Least Squares Methods for Ill-Posed Problems with a Prescribed Bound

Miller, Keith

doi:10.1137/0501006

Cited by 467 publications

(242 citation statements)

References 8 publications

(9 reference statements)

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“…The procedure represented in graphical form is referred to as the L-curve criterion (Hansen, 1994(Hansen, , 1998. The use of such a criterion in connection with ill-posed Least Squares problems goes back to Miller (1970), and Lawson and Hanson (1974). The L-curve criterion is clearly illustrated and extensively applied to the analysis of discrete ill-posed problems by Hansen (1992).…”

Section: The Unfolding Methodsmentioning

confidence: 99%

Application of the Monte Carlo method to analyze materials used in flat panel detectors to obtain X-ray spectra

Gallardo

Pozuelo

Querol

et al. 2014

SNA + MC 2013 - Joint International Conference on Supercomputing in Nuclear Applications + Monte Carlo

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Abstract. An accurate knowledge of the photon spectra emitted by X-ray tubes in radiodiagnostic is essential to better estimate the imparted dose to patients and to improve the quality image obtained with these devices. In this work, it is proposed the use of a flat panel detector together with a PMMA wedge to estimate the actual X-ray spectrum using the Monte Carlo method and unfolding techniques. The MCNP5 code has been used to model different flat panels (based on indirect and direct methods to produce charge carriers from absorbed X-rays) and to obtain the dose curves and system response functions. Most of the actual flat panel devices use scintillator materials that present K-edge discontinuities in the mass energy-absorption coefficient, which strongly affect the response matrix. In this paper, the applicability of different flat panels for reconstructing X-ray spectra is studied. The effect of the mass energy-absorption coefficient of the scintillator material has been studied on the response matrix and consequently, in the reconstructed spectra. Different unfolding methods are tested to reconstruct the actual X-ray spectrum knowing the dose curve and the response function. It has been concluded that the regularization method Modified Truncated Singular Value Decomposition (MTSVD) is appropriate to unfold Xray spectra in all the scintillators studied.

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Section: The Unfolding Methodsmentioning

confidence: 99%

Application of the Monte Carlo method to analyze materials used in flat panel detectors to obtain X-ray spectra

Gallardo

Pozuelo

Querol

et al. 2014

SNA + MC 2013 - Joint International Conference on Supercomputing in Nuclear Applications + Monte Carlo

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“…Therefore, the retained α * 's are chosen by dichotomy to verify V(α * ) = 0.1. It can be noted that the proposed procedure for choosing α differs from the so-called Miller criterion (Miller, 1970). The Miller criterion sets α as α * = 2 /E 2 , where stands for the upper bounds of u − Wc 2 and E for the upper bound of c 2 .…”

Section: Determination Of the Regularization Parametermentioning

confidence: 99%

A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part I. Theoretical material

et al. 2009

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Abstract. The present paper addresses the inverse problem of time-resolved (fluorescence) diffuse optical tomography from temporal moments of the measurements.A methodology that enables to provide fairly comparable reconstructions is presented.The proposed reconstruction methodology is applied to infinite medium synthetic phantoms in the transmission geometry. Reconstructions are performed for moment orders increasing from 0 to 3. The reconstruction quality is shown to be increasing when higher moment orders are added. However, the value of the highest useful moments order strongly depends on the number of photons that can be acquired. In particular, it can be considered that the benefit of using higher order moments vanishes when fewer than 10 8 photons are detected. The evolution of the reconstruction quality with respect to the optical properties of the medium and fluorescence lifetime is also shown.

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“…Tikhonov regularization, which was first proposed and studied extensively in the1960's and 1970's [63,78,94,95,96], is perhaps the most well known approach to regularizing ill-posed problems. L is typically chosen to be the identity matrix, or a discrete approximation to a derivative operator, such as the Laplacian.…”

Section: Regularizationmentioning

confidence: 99%

Iterative Methods for Image Restoration

Berisha

Nagy

2014

Academic Press Library in Signal Processing: Volume 4 - Image, Video Processing and Analysis, Hardware, Audio, Acoustic and Spe

104

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Although image restoration methods based on spectral filtering techniques are very efficient, they can be applied only to problems with fairly simple spatially invariant blurring operators. Iterative methods, however, are much more flexible; they can be very efficient for spatially invariant as well as spatially variant blurs, they can incorporate a variety of regularization techniques and boundary conditions, and they can more easily incorporate additional constraints, such as nonnegativity. This chapter describes a variety of iterative methods used in image restoration, with a particular emphasis on efficiency, convergence behavior, and implementation. Discussion of MATLAB software implementing the methods is also provided.

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Least Squares Methods for Ill-Posed Problems with a Prescribed Bound

Cited by 467 publications

References 8 publications

Application of the Monte Carlo method to analyze materials used in flat panel detectors to obtain X-ray spectra

Application of the Monte Carlo method to analyze materials used in flat panel detectors to obtain X-ray spectra

A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part I. Theoretical material

Iterative Methods for Image Restoration

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