Imaging studies are finding increasing applications in clinical medicine. Fueling this have been advances in technology that have expanded the range of applications and have improved their clinical performance, (i.e., diagnostic sensitivity and specificity). In addition, in many cases the risks associated with imaging procedures are minimal. The combined effect has been a tilt in the cost-benefit balance in favor of expanded use. Recent economic pressures, however, are adding increasing weight to cost considerations, requiring many to rethink their approach to the development of improved or fundamentally new imaging technologies. In this report, we examine some fundamental and practical issues that can be expected to influence future development of imaging technologies. We also discuss approaches that we are taking to develop NIR imaging methods, which we believe have significant potential for improving the performance for certain applications while maintaining high costeffectiveness.

This paper addresses the problem of constructing electrical conductivity models of the earth from surface electro- magnetic measurements. The construction of these models is a nonlinear inverse problem that can be approached by linearizing the corresponding functional and applying iterative techniques such as Tikhonov's regularization. The standard application of these techniques usually leads to smooth models that represent a continuous variation of conductivity with depth. In this work we describe how these methods can be modified to incorporate what is known in Computer Vision as the line process (LP) decoupling technique, which has the ability to include discontinuities in the models. This results in piecewise smooth models which are often more adequate for representing stratified media. We implemented a relaxation technique to construct these types of models and present numerical experiments and applications to field data. These examples illustrate the performance of the combined LP and Tikhonov's regularization method.

The authors have recently introduced a novel imaging algorithm which they call the Elliptic Systems Method (ESM). The performance of the ESM for experimental data using only one source and seven detectors is discussed in this paper. Two cases are considered: one in which the background medium is known, and one in which there is an error of 30%.

This paper is concerned with the estimation of underlying discontinuous functions from error-contaminated and blurred observations. Such problems occur in a number of important applications, particularly in inverse problems and signal and image reconstruction. We compare three different regularization techniques for estimating discontinuous signals.

We consider the problem of deconvolving the expression f(y, t) = ix g(x, y)h(x, t)dX, where it is required to estimate h(z, t) from the square-integrable functions g(z, y) and f(y, t), and the latter function is a noise corrupted version of f(y, t). This problem a.rises in the setting defined by a linear elliptic partial differential equation, where the unknown time-varying inhomogeneous term (the source) is to be computed from incomplete field data. Performance of such a deconvolution constitutes the Inverse Problem of Electrocardiography, for example. The standard approach to this problem constructs a global solution by collecting individually regularized solutions to the ill-posed problems defined by the above first kind Fredholm equations for different fixed values of para.meter t. That approach is shown to contain a. flaw. The corrected approach leads to more accurate deconvolution, particularly evident in the setting of Gaussian noise (a. theorem). In the presence of dominating geometric noise (noise generated by rotation, translation, or other systematic imprecision in the knowledge of g(x, y)), it may be expected that the accuracy advantage will be diminished, but there will be persisting benefits relating to superior time stability of the solutions. Keywords: inverse problem of electrocardiography, first kind Fredholm integral equations, deconvolution, compact opera.tors, cardiac electrical source imaging

In current public discussions, much attention is paid to the danger of breast cancer. X-ray methods for cancer detection like mammography are sometimes considered inappropriate. Recently, novel optical and acoustical methods are being examined. One of these is image reconstruction from time- harmonic ultrasound data, which requires determining the coefficient f in the Helmholtz equation (Delta) u + k²(l + f)u equals 0 from boundary measurement of various solutions u of the equation for different boundary conditions. In this equation, f determines the physical properties of the scatterer, while k is the wave number and u is the time-harmonic sound wave. Born- and Rytov- approximations have been used for some time to accomplish this task, but are not precise enough. All functions may be either a 2D or a 3D function, depending on the application. To overcome the complexity problems, an iterative algorithm that works on finite data and incorporates finite boundary conditions instead of radiation condition will be presented. Two main ingredients are used to accomplish this. First, an adjoint-field type iteration scheme is employed. Second, an initial value solver is used to solve the direct problem of computing a sound wave, given boundary conditions and a scatterer. The algorithm allows us to solve the problem with realistic parameters for simulated data on regular workstations in a few minutes.

Many methods for deconvolving images assume either that the entire convolution is available, or that the convolution is adequately modelled as a circular convolution. In reality, neither is usually the case, and only a section of a much larger blurred (and contaminated) image is observed. The truncation gives rise to null objects in reconstructions obtained by deconvolution methods. It is possible to formulate the problem as exact, though underdetermined, and to apply singular value decomposition to deriving an inverse operator. We compare different practical methods for performing deconvolution with a scanning finite impulse response filter derived in this manner.

We describe a method for generating a high resolution image or target identifier from limited sampled noisy Fourier data. These data are assumed to be collected from either limited-angle tomographic data or limited-angle scattering data. In extreme cases of, for example, only backscatter data being measured, little useful image information can be deduced directly from the available measurements. However, if some a prior knowledge about the support and/or internal features of the object can be assumed, this information can be incorporated into a spectral estimation technique we refer to as the PDFT. This technique has a closed form solution that estimates the image and is easily regularized in the Miller-Tikhonov sense. We have studied how the availability of less and less data affects the resulting image quality when using the PDFT. We explain the trade-offs between reducing the number of measurements (e.g. for time or radiation exposure considerations) and the resulting image fidelity. We also discuss how little data need be measured to be able to identify a given object when good a priori information is available. Examples using real data are presented.

