Search

Unmixing is an important issue of hyperspectral images. Most unmixing methods adopt linear mixing models for simplicity. However, multiple scattering usually occurs between vegetation and soil in a bilinear scene. Thus, nonlinear mixing problems which are difficult to be solved should be taken into consideration under this circumstance. In practice, both linear and nonlinear spectral mixtures exist in hyperspectral scenes. Considering the characteristics of different regions in images, we propose a hybrid unmixing algorithm for hyperspectral images based on region adaptive segmentation. Our method uses a standard K-means clustering algorithm to obtain different regions, including homogeneous regions and detailed regions. The model of the homogeneous regions is assumed to be linear, which will be pursued using the method of sparse-constrained nonnegative matrix factorization (NMF), and the mixing in the detailed regions is assumed to be based on a nonlinear model. We also propose a new nonlinear unmixing method, called graph-regularized semi-NMF, which considers the manifold structure of hyperspectral data as the unmixing method to deal with the detailed regions. Finally, by combining the two regions, we obtain the abundance of the whole hyperspectral image. The proposed method can not only achieve more precise abundance but also be good at keeping the edge information of the bilinear abundance. The experimental results on both synthetic and real data also show that the proposed method is effective for improving the unmixing accuracy of hyperspectral remote-sensing images.

The volume integral equation method is applied in electromagnetic scattering from arbitrarily shaped three-dimensional inhomogeneous objects. The properties of the volume electric and magnetic field integral equations (VEFIE and VMFIE) are investigated. Numerical experiments show that if the Galerkin's method with the lowest mixed-order basis functions is used to discretize the equations the accuracy of the VMFIE can be significantly poorer than the accuracy of the VEFIE, in particular, for high-contrast objects at high frequencies. The accuracy of the VMFIE can be essentially improved with full first order (linear) basis functions. The linear basis functions are found to be useful also when a single volume integral equation is used to model a general scatterer where both permittivity and permeability differ from the background.

The traditional thin dielectric sheet approximation (TDS) method models the electromagnetic scattering by thin dielectric sheet using surface integral equation. In this approximation, only the tangential current within the dielectric sheet is considered, and the vertical component is ignored. This article proposes a method that uses the volume integral equation solution to predict the vertical volume current and hence to improve the overall solution accuracy. The formulation is based on the hybrid surface (SIE) and volume integration equation (VIE) in which, the thin dielectric is modeled with SIE, and the thin layers are modeled by VIE. It is shown that the inclusion of the vertical current component does not increase memory and solution time. Numerical examples will be provided to demonstrate the effectiveness of this approach. © Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 1978–1982, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23518

In practical applications, frequency-selective surfaces (FSSs) are finite, and sometimes even curved. In this paper, we present a hybrid volume-surface integral-equation approach to analyze the transmission and reflection characteristics of finite and curved FFS structures. The hybrid integral equations are established using the surface- and volume-equivalent principles. This approach has two advantages. One is the capability of modeling arbitrarily shaped FSS structures in detail, the other one allows us to easily apply the multilevel fast multiple algorithm to speed up the solution process. The scattering characteristics and frequency responses of several FSSs are analyzed. The simulation results show that for a finite-sized FSS, reducing the radius of curvature causes amplitude variation, frequency shift, and bandwidth change in the reflection and transmission responses. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 45: 107–112, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20738

A high-order locally corrected Nystro/spl uml/m (LCN) method employing the mixed-order basis functions proposed by C$80al/spl iota/s$80kan and Peterson is presented for the electromagnetic scattering by targets composed of both dielectric and conducting bodies. An integral operator based on a combined field formulation for conducting surfaces and a Mu/spl uml/ller formulation for dielectric surfaces is used. It is found that for general scattering objects, mixed-order basis functions accelerate the convergence of the LCN solution, can eliminate spurious charges, and can significantly reduce the condition number of the impedance matrix.

