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39
New Multiscale Transforms, Minimum Total Variation Synthesis: Applications to EdgePreserving Image Reconstruction
, 2001
"... This paper describes newly invented multiscale transforms known under the name of the ridgelet [6] and the curvelet transforms [9, 8]. These systems combine ideas of multiscale analysis and geometry. Inspired by some recent work on digital Radon transforms [1], we then present very effective and acc ..."
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Cited by 76 (11 self)
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This paper describes newly invented multiscale transforms known under the name of the ridgelet [6] and the curvelet transforms [9, 8]. These systems combine ideas of multiscale analysis and geometry. Inspired by some recent work on digital Radon transforms [1], we then present very effective and accurate numerical implementations with computational complexities of at most N log N. In the second part of the paper, we propose to combine these new expansions with the Total Variation minimization principle for the reconstruction of an object whose curvelet coefficients are known only approximately: quantized, thresholded, noisy coefficients, etc. We set up a convex optimization problem and seek a reconstruction that has minimum Total Variation under the constraint that its coefficients do not exhibit a large discrepancy from the the data available on the coefficients of the unknown object. We will present a series of numerical experiments which clearly demonstrate the remarkable potential of this new methodology for image compression, image reconstruction and image ‘denoising.’
Fast slant stack: A notion of Radon transform for data in a Cartesian grid which is rapidly computible, algebraically exact, geometrically faithful and invertible
 SIAM J. Sci. Comput
, 2001
"... Abstract. We define a notion of Radon Transform for data in an n by n grid. It is based on summation along lines of absolute slope less than 1 (as a function either of x or of y), with values at nonCartesian locations defined using trigonometric interpolation on a zeropadded grid. The definition i ..."
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Cited by 48 (11 self)
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Abstract. We define a notion of Radon Transform for data in an n by n grid. It is based on summation along lines of absolute slope less than 1 (as a function either of x or of y), with values at nonCartesian locations defined using trigonometric interpolation on a zeropadded grid. The definition is geometrically faithful: the lines exhibit no ‘wraparound effects’. For a special set of lines equispaced in slope (rather than angle), we describe an exact algorithm which uses O(N log N) flops, where N = n2 is the number of pixels. This relies on a discrete projectionslice theorem relating this Radon transform and what we call the Pseudopolar Fourier transform. The Pseudopolar FT evaluates the 2D Fourier transform on a nonCartesian pointset, which we call the pseudopolar grid. Fast Pseudopolar FT – the process of rapid exact evaluation of the 2D Fourier transform at these nonCartesian grid points – is possible using chirpZ transforms. This Radon transform is onetoone and hence invertible on its range; it is rapidly invertible to any degree of desired accuracy using a preconditioned conjugate gradient solver. Empirically, the numerical conditioning is superb; the singular value spread of the preconditioned Radon transform turns out numerically to be less than 10%, and three iterations of the conjugate gradient solver typically suffice for 6 digit accuracy. We also describe a 3D version of the transform.
Rotational motion deblurring of a rigid object from a single image
 In ICCV
, 2007
"... Most previous motion deblurring methods restore the degraded image assuming a shiftinvariant linear blur filter. These methods are not applicable if the blur is caused by spatially variant motions. In this paper, we model the physical properties of a 2D rigid body movement and propose a practical ..."
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Cited by 36 (4 self)
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Most previous motion deblurring methods restore the degraded image assuming a shiftinvariant linear blur filter. These methods are not applicable if the blur is caused by spatially variant motions. In this paper, we model the physical properties of a 2D rigid body movement and propose a practical framework to deblur rotational motions from a single image. Our main observation is that the transparency cue of a blurred object, which represents the motion blur formation from an imaging perspective, provides sufficient information in determining the object movements. Comparatively, single image motion deblurring using pixel color/gradient information has large uncertainties in motion representation and computation. Our results are produced by minimizing a new energy function combining rotation, possible translations, and the transparency map using an iterative optimizing process. The effectiveness of our method is demonstrated using challenging image examples. anteed since the convolution with a blur kernel is noninvertible. To tackle this problem, additional image priors, such as the global gradient distribution from clear images [7], are proposed. Some approaches use multiple images or additional visual cues [2, 20] to constrain the kernel estimation. (a) (b)
Multidigit Multiplication For Mathematicians
"... . This paper surveys techniques for multiplying elements of various commutative rings. It covers Karatsuba multiplication, dual Karatsuba multiplication, Toom multiplication, dual Toom multiplication, the FFT trick, the twisted FFT trick, the splitradix FFT trick, Good's trick, the SchonhageStrass ..."