We discuss the problem associated with obtaining an image of an object from the magnitude of its Fourier transform. This problem arises in many imaging applications. We discuss some new ideas developed to address this problem and describe constraints on the object function that can lead to its Fourier transform having only real zeros, thereby eliminating the phase retrieval problem.

Currently available tomographic image reconstruction schemes for photon migration tomography are mostly based on the limiting assumptions of small perturbations and a priori knowledge of the optical properties of a reference medium. In this work a model-based iterative image reconstruction (MOBIIR) method is presented, which does not require the knowledge of a reference medium or that the encountered heterogeneities are small perturbations. The code consists of three major parts: (1) a finite-difference, time- resolved, diffusion forward model is used to predict detector readings based on the spatial distribution of optical properties; (2) an objective function that describes the difference between predicted and measured data; (3) an updating scheme that uses the gradient of the objective function to provide subsequent guesses of the spatial distribution of the optical properties for the forward model. The reconstruction of these properties is completed, once a minimum of this objective function is found. After a review of the previously published mathematical background, the clinically relevant examples of breast cancer detection and brain imaging are discussed. It is shown that cysts and tumors can be distinguished using the MOBIIR technique, even in a heterogeneous background.

A weighted distorted Born iterative method is presented for reconstruction optical diffusion images from scattering data. A generalization of the distorted Born iterative method that uses a preconditioned cost function and an elliptical constraint allows a weighting matrix to be applied to the gradient term in the iterative algorithm according to the reconstruction history. The proposed algorithm shows stable and fast convergence for reconstruction of high contrast inhomogeneities.

A standard method for noninvasively computing neurocortical potentials from potentials measured on the scalp surface is to solve the problem on a generalized geometry and map the results back to the true model. This solution to the inverse EEG problem has been employed using spherical and, more recently, generic cranial models as templates. In the case of the most complex spherical models, the patient's skin, bone, cerebrospinal fluid, gray matter and white matter surfaces are mapped onto concentric spheres. The simplicity of the spherical domain allows for an analytic solution to the surface mapping inverse problem; however, the inaccuracy of such a solution challenges its clinical value. Similarly, solving the problem on a predefined generic model also holds computational allure--the generic model can be hand-picked to reduce the ill-conditioning of the problem. However, we suggest that such results from generic models are still not sufficiently accurate to be of general clinical use. In our paper, we evaluate the impact of varying both model accuracy and model complexity on the inverse cortical mapping. Small modeling perturbations (as might be introduced from noisy or under-sampled data) are shown to have large and detrimental effects on the quality of the solution.

This paper proposes a new Wiener estimation method for reconstructing current distribution from biomagnetic measurement. The present method is an improved version of the method proposed by Sekihara et al., which is based on estimating the source-current correlation matrix from the measured-data correlation matrix. In our method, the regularization term used in pseudo inversion of the lead field matrix is modified so as to reduce the artifact due to noise more efficiently. The parameters in the modified regularization term are determined, not empirically, but optimally based on the measured data. Computer simulation demonstrates the effectiveness of the proposed method.

In this report we consider the merits of optical methods for use as a nonionizing diagnostic imaging tool, in relation to current imaging methods, and we discuss critical issues needing further development. Evidence of feasibility of provided by recovery of inclusions embedded in an anatomically accurate 3D map of the breast derived from MRI data. In addition, progress toward developing a practical imaging system is given by description of a general-purpose imaging device. Also discussed is the potential for expanding the utility of optical methods by development of optical contrast agents sensitive to different metabolic states, and for extending the utility of optical methods beyond diagnosis to create new strategies for rational therapeutic intervention.

In Electrical Impedance Tomography (EIT) an approximation for the internal resistivity distribution is computed based on the knowledge of the injected currents and measured voltages on the surface of the body. It is often assumed that the injected currents are confined to the 2D electrode plane and the reconstruction is based on the 2D assumptions. However, the currents spread out in three dimensions and therefore off-plane structures have significant effect on the reconstructed images. In this paper we propose a finite element-based method for the reconstruction of 3D resistivity distributions. Both the forward and the inverse problems are discussed and the results from reconstructions with simulated an real measurement data are presented. The proposed method is based on the so-called complete electrode model that takes into account the presence of the electrodes and the contact impedances. This makes it possible to apply the proposed method also for static EIT with complicated geometries.

The article discusses the electrical impedance imaging problem (EIT) from a Bayesian point of view. We discuss two essentially different EIT problems: The first one is the static problem of estimating the resistivity distribution of a body from the static current/voltage measurements on the surface of the body. The other problem is a gas temperature distribution retrieval problem by resistivity measurements of metal filaments placed in the gas funnel. In these examples, the prior information contains inequality constraints and non-smooth functionals. Consequently, gradient-based maximum likelihood search algorithms converge poorly. To overcome this difficulty, we study the possibility of using a Markov chain Monte Carlo algorithm to explore the posterior distribution.

