Algebraic aspects of discrete tomography

Hajdu, Lajos; Tijdeman, R.

doi:10.1515/crll.2001.037

Cited by 58 publications

(81 citation statements)

References 4 publications

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“…There is a wide literature concerning (weakly) bad configurations, which highlights their central role in important issues such as ambiguity in the reconstruction problem, or, on the contrary, uniqueness (see, for instance, [24], [25], or [3] and the references given there). For instance, as mentioned above, a set is S-unique if and only if it has no S-bad configuration.…”

Section: Resultsmentioning

confidence: 99%

“…A related intriguing problem is to find non-additive sets of uniqueness (see [27]). A complete characterization of bad configurations (weakly or not weakly) has been obtained in [25] with a new algebraic approach employed then in several papers (see for instance [27][28][29][30]). S-bad configurations, with the extra condition of convexity, are known as S-polygons, and reveal to be useful both in geometric tomography and in discrete tomography (see for instance [31,32], and [33] for an algorithmic approach), as well as very interesting also from a purely geometric point of view (see for instance [34][35][36]).…”

Section: Resultsmentioning

confidence: 99%

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Ghosts in Discrete Tomography

et al. 2015

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Switching components, also named bad configurations, interchanges, and ghosts (according to different scenarios) play a key role in the study of ambiguous configurations, which often appear in Discrete Tomography and in several other areas of research. In this paper we give an upper bound for the minimal size bad configurations associated to a given set S of lattice directions. In the special but interesting case of four directions, we show that the general argument can be considerably improved, and we present an algebraic method which provides such an improvement. Moreover, it turns out that finding bad configurations is in fact equivalent to finding multiples of a suitable polynomial in two variables, having only coefficients from the set {−1, 0, 1}. The general problem of describing all polynomials having such multiples seems to be very hard [1]. However, in our particular case, it is hopeful to give some kind of solution. In the context of Digital Image Analysis, it represents an explicit method for the construction of ghosts, and consequently might be

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Ghosts in Discrete Tomography

et al. 2015

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“…We therefore consider a different approach. In 2001, Hajdu and Tijdeman [9] introduced an algebraic approach to discrete tomography, which shows how ghosts can be treated in terms of polynomials.…”

Section: Valid Directionsmentioning

confidence: 99%

Stability results for sets of uniqueness in binary tomography

Dulio

Pagani

2016

MATEC Web Conf.

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Abstract. The recovery of an unknown density function from the knowledge of its projections is the aim of tomography. In many cases, considering the problem from a discrete perspective is more convenient than employing a continuous approach: discrete tomography, and in particular binary tomography, is therefore exploited. One of the main goals of tomography is guaranteeing that the produced output coincides with the scanned object, namely, one wants to achieve uniqueness of reconstruction, even when only a few directions, from which projections are taken, are employed. Relying on a theoretical result stating that special sets of just four lattice directions are enough to uniquely reconstruct a binary grid, we prove that such sets are stable, in the sense that a small discrete perturbation of the components of the directions returns sets which again ensure uniqueness of reconstruction. Examples are provided.

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“…L.Hajdu and R.Tijdman [14,Lemma 3.1] showed that if a function h : A → Z has zero line sums along the lines taken in the directions in S, then F S (x, y) divides G h (x, y) over Z ([14, Lemma 3.1]). In other words, since a weakly bad configuration is algebrically determined by h, we can reformulate the result as follows:…”

Section: Weakly Bad Configurationsmentioning

confidence: 99%

On the Non-additive Sets of Uniqueness in a Finite Grid

Brunetti

Dulio

Peri

2013

Discrete Geometry for Computer Imagery

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Abstract. In Discrete Tomography there is a wide literature concerning (weakly) bad configurations. These occur in dealing with several questions concerning the important issues of uniqueness and additivity. Discrete lattice sets which are additive with respect to a given set S of lattice directions are uniquely determined by X-rays in the direction of S. These sets are characterized by the absence of weakly bad configurations for S. On the other side, if a set has a bad configuration with respect to S, then it is not uniquely determined by the X-rays in the directions of S, and consequently it is also non-additive. Between these two opposite situations there are also the non-additive sets of uniqueness, which deserve interest in Discrete Tomography, since their unique reconstruction cannot be derived via the additivity property. In this paper we wish to investigate possible interplays among such notions in a given lattice grid A, under X-rays taken in directions belonging to a set S of four lattice directions.

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Algebraic aspects of discrete tomography

Cited by 58 publications

References 4 publications

Ghosts in Discrete Tomography

Ghosts in Discrete Tomography

Stability results for sets of uniqueness in binary tomography

On the Non-additive Sets of Uniqueness in a Finite Grid

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