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Implementation of a computationally efficient least‐squares algorithm for highly under‐determined three‐dimensional diffuse optical tomography problems
Author(s) -
Yalavarthy Phaneendra K.,
Lynch Daniel R.,
Pogue Brian W.,
Dehghani Hamid,
Paulsen Keith D.
Publication year - 2008
Publication title -
medical physics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.473
H-Index - 180
eISSN - 2473-4209
pISSN - 0094-2405
DOI - 10.1118/1.2889778
Subject(s) - algorithm , diffuse optical imaging , tikhonov regularization , levenberg–marquardt algorithm , iterative reconstruction , computation , minification , mathematics , inverse problem , regularization (linguistics) , computer science , mathematical optimization , artificial neural network , mathematical analysis , artificial intelligence
Three‐dimensional (3D) diffuse optical tomography is known to be a nonlinear, ill‐posed and sometimes under‐determined problem, where regularization is added to the minimization to allow convergence to a unique solution. In this work, a generalized least‐squares (GLS) minimization method was implemented, which employs weight matrices for both data‐model misfit and optical properties to include their variances and covariances, using a computationally efficient scheme. This allows inversion of a matrix that is of a dimension dictated by the number of measurements, instead of by the number of imaging parameters. This increases the computation speed up to four times per iteration in most of the under‐determined 3D imaging problems. An analytic derivation, using the Sherman–Morrison–Woodbury identity, is shown for this efficient alternative form and it is proven to be equivalent, not only analytically, but also numerically. Equivalent alternative forms for other minimization methods, like Levenberg–Marquardt (LM) and Tikhonov, are also derived. Three‐dimensional reconstruction results indicate that the poor recovery of quantitatively accurate values in 3D optical images can also be a characteristic of the reconstruction algorithm, along with the target size. Interestingly, usage of GLS reconstruction methods reduces error in the periphery of the image, as expected, and improves by 20% the ability to quantify local interior regions in terms of the recovered optical contrast, as compared to LM methods. Characterization of detector photomultiplier tubes noise has enabled the use of the GLS method for reconstructing experimental data and showed a promise for better quantification of target in 3D optical imaging. Use of these new alternative forms becomes effective when the ratio of the number of imaging property parameters exceeds the number of measurements by a factor greater than 2.