Application of the regularization method to the inverse black bodyradiation problem

The inverse black body radiation problem is concerned with the determination of the area temperature distribution of a black body source from spectral measurements of its radiation. Although several inversion approaches have been developed, none of them has overcome the problem of ill-posedness. In this study, Tikhonov's regularization method is applied to the solution of the inverse black body radiation problem. A very simple implementation of this approach together with applications of this method are presented. The effect of the regularization parameter and operator on the results is discussed     

An application of Marquardt's procedure to the seismic inverse problem

The seismic inverse problem is to infer characteristics of the subsurface from measurements of the wave field at the surface. The Marquardt procedure offers one approach to this problem. In applying this procedure, a linear relationship is developed between the wave field and some parameter which describes a physical property of the subsurface. Then a selection criterion is designed to choose the subsurface parameter which provides the best match for the observed seismic data.     

A pixel-based regularization approach to inverse lithography

Inverse lithography attempts to synthesize the input mask which leads to the desired output wafer pattern by inverting the forward model from mask to wafer. In this article, we extend our earlier framework for image prewarping to solve the mask design problem for coherent, incoherent, and partially coherent imaging systems. We also discuss the synthesis of three variants of phase shift masks (PSM); namely, attenuated (or weak) PSM, 100% transmission PSM, and strong PSM with chrome. A new two-step optimization strategy is introduced to promote the generation and placement of assist bar features. The regularization framework is extended to guarantee that the estimated PSM have only two or three (allowable) transmission values, and the aerial-image penalty term is introduced to boost the aerial image contrast and keep the side-lobes under control. Our approach uses the pixel-based mask representation, a continuous function formulation, and gradient-based iterative optimization techniques to solve the inverse problem. The continuous function formulation allows analytic calculation of the gradient in O(MNlog (MN)) operations for an M × N pattern making it practically feasible. We also present some results for coherent and incoherent imaging systems with very low k₁ values to demonstrate the effectiveness of our approach.     

Inverse Jury-Blanchard table and its applications to the inverse stability problem

This letter shows that the coefficients of the characteristic equation of a discrete time system can be obtained in a simple manner if the elements of the first column of the Jury-Blanchard table are given.     

Efficient algorithm for nonconvex minimization and its application to PM regularization

In image processing, nonconvex regularization has the ability to smooth homogeneous regions and sharpen edges but leads to challenging computation. We propose some iterative schemes to minimize the energy function with nonconvex edge-preserving potential. The schemes are derived from the duality-based algorithm proposed by Bermúdez and Moreno and the fixed point iteration. The convergence is proved for the convex energy function with nonconvex potential and the linear convergence rate is given. Applying the proposed schemes to Perona and Malik's nonconvex regularization, we present some efficient algorithms based on our schemes, and show the approximate convergence behavior for nonconvex energy function. Experimental results are presented, which show the efficiency of our algorithms, including better denoised performance of nonconvex regularization, faster convergence speed, higher calculation precision, lower calculation cost under the same number of iterations, and less implementation time under the same peak signal noise ratio level.     

Genus zero surface conformal mapping and its application to brain surface mapping

We developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one-forms for surfaces with or without boundaries (Gu and Yau, 2002), (Gu and Yau, 2003). For genus zero surfaces, our algorithm can find a unique mapping between any two genus zero manifolds by minimizing the harmonic energy of the map. In this paper, we apply the algorithm to the cortical surface matching problem. We use a mesh structure to represent the brain surface. Further constraints are added to ensure that the conformal map is unique. Empirical tests on magnetic resonance imaging (MRI) data show that the mappings preserve angular relationships, are stable in MRIs acquired at different times, and are robust to differences in data triangulation, and resolution. Compared with other brain surface conformal mapping algorithms, our algorithm is more stable and has good extensibility.     

Self-affine mapping system and its application to object contourextraction

A self-affine mapping system which has conventionally been used to produce fractal images is used to fit rough lines to contours. The self-affine map's parameters are detected by analyzing the blockwise self-similarity of a grayscale image using a simplified algorithm in fractal encoding. The phenomenon that edges attract mapping points in self-affine mapping is utilized in the proposed method. The boundary of the foreground region of an alpha mask is fitted by mapping iterations of the region. It is shown that the proposed method accurately produces not only smooth curves but also sharp corners, and has the ability to extract both distinct edges and blurred edges using the same parameter. It is also shown that even large gaps between the hand-drawn line and the contour can be fitted well by the recursive procedure of the proposed algorithm, in which the block size is progressively decreased. These features reduce the time required for drawing contours by hand.     

