Ram Mohan Vasu

• Indian Institute of Science Bangalore

Ultrasound-Mediated Imaging in scattering media has deeper roots than hitherto imagined, roots in condensed matter physics. Professors Vasu and Roy discuss these in order to establish a firm connection between measurement and a host of parameters in the medium, mechanical and optical. These connections are verified through recovery of parameters us...

This paper presents the development of a novel wavefront measuring camera capable of detecting both the amplitude and phase of the captured light wave simultaneously. The main objective of the present work is to develop a simple "aim and shoot" camera system for quantitative estimation of density variations in high-speed gas flow fields. The interr...

A method for quantitative estimation of density variation in high-speed flow, which uses light as an interrogating tool, is described. The wavefront distortion of the interrogating beam induced by the compressible flow field is estimated quantitatively, in which the density gradient of the flow field is seen as refractive-index gradient by the prob...

A pseudo-time, sub-optimal stochastic filtering approach based on derivative free variant of ensemble Kalman filter for solving inverse problem of Ultrasound Modulated Optical Tomography is developed. The proposed scheme enhanced the contrast of reconstructed images.

We obtain vibro-acoustic (VA) spectral signatures of a remotely palpated region in tissue or tissue-like objects through diffusing-wave spectroscopy (DWS) measurements. Remote application of force is through focused ultrasound, and the spectral signatures correspond to vibrational modes of the focal volume (also called the ROI) excited through ultr...

In this manuscript a handheld camera is proposed that is built around a modified CMOS image sensor chip. The camera can detect both light amplitude and variation of light phase quantitatively. The refractive material of interest is interrogated by passing light through it, and the exit light from the test object is captured by the proposed camera....

Nonintrusive optical schemes are preferred for quantitative flow diagnostics. Shadow-casting scheme is one of the flow diagnostic techniques in which the projected shadow of a random-dot pattern mask reveals the flow information under the observation of the interrogating beam. Any variation in the flow field is seen as a change in position of the c...

Optical schemes based on light as an interrogating tool are in demand for quantitative flow visualization. The associated phase of the interrogating beam changes on passing through the flow field. The phase change of the light beam cannot be detected by a regular photodetector or by a camera as they can detect intensity only. In this manuscript, an...

We demonstrate a non-invasive technique, based on the modal frequency shift of a region insonified by a dual-beam ultrasound (US) transducer (region of interest, ROI), to remotely assess the temperature of the region in a tissue-mimicking object. The application is in ultrasound hyperthermia systems for controlled maintenance of tumour temperature...

The experimental results are obtained for accelerated particle filtering for Diffuse Optical Tomography. Broyden updates are used to accomplish computation of Jacobian. The proposed scheme has shown remarkable efficacy for reconstruction with data.

We demonstrate a simple and computationally efficient method to recover the shear modulus pertaining to the focal volume of an ultrasound transducer from the measured vibro-acoustic spectral peaks. A model that explains the transport of local deformation information with the acoustic wave acting as a carrier is put forth. It is also shown that the...

We present, perhaps for the first time, a stochastic search algorithm in quantitative photoacoustic tomography (QPAT) for a one-step recovery of the optical absorption map from time-resolved photoacoustic signals. Such a direct recovery is free of the numerical inaccuracies inherent in conventional two-step approaches that depend on an accurate est...

A simple noninterferometric optical probe is developed to estimate wavefront distortion suffered by a plane wave in its passage through density variations in a hypersonic flow obstructed by a test model in a typical shock tunnel. The probe has a plane light wave trans-illuminating the flow and casting a shadow of a continuous-tone sinusoidal gratin...

Ultrasound modulated optical tomography (UMOT) combines high optical contrast with high ultrasound resolution to image soft tissues. A focused ultrasound beam introduced to a specific region of interest (ROI) in the object modulates the mean position of the scattering centers locally. This in turn modulates the overall decay of the specific intensi...

The irradiation of selective regions in a polymer gel dosimeter results in an increase in optical density and refractive index (RI) at those regions. An optical tomography-based dosimeter depends on rayline path through the dosimeter to estimate and reconstruct the dose distribution. The refraction of light passing through a dose region results in...

