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A framework for discrete integral transformations II – the 2D 31 Radon transform
"... This paper is dedicated to the memory of Professor Moshe Israeli 19402007, who passed away on February 18. Computing the Fourier transform of a function in polar coordinates is an important building block in many scientific disciplines and numerical schemes. In this paper we present the pseudopola ..."
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Cited by 20 (10 self)
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This paper is dedicated to the memory of Professor Moshe Israeli 19402007, who passed away on February 18. Computing the Fourier transform of a function in polar coordinates is an important building block in many scientific disciplines and numerical schemes. In this paper we present the pseudopolar Fourier transform that samples the Fourier transform on the pseudopolar grid, also known as the concentric squares grid. The pseudopolar grid consists of equally spaced samples along rays, where different rays are equally spaced and not equally angled. The pseudopolar Fourier transform Fourier transform is shown to be fast (the same complexity as the FFT), stable, invertible, requires only
Fast and accurate Polar Fourier transform
 Appl. Comput. Harmon. Anal.
, 2006
"... In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is pr ..."
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Cited by 17 (1 self)
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In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is problematic. In this article we develop a fast high accuracy Polar FFT. For a given twodimensional signal of size N × N, the proposed algorithm’s complexity is O(N^2 log N), just like in a Cartesian 2DFFT. A special feature of our approach is that it involves only 1D equispaced FFT’s and 1D interpolations. A central tool in our method is the pseudoPolar FFT, an FFT where the evaluation frequencies lie in an oversampled set of nonangularly equispaced points. We describe the concept of pseudoPolar domain, including fast forward and inverse transforms. For those interested primarily in Polar FFT’s, the pseudoPolar FFT plays the role of a halfway point—a nearlyPolar system from which conversion to Polar coordinates uses processes relying purely on 1D FFT’s and interpolation operations. We describe the conversion process, and give an error analysis of it. We compare accuracy results obtained by a Cartesianbased unequallysampled FFT method to ours, both algorithms using a smallsupport interpolation and no precompensating, and show marked advantage to the use of the pseudoPolar initial grid.
Using NFFT 3  a software library for various nonequispaced fast Fourier transforms
, 2008
"... NFFT 3 is a software library that implements the nonequispaced fast Fourier transform (NFFT) and a number of related algorithms, e.g. nonequispaced fast Fourier transforms on the sphere and iterative schemes for inversion. This is to provide a survey on the mathematical concepts behind the NFFT and ..."
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Cited by 12 (8 self)
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NFFT 3 is a software library that implements the nonequispaced fast Fourier transform (NFFT) and a number of related algorithms, e.g. nonequispaced fast Fourier transforms on the sphere and iterative schemes for inversion. This is to provide a survey on the mathematical concepts behind the NFFT and its variants, as well as a general guideline for using the library. Numerical examples for a number of applications are given.
Discrete Analytical Ridgelet Transform
 Signal Processing
, 2004
"... In this paper, we propose an implementation of the 3D ridgelet transform: The 3D Discrete Analytical Ridgelet Transform (3D DART). This transform uses the Fourier strategy for the computation of the associated 3D discrete Radon transform. The innovative step is the definition of a discrete 3D t ..."
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Cited by 5 (0 self)
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In this paper, we propose an implementation of the 3D ridgelet transform: The 3D Discrete Analytical Ridgelet Transform (3D DART). This transform uses the Fourier strategy for the computation of the associated 3D discrete Radon transform. The innovative step is the definition of a discrete 3D transform with the discrete analytical geometry theory by the construction of 3D discrete analytical lines in the Fourier domain. We propose two types of 3D discrete lines: 3D discrete radial lines going through the origin defined from their orthogonal projections and 3D planes covered with 2D discrete line segments. These discrete analytical lines have a parameter called arithmetical thickness, allowing us to define a 3D DART adapted to a specific application. Indeed, the 3D DART representation is not orthogonal, It is associated with a flexible redundancy factor. The 3D DART has a very simple forward/inverse algorithm that provides an exact reconstruction without any iterative method. In order to illustrate the potentiality of this new discrete transform, we apply the 3D DART and its extension to the LocalDART (with smooth windowing) to the denoising of 3D image and colour video. These experimental results show that the simple thresholding of the 3D DART coefficients is efficient.
