Binary vectors partially determined by linear equation systems

Aharoni, Ron; Herman, G. T.; Kuba, Attila

doi:10.1016/s0012-365x(96)00068-4

Cited by 33 publications

(48 citation statements)

References 11 publications

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“…The task of finding the solution to (4) can be expressed in terms of many optimization problems such as Linear Programming (LP) [1,17], Best Inner Fit (BIF) [23,60,61], Non-Linear Integer Programming (NLIP), or Non-Linear Relaxed Programming (NLRP) [55].…”

Section: Methodsmentioning

confidence: 99%

“…This is a scalable function, so δ can be readily merged with the regularization parameter. Its convexity depends on parameter p, but usually p ∈ [1,2]. For p = 2, the function simplifies to the Gaussian one which tends to over-smooth edges of sharp objects, but for p = 1 we get the Besag (Laplacian) function [4] that tends to enforce sparse objects.…”

Section: Methodsmentioning

confidence: 99%

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On image reconstruction algorithms for binary electromagnetic geotomography

Zdunek

2008

Theoretical Computer Science

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a b s t r a c tIn this paper we discuss the selected image reconstruction methods of binary tomography in the context of their application to geophysical imaging. We restrict our considerations to a discrete version of high-frequency electromagnetic geotomography, which we label as Binary Electromagnetic Geotomography (BEG). Basically, such an imaging technique may be applied to detect subsurface anomalies (air-filled voids) whose attenuation coefficient is very low (nearly zero-value) and considerably different from that for the background.The assumption for a binary representation of the image to be reconstructed substantially relaxes image reconstruction problems related to ill-posedness that comes from an intrinsic limitation of an angular range of projections. We test two algorithms for binary tomography, where the penalty term is based on the Markov Random Field (MRF) model. The mean-field reference distribution and mean-field annealing are applied to estimate the global maximizer of the Gibbs-Boltzmann distribution associated with the objective function. We also apply the projected gradient algorithm that uses a binary steering. Very efficient implementations of the algorithms are also given. The numerical results are presented for noise-free, noisy, and real data.

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Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

On image reconstruction algorithms for binary electromagnetic geotomography

Zdunek

2008

Theoretical Computer Science

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“…numbers, it is well known (see [6,Section 16.4]) that whenever we are given any ÿnite number of views of f (a view is deÿned as the integrals of f along all lines parallel to a given direction), there will be another image that di ers from f in an arbitrarily large manner in a region around any speciÿed point in the domain, which will have the same views as f. (Note that in spite of this seemingly devastating result, continuous tomography is extremely useful in practice, as is evidenced by the reliance of modern radiology on computed tomography scanners.) The situation in binary tomography (as presented in this paper) is better: there are well-understood conditions under which the values of a binary image f are uniquely determined from its (noiseless) projections data on an M -grid (whose deÿnition was given in Section 1.2) at certain (well-speciÿed) points of its domain or even over all of its domain [1]. However, such uniqueness cannot always be guaranteed, even if we combine the information in the projection data with the demand that H (f) be minimized (see Section 1.2), as is illustrated in Fig.…”

Section: Methodsmentioning

confidence: 99%

Automated estimation of the parameters of Gibbs priors to be used in binary tomography

Liao

Herman

2004

Discrete Applied Mathematics

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Image modeling using Gibbs priors was previously shown, based on experiments, to be e ective in image reconstruction problems. This motivated us to evaluate three methods for estimating the priors. Two of them accurately recover the parameters of the priors; however, all of them are useful for binary tomography. This is demonstrated by two sets of experiments: in one the images are from a Gibbs distribution and in the other they are typical cardiac phantom images. ?

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“…Therefore, in order to deal with this problem one can transform (10) by replacing the integrality constraint with an interval constraint, that is, z ∈ [0, 1] m×n . This approach has been introduced by Aharoni et al in [1] and then studied by several authors (see, for instance [12,21,22]). To our aim, we can follow the method described in [7].…”

Section: Non-additive Set Of Uniqueness;mentioning

confidence: 99%

“…The inversion of DRT aims to deduce the local atomic structure from the collected counting data. The original motivation came from High-Resolution Transmission Electron Microscopy (HRTEM) which is able to obtain images with atomic resolution and provides quantitative information on the number of atoms that lie in single atomic columns in crystals choosing main X-ray directions such as (0, 1), (1,0), (1,1), (1,2), . .…”

Section: Introductionmentioning

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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Binary vectors partially determined by linear equation systems

Cited by 33 publications

References 11 publications

On image reconstruction algorithms for binary electromagnetic geotomography

On image reconstruction algorithms for binary electromagnetic geotomography

Automated estimation of the parameters of Gibbs priors to be used in binary tomography

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

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