Optical tomography of gel dosimeters is used to verify radiotherapy plans. Refraction of light results in distortion of reconstructed dose profiles. Refraction correction provides distortion free reconstructions and improves dose verification.

A framework is proposed for pseudo-time particle filtering for Diffuse Optical Tomography reconstruction. A fictitious ‘measurement’ equation is employed based on the quasi-Newton steps. The proposed scheme, accelerates convergence and yields substantially reduced sample variance.

An iterative Gauss-Newton algorithm which uses normal derivative of intensity(without estimating phase) as the measurement, to reconstruct the cross-sectional refractive index profile of a photonic crystal fiber.

Building upon our earlier work of a martingale approach to global optimization, a powerful stochastic search scheme for the global optimum of cost functions is proposed on the basis of change of measures on the states that evolve as diffusion processes and splitting of the state-space along the lines of a Bayesian game. To begin with, the efficacy...

Transport of intensity phase imaging suffers from phase aberration due to presence of noise in experimental data and aberration induced by optical system. This induced low frequency aberration corrupts the original phase and makes it difficult to use for any further quantitative processing. In this paper an aberration correction scheme is proposed....

Using coherent light interrogating a turbid object perturbed by a focused ultrasound (US) beam, we demonstrate localized measurement of dynamics in the focal region, termed the region-of-interest (ROI), from the decay of the modulation in intensity autocorrelation of light. When the ROI contains a pipe flow, the decay is shown to be sensitive to th...

A global optimization framework, COMBEO (Change Of Measure Based Evolutionary Optimization), is proposed. An important aspect in the development is a set of derivative-free additive directional terms, obtainable through a change of measures en route to the imposition of any stipulated conditions aimed at driving the realized design variables (parti...

Based on an ultrasound-modulated optical tomography experiment, a direct, quantitative recovery of Young’s modulus () is achieved from the modulation depth () in the intensity autocorrelation. The number of detector locations is limited to two in orthogonal directions, reducing the complexity of the data gathering step whilst ensuring against an im...

Starting with a micropolar formulation, known to account for nonlocal microstructural effects at the continuum level, a generalized Langevin equation (GLE) for a particle, describing the predominant motion of a localized region through a single displacement degree-of-freedom (DOF), is derived. The GLE features a memory dependent multiplicative or i...

Optical computed tomography is extensively used for quantitative flow visualization studies [1, 2]. Based on detection methodology it is classified as phase and deflection tomography. Here, we are considering deflection tomography method.

A simple method employing an optical probe is presented to measure density variations in a hypersonic flow obstructed by a test model in a typical shock tunnel. The probe has a plane light wave trans-illuminating the flow and casting a shadow of a random dot pattern. Local slopes of the distorted wavefront are obtained from shifts of the dots in th...

A new global stochastic search, guided mainly through derivative-free directional information computable from the sample statistical moments of the design variables within a Monte Carlo setup, is proposed. The search is aided by imparting to the directional update term additional layers of random perturbations referred to as ‘coalescence’ and ‘scra...

Despite the numerous applications that may be expeditiously modelled by counting processes, stochastic filtering strategies involving Poisson-type observations still remain somewhat poorly developed. In this work, we propose a Monte Carlo stochastic filter for recursive estimation in the context of linear/nonlinear dynamical systems with Poisson-ty...

We demonstrate diffusing-wave spectroscopy (DWS) in a localized region of a viscoelastically inhomogeneous object by measurement of the intensity autocorrelation [g_{2}(τ)] that captures only the decay introduced by the temperature-induced Brownian motion in the region. The region is roughly specified by the focal volume of an ultrasound transducer...

We have developed a graphics processor unit (GPU-) based high-speed fully 3D system for diffuse optical tomography (DOT). The reduction in execution time of 3D DOT algorithm, a severely ill-posed problem, is made possible through the use of (1) an algorithmic improvement that uses Broyden approach for updating the Jacobian matrix and thereby updati...

We develop iterative diffraction tomography algorithms, which are similar to the distorted Born algorithms, for inverting scattered intensity data. Within the Born approximation, the unknown scattered field is expressed as a multiplicative perturbation to the incident field. With this, the forward equation becomes stable, which helps us compute nea...