Iterative regularization and nonlinear inverse scale space applied to wavelet-based denoising.

In this paper, we generalize the iterative regularization method and the inverse scale space method, recently developed for total-variation (TV) based image restoration, to wavelet-based image restoration. This continues our earlier joint work with others where we applied these techniques to variational-based image restoration, obtaining significant improvement over the Rudin-Osher-Fatemi TV-based restoration. Here, we apply these techniques to soft shrinkage and obtain the somewhat surprising result that a) the iterative procedure applied to soft shrinkage gives firm shrinkage and converges to hard shrinkage and b) that these procedures enhance the noise-removal capability both theoretically, in the sense of generalized Bregman distance, and for some examples, experimentally, in terms of the signal-to-noise ratio, leaving less signal in the residual.     

Convexly constrained linear inverse problems: iterativeleast-squares and regularization

We consider robust inversion of linear operators with convex constraints. We present an iteration that converges to the minimum norm least squares solution; a stopping rule is shown to regularize the constrained inversion. A constrained Laplace inversion is computed to illustrate the proposed algorithm     

A critical analysis of linear inverse solutions to theneuroelectromagnetic inverse problem

This paper explores the possibilities of using linear inverse solutions to reconstruct arbitrary current distributions within the human brain. The authors formally prove that due to the underdetermined character of the problem, the only class of measurable current distributions that can be totally retrieved are those of minimal norm. The reconstruction of smooth or averaged versions of the currents is also explored. A solution that explicitly attempts to reconstruct averages of the current is proposed and compared with the minimum norm and the minimum Laplacian solution. In contrast to the majority of previous analysis carried out in the field, in the comparisons, the authors avoid the use of measures designed for the case of dipolar sources. To allow for the evaluation of distributed solutions in the case of arbitrary current distributions the authors use the concept of resolution kernels. Two summarizing measures, source identifiability and source visibility, are proposed and applied to the comparison. From this study can be concluded: (1) linear inverse solutions are unable to produce adequate estimates of arbitrary current distributions at many brain sites and (2) averages or smooth solutions are better than the minimum norm solution estimating the position of single point sources. However, they systematically underestimate their amplitude or strength especially for the deeper brain areas. Based on these result, it appears unlikely that a three-dimensional (3-D) tomography of the brain electromagnetic activity can be based on linear reconstruction methods without the use of a significant amount of a priori information     

Antenna synthesis and solution of inverse problems by regularization methods

Antenna pattern synthesis is discussed as an example of "improperly posed" problems. This serves the purpose of introducing a concept that is useful in many other applications: remote sensing, inverse scattering, etc. It also suggests that regulation methods that have been devised to "solve" improperly posed problems can be applied to antenna synthesis and the aforementioned problems. This gives systematic methods for solving the pattern synthesis problem even when the element patterns are arbitrary.     

Analytical and numerical solution to the two-potentialZakharov-Shabat inverse scattering problem

An analytical and a numerical method are presented in order to solve the inverse scattering problem associated with the two-potential Zakharov-Shabat coupled mode equations. The numerical solution, which uses leapfrogging in space and time, represents a direct numerical solution to the coupled Gel'fand-Levitan-Marchenko (GLM) integral equations as an extension of the authors' previous work on GLM equations of simpler form. The analytical method, which is applied for one-pole reflection coefficients, consists in constructing appropriate differential operators which transform the coupled GLM equations to ordinary linear differential equations. An application of these methods for nonuniform transmission line synthesis is presented     

Resolving the hemodynamic inverse problem

The "hemodynamic inverse problem" is the determination of arterial system properties from pressures and flows measured at the entrance of an arterial system. Conventionally, investigators fit reduced arterial system models to data, and the resulting model parameters represent putative arterial properties. However, no unique solution to the inverse problem exists-an infinite number of arterial system topologies result in the same input impedance (Zin) and, therefore, the same pressure and flow. Nevertheless, there are exceptions to this theoretical limitation; total peripheral resistance (Rtot), total arterial compliance (Ctot), and characteristic impedance (ZO) can be uniquely determined from input pressure and flow. Zin is determined completely by Ctot and Rtot at low frequencies, Zo at high frequencies, and arterial topology and reflection effects at intermediate frequencies. We present a novel method to determine the relative contribution of Zo, Ctot, Rtot and arterial topology/reflection to Zin without assuming a particular reduced model. This method is tested with a large-scale distributed model of the arterial system, and is applied to illustrative cases of measured pressure and flow. This work, thus, lays the theoretical foundation for determining the arterial properties responsible for increased pulse pressure with age and various arterial system pathologies.     

