Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 2 (9 October 1995); doi: 10.1117/12.224149
One of the fundamental problems in electroencephalography can be characterized by an inverse problem. Given a subset of electrostatic potentials measured on the surface of the scalp and the geometry and conductivity properties within the head, calculate the current vectors and potential fields within the cerebrum. Mathematically the generalized EEG problem can be stated as solving Poisson's equation of electrical conduction for the primary current sources. The resulting problem is mathematically ill-posed i.e., the solution does not depend continuously on the data, such that small errors in the measurement of the voltages on the scalp can yield unbounded errors in the solution, and, for the general treatment of a solution of Poisson's equation, the solution is non-unique. However, if accurate solutions the general treatment of a solution of Poisson's equation, the solution is non-unique. However, if accurate solutions to such problems could be obtained, neurologists would gain noninvasive accesss to patient-specific cortical activity. Access to such data would ultimately increase the number of patients who could be effectively treated for pathological cortical conditions such as temporal lobe epilepsy. In this paper, we present the effects of spatial adaptive refinement on the inverse EEG problem and show that the use of adaptive methods allow for significantly better estimates of electric and potential fileds within the brain through an inverse procedure. To test these methods, we have constructed several finite element head models from magneteic resonance images of a patient. The finite element meshes ranged in size from 2724 nodes and 12,812 elements to 5224 nodes and 29,135 tetrahedral elements, depending on the level of discretization. We show that an adaptive meshing algorithm minimizes the error in the forward problem due to spatial discretization and thus increases the accuracy of the inverse solution.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 19 (9 October 1995); doi: 10.1117/12.224165
The presence of leukemia in bone marrow causes an increase in the electric permittivity and a decrease in the conductivity of the marrow. This suggests the possibility of detecting leukemia by electromagnetic imaging. We show how this can be done for the case of an absorbing host medium (i.e. water) and provide numerical experiments using synthetic data for detecting proliferated tissue at localized portions of the bone marrow. We do not assume that the refractive index of the fat, bone, and muscle are known but will instead recover these values as part of the imaging process.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 27 (9 October 1995); doi: 10.1117/12.224175
The limitations of weak and distorted wave inverse scattering methods can be oversome using a signal processing based technique involving filtering of backpropagated field data in the differential cepstral domain. The purpose of this approach is to extract information about the scattering function from sets of products of this function with unknown field distributions. Being a nonlinear filtering method, there are difficulties in determining the optimal filter function for any given problem. This paper addresses this concern by presenting a systematic study of the information content of the differential cepstrum for a series of scatterers.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 38 (9 October 1995); doi: 10.1117/12.224184
A measurement technique for collecting scattered field data around a small solid angle in the backscatter direction is described. Given these highpass filtered data, a number of methods that can recover an image of the target, depending on whether the scattering target is strongly scattering or not, are briefly reviewed. Due to the nature of these limited frequency data, a method is described which allows one to incorporate prior knowledge about the target, in order to estimate a reconstruction. The performance of the method is compared against traditional Fourier methods.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 50 (9 October 1995); doi: 10.1117/12.224185
An explicit inverse radiative transfer algorithm has been developed to estimate the spatial distribution of a radiant energy source, embedded within a homogeneous plane-parallel medium that can both absorb and scatter light, from measurements of the radiance at its boundaries. The algorithm can be used with fluorescent sources for applications in a variety of fields such as medical imaging and ocean optics. Since the source estimation is done explicitly this algorithm could be used as a starting condition for iterative schemes. The algorithm is intended for use with a beam-expanded laser normally-incident on a target possessing a flat surface, exciting fluorescence at a different wavelength. The algorithm requires that the angle- dependent radiance distribution be measured over incident and outward directions at both boundaries of a slab and that the optical properties of the medium are known a priori. General boundary conditions as well as anisotropic sources can be treated. The algorithm is presented along with some numerical results for a variety of source distributions with a medium modeled as tissue. The results suggest that this algorithm provides a promising way to explicitly estimate the spatial distribution of an embedded source in a participating medium.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 59 (9 October 1995); doi: 10.1117/12.224186