A new global stochastic search, guided mainly through derivative-free directional information computable from the sample statistical moments of the design variables within a Monte Carlo setup, is proposed. The search is aided by imparting to a directional update term, which parallels the conventional Gateaux derivative used in a local search for th...

A Monte Carlo filter, based on the idea of averaging over characteristics and fashioned after a particle-based time-discretized approximation to the Kushner–Stratonovich (KS) nonlinear filtering equation, is proposed. A key aspect of the new filter is the gain-like additive update, designed to approximate the innovation integral in the KS equation...

This report addresses the assessment of variation in elastic property of soft biological tissues non-invasively using laser speckle contrast measurement. The experimental as well as the numerical (Monte-Carlo simulation) studies are carried out. In this an intense acoustic burst of ultrasound (an acoustic pulse with high power within standard safet...

3-Dimensional Diffuse Optical Tomographic (3-D DOT) image reconstruction algorithm is computationally complex and requires excessive matrix computations and thus hampers reconstruction in real time. In this paper, we present near real time 3D DOT image reconstruction that is based on Broyden approach for updating Jacobian matrix. The Broyden method...

We propose a novel form of nonlinear stochastic filtering based on an iterative evaluation of a Kalman-like gain matrix computed within a Monte Carlo scheme as suggested by the form of the parent equation of nonlinear filtering (Kushner–Stratonovich equation) and retains the simplicity of implementation of an ensemble Kalman filter (EnKF). The nume...

Impoverishment of particles, i.e. the discretely simulated sample paths of the process dynamics, poses a major obstacle in employing the particle filters for large dimensional nonlinear system identification. A known route of alleviating this impoverishment, i.e. of using an exponentially increasing ensemble size vis-à-vis the system dimension, rem...

We demonstrate a direct recovery of elasticity distribution from ultrasound-modulated optical tomography data gathered at a single detector. The reconstructions are seen to be of good quality and the convergence of the algorithm quick. We have en route devised a means to estimate the Jacobian needed for this reconstructions which uses both the equa...

We propose a novel numerical method based on a generalized eigenvalue decomposition for solving the diffusion equation governing the correlation diffusion of photons in turbid media. Medical imaging modalities such as diffuse correlation tomography and ultrasound-modulated optical tomography have the (elliptic) diffusion equation parame- terized by...

A global stochastic search method, which is strictly derivative-free yet directed through a gain-based additive update term, is proposed and applied to the inverse problem of ultrasound modulated optical tomography (UMOT). The additive update term results from a change of measure that is effected to drive appropriately derived observation-predictio...

Nearly pollution-free solutions of the Helmholtz equation for k-values corresponding to visible light are demonstrated and verified through experimentally measured forward scattered intensity from an optical fiber. Numerically accurate solutions are, in particular, obtained through a novel reformulation of the H1 optimal Petrov-Galerkin weak form o...

Using a Girsanov change of measures, we propose novel variations within a particle-filtering algorithm, as applied to the inverse problem of state and parameter estimations of nonlinear dynamical systems of engineering interest, toward weakly correcting for the linearization or integration errors that almost invariably occur whilst numerically prop...

A novel form of nonlinear stochastic filtering employing an annealing-type iterative update scheme, aided by the introduction of an artificial diffusion parameter and based on the Gaussian sum approximations of the prior and posterior densities, is presented. The proposed Monte Carlo filter bank conforms in structure to the parent nonlinear filteri...

A novel form of nonlinear stochastic filtering employing an annealing-type iterative update scheme, aided by the introduction of an artificial diffusion parameter and based on the Gaussian sum approximations of the prior and posterior densities, is presented. The proposed Monte Carlo filter bank conforms in structure to the parent nonlinear filteri...

The Girsanov linearization method (GLM), proposed earlier in Saha, N., and Roy, D., 2007, "The Girsanov Linearisation Method for Stochastically Driven Nonlinear Oscillators," J. Appl. Mech., 74, pp. 885-897, is reformulated to arrive at a nearly exact, semianalytical, weak and explicit scheme for nonlinear mechanical oscillators under additive stoc...

A Monte Carlo filter, based on the idea of averaging over characteristics and fashioned after a particle-based time-discretized approximation to the Kushner-Stratonovich (KS) nonlinear filtering equation, is proposed. A key aspect of the new filter is the gain-like additive update, designed to approximate the innovation integral in the KS equation...

