effects of measurement precision and finite numbers of electrodes on linear impedance imaging algorithms impedance imaging systems reconstruct approximations to the conductivity inside a body from electrical measurements made on the boundary the linearized problem was studied by calderon who showed that if the conductivity differs little from a constant then it can be reconstructed approximately in his work calderon assumed that arbitrary current densities can be applied anywhere on the body s surface and the resulting voltages perfectly measured in practice however imaging systems apply currents through a finite number of electrodes on the surface and measure the resulting voltages with a limited precision in this paper linear algorithms are described for approximating the electrical conductivity in a disk from finitely many limited precision measurements the effects of finite precision and electrode number on a two dimensional version of calderon s method are studied it is shown that as the measurement errors go to zero and the electrodes proliferate in a space filling manner the finite electro

spectral properties of hankel matrices and numerical solutions of finite moment problems abstract after proving that any hankel matrix generated by moments of positive functions is conditioned essentially the same as the hilbert matrix of the same size we show a preconditioning technique i e a congruence transform of the original hankel matrix that drastically reduces its ill conditioning applications of this result to classical orthogonal polynomial sequences and to modified moment problems are given also we outline an efficient algorithm for the computation of the function f x w x exp p x where w x is positive and p x is a polynomial of degree n 1 from the knowledge of its first n moments

hyperbolic geometry and local dirichlet neumann map abstract we show in three dimensions that if we know the dirichlet to neumann dn map associated to the schrodinger equation with a potential around the intersection of a sphere centered outside the convex hull of a bounded domain with smooth boundary we can determine uniquely the corresponding spherical mean of the potential with an appropriate measure this allows one to determine the potential near the boundary where the dn map is measured similar results are valid for electrical impedance tomography

regularized reconstruction in electrical impedance tomography using a variance uniformization constraint this paper describes a new approach to reconstruction of the conductivity field in electrical impedance tomography our goal is to improve the tradeoff between the quality of the images and the numerical complexity of the reconstruction method in order to reduce the computational load we adopt a linearized approximation to the forward problem that describes the relationship between the unknown conductivity and the measurements in this framework we focus on finding a proper way to cope with the ill posed nature of the problem mainly caused by strong attenuation phenomena this is done by devising regularization techniques well suited to this particular problem first we propose a solution which is based on tikhonov regularization of the problem second we introduce an original regularized reconstruction method in which the regularization matrix is determined by space uniformization of the variance of the reconstructed conductivities both methods are nonsupervised i e all tuning parameters are automatically determined from the measured data tests performed on simulated and real data indicate that tikhonov regularization provides results similar to those obtained with iterative methods but with a much smaller amount of computations regularization using a variance uniformization constraint yields further improvements particularly in the central region of the unknown object where attenuation is most severe we anticipate that the variance uniformization approach could be adapted to iterative methods that preserve the nonlinearity of the forward problem more generally it appears as a useful tool for solving other severely ill posed reconstruction problems such as eddy current tomography

integral geometry in hyperbolic spaces and electrical impedance tomography we study the relation between convolution operators and the totally geodesic radon transform on hyperbolic spaces as an application we show that the linearized inverse conductivity problem in the

a real time algorithm for the location search of discontinuous conductivities with one measurement we consider an inverse problem for finding the anomaly of discontinuous electrical conductivity by one current voltage observation we develop a real time algorithm for determining the location of the anomaly this new idea is based on the observation of the pattern of a simple weighted combination of the input current and the output voltage combined with the size estimation result this algorithm gives a good initial guess for newton type schemes we give the rigorous proof for the location search algorithm both the mathematical analysis and its numerical implementation indicate our location search algorithm is very fast stable and efficient 2001 john wiley sons inc

on the construction of a potential from cauchy data abstract we investigate the problem of recovering a potential q y in the differential equation u q y u 0 x y 0 1 0 1 u 0 y u 1 y u x 0 0 u x 1 f x u y x 1 g x the method of separation of variables reduces the recovery of q y to a nonstandard inverse sturm liouville problem an asymptotic formula is developed that suggests that under appropriate conditions on the cauchy pair f g q y is uniquely determined up to the mean moreover the recovery of q y is comparable to finding a function from its polynomial moments a reconstruction scheme is suggested and numerical examples are considered

a backprojection algorithm for electrical impedance imaging the authors study a two dimensional reconstruction algorithm due to d c barber and b h brown applied to a linearized electrostatic inverse problem first the authors demonstrate how this algorithm fits within the framework of inverses of generalized radon transforms studied by g beylkin second an iterative improvement of the barber brown algorithm is constructed based on the conjugate residual method several numerical results obtained with this iterative algorithm are presented

a fidelity embedded regularization method for robust electrical impedance tomography electrical impedance tomography eit provides functional images of an electrical conductivity distribution inside the human body since the 1980s many potential clinical applications have arisen using inexpensive portable eit devices eit acquires multiple trans impedance measurements across the body from an array of surface electrodes around a chosen imaging slice the conductivity image reconstruction from the measured data is a fundamentally ill posed inverse problem notoriously vulnerable to measurement noise and artifacts most available methods invert the ill conditioned sensitivity or the jacobian matrix using a regularized least squares data fitting technique their performances rely on the regularization parameter which controls the trade off between fidelity and robustness for clinical applications of eit it would be desirable to develop a method achieving consistent performance over various uncertain data regardless of the choice of the regularization parameter based on the analysis of the structure of the jacobian matrix we propose a fidelity embedded regularization fer method and a motion artifact reduction filter incorporating the jacobian matrix in the regularization process the new fer method with the motion artifact reduction filter offers stable reconstructions of high fidelity images from noisy data by taking a very large regularization parameter value the proposed method showed practical merits in experimental studies of chest eit imaging

