Using state-of-the-art inverse problem techniques to develop reconstruction methods for fluorescence diffuse optical

Resumen

An inverse problem is a mathematical framework that is used to obtain info about a physical object or system from observed measurements. It usually appears when we wish to obtain information about internal data from outside measurements and has many applications in science and technology such as medical imaging, geophysical imaging, image deblurring, image inpainting, electromagnetic scattering, acoustics, machine learning, mathematical finance, physics, etc. The main goal of this PhD thesis was to use state-of-the-art inverse problem techniques to develop modern reconstruction methods for solving the fluorescence diffuse optical tomography (fDOT) problem. fDOT is a molecular imaging technique that enables the quantification of tomographic (3D) bio-distributions of fluorescent tracers in small animals. One of the main difficulties in fDOT is that the high absorption and scattering properties of biological tissues lead to an ill-posed inverse problem, yielding multiple nonunique and unstable solutions to the reconstruction problem. Thus, the problem requires regularization to achieve a stable solution. The so called “non-contact fDOT scanners” are based on using CCDs as virtual detectors instead of optic fibers in contact with the sample. These non-contact systems generate huge datasets that lead to computationally demanding inverse problem. Therefore, techniques to minimize the size of the acquired datasets without losing image performance are highly advisable. The first part of this thesis addresses the optimization of experimental setups to reduce the dataset size, by using l?–based regularization techniques. The second part, based on the success of l? regularization techniques for denoising and image reconstruction, is devoted to advanced regularization problem using l?–based techniques, and the last part introduces compressed sensing (CS) theory, which enables further reduction of the acquired dataset size. The main contributions of this thesis are: 1) A feasibility study (the first one for fDOT to our knowledge) of the automatic Ucurve method to select the regularization parameter (l?–norm). The U-curve method has shown to be an excellent automatic method to deal with large datasets because it reduces the regularization parameter search to a suitable interval. 2) Once we found an automatic method to choose the l? regularization parameter for fDOT, singular value analysis (SVA) of fDOT forward matrix was used to maximize the information content in acquired measurements and minimize the computational cost. It was shown for the first time that large meshes can be reduced in the z direction, without any loss in imaging performance but reducing computational times and memory requirements. 3) Dealing with l?–based regularization techniques, we presented a novel iterative algorithm, ART-SB, that combines the advantage of Algebraic reconstruction method (ART) in handling large datasets with Split Bregman (SB) denoising, an approach which has been shown to be optimum for Total Variation (TV) denoising. SB has been implemented in a cost-efficient way to handle large datasets. This makes ART-SB more computationally efficient than previous TV-based reconstruction algorithms and most splitting approaches. 4) Finally, we proposed a novel approach to CS for fDOT, named the SB-SVA iterative method. This approach is based on the analysis-based co-sparse representation model, where an analysis operator multiplies the image transforming it in a sparse one. Taking advantage of the CS-SB algorithm, we restrict the solution reached at each CS-SB iteration to a certain space where the singular values of the forward matrix and the sparsity structure combine in beneficial manner. In this way, SB-SVA forces indirectly the wellconditioninig of the forward matrix while designing (learning) the analysis operator and finding the solution. Furthermore, SB-SVA outperforms the CS-SB algorithm in terms of image quality and needs fewer acquisition parameters. The approaches presented here have been validated with experimental. -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------