Inverse Problem Of Diffuse Optical Tomography Research Articles

Deep learning-enabled high-speed, multi-parameter diffuse optical tomography.

Frequency-domain diffuse optical tomography (FD-DOT) could enhance clinical breast tumor characterization. However, conventional diffuse optical tomography (DOT) image reconstruction algorithms require case-by-case expert tuning and are too computationally intensive to provide feedback during a scan. Deep learning (DL) algorithms front-load computational and tuning costs, enabling high-speed, high-fidelity FD-DOT. We aim to demonstrate a simultaneous reconstruction of three-dimensional absorption and reduced scattering coefficients using DL-FD-DOT, with a view toward real-time imaging with a handheld probe. A DL model was trained to solve the DOT inverse problem using a realistically simulated FD-DOT dataset emulating a handheld probe for human breast imaging and tested using both synthetic and experimental data. Over a test set of 300 simulated tissue phantoms for absorption and scattering reconstructions, the DL-DOT model reduced the root mean square error by and , increased the spatial similarity by and , increased the anomaly contrast accuracy by ( ), and reduced the crosstalk by and , respectively, compared with model-based tomography. The average reconstruction time was reduced from 3.8min to 0.02s for a single reconstruction. The model was successfully verified using two tumor-emulating optical phantoms. There is clinical potential for real-time functional imaging of human breast tissue using DL and FD-DOT.

Sparsity promoting reconstructions via hierarchical prior models in diffuse optical tomography

Diffuse optical tomography (DOT) is a severely ill-posed nonlinear inverse problem that seeks to estimate optical parameters from boundary measurements. In the Bayesian framework, the ill-posedness is diminished by incorporating a priori information of the optical parameters via the prior distribution. In case the target is sparse or sharp-edged, the common choice as the prior model are non-differentiable total variation and $ \ell^1 $ priors. Alternatively, one can hierarchically extend the variances of a Gaussian prior to obtain differentiable sparsity promoting priors. By doing this, the variances are treated as unknowns allowing the estimation to locate the discontinuities.In this work, we formulate hierarchical prior models for the nonlinear DOT inverse problem using exponential, standard gamma and inverse-gamma hyperpriors. Depending on the hyperprior and the hyperparameters, the hierarchical models promote different levels of sparsity and smoothness. To compute the MAP estimates, the previously proposed alternating algorithm is adapted to work with the nonlinear model. We then propose an approach based on the cumulative distribution function of the hyperpriors to select the hyperparameters. We evaluate the performance of the hyperpriors with numerical simulations and show that the hierarchical models can improve the localization, contrast and edge sharpness of the reconstructions.

A Model-Based Iterative Learning Approach for Diffuse Optical Tomography.

Diffuse optical tomography (DOT) utilises near-infrared light for imaging spatially distributed optical parameters, typically the absorption and scattering coefficients. The image reconstruction problem of DOT is an ill-posed inverse problem, due to the non-linear light propagation in tissues and limited boundary measurements. The ill-posedness means that the image reconstruction is sensitive to measurement and modelling errors. The Bayesian approach for the inverse problem of DOT offers the possibility of incorporating prior information about the unknowns, rendering the problem less ill-posed. It also allows marginalisation of modelling errors utilising the so-called Bayesian approximation error method. A more recent trend in image reconstruction techniques is the use of deep learning, which has shown promising results in various applications from image processing to tomographic reconstructions. In this work, we study the non-linear DOT inverse problem of estimating the (absolute) absorption and scattering coefficients utilising a 'model-based' learning approach, essentially intertwining learned components with the model equations of DOT. The proposed approach was validated with 2D simulations and 3D experimental data. We demonstrated improved absorption and scattering estimates for targets with a mix of smooth and sharp image features, implying that the proposed approach could learn image features that are difficult to model using standard Gaussian priors. Furthermore, it was shown that the approach can be utilised in compensating for modelling errors due to coarse discretisation enabling computationally efficient solutions. Overall, the approach provided improved computation times compared to a standard Gauss-Newton iteration.

