Feasibility of U-curve method to select the regularization parameter for fluorescence diffuse optical tomography in phantom and small animal studies

Regularization parameter based on incomplete variables for X-ray luminescence computed tomography

X-ray luminescence computed tomography (XLCT) is an emerging molecular imaging technique for biological application. However, it is still a challenge to get a stable and accurate solution of the reconstruction of XLCT. This paper presents a regularization parameter selection strategy based on incomplete variables frame for XLCT. A residual information, which is derived from Karush-Kuhn-Tucker (KKT) equivalent condition, is employed to determine the regularization parameter. This residual contains the relevant information about the solution norm and gradient norm, which improved the recovered results. Simulation and phantom experiments are designed to test the performance of the algorithm.Clinical Relevance- The results have not yet been used in clinical relevance currently, we believed that this strategy will facilitate the development of the preclinical applications in FMT.

A review of advances in imaging methodology in fluorescence molecular tomography

Objective. Fluorescence molecular tomography (FMT) is a promising non-invasive optical molecular imaging technology with strong specificity and sensitivity that has great potential for preclinical and clinical studies in tumor diagnosis, drug development and therapeutic evaluation. However, the strong scattering of photons and insufficient surface measurements make it very challenging to improve the quality of FMT image reconstruction and its practical application for early tumor detection. Therefore, continuous efforts have been made to explore more effective approaches or solutions in the pursuit of high-quality FMT reconstructions. Approach. This review takes a comprehensive overview of advances in imaging methodology for FMT, mainly focusing on two critical issues in FMT reconstructions: improving the accuracy of solving the forward physical model and mitigating the ill-posed nature of the inverse problem from a methodological point of view. More importantly, numerous impressive and practical strategies and methods for improving the quality of FMT reconstruction are summarized. Notably, deep learning methods are discussed in detail to illustrate their advantages in promoting the imaging performance of FMT thanks to large datasets, the emergence of optimized algorithms and the application of innovative networks. Main results. The results demonstrate that the imaging quality of FMT can be effectively promoted by improving the accuracy of optical parameter modeling, combined with prior knowledge, and reducing dimensionality. In addition, the traditional regularization-based methods and deep neural network-based methods, especially end-to-end deep networks, can enormously alleviate the ill-posedness of the inverse problem and improve the quality of FMT image reconstruction. Significance. This review aims to illustrate a variety of effective and practical methods for the reconstruction of FMT images that may benefit future research. Furthermore, it may provide some valuable research ideas and directions for FMT in the future, and could promote, to a certain extent, the development of FMT and other methods of optical tomography.

Comparison of parameter optimization methods for quantitative susceptibility mapping

PURPOSE

Quantitative Susceptibility Mapping (QSM) is usually performed by minimizing a functional with data fidelity and regularization terms. A weighting parameter controls the balance between these terms. There is a need for techniques to find the proper balance that avoids artifact propagation and loss of details. Finding the point of maximum curvature in the L-curve is a popular choice, although it is slow, often unreliable when using variational penalties, and has a tendency to yield overregularized results.

METHODS

We propose 2 alternative approaches to control the balance between the data fidelity and regularization terms: 1) searching for an inflection point in the log-log domain of the L-curve, and 2) comparing frequency components of QSM reconstructions. We compare these methods against the conventional L-curve and U-curve approaches.

RESULTS

Our methods achieve predicted parameters that are better correlated with RMS error, high-frequency error norm, and structural similarity metric-based parameter optimizations than those obtained with traditional methods. The inflection point yields less overregularization and lower errors than traditional alternatives. The frequency analysis yields more visually appealing results, although with larger RMS error.

CONCLUSION

Our methods provide a robust parameter optimization framework for variational penalties in QSM reconstruction. The L-curve-based zero-curvature search produced almost optimal results for typical QSM acquisition settings. The frequency analysis method may use a 1.5 to 2.0 correction factor to apply it as a stand-alone method for a wider range of signal-to-noise-ratio settings. This approach may also benefit from fast search algorithms such as the binary search to speed up the process.

