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Title:  Development Of Deterministic And Stochastic Algorithms For Inverse Problems Of Optical Tomography 
Authors:  Gupta, Saurabh 
Advisors:  Vasu, Ram Mohan Roy, Debasish 
Keywords:  Optical Tompgraphy Diffuse Optical Tomography (DOT) Ultrasound Modulated Optical Tomography (UMOT) Inverse Problems GaussNewton Algorithm PseuoDynamic Ensemble Kalman Filter Stochastic Algorithms Stochastic Filtering Algorithms Medical Imaging 
Submitted Date:  Jul2013 
Series/Report no.:  G26351 
Abstract:  Stable and computationally efficient reconstruction methodologies are developed to solve two important medical imaging problems which use nearinfrared (NIR) light as the source of interrogation, namely, diffuse optical tomography (DOT) and one of its variations, ultrasoundmodulated optical tomography (UMOT). Since in both these imaging modalities the system matrices are illconditioned owing to insufficient and noisy data, the emphasis in this work is to develop robust stochastic filtering algorithms which can handle measurement noise and also account for inaccuracies in forward models through an appropriate assignment of a process noise.
However, we start with demonstration of speeding of a GaussNewton (GN) algorithm for DOT so that a videorate reconstruction from data recorded on a CCD camera is rendered feasible. Towards this, a computationally efficient linear iterative scheme is proposed to invert the normal equation of a GaussNewton scheme in the context of recovery of absorption coefficient distribution from DOT data, which involved the singular value decomposition (SVD) of the Jacobian matrix appearing in the update equation. This has sufficiently speeded up the inversion that a video rate recovery of time evolving absorption coefficient distribution is demonstrated from experimental data. The SVDbased algorithm has made the number of operations in image reconstruction to be rather than. 2()ONN3()ONN
The rest of the algorithms are based on different forms of stochastic filtering wherein we arrive at a meansquare estimate of the parameters through computing their joint probability
distributions conditioned on the measurement up to the current instant. Under this, the first algorithm developed uses a Bootstrap particle filter which also uses a quasiNewton direction within. Since keeping track of the Newton direction necessitates repetitive computation of the Jacobian, for all particle locations and for all time steps, to make the recovery computationally feasible, we devised a faster update of the Jacobian. It is demonstrated, through analytical reasoning and numerical simulations, that the proposed scheme, not only accelerates convergence but also yields substantially reduced sample variance in the estimates visàvis the conventional BS filter. Both accelerated convergence and reduced sample variance in the estimates are demonstrated in DOT optical parameter recovery using simulated and experimental data.
In the next demonstration a derivative free variant of the pseudodynamic ensemble Kalman filter (PDEnKF) is developed for DOT wherein the size of the unknown parameter is reduced by representing of the inhomogeneities through simple geometrical shapes. Also the optical parameter fields within the inhomogeneities are approximated via an expansion based on the circular harmonics (CH) (Fourier basis functions). The EnKF is then used to recover the coefficients in the expansion with both simulated and experimentally obtained photon fluence data on phantoms with inhomogeneous inclusions. The process and measurement equations in the PseudoDynamic EnKF (PDEnKF) presently yield a parsimonious representation of the filter variables, which consist of only the Fourier coefficients and the constant scalar parameter value within the inclusion. Using fictitious, lowintensity Wiener noise processes in suitably constructed ‘measurement’ equations, the filter variables are treated as pseudostochastic processes so that their recovery within a stochastic filtering framework is made possible. In our numerical simulations we have considered both elliptical inclusions (two inhomogeneities) and those with more complex shapes ( such as an annular ring and a dumbbell) in 2D objects which are crosssections of a cylinder with background absorption and (reduced) scattering coefficient chosen as = 0.01 mm1 and = 1.0 mm1respectively. We also assume=0.02 mm1 within the inhomogeneity (for the single inhomogeneity case) and=0.02 and 0.03 mm1 (for the two inhomogeneities case). The reconstruction results by the PDEnKF are shown to be consistently superior to those through a deterministic and explicitly regularized GaussNewton algorithm. We have also estimated the unknown from experimentally gathered fluence data and verified the reconstruction by matching the experimental data with the computed one.
The superiority of a modified version of the PDEnKF, which uses an ensemble square root filter, is also demonstrated in the context of UMOT by recovering the distribution of meansquared amplitude of vibration, related to the Young’s modulus, in the ultrasound focal volume. Since the ability of a coherent light probe to pickup the overall optical pathlength change is limited to modulo an optical wavelength, the individual displacements suffered owing to the US forcing should be very small, say within a few angstroms. The sensitivity of modulation depth to changes in these small displacements could be very small, especially when the ROI is far removed from the source and detector. The contrast recovery of the unknown distribution in such cases could be seriously impaired whilst using a quasiNewton scheme (e.g. the GN scheme) which crucially makes use of the derivative information. The derivativefree gainbased Monte Carlo filter not only remedies this deficiency, but also provides a regularization insensitive and computationally competitive alternative to the GN scheme. The inherent ability of a stochastic filter in accommodating the model error owing to a diffusion approximation of the correlation transport may be cited as an added advantage in the context of the UMOT inverse problem.
Finally to speed up forward solve of the partial differential equation (PDE) modeling photon transport in the context of UMOT for which the PDE has time as a parameter, a spectral decomposition of the PDE operator is demonstrated. This allows the computation of the time dependent forward solution in terms of the eigen functions of the PDE operator which has speeded up the forward solution, which in turn has rendered the UMOT parameter recovery computationally efficient. 
Abstract file URL:  http://etd.ncsi.iisc.ernet.in/abstracts/3422/G26351Abs.pdf 
URI:  http://hdl.handle.net/2005/2608 
Appears in Collections:  Instrumentation and Applied Physics (iap)

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