Problem Of Diffuse Optical Tomography Research Articles

Inverse problem for a nonlocal diffuse optical tomography equation

In this article a nonlocal analogue of an inverse problem in diffuse optical tomography is considered. We show that whenever one has given two pairs of diffusion and absorption coefficients , , such that there holds in the measurement set W and they generate the same DN data, then they are necessarily equal in and Ω, respectively. Additionally, we show that the condition is optimal in the sense that without this restriction one can construct two distinct pairs , generating the same DN data.

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.

Fast linear inversion for highly overdetermined inverse scattering problems

In this paper we present a fast numerical method for solving large-scale inverse scattering problems. The computational efficiency of the proposed method stems from the utilization of the special structure of the linear forward scattering operator, and does not require or assume any symmetries of the measurement geometry. The described approach is especially useful for inverse problems involving large data sets. As an illustration, we have performed direct numerical inversions for the problem of diffuse optical tomography in measurement geometries with up to ∼108 independent data points and ∼ unknowns.

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.

A regularization method based on level sets and augmented Lagrangian for parameter identification problems with piecewise constant solutions

We propose and analyse a regularization method for parameter identification problems modeled by ill-posed nonlinear operator equations, where the parameter to be identified is a piecewise constant function taking known values.Following (De Cezaro et al 2013 Inverse Problems 29 015003), a piecewise constant level set approach is used to represent the unknown parameter, and a corresponding Tikhonov functional is defined on an appropriated space of level set functions. Additionally, a suitable constraint is enforced, resulting that minimizers of our Tikhonov functional belong to the set of piecewise constant level set functions. In other words, the original parameter identification problem is rewritten in the form of a constrained optimization problem, which is solved using an augmented Lagrangian method.We prove the existence of zero duality gaps and the existence of generalized Lagrangian multipliers. Moreover, we extend the analysis in De Cezaro et al’s work (2013 Inverse Problems 29 015003), proving convergence and stability of the proposed parameter identification method.A primal-dual algorithm is proposed to compute approximate solutions of the original inverse problem, and its convergence is proved. Numerical examples are presented: this algorithm is applied to a 2D diffuse optical tomography problem. The numerical results are compared with the ones in Agnelli et al (2017 ESAIM: COCV 23 663–83) demonstrating the effectiveness of this primal-dual algorithm.

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.

Determination of space-dependent radiative properties in diffuse optical tomography using a wavelet multi-scale method

This paper deals with the estimation of radiative property distributions of participating media from a set of light sources and sensors located on the boundaries of a medium. This is the so-called diffuse optical tomography problem. Such a non-linear ill-posed inverse problem is solved through the minimization of a cost function which depends on the discrepancy, in a least-square sense, between some measurements and associated predictions. In the present case, predictions are based on the diffuse approximation model in the frequency domain while the optimization problem is solved by the L-BFGS algorithm. To cope with the local convergence property of the optimizer and the presence of numerous local minima in the cost function, a wavelet multi-scale method associated with the L-BFGS method is designed.

Space-Dependent Sobolev Gradients as a Regularization for Inverse Radiative Transfer Problems

Diffuse optical tomography problems rely on the solution of an optimization problem for which the dimension of the parameter space is usually large. Thus, gradient-type optimizers are likely to be used, such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, along with the adjoint-state method to compute the cost function gradient. Usually, theL2-inner product is chosen within the extraction procedure (i.e., in the definition of the relationship between the cost function gradient and the directional derivative of the cost function) while alternative inner products that act as regularization can be used. This paper presents some results based on space-dependent Sobolev inner products and shows that this method acts as an efficient low-pass filter on the cost function gradient. Numerical results indicate that the use of Sobolev gradients can be particularly attractive in the context of inverse problems, particularly because of the simplicity of this regularization, since a single additional diffusion equation is to be solved, and also because the quality of the solution is smoothly varying with respect to the regularization parameter.