An automatic estimation of the ridge parameter for extreme learning machine

Extreme learning machine (ELM) is an emerging learning method with a single-hidden layer feed-forward neural network that involves obtaining a solution to the system of linear equations. Unlike traditional gradient-based back-propagating neural networks, ELM is computationally efficient with fast training speed and good generalization capability. However, most of the time when applied to real-time problems, the linear system becomes ill-posed in the structure and needs the inclusion of a ridge parameter to obtain a reliable solution, and hence, the selection of the ridge parameter (C) is a crucial task. The ridge parameter is chosen heuristically from a predefined set. The generalized cross-validation is a widely used technique for the automatic estimation of the same, which is computationally expensive as it involves inversion of large matrices. The focus of the proposed work is on pragmatic aspects of the time-efficient automatic estimation of ridge parameter that result in a better generalization performance. In this work, methods are proposed that use the L-curve and U-curve techniques to automatically estimate the ridge parameter, and these methods are effective in the estimation of the ridge parameter even for systems with larger data. Through extensive numerical results, it is shown that the proposed methods outperform the existing ones in terms of accuracy, precision, sensitivity, specificity, F₁-score, and computational time on various benchmark binary as well as multiclass classification data sets. Finally, the proposed methods are statistically analyzed using the nonparametric Friedman ranking test, which is also proving the effectiveness of the proposed method as it is providing a better rank for the same over existing methods.

Considering New Regularization Parameter-Choice Techniques for the Tikhonov Method to Improve the Accuracy of Electrocardiographic Imaging

The electrocardiographic imaging (ECGI) inverse problem highly relies on adding constraints, a process called regularization, as the problem is ill-posed. When there are no prior information provided about the unknown epicardial potentials, the Tikhonov regularization method seems to be the most commonly used technique. In the Tikhonov approach the weight of the constraints is determined by the regularization parameter. However, the regularization parameter is problem and data dependent, meaning that different numerical models or different clinical data may require different regularization parameters. Then, we need to have as many regularization parameter-choice methods as techniques to validate them. In this work, we addressed this issue by showing that the Discrete Picard Condition (DPC) can guide a good regularization parameter choice for the two-norm Tikhonov method. We also studied the feasibility of two techniques: The U-curve method (not yet used in the cardiac field) and a novel automatic method, called ADPC due its basis on the DPC. Both techniques were tested with simulated and experimental data when using the method of fundamental solutions as a numerical model. Their efficacy was compared with the efficacy of two widely used techniques in the literature, the L-curve and the CRESO methods. These solutions showed the feasibility of the new techniques in the cardiac setting, an improvement of the morphology of the reconstructed epicardial potentials, and in most of the cases of their amplitude.

Recent methodology advances in fluorescence molecular tomography

Molecular imaging (MI) is a novel imaging discipline that has been continuously developed in recent years. It combines biochemistry, multimodal imaging, biomathematics, bioinformatics, cell & molecular physiology, biophysics, and pharmacology, and it provides a new technology platform for the early diagnosis and quantitative analysis of diseases, treatment monitoring and evaluation, and the development of comprehensive physiology. Fluorescence Molecular Tomography (FMT) is a type of optical imaging modality in MI that captures the three-dimensional distribution of fluorescence within a biological tissue generated by a specific molecule of fluorescent material within a biological tissue. Compared with other optical molecular imaging methods, FMT has the characteristics of high sensitivity, low cost, and safety and reliability. It has become the research frontier and research hotspot of optical molecular imaging technology. This paper took an overview of the recent methodology advances in FMT, mainly focused on the photon propagation model of FMT based on the radiative transfer equation (RTE), and the reconstruction problem solution consist of forward problem and inverse problem. We introduce the detailed technologies utilized in reconstruction of FMT. Finally, the challenges in FMT were discussed. This survey aims at summarizing current research hotspots in methodology of FMT, from which future research may benefit.

Automatic selection of regularization parameters for dynamic fluorescence molecular tomography: a comparison of L-curve and U-curve methods

Dynamic fluorescence molecular tomography (FMT) is a promising technique for the study of the metabolic process of fluorescent agents in the biological body in vivo, and the quality of the parametric images relies heavily on the accuracy of the reconstructed FMT images. In typical dynamic FMT implementations, the imaged object is continuously monitored for more than 50 minutes. During each minute, a set of the fluorescent measurements is acquired and the corresponding FMT image is reconstructed. It is difficult to manually set the regularization parameter in the reconstruction of each FMT image. In this paper, the parametric images obtained with the L-curve and U-curve methods are quantitatively evaluated through numerical simulations, phantom experiments and in vivo experiments. The results illustrate that the U-curve method obtains better accuracy, stronger robustness and higher noise-resistance in parametric imaging. Therefore, it is a promising approach to automatic selection of the regularization parameters for dynamic FMT.

