Sparsity and conditioning of impedance matrices obtained withsemi-orthogonal and bi-orthogonal wavelet bases

Wavelet and wavelet packet transforms are often used to sparsify dense matrices arising in the discretization of CEM integral equations. This paper compares orthogonal, semi-orthogonal, and bi-orthogonal wavelet and wavelet packet transforms with respect to the condition numbers, matrix sparsity, and number of iterations for the transformed systems. The best overall results are obtained with the orthogonal wavelet packet transforms that produce highly sparse matrices requiring fewest iterations. Among wavelet transforms the semi-orthogonal wavelet transforms lead to the sparsest matrices, but require too many iterations due to high condition numbers. The bi-orthogonal wavelets produce very poor sparsity and require many iterations and should not be used in these applications     

An effective wavelet matrix transform approach for efficientsolutions of electromagnetic integral equations

An effective numerical method based on wavelet matrix transforms for efficient solution of electromagnetic (EM) integral equations is proposed. Using the wavelet matrix transform produces highly sparse moment matrices which can be solved efficiently. A fast construction method for various orthonormal or nonorthonormal wavelet basis matrices is also given. It has been found that using nonsimilarity wavelet matrix transforms such as nonsimilarity nonorthonormal cardinal spline wavelet (NSNCSW) transform, one can obtain a much higher compression rate and much better accuracy of the approximate solutions than using similarity wavelet transforms such as Daubechies' (1992) orthonormal wavelet (DOW) transform. Numerical examples are given to show the validity and effectiveness of the method     

Automatic generation of fast discrete signal transforms

This paper presents an algorithm that derives fast versions for a broad class of discrete signal transforms symbolically. The class includes but is not limited to the discrete Fourier and the discrete trigonometric transforms. This is achieved by finding fast sparse matrix factorizations for the matrix representations of these transforms. Unlike previous methods, the algorithm is entirely automatic and uses the defining matrix as its sole input. The sparse matrix factorization algorithm consists of two steps: first, the “symmetry” of the matrix is computed in the form of a pair of group representations; second, the representations are stepwise decomposed, giving rise to a sparse factorization of the original transform matrix. We have successfully demonstrated the method by computing automatically efficient transforms in several important cases: for the DFT, we obtain the Cooley-Tukey (1965) FFT; for a class of transforms including the DCT, type II, the number of arithmetic operations for our fast transforms is the same as for the best-known algorithms. Our approach provides new insights and interpretations for the structure of these signal transforms and the question of why fast algorithms exist. The sparse matrix factorization algorithm is implemented within the software package AREP     

稀疏随机矩阵的观测次数下界

压缩感知理论中的稀疏重构问题,要将一个高维信号从它的低维投影中恢复出来,通常选用稠密随机矩阵作为观测矩阵来解决这一问题。而某些稀疏随机矩阵作为观测矩阵也可以达到这一目的。稀疏随机矩阵的特点是,在编码和重构过程中都具有较低的计算复杂度,更新方便,且对存储容量的要求较低。该文基于压缩感知理论,分别对列重固定、行重固定以及一般的稀疏随机矩阵进行了研究,当这些稀疏随机矩阵满足有限等距性质时,推导了观测次数应满足的下界条件,并对三种矩阵的性能进行了分析。以二值稀疏随机矩阵为特例,进行了仿真实验。实验结果显示,结论给出的观测次数下界是比较紧的,并验证了列重固定、行重固定的稀疏随机矩阵作为观测矩阵的可行性和实用性。      

基于Gabor变换与自组织稀疏RAM的N-tuple神经网络的人脸识别方法

本文提出了一种采用首先对人脸图像进行Gabor变换，然后由自组织稀疏RAM的N-tuple神经网络进行训练识别的方法，通过大量实验证明，该方法在较少训练样本下条件下，能够取得较高的识别率。     

Eigenvalues and eigenvectors of generalized DFT, generalized DHT,DCT-IV and DST-IV matrices

In this paper, the eigenvalues and eigenvectors of the generalized discrete Fourier transform (GDFT), the generalized discrete Hartley transform (GDHT), the type-IV discrete cosine transform (DCT-IV), and the type-IV discrete sine transform (DST-IV) matrices are investigated in a unified framework. First, the eigenvalues and their multiplicities of the GDFT matrix are determined, and the theory of commuting matrices is applied to find the real, symmetric, orthogonal eigenvectors set that constitutes the discrete counterpart of Hermite Gaussian function. Then, the results of the GDFT matrix and the relationships among these four unitary transforms are used to find the eigenproperties of the GDHT, DCT-IV, and DST-IV matrices. Finally, the fractional versions of these four transforms are defined, and an image watermarking scheme is proposed to demonstrate the effectiveness of fractional transforms     

Linearly independent ternary arithmetic helix transforms, their properties and relations

New classes of linearly independent ternary arithmetic transforms in standard algebra called ternary arithmetic helix transforms are introduced. Four types of helix transform matrices with detailed recursive equations are shown. Various properties, mutual relationships among transform matrices and spectra as well as results of helix transforms for some special cases of ternary logic functions are discussed. Computational costs for the calculation of new transforms are also presented.     