Regularization opportunities for the diffuse optical tomography problem

In optical tomography, the radiative properties of the medium under investigation are estimated from information contained in measurements provided by a set of light sources and sensors located on the frontier of the probed medium. Such a non-linear ill-posed inverse problem is usually solved through optimization with the help of gradient-type methods. Since it is well known that such inverse problems are ill-posed, regularization is required. This paper compares three distinct regularization strategies for two different optimization algorithms, namely the damped Gauss–Newton and BFGS algorithms, for the two-dimensional diffuse approximation to the radiative transfer equation in the frequency domain. More specifically, the mesh-based regularization is combined with the Tikhonov regularization in the damped Gauss–Newton algorithm. For the BFGS algorithm, the mesh-based regularization is combined with the Sobolev gradients method. Moreover, a space-dependent Sobolev gradients method is proposed for the first time. The performance of the proposed algorithms are compared by utilizing synthetic data. The deviation factor and correlation coefficient are used to quantitatively compare the final reconstructions. Also, three levels of noise are considered to characterize the behaviour of the proposed methods against measurement noises. Numerical results indicate that the BFGS algorithm outperforms the damped Gauss–Newton in many aspects.

A wavelet multi-scale method for the inverse problem of diffuse optical tomography

This paper deals with the estimation of optical property distributions of participating media from a set of light sources and sensors located on the boundaries of the medium. This is the so-called diffuse optical tomography problem. Such a non-linear ill-posed inverse problem is solved through the minimization of a cost function which depends on the discrepancy, in a least-square sense, between some measurements and associated predictions. In the present case, predictions are based on the diffuse approximation model in the frequency domain while the optimization problem is solved by the L-BFGS algorithm. To cope with the local convergence property of the optimizer and the presence of numerous local minima in the cost function, a wavelet multi-scale method associated with the L-BFGS method is developed, implemented, and validated. This method relies on a reformulation of the original inverse problem into a sequence of sub-inverse problems of different scales using wavelet transform, from the largest scale to the smallest one. Numerical results show that the proposed method brings more stability with respect to the ordinary L-BFGS method and enhances the reconstructed images for most of initial guesses of optical properties.

Performance evaluation of typical approximation algorithms for nonconvex ℓ_p-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.

Fast multipole boundary element method for diffuse optical tomography

The forward problem of diffuse optical tomography (DOT) is commonly solved by the finite element method (FEM) currently. However, with the increase of the model scale, the computational complexity of FEM increases significantly; while the boundary element method (BEM) attracts much attention because of its reduction in calculated dimensions. In this paper, the fast multipole boundary element method (FMBEM) for DOT is studied using a model of highly scattering homogenous medium. In FMBEM, by the multipole expansions of kernel functions, the product of matrix coefficient and iterative vector can be equivalent to the recursion of a quadtree; and then a generalized minimal residual method is used to solve the BEM equation iteratively. The calculations of the FMBEM are compared with Monte Carlo simulations. The results show that the calculations of the FMBEM are in good agreement with Monte Carlo simulations. This demonstrates the feasibility of FMBEM in DOT and indicates that the FMBEM has a bright future for large-scale and real-time imaging.

A pseudo-time EnKF incorporating shape based reconstruction for diffuse optical tomography