The solution of the forward equation that models the transport of light through a highly scattering tissue material in diffuse optical tomography (DOT) using the finite element method gives flux density (Φ) at the nodal points of the mesh. The experimentally measured flux (Umeasured) on the boundary over a finite surface area in a DOT system has to...

Objects viewed through transparent sheets with residual non-parallelism and irregularity appear shifted and distorted. This distortion is measured in terms of angular and binocular deviation of an object viewed through the transparent sheet. The angular and binocular deviations introduced are particularly important in the context of aircraft windsc...

We demonstrate quantitative optical property and elastic property imaging from ultrasound assisted optical tomography data. The measurements, which are modulation depth M and phase ϕ of the speckle pattern, are shown to be sensitively dependent on these properties of the object in the insonified focal region of the ultrasound (US) transducer. We de...

We demonstrate the phase fluctuation introduced by oscillation of scattering centers in the focal volume of an ultrasound transducer in an optical tomography experiment has a nonzero mean. The conditions to be met for the above are: (i) the frequency of the ultrasound should be in the vicinity of the most dominant natural frequency of vibration of...

The smooth DMS‐FEM, recently proposed by the authors, is extended and applied to the geometrically nonlinear and ill‐posed problem of a deformed and wrinkled/slack membrane. A key feature of this work is that three‐dimensional nonlinear elasticity equations corresponding to linear momentum balance, without any dimensional reduction and the associat...

We have developed an efficient fully three-dimensional (3D) reconstruction algorithm for diffuse optical tomography (DOT). The 3D DOT, a severely ill-posed problem, is tackled through a pseudodynamic (PD) approach wherein an ordinary differential equation representing the evolution of the solution on pseudotime is integrated that bypasses an explic...

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 tum...

Diffuse optical tomography (DOT) is one of the ways to probe highly scattering media such as tissue using low-energy near infra-red light (NIR) to reconstruct a map of the optical property distribution. The interaction of the photons in biological tissue is a non-linear process and the phton transport through the tissue is modelled using diffusion...

A Monte Carlo model of ultrasound modulation of multiply scattered coherent light in a highly scattering media has been carried out for estimating the phase shift experienced by a photon beam on its transit through US insonified region. The phase shift is related to the tissue stiffness, thereby opening an avenue for possible breast tumor detection...

The aim of this article is to study the mathematical analysis for an inverse problem and its numerical implementation associated with diffuse correlation tomography. The coefficients of the diffusion equation governing the propagation of field autocorrelation through a turbid medium (tissue-like) depend on both the optical and mechanical properties...

The preferred tool for studying hypersonic flow dynamics is that the technique which uses the noninvasive, light-based flow visualization techniques. Various strategies leading to qualitative as well as quantitative flow visualization such as Shadowgraphy, Schlieren imaging, background oriented Schlieren (BOS), interferometry and laser induced fluo...

We demonstrate a method to recover the Young’s modulus (E) of a tissue-mimicking phantom from measurements of ultrasound modulated optical tomography (UMOT). The object is insonified by a dual-beam, confocal ultrasound transducer (US) oscillating at frequencies f0 and f0 + Δf and the variation of modulation depth (M) in the autocorrelation of light...

We address a certain inverse problem in ultrasound-modulated optical tomography: the recovery of the amplitude of vibration of scatterers [ p ( r ) ] in the ultrasound focal volume in a diffusive object from boundary measurement of the modulation depth (M) of the amplitude autocorrelation of light [ ϕ ( r , τ ) ] traversing through it. Since M is d...

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 (...

We explore a pseudodynamic form of the quadratic parameter update equation for diffuse optical tomographic reconstruction from noisy data. A few explicit and implicit strategies for obtaining the parameter updates via a semianalytical integration of the pseudodynamic equations are proposed. Despite the ill-posedness of the inverse problem associate...

Scattering of coherent light from scattering particles causes phase shift to the scattered light. The interference of unscattered and scattered light causes the formation of speckles. When the scattering particles, under the influence of an ultrasound (US) pressure wave, vibrate, the phase shift fluctuates, thereby causing fluctuation in speckle in...

This paper attempts to model the forward problem in photoacoustic tomography (PAT) using discontinuous Galerkin (DG) method. Numerical experiments show that DG solutions are comparable with those obtained by finite element method (FEM).