Fast clifford fourier transformation for unstructured vector field data
 In Proc. Intl. Conf. Numerical Grid Generation in Computational Field Simulations
, 2005
"... Vector fields play an important role in many areas of computational physics and engineering. For effective visualization of vector fields it is necessary to identify and extract important features inherent in the data, defined by filters that characterize certain “patterns”. Our prior approach for v ..."
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Cited by 4 (1 self)
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Vector fields play an important role in many areas of computational physics and engineering. For effective visualization of vector fields it is necessary to identify and extract important features inherent in the data, defined by filters that characterize certain “patterns”. Our prior approach for vector field analysis used the Clifford Fourier transform for efficient pattern recognition for vector field data defined on regular grids [1,2]. Using the frequency domain, correlation and convolution of vectors can be computed as a Clifford multiplication, enabling us to determine similarity between a vector field and a predefined pattern mask (e.g., for critical points). Moreover, compression and spectral analysis of vector fields is possible using this method. Our current approach only applies to rectilinear grids. We combine this approach with a fast Fourier transform to handle unstructured scalar data [6]. Our extension enables us to provide a featurebased visualization of vector field data defined on unstructured grids, or completely scattered data. Besides providing the theory of Clifford Fourier transform for unstructured vector data, we explain how efficient pattern matching and visualization of various selectable features can be performed efficiently. We have tested our method for various vector data sets.
Numerical stability of nonequispaced fast Fourier transforms
"... Dedicated to Franz Locher in honor of his 65th birthday This paper presents some new results on numerical stability for multivariate fast Fourier transform of nonequispaced data (NFFT). In contrast to fast Fourier transform (of equispaced data), the NFFT is an approximate algorithm. In a worst case ..."
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Cited by 3 (1 self)
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Dedicated to Franz Locher in honor of his 65th birthday This paper presents some new results on numerical stability for multivariate fast Fourier transform of nonequispaced data (NFFT). In contrast to fast Fourier transform (of equispaced data), the NFFT is an approximate algorithm. In a worst case study, we show that both approximation error and roundoff error have a strong influence on the numerical stability of NFFT. Numerical tests confirm the theoretical estimates of numerical stability.
DETECTION OF EDGES FROM NONUNIFORM FOURIER DATA
"... Abstract. Edge detection is important in a variety of applications. While there are many algorithms available for detecting edges from pixelated images or equispaced Fourier data, much less attention has been given to determining edges from nonuniform Fourier data. There are applications in sensing ..."
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Cited by 1 (0 self)
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Abstract. Edge detection is important in a variety of applications. While there are many algorithms available for detecting edges from pixelated images or equispaced Fourier data, much less attention has been given to determining edges from nonuniform Fourier data. There are applications in sensing (e.g. MRI) where the data is given in this way, however. This paper introduces a method for determining the locations of jump discontinuities, or edges, in a onedimensional periodic piecewisesmooth function from nonuniform Fourier coefficients. The technique employs the use of Fourier frames. Numerical examples are provided. 1.
A Clifford Fourier Transform for Vector Field Analysis and Visualization
"... Vector fields arise in many areas of computational science and engineering. For effective visualization of vector fields it is necessary to identify and extract important features inherent in the data, defined by filters that characterize certain patterns. Our prior approach for vector field analysi ..."
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Vector fields arise in many areas of computational science and engineering. For effective visualization of vector fields it is necessary to identify and extract important features inherent in the data, defined by filters that characterize certain patterns. Our prior approach for vector field analysis used the Clifford Fourier transform for efficient pattern recognition for vector field data defined on regular grids [1, 2]. Using the frequency domain, correlation and convolution of vectors can be computed as a Clifford multiplication, enabling us to determine similarity between a vector field and a predefined pattern mask (e.g., for critical points). Moreover, compression and spectral analysis of vector fields is possible using this method. Our approach in its current form only applies to rectilinear grids. We combine this approach with a fast Fourier transform to handle scalar data on arbitrary grids [3]. Our extension enables us to provide a featurebased visualization of vector field data defined on arbitrary grids, or completely scattered data. Besides providing the theory of Clifford Fourier transform for unstructured vector data, we explain how efficient pattern matching and visualization of various selectable features can be performed efficiently.