Variable projection for nonlinear least squares problems

O’Leary, Dianne P.; Rust, Bert W.

doi:10.1007/s10589-012-9492-9

Cited by 146 publications

(117 citation statements)

References 12 publications

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“…The special structure of our cost function allows us to use the variable projection optimization algorithm by O’Leary and Rust (2013) which separates the optimization problem into linear and nonlinear parts to obtain more accurate parameter estimation. Rearranging of S υ, i in the form of

S_{υ, i} = \sum_{j = 1}^{2} c_{υ, j} Φ_{j} false(θ_{j}, k_{j} false),

we obtain the least squares problem of the form

\sum_{i = 1}^{N} {true(M_{υ, i} - \sum_{j = 1}^{2} c_{υ, j} normalΦ false(θ_{j}, k_{j} false) true)}^{2}

where c υ,1 = So υ × f υ , c υ,2 = So υ × (1 − f υ ) are the linear parameters and θ j , k j are the nonlinear parameters.…”

Section: Theorymentioning

confidence: 99%

Spatially-constrained probability distribution model of incoherent motion (SPIM) for abdominal diffusion-weighted MRI

Kurugol

Freiman

Afacan

et al. 2016

Medical Image Analysis

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Quantitative diffusion-weighted MR imaging (DW-MRI) of the body enables characterization of the tissue microenvironment by measuring variations in the mobility of water molecules. The diffusion signal decay model parameters are increasingly used to evaluate various diseases of abdominal organs such as the liver and spleen. However, previous signal decay models (i.e., mono-exponential, bi-exponential intra-voxel incoherent motion (IVIM) and stretched exponential models) only provide insight into the average of the distribution of the signal decay rather than explicitly describe the entire range of diffusion scales. In this work, we propose a probability distribution model of incoherent motion that uses a mixture of Gamma distributions to fully characterize the multi-scale nature of diffusion within a voxel. Further, we improve the robustness of the distribution parameter estimates by integrating spatial homogeneity prior into the probability distribution model of incoherent motion (SPIM) and by using the fusion bootstrap solver (FBM) to estimate the model parameters. We evaluated the improvement in quantitative DW-MRI analysis achieved with the SPIM model in terms of accuracy, precision and reproducibility of parameter estimation in both simulated data and in 68 abdominal in-vivo DW-MRIs. Our results show that the SPIM model not only substantially reduced parameter estimation errors by up to 26%; it also significantly improved the robustness of the parameter estimates (paired Students t-test, p < 0.0001) by reducing the coefficient of variation (CV) of estimated parameters compared to those produced by previous models. In addition, the SPIM model improves the parameter estimates reproducibility for both intra- (up to 47%) and inter-session (up to 30%) estimates compared to those generated by previous models. Thus, the SPIM model has the potential to improve accuracy, precision and robustness of quantitative abdominal DW-MRI analysis for clinical applications.

show abstract

S_{υ, i} = \sum_{j = 1}^{2} c_{υ, j} Φ_{j} false(θ_{j}, k_{j} false),

we obtain the least squares problem of the form

\sum_{i = 1}^{N} {true(M_{υ, i} - \sum_{j = 1}^{2} c_{υ, j} normalΦ false(θ_{j}, k_{j} false) true)}^{2}

where c υ,1 = So υ × f υ , c υ,2 = So υ × (1 − f υ ) are the linear parameters and θ j , k j are the nonlinear parameters.…”

Section: Theorymentioning

confidence: 99%

Spatially-constrained probability distribution model of incoherent motion (SPIM) for abdominal diffusion-weighted MRI

Kurugol

Freiman

Afacan

et al. 2016

Medical Image Analysis

View full text Add to dashboard Cite

show abstract

“…Hence, the optimization algorithm may converge to a local minimum. However, the literature [32] suggests that the VP algorithm is more likely to converge to the global minimum than other algorithms such as block-coordinate descent algorithms. Our computer-simulation studies revealed that the VP algorithm consistently converged to accurate solutions, suggesting that utilizing proper regularization methods and good initial guesses will improve the ability of the algorithm to avoid local minima.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…Note that Line-7 can be computed much more efficiently than the problem arg min θ ≥0 φ ( θ , h k ). In addition, VP algorithms have been reported to possess faster convergence rates and may be less likely to be become trapped by local minima as compared to block-coordinate descent algorithms [32]. …”

Section: Pact Image Reconstruction Without Accurate Knowledge Ofmentioning

confidence: 99%

A Constrained Variable Projection Reconstruction Method for Photoacoustic Computed Tomography Without Accurate Knowledge of Transducer Responses

Sheng

Wang

Matthews

et al. 2015

IEEE Trans. Med. Imaging

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Photoacoustic computed tomography (PACT) is an emerging computed imaging modality that exploits optical contrast and ultrasonic detection principles to form images of the absorbed optical energy density within tissue. When the imaging system employs conventional piezoelectric ultrasonic transducers, the ideal photoacoustic (PA) signals are degraded by the transducers’ acousto-electric impulse responses (EIRs) during the measurement process. If unaccounted for, this can degrade the accuracy of the reconstructed image. In principle, the effect of the EIRs on the measured PA signals can be ameliorated via deconvolution; images can be reconstructed subsequently by application of a reconstruction method that assumes an idealized EIR. Alternatively, the effect of the EIR can be incorporated into an imaging model and implicitly compensated for during reconstruction. In either case, the efficacy of the correction can be limited by errors in the assumed EIRs. In this work, a joint optimization approach to PACT image reconstruction is proposed for mitigating errors in reconstructed images that are caused by use of an inaccurate EIR. The method exploits the bi-linear nature of the imaging model and seeks to refine the measured EIR during the process of reconstructing the sought-after absorbed optical energy density. Computer-simulation and experimental studies are conducted to investigate the numerical properties of the method and demonstrate its value for mitigating image distortions and enhancing the visibility of fine structures.

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“…The minimization problem is implemented in Matlab making use of the built-in trust region minimization algorithm and the VARPRO implementation given by [18]. The Bloch equation simulator is implemented in C [19] and was adapted to include slice profile response, off-resonance effects and

B_{1}^{+}

inhomogeneities.…”

Section: Methodsmentioning

confidence: 99%

Fast quantitative MRI as a nonlinear tomography problem

Sbrizzi

Heide

Cloos

et al. 2018

Magnetic Resonance Imaging

101

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Quantitative Magnetic Resonance Imaging (MRI) is based on a two-steps approach: estimation of the magnetic moments distribution inside the body, followed by a voxel-by-voxel quantification of the human tissue properties. This splitting simplifies the computations but poses several constraints on the measurement process, limiting its efficiency. Here, we perform quantitative MRI as a one step process; signal localization and parameter quantification are simultaneously obtained by the solution of a large scale nonlinear inversion problem based on first-principles. As a consequence, the constraints on the measurement process can be relaxed and acquisition schemes that are time efficient and widely available in clinical MRI scanners can be employed. We show that the nonlinear tomography approach is applicable to MRI and returns human tissue maps from very short experiments.

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Variable projection for nonlinear least squares problems

Cited by 146 publications

References 12 publications

Spatially-constrained probability distribution model of incoherent motion (SPIM) for abdominal diffusion-weighted MRI

Spatially-constrained probability distribution model of incoherent motion (SPIM) for abdominal diffusion-weighted MRI

A Constrained Variable Projection Reconstruction Method for Photoacoustic Computed Tomography Without Accurate Knowledge of Transducer Responses

Fast quantitative MRI as a nonlinear tomography problem

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