Variational image segmentation using boundary functions

A general variational framework for image approximation and segmentation is introduced. By using a continuous "line-process" to represent edge boundaries, it is possible to formulate a variational theory of image segmentation and approximation in which the boundary function has a simple explicit form in terms of the approximation function. At the same time, this variational framework is general enough to include the most commonly used objective functions. Application is made to Mumford-Shah type functionals as well as those considered by Geman and others. Employing arbitrary L(p) norms to measure smoothness and approximation allows the user to alternate between a least squares approach and one based on total variation, depending on the needs of a particular image. Since the optimal boundary function that minimizes the associated objective functional for a given approximation function can be found explicitly, the objective functional can be expressed in a reduced form that depends only on the approximating function. From this a partial differential equation (PDE) descent method, aimed at minimizing the objective functional, is derived. The method is fast and produces excellent results as illustrated by a number of real and synthetic image problems.     

Field Solution, Polarization, and Eigenmodes of Shielded Microstrip Transmission Line

Application of the reciprocity theorem leads to a variational expression for the propagation constant of the fields inside shielded microstrip-like transmission lines. The resulting equation involves both the propagation constant and the tangential fields at the air-dielectric interface. Using the Rayleigh-Ritz optimization technique, both the propagation constant and the fields are completely determined. The calculated results of the propagation constant compare well with other available data. Moreover, the field solution obtained is presented in the form of a polarization ratio relating the axial to the transverse electric field. Results cover both low and high frequencies, and the technique proves valid at both frequency ranges. The method may be extended to other configurations of planar striplines by proper adjustment of the integration limits.     

重建散射位势的变分公式

本文研究一维Ｓｃｈｒｏｄｉｎｇｅｒ方程散射位势的重建。从著名的Ｇｅｌ＇ｆｎａｄ Ｌｅｖｉｔａｎ（Ｇ－Ｌ）方程出发，本文构造了一个重建散射位势的变分公式。由于该公式的变分特性，加之它充分利用了Ｇ－Ｌ方程近似解的全部信息，因此能够用于提高一些基于求解Ｇ－Ｌ方程的近似重建方法的计算效率。文中还给出该公式应用于迭代法近似解的一些例子。     

Vector Variational Solutions of Inhomogeneously Loaded Cylindrical Waveguide Structures

A vector variational formulation of the Maxwell curl equations in terms of all six electromagnetic field components is developed for circular waveguides symmetrically loaded with dialectric. A guide-wall boundary integral term, which renders the electric Dirichlet boundary conditions natural, is included in the variational expression. It is, therefore, unnecessary for the trial electromagnetic field functions to satisfy any guide-wall boundary conditions. This greatly extends the class of problems for which the vector variational-solution procedure is applicable and permits simple expansion functions for the trial fields, thus facilitating integral calculations. Solution results for several hybrid-mode circulator-waveguide structures are presented and compared with available experimental data.     

Numerical Solution of Dielectric Loaded Waveguides: I-Finite-Element Analysis

A variational expression of the electromagnetic fields in dielectric loaded waveguides is derived. This expression is discretized using the finite-element method and an electromagnetic coupling matrix is derived and evaluated. No restriction is placed on the shapes of the triangular elements or the order of the polynomial approximation. A general finite-element computer program is described and dispersion curves and field plots of some dielectric loaded waveguides are presented.     

Time domain modeling of nonlinear loads

The behavior of an antenna or scatterer when loaded with a nonlinear element can be changed greatly from that observed under linear conditions. In some cases, the nonlinearity causes effects such as the intermodulation products arising from the nonlinear mixing of two frequencies. In other cases, the nonlinearity may be exploited, for example, to reduce late-time ringing on a pulse-excited antenna. A procedure for treating general nonlinear loads is described and illustrated. This procedure is applied using a computer to several specific types of nonlinearities. The treatment within the framework of the thin-wire approximation to the electric field integral equation is developed. As such, the treatment can be applied to the large class of objects modeled by wires. The nonlinear load types that are considered include those with piecewise-linear voltage-current curves (e.g., one or more diodes), a load with a time varying resistance (which permits modulating the scattered fields), and a general nonlinear load represented by specified voltage-current functions.     