Compared to the TM case, the inverse scattering problem for the TE incident field is more complicated due to its stronger nonlinearity. This work provides an effective method for the reconstruction of two-dimensional (2-D) inhomogeneous dielectric objects from TE scattering data. The algorithm applies the distorted Born iterative method to the integral equation of strong permittivity fluctuation to reconstruct scatterers with high-permittivity contrast. Numerical simulations are performed and the results show that the distorted Born iterative method (DBIM) for strong permittivity fluctuation (SPF-DBIM) converges faster and can obtain better reconstructions for objects with larger dimensions and higher contrasts in comparison with ordinary DBIM. A frequency hopping technique is also applied to further increase the contrast.

We present an accurate method of moments (MoM) solution of the combined field integral equation (CFIE) using the multilevel fast multipole algorithm (MLFMA) for scattering by large, three-dimensional (3-D), arbitrarily shaped, homogeneous objects. We first investigate several different MoM formulations of the CFIE and propose a new formulation, which is both accurate and free of interior resonances. We then employ the MLFMA to significantly reduce the memory requirement and computational complexity of the MoM solution. Numerical results are presented to demonstrate the accuracy and capability of the proposed method. The method can be extended in a straightforward manner to scatterers composed of different homogeneous dielectric and conducting objects.

The fast multipole method (FMM) and multilevel fast multipole algorithm (MLFMA) are reviewed. The number of modes required, block-diagonal preconditioner, near singularity extraction, and the choice of initial guesses are discussed to apply the MLFMA to calculating electromagnetic scattering by large complex objects. Using these techniques, we can solve the problem of electromagnetic scattering by large complex three-dimensional (3-D) objects such as an aircraft (VFY218) on a small computer.

The generalized network formulation is applied in combination with the method of moments to calculate the electromagnetic scattering from conducting objects containing near-resonant open-ended cavities. The presence of the cavity increases the iteration number in conjugate gradient iterations due to the near-resonant modes and the multiple wave bounces inside the cavity. The equivalence principle is used to separate the cavity region from the rest of the object, allowing an independent solution of the cavity problem by a direct inversion algorithm using a connection scheme. The solution of the cavity is then represented by a generalized admittance matrix. Numerical results for two-dimensional (2-D) composite targets show that the iteration number can be reduced significantly for objects containing long cavities. This algorithm can also be applied to three-dimensional problems.

The recursive aggregate interaction matrix algorithm (RAIMA) for calculating a wave scattering solution is developed. This algorithm combines the strength of both the recursive aggregate T matrix algorithm (RATMA) and the recursive interaction matrix algorithm (RIMA) that have been previously developed. The resultant algorithm is robust for scattering problems involving highly singular Green's functions by avoiding the violation of the addition theorem. It also has reduced computational complexity for inverting the volume integral equation of scattering. The computational complexity of RAIMA is O(N/sup 7/3/) in three dimensions and O(N/sup 2/) in two dimensions.

In the solution of an integral equation using the conjugate gradient (CG) method, the most expensive part is the matrix-vector multiplication, requiring O(N2) floating-point operations. The fast multipole method (FMM) reduced the operation to O(N15). In this article we apply a multilevel algorithm to this problem and show that the complexity of a matrix-vector multiplication is proportional to N (log(N))2. © 1994 John Wiley & Sons, Inc.

The volume integral equation method is used to analyze electromagnetic scattering from arbitrarily shaped three-dimensional homogeneous objects whose permittivity and permeability differ from those of background medium. We consider three different types of formulations, the DB-formulation with flux densities as unknowns, the EH-formulation with fields as unknowns, and EE- and HH-formulations that contain only single unknown, either electric or magnetic field. The properties of the formulations are investigated with numerical examples. Numerical results show that at least fully linear curl conforming basis functions must be used when applying the single unknown volume intergal equation for general scatterers.

A truncated, nonuniform, finite array of strips does not have a closed form solution. Using translational symmetry, a recursive algorithm that calculates the scattering solution with N log[sup 2] N computational complexity is described. First, the algorithm is validated with the method of moments for both the TM-to-z and TE-to-z polarizations. Then the scattering solution from a large strip is calculated for both TE and TM polarizations. The current distribution for the TE polarization shows small-length-scale oscillations not present in the TM polarization. 26 refs., 10 figs.