Inverse Routh-Hurwitz array solution to the inverse stability problem

In this letter, a general method will be given whereby the inverse Routh-Hurwitz array may be constructed. This gives a solution to the inverse stability problem as first posed by Jarominek.     

A generalized eigensystem approach to the inverse problem ofelectrocardiography

The authors develop a new approach to the ill-conditioned inverse problem of electrocardiography which employs finite element techniques to generate a truncated eigenvector expansion to stabilize the inversion. Ordinary three-dimensional isoparametric finite elements are used to generate the conductivity matrix for the body. The authors introduce a related eigenproblem, for which a special two-dimensional isoparametric area matrix is used, and solve for the lowest eigenvalues and eigenvectors. The body surface potentials are expanded in terms, of the eigenvectors, and a least squares fit to the measured body surface potentials is used to determine the coefficients of the expansion. This expansion is then used directly to determine the potentials on the surface of the heart. The number of measurement points on the surface of the body can be less than the number of finite element nodes on the body surface, and the number of modes employed in the expansion can be adjusted to reduce errors due to noise     

Effects of experimental and modeling errors on electrocardiographic inverse formulations

The inverse problem of electrocardiology aims to reconstruct the electrical activity occurring within the heart using information obtained noninvasively on the body surface. Potentials obtained on the torso surface can be used as input for the inverse problem and an electrical image of the heart obtained. There are a number of different inverse algorithms currently used to produce electrical images of the heart. The relative performances of these inverse algorithms at this stage is largely unknown. Although there have been many simulation studies investigating the accuracy of each of these algorithms, to date, there has been no comprehensive study which compares a wide variety of inverse methods. By performing a detailed simulation study, we compare the performances of epicardial potential [Tikhonov, Truncated singular value decomposition (TSVD), and Greensite] and myocardial activation-based (critical point) inverse simulations along with different methods of choosing the appropriate level of regularization (optimal, L-curve, composite residual and smoothing operator, zero-crossing) to apply to each of these inverse methods. We also examine the effects of a variety of signal error, material property error, geometric error and a combination of these errors on each of the electrocardiographic inverse algorithms. Results from the simulation study show that the activation-based method is able to produce solutions which are more accurate and stable than potential-based methods especially in the presence of correlated errors such as geometric uncertainty. In general, the Greensite-Tikhonov method produced the most realistic potential-based solutions while the zero-crossing and L-curve were the preferred method for determining the regularization parameter. The presence of signal or material property error has little effect on the inverse solutions when compared with the large errors which resulted from the presence of any geometric error. In the presence of combined Gaussian and correlated errors representing conditions which may be encountered in an experimental or clinical environment, there was less variability between potential-based solutions produced by each of the inverse algorithms.     

Finite-element-based discretization and regularization strategies for 3-D inverse electrocardiography

We consider the inverse electrocardiographic problem of computing epicardial potentials from a body-surface potential map. We study how to improve numerical approximation of the inverse problem when the finite-element method is used. Being ill-posed, the inverse problem requires different discretization strategies from its corresponding forward problem. We propose refinement guidelines that specifically address the ill-posedness of the problem. The resulting guidelines necessitate the use of hybrid finite elements composed of tetrahedra and prism elements. Also, in order to maintain consistent numerical quality when the inverse problem is discretized into different scales, we propose a new family of regularizers using the variational principle underlying finite-element methods. These variational-formed regularizers serve as an alternative to the traditional Tikhonov regularizers, but preserves the L(2) norm and thereby achieves consistent regularization in multiscale simulations. The variational formulation also enables a simple construction of the discrete gradient operator over irregular meshes, which is difficult to define in traditional discretization schemes. We validated our hybrid element technique and the variational regularizers by simulations on a realistic 3-D torso/heart model with empirical heart data. Results show that discretization based on our proposed strategies mitigates the ill-conditioning and improves the inverse solution, and that the variational formulation may benefit a broader range of potential-based bioelectric problems.     

The tapered antenna and its application to the junction problem for thin wires

When electrically thin conductors of different cross sectional size meet, the continuity of current is assured by Kirchhoff's current law. Additional conditions must be imposed on the derivatives of the currents or the charges per unit length. The nature of the required conditions is determined from an analysis of the tapered antenna.     

等几何分析法应用于偏心柱面静电场问题

利用等几何分析思想通过加权余量法对静电场控制方程进行弱化,推导出静电场的等几何分析方程,提出了一种分析静电场问题的高精度方法。将此方法应用于求解偏心柱面静电场问题,并对计算结果与解析解和经典的有限元方法比较,结果表明:此方法具有自由度少、精度高、收敛速度快的优点。