Family of generalised multi-polarity complex Hadamard transforms

A family of generalised complex Hadamard transforms using the concept of polarity is introduced. Forward and inverse transformation kernels and methods of recursive generation of transform matrices using Kronecker products of elementary matrices are shown. Mutual relationships among transform matrices and spectra for arbitrary polarities are presented. Efficient ways of calculating spectra for logic functions through decision diagrams are also shown. The half-spectrum property is used to reduce further the computational requirements for both fast transforms and decision diagram based calculations     

Walsh Summing and Differencing Transforms

Analogous to Fourier frequency transforms of the integration and differentiation of a continuous-time function, Walsh sequency transforms of the summing and differencing of an arbitrary discrete-time function have been derived. These transforms can be represented numerically in the form of matrices of simple recursive structure. The matrices are not orthogonal, but they are the inverse of each other, and the value of their determinants is one.     

A performance comparison of measurement matrices in compressive sensing

Compressive sensing involves 3 main processes: signal sparse representation, linear encoding or measurement collection, and nonlinear decoding or sparse recovery. In the measurement process, a measurement matrix is used to sample only the components that best represent the signal. The choice of the measurement matrix has an important impact on the accuracy and the processing time of the sparse recovery process. Hence, the design of accurate measurement matrices is of vital importance in compressive sensing. Over the last decade, a number of measurement matrices have been proposed. Therefore, a detailed review of these measurement matrices and a comparison of their performances are strongly needed. This paper explains the foundation of compressive sensing and highlights the process of measurement by reviewing the existing measurement matrices. It provides a 3‐level classification and compares the performance of 8 measurement matrices belonging to 4 different types using 5 evaluation metrics: the recovery error, processing time, recovery time, covariance, and phase transition diagram. The theoretical performance comparison is validated with experimental results, and the results show that the Circulant, Toeplitz, and Hadamard matrices outperform the other measurement matrices.     

A Class of Deterministic Sensing Matrices and Their Application in Harmonic Detection

In this paper, a class of deterministic sensing matrices are constructed by selecting rows from Fourier matrices. These matrices have better performance in sparse recovery than random partial Fourier matrices. The coherence and restricted isometry property of these matrices are given to evaluate their capacity as compressive sensing matrices. In general, compressed sensing requires random sampling in data acquisition, which is difficult to implement in hardware. By using these sensing matrices in harmonic detection, a deterministic sampling method is provided. The frequencies and amplitudes of the harmonic components are estimated from under-sampled data. The simulations show that this under-sampled method is feasible and valid in noisy environments.     

Non-binary Sparse Measurement Matrices for Binary Signal Recovery

Binary sparse measurement matrices are widely used in compressed sensing (CS) due to their low computational complexity. However, binary sparse measurement matrices perform well in CS-based binary signal recovery only when the source signals are very sparse (e.g., k/n=0.1, where k is the sparsity of the source signal, n is the length of the source signal). In this paper, we propose to construct a non-binary sparse measurement matrix to recover binary source signals which are not so sparse (e.g., k/n=0.2) accurately with few measurements. The novel measurement matrix enables us to design a suboptimal and effective recovery algorithm by fully exploiting the structural features. Moreover, we analyze and estimate the un-recovery probability based on the tree structure to evaluate the recovery performance. The simulation results validate that non-binary sparse measurement matrices can be used to recover binary source signals which are not so sparse, the recovery performance of non-binary sparse measurement matrices is better than that of binary sparse measurement matrices in terms of the un-recovery probability.     

Infinity-Norm Rotation Transforms

A new general paradigm of dynamic-range-preserving one-to-one mapping-infinity-norm rotations, analogous to the general 2-norm rotations, are proposed in this paper. Analogous to the well-known discrete cosine transforms, the linear 2-norm rotation transforms which preserve the 2-norm of the rotated vectors, the proposed infinity-norm rotation transforms are piecewise linear transforms which preserve the infinity-norm of vectors. Besides the advantages of perfect reversibility, in-place calculation and dynamic range preservation, the infinity-norm rotation transforms also have good energy-compact ability, which is suitable for signal compression and analysis. It can be implemented by shear transforms based on the 2-D rotation factorization of similar orthogonal transform matrices, such as DCT matrices. The performance of the new transforms is illustrated with 2-D patterns and histograms. Its good performance in lossy and lossless image compression, compared with other integer reversible transforms, is demonstrated in the experiments.     