A feast recursive algorithm is used to compute the scattering properties of a finite array of strip gratings on a dielectric slab. this algorithm has a computational complexity of O(N log/sup 2/ N) for one incident angle and O(N/sup 2/ log N) for N incident angles. It uses plane wave basis for expanding the incident wave and the scattered wave. The scattered wave is expanded in terms of a Sommerfeld-type integral with spectral distribution along a vertical branch cut, rendering its expansion very efficient. To validate the scattering solution obtained using the recursive algorithm, comparisons with the method of moments are illustrated. The current distributions on the strips and scattering patterns are both presented. Since this algorithm has reduced computational complexity and is fast compared to other conventional methods, it can be used to analyze very large strip arrays. Scattering solution of a 50-wavelength wide strip is illustrated. >

A coupled surface-volume integral equation approach is presented fur the calculation of electromagnetic scattering from conducting objects coated with materials. Free-space Green's function is used in the formulation of both integral equations. In the method of moments (MoM) solution to the integral equations, the target is discretized using triangular patches for conducting surfaces and tetrahedral cells for dielectric volume. General roof-top basis functions are used to expand the surface and volume currents, respectively. This approach is applicable to inhomogeneous material coating, and, because of the use of free-space Green's function, it can be easily accelerated using fast solvers such as the multilevel fast multipole algorithm.

This paper presented numerical simulation of imaging reconstruction of objects confined within a dielectric wall. The image for the unknown region can be improved if the scattering properties of the wall are known. This improvement is achieved by subtracting the scattered field of the wall. From the total scattered field. The scattered field for wall alone case is obtained by solving a forward scattering problem. Whether this approach can be applied to the partially known background needs to be further investigated.

The electromagnetic scattering by dielectric material sheet can be modeled in several different ways. In the integral equation approach to solve such a problem, there are surface equivalence method and volume equivalence method. When the thickness of a material sheet is much smaller than a wavelength, the thin dielectric sheet approximation (TDS) is often used. In this approximation, a volume equivalent problem is transferred to a surface integral equation for the material sheet, and hence number of unknowns is reduced. A major source of error in TDS is the lack of the vertical electric field component that is perpendicular to the sheet. This paper proposes a modification to TDS by adding the contribution of the vertical electrical field without increasing the computational cost. Numerical simulation results show that the accuracy is improved when this modification is employed.

The mixed mesh discretization scheme has been applied to solve hybrid integral equations for electromagnetic scattering and radiation. The mixed mesh scheme makes it easier to model complex objects without significantly increasing the number of unknowns. The program from this mixed scheme algorithm has been validated using exact solutions for spherical objects. When accelerated by MLFMA, this algorithm has the potential to solve many realistic problems as demonstrated by the examples shown in this paper

We investigate preconditioned iterative solutions of large dense complex valued matrices arising from discretizing the hybrid integral equations of electromagnetic scattering. Due to the specific data pattern, we propose to construct two-level (TL) preconditioners to treat the conducting and dielectric parts more efficiently. The main purpose of this study is to show that the TL preconditioning is effective with the MLFMA and can reduce the number of Krylov iterations substantially by using subdivided domain of the near part matrix for each level. Our experimental results indicate that the TL preconditioning maintains the computational complexity of the MLFMA, but converges much faster, thus effectively reduces the overall simulation time for the coated material objects.

The algorithms based on the iterative solution of the integral equations and accelerated by fast solvers to provide efficient and accurate ways to calculate the scattering by large and complex objects. To further increase the efficiency in producing multi-angle and multi-frequency scattering data, attentions are focused on (a) developing precondition techniques to reduce the number of iterations for a converged solution and (b) developing advanced post processing methods which uses the information of the available solution to predict as much scattering data as possible. In this paper, an approximate extrapolation method is introduced to rapidly fill the angular and frequency far-field points using the solutions at sparsely sampled points.

A high-order method of moment solution for the electromagnetic scattering by complex objects with composite materials is presented. The generalized technique can analyze material scattering by homogeneous and inhomogeneous material and conducting objects through both surface and volume equivalent currents. The solution employs high-order basis functions and a quadrature point-based discretization. It is shown that the solution leads to high-order convergence for the electromagnetic scattering by heterogeneous scatterers. This is validated here through the study of canonical scattering structures.

