S. R. Deans, The Radon Transform and Some of Its Applications.New York: Wiley, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....optical flow method. The aim of this paper is to show that these gradient based methods can be implemented in the Radon transform domain to yield fast, and accurate, estimates of the motion parameters. The Radon transform (projection) of an image is defined as line integrals across the image [10]. It is well known that pure translational motion in an image results in translation of the projections [10] along the direction of projection. This property has been used successfully in the past to estimate motion using projections [1, 2, 8, 9, 15, 18, 20 22, 24 26] More recently, we have ....

....in the Radon transform domain to yield fast, and accurate, estimates of the motion parameters. The Radon transform (projection) of an image is defined as line integrals across the image [10] It is well known that pure translational motion in an image results in translation of the projections [10] along the direction of projection. This property has been used successfully in the past to estimate motion using projections [1, 2, 8, 9, 15, 18, 20 22, 24 26] More recently, we have unified much of the (mostly ad hoc) work in this area and proposed a model of more general motion vector fields ....

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S. R. Deans, The Radon Transform and Some of its Applications, John Wiley and Sons: New York, 1983.

....acoustoelectric transfer function convolved with itself. The echo from a sloping plane surface is estimated and reasonable agreement is found with the measured echo. The theory can be expanded by using the Radon transform instead of the Abel transform, which allows non axisymmetrical transducers [3]. ....

Deans, Stanley R.; The Radon transform and some of its applications, John Wiley & Sons, 1983.

....see [7] This algorithm decreases the misalignment distortion and thus improves the measurements of g(p) One other thing that could give erroneous results is that the transducer is not circular symmetric. If this is the case, one should use the Radon transform instead of the Abel transform [4]. The latter assumes circular symmetry while the former is a more general form of tomography. The measurements should be extended to also record the angular dependency of the echo. This means that the measured echo g(p, t) should not only be a function of the distance p from the focal point to the ....

S.R. Deans; The Radon transform and some of its applications, John Wiley & Sons, 1983.

....approach [9] This method gives us an independent estimation of all single point echoes in the field simultaneously. In tomography, the line integral of a two dimensional function is normally measured. Methods for inverting these measurements to the original function are known, at least in theory [7]. In practice, restrictions like a finite number of projections, limit the accuracy of these inversions [17] However, we chose instead to mea sure the echoes from halfplanes, i.e. the integrals of the line echoes. The reason for this is a better SNR and to avoid the spatial smoothing that will ....

S. Deans, The Radon transform and some of its applica- tions, John Wiley & Sons, 1983.

....To overcome the weakness of wavelets in higher dimensions, Cands and Donoho [6] 7] re cently pioneered a new system of representations named ridgelets which deal effectively with line singularities in 2 D. The idea is to map a line singularity into a point singularity using the Radon transform [8]. Then, the wavelet transform can be used to effectively handle the point singularity in the Radon domain. Their initial proposal was intended for functions defined in the continuous 2 space. For practical applications, the development of discrete versions of the ridgelet transform that lead to ....

S.R. Deans, The Radon transform and some of its applications, John Wiley Sons, 1983.

.... of the century (Radon proved his inversion formula in 1917) but that has become especially active in the last two decades, as new technologies and the spreading of computers have made possible many ways of gathering data about bodies as well a powerful capability of handling the information [10, 22, 23, 24, 25, 30]. When density functions are replaced by geometric objects (for example we have a convex polytope instead of a body with non constant internal density) then geometric information shape, measure, is the detected data, and we arrive to the area of Geometric Tomography [14, 15] a topic ....

S. Deans, The Radon Transform and Some of Its Applications, John Wiley &Sons, New York, 1983.

....body and dynamic PET. The direct application of FBP method with ramp filter to PET emission and transmission data results in unacceptably noisy images. Windowing or reducing the cut o# frequency of the ramp filter used in the FBP method reduces the amount of noise but results in loss of resolution [23, 50, 80]. Non stationary sinogram processing [30, 63] and image post processing methods [73] have shown some promise to improve image quality. In the absence of the e#ects of random coincidences, PET transmission and emission measurements are well modeled as Poisson random variables [102] Statistical ....

