Ghosts in Discrete Tomography

Brunetti, Sara; Dulio, Paolo; Hajdu, Lajos; Peri, Carla

doi:10.1007/s10851-015-0571-2

Cited by 9 publications

(5 citation statements)

References 44 publications

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“…In this case no multiple point belongs to G = (G + , G − ), which is called bad configuration. See also [6,32,33,34] for recent results concerning ghosts in discrete tomography. Remark 1.…”

Section: Preliminariesmentioning

confidence: 99%

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A rounding theorem for unique binary tomographic reconstruction

Dulio

Pagani

2019

Discrete Applied Mathematics

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Discrete tomography deals with the reconstruction of images from projections collected along a few given directions. Different approaches can be considered, according to different models. In this paper we adopt the grid model, where pixels are lattice points with integer coordinates, X-rays are discrete lattice lines, and projections are obtained by counting the number of lattice points intercepted by X-rays taken in the assigned directions.We move from a theoretical result that allows uniqueness of reconstruction in the grid with just four suitably selected X-ray directions. In this framework, the structure of the allowed ghosts is studied and described. This leads to a new result, stating that the unique binary solution can be explicitly and exactly retrieved from the minimum Euclidean norm solution by means of a rounding method based on some special entries, which are precisely determined. A corresponding iterative algorithm has been implemented, and tested on a few phantoms having different characteristics and structure.

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Section: Preliminariesmentioning

confidence: 99%

“…5 Compute the enlarging region E (see Definition 8). 6 Compute round(x * κ (λ 0 + u)) for all u ∈ E. 7 Compute the weights α * u = x * κ (λ 0 + u) − round(x * κ (λ 0 + u)) for all u ∈ E (see Equation (12)). 8 Compute E − (ξ, η), E + (ξ, η) for all (ξ, η) ∈ H (see Definition 8 and Equation (11)).…”

mentioning

confidence: 99%

A rounding theorem for unique binary tomographic reconstruction

Dulio

Pagani

2019

Discrete Applied Mathematics

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“…From a bad configuration (Z, W ) ghosts can be easily obtained, by giving opposite sign to the weights of the pixels of Z and W , and by setting all the other pixels to value zero. Recent results concerning ghosts in Discrete Tomography can be found in [1]. A grid was proven to be uniquely determined by a set S of directions if and only if it contains no bad configurations along the directions in S (see [6]).…”

Section: Definition and Known Resultsmentioning

confidence: 99%

Geometrical Characterization of the Uniqueness Regions Under Special Sets of Three Directions in Discrete Tomography

Dulio

Frosini

Pagani

2016

Discrete Geometry for Computer Imagery

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The faithful reconstruction of an unknown object from projections, i.e., from measurements of its density function along a finite set of discrete directions, is a challenging task. Some theoretical results prevent, in general, both to perform the reconstruction sufficiently fast, and, even worse, to be sure to obtain, as output, the unknown starting object. In order to reduce the number of possible solutions, one tries to exploit some a priori knowledge. In the present paper we assume to know the size of a lattice grid A containing the object to be reconstructed. Instead of looking for uniqueness in the whole grid A, we want to address the problem from a local point of view. More precisely, given a limited number of directions, we aim in showing, first of all, which is the subregion of A where pixels are uniquely reconstructible, and then in finding where the reconstruction can be performed quickly (in linear time). In previous works we have characterized the shape of the region of uniqueness (ROU) for any pair of directions. In this paper we show that the results can be extended to special sets of three directions, by splitting them in three different pairs. Moreover, we show that such a procedure cannot be employed for general triples of directions. We also provide applications concerning the obtained characterization of the ROU, and further experiments which underline some regularities in the shape of the ROU corresponding to sets of three not-yet-considered directions.

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“…Later work on switching components includes [37,84,105,115,120,123]. Computational investigations related to the explicit construction of switching components can be found in [80]. Switching components for other projection models are considered in [127,128,129,130,136].…”

Section: Theorem 4 ([120]mentioning

confidence: 99%

13. On the reconstruction of static and dynamic discrete structures

Alpers¹,

Gritzmann²

2019

The Radon Transform

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We study inverse problems of reconstructing static and dynamic discrete structures from tomographic data (with a special focus on the 'classical' task of reconstructing finite point sets in R d ). The main emphasis is on recent mathematical developments and new applications, which emerge in scientific areas such as physics and materials science, but also in inner mathematical fields such as number theory, optimization, and imaging. Along with a concise introduction to the field of discrete tomography, we give pointers to related aspects of computerized tomography in order to contrast the worlds of continuous and discrete inverse problems.

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Ghosts in Discrete Tomography

Cited by 9 publications

References 44 publications

A rounding theorem for unique binary tomographic reconstruction

A rounding theorem for unique binary tomographic reconstruction

Geometrical Characterization of the Uniqueness Regions Under Special Sets of Three Directions in Discrete Tomography

13. On the reconstruction of static and dynamic discrete structures

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