Manuel Freiberger

Total variation regularization for nonlinear fluorescence tomography with an augmented Lagrangian splitting approach

Fluorescence tomography is an imaging modality that seeks to reconstruct the distribution of fluorescent dyes inside a highly scattering sample from light measurements on the boundary. Using common inversion methods with L(2) penalties typically leads to smooth reconstructions, which degrades the obtainable resolution. The use of total variation (TV) regularization for the inverse model is investigated. To solve the inverse problem efficiently, an augmented Lagrange method is utilized that allows separating the Gauss-Newton minimization from the TV minimization. Results on noisy simulation data provide evidence that the reconstructed inclusions are much better localized and that their half-width measure decreases by at least 25% compared to ordinary L(2) reconstructions.

The Agile Library for Biomedical Image Reconstruction Using GPU Acceleration

A cheap way to speed up image-reconstruction software is to use modern graphics hardware that can execute algorithms in a massively parallel manner. Here, the authors discuss Agile, an open source library designed for image reconstruction in biomedical sciences. Its modular, object-oriented, and templated design eases the integration of the library into user code.

High-performance image reconstruction in fluorescence tomography on desktop computers and graphics hardware

Image reconstruction in fluorescence optical tomography is a three-dimensional nonlinear ill-posed problem governed by a system of partial differential equations. In this paper we demonstrate that a combination of state of the art numerical algorithms and a careful hardware optimized implementation allows to solve this large-scale inverse problem in a few seconds on standard desktop PCs with modern graphics hardware. In particular, we present methods to solve not only the forward but also the non-linear inverse problem by massively parallel programming on graphics processors. A comparison of optimized CPU and GPU implementations shows that the reconstruction can be accelerated by factors of about 15 through the use of the graphics hardware without compromising the accuracy in the reconstructed images.

Nonlinear Inversion Schemes for Fluorescence Optical Tomography

Fluorescence optical tomography is a noninvasive imaging modality that employs the absorption and reemission of light by fluorescent dyes. The aim is to reconstruct the fluorophore distribution in a body from measurements of light intensities at the boundary. Due to the diffusive nature of light propagation in tissue, fluorescence tomography is a nonlinear and severely ill-posed problem, and some sort of regularization is required for a stable solution. In this paper, we investigate reconstruction methods based on Tikhonov regularization with nonlinear penalty terms, namely total-variation regularization and a levelset-type method using a nonlinear parameterization of the unknown function. Moreover, we use the full 3-D nonlinear forward model, which arises from the governing system of partial differential equations. We discuss the numerical realization of the regularization schemes by Newton-type iterations, present some details of the discretization by finite-element methods, and outline the efficient implementation of sensitivity systems via adjoint methods. As we will demonstrate in numerical tests, the proposed nonlinear methods provide better reconstructions than standard methods based on linearized forward models and linear penalty terms. We will additionally illustrate that the careful discretization of the methods derived on the continuous level allows us to obtain reliable, mesh-independent reconstruction algorithms.

Adaptation and focusing of optode configurations for fluorescence optical tomography by experimental design methods

Fluorescence tomography excites a fluorophore inside a sample by light sources on the surface. From boundary measurements of the fluorescent light, the distribution of the fluorophore is reconstructed. The optode placement determines the quality of the reconstructions in terms of, e.g., resolution and contrast-to-noise ratio. We address the adaptation of the measurement setup. The redundancy of the measurements is chosen as a quality criterion for the optodes and is computed from the Jacobian of the mathematical formulation of light propagation. The algorithm finds a subset with minimum redundancy in the measurements from a feasible pool of optodes. This allows biasing the design in order to favor reconstruction results inside a given region. Two different variations of the algorithm, based on geometric and arithmetic averaging, are compared. Both deliver similar optode configurations. The arithmetic averaging is slightly more stable, whereas the geometric averaging approach shows a better conditioning of the sensitivity matrix and mathematically corresponds more closely with entropy optimization. Adapted illumination and detector patterns are presented for an initial set of 96 optodes placed on a cylinder with focusing on different regions. Examples for the attenuation of fluorophore signals from regions outside the focus are given.