Novel regularization method for diffuse optical tomography inverse problem

Diffuse optical tomography (DOT) is an optical imaging technique using near-infrared spectroscopy (NIRS). Because the NIR light is highly scattered in biological tissue, the photon density drops rapidly with increasing depth. This makes the measurement accuracy of DOT in deep tissue significantly lower than in superficial tissue. Classical Regularized Least Square (RLS) based regularization methods with different penalty term (l1-norm, l2-norm, LASSO, Ridge, and Elastic Net etc.) come up short in finding the inclusions when the inclusion number is increased or one of the inclusion located on the upper surface and the other one located on the deeper surface of the tissue. This is the depth problem encountered in DOT. In this paper, a new least square based regularization method is proposed. Within the scope of the proposed method, both the weight matrix and penalty terms are modified in order to solve this problem. In this process, firstly, the A matrix is changed using the diagonal value of AAT based on DCA (Depth Compensation Algorithm). Then, a new penalty term using modified A matrix is added. It has been observed that when the noise level increases, proposed method give higher accuracy and more robust according to other RLS Method on synthetic data. Experimental results on synthetic data show that the new method improves inversion accuracy, reduces errors, and the lowers inversion instability compared to the state-of-the-art methods.

Regression-based neural network for improving image reconstruction in diffuse optical tomography.

Diffuse optical tomography (DOT) is a non-invasive imaging technique utilizing multi-scattered light at visible and infrared wavelengths to detect anomalies in tissues. However, the DOT image reconstruction is based on solving the inverse problem, which requires massive calculations and time. In this article, for the first time, to the best of our knowledge, a simple, regression-based cascaded feed-forward deep learning neural network is derived to solve the inverse problem of DOT in compressed breast geometry. The predicted data is subsequently utilized to visualize the breast tissues and their anomalies. The dataset in this study is created using a Monte-Carlo algorithm, which simulates the light propagation in the compressed breast placed inside a parallel plate source-detector geometry (forward process). The simulated DL-DOT system's performance is evaluated using the Pearson correlation coefficient (R) and the Mean squared error (MSE) metrics. Although a comparatively smaller dataset (50 nos.) is used, our simulation results show that the developed feed-forward network algorithm to solve the inverse problem delivers an increment of ∼30% over the analytical solution approach, in terms of R. Furthermore, the proposed network's MSE outperforms that of the analytical solution's MSE by a large margin revealing the robustness of the network and the adaptability of the system for potential applications in medical settings.

Study on the inverse problem of diffuse optical tomography based on improved stacked auto-encoder

The inverse problem of diffuse optical tomography (DOT) is ill-posed. Traditional method cannot achieve high imaging accuracy and the calculation process is time-consuming, which restricts the clinical application of DOT. Therefore, a method based on stacked auto-encoder (SAE) was proposed and used for the DOT inverse problem. Firstly, a traditional SAE method is used to solved the inverse problem. Then, the output structure of SAE neural network is improved to a single output SAE, which reduce the burden on the neural network. Finally, the improved SAE method is used to compare with traditional SAE method and traditional levenberg-marquardt (LM) iterative method. The result shows that the average time to solve the inverse problem of the method proposed in this paper is only 1.67% of the LM method. The mean square error (MSE) value is 46.21% lower than the traditional iterative method, 61.53% lower than the traditional SAE method, and the image correlation coefficient(ICC) value is 4.03% higher than the traditional iterative method, 18.7% higher than the traditional SAE method and has good noise immunity under 3% noise conditions. The research results in this article prove that the improved SAE method has higher image quality and noise resistance than the traditional SAE method, and at the same time has a faster calculation speed than the traditional iterative method, which is conducive to the application of neural networks in DOT inverse problem calculation.

Image Reconstruction in Diffuse Optical Tomography Using Adaptive Moment Gradient Based Optimizers: A Statistical Study

Diffuse optical tomography (DOT) is an emerging modality that reconstructs the optical properties in a highly scattering medium from measured boundary data. One way to solve DOT and recover the quantities of interest is by an inverse problem approach, which requires the choice of an optimization algorithm for the iterative approximation of the solution. However, the well-established and proven fact of the no free lunch principle holds in general. This paper aims to compare the behavior of three gradient descent-based optimizers on solving the DOT inverse problem by running randomized simulation and analyzing the generated data in order to shade light on any significant difference—if existing at all—in performance among these optimizers in our specific context of DOT. The major practical problems when selecting or using an optimization algorithm in a production context for a DOT system is to be confident that the algorithm will have a high convergence rate to the true solution, reasonably fast speed and high quality of the reconstructed image in terms of good localization of the inclusions and good agreement with the true image. In this work, we harnessed carefully designed randomized simulations to tackle the practical problem of choosing the right optimizer with the right parameters in the context of practical DOT applications, and derived statistical results concerning rate of convergence, speed, and quality of image reconstruction. The statistical analysis performed on the generated data and the main results for convergence rate, reconstruction speed, and quality between three optimization algorithms are presented in the paper at hand.

