A comparison of computations for spatial frequency filtering

Three computational algorithms for performing spatial frequency filtering are compared and tradeoffs developed. Although each method is defined by a convolution relation, the convolution computations are different. Equal filter point-spread functions are assumed to effect the comparison. If the filter point-spread function is nonzero only over a small area, then the computation tradeoff is simply the well-known comparison between direct convolution and the fast Fourier trsnsform (FFT). If the filter point-spread function is nonzero over a large area, then a recursive filter is competitive with the FFT. Core memory requirements for this case are smallest with the recursive filter. Experimental examples are given to illustrate the subjective evaluation problem.     

Fast anisotropic Gauss filtering

We derive the decomposition of the anisotropic Gaussian in a one-dimensional (1-D) Gauss filter in the x-direction followed by a 1-D filter in a nonorthogonal direction /spl phi/. So also the anisotropic Gaussian can be decomposed by dimension. This appears to be extremely efficient from a computing perspective. An implementation scheme for normal convolution and for recursive filtering is proposed. Also directed derivative filters are demonstrated. For the recursive implementation, filtering an 512 /spl times/ 512 image is performed within 40 msec on a current state of the art PC, gaining over 3 times in performance for a typical filter, independent of the standard deviations and orientation of the filter. Accuracy of the filters is still reasonable when compared to truncation error or recursive approximation error. The anisotropic Gaussian filtering method allows fast calculation of edge and ridge maps, with high spatial and angular accuracy. For tracking applications, the normal anisotropic convolution scheme is more advantageous, with applications in the detection of dashed lines in engineering drawings. The recursive implementation is more attractive in feature detection applications, for instance in affine invariant edge and ridge detection in computer vision. The proposed computational filtering method enables the practical applicability of orientation scale-space analysis.     

Inherent matrix identities on ARMA lattice filter realizationalgorithm and their application

Presents the matrix identities that are inherent in the solution of the normal equations for an ARMA lattice filter. This derivation also makes clear the relationship between the recursive least squares (RLS) method and the ARMA lattice filter realization algorithm. Further, as an application of the matrix identities, a new method for model identification with frequency weighting (MIFW) is presented     

A new recursive pseudo least squares algorithm for ARMA filteringand modeling. II

For pt.I see ibid., vol.40, no.11, p.2766-74 (Nov. 1992). A recursive algorithm for ARMA (autoregressive moving average) filtering has been developed in a companion paper. These recursions are seen to have a lattice-like filter structure. The ARMA parameters, however, are not directly available from the coefficients of this filter. The problem of identification of the ARMA model from the coefficients of this filter is addressed here. Two new update relations for certain pseudoinverses are derived and used to obtain a recursive least squares algorithm for AR parameter estimation. Two methods for the estimation of the MA parameters are also presented. Numerical results demonstrate the usefulness of the proposed algorithms     

磁共振截断频谱图像边缘检测新技术

本文提出了一种适合于含有截断伪影磁共振图像(磁共振截断频谱图像)的边缘检测新算法.本方法中,把任何有截断伪影的信号表示为以奇异点为参量的截断奇异函数的加权和,奇异点和加权系数由该信号决定,而计算出的奇异点就是图像的边缘,从而剔除了由截断伪影而引入的虚假边缘.实际和仿真结果表明这种方法效果高于现有方法.     

A fast algorithm for FIR digital filtering with a sum-of-triangles weighting function

A doubly recursive algorithm for time domain convolution with a piecewise linear weighting function is presented that combines the speed of a recursive (IIR) digital filter with the flexibility and ease of design of a nonrecursive (FIR) digital filter. The approach approximates the desired FIR weighting function by a sum-of-triangles weighting function. ForL triangles (or triangle pairs for a linear phase filter) the algorithm is of orderLN. The approximation improves with the number of triangles. A significant advantage of the algorithm compared to FFT filtering or direct convolution is that there is no necessity of a tradeoff between frequency response accuracy and computation time per output point as the data spacing decreases in the filtered signal. The computational complexity is dependent on the number of triangles chosen, not the width of the weighting function, so the algorithm is especially effective for filters with an inherently wide FIR weighting function.     

