Algorithms for linear time reconstruction by discrete tomography II

Ceko, Matthew; Pagani, Silvia Maria Carla; Tijdeman, R.

doi:10.1016/j.dam.2021.03.008

Cited by 3 publications

(18 citation statements)

References 14 publications

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“…The algorithm is extended in the paper [1] of the present authors. Theorems 5 and 6 of that paper state that given a function f : A → R of which the line sums in the directions of D are known, a function f * : A → R can be computed in linear time which has the same line sums as f in the directions of D. Hence, by using the theory in [7], the set of all functions F : A → R which have the same line sums as f in the directions of D are known.…”

Section: Results In the Two-dimensional Casementioning

confidence: 99%

“…The extension in [1] means that not only the f -values of the integer points outside C can be computed, but also the f -values of the points inside C which are not in T . Ceko, Petersen, Svalbe and Tijdeman [2] studied socalled boundary ghosts which are switching functions defined on c n points with 1 < c < 2 enclosing a set of almost 2 n points, for any n. The new algorithm enables one to compute the f -values of those interior points in linear time.…”

Section: Results In the Two-dimensional Casementioning

confidence: 99%

“…The border points are (P 0 , Q 0 ) = (8, 0), (P 1 , Q 1 ) = (5, 2), (P 2 , Q 2 ) = (1, 5), (P 3 , Q 3 ) = (0, 7). For every H the line through (P H−1 , Q H−1 ) and (P H , Q H ) is an edge of triangle V H , and intersects each other triangle V h , since the slopes increase with increasing h by the ordering in (1).…”

Section: Extension To Non-primitive Directionsmentioning

confidence: 99%

“…Therefore the intersection of C and the plane z = o − 1 is a shift of the union of all the switching domains of the directions (a h , b h , 0) ∈ D. Since the directions in D are primitive, the corresponding directions (a h , b h ) are primitive. Hence we can apply the algorithm in [1] to compute the (remaining) f * -values of all the points of the form (p, q, o − 1). In this process there is free choice of…”

Section: Determination Of the Set Of Solutionsmentioning

confidence: 99%

“…The coalescence of these results yielded an algorithm provided by the authors which allows for reconstruction of all solutions from the line sums in linear time in terms of the product of the number of directions and the size of the product [1]. It implies the reconstruction of any unknown function f from its line sums, provided f takes values in a unique factorization domain and the line sums are noise free.…”

Section: Introductionmentioning

confidence: 99%

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A linear time approach to three-dimensional reconstruction by discrete tomography

Ceko¹,

Pagani²,

Tijdeman³

2020

Preprint

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The goal of discrete tomography is to reconstruct an unknown function f via a given set of line sums. In addition to requiring accurate reconstructions, it is favourable to be able to perform the task in a timely manner. This is complicated by the presence of switching functions, or ghosts, which allow many solutions to exist in general. Previous work has shown that it is possible to determine all solutions in linear time (with respect to the number of directions and grid size) regardless of whether the solution is unique. In this work, we show that a similar linear algorithm exists in three dimensions. This is achieved by viewing the three-dimensional grid along each 2D coordinate plane, effectively solving the problem with a series of 2D linear algorithms. By that, it is possible to solve the problem of 3D discrete tomography in linear time.

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Section: Results In the Two-dimensional Casementioning

confidence: 99%

Section: Results In the Two-dimensional Casementioning

confidence: 99%

Section: Extension To Non-primitive Directionsmentioning

confidence: 99%

Section: Determination Of the Set Of Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A linear time approach to three-dimensional reconstruction by discrete tomography

Ceko¹,

Pagani²,

Tijdeman³

2020

Preprint

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Error Correction for Discrete Tomography

Ceko

Hajdu

Tijdeman

2023

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Discrete tomography focuses on the reconstruction of functions from their line sums in a finite number d of directions. In this paper we consider functions f : A → R where A is a finite subset of ℤ2 and R an integral domain. Several reconstruction methods have been introduced in the literature. Recently Ceko, Pagani and Tijdeman developed a fast method to reconstruct a function with the same line sums as f. Up to here we assumed that the line sums are exact. Some authors have developed methods to recover the function f under suitable conditions by using the redundancy of data. In this paper we investigate the case where a small number of line sums are incorrect as may happen when discrete tomography is applied for data storage or transmission. We show how less than d/2 errors can be corrected and that this bound is the best possible. Moreover, we prove that if it is known that the line sums in k given directions are correct, then the line sums in every other direction can be corrected provided that the number of wrong line sums in that direction is less than k/2.

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Error Correction for Discrete Tomography

Ceko¹,

Hajdu²,

Tijdeman³

2022

Preprint

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Discrete tomography focuses on the reconstruction of functions f : A → R from their line sums in a finite number d of directions, where A is a finite subset of Z 2 . Consequently, the techniques of discrete tomography often find application in areas where only a small number of projections are available. In 1978 M.B. Katz gave a necessary and sufficient condition for the uniqueness of the solution. Since then, several reconstruction methods have been introduced. Recently Pagani and Tijdeman developed a fast method to reconstruct f if it is uniquely determined. Subsequently Ceko, Pagani and Tijdeman extended the method to the reconstruction of a function with the same line sums of f in the general case. Up to here we assumed that the line sums are exact. In this paper we investigate the case where a small number of line sums are incorrect as may happen when discrete tomography is applied for data storage or transmission. We show how less than d/2 errors can be corrected and that this bound is the best possible.

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Algorithms for linear time reconstruction by discrete tomography II

Cited by 3 publications

References 14 publications

A linear time approach to three-dimensional reconstruction by discrete tomography

A linear time approach to three-dimensional reconstruction by discrete tomography

Error Correction for Discrete Tomography

Error Correction for Discrete Tomography

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