A spline-based forward model for Optical Diffuse Tomography

Reconstruction algorithms for Optical Diffuse Tomography (ODT) rely heavily on fast and accurate forward models. Arbitrary geometries and boundary conditions need to be handled rigorously since they are the only input to the inverse problem. From this perspective, Finite Element Methods (FEM) are good candidates to implement a forward model. However, these methods require to mesh the domain of interest, which is impractical on a routine basis. The other downside of the FEM is that the basis functions are often not compatible with the ones used for solving the inverse problem, which typically have less degrees of freedom. In this work, we tackle the 2D problem, and propose a forward model that uses a mesh-free discretization based on linear B-Splines. It combines the advantages of the FEM, while offering a fast and much simpler way of handling complex geometries. Another motivation for this work is that the underlying B-spline model is equally suitable for the subsequent reconstruction part of the process (solving the inverse problem). In particular, it is compatible with wavelets and multiresolution-type signal representations.

A Spline-Based Forward Model for Optical Diffuse Tomography

Jean-Charles Baritaux, Michaël Unser

IEEE2008

Sparsity-Inducing Reconstruction Methods for Fluorescence Diffuse Optical Tomography

Jean-Charles Baritaux

The recent advances in fluorescent molecular probes, photon detection instrumentation, and photon propagation models in tissue, have facilitated the emergence of innovative molecular imaging technologies such as Fluorescence Diffuse Optical Tomography (FDOT). FDOT has already successfully been applied on its own to resolve specific molecular targets and pathways in vivo. Recent attempts have tried to integrate FDOT with structural imaging technology into a multi-modal system. This thesis is amongst the first academic-industrial partnerships to develop a hybrid X-Ray Computed Tomography/FDOT platform for small animal imaging. In this study, we focus on image reconstruction algorithms for FDOT, and consider three main research directions. First, we investigate innovative discretization techniques for FDOT, in the context of variational reconstruction methods. Our goal is to treat in a consistent manner the propagation equation, and the regularization, needed in the reconstruction algorithm. This study uses the diffusion approximation for modeling the propagation of photons in tissue. We solve the diffusion equation with a novel Finite Volumes Method that is specifically designed to be compatible with the sparsity-inducing regularization developed in the second part of this work. Our approach results in a computationally efficient implementation of the forward model. Next, we study a novel regularization approach for FDOT which is based on sparsity-promoting regularization. This approach is expected to enhance the reconstruction when the fluorescence distribution presents a sparsity pattern in a suitable representation, or after linear transformation. We propose a variational framework based on ℓp-norm penalization in order to incorporate the sparsity a priori in the inversion algorithm. We demonstrate this approach on experimental data, using computationally efficient algorithms relying on convex optimization principles. Finally, we focus on hybrid imaging systems that combine FDOT with another imaging technique. The latter is employed to acquire a high-resolution image of the anatomy of the specimen under investigation, which provides valuable information for the FDOT reconstruction. We propose a new regularization method that allows the incorporation of the anatomical information in the FDOT reconstruction. Our method is based on the concept of group sparsity, and implemented in practice using ℓ2,1-norm penalization. We extend our sparsity-inducing algorithms to this case, and study the performance of the resulting implementation on experimental data.

EPFL2012

Differential Phase-Contrast X-Ray Computed Tomography: From Model Discretization To Image Reconstruction

Masih Nilchian, Michaël Unser

Our contribution in this paper is two fold. First, we propose a novel discretization of the forward model for differential phase-contrast imaging that uses B-spline basis functions. The approach yields a fast and accurate algorithm for implementing the forward model, which is based on the first derivative of the Radon transform. Second, as an alternative to the FBP-like approaches that are currently used in practice, we present an iterative reconstruction algorithm that remains more faithful to the data when the number of projections dwindles. Since the reconstruction is an ill-posed problem, we impose a total-variation (TV) regularization constraint. We propose to solve the reconstruction problem using the alternating direction method of multipliers (ADMM). A specificity of our system is the use of a preconditioner that improves the convergence rate of the linear solver in ADMM. Our experiments on test data suggest that our method can achieve the same quality as the standard direct reconstruction, while using only one-third of the projection data. We also find that the approach is much faster than the standard algorithms (ISTA and FISTA) that are typically used for solving linear inverse problems subject to the TV regularization constraint.

Ieee2012

Sparsity-Inducing Reconstruction Methods for Fluorescence Diffuse Optical Tomography

Jean-Charles Baritaux

EPFL2012