Discrete models and singular-value decompositions of single-slice imagers with orthogonal detectors

Varatharajah, Paramanathan; Tankersley, Barbara; Aarsvold, John N.

doi:10.1109/nssmic.1998.774371

“…1 we can see that the some singular vectors occur in pairs with the same singular value (doublets), while others occur by themselves (singlets). The existence of this type of pattern would have been predicted from an analysis that made use of the full symmetry group of the system, which would include reflections as well as rotations [11–13]. By replacing the cyclic group of rotations used in this work with the dihedral group of rotations and reflections, we can show that the spaces spanned by the singular vectors that share a singular value are predicted to be either one-dimensional or two-dimensional.…”

Section: Examplesmentioning

confidence: 94%

SVD for imaging systems with discrete rotational symmetry

Clarkson

¹

,

Palit

²

,

Kupinski

³

2010

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The singular value decomposition (SVD) of an imaging system is a computationally intensive calculation for tomographic imaging systems due to the large dimensionality of the system matrix. The computation often involves memory and storage requirements beyond those available to most end users. We have developed a method that reduces the dimension of the SVD problem towards the goal of making the calculation tractable for a standard desktop computer. In the presence of discrete rotational symmetry we show that the dimension of the SVD computation can be reduced by a factor equal to the number of collection angles for the tomographic system. In this paper we present the mathematical theory for our method, validate that our method produces the same results as standard SVD analysis, and finally apply our technique to the sensitivity matrix for a clinical CT system. The ability to compute the full singular value spectra and singular vectors could augment future work in system characterization, image-quality assessment and reconstruction techniques for tomographic imaging systems.

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“…1 we can see that the some singular vectors occur in pairs with the same singular value (doublets), while others occur by themselves (singlets). The existence of this type of pattern would have been predicted from an analysis that made use of the full symmetry group of the system, which would include reflections as well as rotations [11][12][13]. By replacing the cyclic group of rotations used in this work with the dihedral group of rotations and reflections, we can show that the spaces spanned by the singular vectors that share a singular value are predicted to be either one-dimensional or two-dimensional.…”

Section: Examplesmentioning

confidence: 98%

SVD for Imaging Systems with Discrete Rotational Symmetry

Clarkson

¹

,

Palit

²

,

Kupinski

³

2011

Frontiers in Optics 2011/Laser Science XXVII

0

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The singular value decomposition (SVD) of an imaging system is a computationally intensive calculation for tomographic imaging systems due to the large dimensionality of the system matrix. The computation often involves memory and storage requirements beyond those available to most end users. We have developed a method that reduces the dimension of the SVD problem towards the goal of making the calculation tractable for a standard desktop computer. In the presence of discrete rotational symmetry we show that the dimension of the SVD computation can be reduced by a factor equal to the number of collection angles for the tomographic system. In this paper we present the mathematical theory for our method, validate that our method produces the same results as standard SVD analysis, and finally apply our technique to the sensitivity matrix for a clinical CT system. The ability to compute the full singular value spectra and singular vectors could augment future work in system characterization, image-quality assessment and reconstruction techniques for tomographic imaging systems.

show abstract