The use of the least-squares minimal residual algorithm for fast and regularized attenuation compensation in SPECT

La, Valérie; Grangeat, Pierre; Iovleff, Serge; Mallon, Alexis; Sire, Pascal

doi:10.1109/nssmic.1996.587944

Cited by 3 publications

(11 citation statements)

References 6 publications

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“…In SPECT attenuation correction, the minimal residual algorithm (Axelsson 1980) is used for solving the following inverse problem (La et al 1996, La andGrangeat 1998):…”

Section: The Minimal Residual Algorithmmentioning

confidence: 99%

“…It has been shown that the operators used in the iterative filtered backprojections (Walters et al 1981) are good candidates for the R * operator when using the minimal residual algorithm (La et al 1996, La andGrangeat 1998), but they do not ensure regularization. A spatially adaptative filtering technique has already been proposed for the regularization of MR (La et al 1996, La andGrangeat 1998) and applied to brain (Almeida et al 1997) and cardiac (La and Grangeat 1998) SPECT reconstruction: each estimated activity reconstruction is convolved with a 3D Laplacian kernel and then multiplied by a regularization parameter λ, which differs from one region to another, thus requiring a preliminary segmentation of the images.…”

Section: Regularization Of the Estimation Stepsmentioning

confidence: 99%

“…the attenuation is modelled in the projector and not in the backprojector). Unlike the CG algorithm, the MR algorithm does not necessarily converge to a global minimum solution (Axelsson 1980, Greenbaum 1997), yet it has been shown to give similar results to those obtained with CG for attenuation correction in SPECT, using four times fewer iterations (La et al 1996, La andGrangeat 1998). MR presents about the same sensitivity to noise as CG (e.g.…”

Section: Introductionmentioning

confidence: 97%

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A spline-regularized minimal residual algorithm for iterative attenuation correction in SPECT

et al. 1999

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In SPECT, regularization is necessary to avoid divergence of the iterative algorithms used for non uniform attenuation compensation. In this paper, we propose a spline-based regularization method for the minimal residual algorithm. First, the acquisition noise is filtered using a statistical model involving spline smoothing so that the filtered projections belong to a Sobolev space with specific continuity and derivability properties. Then, during the iterative reconstruction procedure, the continuity of the inverse Radon transform between Sobolev spaces is used to design a spline-regularized filtered backprojection method, by which the known regularity properties of the projections determine those of the corresponding reconstructed slices. This ensures that the activity distributions estimated at each iteration present regularity properties, which avoids computational noise amplification, thus stabilizing the iterative process. Analytical and Monte Carlo simulations are used to show that the proposed spline-regularized minimal residual algorithm converges to a satisfactory stable solution in terms of restored activity and homogeneity, using at most 25 iterations, whereas the non regularized version of the algorithm diverges. Choosing the number of iterations is therefore no longer a critical issue for this reconstruction procedure.

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“…In SPECT attenuation correction, the minimal residual algorithm (Axelsson 1980) is used for solving the following inverse problem (La et al 1996, La andGrangeat 1998):…”

Section: The Minimal Residual Algorithmmentioning

confidence: 99%

Section: Regularization Of the Estimation Stepsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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A spline-regularized minimal residual algorithm for iterative attenuation correction in SPECT

et al. 1999

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“…The closer P X µ is to the identity matrix within a normalization factor, the faster the convergence rate is. Possible expressions for P are discretized analytical inversion formulae for unattenuated projection data (La et al 1996). System (3) can also be solved using CG.…”

Section: Brief Review Of Algebraic Attenuation Correction Algorithmsmentioning

confidence: 99%

“…The proposed reconstruction method is an algebraic method based on the application of the least-squares (LS) minimal residual (MR) minimization algorithm to a preconditioned system. Unlike conjugate gradient (CG) based reconstruction techniques, the proposed MR-based algorithm solves directly a quasisymmetric linear system (Axelsson 1980, La 1997, La et al 1996. By avoiding the normal equations, it improves the convergence rate.…”

Section: Introductionmentioning

confidence: 99%

Minimal residual cone-beam reconstruction with attenuation correction in SPECT

La¹,

Grangeat²

1998

Phys. Med. Biol.

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This paper presents an iterative method based on the minimal residual algorithm for tomographic attenuation compensated reconstruction from attenuated cone-beam projections given the attenuation distribution. Unlike conjugate-gradient based reconstruction techniques, the proposed minimal residual based algorithm solves directly a quasisymmetric linear system, which is a preconditioned system. Thus it avoids the use of normal equations, which improves the convergence rate. Two main contributions are introduced. First, a regularization method is derived for quasisymmetric problems, based on a Tikhonov-Phillips regularization applied to the factorization of the symmetric part of the system matrix. This regularization is made spatially adaptive to avoid smoothing the region of interest. Second, our existing reconstruction algorithm for attenuation correction in parallel-beam geometry is extended to cone-beam geometry. A circular orbit is considered. Two preconditioning operators are proposed: the first one is Grangeat's inversion formula and the second one is Feldkamp's inversion formula. Experimental results obtained on simulated data are presented and the shadow zone effect on attenuated data is illustrated.

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Evaluation of two conjugate gradient based algorithms for quantitation in cardiac SPECT imaging

Sire

Grangeat²,

Iovleff³

et al.

1996 IEEE Nuclear Science Symposium. Conference Record

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The use of the least-squares minimal residual algorithm for fast and regularized attenuation compensation in SPECT

Cited by 3 publications

References 6 publications

A spline-regularized minimal residual algorithm for iterative attenuation correction in SPECT

A spline-regularized minimal residual algorithm for iterative attenuation correction in SPECT

Minimal residual cone-beam reconstruction with attenuation correction in SPECT

Evaluation of two conjugate gradient based algorithms for quantitation in cardiac SPECT imaging

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