Reduced inverse Born series: a computational study.

We investigate the inverse scattering problem for scalar waves. We report conditions under which the terms in the inverse Born series cancel in pairs, leaving only one term at each order. We refer to the resulting expansion as the reduced inverse Born series. The reduced series can also be derived from a nonperturbative inversion formula. Our results are illustrated by numerical simulations that compare the performance of the reduced series to the full inverse Born series and the Newton-Kantorovich method.

Algorithms and instrumentation for rapid spatial frequency domain fluorescence diffuse optical imaging.

Significance: Rapid estimation of the depth and margins of fluorescence targets buried below the tissue surface could improve upon current image-guided surgery techniques for tumor resection. Aim: We describe algorithms and instrumentation that permit rapid estimation of the depth and transverse margins of fluorescence target(s) in turbid media; the work aims to introduce, experimentally demonstrate, and characterize the methodology. Approach: Spatial frequency domain fluorescence diffuse optical tomography (SFD-FDOT) technique is adapted for rapid and computationally inexpensive estimation of fluorophore target depth and lateral margins. The algorithm utilizes the variation of diffuse fluorescence intensity with respect to spatial-modulation-frequency to compute target depth. The lateral margins are determined via analytical inversion of the data using depth information obtained from the first step. We characterize method performance using fluorescent contrast targets embedded in tissue-simulating phantoms. Results: Single and multiple targets with significant lateral size were imaged at varying depths as deep as 1 cm. Phantom data analysis showed good depth-sensitivity, and the reconstructed transverse margins were mostly within ∼30 % error from true margins. Conclusions: The study suggests that the rapid SFD-FDOT approach could be useful in resection surgery and, more broadly, as a first step in more rigorous SFD-FDOT reconstructions. The experiments permit evaluation of current limitations.

Image reconstruction in non-reciprocal broken-ray tomography.

Optical methods of biomedical tomographic imaging are of considerable interest due to their non-invasive nature and sensitivity to physiologically important markers. Similarly to other imaging modalities, optical methods can be enhanced by utilizing extrinsic contrast agents. Typically, these are fluorescent molecules, which can aggregate in regions of interest due to various mechanisms. In the current approaches to imaging, the intrinsic (related to the tissue) and extrinsic (related to the contrast agent) optical parameters are determined separately. This can result in errors, in particular, due to using simplified heuristic models for the spectral dependence of the optical parameters. Recently, we have developed the theory of non-reciprocal broken-ray tomography (NRBRT) for fluorescence imaging of weakly scattering systems. NRBRT enables simultaneous reconstruction of the fluorophore concentration as well as of the intrinsic optical attenuation coefficient at both the excitation and the emission wavelengths. Importantly, no assumption about the spectral dependence of the tissue optical properties is made in NRBRT. In this study, we perform numerical validation of NRBRT under realistic conditions using the Monte Carlo method to generate forward data. We demonstrate that NRBRT can be used for tomographic imaging of samples of up to four scattering lengths in size. The effects of physical characteristics of the detectors such as the area and the acceptance angle are also investigated.

Maxwell Garnett approximation in random media: tutorial.

In the third part of the tutorial, we develop a Maxwell Garnett approximation, which is applicable, in particular, to disordered random media with non-spherical inclusions and to multicomponent mixtures. We are especially interested in the case when the inclusions are non-spherical and randomly distributed and oriented in space, but the composite is isotropic on average. The effective medium formula applicable to such media cannot be obtained by direct generalization of the results that were derived in the first two parts of the tutorial. We show that the Maxwell Garnett effective medium approximation can be stated in a very general form. The results derived by us previously as well as a result applicable to the medium described above can be obtained as special cases of this general result.

Investigation of the effect of super-resolution in nonlinear inverse scattering.

The idea that a solution to a nonlinear inverse scattering problem (ISP) can contain information about the target on a subwavelength scale and thus allow one to achieve super-resolution (spatial resolution beyond the diffraction limit) has been around since the 1990s. However, a solid mathematical theory of super-resolution in nonlinear image reconstruction is still lacking. In this paper, we investigate the effect of super-resolution in nonlinear ISPs (both analytically and numerically) by analyzing several inverse problems in which the limit of spatial resolution can be defined precisely. The conclusions we obtain are not optimistic. Although it is possible to create examples of exactly solvable models in which account of nonlinearity in the ISP results in additional mathematically independent equations (one such example is shown herein), our results indicate that super-resolution is not achievable in any practical sense. Rather, we find that the linear subspace of possible solutions to a band-limited linearized ISP is transformed into a more general curved manifold due to the effects of nonlinearity. In the one-dimensional problem with realistic interaction that we have considered, the manifold can have a slightly smaller dimensionality that the subspace of solutions to the linearized problem but it does not contract to a point and the effect is practically insignificant.

External versus induced and free versus bound electric currents and related fundamental questions of the classical electrodynamics of continuous media: discussion.