Mathematical and numerical challenges in optical screening of female breast.

Diffuse optical tomography (DOT) is an emerging imaging technique which uses light for diagnostic purposes in a non-invasive and non-ionizing way. In this paper, we focus on DOT application to female breast screening, where the surface of the breast is illuminated by light sources and the outgoing light is collected on the surface. The comparison of measured light data with the equivalent field obtained from a relevant mathematical model yields the DOT inverse problem whose solution provides an estimate of the optical coefficients of the tissue. These latter, in turn, can be related to clinical markers for cancer detection. The goal of this work is to propose a mathematical and computational approach tailored to the concept of a DOT imaging device able to perform fast and accurate screenings at an affordable cost. Namely, we address two original points about the crucial issue of the solution of the severely ill-conditioned DOT inverse problem: (a) a computational approach based on Green's functions which do not require the exact knowledge of the tissue geometry, proposed here in the declination of the Method of Fundamental Solutions, which allows to enforce correct boundary conditions; (b) the elastic net regularization technique that shares the desirable properties of both the ℓ2 - and ℓ1 -norm penalization approaches and opens the possibility for sparsity recognition in the optical coefficients field and refinement procedures.

Modified CS-MUSIC for diffuse optical tomography using joint sparsity

In this paper, a modified compressive multiple signal classification (CS-MUSIC) algorithm has been proposed for diffuse optical tomography (DOT) reconstruction. The basic principle involved in DOT is that it illuminates the biological tissue using near infrared light and reconstructs the optical parameters of the tissue from the boundary measurements. The inverse problem of diffuse optical tomography is non-linear and severely ill-conditioned due to the zig-zag nature of light propagation by photons that diffuses through the tissue. Although nonlinear iterative methods are commonly used to solve this problem, they are computationally expensive since the forward problem has to be solved iteratively as well as they do not perform well for complex geometries. Recently, the DOT with compressive sensing (CS) has received a great attention due to its efficient possible reconstructions in DOT imaging. In this, the DOT inverse problem has been formulated as an multiple measurement vector (MMV) problem by using joint sparsity and CS frame work. The modified CS-MUSIC is a novel, non-iterative, and exact algorithm to reconstruct the absorption parameter change for Δα from the boundary data. In addition, this algorithm takes hybridization of sensor array signal processing and probabilistic compressive sensing. The experimental validation of the proposed algorithm has been done on a paraffin wax rectangular phantom through a DOT imaging setup. The performance metrics such as structural similarity index (SSIM), mean square error (MSE), normalized mean square error (NMSE) have been used to evaluate the performance of the reconstruction in this paper. Extensive numerical simulations show that the modified CS-MUSIC algorithm outperforms the current state-of-the-art algorithms and reliably reconstructs the absorption change in DOT.

Greedy algorithms for diffuse optical tomography reconstruction

Diffuse optical tomography (DOT) is a noninvasive imaging modality that reconstructs the optical parameters of a highly scattering medium. However, the inverse problem of DOT is ill-posed and highly nonlinear due to the zig-zag propagation of photons that diffuses through the cross section of tissue. The conventional DOT imaging methods iteratively compute the solution of forward diffusion equation solver which makes the problem computationally expensive. Also, these methods fail when the geometry is complex. Recently, the theory of compressive sensing (CS) has received considerable attention because of its efficient use in biomedical imaging applications. The objective of this paper is to solve a given DOT inverse problem by using compressive sensing framework and various Greedy algorithms such as orthogonal matching pursuit (OMP), compressive sampling matching pursuit (CoSaMP), and stagewise orthogonal matching pursuit (StOMP), regularized orthogonal matching pursuit (ROMP) and simultaneous orthogonal matching pursuit (S-OMP) have been studied to reconstruct the change in the absorption parameter i.e, Δα from the boundary data. Also, the Greedy algorithms have been validated experimentally on a paraffin wax rectangular phantom through a well designed experimental set up. We also have studied the conventional DOT methods like least square method and truncated singular value decomposition (TSVD) for comparison. One of the main features of this work is the usage of less number of source–detector pairs, which can facilitate the use of DOT in routine applications of screening. The performance metrics such as mean square error (MSE), normalized mean square error (NMSE), structural similarity index (SSIM), and peak signal to noise ratio (PSNR) have been used to evaluate the performance of the algorithms mentioned in this paper. Extensive simulation results confirm that CS based DOT reconstruction outperforms the conventional DOT imaging methods in terms of computational efficiency. The main advantage of this study is that the forward diffusion equation solver need not be repeatedly solved.

