Image reconstruction of fluorescent molecular tomography based on the tree structured Schur complement decomposition

The Newsletter is a key element in further enhancing the value of the journal to the inverse problems community. So why not be a part of this exciting forum by sending to our Bristol office material suitable for inclusion under any of the categories mentioned above. Your contributions will be very welcome. Book review Introduction to Inverse Problems in Imaging M Bertero and P Boccacci 1998 Bristol: Institute of Physics 362 pp ISBN 0-7503-0435-9 (pbk) £25.00, $49.00 This book shows that several problems in imaging are in fact linear inverse problems. In general inverse problems are ill-posed in the sense that small perturbations in the (measured) data have a significant influence on the output. Consequently, numerical methods developed in a general setting of inverse problems which cure the ill-posedness (such methods are called regularization methods) can be applied to solve problems in imaging. This is an excellent textbook on the principle of linear inverse problems, methods of their numerical solution and practical applications in imaging. It introduces basic ideas and methods for the solution of inverse and ill-posed problems while avoiding the mathematics of functional treatment of operator equations. This goal is achieved by a bottom-up presentation: the authors focus on the two problems of image deconvolution and tomography (bottom) and develop regularization theory specifically for these two problems. Most other books which are concerned with the numerical solution of inverse problems present a top-down approach. That is, a general regularization theory (top) is developed which is then applied to particular problems. The distinguished presentation makes the book very useful to students in applied mathematics, practitioners in engineering science and image processing. This book has the best prerequisites to establish a close link between the mathematical fields of imaging and inverse problems. Mainly the book focuses on two topics: image deconvolution and linear imaging systems. 1. In image deconvolution the goal is to restore a space invariant blurred image. This part of the book contains mathematical tools in image deconvolution, examples of space invariant imaging systems, a comparison of image deconvolution techniques and low-pass filtering techniques, the analysis of the ill-posedness of the deconvolution problem, regularization methods for deconvolution such as constrained least squares regularization, Tikhonov regularization, iterative regularization and statistical methods. Since the authors focus on particular problems they are able to analyse various methods for deconvolution on a very concrete level. As an example I would like to mention that the authors analyse Fourier methods for deconvolution. Thus they provide an optimal starting point to the theory of inverse problems for people working in the area of signal and image processing. 2. The second part of the book considers linear imaging systems that are not of a convolution type such as tomography problems, diffraction problems and inverse scattering problems. Efficient numerical methods for the solution of this class of problems such as singular value decomposition, Tikhonov regularization methods and Fourier-based methods are presented. O Scherzer Universität Linz

We propose an adaptive regularization scheme in a variational framework where a convex composite energy functional is optimized. We consider a number of imaging problems including denoising, segmentation and motion estimation, which are considered as optimal solutions of the energy functionals that mainly consist of data fidelity, regularization and a control parameter for their trade-off. We presents an algorithm to determine the relative weight between data fidelity and regularization based on the residual that measures how well the observation fits the model. Our adaptive regularization scheme is designed to locally control the regularization at each pixel based on the assumption that the diversity of the residual of a given imaging model spatially varies. The energy optimization is presented in the alternating direction method of multipliers (ADMM) framework where the adaptive regularization is iteratively applied along with mathematical analysis of the proposed algorithm. We demonstrate the robustness and effectiveness of our adaptive regularization through experimental results presenting that the qualitative and quantitative evaluation results of each imaging task are superior to the results with a constant regularization scheme. The desired properties, robustness and effectiveness, of the regularization parameter selection in a variational framework for imaging problems are achieved by merely replacing the static regularization parameter with our adaptive one.