The inverse problem in photoacoustic tomography (PAT) seeks to obtain the absorbed energy map from the boundary pressure measurements for which computationally in- tensive iterative algorithms exist. The computational challenge is heightened when the reconstruction is done using boundary data split into its frequency spectrum to improve source loca...

Fast reconstruction of interior optical parameter distribution using a new approach called Broyden-based model iterative image reconstruction (BMOBIIR) and adjoint Broyden-based MOBIIR (ABMOBIIR) of a tissue and a tissue mimicking phantom from boundary measurement data in diffuse optical tomography (DOT). DOT is a nonlinear and ill-posed inverse pr...

A regional information based multiple regularization is studied for solving biological inverse problem. Inverse problems are usually optimized and solved by Newtons method and its variants. Optimization based on calculus is extremely localized. The regional gradient in calculus is more important for local physiological changes. A sub-block based mu...

We present the development of an experimental three-dimensional (3-D) diffuse optical tomographic image reconstruction system for imaging of highly scattering tissue-mimicking material using near infrared laser light and the validation of 3-D reconstruction algorithms. The algorithm reconstructs the spatial distribution of the optical parameters of...

The static response of thin, wrinkled membranes is studied using both a tension field approximation based on plane stress conditions and a 3D nonlinear elasticityformulation, discretized through 8-noded Cosserat point elements. While the tension field approach only obtains the wrinkled/slack regions and at best a measure of the extent of wrinklines...

Purpose To assess the effect of ultrasound modulation of near infrared (NIR) light on the quantification of scattering coefficient in tissue‐mimicking biological phantoms. Methods A unique method to estimate the phase of the modulated NIR light making use of only time averaged intensity measurements using a charge coupled device camera is used in...

An adaptive regularization algorithm that combines elementwise photon absorption and data misfit is proposed to stabilize the non-linear ill-posed inverse problem. The diffuse photon distribution is low near the target compared to the normal region. A Hessian is proposed based on light and tissue interaction, and is estimated using adjoint metho...

The problem of reconstruction of a refractive-index distribution (RID) in optical refraction tomography (ORT) with optical path-length difference (OPD) data is solved using two adaptive-estimation-based extended-Kalman-filter (EKF) approaches. First, a basic single-resolution EKF (SR-EKF) is applied to a state variable model describing the tomograp...

An adaptive regularization algorithm that combines elementwise photon absorption and data misfit is proposed to stabilize the non-linear ill-posed inverse problem. The diffuse photon distribution is low near the target compared to the normal region. A Hessian is proposed based on light and tissue interaction, and is estimated using adjoint method b...

We propose a self-regularized pseudo-time marching scheme to solve the ill-posed, nonlinear inverse problem associated with diffuse propagation of coherent light in a tissuelike object. In particular, in the context of diffuse correlation tomography (DCT), we consider the recovery of mechanical property distributions from partial and noisy boundary...

A computationally efficient algorithm (linear iterative type) based on singular value decomposition (SVD) of the Jacobian has been developed that can be used in rapid dynamic near-infrared (NIR) diffuse optical tomography. Numerical and experimental studies have been conducted to prove the computational efficacy of this SVD-based algorithm over con...

We explore the application of pseudo time marching schemes, involving either deterministic integration or stochastic filtering, to solve the inverse problem of parameter identification of large dimensional structural systems from partial and noisy measurements of strictly static response. Solutions of such non-linear inverse problems could provide...

The study of non-invasive characterization of elastic properties of soft biological tissues has been a focus of active researches since recent years. Light is highly scattered by biological tissues and hence, sophisticated reconstruction algorithms are required to achieve good imaging depth and a reasonable resolution. Ultrasound (US), on the other...

We propose a self-regularized pseudo-time marching strategy for ill-posed, nonlinear inverse problems involving recovery of system parameters given partial and noisy measurements of system response. While various regularized Newton methods are popularly employed to solve these problems, resulting solutions are known to sensitively depend upon the n...

A computationally efficient pseudodynamical filtering setup is established for elasticity imaging (i.e., reconstruction of shear modulus distribution) in soft-tissue organs given statically recorded and partially measured displacement data. Unlike a regularized quasi-Newton method (QNM) that needs inversion of ill-conditioned matrices, the authors...

