In this paper, we analyze the error resulting from the discretization of the forward and inverse problems in simultaneously
reconstructed optical absorption and scattering images. Our analysis indicates the mutual dependence
of the forward and inverse problems, the number of sources and detectors, their configuration and the location
of optical heterogeneities with respect to sources and detectors affect the extent of the error in the reconstructed
optical images resulting from discretization. One important implication of the error analysis is that poor discretization
of one optical coefficient results in error in the other, resulting in inter-parameter "cross-talk" due
entirely to discretization.
In fluorescence diffuse optical tomography, the error due to
discretization of the forward and inverse problems leads to an error
in the reconstructed image. Using a Galerkin formulation, we
consider zeroth and first order Tikhonov regularization terms and
analyze the forward and inverse problems under an optimization
formulation which incorporates a priori information. We
derive error estimates to describe the impact that discretization of
the forward and inverse problems due to finite element method has on
the accuracy of the reconstructed optical absorbtion image.
In this work, we discuss the incorporation of a priori information into the inverse problem formulation for
fluorescence optical tomography. In this respect, we first formulate the inverse problem in the optimization
framework which allows the incorporation of a priori information about the solution and its gradient. Then, we
consider the variational problem, which is equivalent to the optimization problem and prove the existence and
uniqueness of the solution. Finally, we discuss the design of the functions that incorporate the a priori information
into the inverse problem formulation and present a model problem to illustrate the design procedure.
Diffuse Optical Tomography (DOT) image reconstruction is a
challenging 3D problem with a relatively large number of unknowns.
DOT poses a typical ill-posed problem usually plagued by
under-determination, which complicates the inverse problem.
Conventional image reconstruction algorithms can not provide high
spatial resolution and may become computationally expensive and
unreliable especially in the presence of noise.
In this work, we extend our previous formulation for the 3D
inverse DOT problem, where we focus to improve the spatial
resolution and quantitative accuracy of 3D DOT images by using
anatomical a priori information, which is specific to the medium
of interest. Maximum A Posteriori (MAP) estimate of the image is
formed based on the formulation of the image's probability density
function, which is extracted from the available a priori
anatomical information. An ``alternating minimization'' algorithm,
which sequentially updates the unknown parameters, is used to
solve the resulting optimization problem. Proposed method is
evaluated in a 3D simulation experiment. Results demonstrate
that the proposed method leads to significantly improved spatial
resolution, quantitative accuracy and faster convergence than
standard and regularized least squares solutions even in the
presence of noise. As a result, the approach demonstrated in this
paper both addresses the ill-posedness and balances the
computation complexity vs. image quality trade-off in the 3D DOT
Diffuse optical imaging is an emerging modality that uses Near Infrared (NIR) light to reveal structural and functional information of deep biological tissue. It provides contrast mechanisms for molecular, chemical, and anatomical imaging that is not available from other imaging modalities. Diffuse Optical Tomography (DOT) deals with 3D reconstruction of optical properties of tissue given the measurements and a forward model of photon propagation. DOT has inherently low spatial resolution due to diffuse nature of photons. In this work, we focus to improve the spatial resolution and the quantitative accuracy of DOT by using a priori anatomical information specific to unknown image. Such specific a priori information can be obtained from a secondary high-resolution imaging modality such as Magnetic Resonance (MR) or X-ray. Image reconstruction is formulated within a Bayesian framework to determine the spatial distribution of the absorption coefficients of the medium. A spatially varying a priori probability density function is designed based on the segmented anatomical information. Conjugate gradient method is utilized to solve the resulting optimization problem. Proposed method is evaluated using simulation and phantom measurements collected with a novel time-resolved optical imaging system. Results demonstrate that the proposed method leads to improved spatial resolution, quantitative accuracy and faster convergence than standard least squares approach.