We review a set of algorithms and methodologies developed recently for the numerical solution of problems of scattering by complex bodies in three-dimensional space. These methods, which are based on integral equations, high-order integration, Fast Fourier Transforms and highly accurate high-frequency integrators, can be used in the solution of problems of electromagnetic and acoustic scattering by surfaces and penetrable scatterers — even in cases in which the scatterers contain geometric singularities such as comers and edges. All of the solvers presented here exhibit high-order convergence, they run on low memories and reduced operation counts, and they result in solutions with a high degree of accuracy. In particular, our approach to direct solution of integral equations results in algorithms that can evaluate accurately in a personal computer scattering from hundred-wavelength-long objects — a goal, otherwise achievable today only by super-computing. The high-order high-frequency methods we present, in turn, are efficient where our direct methods become costly, thus leading to an overall computational methodology which is applicable and accurate throughout the electromagnetic spectrum.

Fast, High-Order, High-Frequency Integral Methods for Computational Acoustics and Electromagnetics

Acoustic scattering problems are considered when the material parameters (density and speed of sound) are functions of position within a bounded region. An integro-differential equation for the pressure in this region is obtained. It is proved that solving this equation is equivalent to solving the scattering problem. Problems of this kind are often solved by regarding the effects of the inhomogeneity as an unknown source term driving a Helmholtz equation, leading to an equation of Lippmann--Schwinger type. It is shown that this approach is incomplete when the density is discontinuous. Analogous scattering problems for elastic waves and for electromagnetic waves are also discussed briefly.

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Acoustic Scattering by Inhomogeneous Obstacles

Frequency analysis using DFT (discrete Fourier transform) or its faster computational technique (FFT) is an obvious choice for the entire image and signal processing domain where spectral leakage or picket fence effect is a major problem. Earlier works describe the software and ROM-based implementation of windowing functions to overcome the above-mentioned problems during spectral analysis. In this work we have proposed a CORDIC (co-ordinate rotation digital computer)-based unified windowing architecture to remove the spectral leakage, picket fence effect and resolution problems with different tradeoff between mainlobe and sidelobe in the frequency domain. A parallel-pipelined architecture has been adopted for the present design to ensure high throughput for real-time applications with the latency equal to twice of CORDIC length plus three extra cycles. This unified architecture includes a combination of linear CORDIC and circular CORDIC with FIFO and a few multiplexers where the selection of window and its length are user defined. We have synthesised this architecture with 0.18 μm CMOS technology using Synopsys Design Analyser. The total estimated dynamic power was found to be 350 mW with an operating frequency of 125 MHz and total cell area 11 mm2 (approximately).

CORDIC-based unified VLSI architecture for implementing window functions for real time spectral analysis

This paper provides a theoretical analysis of a higher-order, FFT-based integral equation method introduced recently [IEEE Trans. Antennas and Propagation, 48 (2000), pp. 1862--1864] for the evaluation of transverse electric--polarized electromagnetic scattering from a bounded, penetrable inhomogeneity in two-dimensional space. Roughly speaking, this method is based on Fourier smoothing of the integral operator and the refractive index n(x). Here we prove that the solution of the resulting integral equation approximates the solution of the exact integral equation with higher-order accuracy, even when n(x) is a discontinuous function---as suggested by the numerical experiments contained in the paper mentioned above. In detail, we relate the convergence rates of the computed interior and exterior fields to the regularity of the scatterer, and we demonstrate, with a few numerical examples, that the predicted convergence rates are achieved in practice.

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Higher-Order Fourier Approximation in Scattering by Two-Dimensional, Inhomogeneous Media

The most widely used method for weed control is to use agricultural chemicals (herbicides products). This heavy reliance on chemicals raises many environmental and economic concerns, causing many plantation companies to seek alternatives for weed control in order to reduce chemical usage in their plantation. Since manual labor is costly and expensive, an automated weed control system may be economically feasible. A machine vision precision automated weed control system could also reduce the usage of chemicals. In this research, an intelligent real-time system for automatic weeding strategy in oil palm plantation using image processing has been developed to identify and discriminate the weed types namely as narrow and broad. In machine vision technology, the main component of the system is image processing to recognize type of weeds. Three techniques of image processing, involving statistical approach GLCM and structural approach FFT and SIFT, have been used and compared to find the best solution of weed recognition for classification. The developed machine vision system consists of a mechanical structure, which includes a sprayer, a Logitech web digital camera, 12v motor coupled with a pump system and a small size CPU as a processor. Offline images and recorded video has been tested to the system and classification result of weed shows the successful rate is above 80%.

Machine Vision System for Automatic Weeding Strategy using Image Processing Technique

This paper presents the application of the conjugate-gradient (CG) fast Fourier transform (FFT) (CG-FFT) method and the CG nonuniform FFT (CG-NUFFT) method for the integral equation arising from acoustic scattering problems. In the conventional method of moments (MoM) for integral equations, the CPU and memory requirements are O(N/sup 3/) and O(N/sup 2/), respectively, where N is the number of unknowns in the problem. The CG-FFT method, which combines the iterative conjugate-gradient method with FFT, reduces these requirements to O(KN log/sub 2/N) and O(N), respectively, where K is the number of CG iterations. The CG-NUFFT method differs from the CG-FFT method in that it makes use of nonuniform FFT algorithms instead of FFT to allow a nonuniform discretization. Therefore, the CG-NUFFT method can solve the integral equation with both uniform and nonuniform grid while retaining the efficiency of the CG-FFT method. These two methods are applied to solve for two-dimensional constant density acoustic scattering problems. Numerical. results demonstrate that they can solve much larger problems than the MoM.

X.M. Xu

Papers

Fast spectral-domain method for acoustic scattering problems