Two one-speed radiation transport equations coupled by a dynamic equation for the distribution of fluorophore electronic states were used to model the migration of excitation photons and emitted fluorescent photons. The conditions for producing appreciable levels of the fluorophore in the excited state were studied, and we concluded that under the conditions applicable to tissue imaging, minimal saturation occurs. This simplified the derivation of the frequency response for a time-harmonic excitation source and of the imaging operator. Several factors known to influence the fluorescence response have been examined. Among these are the concentration, mean lifetime, and quantum yield of the fluorophore, and the modulation frequency of the excitatory source. The fluorescence source strength was calculated as a function of the mean lifetime and modulation frequencies in the 50-200 MHz range. The dependence of demodulation of the fluorescent signal on the above factors was also examined. Results showed that demodulation increases at longer lifetimes and higher modulation frequencies. In additional studies, tomographic imaging operators based on transport theory were derived for imaging fluorophore concentrations embedded in a highly scattering medium. Experimental data were collected by irradiating a cylindrical phantom containing one or two fluorophore-filled balloons with CW laser lihgt. The reconstruction results show that good quality images can be obtained, with embedded objects accurately located.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 74 (9 October 1995); doi: 10.1117/12.224150
The importance of a priori assumptions of a geometrical nature in ill-posed inverse problems is presented. It is illustrated by examples, dealing either with basic questions as uniqueness and stability, or with the way of constructing algorithms that stick at the physics of the problem. At least one of these examples, which may be called 'patchwork analysis', seems new.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 83 (9 October 1995); doi: 10.1117/12.224151
We present a fundamentally novel mathematical algorithm for reconstruction of small inclusion hidden in the diffuse background. This is a numerical method with an a priori guaranteed global convergence. Our theory, which we call Carleman's Weight Method, assures that this technique should provide images with the finest possible resolution. Work on numerical testing is in progress. We also provide a brief historical survey for inverse versus forward problems.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 96 (9 October 1995); doi: 10.1117/12.224152
In this paper, we present a Born Iterative Method for imaging optical properties of turbid media using frequency-domain data. In each iteration, the incident field and the associated weight matrix are first recalculated based on the previous reconstructed image. A new estimate is then obtained by a multigrid finite difference method. The inversion is carried out through a Tikhonov regularized optimization process using the conjugate gradient descent. Using this method, the distribution of the complex wavenumbers in a test medium is first reconstructed, from which the absorption and scattering distributions are then derived. Simulation results have shown that this method can yield quantitatively quite accurate reconstruction even when a strong perturbation exists between the actual medium and an assumed homogeneous background medium, in which case the Born approximation cannot work well. Both full-angle and limited angle measurement schemes have been simulated to understand the effect of the location of detectors and sources.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 108 (9 October 1995); doi: 10.1117/12.224153
We present an algorithm-independent theory of statistical accuracy attainable in emission tomography. Let f denote the tracer density as a function of position (i.e., f is the underlying image). We consider the problem of estimating (Phi) (f) equalsV (integral) (phi) (x)f(x)dx, where (phi) is a smooth function, given n independent observations distributed according to the Radon transform of f. Assuming only that f is bounded above and below away from 0, we construct minimum-variance unbiased (MVU) estimators for (Phi) (f). By definition, the variavnce of the MVU estimator is a best-possible lower bound (depending on (phi) and f) on the variance of unbiased estimators of (Phi) (f). The analysis gives a geometrical explanation of when and by how much estimators based on the standard filtered-backpropagation reconstruction algorithm may be improved.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 120 (9 October 1995); doi: 10.1117/12.224154
In many inverse problems with prior information, the measurement residual and the reconstruction error are two natural metrics for reconstruction quality, where the measurement residual is defined as the weighted sum of the squared differences between the data actually measured and the data predicted by the reconstructed model, and the reconstruction error is defined as the sum of the squared differences between the reconstruction and the truth, averaged over some a priori probability space of possible solutions. A reconstruction method that minimizes only one of these cost functions may produce unacceptable results on the other. This paper develops reconstruction methods that control both residual and error, achieving the minimum residual for any fixed error or vice versa. These jointly optimal estimators can be obtained by minimizing a weighted sum of the residual and the error; the weights are determined by the slope of the tradeoff curve at the desired point and may be determined iteratively. These results generalize to other cost functions, provided that the cost functions are quadratic and have unique minimizers; some results are obtained under the weaker assumption that the cost functions are convex.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 134 (9 October 1995); doi: 10.1117/12.224155
