Sparsity-driven reconstruction for FDOT with anatomical priors

Development of tomographic reconstruction for three-dimensional optical imaging: From the inversion of light propagation to artificial intelligence

Optical molecular tomography (OMT) is an imaging modality which uses an optical signal, especially near-infrared light, to reconstruct the three-dimensional information of the light source in biological tissue. With the advantages of being low-cost, noninvasive and having high sensitivity, OMT has been applied in preclinical and clinical research. However, due to its serious ill-posedness and ill-condition, the solution of OMT requires heavy data analysis and the reconstruction quality is limited. Recently, the artificial intelligence (commonly known as AI)-based methods have been proposed to provide a different tool to solve the OMT problem. In this paper, we review the progress on OMT algorithms, from conventional methods to AI-based methods, and we also give a prospective towards future developments in this domain.

First-order approximation of fluorescence excitation-transmission to accelerate fluorescence molecular tomography image reconstruction

We present a method to accelerate image reconstruction in fluorescence molecular tomography based on the historical path fluorescence Monte Carlo model. The method exploits a first-order approximation expression during the fluorescence excitation-transmission process to merge the path and state information of the photon in a voxel. The experiments show that our method not only greatly reduces the amount of data required for storage in the hard disk and accelerates image reconstruction, but also maintains the quantitative and positioning accuracy of the conventional method.

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 unstructured discretization of complex 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 TV regularization 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 (FEM)- 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 L1 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.

Sparsity-regularized approaches to directly reconstructing hemodynamic response in brain functional diffuse optical tomography

In brain functional diffuse optical tomography, conventional indirect approaches first separately reconstruct the spatial changes in the absorption coefficients at every time point and then calculate the spatial excited levels in terms of hemodynamic models. Direct approaches combine the two steps necessary in the indirect approaches and obtain the spatial excited levels directly. Although reconstruction quality has been improved by the direct approaches to some extent, they still lack sharp edges and suffer from low spatial resolution because of the ill-posedness of the inverse problems. In this paper, a priori sparsity is introduced to obtain the sparse solutions and further improve reconstruction quality. Simulation experiments are conducted to illustrate the expected performance improvements of the proposed approaches.

L1-norm based nonlinear reconstruction improves quantitative accuracy of spectral diffuse optical tomography

Spectrally constrained diffuse optical tomography (SCDOT) is known to improve reconstruction in diffuse optical imaging; constraining the reconstruction by coupling the optical properties across multiple wavelengths suppresses artefacts in the resulting reconstructed images. In other work, L1-norm regularization has been shown to improve certain types of image reconstruction problems as its sparsity-promoting properties render it robust against noise and enable the preservation of edges in images, but because the L1-norm is non-differentiable, it is not always simple to implement. In this work, we show how to incorporate L1 regularization into SCDOT. Three popular algorithms for L1 regularization are assessed for application in SCDOT: iteratively reweighted least square algorithm (IRLS), alternating directional method of multipliers (ADMM), and fast iterative shrinkage-thresholding algorithm (FISTA). We introduce an objective procedure for determining the regularization parameter in these algorithms and compare their performance in simulated experiments, and in real data acquired from a tissue phantom. Our results show that L1 regularization consistently outperforms Tikhonov regularization in this application, particularly in the presence of noise.

Nonlinear greedy sparsity-constrained algorithm for direct reconstruction of fluorescence molecular lifetime tomography

In order to improve the spatial resolution of time-domain (TD) fluorescence molecular lifetime tomography (FMLT), an accelerated nonlinear orthogonal matching pursuit (ANOMP) algorithm is proposed. As a kind of nonlinear greedy sparsity-constrained methods, ANOMP can find an approximate solution of L0 minimization problem. ANOMP consists of two parts, i.e., the outer iterations and the inner iterations. Each outer iteration selects multiple elements to expand the support set of the inverse lifetime based on the gradients of a mismatch error. The inner iterations obtain an intermediate estimate based on the support set estimated in the outer iterations. The stopping criterion for the outer iterations is based on the stability of the maximum reconstructed values and is robust for problems with targets at different edge-to-edge distances (EEDs). Phantom experiments with two fluorophores at different EEDs and in vivo mouse experiments demonstrate that ANOMP can provide high quantification accuracy, even if the EED is relatively small, and high resolution.

Fast multislice fluorescence molecular tomography using sparsity-inducing regularization

Fluorescence molecular tomography (FMT) is a rapidly growing imaging method that facilitates the recovery of small fluorescent targets within biological tissue. The major challenge facing the FMT reconstruction method is the ill-posed nature of the inverse problem. In order to overcome this problem, the acquisition of large FMT datasets and the utilization of a fast FMT reconstruction algorithm with sparsity regularization have been suggested recently. Therefore, the use of a joint L1/total-variation (TV) regularization as a means of solving the ill-posed FMT inverse problem is proposed. A comparative quantified analysis of regularization methods based on L1-norm and TV are performed using simulated datasets, and the results show that the fast composite splitting algorithm regularization method can ensure the accuracy and robustness of the FMT reconstruction. The feasibility of the proposed method is evaluated in an in vivo scenario for the subcutaneous implantation of a fluorescent-dye-filled capillary tube in a mouse, and also using hybrid FMT and x-ray computed tomography data. The results show that the proposed regularization overcomes the difficulties created by the ill-posed inverse problem.