An adaptive ARMA four-line lattice filter for spectral estimationwith frequency weighting

A method for designing an adaptive four-line lattice filter which can perform frequency-weighting spectral estimation, which provides more accurate spectral estimation for some frequency bands than for others, is proposed. Using a suitable frequency-weighting function, denoted as an ARMA (autoregressive moving-average) model, an estimated spectrum is obtained by arbitrarily weighing some frequency bands more heavily than others. if the frequency-weighting function has the property of a low-pass filter, the spectrum of the reference model can be estimated accurately with a reduced ARMA order in the low-frequency band. Spectra of time-varying models can be estimated with an exponentially weighted sliding window, and the input signal of the reference model can be estimated by assumption. The order-update and the time-update recursive formulas and the frequency-weighting method for the filter are described. The algorithm is verified by experimental results     

Effects of truncated Gaussian beam on the performance of fiber optical synthetic aperture system

In the fiber optical synthetic aperture (FOSA) system, the diffraction of the Gaussian beam limited by the aperture in exit pupil plane of fiber collimator is studied theoretically, and the axial and transverse irradiance distributions are obtained. The point spread function (PSF) and modulation transfer function (MTF) of the truncated Gaussian beam array are computed numerically with different truncation factors. The results show that the diffraction of the truncated Gaussian beam array agrees with the uniform-beam Rayleigh diffraction when the truncation factor is less than 0.5, but little power is transmitted. The PSF and MTF are degraded, but more power can be contained when the truncation factor is larger. The selection of the truncation factor is a trade-off between the loss of transmission and the qualities of PSF and MTF in practical application.     

Convolution-Based Trigonometric Interpolation of Band-Limited Signals

This paper presents a method to obtain a trigonometric polynomial that accurately interpolates a given band-limited signal from a finite sequence of samples. The polynomial delivers accurate approximations in the range covered by the sequence, except for a short frame close to the range limits. Besides, its accuracy increases exponentially with the frame width. The method is based on using a band-limited window in order to reduce the truncation error of a convolution series. It is shown that the polynomial can be efficiently constructed and evaluated using algorithms designed for the discrete Fourier transform (DFT). Specifically, two basic procedures are presented, one based on the fast Fourier transform (FFT), and another based on a recursive update algorithm for the short-time FFT. The paper contains three applications. The first is a variable fractional delay (VFD) filter, which consists of a short-time FFT combined with the evaluation of a trigonometric polynomial. This filter has low complexity and can be implemented using CORDIC rotations. The second is the interpolation of nonuniform Fourier summations, where the proposed method eliminates the need to interpolate any kernel sample. Finally, the third can be viewed as a generalization of the FFT convolution algorithm and makes it possible to interpolate the output of an finite-impulse-response (FIR) filter efficiently.      

Sampling rate conversion systems using a new generalized form ofthe discrete Fourier transform

A recursive factorization of the polynomial 1-z^N leads to an efficient algorithm for the computation of the discrete Fourier transform (DFT) and the cyclic convolution. The paper introduces a new recursive polynomial factorization of the polynomial when N is highly composite. The factorization is used to define a generalized form of the DFT and to derive an efficient algorithm for the computation. The generalized form of the DFT is shown to be closely related to the polyphase decomposition of a sequence, and is applied for the design of sampling rate conversion systems, it gives not only alternative derivations for the polyphase interpolation and the polyphase decimation by an integer factor, but also a new sampling rate conversion system by a rational factor, which is more efficient than the known rational polyphase implementation when the filter length is large     