Fluorescence molecular tomography in the near-infrared region is becoming a powerful modality for mapping the three-dimensional quantitative distributions of fluorochromes in live small animals. However, wider application of fluorescence molecular tomography still requires more accurate and stable reconstruction tools. We propose a shape-based reconstruction method that uses spherical harmonics parameterization, where fluorophores are assumed to be distributed as piecewise constants inside disjointed subdomains and the remaining background. The inverse problem is then formulated as a constrained nonlinear least-squares problem with respect to shape parameters, which decreases ill-posedness because of the significantly reduced number of unknowns. Since different shape parameters contribute differently to the boundary measurements, a two-step and modified block coordinate descent optimization algorithm is introduced to stabilize the reconstruction. We first evaluated our method using numerical simulations under various conditions for the noise level and fluorescent background; it showed significant superiority over conventional voxel-based methods in terms of the spatial resolution, reconstruction accuracy with regard to the morphology and intensity, and robustness against the initial estimated distribution. In our phantom experiment, our method again showed better spatial resolution and more accurate intensity reconstruction. Finally, the results of an in vivo experiment demonstrated its applicability to the imaging of mice.

Fluorescence molecular tomography (FMT) is an important tool for life science, which can noninvasive real-time three-dimensional (3-D) visualization for fluorescence source location. FMT is widely used in tumor research due to its high-sensitive and low cost. However, the reconstruction of FMT is difficult. Although the reconstruction methods of FMT have developed rapidly in recent years, the morphological reconstruction of FMT is still a challenge problem. Thus, the purpose of this study is to realize the morphological reconstruction performance of FMT in glioma research. In this study, group sparsity was used as a new priori information for FMT. Besides sparsity, group sparsity also takes the group structure of the fluorescent sources, which can maintain the morphological information of the sources. Fused LASSO method (FLM) was proved it can efficiently model the group sparsity prior. Thus, we utilize FLM to reconstruct the morphological information of glioma. Furthermore, to reduce the influence of the high scattering of skull, we modified the FLM for improving the accuracy of morphological reconstruction. Glioma numerical simulation model and in vivo glioma model were established to evaluate the performance of morphological reconstruction of the proposed method. The results demonstrated that the proposed method was efficient to reconstruct the morphological information of glioma. Group sparsity priori can effectively improve the morphological accuracy of FMT reconstruction. Group sparsity can maintain the morphological information of fluorescent sources effectively, which has great application potential in FMT. The group sparsity based methods can realize the morphological reconstruction, which is of great practical significance in tumor research.

In most full-waveform inversion (FWI) problems, sufficient prior information is available to constrain the velocity of certain parts of the model, e.g., the water column or, in some cases, near-surface velocities. We take advantage of this situation and develop a fast Schur-complement-based forward modeling and inversion approach by partitioning the velocity model into two parts. The first part consists of the constrained zone that does not change during the inversion, whereas the second part is the anomalous zone to be updated during the inversion. For this decomposition, we partially factorize the governing system of linear equations by computing a Schur complement for the anomalous zone. The Schur complement system is then solved to compute the fields in the anomalous zone, which are then back substituted to compute the fields in the constrained region. For each successive modeling steps with new anomalous zone velocities, the corresponding Schur complement is easily computed using simple algebra. Because the anomalous part of the model is comparatively smaller than the whole model, considerable computational savings can be achieved using our Schur approach. Additionally, we showed that the Schur complement method maintains the accuracy of standard frequency-domain finite difference formulations, but this comes at a slightly higher peak memory requirement. Our FWI workflow shows reduced runtime by 15%–57% depending upon the depth of the water column without losing any accuracy compared to the standard method.

Fluorescence Molecular Tomography (FMT) is one of the most important preclinical research techniques, which can obtain three-dimensional reconstruction of tumors in mouse in vivo. However, the ill-posedness of FMT makes its reconstruction a challenging problem. Therefore, more effective, robust, and accurate reconstruction methods are needed to be developed to solve the FMT reconstruction problem. In this paper, a reconstruction method named multipath subspace pursuit (MSP) is applied to solve the FMT problem. At the end of an iteration, the MSP method creates several candidate support set. Through evaluating the normal of final residual vector, the best candidate can be selected as the final support set. Then the support set is used for reconstructing sense matrix to achieve the goal of FMT reconstruction. In order to verity the reconstruction result of the proposed MSP method, the simulated experiment of triple fluorescent sources and quantitative analyses of position error and relative intensity error for the experiment have been conducted. The MSP method obtains satisfactory results, and the source position error is below 1 mm. Moreover, the computation time of the MSP method is about one order of magnitude less than iterated shrinkage with the L1-norm (ISlL1) method. The MSP method not only can obtain the result of robustness but also can reduce the artifacts in the background. The above results revealed the MSP method for the potential FMT application.