An inverse problem for the one-speed, time-independent, homogeneous radiative transfer equation (RTE) for the intensity of light propagating in a strongly-scattering medium is investigated. We assume that the equation has a rotationally invariant scattering kernel and the diffusion approximation of the spherical harmonics method can be applied. The problem consists in simultaneous determination of a solution to the RTE and a combination of functions describing media absorption and scattering characteristics. It is shown that strictly following the diffusion approximation assumption brings about a general solution to the problem which, in turn, demonstrates that for any limited domain D, the only data we need to find a unique solution to the inverse problem are angular distributions of light intensity in four points belonging to the boundary (delta) D of the domain D and an angular mean value of the intensity along a curve on (delta) D. Explicit formulas for calculating a solution to the inverse problem are obtained.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 145 (9 October 1995); doi: 10.1117/12.224156
X-ray fluoroscopic images are degraded by x-ray scattering within the subject and veiling glare in the image intensifer. Densitometric accuracy is further degraded by beam hardening. Scattering, veiling glare, or both are modeled as a blurred representation of the primary image plus an offset. If the image can be represented by convolution of the primary with a known response function, then an estimate of the primary component of the image can be computed by deconvolution. We describe a technique for estimating a parameterized response function so that a good estimate of the subject density profile can be recovered even if the response function parameters are not known in advance. This is important for x-ray imaging (particularly fluoroscopy) since the acquisition parameters are variable. A reference object designed to be uncorrelated with the subject is imaged in superposition with the subject. The unknown parameters are then adjusted to minimize a cost function subject to the constraint that the correlation between the known reference density and the estimated subject density be zero. The method can be extended to include a correction for beam hardening.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 332 (9 October 1995); doi: 10.1117/12.224157
The method of singular regularization for inverse ill-posed solving with entropy choosing the regularization parameter is purposed. As an application the scheme of backscattered microwave tomography of biological tissue with double focusing is considered.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 163 (9 October 1995); doi: 10.1117/12.224158
Single photon emission computed tomography (SPECT) images are significantly degraded by scattering, attenuation, and intrinsic resolution of collimator. The result is a loss of image quality and quantitative accuracy. These images can still give some useful information, when compared with MR or CT images, so that some patients are periodically monitored by morphological (MR or CT) and functional (SPECT) investigations and these images are matched to determine a space correlation. A method to deconvolve the SPECT images, using the 'a priori' information coming from morphological images, is proposed. This information may be seen as a deterministic constraint to be imposed on the solution of the deconvolution problem and incorporated in the numerical algorithms using projection operators.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 12 (9 October 1995); doi: 10.1117/12.224159
The statement and some questions of the solution to the inverse problem of diffusion tomography are considered. Some numerical results of the solution and the regularization technique are briefly discussed. The main characteristics of this approach is taking account of the circular symmetry in the statement. The algorithm proposed for the solution of this problem in such symmetric statement showed high performance and sufficient accuracy.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 174 (9 October 1995); doi: 10.1117/12.224160
Singular value decomposition has served as a diagnostic tool in optical computed tomography by using its capability to provide insight into the condition of ill-posed inverse problems. Various tomographic geometries are compared to one another through the singular value spectrum of their weight matrices. The number of significant singular values in the singular value spectrum of a weight matrix is a quantitative measure of the condition of the system of linear equations defined by a tomographic geometery. The analysis involves variation of the following five parameters, characterizing a tomographic geometry: 1) the spatial resolution of the reconstruction domain, 2) the number of views, 3) the number of projection rays per view, 4) the total observation angle spanned by the views, and 5) the selected basis function. Five local basis functions are considered: the square pulse, the triangle, the cubic B-spline, the Hanning window, and the Gaussian distribution. Also items like the presence of noise in the views, the coding accuracy of the weight matrix, as well as the accuracy of the accuracy of the singular value decomposition procedure itself are assessed.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 186 (9 October 1995); doi: 10.1117/12.224161