A deterministic approach to the adapted optode placement for illumination of highly scattering tissue

A novel approach is presented for computing optode placements that are adapted to specific geometries and tissue characteristics, e.g., in optical tomography and photodynamic cancer therapy. The method is based on optimal control techniques together with a sparsity-promoting penalty that favors pointwise solutions, yielding both locations and magnitudes of light sources. In contrast to current discrete approaches, the need for specifying an initial set of candidate configurations as well as the exponential increase in complexity with the number of optodes are avoided. This is demonstrated with computational examples from photodynamic therapy.

Sensor optimization for fluorescence optical tomography by experimental design methods

With the increasing importance of molecular imaging fluorescence based methods are continuously gaining impact. In fluorescence optical tomography excitation light is injected into the tissue where the fluorophore converts it to radiation of another wavelength. From the emitted light reaching the boundary the 3-D distribution of the fluorophore is reconstructed. This paper aims at finding the optimal spatial distribution of optodes in order to keep their number (hardware costs) low while gaining maximum information from the target object. The implemented algorithm starts with an arbitrary pool of feasible optodes. The optimal subset is searched by minimizing the mutual information between the different measurements. This goal is reached by subsequently removing those sources and detectors which add the least independent information until a stopping criterion is reached. Mutual information is estimated by calculating the inner products between the rows of the sensitivity matrix i.e. the first derivative of the forward mapping with respect to the optical parameters to be reconstructed. We assembled this matrix with a finite element implementation of the diffusion approximation of light propagation in scattering tissues. When starting with an initial pool of 96 optodes regularly spaced on a cylindrical surface and focusing on different target regions within the cylinder, the algorithm always converged towards physically reasonable optimal sets. Optimal source/detector patterns are be presented graphically and numerically.

Uncertainty Analysis for Fluorescence Tomography with Monte Carlo Method

Fluorescence tomography seeks to image an inaccessible fluorophore distribution inside an object like a small animal by injecting light at the boundary and measuring the light emitted by the fluorophore. Optical parameters (e.g. the conversion efficiency or the fluorescence life-time) of certain fluorophores depend on physiologically interesting quantities like the pH value or the oxygen concentration in the tissue, which allows functional rather than just anatomical imaging. To reconstruct the concentration and the life-time from the boundary measurements, a nonlinear inverse problem has to be solved. It is, however, difficult to estimate the uncertainty of the reconstructed parameters in case of iterative algorithms and a large number of degrees of freedom. Uncertainties in fluorescence tomography applications arise from model inaccuracies, discretization errors, data noise and a priori errors. Thus, a Markov chain Monte Carlo method (MCMC) was used to consider all these uncertainty factors exploiting Bayesian formulation of conditional probabilities. A 2-D simulation experiment was carried out for a circular object with two inclusions. Both inclusions had a 2-D Gaussian distribution of the concentration and constant life-time inside of a representative area of the inclusion. Forward calculations were done with the diffusion approximation of Boltzmanns transport equation. The reconstruction results show that the percent estimation error of the lifetime parameter is by a factor of approximately 10 lower than that of the concentration. This finding suggests that lifetime imaging may provide more accurate information than concentration imaging only. The results must be interpreted with caution, however, because the chosen simulation setup represents a special case and a more detailed analysis remains to be done in future to clarify if the findings can be generalized.

3D Optical Imaging of Fluorescent Agents in Biological Tissues

Fluorescence diffuse optical tomography (fDOT) is an imaging modality which goes beyond well-established techniques such as 2D fluorescence imaging and fluorescence microscopy. Being a 3D tomographic modality, it seeks to overcome limitations of 2D systems as are: (i) the determination of the depth of fluorescent objects and (ii) a correction of the broadening of the fluorescence signal due to the massive scattering of photons.