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Cited by 27 (9 self)
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. This paper surveys techniques for multiplying elements of various commutative rings. It covers Karatsuba multiplication, dual Karatsuba multiplication, Toom multiplication, dual Toom multiplication, the FFT trick, the twisted FFT trick, the splitradix FFT trick, Good's trick, the SchonhageStrassen trick, Schonhage's trick, Nussbaumer's trick, the cyclic SchonhageStrassen trick, and the CantorKaltofen theorem. It emphasizes the underlying ring homomorphisms. 1.
Smoothness of scale functions for spectrally negative Lévy processes
, 2006
"... Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that stan ..."
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Cited by 25 (8 self)
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Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lévy measure. We place particular emphasis on spectrally negative Lévy processes with a Gaussian component and processes of bounded variation. An additional motivation is the very intimate relation of scale functions to renewal functions of subordinators. The results obtained for scale functions have direct implications offering new results concerning the smoothness of such renewal functions for which there seems to be very little existing literature on this topic.
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
Iterative Solution Of The Helmholtz Equation By A SecondOrder Method
 SIAM J. Matrix Anal. Appl
, 1996
"... . The numerical solution of the Helmholtz equation subject to nonlocal radiation boundary conditions is studied. The specific problem is the propagation of hydroacoustic waves in a twodimensional curvilinear duct. The problem is discretized with a secondorder accurate finitedifference method, resu ..."
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Cited by 17 (6 self)
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. The numerical solution of the Helmholtz equation subject to nonlocal radiation boundary conditions is studied. The specific problem is the propagation of hydroacoustic waves in a twodimensional curvilinear duct. The problem is discretized with a secondorder accurate finitedifference method, resulting in a linear system of equations. To solve the system of equations, a preconditioned Krylov subspace method is employed. The preconditioner is based on fast transforms, and yields a direct fast Helmholtz solver for rectangular domains. Numerical experiments for curved ducts demonstrate that the rate of convergence is high. Compared with band Gaussian elimination the preconditioned iterative method shows a significant gain in both storage requirement and arithmetic complexity. This research was supported by the U. S. National Science Foundationunder grant ASC8958544 and by the Swedish National Board for Industrial and Technical Development (NUTEK). y Department of Scientific Computi...
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.
T.: Methods for efficient, high quality volume resampling in the frequency domain
 In Proceedings of IEEE Visualization (2004
"... Resampling is a frequent task in visualization and medical imaging. It occurs whenever images or volumes are magnified, rotated, translated, or warped. Resampling is also an integral procedure in the registration of multimodal datasets, such as CT, PET, and MRI, in the correction of motion artifact ..."
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Cited by 15 (2 self)
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Resampling is a frequent task in visualization and medical imaging. It occurs whenever images or volumes are magnified, rotated, translated, or warped. Resampling is also an integral procedure in the registration of multimodal datasets, such as CT, PET, and MRI, in the correction of motion artifacts in MRI, and in the alignment of temporal volume sequences in fMRI. It is well known that the quality of the resampling result depends heavily on the quality of the interpolation filter used. However, highquality filters are rarely employed in practice due to their large spatial extents. In this paper, we explore a new resampling technique that operates in the frequencydomain where high quality filtering is feasible. Further, unlike previous methods of this kind, our technique is not limited to integerratio scaling factors, but can resample image and volume datasets at any rate. This would usually require the application of slow Discrete Fourier Transforms (DFT) to return the data to the spatial domain. We studied two methods that successfully avoid these delays: the chirpz transform and the FFTW package. We also outline techniques to avoid the ringing artifacts that may occur with frequencydomain filtering. Thus, our method can achieve highquality interpolation at speeds that are usually associated with spatial filters of far lower quality.
On the computation of the polar FFT
 Appl. Comput. Harmon. Anal
, 2007
"... We show that the polar as well as the pseudopolar FFT can be computed very accurately and efficiently by the well known nonequispaced FFT. Furthermore, we discuss the reconstruction of a 2d signal from its Fourier transform samples on a (pseudo)polar grid by means of the inverse nonequispaced FFT. ..."
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Cited by 9 (7 self)
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We show that the polar as well as the pseudopolar FFT can be computed very accurately and efficiently by the well known nonequispaced FFT. Furthermore, we discuss the reconstruction of a 2d signal from its Fourier transform samples on a (pseudo)polar grid by means of the inverse nonequispaced FFT.