Graph- and finite element-based total variation models for the inverse problem in diffuse optical tomography

Total variation (TV) is a powerful regularization method that has been widely applied in different imaging applications, but is difficult to apply to diffuse optical tomography (DOT) image reconstruction (inverse problem) due to complex and unstructured geometries, non-linearity of the data fitting and regularization terms, and non-differentiability of the regularization term. We develop several approaches to overcome these difficulties by: i) defining discrete differential operators for unstructured geometries using both finite element and graph representations; ii) developing an optimization algorithm based on the alternating direction method of multipliers (ADMM) for the non-differentiable and non-linear minimization problem; iii) investigating isotropic and anisotropic variants of TV regularization, and comparing their finite element- and graph-based implementations. These approaches are evaluated on experiments on simulated data and real data acquired from a tissue phantom. Our results show that both FEM and graph-based TV regularization is able to accurately reconstruct both sparse and non-sparse distributions without the over-smoothing effect of Tikhonov regularization and the over-sparsifying effect of L$_1$ regularization. The graph representation was found to out-perform the FEM method for low-resolution meshes, and the FEM method was found to be more accurate for high-resolution meshes.Comment: 24 pages, 11 figures. Reviced version includes revised figures and improved clarit

Parametric Level Set Methods for Inverse Problems

In this paper, a parametric level set method for reconstruction of obstacles in general inverse problems is considered. General evolution equations for the reconstruction of unknown obstacles are derived in terms of the underlying level set parameters. We show that using the appropriate form of parameterizing the level set function results a significantly lower dimensional problem, which bypasses many difficulties with traditional level set methods, such as regularization, re-initialization and use of signed distance function. Moreover, we show that from a computational point of view, low order representation of the problem paves the path for easier use of Newton and quasi-Newton methods. Specifically for the purposes of this paper, we parameterize the level set function in terms of adaptive compactly supported radial basis functions, which used in the proposed manner provides flexibility in presenting a larger class of shapes with fewer terms. Also they provide a "narrow-banding" advantage which can further reduce the number of active unknowns at each step of the evolution. The performance of the proposed approach is examined in three examples of inverse problems, i.e., electrical resistance tomography, X-ray computed tomography and diffuse optical tomography

Edge-promoting reconstruction of absorption and diffusivity in optical tomography

In optical tomography a physical body is illuminated with near-infrared light and the resulting outward photon flux is measured at the object boundary. The goal is to reconstruct internal optical properties of the body, such as absorption and diffusivity. In this work, it is assumed that the imaged object is composed of an approximately homogeneous background with clearly distinguishable embedded inhomogeneities. An algorithm for finding the maximum a posteriori estimate for the absorption and diffusion coefficients is introduced assuming an edge-preferring prior and an additive Gaussian measurement noise model. The method is based on iteratively combining a lagged diffusivity step and a linearization of the measurement model of diffuse optical tomography with priorconditioned LSQR. The performance of the reconstruction technique is tested via three-dimensional numerical experiments with simulated measurement data.Comment: 18 pages, 6 figure

Image reconstruction in fluorescence molecular tomography with sparsity-initialized maximum-likelihood expectation maximization

We present a reconstruction method involving maximum-likelihood expectation maximization (MLEM) to model Poisson noise as applied to fluorescence molecular tomography (FMT). MLEM is initialized with the output from a sparse reconstruction-based approach, which performs truncated singular value decomposition-based preconditioning followed by fast iterative shrinkage-thresholding algorithm (FISTA) to enforce sparsity. The motivation for this approach is that sparsity information could be accounted for within the initialization, while MLEM would accurately model Poisson noise in the FMT system. Simulation experiments show the proposed method significantly improves images qualitatively and quantitatively. The method results in over 20 times faster convergence compared to uniformly initialized MLEM and improves robustness to noise compared to pure sparse reconstruction. We also theoretically justify the ability of the proposed approach to reduce noise in the background region compared to pure sparse reconstruction. Overall, these results provide strong evidence to model Poisson noise in FMT reconstruction and for application of the proposed reconstruction framework to FMT imaging

Linearized Reconstruction for Diffuse Optical Spectroscopic Imaging

In this paper, we present a novel reconstruction method for diffuse optical spectroscopic imaging with a commonly used tissue model of optical absorption and scattering. It is based on linearization and group sparsity, which allows recovering the diffusion coefficient and absorption coefficient simultaneously, provided that their spectral profiles are incoherent and a sufficient number of wavelengths are judiciously taken for the measurements. We also discuss the reconstruction for imperfectly known boundary and show that with the multi-wavelength data, the method can reduce the influence of modelling errors and still recover the absorption coefficient. Extensive numerical experiments are presented to support our analysis.Comment: 18 pages, 7 figure

Quantitative photoacoustic imaging in radiative transport regime

The objective of quantitative photoacoustic tomography (QPAT) is to reconstruct optical and thermodynamic properties of heterogeneous media from data of absorbed energy distribution inside the media. There have been extensive theoretical and computational studies on the inverse problem in QPAT, however, mostly in the diffusive regime. We present in this work some numerical reconstruction algorithms for multi-source QPAT in the radiative transport regime with energy data collected at either single or multiple wavelengths. We show that when the medium to be probed is non-scattering, explicit reconstruction schemes can be derived to reconstruct the absorption and the Gruneisen coefficients. When data at multiple wavelengths are utilized, we can reconstruct simultaneously the absorption, scattering and Gruneisen coefficients. We show by numerical simulations that the reconstructions are stable.Comment: 40 pages, 13 figure