Reconstruction of a discontinuous function from a few fourier coefficients using bayesian estimation

The goal of this paper is the application of spectral methods to the numerical solution of conservation law equations. Spectral methods furnish estimates of the firstn Fourier coefficients of the solution. But since the solutions of conservation law equations can have discontinuities, the estimate of the solution by summing the firstn terms of the Fourier series will haveO(1/n) error, even if the Fourier coefficients are known to high accuracy. But if the solution could be accurately reconstructed from its Fourier coefficients, spectral methods could be used effectively in these problems. A method for doing this is to assume a probability distribution for functions. Functions which are smooth away from the discontinuity are assumed to be likely, and those which are not smooth away from the discontinuity are assumed to be unlikely. Then a reconstruction algorithm is chosen by minimizing the expected error over all algorithms. It is possible to put the smoothness assumptions mentioned earlier into an infinite-dimensional Gaussian probability distribution, and then the minimum-error algorithm is well-known and fairly simple to construct and apply. If the Fourier coefficients of the reconstructed function are known exactly, then this approach gives very good results. But when used with Fourier coefficients obtained from a spectral approximation to Burgers' equation, the results were much less impressive, probably because the coefficients were not known very accurately. It is possible to construct filters that reconstruct a function using Legendre or Chebyshev coefficients for information instead Fourier coefficients. It is found that the performance of these filters is similar to the Fourier case.     

一种Zynq SoC片内硬件加速的二维傅里叶变换

由于二维傅里叶变换计算量大,会导致在嵌入式应用过程中速度过慢.为此本文实验了一种基于Xilinx Zynq芯片的片内硬件加速实现方式,主要利用片内的可编程逻辑资源来完成变换过程中的大量计算,利用片内的处理器系统完成整个算法实现过程中的数据传输与调度.在获得FPGA提供的并行计算的速度优势同时,又保留了处理器系统软件开发的灵活性.借助于Xilinx提供的一维快速傅里叶变换IP核与Xillybus提供的总线方案,本文的实验通过软硬件结合的方式实现了二维傅里叶变换算法,与OpenCV计算比较,计算速度显著提高.     

基于算术傅里叶变换的离散Hartley变换的快速算法

离散Hartley变换是一种有用的实值正交变换。文中对其快速算法进行研究,首先介绍利用算术傅里叶变换（AFT）计算离散傅里叶变换（DFT）可使其乘法计算量仅为O（N）,然后文章根据这一特点,分析离散Hartley变换（DHT）的结构特征,通过DFT将AFT和DHT建立了直接联系,提出了一种新的快速DHT算法。算法的计算复杂度能够达到线性O（N）,且算法结构简单,公式统一且易于实现,并与其他快速算法进行了比较,分析可知在数据长度不是2的幂次方时,文中提出的算法的计算时间明显比其他算法的计算时间要小。实验结果也验证了文中算法的有效性,从而为DHT的快速计算开辟了新的思路和途径。     

The Partial Fast Fourier Transform

An efficient algorithm for computing the one-dimensional partial fast Fourier transform \(f_j=\sum _{k=0}^{c(j)}e^{2\pi ijk/N} F_k\) is presented. Naive computation of the partial fast Fourier transform requires \({\mathcal O}(N^2)\) arithmetic operations for input data of length N. Unlike the standard fast Fourier transform, the partial fast Fourier transform imposes on the frequency variable k a cutoff function c(j) that depends on the space variable j; this prevents one from directly applying standard FFT algorithms. It is shown that the space–frequency domain can be partitioned into rectangular and trapezoidal subdomains over which efficient algorithms can be developed. As in the previous work of Ying and Fomel (Multiscale Model Simul 8(1):110–124, 2009), the contribution from rectangular regions can be reduced to a series of fractional-phase Fourier transforms over squares, each of which can be reduced to a convolution. In this work, we demonstrate that the partial Fourier transform over trapezoidal domains can also be reduced to a convolution. Since the computational complexity of a dealiased convolution of N inputs is \({\mathcal O}(N\log N)\), a fast algorithm for the partial Fourier transform is achieved, with a lower overall coefficient than obtained by Ying and Fomel.     