In this paper, we present a wavelet based multigrid approach to solve the perturbation equation encountered in optical tomogrpahy. With this scheme, the unkown image, the data, as well as weight matrix are all represented by wavelet expansions, and thus yielding a multiresolution representation of the original perturbation equation in the wavelet domain. This transformed equation is then solved using multigrid scheme, by which an increasing portion of wavelet coefficients of the unknown image are solved in successive approximations. One can also quickly identify regions of interest from a coarse level reconstruction and restrict the reconstruction in the following fine resolutions to those regions. At each resolution level, a regularized least squares solution is obtained using a conjugate gradient descent method. Compared to a previously reported one grid algorithm, the multigrid method requires substantially shorter computation time under the same reconstruction quality criterion.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 197 (9 October 1995); doi: 10.1117/12.224162
In these resolution studies, a sophisticated version of the finite element method was used to model the time evolution of a laser pulse within a simulated medium. An array of 'detectors', placed in the medium, were used to measure the pulse intensity at a discrete set of points. As we will show in this report, the 'detectors' were able to resolve distortions in the pulse owing to two 1 mm diameter inclusions embedded in this otherwise homogeneous medium and separated by as little as 1 mm. Thus, the data from these detectors could, in principle, be employed by an algorithm designed to solve the inverse problem: the imaging, at 1 mm resolution, of inclusions based upon data from detectors placed, or surrounding, the medium.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 209 (9 October 1995); doi: 10.1117/12.224163
This paper is concerned with the real invserion of a Laplace tranform function F(s) equals L[f(t)]. The real inversion problem is that of reconstructing f(t) from known values of F(s), given only at real points. Numerical inversion of a Laplace transform is a very difficult problem to be solved in general, as shown in the survey written by Davies and Martin in 1979. Actually, this is still true. It is well known that the numerical solution of the real inversion problem is much more difficult than that of the complex one. Briefly, this is due to an intrinsic ill-posedness of the real inversion problem in the sense that small changes in data can cause arbitrary large changes in the solutionl. This is reflected in ill-conditioning of the discrete model. We propose a numerical method for the real inversion problem based on a Fourier series expansion of f(t). Introducing some kind of regularization we show how it is possible to approximate the Fourier coefficients of f(t), and then to compute the inverse Laplace function, only using the knowledge of the restriction of F(s) on the real axes.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 219 (9 October 1995); doi: 10.1117/12.224164
Formulas for the response of a detector on the surface of a highly scattering random medium to a change in physical properties within the medium were deduced from a transport-theory- based linear perturbation model. These expressions are called weight functions. Position- dependent intensities and fluxes were computed for the interior of a twenty mean free pathlength diameter, infinitely long, nonabsorbing, homogeneous cylindrical medium. Numerical values for the weight functions were calculated from these data. Surface detector readings were computed for the homogeneous reference medium and for six target media, each of which contained an array of twelve or thirteen thin, infinitely long rods embedded in the cylinder as a perturbation. The rods had a positive absorption cross section and a smaller scattering cross section than the reference, such that the mean free pathlength was constant throughout each medium. The directly computed detector readings perturbations were compared to those calculated from the linear perturbation model; as expected, the agreement was very good for the weakest cross section perturbations and became steadily poorer as the perturbations increased. Two iterative algebraic image reconstruction algorithms are described; both were used to compute images of the six target media. One algorithm tends to correctly identify the location of the rods lying closest to the surface, but places the deeper ones bunched too closely together near the cylinder axis. The other tends to place the superficial rods too close to the surface. In addition, while it appears to identify heterogeneities on the cylinder axis correctly after relatively few iterations, the estimated cross section perturbation along the axis gradually goes to zero as the number of iterations increases. Still in all, the performance of these algorithms is probably as good as can be achieved using a first-order Born reconstruction (i.e., no update of forward problem).
