Three-dimensional myocardial strain reconstruction from tagged MRI using a cylindrical B-spline model

In this paper, we present a new method for reconstructing three-dimensional (3-D) left ventricular myocardial strain from tagged magnetic resonance (MR) image data with a 3-D B-spline deformation model. The B-spline model is based on a cylindrical coordinate system that more closely fits the morphology of the myocardium than previously proposed Cartesian B-spline models and does not require explicit regularization. Our reconstruction method first fits a spatial coordinate B-spline displacement field to the tag line data. This displacement field maps each tag line point in the deformed myocardium back to its reference position (end-diastole). The spatial coordinate displacement field is then converted to material coordinates with another B-spline fit. Finally, strain is computed by analytically differentiating the material coordinate B-spline displacement field with respect to space. We tested our method with strains reconstructed from an analytically defined mathematical left ventricular deformation model and ten human imaging studies. Our results demonstrate that a quadratic cylindrical B-spline with a fixed number of control points can accurately fit a physiologically realistic range of deformations. The average 3-D reconstruction computation time is 20 seconds per time frame on a 450 MHz Sun Ultra80 workstation.     

Parametric shape representation by a deformable NURBS model for cardiac functional measurements

This paper proposes a method of parametric representation and functional measurement of 3-D cardiac shapes in a deformable nonuniform rational B-splines (NURBS) model. This representation makes it very easy to automatically evaluate the functional parameters and myocardial kinetics of the heart, since quantitative analysis can be followed in a simple way. In the model, local deformation and motion on the cardiac shape are expressed in adjustable parameters. Especially, an effective integral algorithm is used for volumetric measurement of a NURBS shape since the volume is the most basic parameter in cardiac functional analysis. This method promises the numerical computation to be very convenient, efficient, and accurate, in comparison with traditional methods. Practical experiments are carried out, and results show that the algorithm can get satisfactory measurement accuracy and efficiency. The parametric NURBS model in cylindrical coordinates is not only very suitable to fit the anatomical surfaces of a cardiac shape, but also easy for geometric transformation and nonrigid registration, and able to represent local dynamics and kinetics, and thus, can easily be applied for quantitative and functional analysis of the heart.     

Myocardial kinematics from tagged MRI based on a 4-D B-spline model

Current research investigating the modeling of left ventricular dynamics for accurate clinical assessment of cardiac function is extensive. Magnetic resonance (MR) tagging is a functional imaging method which allows for encoding of a grid of signal voids on cardiac MR images, providing a mechanism for noninvasive measurement of intramural tissue deformations, in vivo. We present a novel technique of employing a four-dimensional (4-D) B-spline model which permits concurrent determination of myocardial beads and myocardial strains. The method entails fitting the knot planes of the 4-D B-spline model for fixed times to a sequence of triplets of orthogonal sets of tag surfaces for all imaged volumetric frames within the constraints of the model's spatio-temporal internal energy. From a three-dimensional (3-D) displacement field, the corresponding long and short-axis Lagrangian normal, shear, and principal strain maps are produced. As an important byproduct, the points defined by the 3-D intersections of the triplets of orthogonal tag planes, which we refer to as myocardial beads, can easily be determined by our model. Displaying the beads as a movie loop allows for the visualization of the nonrigid movement of the left ventricle in 3-D.     

Amplitude- and Time-Limited Functions With Minimum Out-of-Band Energy

The theory of time-limited functions with minimum out-of-band energy is extended to apply to functions whose amplitude is held constant and hence, are limited to phase modulation. In the absence of the amplitude constraint, the desired functions are prolate spheroidal functions of time, of order zero, with time-bandwidth product as a parameter. With the amplitude constraint, the desired functions can be expressed in terms of sums of even-order prolate spheroidal functions. An algorithm suitable for computations of the amplitude constrained functions is derived.     