We discuss the inverse problem associated with the propagation of the field autocorrelation of light through a highly scattering object like tissue. In the first part of the work, we reconstruct the optical absorption coefficient μ a and particle diffusion coefficient D B from simulated measurements which are integrals of a quantity computed from t...

A pseudo-dynamical approach for a class of inverse problems involving static measurements is proposed and explored. Following linearization of the minimizing functional associated with the underlying optimization problem, the new strategy results in a system of linearized ordinary differential equations (ODEs) whose steady-state solutions yield the...

Near-infrared diffuse optical tomography (DOT) technique has the capability of providing good quantitative reconstruction of tissue absorption and scattering properties with additional inputs such as input and output modulation depths and correction for the photon leakage. We have calculated the two-dimensional (2D) input modulation depth from thre...

We propose a pseudo-dynamic form of a sub-optimal Kalman filter for elastography of plane-strain models of soft tissues under strictly static deformations and partial measurements. Since the tissue material is nearly incompressible and is thus prone to volumetric locking via standard displacement-based finite element formulations, we use a Cosserat...

We discuss the issue of separating contributions from mechanical and optical properties of a moderately scattering tissue phantom to the modulation depth (M) of intensity autocorrelation measured in an ultrasound-assisted optical tomography system using axial and transverse illuminations. For axial illumination, M is affected by both the displaceme...

Diffuse optical tomography (DOT) using near-infrared (NIR) light is a promising tool for noninvasive imaging of deep tissue. This technique is capable of quantitative reconstructions of absorption coefficient inhomogeneities of tissue. The motivation for reconstructing the optical property variation is that it, and, in particular, the absorption co...

We describe a noniterative method for recovering optical absorption coefficient distribution from the absorbed energy map reconstructed using simulated and noisy boundary pressure measurements. The source reconstruction problem is first solved for the absorbed energy map corresponding to single- and multiple-source illuminations from the side of th...

Light transmission data collected around the object show large variation with source-detector separation owing to the presence of single- or multiple inhomogeneous regions in the object. This variation in the measured intensity is made use of to reconstruct regions of the inhomogeneous inclusions. The reconstructed region is found to always contain...

We obtain the reconstruction of the refractive index distribution of body based on the intensity and normal derivative of the intensity measurements. The Helmholtz equation is inverted either directly or indirectly through repeated implementation of the forward operator and its adjoint, for recovering the complex refractive index distribution. We d...

We present an optical tomographic reconstruction method to recover the complex refractive index distribution from boundary measurements based on intensity, which are the logarithm of intensity and normal derivative of intensity. The method, which is iterative, repeatedly implements the forward propagation equation for light amplitude, the Helmholtz...

In ultrasound assisted optical elastography (UAOE) the amplitude of vibration inside the object introduced by an ultrasound (US) beam is read out by a coherent light beam. The measurement is the depth of modulation in the intensity autocorrelation of light that intercepted the insonified region and detected at the boundary. It is observed that the...

In this paper, we present a wavelet-based approach to solve the reconstruction problem encountered in diffuse optical tomography (DOT). The DOT reconstruction problem using model based iterative image reconstruction (MoBIIR) procedure involves repeated implementation of the three steps: (i) solution to the diffusion equation (DE) to generate the si...

We use a focused ultrasound beam to load a region of interest (ROI) in a tissue-mimicking phantom and read out the vibration amplitude of phantom particles from the modulation depth in the intensity autocorrelation of a coherent light beam that intercepted the ROI. The modulation depth, which is also affected by the local light absorption coefficie...

Diffusing wave spectroscopy (DWS), without the use of tracer particles, has been used to study the internal dynamics of polyvinyl alcohol (PVA) phantoms, which mimic the properties of normal and malignant breast tissues. From the measured intensity autocorrelations, the mean square displacement (MSD) of phantom meshing is estimated, leading to the...

An iterative reconstruction procedure is used to invert intensity data from both single- and phase-correlated dual-source illuminations for absorption inhomogeneities. The Jacobian for the dual source is constructed by an algebraic addition of the Jacobians estimated for the two sources separately. By numerical simulations, it is shown that the dua...