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 236 (9 October 1995); doi: 10.1117/12.224166
The existing technology for detecting chemical species is based on spectral decomposition of light from a multispecies sample. This approach is limited because of line overlapping from different species and is getting increasingly unreliable when the number of species is large. We propose a holographic species analyzer, which decomposes an input in terms of a prespecified set of species. The innovation is to replace a diffraction grating with a holographic memory element designed to recognize the whole light patterns of different species and generate in an output plane a position sensitive map of species present. The holographic element can be optically or computer generated. The proposed technology is shown using computer models to be both highly wavelength and position sensitive.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 242 (9 October 1995); doi: 10.1117/12.224167
We consider the application of Tikhonov type regularization methods for computing a cubic spline approximation to the solution of a particular Fredholm integral equation of the first kind which arises in laser Doppler anemometry experiments. The method of generalized cross validation is used to calculate an unbiased estimate to the value of the regularization parameter controlling the trade-off between the smoothness of the approximation and the fidelity of the tranformed approximation to the data, which are assumed to be contaminated by 'white noise' error. Numerical results are presented, for zero order regularization on simulated laser anemometry data, which demonstrate that the success of the method is dependent on the positioning of the knots of the spline. Proposed extensions to this work are discussed, which include techniques for incorporating cross validation with higher orders of regularization and the addition of an automatic knot selection algorithm.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 254 (9 October 1995); doi: 10.1117/12.224168
In this paper, a reconstruction algorithm for frequency-domain optical tomography in human tissue is presented. A fast and efficient multigrid finite difference (MGFD) method is adopted as a forward solver to obtain the simulated detector responses and the required imaging operator. The solutions obtained form MGFD method for 3D problems with weakly discontinuous cocoefficients are compared with analyzed solutions to determine the accuracy of the numerical method. Simultaneous reconstruction of both absorption and scattering coefficients for tissue-like media is accomplished by solving a perturbation equation using the Born approximation. This solution is obtained by a conjugate gradient descent method with Tikhonov regularization. Two examples are given to show the quality of the reconstruction results. Both involve the examination of anatomically accurate optical models of tissue derived from segmented 3D magnetic resonance images to which have been assigned optical coefficients to the designated tissue types. One is a map of a female breast containing two small 'added pathologies', such as tumors. The other is a map of the brain containing a 'local bleeding' area, representing a hemorrhage. The reconstruction results show that the algorithm is computationally practical and can yield qualitatively correct geometry of the objects embedded in the simulated human tissue. Acceptable results are obtaiend even when 10% noise is present in the data.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 267 (9 October 1995); doi: 10.1117/12.224169
Deconvolution of images of the same object from multiple sensors with different point spread functions (PSF), as shown by Berenstein and Patrick, can be a well-posed problem in the sense of distributions if the PSF satisfy some suitable conditions. More precisely, if these operators are represented by compactly supported distributions, a corresponding set of deconvolvers, also given by compactly supported distributions, may exist. Nevertheless, it must be observed that this inverse operator is not particularly useful if the multiple images which must be deconvolved are affected by noise, because continuity in the sense of distributions is too weak. This is the reason why a more effective approach is provided by the inverse methods typical of regularization theory. We have considered the case described by Berenstein and Patrick, in which the input function consists of the sum of two Gaussian pulses and the PSF are the characteristic functions of the intervals (-1, 1) and (- (root)2, 2). The two images we have obtained have been affected by Gaussian noise and then simulated data have been inverted by using various regularization techniques; in particular, in the case of iterative methods, it has also been possible to introduce the positivity constraint. The comparison between the reconstructions we have obtained and the input function allows to estimate the greater efficiency of the regularized multiple operators deconvolution, compared with the inversion of a single image, when linear filtering is applied. On the contrary the performance of the nonlinear constrained iterative method seems not to be particularly sensitive to the use of two images instead of one. An explanation of this fact is given and an example, where the use of multiple images can be advantageous, is presented.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 276 (9 October 1995); doi: 10.1117/12.224170
Phase retrieval for reconstructing symmetric molecules from x-ray crystallographic data is investigated. The additional information contained in the symmetry should be sufficient to render the solution to such problems unique in many cases. This is supported by the application of iterative phase retrieval algorithms to simulated crystallographic data from a model of an icosahedral virus particle.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 282 (9 October 1995); doi: 10.1117/12.224171