The authors aim at developing a pseudo-time, sub-optimal stochastic filtering approach based on a derivative free variant of the ensemble Kalman filter (EnKF) for solving the inverse problem of diffuse optical tomography (DOT) while making use of a shape based reconstruction strategy that enables representing a cross section of an inhomogeneous tumor boundary by a general closed curve. The optical parameter fields to be recovered are approximated via an expansion based on the circular harmonics (CH) (Fourier basis functions) and the EnKF is used to recover the coefficients in the expansion with both simulated and experimentally obtained photon fluence data on phantoms with inhomogeneous inclusions. The process and measurement equations in the pseudo-dynamic EnKF (PD-EnKF) presently yield a parsimonious representation of the filter variables, which consist of only the Fourier coefficients and the constant scalar parameter value within the inclusion. Using fictitious, low-intensity Wiener noise processes in suitably constructed "measurement" equations, the filter variables are treated as pseudo-stochastic processes so that their recovery within a stochastic filtering framework is made possible. In our numerical simulations, we have considered both elliptical inclusions (two inhomogeneities) and those with more complex shapes (such as an annular ring and a dumbbell) in 2-D objects which are cross-sections of a cylinder with background absorption and (reduced) scattering coefficient chosen as μ(a) (b)=0.01mm(-1) and μ(s) ('b)=1.0mm(-1), respectively. We also assume μ(a) = 0.02 mm(-1) within the inhomogeneity (for the single inhomogeneity case) and μ(a) = 0.02 and 0.03 mm(-1) (for the two inhomogeneities case). The reconstruction results by the PD-EnKF are shown to be consistently superior to those through a deterministic and explicitly regularized Gauss-Newton algorithm. We have also estimated the unknown μ(a) from experimentally gathered fluence data and verified the reconstruction by matching the experimental data with the computed one. The PD-EnKF, which exhibits little sensitivity against variations in the fictitiously introduced noise processes, is also proven to be accurate and robust in recovering a spatial map of the absorption coefficient from DOT data. With the help of shape based representation of the inhomogeneities and an appropriate scaling of the CH expansion coefficients representing the boundary, we have been able to recover inhomogeneities representative of the shape of malignancies in medical diagnostic imaging.

Pseudo-time particle filtering for diffuse optical tomography

We recast the reconstruction problem of diffuse optical tomography (DOT) in a pseudo-dynamical framework and develop a method to recover the optical parameters using particle filters, i.e., stochastic filters based on Monte Carlo simulations. In particular, we have implemented two such filters, viz., the bootstrap (BS) filter and the Gaussian-sum (GS) filter and employed them to recover optical absorption coefficient distribution from both numerically simulated and experimentally generated photon fluence data. Using either indicator functions or compactly supported continuous kernels to represent the unknown property distribution within the inhomogeneous inclusions, we have drastically reduced the number of parameters to be recovered and thus brought the overall computation time to within reasonable limits. Even though the GS filter outperformed the BS filter in terms of accuracy of reconstruction, both gave fairly accurate recovery of the height, radius, and location of the inclusions. Since the present filtering algorithms do not use derivatives, we could demonstrate accurate contrast recovery even in the middle of the object where the usual deterministic algorithms perform poorly owing to the poor sensitivity of measurement of the parameters. Consistent with the fact that the DOT recovery, being ill posed, admits multiple solutions, both the filters gave solutions that were verified to be admissible by the closeness of the data computed through them to the data used in the filtering step (either numerically simulated or experimentally generated).

Image reconstruction in diffuse optical tomography using the coupled radiative transport–diffusion model

The coupled radiative transport–diffusion model can be used as light transport model in situations in which the diffusion equation is not a valid approximation everywhere in the domain. In the coupled model, light propagation is modelled with the radiative transport equation in sub-domains in which the approximations of the diffusion equation are not valid, such as within low-scattering regions, and the diffusion approximation is used elsewhere in the domain. In this paper, an image reconstruction method for diffuse optical tomography based on using the coupled radiative transport–diffusion model is developed. In the approach, absorption and scattering distributions are estimated by minimising a regularised least-squares error between the measured data and solution of the coupled model. The approach is tested with simulations. Reconstructions from different cases including domains with low-scattering regions are shown. The results show that the coupled radiative transport–diffusion model can be utilised in image reconstruction problem of diffuse optical tomography and that it produces as good quality reconstructions as the full radiative transport equation also in the presence of low-scattering regions.

