Maximum-likelihood transmission image reconstruction for overlapping transmission beams

Dong, Yu; Fessler, Jeffrey A.; Ficaro, Edward P.

doi:10.1109/42.896785

Cited by 54 publications

(47 citation statements)

References 15 publications

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“…Further examples can be found in in [16,17,22,24]). The interest in maximum likelihood estimation for tomographic image reconstruction subsequently led to many examples of EM, and more general MM algorithms, in image processing (e.g., [34,23,7,8,9,38,36]). MM has also received considerable attention in the signal processing literature, including [28,4,21,26,5].…”

Section: Introductionmentioning

confidence: 99%

An Expanded Theoretical Treatment of Iteration-Dependent Majorize-Minimize Algorithms

Jacobson

Fessler

2007

IEEE Trans. on Image Process.

112

111

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The Majorize-Minimize (MM) optimization technique has received considerable attention in signal and image processing applications, as well as in the statistics literature. At each iteration of an MM algorithm, one constructs a tangent majorant function that majorizes the given cost function and is equal to it at the current iterate. The next iterate is obtained by minimizing this tangent majorant function, resulting in a sequence of iterates that reduces the cost function monotonically. A wellknown special case of MM methods are Expectation-Maximization (EM) algorithms. In this paper, we expand on previous analyses of MM, due to [14,15], that allowed the tangent majorants to be constructed in iteration-dependent ways. Also, in [15], there was an error in one of the steps of the convergence proof that this paper overcomes.There are three main aspects in which our analysis builds upon previous work. Firstly, our treatment relaxes many assumptions related to the structure of the cost function, feasible set, and tangent majorants. For example, the cost function can be non-convex and the feasible set for the problem can be any convex set. Secondly, we propose convergence conditions, based on upper curvature bounds, that can be easier to verify than more standard continuity conditions. Furthermore, these conditions allow for considerable design freedom in the iteration-dependent behavior of the algorithm. Finally, we give an original characterization of the local region of convergence of MM algorithms based on connected (e.g., convex) tangent majorants. For such algorithms, cost function minimizers will locally attract the iterates over larger neighborhoods than is guaranteed typically with other methods. This expanded treatment widens the scope of MM algorithm designs that can be considered for signal and image processing applications, allows us to verify the convergent behavior of previously published algorithms, and gives a fuller understanding overall of how these algorithms behave.

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Section: Introductionmentioning

confidence: 99%

An Expanded Theoretical Treatment of Iteration-Dependent Majorize-Minimize Algorithms

Jacobson

Fessler

2007

IEEE Trans. on Image Process.

112

111

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“…A surrogate must satisfy the following conditions: By way of two intermediate surrogate functions, using methods similar to those in [4] and "tricks" given by De Pierro in [6], we obtain where l is the logged measured data, and…”

Section: Methodsmentioning

confidence: 99%

“…Emulating the derivation in Yu, Fessler, and Ficaro Yi (x) = L bije-L k ajkXk + ri, (4) j where bij is a weighting factor comprised of the blank scan intensity for beam let j on sinogram location i, ajk is the distance traversed by beamlet j within voxel k, and ri is a deterministic nonnegative constant that can represent scatter, crosstalk, and background noise.…”

Section: Methodsmentioning

confidence: 99%

An algorithm for modeling non-linear system effects in iterative CT reconstruction

Little

Rivière

2012

2012 IEEE Nuclear Science Symposium and Medical Imaging Conference Record (NSS/MIC)

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With continued development and clinical deploy ment of iterative algorithms for CT, there remain many openquestions about the best way to model various physical effects that degrade CT images, including geometric effects (finite sized detectors and sources), spectral effects due to the use of polychromatic spectra, scattered radiation, and noise. In this paper, we focus on the question of geometric modeling. Some recent investigations have concluded that geometric modeling may be unnecessary, with simple point source and point detector models yielding good results as measured near the isocenter of images. We further investigate this question in the present work, paying particular attention to peripheral image quality, where we find the effects of modeling to be more significant than at the isocenter. We also seek to evaluate the question of whether non linear modeling, which more closely captures the way CT data are acquired, offers any advantage over the traditional linear modeling used in iterative CT algorithms. In order to more accurately model the averaging in the transmitted intensity do main, we derive an update equation using separable paraboloidal surrogates (SPS) that is able to model a finite source, finite detectors, and data acquired over a small angular trajectory.We model these effects by incorporating many "beamlets" into the imaging model. In this work, we compare reconstructions made with additional modeling and without modeling in an SPS algorithm. We find that while modeling geometry makes a significant impact on the reconstructed image, a non-linear model may not be worth the additional computational cost.

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“…2) Updating the Attenuation Map: The update of the attenuation map is slightly more complicated. It involves recognizing that with considered fixed, (1) has the same form as the overlapping transmission source beams imaging equation considered by Yu et al [24]. To see this, define (21) and (22) Then (1) can be rewritten (23) which has the same form as the imaging equation studied by Yu et al In seeking to derive a monotonic penalized-likelihood algorithm, they derived a sequence of surrogate functions leading to a paraboloidal surrogates coordinate-ascent algorithm [24].…”

Section: Alternating Maximization Strategymentioning

confidence: 99%

Monotonic penalized-likelihood image reconstruction for X-ray fluorescence computed tomography

Rivière

Vargas

2006

IEEE Trans. Med. Imaging

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In this paper, we derive a monotonic penalized-likelihood algorithm for image reconstruction in X-ray fluorescence computed tomography (XFCT) when the attenuation maps at the energies of the fluorescence X-rays are unknown. In XFCT, a sample is irradiated with pencil beams of monochromatic synchrotron radiation that stimulate the emission of fluorescence X-rays from atoms of elements whose K- or L-edges lie below the energy of the stimulating beam. Scanning and rotating the object through the beam allows for acquisition of a tomographic dataset that can be used to reconstruct images of the distribution of the elements in question. XFCT is a stimulated emission tomography modality, and it is thus necessary to correct for attenuation of the incident and fluorescence photons. The attenuation map is, however, generally known only at the stimulating beam energy and not at the energies of the various fluorescence X-rays of interest. We have developed a penalized-likelihood image reconstruction strategy for this problem. The approach alternates between updating the distribution of a given element and updating the attenuation map for that element's fluorescence X-rays. The approach is guaranteed to increase the penalized likelihood at each iteration. Because the joint objective function is not necessarily concave, the approach may drive the solution to a local maximum. To encourage the algorithm to seek out a reasonable local maximum, we include in the objective function a prior that encourages a relationship, based on physical considerations, between the fluorescence attenuation map and the distribution of the element being reconstructed.

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Maximum-likelihood transmission image reconstruction for overlapping transmission beams

Cited by 54 publications

References 15 publications

An Expanded Theoretical Treatment of Iteration-Dependent Majorize-Minimize Algorithms

An Expanded Theoretical Treatment of Iteration-Dependent Majorize-Minimize Algorithms

An algorithm for modeling non-linear system effects in iterative CT reconstruction

Monotonic penalized-likelihood image reconstruction for X-ray fluorescence computed tomography

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