Analysis of 3-D myocardial motion in tagged MR images using nonrigid image registration

Tagged magnetic resonance imaging (MRI) is unique in its ability to noninvasively image the motion and deformation of the heart in vivo, but one of the fundamental reasons limiting its use in the clinical environment is the absence of automated tools to derive clinically useful information from tagged MR images. In this paper, we present a novel and fully automated technique based on nonrigid image registration using multilevel free-form deformations (MFFDs) for the analysis of myocardial motion using tagged MRI. The novel aspect of our technique is its integrated nature for tag localization and deformation field reconstruction using image registration and voxel based similarity measures. To extract the motion field within the myocardium during systole we register a sequence of images taken during systole to a set of reference images taken at end-diastole, maximizing the normalized mutual information between the images. We use both short-axis and long-axis images of the heart to estimate the full four-dimensional motion field within the myocardium. We also present validation results from data acquired from twelve volunteers.     

Comparison of the absorption characteristics of block and prolate spheroidal models of man exposed to near fields of short electric dipole

It is shown that the average specific absorption rates (SAR's) in a prolate spheroidal model of man are essentially the same as those for a block model of man irradiated by a short electric dipole at 27.12 MHz, even in the near fields. For purposes of average SAR, this allows use of the simpler and less expensive prolate spheroidal calculations.     

零阶角长球面波函数与长球面波函数的研究

本文深入研究了零阶角长球面波函数与长球面波函数,给出描述零阶角长球面波函数的微分方程与描述长球面波函数的积分方程不是相对应的关系。给出亥姆霍兹方程在球坐标系中与长旋转椭球坐标系中微分方程解之间的关系,可对零阶角长球面波函数了解得更清楚。对由信号最优波形设计法导出的长球面波函数积分方程,给出一种理论综合分析法,可以解决信号最优波形设计的具体实现。对长球面波函数命名尝试给出一种解释。      

Improved Pseudospectral Mode Solver by Prolate Spheroidal Wave Functions for Optical Waveguides With Step-Index

We propose a set of more efficient basis functions prolate spheroidal wave functions of order zero, comparing with the conventionally used Chebyshev polynomials expanding the optical fields in interior subdomains, to improve significantly the convergence rate of pseudospectral mode solver for solving dielectric optical waveguides with step-index. The bandwidth parameter, which influences strongly the computational accuracy, in prolate spheroidal wave functions is also optimized in this work. First, the numerical examples of two-dimensional waveguides show that the new basis functions achieve faster convergence than Chebyshev polynomials. Furthermore, a 3-D rib waveguide based on the full-vectorial formulations demonstrates that the proposed approach reduces 23% of computational time and 25% of memory storage for obtaining the convergent values of the effective index for the tenth order quasi-TE and quasi-TM modes.     

Prolate spheroidal antennas in isotropic plasma media

The problems of a gap-excited finite-sized prolate spheroidal antenna, with and without a confocal prolate spheroidal vacuum sheath, operated in a uniform cold and in a uniform warm lossy plasma medium, have been considered by applying boundary value problem techniques. The dependence of the input admittance, the current distribution, and the radiation pattern, upon the collision frequency, the electron temperature, the length of the antenna, the length-to-radius ratio for the antenna, the ratio of the plasma frequency to the operating frequency, and the thickness of a vacuum sheath have been investigated. Admittance curves are given from which the input admittance of a finite linear antenna can be estimated for a wide range of operating conditions and length-to-radius ratios. Conductance and susceptance maxima, dependent upon the vacuum sheath thickness, have been observed when using the cold plasma model foromega < omega_{p}. When using the warm plasma model in conjunction with the boundary condition that the normal component of the fluid velocity is zero at the antenna surface and at the vacuum sheath surface, the radiation resistance of the antenna is increased significantly for the parameters considered.     

Prolate spheroidal wave functions of large frequency parametersc=kfand their applications in electromagnetic theory

Maxwell's equations in prolate spheroidal coordinates have been separated into three second-order differential equations. These differential equations, satisfied byxiandeta, were solved for the frequency parameter c > 10 and the separation constant m = 1, i,e., the uniform circumferential excitations. The asymptotic solutions of the equations for any positive integer m > 1 and c > 10 are given. Thereby, the radiation fields of the metallic prolate spheroid of any length excited by an arbitrary source can be calculated.     