In 1994, to overcome the difficulties encountered by the maximum entropy method (MEM) to restore images containing both high and low frequencies, Bonteoke et al. introduced the Pyramid Maximum Entropy Deconvolution. However, this method presents several drawbacks such as parameters estimation (model, alpha). Furthermore, in their method they don't minimize any functional. Following these ideas, we propose the multiresolution maximum entropy method which is based on the concept of multiscale entropy derived from the wavelet decomposition of a signal into different frequency bands. It leads to a method which is flux conservative, and the use of a multiresolution support solves the problem of MEM to choose the (alpha) parameter, i.e. relative weight between the goodness-of-fit and the entropy. We also show that our algorithm is efficient for filtering astronomical images.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 338 (9 October 1995); doi: 10.1117/12.224172
During ill-posed inverse problem solution authors discussed consecutive technology of account known a priori information. In the present paper they realized this technology as the unique filtration procedures of singular components or CK-components. Sample correlation and distortion operator are used for singular codebook calculation. Correlation and level of noise are used for regularization. The application of this technique for more complex stochastic objects will be subject in future investigations.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 347 (9 October 1995); doi: 10.1117/12.224173
Restoration of light intensity fluctuations from photon-counting data provides typical ill-posed inverse problem for non-Hermitian transforming operator. Such problems cannot be solved with the help of eigenfunction representation which diagonalizes operator and simplifies the regularization of quasi-solution. We describe an analytical method to solve this inverse problem which is based on the Poisson transform operator representation in mixed basis. The last one is determined by eigenfunctions of left and right iterated operators and diagonlize Poisson transform. On the base of proposed method we have performed the set of numerical experiments with typical intensity distributions. It has been shown that restoration procedure has necessary stability and may be used when the level of statistical errors is relatively high. In conclusion we describe the results of inversion procedure appreciation to the processing of photon counts statistics experimental data.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 359 (9 October 1995); doi: 10.1117/12.224174
In the present paper, we consider that a planar laser pulse falls obliquely on the surface of a stratified tissue slab. The Green function approach is developed to reconstruct two optical parameters of the stratified tissue simultaneously. By analyzing the strucutre of the fundamental solution, the transport equation of the Green function and the explicit form of the initial condition are derived. The initial condition of Green function is found that correlates to the parameters of tissue and incident angle. By the characteristic method, the measurable boundary data are propagated to the initial value layer by layer for each incident angle. The scattering and absorption coefficient of tissue are then reconstructed simultaneously layer by layer using the explicit expression of the initial condition of Green fucntion. A detailed algorithm for the reconstruction is presented and an imaging picture as a numerical examples is given.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 366 (9 October 1995); doi: 10.1117/12.224176
The projection algorithm for solving systems of linear algebraic equations together with method of statistical regularization were used for solving ill-posed inverse problems of processing experimental data. That problems were smoothing of differentiation of experimental data, apparatus distortion, decomposition of complex signal on elementary components and Radon tranforms (problems of computer tomography). Our method gives it possible to use maximum known a prior information, what is important for solving ill-posed problems.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 373 (9 October 1995); doi: 10.1117/12.224177
Theory is developed to describe diffraction, in the weak scattering regime, from finite distorted lattices. Correlations between lattice distortions are modeled using an imposed correlation field. The validity of the imposed correlation field is examined. Expressions for the intensity, and the circularly averaged intensity, diffracted from an ensemble of distorted lattices are derived. Calculated diffraction patterns are used to examine the relationship between diffraction and the kind and degree of disorder in a specimen. Implications of this work for inverse problems in x-ray crystallography are discussed.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 385 (9 October 1995); doi: 10.1117/12.224178
Recovery of an image from a finite set of irregularly spaced samples is considered. A method of ranking the quality of such sampling sets described by Chen and Alleback is extended to the case of unequal numbers of samples. Calculations provide information on the relative effects of sample number, density and nonuniformity on reconstruction error.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 292 (9 October 1995); doi: 10.1117/12.224179