A matching Schur complement preconditioning technique for inverse source problems

Fluorescence Molecular Tomography is an optical imaging technique which aims at reconstructing the 3D distribution of fluorescent markers in bio-tissues based on surface measurements of emitted photons and a model of light propagation. The gold standard of accuracy in creating this light propagation model is the Monte Carlo method (MC), which simulates the path of photon packets through a discretized model of the tissue. One drawback of MC is the computational burden associated with its stochastic nature. Mesh based MC are computational implementations of MC techniques with favorable computational costs. Herein, we investigate the effects of locally refining a mesh discretization on reconstruction accuracy in mesh based Fluorescence Molecular Tomography. Using a mouse model created from μCT data and average murine optical properties, we are investigating the performances of mesh refinement strategies in reconstructing an 48.9 mm 3 fluorescence inclusion in the center of the model. Iterative mesh optimization is applied to the inverse problem in which after each reconstruction, the mesh is refined in the area of interest. Performance of the method is evaluated in terms of in volume and center of mass position of the inclusion compared to the ground truth. Our preliminary results indicate that accuracy improves with each refinement until convergence. Moreover, a method for rescaling analytically the forward model to fit each new mesh is also proposed in order to reduce the computational expense of the procedure while maintaining the improvements in accuracy

Fluorescence molecular tomography (FMT) plays an important role in in vivo small animal imaging. However, due to the diffusive nature of photon propagation in biological tissues, FMT suffers from a low spatial resolution, which limits its capability of resolving the distribution of fluorescent biomarkers. In this paper, we investigate the effect of functional and structural a priori information on the accuracy of FMT reconstruction by a hybrid FMT and X-ray computed tomography imaging system. The results from numerical simulation and phantom experiments suggest that the fluorescence targets embedded in heterogeneous medium can be localized when structural a priori information is utilized to constrain the reconstruction process. In addition, both the functional and structural a priori information are essential for the recovery of fluorophore concentration.

Fluorescence molecular tomography (FMT) is a promising imaging technique to noninvasively resolve the three-dimensional spatial distributions of fluorescent targets associated with molecular and cellular functions. The reconstruction quality, including the localization and quantitative accuracy, are greatly improved by the use of normalized Born ratio method. However, the reconstruction quality is still far from reaching their potential of accurately visualizing biological information with a high localization and quantitative accuracy. It is because the inverse problem of FMT is severely ill-posed and is easily affected by the noise in the measurement data. Here we provided a scaling normalization method for FMT to restrain the influence of excitation noise on fluorescence molecular tomography reconstruction. In this method, we use the total intensity of the excitation data detected by CCD camera to normalize the fluorescence measurements. It's feasible to suppress excitation noise when compared with the normalized Born ration method. Furthermore, by applying singular-value decomposition to the forward matrices, we find that the method we proposed has more useful singular values than the normalized Born ratio method which means that it reduces the ill-posedness of FMT inverse problem. To examine its performance, we perform phantom experiments with a hybrid FMT-XCT system. To check the robustness of our method to excitation noise, we add 10% Gaussian white noise to the measured excitation data. The reconstruction results are compared with those of the normalized Born ratio method. The results indicate that our method has a better reconstruction results and a better noise robustness.

