecture for CGA Operations and Its Application to a Simulated Robotic Arm
Conformal geometric algebra (CGA) is a new geometric computation tool that is attracting growing attention in many research fields, such as computer graphics, robotics, and computer vision. Regarding the robotic applications, new approaches based on CGA have been proposed to efficiently solve problems as the inverse kinematics and grasping of a robotic arm. The hardware acceleration of CGA operations is required to meet real-time performance requirements in embedded robotic platforms. In this paper, we present a novel embedded coprocessor for accelerating CGA operations in robotic tasks. Two robotic algorithms, namely, inverse kinematics and grasping of a human-arm-like kinematics chain, are used to prove the effectiveness of the proposed approach. The coprocessor natively supports the entire set of CGA operations including both basic operations (products, sums/differences, and unary operations) and complex operations as rigid body motion operations (reflections, rotations, translations, and dilations). The coprocessor prototype is implemented on the Xilinx ML510 development platform as a complete system-on-chip (SoC), integrating both a PowerPC processing core and a CGA coprocessing core on the same Xilinx Virtex-5 FPGA chip. Experimental results show speedups of 78x and 246x for inverse kinematics and grasping algorithms, respectively, with respect to the execution on the PowerPC processor
The Color Clifford Hardy Signal: Application to Color Edge Detection and Optical Flow
This paper introduces the idea of the color Clifford Hardy signal, which can
be used to process color images. As a complex analytic function's
high-dimensional analogue, the color Clifford Hardy signal inherits many
desirable qualities of analyticity. A crucial tool for getting the color and
structural data is the local feature representation of a color image in the
color Clifford Hardy signal. By looking at the extended Cauchy-Riemann
equations in the high-dimensional space, it is possible to see the connection
between the different parts of the color Clifford Hardy signal. Based on the
distinctive and important local amplitude and local phase generated by the
color Clifford Hardy signal, we propose five methods to identify the edges of
color images with relation to a certain color. To prove the superiority of the
offered methodologies, numerous comparative studies employing image quality
assessment criteria are used. Specifically by using the multi-scale structure
of the color Clifford Hardy signal, the proposed approaches are resistant to a
variety of noises. In addition, a color optical flow detection method with
anti-noise ability is provided as an example of application.Comment: 13 page
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