基于DST的实值离散Gabor变换

Gabor变换已被公认为是通信和信号处理中信号与图像表示的最好的方法之一,一直以来对Gabor变换的研究和应用实际上是基于Fourier变换的复值Gabor变换,因此这里对实值Gabor变换进行了研究。采用双正交分析方法,定义了一种基于离散正弦变换(DST)的实值离散Gabor变换(RDGT),该变换不仅适用于临界抽样条件而且适用于过抽样条件,并证明了变换的完备性条件(即该变换中综合窗与分析窗的双正交条件),该实验结果也验证了变换的完备性。针对实值信号,该变换由于仅涉及实值运算,并可利用快速DSTI、DST算法来加速变换,因此比传统复值离散Gabor变换在计算、实现方面更为简单。在实际应用中,将更方便于软件和硬件的实现。     

Univied Parallel Lattice Structures for Block Time—Recursive Real—Valued Duiscrete Gabor Transforms

In this paper,the 1-D real-valued discrete Gabor transform(RDGT)proposed in the previous work and its relationship with the complex-valued discrete Gabor transform(CDGT)are briefly reviewed.Block time-recursive RDGT algorithms for the efficient and fast computation of the 1-D RDGT coefficients and for the fast reconstruction of the original signal from the coefficients are developed in both critical sampling and oversampling cases.Unified parallel lattice structuires for the implementation of the algorithms are studied.And the computational complexity analysis and comparison show that the proposed algorithms provide a more efficient and faster approach to the computation of the discrete Gabor transforms.     

Hardware accelerator for solving 0–1 knapsack problems using binary harmony search

The 0–1 knapsack problem (KP) is a well-known intractable optimization problem with wide range of applications. Harmony Search (HS) is one of the most popular metaheuristic algorithms to successfully solve 0–1 KPs. Nevertheless, metaheuristic algorithms are generally compute intensive and slow when implemented in software. In this paper, we present an FPGA-based pipelined hardware accelerator to reduce computation time for solving large dimension 0–1 KPs using Binary Harmony Search algorithm. The proposed architecture exploits the intrinsic parallelism of population based metaheuristic algorithm and the flexibility and parallel processing capabilities of FPGAs to perform the computation concurrently thus enhancing performance. To validate the efficiency of the proposed hardware accelerator, experiments were conducted using a large number of 0–1 KPs. Comparative analysis on experimental results reveals that the proposed approach offers promising speedups of 51× – 111× as compared with a software implementation and 2× – 5× as compared with a hardware implementation of Binary Particle Swarm Optimization algorithm.     

Quantum Fourier transform in computational basis

The quantum Fourier transform, with exponential speed-up compared to the classical fast Fourier transform, has played an important role in quantum computation as a vital part of many quantum algorithms (most prominently, Shor’s factoring algorithm). However, situations arise where it is not sufficient to encode the Fourier coefficients within the quantum amplitudes, for example in the implementation of control operations that depend on Fourier coefficients. In this paper, we detail a new quantum scheme to encode Fourier coefficients in the computational basis, with fidelity \(1 - \delta \) and digit accuracy \(\epsilon \) for each Fourier coefficient. Its time complexity depends polynomially on \(\log (N)\), where N is the problem size, and linearly on \(1/\delta \) and \(1/\epsilon \). We also discuss an application of potential practical importance, namely the simulation of circulant Hamiltonians.     

Adjustable SAD matching algorithm using frequency domain

Fast Fourier transforms (FFTs) which are O(N logN) algorithms to compute a discrete Fourier transform (DFT) of size N have been called one of the ten most important algorithms of the twentieth century. However, even though many algorithms have been developed to speed up the computation the sum of absolute difference (SAD) matching, they are exclusively designed in the spatial domain. In this paper, we propose a fast frequency algorithm to speed up the process of (SAD) matching. We use a new approach to approximate the SAD metric by cosine series which can be expressed in correlation terms. These latter can be computed using FFT algorithms. Experimental results demonstrate the effectiveness of our method when using only the first correlation terms for block and template matching in terms of accuracy and speed. The proposed algorithm is suitable for software implementations and has a deterministic execution time unlike the existing fast algorithms for SAD matching.

