Estimating an even spherical measure from its sine transform

Hoffmann, Lars Michael

doi:10.1007/s10492-009-0005-9

Cited by 2 publications

(2 citation statements)

References 13 publications

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“…To be able to determine the rose of directions numerically, they restrict their considerations to atomic measures. Hoffmann [2] used in 2007 also a least square estimator to invert the sine transform. There exist convergence results and a proof of consistency ( [3]) for these algorithms.…”

Section: Introductionmentioning

confidence: 99%

Inversion algorithms for the spherical Radon and cosine transform

et al. 2011

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We consider two integral transforms which are frequently used in integral geometry and related fields, namely the cosine and the spherical Radon transform. Fast algorithms are developed which invert the respective transforms in a numerically stable way. So far, only theoretical inversion formulas or algorithms for atomic measures have been derived, which are not so important for applications. We focus on the two and threedimensional case, where we also show that our method leads to a regularization. Numerical results are presented and show the validity of the resulting algorithms. First, we use synthetic data for the inversion of the Radon transform. Then we apply the algorithm for the inversion of the cosine transform to reconstruct the directional distribution of line processes from finitely many intersections of their lines with test lines (2D) or planes (3D), respectively. Finally we apply our method to analyze a series of microscopic two-and three-dimensional images of a fibre system.

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Section: Introductionmentioning

confidence: 99%

Inversion algorithms for the spherical Radon and cosine transform

et al. 2011

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“…More recently, inversion formulas were found for L p densities ( [7]). In [5,3,6,4] various numerical reconstructions of the measure are derived.…”

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confidence: 99%

Reconstructing curves from lengths of projections onto lines

Vargo¹

2013

Preprint

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In this paper, we address the problem of reconstructing a curve from the lengths of its projections onto lines. We first note that the curve itself is not uniquely determined from these measurements. However, we find that a curve determines a measure on projective space which, as a function on Borel subsets of projective space, returns the length of curve parallel to elements of the set. We show that the projected length data can be expressed as the cosine transform of this measure on projective space. The cosine transform is a well studied integral transform on the sphere which is known to be injective. We conclude that the measured length data uniquely determines the associated measure on projective space. We then characterize the class of curves that produce a common measure by starting with the case of piecewise linear curves and then passing to limits to obtain results for more general curves.

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Estimating an even spherical measure from its sine transform

Cited by 2 publications

References 13 publications

Inversion algorithms for the spherical Radon and cosine transform

Inversion algorithms for the spherical Radon and cosine transform

Reconstructing curves from lengths of projections onto lines

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