electrical impedance tomography by elastic deformation this paper presents a new algorithm for conductivity imaging our idea is to extract more information about the conductivity distribution from data that have been enriched by coupling impedance electrical measurements to localized elastic perturbations using asymptotics of the fields in the presence of small volume inclusions we relate the pointwise values of the energy density to the measured data through a nonlinear pde our algorithm is based on this pde and takes full advantage of the enriched data we give numerical examples that illustrate the performance and the accuracy of our approach

inverse problems and derivatives of determinants this paper is about a well posed and well conditioned inverse problem instead of measurements at the boundary we go right inside a network all nodes and edges are treated equally by probing the network we find its response to m different source terms from that information about the system the goal is to find the conductances on the m edges the equations are highly nonlinear what is attractive is the appearance of a convex potential function whose gradient is zero at the solution the analysis of the problem becomes an analysis of this potential function

a posteriori error estimate and convergence analysis for conductivity image reconstruction in mreit magnetic resonance electrical impedance tomography mreit takes advantage of internal information to solve its nonlinear inverse problem of recovering a conductivity distribution inside an imaging object when we inject current into the imaging object there occurs a distribution of internal magnetic flux density mathbf b b x b y b z in mreit we utilize a magnetic resonance imaging scanner with its main magnetic field in the z direction to acquire b z data the harmonic b z algorithm was invented in 2001 to reconstruct cross sectional conductivity images from b z data sets subject to multiple injection currents utilizing internal b z data it overcomes the inherent ill posedness in electrical impedance tomography we can set up the inverse problem in mreit as a coefficient identification problem of finding sigma appearing in nabla cdot sigma nabla u 0 from acquired data of the z component of nabla times sigma nabla u the harmonic b z algorithm has shown an excellent perfor

direct reconstructions of conductivities from boundary measurements the problem of reconstructing an unknown electric conductivity from boundary measurements has applications in medical imaging geophysics and nondestructive testing a nachman ann of math 2 143 1996 pp 71 96 proved global uniqueness for the two dimensional inverse conductivity problem using a constructive method of proof based on this proof siltanen mueller and isaacson inverse problems 16 2000 pp 681 699 presented a new numerical reconstruction method that solves the nonlinear problem directly without iteration the method was verified with nonnoisy rotationally symmetric examples in this paper the method is extended by introducing a new regularization scheme which is analyzed theoretically and tested on symmetric and nonsymmetric numerical examples containing computer simulated noise

the determination of a coefficient in an elliptic equation from average flux data abstract we consider the question of recovering the coefficient q from the equation u j q x u j j x subject to homogeneous dirichlet boundary conditions in a bounded domain r 2 the nonhomogeneous source terms j x j 1 form a basis for l 2 it will be proven that a unique determination is possible from data measurements consisting of measurements of the net flux r uj vds 1 leaving a subset of the boundary for each input source j a continuous dependence result and an algorithm that allows efficient numerical reconstruction of q x from finite data is presented

projective electrical impedance reconstruction with two measurements this paper presents a projective reconstruction method of electrical impedance distribution using two boundary measurements of current voltage data the proposed method has a major advantage over existing electrical impedance tomography systems relatively high resolution impedance image in a near surface region underneath the voltage sensing probe that is placed on a skin surface we attach two pairs of electrodes on the border of the probe in such a way that each pair of electrodes generates electrical current flowing underneath the voltage sensing probe and two currents flow across each other a careful analysis of the measured data with respect to local perturbation of admittivity leads to the projective reconstruction method and its rationale numerical simulations show that the proposed method successfully reconstructs high resolution image of local perturbation of admittivity due to object position changes or admittivity value changes or the both in a near surface region underneath the probe

regular article a shape decomposition technique in electrical impedance tomography consider a two dimensional domain containing a medium with unit electrical conductivity and one or more non conducting objects the problem considered here is that of identifying shape and position of the objects on the sole basis of measurements on the external boundary of the domain an iterative technique is presented in which a sequence of solutions of the direct problem is generated by a boundary element method on the basis of assumed positions and shapes of the objects the key new aspect of the approach is that the boundary of each object is represented in terms of fourier coefficients rather than a point wise discretization these fourier coefficients generate the fundamental shapes mentioned in the title in terms of which the object shape is decomposed the iterative procedure consists in the successive updating of the fourier coefficients at every step by means of the levenberg marquardt algorithm it is shown that the fourier decomposition which essentially amounts to a form of image compression enables the algorithm to image the embedded objects with unprecedented accuracy and clarity in a separate paper the method has also been extended to three dimensions with equally good results

electrical impedance tomography using level set representation and total variational regularization in this paper we propose a numerical scheme for the identification of piecewise constant conductivity coefficient for a problem arising from electrical impedance tomography the key feature of the scheme is the use of level set method for the representation of interface between domains with different values of coefficients numerical tests show that our method can recover sharp interfaces and can tolerate a relatively high level of noise in the observation data results concerning the effects of number of measurements noise level in the data as well as the regularization parameters on the accuracy of the scheme are also given

second order optimality conditions using approximations for nonsmooth vector optimization problems under inclusion constraints abstract second order necessary conditions and sufficient conditions for optimality in nonsmooth vector optimization problems with inclusion constraints are established we use approximations as generalized derivatives and avoid even continuity assumptions convexity conditions are not imposed explicitly not all approximations in use are required to be bounded the results improve or include several recent existing ones examples are provided to show that our theorems are easily applied in situations where several known results do not work

smooth and irreducible multigraded hilbert schemes abstract the multigraded hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed hilbert function we prove that any multigraded hilbert scheme is smooth and irreducible when the polynomial ring is z x y which establishes a conjecture of haiman and sturmfels