Parallel algorithms for a hypercomplex discrete fourier transform

Methods for the parallel computation of a multidimensional hypercomplex discrete Fourier transform (HDFT) are considered. The basic idea consists in the application of the properties of the hypercomplex algebra in which this transform is performed. Additional possibilities for increasing the efficiency of the algorithm are provided by the natural parallelism of the multidimensional Cooley-Tukey scheme. Marat Vyacheslavovich Aliev. Born 1978. Graduated from the Adygeya State University in 2000. Received candidate’s degree in physics and mathematics in 2004. Presently he is a senior lecturer at the Department of Applied Mathematics and Information Technologies, Adygeya State University. Scientific interests: image processing, fractals, fast algorithms of discrete transforms, and finite-dimensional algebras. Author of 14 publications, including 7 papers. Member of the Russian Association of Pattern Recognition and Image Analysis. Aleksandr Mikhailovich Belov. Born 1980. Graduated from the Samara State Aerospace University in 2002. In the same year, he entered postgraduate courses with the specialty 05.13.18: mathematical modeling, numerical methods, and program complexes. Presently he is a postgraduate student at the Department of Geoinformatics, Samara State Aerospace University, and a trainee at the Laboratory of Mathematical Methods of Image Processing, Image Processing Systems Institute, Russian Academy of Sciences. Scientific interests: discrete orthogonal transforms, fast algorithms of discrete orthogonal transforms, and theory of canonical systems of calculus. Author of 13 publications, including 5 papers. Member of the Russian Association of Pattern Recognition and Image Analysis. Aleksei Vladimirovich Ershov. Born 1983. In 2000, he graduated from the Samara Lyceum of Economics and entered the Faculty of Mechanics and Mathematics, Samara State University, to specialize in the field of Organization and Technology of Information Security. In 2001, he started his training within an additional educational program and was qualified as a translator in the field of professional communication. Presently he is a fifth-year student at Samara State University. The title of his diploma work is “Control of the Flows of Confidential Information.” He is an active participant in the translation of the monograph Principia Mathematica, Cambridge University Press, 1927, by A. Whitehead and B. Russell. Author of four publications, including two papers. Marina Aleksandrovna Chicheva. Born 1964. Graduated from the Kuibyshev Aviation Institute (now Samara State Aerospace University) in 1987. Received candidate’s degree in Engineering in 1998. Presently she is a senior researcher at the Image Processing Systems Institute, Russian Academy of Sciences. Scientific interests: image processing, compression, and fast algorithms of discrete transforms. Author of more than 50 publications, including 18 papers and 1 monograph. Member of the Russian Association of Pattern Recognition and Image Analysis.     

Efficient Modular Squaring Algorithms for Hardware Implementation in GF(p)

ABSTRACT

Some of the most popular public key encryption algorithms use exponentiation as their core operation, which can be mostly broken into several modular squaring operations. In this paper, we present GF(p) modular squaring algorithms and efficiently implement them on hardware. We present different algorithms, two for squaring and one for reduction combined with the squaring, to provide a general modular squaring algorithm. The algorithms are implemented through datapaths that uses redundant Carry-Save Adders, making the computation time independent form the operands precision. The proposed algorithms are compared with each other as well as with the existing modular squaring algorithms. The experimental results are obtained by synthesizing the hardware designs for FPGA Virtex5 chip (xc5vlx50 – ff1153 technology), which showed interesting results and made our ideas very attractive.     

A Fast Spectral Subtractional Solver for Elliptic Equations

The paper presents a fast subtractional spectral algorithm for the solution of the Poisson equation and the Helmholtz equation which does not require an extension of the original domain. It takes O(N ² log N) operations, where N is the number of collocation points in each direction. The method is based on the eigenfunction expansion of the right hand side with integration and the successive solution of the corresponding homogeneous equation using Modified Fourier Method. Both the right hand side and the boundary conditions are not assumed to have any periodicity properties. This algorithm is used as a preconditioner for the iterative solution of elliptic equations with non-constant coefficients. The procedure enjoys the following properties: fast convergence and high accuracy even when the computation employs a small number of collocation points. We also apply the basic solver to the solution of the Poisson equation in complex geometries.     