S. Deans, The Radon transform and some of its applications, Wiley, New York,

....the near field scenarios and present simulations. 2 Review of fast algorithms Boag and Bresler have proposed a novel fast reprojection algorithm [1] in tomography. Reprojection means to obtain a set of parallel beam projections at view angles ( 10 2 0 3547680 9:9 9 ; given a 2 D image [7]. Direct algorithms for reprojection require operations to generate projections for an image. The fast reprojection algorithm proposed in [1] utilizes the angular bandlimit property of the sinogram [8] 9] Let = 0: be the projection function (Radon transform) of image ....

....sampling rate in ( is also halved. Therefore, to calculate ; projections of the subimages, it is enough to calculate half the number of projections and then angularly interpolate them to ; angles. Reprojection of the whole image is simply a shifted summation of the reprojections of the subimages [7][10] Recursively using the domain decomposition, the computational requirement for fast reprojection is WV XHYZ . original image subimage Figure 1. Original image and its subimages. Basu and Bresler further proposed a fast back projection algorithm [2] 11] based on the fast ....

S. Deans. The Radon Transform and Some of its Applications. Wiley, New York, 1983.

....as follows: T 1 n : f n (s) # Z dsf n (s)#(p s #) 2.1) where p # R 1 and #( is the singular distribution known as the Dirac delta function. The image of the mapping for all the direction vectors on S n 1 and all real numbers p is known as the Radon transform (Helgason (1980) Deans (1993)) or as the plane wave integral (John (1955) The T 1 n transformation in a specific direction represents the projection of the n dimensional function on the hyperplanes that are perpendicular to this direction. The mapping (2.1) is an ST expressed as the integral of the function f n (s) over ....

.... The stability of the ST solutions for random fields can be improved by using frequency filters that smooth out the fluctuations of the field (Jain (1989) Several schemes have been investigated that lead to di#erent choices of ST lines (e.g. Jain (1989) Tompson, Ababou, and Gelhar (1989) Deans (1993)) Ill posedness is due to the numerical implementation of the inverse operator T 3 1 =# 3 1 # (2.9) that involves the second SPACE TRANSFORMATIONS IN POROUS MEDIA 627 Fig. 4. Log log plot of the CPU time (seconds) vs. the grid size. All transformations have been calculated using NL = 200 ....

S. R. Deans (1993), The Radon Transform and Some of Its Applications, Krieger, Malabar, FL.

....at some set of orientations. The noninvasive nature of tomography has made it a very useful technique for a variety of applications, including medical imaging, nondestructive testing and evaluation, synthetic aperture radar (SAR) and electron microscopy based tomography, among others [1]. Yet the pervasiveness of the tomographic formulation in this variety of applications is coupled strongly to some key assumptions, which we will examine in this work. Much of the current research in the field of computerized tomography has focused on the problem of estimating the object given a ....

....by an unknown amount is treated in Section VI. The relationship between our results and the claims made in [9, 10] is examined in Section VII. Conclusions are discussed in Section VIII. II. Problem Formulation Background material on the tomographic problem can be found in many sources, including [1, 11] and the references therein. The image or object f we consider, is an element of the space L 2 (B 2 ) of square integrable real valued functions supported on the closed unit ball B 2 in the plane. Let us define the range of view angles to be# # = #, #] We consider the reconstruction of f from ....

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S. R. Deans, The Radon Transform and Some of Its Applications, Wiley, New York, 1983.

....that are line integrals of that object at some set of known orientations, referred to as view angles. This formulation has found a wide variety of applications including medical imaging, nondestructive testing, synthetic aperture radar (SAR) and electron microscopy based tomography among others [1]. In the case of 2D parallel beam projection tomography, however, we have recently demonstrated that under some fairly general conditions the view angles are in fact uniquely determined by the projection data [2] If the angles can be reliably determined from the projection data in the presence of ....

....the bounds developed previously. Conclusions are discussed in Section VII. II. Problem Formulation In this section, we briefly review the Radon transform, and then introduce the problem of angle recovery. Detailed descriptions of the tomographic problem can be found in many sources, including [1, 9] and the references therein. We consider the imaging of objects f that are elements of the space L 2 (B 2 ) of square integrable real valued functions 1 supported on the closed unit ball B 2 in the plane. 1 We have restricted our study to real objects, but the methods and results in this paper ....

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S. R. Deans, The Radon Transform and Some of Its Applications, Wiley, New York, 1983.