Image reconstruction for diffuse optical tomography based on radiative transfer equation.

Diffuse optical tomography is a novel molecular imaging technology for small animal studies. Most known reconstruction methods use the diffusion equation (DA) as forward model, although the validation of DA breaks down in certain situations. In this work, we use the radiative transfer equation as forward model which provides an accurate description of the light propagation within biological media and investigate the potential of sparsity constraints in solving the diffuse optical tomography inverse problem. The feasibility of the sparsity reconstruction approach is evaluated by boundary angular-averaged measurement data and internal angular-averaged measurement data. Simulation results demonstrate that in most of the test cases the reconstructions with sparsity regularization are both qualitatively and quantitatively more reliable than those with standard L 2 regularization. Results also show the competitive performance of the split Bregman algorithm for the DOT image reconstruction with sparsity regularization compared with other existing L 1 algorithms.

Differential visualisation of a spectrally selective structure of strongly scattering objects

We describe a modification of the algorithm for the fast approximate solution of the diffuse optical tomography inverse problem. In this modification the amount of a priori (auxiliary) information necessary for the visualisation of the internal structure of the object is reduced by using a differential measurement scheme. The experiment is performed at two different wavelengths, and some a priori information, necessary to reconstruct only the spectrally selective component of the internal structure (the difference structure of the spatial distributions of the extinction coefficient at the wavelength employed), is replaced by the data of one of these measurements.

Theoretical limit of spatial resolution in diffuse optical tomography using a perturbation model

We have assessed the limit of spatial resolution of time-domain diffuse optical tomography (DOT) based on a perturbation reconstruction model. From the viewpoint of the structure reconstruction accuracy, three different approaches to solving the inverse DOT problem are compared. The first approach involves reconstruction of diffuse tomograms from straight lines, the second – from average curvilinear trajectories of photons and the third – from total banana-shaped distributions of photon trajectories. In order to obtain estimates of resolution, we have derived analytical expressions for the point spread function and modulation transfer function, as well as have performed a numerical experiment on reconstruction of rectangular scattering objects with circular absorbing inhomogeneities. It is shown that in passing from reconstruction from straight lines to reconstruction using distributions of photon trajectories we can improve resolution by almost an order of magnitude and exceed the accuracy of reconstruction of multi-step algorithms used in DOT.

Joint sparsity-driven non-iterative simultaneous reconstruction of absorption and scattering in diffuse optical tomography

Some optical properties of a highly scattering medium, such as tissue, can be reconstructed non-invasively by diffuse optical tomography (DOT). Since the inverse problem of DOT is severely ill-posed and nonlinear, iterative methods that update Green's function have been widely used to recover accurate optical parameters. However, recent research has shown that the joint sparse recovery principle can provide an important clue in achieving reconstructions without an iterative update of Green's function. One of the main limitations of the previous work is that it can only be applied to absorption parameter reconstruction. In this paper, we extended this theory to estimate the absorption and scattering parameters simultaneously when the background optical properties are known. The main idea for such an extension is that a joint sparse recovery step gives us unknown fluence on the estimated support set, which eliminates the nonlinearity in an integral equation for the simultaneous estimation of the optical parameters. Our numerical results show that the proposed algorithm reduces the cross-talk artifacts between the parameters and provides improved reconstruction results compared to existing methods.

Extended hierarchical Bayesian diffuse optical tomography for removing scalp artifact

Functional near-infrared spectroscopy (fNIRS) can non-invasively measure hemodynamic responses in the cerebral cortex with a portable apparatus. However, the observation signal in fNIRS measurements is contaminated by the artifact signal from the hemodynamic response in the scalp. In this paper, we propose a method to separate the signals from the cortex and the scalp by estimating both hemodynamic changes by diffuse optical tomography (DOT). In the inverse problem of DOT, we introduce smooth regularization to the hemodynamic change in the scalp and sparse regularization to that in the cortex based on the nature of the hemodynamic responses. These appropriate regularization models, with the spatial information of optical paths of many measurement channels, allow three-dimensional reconstruction of both hemodynamic changes. We validate our proposed method through two-layer phantom experiments and MRI-based head-model simulations. In both experiments, the proposed method simultaneously estimates the superficial smooth activity in the scalp area and the deep localized activity in the cortical area.