快速傅立叶变换中的一种倒位序生成法

快速傅立叶变换是离散傅立叶变换(DFT)的一种快速算法,它的出现使DFT的计算大大简化,运算时间可缩短一、二个数量级,从而使得离散傅立叶变换在信号分析与处理领域中得到了广泛的应用。在应用软件和硬件程序设计中要实现快速傅立叶变换算法,均涉及到序列的倒位序排列问题。针对该问题提出倒位序生成法,直接计算各自然顺序位置的倒位序数值,然后通过变址运算完成原数列的倒位序的排列。该方法对任何满足N=2M点的快速傅立叶变换,能很快实现其变换中序列的倒位序排列。该方法只涉及倒位序十进制数和顺序十进制数,不用对二进制数进行转换,简单易行,仿真实验结果证明算法可靠有效。     

Computing Hough transforms on hypercube multicomputers

Efficient algorithms to compute the Hough transform on MIMD and SIMD hypercube multicomputer are developed. Our algorithms can compute p angles of the Hough transform of an N × N image, p N, in 0(p + log N) time on both MIMD and SIMD hypercubes. These algorithms require 0(N ²) processors. We also consider the computation of the Hough transform on MIMD hypercubes with a fixed number of processors. Experimental results on an NCUBE/7 hypercube are presented.This research was supported by the National Science Foundation under grants DCR84-20935 and 86-17374. All correspondence should be mailed to Sanjay Ranka.     

A Fast Hough Transform for the Parametrisation of Straight Lines using Fourier Methods

The Hough transform is a useful technique in the detection of straight lines and curves in an image. Due to the mathematical similarity of the Hough transform and the forward Radon transform, the Hough transform can be computed using the Radon transform which, in turn, can be evaluated using the central slice theorem. This involves a two-dimensional Fourier transform, an x-y to r-θ mapping and a 1D Fourier transform. This can be implemented in specialized hardware to take advantage of the computational savings of the fast Fourier transform. In this paper, we outline a fast and efficient method for the computation of the Hough transform using Fourier methods. The maxima points generated in the Radon space, corresponding to the parametrisation of straight lines, can be enhanced with a post transform convolutional filter. This can be applied as a 1D filtering operation on the resampled data whilst in the Fourier space, so further speeding the computation. Additionally, any edge enhancement or smoothing operations on the input function can be combined into the filter and applied as a net filter function.     

Compact Representation of Spectral BRDFs Using Fourier Transform and Spherical Harmonic Expansion

This paper proposes a compact method to represent isotropic spectral BRDFs. In the first step, we perform a Fourier transform in the wavelength dimension. The resulting Fourier coefficients of the same order depend on three angles: the polar angle of the incident light, and the polar and azimuth angles of the outgoing light. In the second step, given an incident light angle, when the Fourier coefficients of the same order have an insensitive dependency on the outgoing direction, we represent these Fourier coefficients using a linear combination of spherical harmonics. Otherwise, we first decompose these Fourier coefficients into a smooth background that corresponds to diffuse component and a sharp lobe that corresponds to specular component. The smooth background is represented using a linear combination of spherical harmonics, and the sharp lobe using a Gaussian function. The representation errors are evaluated using spectral BRDFs obtained from measurement or generated from the Phong model. While maintaining sufficient accuracy, the proposed representation method has achieved data compression over a hundred of times. Examples of spectral rendering using the proposed method are also shown.     

Optimal Computing the Chessboard Distance Transform on Parallel Processing Systems

