A rounding theorem for unique binary tomographic reconstruction

Dulio, Paolo; Pagani, Silvia Maria Carla

doi:10.1016/j.dam.2019.05.005

Cited by 11 publications

(6 citation statements)

References 36 publications

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“…Their result implies that arbitrary function values can be given to a certain set of points and that thereafter the function values of the other points of A are uniquely determined by the line sums. This was made explicit by Dulio and Pagani [9] and serves as a building block in this paper.…”

Section: Introductionmentioning

confidence: 92%

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Algorithms for linear time reconstruction by discrete tomography II

Ceko,

Pagani,

Tijdeman

2020

Preprint

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The reconstruction of an unknown function f from its line sums is the aim of discrete tomography. However, two main aspects prevent reconstruction from being an easy task. In general, many solutions are allowed due to the presence of the switching functions. Even when uniqueness conditions are available, results about the NP-hardness of reconstruction algorithms make their implementation inefficient when the values of f are in certain sets. We show that this is not the case when f takes values in a unique factorization domain, such as R or Z. We present a linear time reconstruction algorithm (in the number of directions and in the size of the grid), which outputs the original function values for all points outside of the switching domains. Freely chosen values are assigned to the other points, namely, those with ambiguities. Examples are provided.

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Section: Introductionmentioning

confidence: 92%

“…Since M = 23, N = 17, there are six pixels where the choice is free, e.g. the pixels (0, 9), (0, 10),(1,9),(1,10),(2,9),(2,10). Any other 3 by 2 block of dark grey pixels can be chosen instead.…”

mentioning

confidence: 99%

Algorithms for linear time reconstruction by discrete tomography II

Ceko,

Pagani,

Tijdeman

2020

Preprint

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“…There are methods, such as the Hough transform [11,12], with voting bins, which may be more efficient. Another family of methods relies on Dirac lattices and lines of rational slope [13,14] which can be also of interest, for example, for reconstructions with just a few projections [15].…”

Section: Introduction 1radon Transform Use and Algorithms For Its Com...mentioning

confidence: 99%

Central and Periodic Multi-Scale Discrete Radon Transforms

et al. 2021

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The multi-scale discrete Radon transform (DRT) calculates, with linearithmic complexity, the summation of pixels, through a set of discrete lines, covering all possible slopes and intercepts in an image, exclusively with integer arithmetic operations. An inversion algorithm exists and is exact and fast, in spite of being iterative. In this work, the DRT forward and backward pair is evolved to propose two faster algorithms: central DRT, which computes only the central portion of intercepts; and periodic DRT, which computes the line integrals on the periodic extension of the input. Both have an output of size N×4N, instead of 3N×4N, as in the original algorithm. Periodic DRT is proven to have a fast inversion, whereas central DRT does not. An interesting application of periodic DRT is its use as building a block of discrete curvelet transform. Central DRT can provide almost a 2× speedup over conventional DRT, probably becoming the faster Radon transform algorithm available, at the cost of ignoring 15% of the summations in the corners.

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“…Furthermore, they showed that arbitrary values can be assigned to a subset of the switching domain, which allows unique values to be applied to all other points. This fact was exploited by Dulio and Pagani in [6], wherein a rounding theorem was proven which allowed exact and unique binary tomographic reconstructions from the minimum Euclidean norm solution. It also motivated the construction of boundary ghosts by Ceko, Petersen, Svalbe and Tijdeman [2].…”

Section: Introductionmentioning

confidence: 99%

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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A rounding theorem for unique binary tomographic reconstruction

Cited by 11 publications

References 36 publications

Algorithms for linear time reconstruction by discrete tomography II

Algorithms for linear time reconstruction by discrete tomography II

Central and Periodic Multi-Scale Discrete Radon Transforms

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

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