The hybrid volume-surface integral equation (VSIE) approach has been applied to investigate the curved FSS with the dielectric substrates of finite extent. The scattering RCS of a curved structure and a planar finite periodic structure have been presented. An important advantage of the VSIE algorithm is that it can be used to simulate the finite and arbitrary curved structure. When the finite structure size increases, the multilevel fast multipole algorithm can be applied to reduce the computational complexity.

A fast, high-order, hybrid integral equation solver for electromagnetic scattering is presented. The solver is based on a locally corrected Nystrom solution of a hybrid volume integral equation (VIE) and surface integral equation (SIE) formulation. The solution can be accelerated by quadrature-sampled pre-corrected FFT or the multilevel fast multipole method (MLFMM), or a combination of both schemes. The methods are validated in this paper through the study of the electromagnetic scattering of canonical geometries made of a composite of conducting and dielectric materials.

Solving integral equation by iterative methods for cavity scattering problems can encounter slow convergence due to several factors, such as internal resonance, near resonance and multiple wave bounces. A numerical study is presented on the solution of the TETH-type (short for t/spl circ//spl middot/E and t/spl circ//spl middot/H) combined field integral equation in the context of the near-resonance decoupling approach (NRDA), in which the TETH-type formulation is discretized by the method of moments with roof-top basis and testing functions over quadrilateral shaped meshes.

The hybrid volume-surface integral equation approach is implemented to calculate the radiation and scattering by large microstrip arrays on curved substrate and ground plane. The multilevel fast multipole algorithm is applied to reduce the memory and CPU requirements for solving the matrix equation. Numerical results for input impedances of single patch and the scattering pattern for an array are presented.

Dielectric materials are often involved in electromagnetic simulation for real-world problems. Previously, researchers have applied the fast multipole method and a hybrid volume-surface integral equation (VSIE) formulation to solve scattering problems involving both PEC and inhomogeneous dielectric objects. Although the above solver can solve scattering from electrically large objects with optimal computational complexity, it only involves surface and volume basis functions to model currents on PEC surfaces and in dielectric media. In order to solve problems involving additional types of scatterers and radiators, such as wire antennas, thin dielectric sheets, and impedance surfaces, we have enhanced the multilevel fast multipole code in Chao et al. (1998) with the impedance boundary condition (IBC) and the hybrid VSIE formulation. The paper discusses how a variety of basis functions are incorporated into the hybrid VSIE formulation and the multilevel fast multipole algorithm (MLFMA).

A high-order solution of the electric field integral equation (EFIE) for penetrable inhomogeneous material volumes is presented. The solution is based on a high-order method of moment solution with a quadrature point-based discretization. It is shown that the solution leads to high-order convergence for the electromagnetic scattering by inhomogeneous dielectric scatterers. This is validated through the study of canonical scattering objects.

[1] A high-order method of moment solution with quadrature-point-based sampling is presented for the solution of the volume electric field integral equation for the scattering of inhomogeneous dielectric bodies. The proposed scheme efficiently allows for the material profile to be inhomogeneous within a curvilinear cell. It is demonstrated that the method leads to exponential convergence in both the radar cross section (RCS) and the volume current density. It is also demonstrated that the method can be more efficient than surface field integral equation formulations for thin-material scattering.

Our recent inverse scattering work has been to derive inverse scattering theory and algorithms that can be used to process practical experimental data. The theory makes use of computation of the forward scattering solution. Therefore, an efficient forward solver is instrumental to the rapid solution of the inverse scattering problem. The advantage of the more sophisticated theory over a linear theory is that it accounts for multiple scattering effects within the scatterers which often give rise to distortions in an image. A new method to invert strong scatterers, such as metallic scatterers, is presented. © 1996 John Wiley & Sons, Inc.

Compared to the TM case, the inverse scattering problem for the TE incident field is more complicated, due to its stronger nonlinearity. This work provides an effective method for the reconstruction of 2D inhomogeneous dielectric objects from TE scattering data. The algorithm is the combination of the distorted Born iterative method and strong permittivity fluctuation theory. Numerical simulations are performed and the results show that the distorted Born iterative method for strong permittivity fluctuation (SPF-DBIM) converges faster and can obtain better reconstructions for objects with larger dimensions and higher contrast in comparison with the ordinary distorted Born iterative method (DBIM). A frequency hopping technique is also applied to further increase the contrast.