In this paper, we develop a refined version of the mathematical model introduced by Shepp and Vardi for positron emission tomogrpahy. This model replaces the usual finite-dimensional linear system by a nonstandard integral equation in which the data-space is finite-dimensional, but the unknown emission intensities are represented by a mathematical measure on the region of interest. Since our measure might not be representable by a density, this is also a refinement of the integral equation model mentioned (but not analyzed) in the work of Vardi, Shepp, and Kaufman. As in the finite-dimensional model, we obtain an iteration procedure which generates a sequence of functions. Such a functional iteration has already been proposed by other researchers for solving a general class of positive linear ill-posed inverse problems. However, unlike the original finite-dimensional problem, the convergenec of this infinite- dimensional version remains an open question. This paper demonstrates examples in which these iterates are continuous functions, but does not converge to a density. We also discuss a feasible approach for settling the convergence question.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 319 (9 October 1995); doi: 10.1117/12.224180
Traditional successful approaches to inverse problem solutions usually deal with continuous variable domains: image inversion is a key example of such a problem. In the present paper the mathematical base for a novel efficient algorithm for inversion problems is discussed. The proposed procedure, called the numerical Fovea, is a general discrete algorithm tuned to image inversion problems and matches the basic characteristics of the human visual system. In fact distance, working in feedback. This kind of structure allows for the formation of images according to the binocular field of view. The represented algorithm operates in a discrete variable domain. Beyond the advantages in terms of simplicity and computational speed, it agrees with the results of biological observation which reveal that the elements sensitive to light stimuli are finite in number. Thus, the domain of interest can be modeled as a couple of bi-dimensional lattices having the mathematical structure of discrete vector groups, consistently with the geometrical receptor displacement in the human fovea. The receptor multilayer structure can be described by a recursive relation. A numerical example is presented: for every source reconstruction problem an optimal computational precision level can be selected according to the required accuracy.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 151 (9 October 1995); doi: 10.1117/12.224181
Simple and numerically stable approaches to approximate solution of inverse geophysical and potential scattering problems are described. The method we propose consists of two steps. Let v(z) be the inhomogeneity (potential), and let D be its support. First, we find approximations to the zeroth moment (total intensity) v(z)dz and the first moment (center of gravity) zv(z)dz/ v(z)dz of the function v(z). We call this step 'inhomogeneity localization', because in many cases the center of gravity lies inside D or is located close to it. Second, we refine the above moments and find the tensor of the second central moments of v(z). Using this information, we find an ellipsoid D and a real constant v, such that the inhomogeneity (potential) v(z) equals v,z an element of D, and v(z) equals 0,z not an element of D, fits best the scattering data and has the same zeroth, first, and second moments. We call this step 'approximate inversion'. The proposed method does not require any intensive computations, it is very simple to implement and it is relatively stable towards noise in the data.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 307 (9 October 1995); doi: 10.1117/12.224182
In this paper, a novel tactile photoelastic transducer for normal forces is presented. When a normal input force profile is applied to the tranduction medium, stress is generated in the photoelastic layer making it birefringent. Consequently, circularly-polarized input light becomes elliptically polarized at the output due to the introduction of a phase-lead distribution. If a circular-reflection polaridoscope is used, the output light-intensity is a circular function of the total phase-lead distribution. The first part of the paper describes the forward analysis of the transducer using finite-element analysis to determine the stress distribution in the transducer. Then, the phase-lead distribution is determined using the theory of photoelasticity. The second part of the paper describes a technique for the recovery of the phase-lead distribution from the ideal noise-free light-intensity distribution. Also, a verification method is proposed to determine whether a recovered phase-lead distribution is the correct one or not. In the third part of the paper, we consider the nonideal situation, where the light-intensity distribution is no longer noise-free. Quantization errors added to the detected light-intensity distribution are also considered. Recovering the phase-lead distribution under noisy conditions constitutes an ill-posed problem. An algorithm that accurately and effectively determines the phase-lead distribution from a noisy light-intensity distribution is presented. The inverse-tactile problem is solved using an optimization function.
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Proc. SPIE 2570, Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, pg 301 (9 October 1995); doi: 10.1117/12.224183
We consider the problem of imaging the optical absorption of a highly-scatering medium probed by diffusive light. An analytic solution to the image reconstruction problem is presented.
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