Image restoration using truncated SVD filter bank based on an energy criterion

Image restoration is formulated using a truncated singular-value-decomposition (SVD) filter bank. A pair of known data patterns is used for identifying a small convolution operator. This is achieved by matrix pseudo-inversion based on SVD. Unlike conventional approaches, however, here SVD is performed upon a data-pattern matrix that is much smaller than the image size, leading to an enormous saving in computation. Regularisation is realised by first decomposing the operator into a bank of sub-filters, and then discarding some high-order ones to avoid noise amplification. By estimating the noise spectrum, sub-filters that produce noise energy more than that of useful information are abandoned. Therefore high-order components in the spectrum responsible for noise amplification are rejected. With the obtained small kernel, image restoration is implemented by convolution in the space domain. Numerical results are given to show the effectiveness of the proposed technique     

矢量匹配结合递归卷积法计算GIS外壳辐射场

提出了一种计算气体绝缘变电站（GIS）金属外壳暂态辐射场的新方法：首先运用矢量匹配法对用NEC计算的频域格林函数进行拟合,拟合结果在时域内表示为指数函数和的形式;然后利用递归卷积在时域内直接计算实际输入下的辐射场。该方法不仅可以提高计算精度,降低计算时间,而且克服了快速傅立叶变换存在的不可避免的问题。通过对一125kVGIS金属外壳暂态辐射场计算结果与测量结果的比较,验证了方法的正确性与可行性。     

Application of autoregressive moving average parametric modeling in magnetic resonance image reconstruction

The modeling of data is an alternative to conventional use of the fast Fourier transform (FFT) algorithm in the reconstruction of magnetic resonance (MR) images. The application of the FFT leads to artifacts and resolution loss in the image associated with the effective window on the experimentally-truncated phase encoded MR data. The transient error modeling method treats the MR data as a subset of the transient response of an infinite impulse filter (H(z) = B(z)IA(z)). Thus, the data are approximated by a deterministic autoregressive moving average (ARMA) model. The algorithm for calculating the filter coefficients is described. It is demonstrated that using the filter coefficients to reconstruct the image removes the truncation artifacts and improves the resolution. However, determining the autoregressive (AR) portion of the ARMA filter by algorithms that minimize the forward and backward prediction errors (e.g., Burg) leads to significant image degradation. The moving average (MA) portion is determined by a computationally efficient method of solving a finite difference equation with initial values. Special features of the MR data are incorporated into the transient error model. The sensitivity to noise and the choice of the best model order are discussed. MR images formed using versions of the transient error reconstruction (TERE) method and the conventional FFT algorithm are compared using data from a phantom and a human subject. Finally, the computational requirements of the algorithm are addressed.     

Regularization of RIF blind image deconvolution

Blind image restoration is the process of estimating both the true image and the blur from the degraded image, using only partial information about degradation sources and the imaging system. Our main interest concerns optical image enhancement, where the degradation often involves a convolution process. We provide a method to incorporate truncated eigenvalue and total variation regularization into a nonlinear recursive inverse filter (RIF) blind deconvolution scheme first proposed by Kundar, and by Kundur and Hatzinakos (1996, 1998). Tests are reported on simulated and optical imaging problems.     

A Modified Reconstruction Filter for Diverging X-Ray Beams

The 3-D reconstruction of a density function for diverging X-ray beams, based on the direct convolution algorithm, was developed by Lakshminarayanan. In this method, the reconstruction of a section of an object was obtained as the average, over all source positions, of the backprojected data convolved with an appropriate filter function. In this note, a modification of Lakshminarayanan's filter is proposed. The modified filter has less undesirable high-frequency components, yet it retains the convolution property. It is shown that the new filter has less oscillatory response across the edge of highly absorbent structures (e. g., bones). Finally, it is shown how the modified fan-beam filter can be FFT implemented.     