Self-guided reconstruction for time-domain fluorescence molecular lifetime tomography

Fluorescence probes have distinct yields and lifetimes when located in different environments, which makes the reconstruction of fluorescence molecular lifetime tomography (FMLT) challenging. To enhance the reconstruction performance of time-domain (TD) FMLT with heterogeneous targets, a self-guided L 1 regularization projected steepest descent (SGL1PSD) algorithm is proposed. Different from other algorithms performed in time domain, SGL1PSD introduces a time-resolved strategy into fluorescence yield reconstruction. The algorithm consists of four steps. Step 1 reconstructs the initial yield map with full time gate strategy; steps 2–4 reconstruct the inverse lifetime map, the yield map, and the inverse lifetime map again with time-resolved strategy, respectively. The reconstruction result of each step is used as a priori for the reconstruction of the next step. Projected iterated Tikhonov regularization algorithm is adopted for the yield map reconstructions in steps 1 and 3 to provide a solution with iterative refinement and nonnegative constraint. The inverse lifetime map reconstructions in steps 2 and 4 are based on L 1 regularization projected steepest descent algorithm, which employ the L 1 regularization to reduce the ill-posedness of the high-dimensional nonlinear problem. Phantom experiments with heterogeneous targets at different edge-to-edge distances demonstrate that SG

Investigation of different sparsity transforms for the PICCS algorithm in small-animal respiratory gated CT

Respiratory gating helps to overcome the problem of breathing motion in cardiothoracic small-animal imaging by acquiring multiple images for each projection angle and then assigning projections to different phases. When this approach is used with a dose similar to that of a static acquisition, a low number of noisy projections are available for the reconstruction of each respiratory phase, thus leading to streak artifacts in the reconstructed images. This problem can be alleviated using a prior image constrained compressed sensing (PICCS) algorithm, which enables accurate reconstruction of highly undersampled data when a prior image is available. We compared variants of the PICCS algorithm with different transforms in the prior penalty function: gradient, unitary, and wavelet transform. In all cases the problem was solved using the Split Bregman approach, which is efficient for convex constrained optimization. The algorithms were evaluated using simulations generated from data previously acquired on a micro-CT scanner following a high-dose protocol (four times the dose of a standard static protocol). The resulting data were used to simulate scenarios with different dose levels and numbers of projections. All compressed sensing methods performed very similarly in terms of noise, spatiotemporal resolution, and streak reduction, and filtered back-projection was greatly improved. Nevertheless, the wavelet domain was found to be less prone to patchy cartoon-like artifacts than the commonly used gradient domain.

Quantitative surface radiance mapping using multiview images of light-emitting turbid media

A novel method is presented for accurately reconstructing a spatially resolved map of diffuse light flux on a surface using images of the surface and a model of the imaging system. This is achieved by applying a model-based reconstruction algorithm with an existing forward model of light propagation through free space that accounts for the effects of perspective, focus, and imaging geometry. It is shown that flux can be mapped reliably and quantitatively accurately with very low error, <3% with modest signal-to-noise ratio. Simulation shows that the method is generalizable to the case in which mirrors are used in the system and therefore multiple views can be combined in reconstruction. Validation experiments show that physical diffuse phantom surface fluxes can also be reconstructed accurately with variability <3% for a range of object positions, variable states of focus, and different orientations. The method provides a new way of making quantitatively accurate noncontact measurements of the amount of light leaving a diffusive medium, such as a small animal containing fluorescent or bioluminescent markers, that is independent of the imaging system configuration and surface position.

A low memory cost model based reconstruction algorithm exploiting translational symmetry for photoacustic microscopy

A model based reconstruction algorithm that exploits translational symmetries for photoacoustic microscopy to drastically reduce the memory cost is presented. The memory size needed to store the model matrix is independent of the number of acquisitions at different positions. This helps us to overcome one of the main limitations of previous algorithms. Furthermore, using the algebraic reconstruction technique and building the model matrix "on the fly", we have obtained fast reconstructions of simulated and experimental data on both two- and three-dimensional grids using a traditional dark field photoacoustic microscope and a standard personal computer.