SU‐C‐211‐05: Improved Target Localization with Integrated X‐Ray/bioluminescent Tomography

Purpose: Bioluminescence tomography (BLT) is a promising approach to guide focal irradiation of pre‐palpable tumors in small animals, but is limited in resolution. In this study, we investigate the use of x‐ray and diffuse optical tomography tomography to provide spatial and optical information about the object to improve BLT reconstruction. Methods: In a 3‐step approach, x‐ray CBCT is used first to provide the anatomic spatial information about the object. Secondly, photons of different wavelengths are transmitted through the object for detection by a CCD camera. The optical properties of the anatomic regions are then found by solutions of the diffuse optical tomography problem. In the final step, the anatomic and optical “priors” are employed in BLT reconstruction, where optical filters are used again to extract the multi‐spectral information required to derive the position and power of the internal source. The approach was tested in an experimental setup where a 1.5 mm diameter (dia) optical fiber light source was inserted at 5 different lateral positions in a 12 mm dia optical heterogeneity within a 25 mm dia cylindrical tissue‐mimicking phantom. The reconstructed positions and powers were compared with the known inputs. Results: The average error in locating the source centroid was 0.13 mm, range 0.3 mm to 0 mm. The average error in source power estimations was 9%, range 14% to −1%. The results are an improvement over commercial systems that typically have an average localization uncertainty of 1 mm and error in power estimation of more than 20%. Conclusions: The spatial and spectral prior information provided by x‐ray CT and diffuse optical tomography improves the accuracy of BLT. Integrated x‐ray/BL tomographic capability on board new small animal radiation research systems can contribute significantly to accurate biological targeting in realistic animal models.J Wong has a sponsored research agreement with Xstrahl Ltd on small animal radiation research.

Compressive Diffuse Optical Tomography: Noniterative Exact Reconstruction Using Joint Sparsity

Diffuse optical tomography (DOT) is a sensitive and relatively low cost imaging modality that reconstructs optical properties of a highly scattering medium. However, due to the diffusive nature of light propagation, the problem is severely ill-conditioned and highly nonlinear. Even though nonlinear iterative methods have been commonly used, they are computationally expensive especially for three dimensional imaging geometry. Recently, compressed sensing theory has provided a systematic understanding of high resolution reconstruction of sparse objects in many imaging problems; hence, the goal of this paper is to extend the theory to the diffuse optical tomography problem. The main contributions of this paper are to formulate the imaging problem as a joint sparse recovery problem in a compressive sensing framework and to propose a novel noniterative and exact inversion algorithm that achieves the l(0) optimality as the rank of measurement increases to the unknown sparsity level. The algorithm is based on the recently discovered generalized MUSIC criterion, which exploits the advantages of both compressive sensing and array signal processing. A theoretical criterion for optimizing the imaging geometry is provided, and simulation results confirm that the new algorithm outperforms the existing algorithms and reliably reconstructs the optical inhomogeneities when we assume that the optical background is known to a reasonable accuracy.

Analysis and Regularization of Problems in Diffuse Optical Tomography

In this paper we consider the regularization of the inverse problem of diffuse optical tomography by standard regularization methods with quadratic penalty terms. We therefore investigate in detail the properties of the associated forward operators and derive continuity and differentiability results, which are based on derivation of $W^{1,p}$ regularity of solutions for the governing elliptic boundary value problems. We then show that Tikhonov regularization can be applied for a stable solution and that the standard convergence and convergence rates results hold. Our analysis also allows us to ensure convergence of iterative regularization methods, which are important from a practical point of view.

Simulation study on the inverse problem of diffuse optical tomography with near-infrared spectroscopy

Simulation study on the inverse problem of diffuse optical tomography with near-infrared spectroscopy

Approximations of the measurable quantity in diffuse optical problems: theoretical analysis of model deviations

Photon density and photon flux are widely used to model the measurable quantity in diffuse optical tomography problems. However, it is not these two quantities that are actually measured, but rather the radiance accepted by the detection system. We provide a theoretical analysis of the model deviations related to the choice of the measurable quantity-either photon density or flux. By using the diffusion approximation to the radiative transfer equation and its solution with extrapolated boundary conditions, an exact analytical expression of the measurable quantity has been obtained. This expression has been employed as a reference to assess model deviation when considering the photon density or the photon flux as the measurable quantity. For the case of semi-infinite geometry and for both continuous wave and time domains, we show that the photon density approximates the measurable quantity better than the photon flux. We also demonstrate that the validity of this approximation strongly depends on the optical parameters.