Virtex FPGA implementation of a pipelined adaptive LMS predictor for electronic support measures receivers

High-speed field-programmable gate array (FPGA) implementations of an adaptive least mean square (LMS) filter with application in an electronic support measures (ESM) digital receiver, are presented. They employ "fine-grained" pipelining, i.e., pipelining within the processor and result in an increased output latency when used in the LMS recursive system. Therefore, the major challenge is to maintain a low latency output whilst increasing the pipeline stage in the filter for higher speeds. Using the delayed LMS (DLMS) algorithm, fine-grained pipelined FPGA implementations using both the direct form (DF) and the transposed form (TF) are considered and compared. It is shown that the direct form LMS filter utilizes the FPGA resources more efficiently thereby allowing a 120 MHz sampling rate.     

Estimation and setting starting values in ARMA algorithms

For the given observations set of the ARMA (autoregressive moving average) process, the likelihood function depends, not only on model parameters, but on the starting values of the input and output. Therefore, it is called theconditional likelihood function. Theunconditional likelihood function can be obtained in two ways. The first is to set the starting values to zero, as is often done, and the second is to set them to the properly estimated values. The difference between these two types of likelihood functions is significant when the given data sequence is short, and any of the zeros of the moving average part is close to the boundary of the unit circle.In this paper the direct method of starting value estimation and its application to two off-line ARMA estimation algorithms, the maximum likelihood (ML) algorithm and the iterative inverse filtering (ITIF) algorithm, is proposed. Experimental results prove both increased efficiency and stability of these algorithms.The importance of setting the starting values properly is also significant when the recursive algorithm, with previously estimated parameters, has to be restarted. The advantage of the proposed reinitialization method is shown on the recursive lattice algorithm working in the block mode.     

A Frequency-Dependent LOD-FDTD Method and Its Application to the Analyses of Plasmonic Waveguide Devices

Detailed frequency-dependent formulations are presented for several efficient locally one-dimensional finite-difference time-domain methods (LOD-FDTDs) based on the recursive convolution (RC), piecewise linear RC (PLRC), trapezoidal RC (TRC), auxiliary differential equation, and ${mmb Z}$ transform techniques. The performance of each technique is investigated through the analyses of surface plasmon waveguides, the dispersions of which are expressed by the Drude and Drude-Lorentz models. The simple TRC technique requiring a single convolution integral is found to offer the comparable accuracy to the PLRC technique with two convolution integrals. As an application, a plasmonic grating filter is studied using the TRC-LOD-FDTD. The use of an apodized and a chirped grating is found quite effective in reducing sidelobes in the transmission spectrum, maintaining a large bandgap. Furthermore, a plasmonic microcavity is analyzed, in which a defect section is introduced into a grating filter. Varying the air core width is shown to exhibit tunable properties of the resonance wavelength.      

多通道信号自校正分布式融合Kalman滤波器

对带未知参数的多传感器多通道自回归滑动平均(ARMA)信号,采用多维递推辅助变量(MRIV)方法得到自回归模型参数估值,通过Gevers-Wouters算法辨识滑动平均模型参数估值,再用相关方法得到噪声方差的估值。把所有的估值都代入到最优分布式融合信息滤波器中得到自校正分布式融合Kalman信息滤波器。该滤波器具有渐近全局最优性,一个多通道信号仿真例子验证了其有效性。     

On quantization,truncation and jitter errors in the sampling theorem and its generalizations

Any band-limited signal f(t) can, according to the sampling theorem, be exactly reconstructed from its sampled values. If the signal is not necessarily band-limited, an alternative model states that it can be approximately reconstructed from its samples. Such signals can also be approximated by generalized sampling sums which can be interpreted as discretized convolution integrals of Fejér's type. In all three cases the physically realized signal is often only roughly equal to f(t) due to errors caused e.g. by the sampling mechanism.In this paper the following types of errors are treated: (1) round-off error arising when quantized sampled values are used, (2) truncation error, arising when a truncated sum is used for representation, (3) time jitter error, a result of sampling at instants slightly different from the sample values. All proofs employ deterministic methods.