Thedistance transform(DT) is an image computation tool which can be used to extract the information about the shape and the position of the foreground pixels relative to each other. It converts a binary image into a grey-level image, where each pixel has a value corresponding to the distance to the nearest foreground pixel. The time complexity for computing the distance transform is fully dependent on the different distance metrics. Especially, the more exact the distance transform is, the worse execution time reached will be. Nowadays, quite often thousands of images are processed in a limited time. It seems quite impossible for a sequential computer to do such a computation for the distance transform in real time. In order to provide efficient distance transform computation, it is considerably desirable to develop a parallel algorithm for this operation. In this paper, based on the diagonal propagation approach, we first provide anO(N²) time sequential algorithm to compute thechessboard distance transform(CDT) of anN×Nimage, which is a DT using the chessboard distance metrics. Based on the proposed sequential algorithm, the CDT of a 2D binary image array of sizeN×Ncan be computed inO(logN) time on the EREW PRAM model usingO(N²/logN) processors,O(log logN) time on the CRCW PRAM model usingO(N²/log logN) processors, andO(logN) time on the hypercube computer usingO(N²/logN) processors. Following the mapping as proposed by Lee and Horng, the algorithm for the medial axis transform is also efficiently derived. The medial axis transform of a 2D binary image array of sizeN×Ncan be computed inO(logN) time on the EREW PRAM model usingO(N²/logN) processors,O(log logN) time on the CRCW PRAM model usingO(N²/log logN) processors, andO(logN) time on the hypercube computer usingO(N²/logN) processors. The proposed parallel algorithms are composed of a set of prefix operations. In each prefix operation phase, only increase (add-one) operation and minimum operation are employed. So, the algorithms are especially efficient in practical applications.     

A class of discrete orthogonal transforms

For a given N-periodic sequence, a class of log₂N discrete orthogonal transforms ranging from Walsh-Hadamard transform to discrete Fourier transform (DFT) is defined. The power spectra invariant to circular shift of the sampled data for these transforms are developed. Phase spectra, analogous to that of the DFT, for all the discrete transforms are defined and developed. Recursive relations for generating the transform matrices are developed. Generalized expressions for factoring these transform matrices are provided. Based on these matrix factors, efficient algorithms for fast computation of the transform coefficients are developed. By introducing a number of zeros as the elements in the transform matrices, a modified version of the transforms is developed. By using these modified matrices, the power and phase spectra can be computed efficiently. These transforms can be used in the general area of information processing.     

波段排序的高光谱影像3维混合树编码方法

目的 高光谱影像压缩的关键技术是对空间维和光谱维的去相关性。根据高光谱影像数据结构的特点,如何有效去除其空间相关性与谱间相关性是高光谱影像压缩中至关重要的问题。对高光谱影像进行编码时,3维小波变换是极为有效的去除冗余的方法。因此提出了一种通过波段排序并结合3维混合树型结构对高光谱影像3维小波变换系数进行编码的算法。方法 首先,将高光谱影像按照自然波段顺序进行波段分组,并对每组影像进行相邻影像的谱间相关性统计;其次,对相关性较弱的波段组,建立以影像波段序号为顶点、影像相关性系数为边的完全图,对这个完全图求其最大汉密尔顿回路。按照求得的最大汉密尔顿回路顺序对该波段组进行重新排序,从而提高波段组的谱间相关性;在此基础上,对重新排序后的波段组进行3维小波变换,并通过3维混合树结构对3维小波变换系数进行零树编码。结果 通过对大量AVIRIS型高光谱影像数据的仿真实验,验证了本文方法的有效性。对相关性较低的波段组,加入排序算法后,其解码影像与未排序时比,峰值信噪比有了一定的提高。通过实验统计,算法平均用时2.7579s。结论 由于采用了对弱相关性波段组的重新排序机制,使得基于混合树结构的3维零树编码出现了更多有效的零树,在一定程度上提高了编码效率。通过实验统计算法用时,表明该方法以较小的时间代价获得了解码效果的提升。     

Efficient Spectral Coefficient Calculation Using Circuit Output Probabilities

Many problems in the field of digital logic may he solved more efficiently in the spectral domain than in the Boolean domain. However, the primary drawback of spectral techniques is the large complexity associated with the calculation of the spectrum of a Boolean function. We present a new method for the computation of a spectral coefficient that has a complexity equal to O(|E|) where |E| is the number of edges in a binary decision diagram characterizing the circuit. This result is especially significant for techniques that require the calculation of only a few spectral coefficients since it allows the computations to be accomplished very efficiently and does not require storage resources for a large number of values. Furthermore, this method holds for any general spectral transform and does not require the transformation matrix to be recursively defined or sparse.