Implementation of a computationally efficient least‐squares algorithm for highly under‐determined three‐dimensional diffuse optical tomography problems

Abstract

Three-dimensional ͑3D͒ 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 ͑GLS͒ 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 ͑LM͒ 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.

Parametric Diffuse Optical Imaging in Reflectance Geometry

“…The Woodbury formula can be used to find the matrix inverse to reduce calculation time because in an underdetermined problem, as is often the case in DOI, the Jacobian matrix has more columns than rows [34]. …”

Image-guided near infrared spectroscopy using boundary element method: phantom validation

“…In 3-D, Dehghani et al [20] used exact knowledge of boundaries and assumption of homogeneous piece-wise regions in a cylindrical shape in 3–D and showed nearly 100% recovery in experiments. Further work by Yalavarthy et al incorporated structural information in a generalized framework for optical imaging [21,22]. Azar et al [23] have demonstrated co-registration of non-concurrent NIR and MRI images and segmentation of MR images to enable priors to be applied in 3-D and have applied this to phantom models and three clinical cases.…”

Optical Imaging in Cancer Research: Basic Principles, Tumor Detection, and Therapeutic Monitoring

“…To accomplish this goal, the optimization equation is solved to recover the optical properties of the tissue. The procedure is usually based on a least-squares equation (linear or nonlinear) expressing the difference between the measured data and the expected measurements obtained from forward problem solution [55,56] . This last step has a number of challenges.…”

“…These terms are added to the least-squares equation restricting the solution and enhancing the image. Basic Tikhonov zero-order regularization [58] and more complex functions such as truncated singular value decomposition have been implemented to improve the image quality [53,56,59] . The search for the appropriate regularization is in progress in the field of image reconstruction.…”