旋转椭球函数的数值计算

本文给出了计算旋转长椭球和扁椭球函数的程序流图和相应的有用数学公式，并讨论算法，所编ＦＯＲＴＲＡＮ程序具有适应参变量范围较宽和可获较高精度的数值结果。     

Quasi-translational addition expressions of prolate scalar wavefunctions for thin prolate spheroids

The conventional, translational addition theorems for both spherical and prolate spheroidal wave functions applied to incoming or outgoing waves possess a well-known convergence sphere near which the addition series fail to converge rapidly. To overcome this deficiency the quasi-translational addition expressions for prolate wave functions are developed by using a physical-geometrical transformation which allows the separation in the azimuthal direction in the translated prolate spheroidal coordinate system. These quasi-translational addition expressions are valid everywhere and, in particular, exhibit a fast convergence characteristics for thin spheroids which are commonly used in physical and engineering applications     

长椭球介质人头模型中的场分布

文中给出一长椭球介质人头模型中的电磁场全波解。运用并矢格林函数和散射叠加原理 ,求解的电磁场表示为椭球矢波函数。导出了在人头模型中的耦合系数 ,最后给出了数字结果的讨论。     

Magnetoquasistatic response of conducting and permeable prolatespheroid under axial excitation

An analytical solution is presented for the problem of magnetic diffusion into and scattering from a permeable, highly but not perfectly conducting prolate spheroid under axial excitation, expressed in terms of an infinite matrix equation. The spheroid is assumed to be embedded in a homogeneous nonconducting medium as appropriate for low-frequency, high-contrast scattering governed by magnetoquasistatics. The solution is based on separation of variables and matching boundary conditions where the prolate spheroidal wavefunctions with complex wavenumber parameter are expanded in terms of spherical harmonics. For small skin depths, an approximate solution is developed that avoids any reference to the spheroidal wavefunctions. The problem of long spheroids and long circular cylinders is solved by using an infinite cylinder approximation. In some cases, our ability to evaluate the spheroidal wavefunctions breaks down at intermediate frequencies. To deal with this, a general broadband rational function approximation technique is developed and demonstrated. We treat special cases and provide numerical reference data for the induced magnetic dipole moment or, equivalently, the magnetic polarizability factor     

Electromagnetic radiation from a prolate spheroidal antenna enclosed in a confocal spheroidal radome

The solution of electromagnetic radiation from a prolate spheroidal antenna, excited by a delta voltage across an infinitesimally narrow gap and enclosed in a confocal radome, is obtained. The method used is that of separating the scalar wave equation in prolate spheroidal coordinates and representing the solution in terms of prolate spheroidal wave functions. A simplified solution of the electric and magnetic fields, taking into account the symmetry of the antenna in the /spl phi/ direction, is obtained. Boundary conditions are then applied on the tangential fields to obtain a linear system of equations. The system of equations is cast into matrix form and solved using an iterative technique for the unknown expansion coefficients of the fields. Radiation patterns of the antenna are obtained and presented here for the first time.     

Eddy-current and current-channeling response to spheroidal anomalies

We derive the electromagnetic induction (EMI) response to conducting anomalies of spheroidal shape embedded in a weakly conducting host medium. As one application, prolate and oblate spheroids are convenient shapes for modeling buried unexploded ordnance, landmines, and naval mines. Under the assumptions of a perfectly conducting or nonconducting target and the assumption of a uniformly incident field, simple analytical solutions are obtained. Both the eddy-current and the current channeling responses are derived for prolate and oblate spheroids. For example, in a marine environment the current-channeling contribution can be quite large. As expected, these responses differ depending on the direction of the incident field relative to the spheroidal symmetry axis. The problem is solved separately for axial and transverse incidence, with the general case being a superposition of the two. These solutions could be used for fitting a spheroidal model to EMI data in which a small number of parameters, such as the location, size, shape, and orientation of the object, are to be determined.     

Field distributions in a three-layer prolate spheroidal human body model for a loop antenna irradiation

Analysis of the field distributions in a three-layer prolate spheroidal human body model for a loop antenna irradiation is described. The loop radiator is assumed to be placed symmetrically with respect to the spheroid axis. Spheroidal wave functions are employed to expand the field components inside and outside of the prolate spheroid. The boundary conditions on the interface layers are satisfied by using a mode matching procedure. The loop radiator can be very close to the human body. Numerical results are given for the 13 MHz frequency.     