A variational coupled-mode theory for periodic waveguides

A simple variational theorem for the dispersion relations and propagation constants of periodic waveguides is derived. The trial fields used in the variational formula incorporate all the Floquet components correct to first order. This yields a propagation constant in which errors enter only to fourth order in the trial field. The analysis yields a new coupled mode formulation which is shown to yield excellent agreement with exact analytic solutions (in l-D), and numerical simulations (in 2-D), for high-index contrast structures     

Relationship between group delay and stored energy in microwavefilters

In this paper, an expression for the time average stored energy (t.a.s.e.) in a passive lossless two-port is derived in terms of its scattering parameters. In particular, it is shown that the t.a.s.e. in a passive lossless reciprocal symmetrical or antimetrical two-port is proportional to the group delay. One implication of this result is that the t.a.s.e., which is linked to the power-handling capability in many passive filters used in practice, is proportional to the group delay of the filter. This rigorous derivation is based on a variational theorem, which has been used in the past to prove energy storage results for passive lossless one-ports and periodic two-ports     

有限域上独立随机变量和的极限分布定理及其在流密码中的应用

利用有限域上随机变量的特征函数，本文得到了有限域上独立随机变量和的极限分布定理，作为这一定理的应用，本文研究了有限域上非线性组合函数的构造，给出了有限域上一类与任何线性函数都有较小相关性的非线性组合函数的结构。     

The Shannon sampling theorem—Its various extensions and applications: A tutorial review

It has been almost thirty years since Shannon introduced the sampling theorem to communications theory. In this review paper we will attempt to present the various contributions made for the sampling theorems with the necessary mathematical details to make it self-contained. We will begin by a clear statement of Shannon's sampling theorem followed by its applied interpretation for time-invariant systems. Then we will review its origin as Whittaker's interpolation series. The extensions will include sampling for functions of more than one variable, random processes, nonuniform sampling, nonband-limited functions, implicit sampling, generalized functions (distributions), sampling with the function and its derivatives as suggested by Shannon in his original paper, and sampling for general integral transforms. Also the conditions on the functions to be sampled will be summarized. The error analysis of the various sampling expansions, including specific error bounds for the truncation, aliasing, jitter and parts of various other errors will be discussed and summarized. This paper will be concluded by searching the different recent applications of the sampling theorems in other fields, besides communications theory. These include optics, crystallography, time-varying systems, boundary value problems, spline approximation, special functions, and the Fourier and other discrete transforms.     

A unified approach for the coupled-mode analysis of nonlinearoptical couplers

A unified approach for the coupled-mode analysis of nonlinear optical couplers is proposed. This approach is basically an extension of the work of [Haus, vol. 5, p. 16, 1987] to include optical nonlinearity. After the nonlinear coupled-mode theory is established, various basis functions are used as trial fields to derive coupled-mode equations. It is found that two published coupled-mode theories can in fact be deduced from the proposed one with individual linear or nonlinear waveguide modes serving as trial fields, thus making the coupled-mode equations from variational principle and reciprocity theorem equivalent. Coupled-mode equations based on system modes are also presented. Furthermore, analytical and/or numerical methods for solving coupled-mode equations are included. More research and discussion can be conducted based on the knowledge addressed in this paper     

Variational Solution of Integral Equations

A variational solution of the Fredholm integral equation of the first kind resulting from Laplace's equation with Dirichlet boundary conditions is discussed. Positive-definiteness of the integral operator is used to guarantee convergence. The square parallel plate capacitor is given as an example with several different types of trial functions. Special singular functions to handle known field behavior are shown to result in improved accuracy with reduced computing cost. The air-dielectric interface condition is related to a general Neumann-mixed boundary condition for which a variational method with a positive-definite integral operator is presented. Multiple boundary conditions are handled by mutually constraining separate variational expressions for each boundary condition. A T-shaped conductor on a dielectric slab, representative of quasi-static solutions of microstrip discontinuities, is presented as a three-dimensional example with multiple boundary conditions. Generally, it is shown how the finite-element method for the solution of partial differential equations may be extended to handle integral equation formulations.     

Use of the WKB and variational methods to calculate the final stages of charge transfer in a CCD

The WKB method has been used to extract the asymptotic behavior of the solution to the charge transfer equation in a CCD with thermal diffusion and fringing-field drift. This enables an excellent approximation to be calculated for the decay time constant and charge distribution while using only a small number of parameters in a variational function.     

