The angles between the null spaces of X rays

Hamaker, C.; Solmon, Donald C.

doi:10.1016/0022-247x(78)90214-7

Cited by 111 publications

(47 citation statements)

References 6 publications

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“…In our reconstructions we have been using 18 x-ray directions (M = 18) and 10 iterations (m = 10) in which case the running time on the CDC 3300 is about 40 seconds. D. C. Solmon and C. Hamaker [6] have computed the angles Oj and have found that (for a judicious ordering of the projections P t ). However, by applying Theorem (2.2) to a sequence of finite dimensional subspaces invariant under P, they have shown that c < 2/3 (where c is the constant in 2.3).…”

Section: If Each Nj Makes An Angle > Otj With the Intersection Of Thementioning

confidence: 99%

Practical and mathematical aspects of the problem of reconstructing objects from radiographs

Smith¹,

Solmon²,

Wagner³

1977

Bull. Amer. Math. Soc.

317

114

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Section: If Each Nj Makes An Angle > Otj With the Intersection Of Thementioning

confidence: 99%

Practical and mathematical aspects of the problem of reconstructing objects from radiographs

Smith¹,

Solmon²,

Wagner³

1977

Bull. Amer. Math. Soc.

317

114

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“…< θ m < π. This property was mentioned in [8] and follows from Lemma 3.2 there. For the sake of completeness, we sketch the proof here.…”

Section: The Interpolation Theoremmentioning

confidence: 59%

“…Interpolation theorems based on integrals over chords can be used for approximate reconstruction of functions from their Radon transforms. Because of the importance of such recovery methods for applications in tomography, they have been intensively studied (see, for example, [11], [8], [10], [19], [4] and the bibliography therein).…”

mentioning

confidence: 99%

Interpolation by bivariate polynomials based on Radon projections

Bojanov

Georgieva

2004

Studia Math.

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Abstract. For any given set of angles θ 0 < . . . < θ n in [0, π), we show that a set of n+2 2Radon projections, consisting of k parallel X-ray beams in each direction θ k , k = 0, . . . , n, determines uniquely algebraic polynomials of degree n in two variables. Introduction.Most of the methods for approximate reconstruction of a univariate function f are based on sampling values of f at a finite number of points, and the tools used are usually those of interpolation. This is a natural approach to approximation of univariate functions since a table of function values is a standard type of information about f that comes as output in practical problems and processes described by functions in one variable, and in addition, the Lagrange interpolation problem by polynomials is always solvable. In the multivariate case, such an approach encounters serious difficulties. For example, it is well known that pointwise interpolation by multivariate polynomials is no more possible for every choice of nodes. Moreover, there are a lot of practical problems in which information about the relevant function comes as a set of functionals different from point evaluations. In tomography, electronic microscopy, and technics, the data often consists of values of linear integrals over segments. In many situations, a table of mean values of a function of d variables on (d − 1)-dimensional hyperplanes is considered to be the most natural type of data for multivariate functions. Hakopian's famous interpolation formula [6] (see also [7]) is an important reason to take this approach.Hakopian proved that for any given n + 2 distinct points X 0 , . . . , X n+1 on the boundary of a convex body D (say, a disk), the set of integrals of f over all the linear segments [X i , X j ] determines uniquely every polyno-

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“…In dimensions higher than two the situation is quite different. We give an example of a convex set ft C R 3 with smooth boundary and two vectors a 1 , a 2 , such that (1.1) is not closed for any q . On the other hand, if ft C R 3 is convex and the principal curvatures of the boundary are non-vanishing at every point (this condition can be relaxed), then we prove that (1.1) is closed for 1 < q < oo (Theorem 1.8).…”

mentioning

confidence: 99%

“…Shepp and Logan raised the problem in [6]. Hamaker and Solmon [3] treated the case when ft is a disk and q = 2. A more general case was treated by Falconer [2].…”

mentioning

confidence: 99%

On the closure of spaces of sums of ridge functions and the range of the $X$-ray transform

Boman¹

1984

Annales De L’institut Fourier

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The angles between the null spaces of X rays

Cited by 111 publications

References 6 publications

Practical and mathematical aspects of the problem of reconstructing objects from radiographs

Practical and mathematical aspects of the problem of reconstructing objects from radiographs

Interpolation by bivariate polynomials based on Radon projections

On the closure of spaces of sums of ridge functions and the range of the $X$-ray transform

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