A near-field to far-field transformation for spheroidal geometry utilizing an eigenfunction expansion

This work presents a near-field to far-field (NF-FF) transformation for antenna and scatterer radiation evaluation. The transformation allows practical computation by making use of a sampling surface in the near-field that is spheroidal in shape: namely a prolate or oblate spheroid. The resulting vector wave equation does not support orthogonal vector solutions in spheroidal coordinates and instead rectangular field components are solved for using the scalar wave equation in spheroidal coordinates. The new transformation only requires knowledge of the completely-specified near-field electric field along the spheroidal transformation surface and does not need any information associated with the corresponding magnetic field. The benefit of using a spheroidal near-field geometry is its ability to closely conform to both linear and planar radiating structures while still permitting evaluation of the full far-field radiation pattern. Our approach makes use of an eigenfunction expansion of spheroidal wave-harmonics to develop two distinct, yet closely related, NF-FF transformation algorithms for each type of spheroidal surface. The spheroidal NF-FF transformation is validated and performance assessed using a well-characterized radiation structure. By applying the prolate and oblate algorithms to a radiating structure with known analytical near- and far-field electric fields, viz., a filament dipole with sinusoidal current distribution, we are able to setup and conduct multiple numerical tests that serve as a proof-of-concept for the spheroidal NF-FF transformation.     

Accuracy of Block Models for Evaluation of the Deposition of Energy by Electromagnetic Fields

Numerical solutions were made for block models using as many as 3048 cubes to approximate a prolate spheroidal model of man at 100 and 225 MHz. A high-frequency modification of Hohmann's formulation (HFH) gave values of mean absorption within five to seven percent of those with the Iterative Extended Boundary Condition Method (IEBCM). Arrangement of the cells for a best-fit approximation of the spheroid is essential for such high accuracy. Numerical quadratures using the cubical shape of the cells verified the high accuracy of the closed-form expressions used for HFH matrix elements. Quadrature over spheres having the same volume as the cells gave inaccurate values for the matrix elements. Values of average absorption calculated with 296, 560, 1376, and 1944 cell models of the prolate spheroidal model of man differed from each other by no more than 17 percent at frequencies of 10 to 400 MHz, and by 5.0 percent at the resonant frequency of 75 MHz.     

大功率毫米波行波管AM/PM转换特性大信号解析理论的研究

毫米波行波管是未来通信发展的重要方向,由于AM/PM转换是影响毫米波行波管通信的重要非线性特性指标,因此AM/PM转换的研究对实现毫米波行波管线性化具有重要意义．本文基于相位展开的毫米波行波管欧拉大信号注波互作用理论模型,忽略高于二阶的非线性项,建立简化的欧拉大信号注波互作用理论模型．采用逐次逼近法推导其解析解,建立二阶逼近的欧拉大信号解析理论模型．利用欧拉大信号解析理论模型推导AM/PM转换的解析解,并发现AM/PM转换的产生与电子相位直流分量的激励成正比．应用建立的欧拉大信号解析理论和AM/PM转换解析模型对一支Q波段毫米波行波管进行大信号分析,并与拉格朗日理论模型以及传统的欧拉非线性理论模型进行对比．结果表明：欧拉大信号解析解的功率、增益和相移以及AM/PM转换与拉格朗日理论在线性区和中度互作用区十分吻合,饱和增益最大误差小于8.5%．同时,相较于传统的欧拉非线性模型,欧拉大信号解析解对非线性区增益和相移非线性特性的描述更具优势,即不但具有更高的精度,而且表现出传统欧拉非线性模型无法描述的增益压缩和相位失真现象．仿真结果验证了欧拉大信号解析解和AM/PM转换解析模型在线性区到中度互作用区的正确性和有效性．该AM/PM转换解析模型为后续研究毫米波行波管AM/PM转换非线性特性的物理机制与抑制方法奠定了基础．