主从模式编队卫星SAR压缩感知成像算法

为减轻主从模式编队卫星SAR对稀疏目标场景回波信号的采集与传输负担,提出了编队卫星SAR的回波信号稀疏方法。在研究编队卫星SAR回波信号特征的基础上,构建了编队卫星SAR距离向和方位向的稀疏基、测量矩阵和重构矩阵。针对主从模式编队卫星SAR与地面的数据传输特点,提出了低传输负荷下的主从模式编队卫星SAR压缩感知成像方法,并借助于正交匹配追踪算法 (Orthogonal Matching Pursuit,OMP) 对稀疏后的回波信号进行了恢复重构,获得了高质量的编队卫星SAR图像。仿真结果表明,针对稀疏目标场景,本文提出的压缩感知成像方法利用较少的回波数据便能重构出原始目标场景,实现了低负荷下的编队卫星SAR成像。      

Unification of Legendre,Laguerre, Hermite,and binomial discrete transforms using Pascal's matrix

Pascal's matrix plays an important role in the computation of the discrete Legendre, Laguerre, Hermite, and binomial transforms. In particular, Pascal's matrix helps to unify the formulation of these orthogonal transforms and demonstrate the similarity of the computation of the transform matrices. It also allows the identification of the identical computations needed for these transforms. The fundamental finding is based on the discovery of the relationship between Pascal's matrix and the binomial coefficient.     

基于概率稀疏随机矩阵的压缩数据收集方法

测量矩阵设计是应用压缩感知理论解决实际问题的关键。该文针对无线传感器网络压缩数据收集问题设计了一种概率稀疏随机矩阵。该矩阵可在减少参与投影值计算节点个数的同时,让参与投影值计算的节点分布集中化,从而降低数据收集的通信能耗。在此基础上,为提高网络数据重构精度,又提出一种适用于概率稀疏随机矩阵优化的测量矩阵优化算法。仿真实验结果表明,与稀疏随机矩阵和稀疏Toeplitz测量矩阵相比,采用优化的概率稀疏随机矩阵作为压缩数据收集的测量矩阵可显著降低通信能耗,且重构误差更小。     

Wavelet transforms for vector fields using omnidirectionally balanced multiwavelets

Vector wavelet transforms for vector-valued fields can be implemented directly from multiwavelets; however, existing multiwavelets offer surprisingly poor performance for transforms in vector-valued signal-processing applications. In this paper, the reason for this performance failure is identified, and a remedy is proposed. A multiwavelet design criterion known as omnidirectional balancing is introduced to extend to vector transforms the balancing philosophy previously proposed for multiwavelet-based scalar-signal expansion. It is shown that the straightforward implementation of a vector wavelet transform, namely, the application of a scalar transform to each vector component independently, is a special case of an omnidirectionally balanced vector wavelet transform in which filter-coefficient matrices are constrained to be diagonal. Additionally, a family of symmetric-antisymmetric multiwavelets is designed according to the omnidirectional-balancing criterion. In empirical results for a vector-field compression system, it is observed that the performance of vector wavelet transforms derived from these omnidirectionally-balanced symmetric-antisymmetric multiwavelets is far superior to that of transforms implemented via other multiwavelets and can exceed that of diagonal transforms derived from popular scalar wavelets.     

基于Berlekamp-Justesen码的压缩感知确定性测量矩阵的构造

确定性测量矩阵构造是近期压缩感知领域的一个重要研究问题。该文基于Berlekamp-Justesen(B-J)码,构造了两类确定性测量矩阵。首先,给出一类相关性渐近最优的稀疏测量矩阵,从而保证其具有较好的限定等距性(RIP)。接着,构造一类确定性复测量矩阵,这类矩阵可以通过删除部分行列使其大小灵活变化。第1类矩阵具有很高的稀疏性,第2类则是基于循环矩阵,因此它们的存储开销较小,编码和重构复杂度也相对较低。仿真结果表明,这两类矩阵常常有优于或相当于现有的随机和确定性测量矩阵的重建性能。     

A survey of sparse matrix research

This paper surveys the state of the art in sparse matrix research in January 1976. Much of the survey deals with the solution of sparse simultaneous linear equations, including the storage of such matrices and the effect of paging on sparse matrix algorithms. In the symmetric case, relevant terms from graph theory are defined. Band systems and matrices arising from the discretization of partial differential equations are treated as separate cases. Preordering techniques are surveyed with particular emphasis on partitioning (to block triangular form) and tearing (to bordered block triangular form). Methods for solving the least squares problem and for sparse linear programming are also reviewed. The sparse eigenproblem is discussed with particular reference to some fairly recent iterative methods. There is a short discussion of general iterative techniques, and reference is made to good standard texts in this field. Design considerations when implementing sparse matrix algorithms are examined and finally comments are made concerning the availability of codes in this area.     

Properties and relations for fast linearly independent arithmetic transforms

New fast linearly independent arithmetic (LIA) transforms are introduced here which can be used to represent any functions of binary variables. The transforms are grouped into classes where consistent formulas relating forward and inverse transform matrices are obtained. All the presented transforms have the same computational cost, which is lower than the computational cost of the well-known fixed polarity arithmetic transforms. General classifications and fast forward and inverse transform definitions for all the fast LIA transforms are given. Various properties and mutual relations that exist for the different transforms and their corresponding spectra are also shown. The presented relations and properties reduce the computational cost of finding the best LIA polynomial expansion based on the new transforms.