Old and new parameter choice rules for discrete ill-posed problems

Reichel, Lothar; Rodriguez, Giuseppe

doi:10.1007/s11075-012-9612-8

Cited by 177 publications

(117 citation statements)

References 40 publications

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“…Thus, the least-squares estimation becomes highly unreliable and is unable to obtain accurate estimations of the parameters. In order to overcome the ill-posedness of the problem, truncated singular value decomposition (TSVD) [26][27][28] is adopted. TSVD truncates the small singular values that greatly enlarge the variances to improve the least-squares estimation.…”

Section: Estimation Of Pure Volume Coherence From the Complex Interfementioning

confidence: 99%

A TSVD-Based Method for Forest Height Inversion from Single-Baseline PolInSAR Data

Lin

Zhu

et al. 2017

Applied Sciences

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Abstract:The random volume over ground (RVoG) model associates vegetation vertical structure parameters with multiple complex interferometric coherence observables. In this paper, on the basis of the RVoG model, a truncated singular value decomposition (TSVD)-based method is proposed for forest height inversion from single-baseline polarimetric interferometric synthetic aperture radar (PolInSAR) data. In addition, in order to improve the applicability of TSVD for this issue, a new truncation method is proposed for TSVD. Differing from the traditional three-stage method, the TSVD-based inversion method estimates the pure volume coherence directly from the complex interferometric coherence, and estimates the forest height from the estimated pure volume coherence with a least-squares method. As a result, the TSVD-based method can adjust the contributions of the polarizations in the estimation of the model parameters and avoid the null ground-to-volume ratio assumption. The simulated experiments undertaken in this study confirmed that the TSVD-based method performs better than the three-stage method in forest height inversion. The TSVD-based method was also applied to E-SAR P-band data acquired over the Krycklan Catchment, Sweden, which is covered with mixed pine forest. The results showed that the TSVD-based method improves the root-mean-square error by 48.6% when compared to the three-stage method, which further validates the performance of the TSVD-based method.

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Section: Estimation Of Pure Volume Coherence From the Complex Interfementioning

confidence: 99%

A TSVD-Based Method for Forest Height Inversion from Single-Baseline PolInSAR Data

Lin

Zhu

et al. 2017

Applied Sciences

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“…A variety of approaches can be used to determine a suitable value of the regularization parameter µ, including the L-curve, generalized cross validation, and the discrepancy principle; see, e.g., [4,8,11,16,23] for discussions and references. We will use the discrepancy principle in the computed examples of Section 4.…”

Section: 3mentioning

confidence: 99%

“…This typically requires the computation of the solution of (1.4) for several values of µ > 0. For instance, when µ is determined by the discrepancy principle, the L-curve criterion, or the generalized cross validation method, the minimization problem (1.4) has to be solved for several values of µ > 0 to be able to determine a suitable µ-value; see [4,8,11,16,23] for discussions on these and other methods for determining a suitable value of µ. The repeated solution of (1.4) for different µ-values can be carried out fairly inexpensively when A and B are of small to medium size by first computing the generalized singular value decomposition (GSVD) of the matrix pair {A, B}.…”

mentioning

confidence: 99%

A Golub–Kahan-Type Reduction Method for Matrix Pairs

Hochstenbach

Reichel

2015

J Sci Comput

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“…For an elaborate comparison of the L-curve method with other ways of fixing model parameters in a priori under-determined inverse problems, see [35]. To the best of our knowledge the conjugate gradient minimisation of the cost function, associated with (3), together with the subsequent, automated determination of parameter λ using the discrete L-curve has not been used before for tomographic reconstruction.…”

Section: Introductionmentioning

confidence: 99%

TV-based conjugate gradient method and discrete L-curve for few-view CT reconstruction of X-ray in vivo data

Yang¹,

Hofmann²,

Dapp³

et al. 2015

Opt. Express

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High-resolution, three-dimensional (3D) imaging of soft tissues requires the solution of two inverse problems: phase retrieval and the reconstruction of the 3D image from a tomographic stack of two-dimensional (2D) projections. The number of projections per stack should be small to accommodate fast tomography of rapid processes and to constrain X-ray radiation dose to optimal levels to either increase the duration of in vivo time-lapse series at a given goal for spatial resolution and/or the conservation of structure under X-ray irradiation. In pursuing the 3D reconstruction problem in the sense of compressive sampling theory, we propose to reduce the number of projections by applying an advanced algebraic technique subject to the minimisation of the total variation (TV) in the reconstructed slice. This problem is formulated in a Lagrangian multiplier fashion with the parameter value determined by appealing to a discrete L-curve in conjunction with a conjugate gradient method. The usefulness of this reconstruction modality is demonstrated for simulated and in vivo data, the latter acquired in parallel-beam imaging experiments using synchrotron radiation.

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Old and new parameter choice rules for discrete ill-posed problems

Cited by 177 publications

References 40 publications

A TSVD-Based Method for Forest Height Inversion from Single-Baseline PolInSAR Data

A TSVD-Based Method for Forest Height Inversion from Single-Baseline PolInSAR Data

A Golub–Kahan-Type Reduction Method for Matrix Pairs

TV-based conjugate gradient method and discrete L-curve for few-view CT reconstruction of X-ray in vivo data

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