State space regularization in the nonstationary inverse problem for diffuse optical tomography

In this paper, we present a regularization method in the nonstationary inverse problem for diffuse optical tomography (DOT). The regularization is based on a choosing time evolution process such that in a stationary state it has a covariance function which corresponds to a process with similar smoothness properties as the first-order smoothness Tikhonov regularization. The proposed method is computationally more lightweight than the method where the regularization is augmented as a measurement. The method was tested in the case of the inverse problem of DOT. A solid phantom with optical properties similar to tissue was made, incorporating two moving parts that simulate two different physiological processes: a localized change in absorption and a surrounding rotating two-part shell which simulates slow oscillations in the tissue background physiology. A sequence of measurements of the phantom was made and the reconstruction of the image sequence was computed using this method. It allows the recovery of the full time series of images from relatively slow measurements with one source active at a time. In practice, this allows instruments with a larger dynamic range to be applied to the imaging of functional phenomena using DOT.

An EM‐like reconstruction method for diffuse optical tomography

AbstractDiffuse optical tomography (DOT) is an optical imaging modality which provides the spatial distributions of optical parameters inside an object. The forward model of DOT is described by the diffusion approximation of radiative transfer equation, while the DOT is to reconstruct the optical parameters from boundary measurements. In this paper, an EM‐like iterative reconstruction method specifically for the steady state DOT problem is developed. Previous iterative reconstruction methods are mostly based on the assumption that the measurement noise is Gaussian, and are of least‐squares type. In this paper, with the assumption that the boundary measurements have independent and identical Poisson distributions, the inverse problem of DOT is solved by maximizing a log‐likelihood functional with inequality constraints, and then an EM‐like reconstruction algorithm is developed according to the Kuhn–Tucker condition. The proposed algorithm is a variant of the well‐known EM algorithm. The performance of the proposed algorithm is tested with three‐dimensional numerical simulation. Copyright © 2010 John Wiley & Sons, Ltd.

Time-domain diffuse optical tomography using analytic statistical characteristics of photon trajectories

The inverse problem of diffuse optical tomography (DOT) is reduced by the method of photon average trajectories (PAT) to the solution of the integral equation integrated along the conditional mean statistical photon trajectory. The PAT bending near the flat boundary of a scattering medium is estimated analytically. These estimates are used to determine the analytic statistical characteristics of photon trajectories for the flat layer geometry. The inverse DOT problem is solved by using the multiplicative algebraic algorithm modified to improve the convergence of the iteration reconstruction process. The numerical experiment shows that the modified PAT method permits the reconstruction of near-surface optical inhomogeneities virtually without distortions.

Anisotropic diffusion regularization methods for diffuse optical tomography using edge prior information

Diffuse optical tomography (DOT) is a non-invasive functional imaging modality that aims to image the optical properties of biological organs. The forward problem of the light propagation of DOT can be modelled as a diffusion process and is expressed as a differential diffusion equation with boundary conditions. The solution of the DOT inverse problem can be formulated as a minimization of some functional that measures the discrepancy between the measured data and the data produced by representation of the modelled object. The minimization of this term alone would force the solution to be consistent with the data and the solver used for this purpose. But since in practice data are always accompanied with noise and with some jump discontinuities, these will unavoidably yield unsatisfactory solutions, so some regularization to restore the solution is required. In this paper we introduce an anisotropic regularization term using a priori structural information about the object. This term aims to reduce the noise associated with the data and to preserve the edges in the solution by a combined strategy using the a priori edge information and the diffusion flux analysis of the local structures at each iteration. To accelerate the iterative solver we use a particular method called the lagged diffusivity Newton–Krylov method. The whole proposed strategy, which makes use of a priori diffusion information has been developed and evaluated on simulated data.

Imaging the body with diffuse optical tomography

Diffuse optical tomography (DOT) is an ongoing medical imaging modality in which tissue is illuminated by near-infrared light from an array of sources, the multiply-scattered light which emerges is observed with an array of detectors, and then a model of the propagation physics is used to infer the localized optical properties of the illuminated tissue. The three primary absorbers at these wavelengths, water and both oxygenated and deoxygenated hemoglobin, all have relatively weak absorption. This fortuitous fact provides a spectral window through which we can attempt to localize absorption (primarily by the two forms of hemoglobin) and scattering in the tissue. The most important current applications of DOT are detecting tumors in the breast and imaging the brain. We introduce the basic idea of DOT and review the history of optical methods in medicine as relevant to the development of DOT. We then detail the concept of DOT, including a review of the tissue's optical properties, modes of operation for DOT, and the challenges which the development of DOT must overcome. The basics of modelling the DOT forward problem and some critical issues among the numerous implementations that have been investigated for the DOT inverse problem, with an emphasis on signal processing. We summarize with some specific results as examples of the current state of DOT research.