Radar Scattering and Target Imaging Obtained Using Ramp-Response Techniques

The response from targets illuminated by a transient plane wave the time dependence of which takes the form of a ramp function has been used to generate signatures for conductive and penetrable targets. Modified geometrical profile functions of various targets are used to generate their ramp response, which in turn is used to evaluate their scattered fields as a function of frequency. In early time, the ramp response is proportional to the physical cross-sectional area of the target (as a function of time or distance as the wave propagates over the target). Thus, the ramp response can be generated from the target's geometry. This is then Fourier transformed to obtain the spectrum of the ramp response. To obtain the spectrum of the impulse response, this result is simply multiplied by (jomega)², where omega is the angular frequency. These simple steps can be readily performed by anyone with an electrical engineering background to obtain the backscattered fields to a reasonable approximation. These same fundamentals can also be used to approximate a target's image from the measured ramp response. This has been done for axial incidence of rotationally symmetric targets. It can be extended to non-rotationally symmetric targets using an iterative process.     

The transfinite element method for modeling MMIC devices

A numerical procedure called the transfinite element-method is used in conjunction with the planar waveguide model to analyze MMIC devices. By using analytic basis functions together with finite-element approximation functions in a variational technique, the transfinite-element method determines the fields and scattering parameters for a wide variety of stripline and microstrip devices. With minor modification, the method can be applied to waveguide junctions, treating singular points in the junctions very efficiently. Calculations are shown for a rectangular-waveguide two-slot 20-dB coupler, stripline band-elimination filter, and several microstrip discontinuity problems. Good agreement of the numerical results with published values demonstrates the validity of the procedure     

Surface integral equation method for the analysis of an obliquelystratified half-space

The radiation of a source in the presence of an obliquely stratified half-space is studied using the surface integral equation method. A generalized extinction theorem for the obliquely stratified half-space problem is derived, which gives eight surface integral equations for the fields. The Green's functions for the stratified media are found by a semianalytical method. These surface integral equations can be used to solve the rather complicated two-dimensional problem by the one-dimensional finite element method (FEM), which saves computer memory and computation time compared to the conventional two-dimensional FEM. for some special cases, the results obtained by this method are compared with those obtained by the numerical mode matching method and the Fourier integral technique. Excellent agreement has been observed between them. This method is general for any obliquely stratified half-space with arbitrarily many layers in each region     

Generalized Gaussian approximation for single-mode fibers

A variational method is presented that generalizes well-known Gaussian method for single-mode fibers. This method uses only simple elementary functions to approximate the fundamental mode fields. By applying it to practical cases such as step index and clad parabolic index fibers, where exact solutions can be found, it is demonstrated that the method is essentially simple and that it is accurate for the analysis of single-mode fibers and devices. This approximation provides much better eigenvalues and, in particular, evanescent fields than the traditional Gaussian. Significantly, the present approximation's range of applicability covers the whole single-mode range, while being only slightly more complicated than the modified Gaussian method     

Comments on "Proof of Weierstrass approximation theory using band-limited functions"

A simple proof of the Weierstrass approximation theorem is given which uses only elementary Fourier series results. This is an alternative to a recent proof of the subject theorem by Giardina and Chirlian, where Fourier integrals and band-limited functions were employed.     

Variational Bayes inference of spatial mixture models for segmentation

Mixture models are commonly used in the statistical segmentation of images. For example, they can be used for the segmentation of structural medical images into different matter types, or of statistical parametric maps into activating and nonactivating brain regions in functional imaging. Spatial mixture models have been developed to augment histogram information with spatial regularization using Markov random fields (MRFs). In previous work, an approximate model was developed to allow adaptive determination of the parameter controlling the strength of spatial regularization. Inference was performed using Markov Chain Monte Carlo (MCMC) sampling. However, this approach is prohibitively slow for large datasets. In this work, a more efficient inference approach is presented. This combines a variational Bayes approximation with a second-order Taylor expansion of the components of the posterior distribution, which would otherwise be intractable to Variational Bayes. This provides inference on fully adaptive spatial mixture models an order of magnitude faster than MCMC. We examine the behavior of this approach when applied to artificial data with different spatial characteristics, and to functional magnetic resonance imaging statistical parametric maps.