A reconstruction method based on a boundary matching technique is presented. It involves an iterative procedure of matching the scattered wave calculated from a certain refractive-index distribution with the measured scattered- wave. The direct scattering problem is exactly and quickly solved for axisymmetric cylinders. The applicability limit of the reconstruction method for a lossless and lossy circularly symmetric cylinder are numerically shown through computer simulations in noise-free and noisy environments in the both case of E-wave and H-wave incidences. Reconstruction of a nonaxisymmetric two-layered cylinder is also briefly mentioned.

At passive sounding the sources of observable radiation are often distributed casually. In this case the intensity of radiation is main information parameter for the decision of an inverse problem. To reconstruction 2D distribution of own radiation sources density on the basis of angular distribution registration of received radiation intensity the decomposition of known and unknown functions in Fourier series on circular harmonics is applied. In case of poor absorption the problem is reduced to equation of Abel type with Chebyshev polynomials in its kernel. New method is offered, which is based on exponential variable replacement. In this case the initial equation is resulted to the convolution equation that admit the decision by use fast algorithms. Efficiency and stability of the decision have been proved by results of initiation modeling. Opportunity of use of the decision for diagnostics of atmosphere radiating pollution and forest fire moving is shown.

We propose a method for the estimation of fast impedance changes in electrical impedance tomography (EIT) and wire tomography (WT). WT is a novel method for the estimation of the temperature profiles of gas flows: a matrix of metal wires whose impedance depends on the temperature (such as nickel or platinum) are placed in the flow and the resistances of each individual wire are tracked. In the EIT problem the impedance changes are assumed to be so fast that all observations on which the reconstruction is based, can not be assumed to be (even approximately) from the same impedance distribution. The proposed method is based on the formulation of the EIT and WT problems first as dynamical state estimation problems and then solving the mean square estimates for the state (discretized impedance distribution) recursively with the Kalman filter. The appropriate evolution models for EIT include generally random walk-based models and in some applications, e.g. human thorax, compartmental models. The evolution model of WT comes directly from the associated parabolic partial differential equation (heat equation).

Restoration of depth images obtained by serial slicing of the image volume, is complicated by the fact that one part of an opaque 3D object can partially block another part from the view of the imaging lens. This type of obscuration makes the point spread function (PSF) object dependent. We present a general algorithm that adapts a transparent PSF to represent the effects of the opaque or semi-transparent object. Examples of its application are given using simulated annealing and modified iterative backprojection to carry out the iterative restoration.

Since 1982, when it was first proposed by Shepp and Vardi, the Expectation Maximization (EM) algorithm has become very popular among researchers in image reconstruction. Recently, a natural extension of the EM algorithm was proposed in order to handle regularization terms containing `a priori' information for emission computed tomography problems. This new idea was further applied to other regularized maximum likelihood problems in transmission and emission tomography. We present in this article a unified approach to more general regularized ML problems. Our convergence proofs also extend those given in the previous papers allowing more general regularizations. We report on numerical simulations.

We consider the numerical solution of first kind Fredholm integral equations. Such integral equations occur in signal processing and image recovery problems among others. For this numerical study, the kernel k(x,t) is the sinc kernel. This study compares traditional Tikhonov regularization with an extension of Tikhonov regularization which updates the solution found by the usual method. In this work, both the identity, derivative and Laplacian operators are used as regularizers and tests were done with and without error in the image data g(x). The results indicate that the extension can provide a decrease in error of about two orders of magnitude.

It is well known that regularization techniques are often required to obtain stable approximate solutions to ill-posed problems in imaging. Most regularization techniques require the choice of one or more parameters. In previous work, the cross referencing method has proven to be an effective method in obtaining such approximate solutions. It is also true that varying the singular values of the regularization operator independently can provide great improvement in the quality of the regularized solution. In the present paper, we incorporate this idea as an extension to the cross referencing method.

The modified gradient algorithm has been shown to provide a stable method for reconstructing complex refractive indices (acoustic and electromagnetic) of bounded isotropic inhomogeneities in a variety of 2D problems where the size of the inhomogeneity is of the order of one to three wavelengths. The essential features of the method will be summarized including the use of regularization techniques for resolving discontinuities in the refractive index. The method involves the iterative construction of a global optimizer of a functional consisting of the error in satisfying an integral form of the field equation (the Lippmann-Schwinger equation) and the discrepancy between measured and predicted data. The optimizer is a function pair consisting of the refractive index and the field within the inhomogeneity. The extension of this method to 3D problems and multifrequency data is described. While the extension to three dimensions presents no basic theoretical difficulties, the computational problem is much more complicated. A way to ameliorate this complication by adjusting the integral form of the field equation is described. When data is available at more than one frequency the algorithm must be further modified and these modifications are given. In this case it is necessary to have a priori information on the dispersion relation in the inhomogeneity, for example, in electromagnetics an assumption that the medium is Maxwellian. Results of numerical experiments will be presented to illustrate both the strengths and weaknesses of the method.