Standard textbooks on classical electrodynamics frequently operate with the notions of free and bound currents (charges). Alternative terminology of external and induced currents also exists. However, a clear physical definition of these physical objects is rarely given. The term "free current" can refer in some cases to the conductivity current, which is subject to constitutive relations in a material sample. In other cases, free current refers to the current that is completely extrinsic to a given material sample and is assumed to be known a priori or manipulated by the experimentalist at will. Although one can argue that all currents flowing in material media are subject to some constitutive relations, there is a clear distinction in the construction of the classical electrodynamics between the external and induced currents. The aim of this paper is to clarify this distinction while pointing out that the traditional distinction between free and bound currents is arbitrary and can be abandoned. In addition, the paper considers some relevant fundamental questions of classical electrodynamics, including the derivation of macroscopic Maxwell's equations, the properties of the external currents, and the physical interpretation of some auxiliary fields such as the field of polarization P.

Diffuse correlation tomography in the transport regime: A theoretical study of the sensitivity to Brownian motion.

Diffuse correlation tomography (DCT) uses the electric-field temporal autocorrelation function to measure the mean-square displacement of light-scattering particles in a turbid medium over a given exposure time. The movement of blood particles is here estimated through a Brownian-motion-like model in contrast to ordered motion as in blood flow. The sensitivity kernel relating the measurable field correlation function to the mean-square displacement of the particles can be derived by applying a perturbative analysis to the correlation transport equation (CTE). We derive an analytical expression for the CTE sensitivity kernel in terms of the Green's function of the radiative transport equation, which describes the propagation of the intensity. We then evaluate the kernel numerically. The simulations demonstrate that, in the transport regime, the sensitivity kernel provides sharper spatial information about the medium as compared with the correlation diffusion approximation. Also, the use of the CTE allows one to explore some additional degrees of freedom in the data such as the collimation direction of sources and detectors. Our results can be used to improve the spatial resolution of DCT, in particular, with applications to blood flow imaging in regions where the Brownian motion is dominant.

Optimized diffusion approximation.

We show that the diffusion approximation (DA) to the radiative transport equation, which is commonly used in biomedical optics to describe propagation of light in tissues, contains a previously unexplored adjustable parameter. This parameter is related to the rate of exponential decay of the reduced intensity. In conventional theories, there are two distinct choices for this parameter. However, neither of these choices is optimal. When the optimal value for the parameter is used, the resulting DA becomes much more accurate near the medium boundaries, e.g., at the depth of up to a few ℓ*, where ℓ* is the transport mean free path (typically, about 1 mm in tissues). We refer to the new adjustable parameter as the reduced extinction coefficient. The proposed technique can reduce the relative error of the predicted diffuse density of the optical energy from about 30% to less than 1%. The optimized DA can still be inaccurate very close to an interface or in some other physical situations. Still, the proposed development extends the applicability range of the DA significantly. This result can be useful, for instance, in tomographic imaging of relatively shallow (up to a few ℓ* deep) layers of tissues in the reflection geometry.

Nonreciprocal broken ray transforms with applications to fluorescence imaging.

Broken ray transforms (BRTs) are typically considered to be reciprocal, meaning that the transform is independent of the direction in which a photon travels along a given broken ray. However, if the photon can change its energy (or be absorbed and re-radiated at a different frequency) at the vertex of the ray, then reciprocity is lost. In optics, non-reciprocal BRTs are applicable to imaging problems with fluorescent contrast agents. In the case of x-ray imaging, problems with single Compton scattering also give rise to non-reciprocal BRTs. In this paper, we focus on tomographic optical fluorescence imaging and show that, by reversing the path of a photon and using the non-reciprocity of the data function, we can reconstruct simultaneously and independently all optical properties of the medium (the intrinsic attenuation coefficients at the excitation and the fluorescence frequency and the concentration of the contrast agent). Our results are also applicable to inverting BRTs that arise due to single Compton scattering.

Numerical investigation of polarization filtering for direct optical imaging within scattering media.

We investigate the ability of polarization filtering to improve direct imaging of absorbing objects which are buried within scattering environments. We extend on previous empirical investigations by exploiting an efficient perturbation-based formalism, which is applicable to arbitrarily arranged sources and detectors with arbitrary polarizations. From this approach, we are able in some cases to find certain non-trivial linear combinations of polarization measurement channels that maximize the object resolution and visibility.

Reciprocity relation for the vector radiative transport equation and its application to diffuse optical tomography with polarized light.

We derive a reciprocity relation for the 3D vector radiative transport equation that describes propagation of polarized light in multiple-scattering media. We then show how this result, together with translational invariance of a plane-parallel sample, can be used to efficiently compute the sensitivity kernel of diffuse optical tomography by Monte Carlo simulations. Numerical examples of polarization-selective sensitivity kernels are given.

Maxwell Garnett approximation (advanced topics): tutorial.