For the analysis of large-scale electromagnetic scattering problems, high-frequency asymptotic methods are fast but approximate, whereas low-frequency numerical methods are accurate but slow. Neither can produce an efficient and accurate solution to scattering by large bodies containing small structures. A promising approach is to combine the best features of both types of methods to produce a hybrid technique that is sufficiently fast, reasonably accurate, and applicable to a class of unsolvable problems such as the scatterers mentioned above. There are two extremes for this type of hybridization. One is simply to superimpose solutions from asymptotic and numerical methods. The other extreme is to combine an asymptotic and a numerical method in an exact manner. A more practical approach is to develop a technique that can include all significant interactions and neglect all trivial interactions. The resulting hybrid technique can produce sufficient accuracy and can be implemented in a general-purpose computer code. In this paper, we develop a technique that combines the shooting-and-bouncing-ray (SBR) method and the method of moments (MoM) to solve for the scattering by large conducting bodies with small structures mounted on their surfaces. The radar cross-section of a VFY 218 airplane is used as an example.

This paper discusses the formulation and the discretization of the volume-surface integral equation (VSIE), the implementation of multilevel fast multiple algorithm (MLFMA) to speed up the iteration in the conjugate gradient method for solving the matrix equation. Validation numerical examples are also provided to demonstrate the performance of the VSIE-MLFMA algorithm. The derivation is in the frequency domain, and the time factor of exp{-i/spl omega/t} has been suppressed.

The multilevel fast multipole algorithm has been applied for solving volume integral equations. Tetrahedron volume cells are used to discretize the material objects, providing a flexible representation for complex object shapes. Due to the reduction of the computational complexity, the method can be applied to solve electromagnetic scattering problems with electrically large dielectrics.

A truncated, nonuniform, finite array of strips does not have a closed-form solution. Using translational symmetry, a recursive algorithm that calculates the scattering solution with N log2 N computational complexity is described. The current distribution for the Hz polarization shows small-length-scale oscillations not present in the Ez polarization.

A review of various methods to solve electromagnetic wave-scattering Problems efficiently is presented. Electromagnetic scattering problems are divided into surface scattering problems and volume scattering problems. Different methods have to be used depending on the nature of the scatterer. We review several fast methods to solve scattering problems of volumetric type and surface type.

Forward scattering solutions, whereby one predicts the scattered electromagnetic field given the goemetry, find numerous engineering applications in computer aided design. Therefore, a fast way to solve the forward scattering problem will impact a number of areas, like high- speed circuits, integrated optics, antenna analysis, remote sensing, geophysical sensing, and inverse scattering. We will describe several fast methods developed in our group to solve the forward scattering problem rapidly. These methods involve solving the volume integral equation of scattering as well as surface integral equation of scattering. Different strategies are used to accelerate the solutions of these integral equations. Both iterative and direct solution techniques will be considered. The computational complexity and memory requirement of various scattering algorithms will be discussed. In inverse scattering, one reconstructs the physical goemetry of a scatterer from the measured scattered field. Hence, it finds applications in image and profile reconstructions. When using a linear method, multiple scattering within a scatterer is ignored. By using a nonlinear inverse scattering method, such multiple scattering effect is accounted for. Image and profile reconstruction using such nonlinear inverse algorithm can remove artifacts that linear methods would not remove. We will discuss the use of the distorted Born iterative method and local shape function method to reconstruct a scattering object.© (1995) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Wave scattering of a truncated, nonuniform, finite array of strips (with or without dielectric supporter) is investigated using a recursive algorithm that calculates the scattering solution with N log/sup 2/ N computational complexity. The algorithm is validated with the method of moments. The scattering solutions from large strips and strip arrays are calculated for both E/sub z/ and H/sub z/ polarizations. The current distribution for the H/sub z/ polarization shows small-length-scale oscillations not present in the E/sub z/ polarization. >