We examine approaches to the incorporation of anatomic structural information into the inverse problem of fluorescence molecular tomography (FMT). Using an appropriate relationship between anatomic and reconstruction image resolution, we build an inverse problem parameterized along the anatomical segmentation. These values serve as the basis for two new regularization techniques. The first regularizes individual voxels in proportion to the importance of the underlying segments in reducing the residual error. The second is based on a well known statistical interpretation of Tikhonov-type regularization in which the statistical prior is defined implicitly as the solution to a PDE whose structure is based on the anatomical segmentation. Results are shown using both techniques for a simulated experiment within the chest cavity of a mouse.

Accurate small animal surface reconstruction is important for full angle non-contact fluorescence molecular tomography (FMT) systems. In this letter, an optimal surface reconstruction method for FMT is proposed. The proposed method uses a line search method to minimize the mismatch between the reconstructed three-dimensional (3D) surface and the projected object silhouette at different angles. The results show that the mean mismatches of the 3D surfaces generated on three live anesthetized mice are all less than two charge coupled device (CCD) pixels (0.154 mm). With the accurately reconstructed 3D surface, in-vivo FMT is also performed.

Fluorescence molecular tomography (FMT) is an emerging tool for biomedical research. There are two factors that influence FMT reconstruction most effectively. The first one is regularization techniques. Traditional methods such as Tikhonov regularization suffer from low resolution and poor signal to noise ratio. Therefore, sparse regularization techniques have been introduced to improve the reconstruction quality. The second factor is the illumination pattern. A better illumination pattern ensures the quantity and quality of the information content of the data set, thus leading to better reconstructions. In this work, we take advantage of the discrete formulation of the forward problem to give a rigorous definition of an illumination pattern as well as the admissible set of patterns. We add restrictions in the admissible set as different types of regularizers to a discrepancy functional, generating another inverse problem with the illumination pattern as unknown. Both inverse problems of reconstructing the fluorescence distribution and finding the optimal illumination pattern are solved by efficient iterative algorithms. Numerical experiments have shown that with a suitable choice of regularization parameters, the two-step approach converges to an optimal illumination pattern quickly and the reconstruction result is improved significantly regardless of the initial illumination setting.

We study the reconstruction of the missing thermal and mechanical data on an inaccessible part of the boundary in the case of two‐dimensional linear isotropic thermoelastic materials from overprescribed noisy measurements taken on the remaining accessible boundary part. This inverse problem is solved by using the method of fundamental solutions together with the method of particular solutions. The stabilization of this inverse problem is achieved using several singular value decomposition (SVD)‐based regularization methods, such as the Tikhonov regularization method (Tikhonov and Arsenin, Methods for solving ill‐posed problems, Nauka, Moscow, 1986), the damped SVD and the truncated SVD (Hansen, Rank‐deficient and discrete ill‐posed problems: numerical aspects of linear inversion, SIAM, Philadelphia, 1998), whilst the optimal regularization parameter is selected according to the discrepancy principle (Morozov, Sov Math Doklady 7 (1966), 414–417), generalized cross‐validation criterion (Golub et al. Technometrics 22 (1979), 1–35) and Hansen's L‐curve method (Hansen and O'Leary, SIAM J Sci Comput 14 (1993), 1487–503). © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 168–201, 2015

In vivo fluorescence molecular tomography (FMT) has been a popular functional imaging modality in research labs in the past two decades. One of the major difficulties of FMT lies in the ill-posed and ill-conditioned nature of the inverse problem in reconstructing the distribution of fluorophores inside objects. The popular regularization methods based on L2, L1 and total variation (TV ) norms have been applied in FMT reconstructions. The non-convex Lq(0 < q < 1) semi-norm and Log function have also been studied recently. In this paper, we adopt a uniform optimization transfer framework for these regularization methods in FMT and compare their individual, as well as the combined effects on both small, localized targets, such as tumors in the early stage, and large targets, such as liver. Numerical simulation studies and phantom experiments have been carried out, and we found that Lq with q near 1/2 performs the best in reconstructing small targets, while joint L2 and Log performs the best for large targets.