In the second part of this tutorial, we consider several advanced topics related to the Maxwell Garnett approximation.

Solution of the nonlinear inverse scattering problem by T-matrix completion. II. Simulations.

This is Part II of the paper series on data-compatible T-matrix completion (DCTMC), which is a method for solving nonlinear inverse problems. Part I of the series [H. W. Levinson and V. A. Markel, Phys. Rev. E 94, 043317 (2016)10.1103/PhysRevE.94.043317] contains theory and here we present simulations for inverse scattering of scalar waves. The underlying mathematical model is the scalar wave equation and the object function that is reconstructed is the medium susceptibility. The simulations are relevant to ultrasound tomographic imaging and seismic tomography. It is shown that DCTMC is a viable method for solving strongly nonlinear inverse problems with large data sets. It provides not only the overall shape of the object, but the quantitative contrast, which can correspond, for instance, to the variable speed of sound in the imaged medium.

Solution of the nonlinear inverse scattering problem by T-matrix completion. I. Theory.

We propose a conceptually different method for solving nonlinear inverse scattering problems (ISPs) such as are commonly encountered in tomographic ultrasound imaging, seismology, and other applications. The method is inspired by the theory of nonlocality of physical interactions and utilizes the relevant formalism. We formulate the ISP as a problem whose goal is to determine an unknown interaction potential V from external scattering data. Although we seek a local (diagonally dominated) V as the solution to the posed problem, we allow V to be nonlocal at the intermediate stages of iterations. This allows us to utilize the one-to-one correspondence between V and the T matrix of the problem. Here it is important to realize that not every T corresponds to a diagonal V and we, therefore, relax the usual condition of strict diagonality (locality) of V. An iterative algorithm is proposed in which we seek T that is (i) compatible with the measured scattering data and (ii) corresponds to an interaction potential V that is as diagonally dominated as possible. We refer to this algorithm as to the data-compatible T-matrix completion. This paper is Part I in a two-part series and contains theory only. Numerical examples of image reconstruction in a strongly nonlinear regime are given in Part II [H. W. Levinson and V. A. Markel, Phys. Rev. E 94, 043318 (2016)10.1103/PhysRevE.94.043318]. The method described in this paper is particularly well suited for very large data sets that become increasingly available with the use of modern measurement techniques and instrumentation.

Introduction to the Maxwell Garnett approximation: tutorial.

This tutorial is devoted to the Maxwell Garnett approximation and related theories. Topics covered in this first, introductory part of the tutorial include the Lorentz local field correction, the Clausius-Mossotti relation and its role in the modern numerical technique known as the discrete dipole approximation, the Maxwell Garnett mixing formula for isotropic and anisotropic media, multicomponent mixtures and the Bruggeman equation, the concept of smooth field, and Wiener and Bergman-Milton bounds.

Radiative transport and optical tomography with large datasets.

We consider the inverse problem of optical tomography in the radiative transport regime. We report numerical tests of a direct reconstruction method that is suitable for use with large datasets. Reconstructions of experimental data obtained from a noncontact optical tomography system are also reported.

A non-asymptotic homogenization theory for periodic electromagnetic structures.

Homogenization of electromagnetic periodic composites is treated as a two-scale problem and solved by approximating the fields on both scales with eigenmodes that satisfy Maxwell's equations and boundary conditions as accurately as possible. Built into this homogenization methodology is an error indicator whose value characterizes the accuracy of homogenization. The proposed theory allows one to define not only bulk, but also position-dependent material parameters (e.g. in proximity to a physical boundary) and to quantify the trade-off between the accuracy of homogenization and its range of applicability to various illumination conditions.

Nondecaying surface plasmon polaritons in linear chains of silver nanospheroids.

We consider propagation of surface plasmon polaritons in linear chains of equidistant metallic nanospheroids. We show that, for suitably chosen parameters, the propagation is free of spatial decay in spite of the full account of absorptive losses in the metal.

Surface waves in three-dimensional electromagnetic composites and their effect on homogenization.

Reflection and transmission of electromagnetic waves at the boundaries of periodic composites (electromagnetic/optical metamaterials) depends in general on both bulk and surface waves. We investigate the interplay of these two contributions using three-dimensional full-wave numerical simulations and a recently developed non-asymptotic homogenization theory.

Diffuse optical tomography in the presence of a chest wall.

Diffuse optical tomography (DOT) has been employed to derive spatial maps of physiologically important chromophores in the human breast, but the fidelity of these images is often compromised by boundary effects such as those due to the chest wall. We explore the image quality in fast, data-intensive analytic and algebraic linear DOT reconstructions of phantoms with subcentimeter target features and large absorptive regions mimicking the chest wall. Experiments demonstrate that the chest wall phantom can introduce severe image artifacts. We then show how these artifacts can be mitigated by exclusion of data affected by the chest wall. We also introduce and demonstrate a linear algebraic reconstruction method well suited for very large data sets in the presence of a chest wall.