Performance evaluation of typical approximation algorithms for nonconvex ℓp-minimization in diffuse optical tomography

The sparse estimation methods that utilize the ℓp-norm, with p being between 0 and 1, have shown better utility in providing optimal solutions to the inverse problem in diffuse optical tomography. These ℓp-norm-based regularizations make the optimization function nonconvex, and algorithms that implement ℓp-norm minimization utilize approximations to the original ℓp-norm function. In this work, three such typical methods for implementing the ℓp-norm were considered, namely, iteratively reweighted ℓ1-minimization (IRL1), iteratively reweighted least squares (IRLS), and the iteratively thresholding method (ITM). These methods were deployed for performing diffuse optical tomographic image reconstruction, and a systematic comparison with the help of three numerical and gelatin phantom cases was executed. The results indicate that these three methods in the implementation of ℓp-minimization yields similar results, with IRL1 fairing marginally in cases considered here in terms of shape recovery and quantitative accuracy of the reconstructed diffuse optical tomographic images.

Subsurface fluorescence molecular tomography with prior information

Subsurface fluorescence molecular tomography (FMT) is an emerging technique determining fluorescence distribution by tomographic means in reflectance geometry. However, due to the highly diffusive nature of the photon propagation in biological tissues and the influence of nearer source-detector separations, stand-alone subsurface FMT could not accurately reflect the fluorophore distributions. To overcome this drawback, we propose a method to improve the performance of fluorescence imaging by coupling x-ray computed tomography (XCT) and subsurface FMT modalities. A Laplacian-type regularization matrix generated with tissue prior information obtained from XCT images is used to guide the reconstruction of fluorophore distribution. Reconstruction results of both simulation and phantom studies showed that significant improvements in localization and demarcation of fluorescent targets can be obtained with the proposed method compared to the reconstruction method without structural prior information.

MAP estimation with structural priors for fluorescence molecular tomography

Fluorescence molecular tomography (FMT) is an attractive imaging tool for quantitatively and three-dimensionally resolving fluorophore distributions in small animals, but it suffers from low spatial resolution due to its inherent ill-posed nature. Structural priors obtained from a secondary modality system such as x-ray computed tomography or magnetic resonance imaging can help to improve FMT reconstruction results. However, challenge remains in how to fully take advantage of the structural priors while effectively avoid undesirable influence caused by an immoderate usage. In this paper, we propose a new method to resolve the FMT inverse problem based on maximum a posteriori (MAP) estimation with structural priors (MAP-SP) in a Bayesian framework. Instead of imposing the structural priors directly on the reconstruction results, the MAP-SP method utilizes them to constrain the unknown hyperparameters of the prior information model which is essential for the Bayesian framework. Then, a low dimensional inverse problem and an alternating optimization scheme are used to automatically calculate the unknown hyperparameters, which make the FMT reconstruction process self-adaptive. Simulation and phantom results show that the proposed MAP-SP method can effectively make use of the structural priors and leads to improvements in reconstruction quality as compared with traditional regularization methods.

Total variation regularization for 3D reconstruction in fluorescence tomography: experimental phantom studies

Fluorescence tomography (FT) is depth-resolved three-dimensional (3D) localization and quantification of fluorescence distribution in biological tissue and entails a highly ill-conditioned problem as depth information must be extracted from boundary measurements. Conventionally, L2 regularization schemes that penalize the euclidean norm of the solution and possess smoothing effects are used for FT reconstruction. Oversmooth, continuous reconstructions lack high-frequency edge-type features of the original distribution and yield poor resolution. We propose an alternative regularization method for FT that penalizes the total variation (TV) norm of the solution to preserve sharp transitions in the reconstructed fluorescence map while overcoming ill-posedness. We have developed two iterative methods for fast 3D reconstruction in FT based on TV regularization inspired by Rudin-Osher-Fatemi and split Bregman algorithms. The performance of the proposed method is studied in a phantom-based experiment using a noncontact constant-wave trans-illumination FT system. It is observed that the proposed method performs better in resolving fluorescence inclusions at different depths.

Effective contrast recovery in rapid dynamic near-infrared diffuse optical tomography using ℓ(1)-norm-based linear image reconstruction method

Traditional image reconstruction methods in rapid dynamic diffuse optical tomography employ ℓ(2)-norm-based regularization, which is known to remove the high-frequency components in the reconstructed images and make them appear smooth. The contrast recovery in these type of methods is typically dependent on the iterative nature of method employed, where the nonlinear iterative technique is known to perform better in comparison to linear techniques (noniterative) with a caveat that nonlinear techniques are computationally complex. Assuming that there is a linear dependency of solution between successive frames resulted in a linear inverse problem. This new framework with the combination of ℓ(1)-norm-based regularization can provide better robustness to noise and provide better contrast recovery compared to conventional ℓ(2)-based techniques. Moreover, it is shown that the proposed ℓ(1)-based technique is computationally efficient compared to its counterpart (ℓ(2)-based one). The proposed framework requires a reasonably close estimate of the actual solution for the initial frame, and any suboptimal estimate leads to erroneous reconstruction results for the subsequent frames.