....f . Roughly, if s is an integer, kfkB s p;q C means that f is in some sense s times differentiable. When s is not an integer, it says that the [s] th derivative of f has some kind of continuity properties. We recall the definition of the Radon transform Rf of an integrable function f (see Deans, 1983 for details) Rf(u; t) Z u T x=t f(x) dx: The quantity (6.1) has a natural interpretation in terms of the smoothness of the Radon transform. Indeed, for p = q, we have the following equivalence: kfk p R s p;p i Ave u kRf(u; Delta)k p B s (d Gamma1) 2 p;p ; 6.2) 20 where ....

Deans, S. R. (1983). The Radon transform and some of its applications. John Wiley & Sons.

....f . Roughly, if s is an integer, #f#B s p,q # C means that f is in some sense s times di#erentiable. When s is not an integer, it says that the [s] th derivative of f has some kind of continuity properties. We recall the definition of the Radon transform Rf of an integrable function f (see Deans, 1983 for details) Rf(u, t) Z u T x=t f(x) dx. The quantity (6.1) has a natural interpretation in terms of the smoothness of the Radon transform. Indeed, for p = q, we have the following equivalence: #f# p R s p,p # Ave u #Rf(u, # p B s (d 1) 2 p,p , 6.2) 20 where B ....

Deans, S. R. (1983). The Radon transform and some of its applications. John Wiley & Sons.

....sampling step is inversely proportional to the scale. A detailed exposition on the ridgelet construction may be found in [1] We would like to emphasize that there is an important connection between ridgelet analysis and wavelet analysis of the Radon transform. Let R be the Radon transform of f [7] Rf( t) Z f(x 1 ; x 2 ) x 1 cos x 2 sin t) dx 1 dx 2 : Then a ridgelet coecient may be viewed as a kind of wavelet coecient of the Radon transform; i.e. hf; j; k i = hRf( j; j;k i; where j;k (t) 2 j=2 (2 j t k) This fact will be used implicitly throughout the ....

....a slightly di erent de nition of monoscale ridgelets. This will ease the statement of our main approximation theorem which follows. We add an extra layer of coarse scale coecients to eliminate various artifacts. Consider a standard multiresolution analysis that is adapted to the unit square [7] so that the set of translates f2 s (2 s k)g, k = k 1 ; k 2 ) k i = 0; 1; 2 s 1 is orthonormal (underlying there is, of course, a wavelet orthobasis of L 2 [0; 1] 2 f2 j (2 j k)g, k = k 1 ; k 2 ) k i = 0; 1; 2 j 1) Let P s be the orthogonal projector onto ....

S. R. Deans, \The Radon transform and some of its applications," John Wiley & Sons, 1983.

....transform to the slices Rf( In this section we develop a digital procedure which is inspired by this viewpoint, and is realizable on n by n numerical arrays, 3. 1 Fourier Strategy for Digital Radon Transform A fundamental fact about the Radon transform is the projection slice formula [13]: f( cos ; sin ) Z Rf(t; e i t dt: 6 This says that the Radon transform can be obtained by applying the one dimensional inverse Fourier transform to the two dimensional Fourier transform restricted to radial lines going through the origin. This of course suggests that ....

S. R. Deans. The Radon transform and some of its applications. John Wiley & Sons, 1983.

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S. R. Deans, The Radon Transform and Some of Its Applications.New York: Wiley, 1983.

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S.R. Deans, The Radon Transform and Some of its Applications. New York: Wiley, 1983.

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S. R. Deans, The Radon Transform and Some of its Applications.New York: Wiley, 1983.

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S.R. Deans. The Radon Transform and Some of Its Applications. J. Wiley, 1983.

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S. R. Deans. The Radon transform and some of its applications. John Wiley, 1983.

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S. R. Deans, The Radon Transform and some of its Applications, Wiley, New York, 1983.

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S. R. Deans, The Radon Transform and Some of Its Applications.New York: Wiley, 1983.

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S. R. Deans, The Radon transform and some of its applications. New York: Wiley, 1983.

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S. R. Deans. The Radon Transform and Some of Its Applications. Krieger Publishing Company, Malabar, Florida, 1993.

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S. R. Deans, The Radon transform and some of its applications, John Wiley & Sons, 1983.

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