基于部分三角函数变换矩阵的块压缩感知测量方法

为了提高块压缩感知的测量效率和重构性能,根据离散余弦变换和离散正弦变换具有汇聚信号能量的特性,提出了基于重复块对角结构的部分离散余弦变换partial discrete cosine transform in repeated block diagonal structure,简称PDCT-RBDS和部分离散正弦变换partial discrete sine transform in repeated block diagonal structure简称PDST-RBDS的两种压缩感知测量方法.所采用的测量矩阵是一种低复杂度的结构化确定性矩阵, 满足受限等距性质.并得到一个与采样能量有关的受限等距常数和精确重构的测量数下限.通过与采用重复块对角结构的部分随机高斯矩阵和部分贝努利矩阵的图像压缩感知对比,结果表明PDCT-RBDS和PDST-RBDS重构的PSNR大约提高1---5dBSSIM提高约0.05, 所需的重构时间和测量矩阵的存储空间大大减少.该方法特别适合大规模图像压缩及实时视频数据处理场合.     

离散余弦变换的一种推广

离散余弦变换(DCT)在数字信号、图像处理、频谱分析、数据压缩和信息隐藏等领域有着广泛的应用.推广离散余弦变换,给出一个包含三个参数的统一表达式,并证明在许多情形新变换是正交变换.最后给出一种新型离散余弦变换,并证明它是正交变换.     

Fast algorithms for discrete Chebyshev-Vandermonde transforms and applications

Applying orthogonal polynomials, the discrete Chebyshev-Vandermonde transform (DCVT) is introduced as a special almost orthogonal transform. An important example of DCVT is the discrete cosine transform (DCT). Using the divide-and-conquer technique and the d'Alembert functional equation, fast DCT-algorithms are described. By the help of these results we present for the first time fast, numerically stable algorithms for simultaneous polynomial approximation and for collocation method for the airfoil equation, a special Cauchytype singular integral equation.     

MULTIVARIATE FOURIER TRANSFORM METHODS OVER SIMPLEX AND SUPER-SIMPLEX DOMAINS

In this paper we propose the well-known Fourier method on some non-tensor productdomains in R~d, inclding simplex and so-called super-simplex which consists of (d 1)!simplices. As two examples, in 2-D and 3-D case a super-simplex is shown as a parallelhexagon and a parallel quadrilateral dodecahedron, respectively. We have extended mostof concepts and results of the traditional Fourier methods on multivariate cases, such asFourier basis system, Fourier series, discrete Fourier transform (DFT) and its fast algorithm(FFT) on the super-simplex, as well as generalized sine and cosine transforms (DST, DCT)and related fast algorithms over a simplex. The relationship between the basic orthogonalsystem and eigen-functions of a Laplacian-like operator over these domains is explored.     

Numerical stability of fast cosine transforms

In this paper we consider the numerical stability of fast algorithms for discrete cosine transform (DCT) of type III and II, respectively. We show that various fast DCTs can possess a very different behaviour of numerical stability. By matrix factorizations we find that a complex fast DCT which is based mainly on a fast Fouier transform has a better numerical stability than a real fast DCT despite its larger arithmetical complexity. Numerical tests illustrate our theoretical results.     

A new approach to fuzzy wavelet system modeling

In this paper, we propose simple but effective two different fuzzy wavelet networks (FWNs) for system identification. The FWNs combine the traditional Takagi–Sugeno–Kang (TSK) fuzzy model and discrete wavelet transforms (DWT). The proposed FWNs consist of a set of if–then rules and, then parts are series expansion in terms of wavelets functions. In the first system, while the only one scale parameter is changing with it corresponding rule number, translation parameter sets are fixed in each rule. As for the second system, DWT is used completely by using wavelet frames. The performance of proposed fuzzy models is illustrated by examples and compared with previously published examples. Simulation results indicate the remarkable capabilities of the proposed methods. It is worth noting that the second FWN achieves high function approximation accuracy and fast convergence.     

Fast Wavelet Transform for Toeplitz Matrices and Property Analysis

Fast wavelet transform algorithms for Toeplitz matrices are proposed in this paper. Distinctive from the well known discrete trigonometric transforms, such as the discrete cosine transform （DCT） and the discrete Fourier transform （DFT） for Toeplitz matrices, the new algorithms are achieved by compactly supported wavelet that preserve the character of a Toeplitz matrix after transform, which is quite useful in many applications involving a Toeplitz matrix. Results of numerical experiments show that the proposed method has good compression performance similar to using wavelet in the digital image coding. Since the proposed algorithms turn a dense Toeplitz matrix into a band-limited form, the arithmetic operations required by the new algorithms are O（N） that are reduced greatly compared with O（N log N） by the classical trigonometric transforms.     

Continuous vs. discrete fractional Fourier transforms

We compare the finite Fourier (-exponential) and Fourier–Kravchuk transforms; both are discrete, finite versions of the Fourier integral transform. The latter is a canonical transform whose fractionalization is well defined. We examine the harmonic oscillator wavefunctions and their finite counterparts: Mehta's basis functions and the Kravchuk functions. The fractionalized Fourier–Kravchuk transform was proposed in J. Opt. Soc. Amer. A (14 (1997) 1467–1477) and is here subject of numerical analysis. In particular, we follow the harmonic motions of coherent states within a finite, discrete optical model of a shallow multimodal waveguide.     

An image adaptive, wavelet-based watermarking of digital images

In digital management, multimedia content and data can easily be used in an illegal way—being copied, modified and distributed again. Copyright protection, intellectual and material rights protection for authors, owners, buyers, distributors and the authenticity of content are crucial factors in solving an urgent and real problem. In such scenario digital watermark techniques are emerging as a valid solution. In this paper, we describe an algorithm—called WM2.0—for an invisible watermark: private, strong, wavelet-based and developed for digital images protection and authenticity. Using discrete wavelet transform (DWT) is motivated by good time-frequency features and well-matching with human visual system directives. These two combined elements are important in building an invisible and robust watermark. WM2.0 works on a dual scheme: watermark embedding and watermark detection. The watermark is embedded into high frequency DWT components of a specific sub-image and it is calculated in correlation with the image features and statistic properties. Watermark detection applies a re-synchronization between the original and watermarked image. The correlation between the watermarked DWT coefficients and the watermark signal is calculated according to the Neyman–Pearson statistic criterion. Experimentation on a large set of different images has shown to be resistant against geometric, filtering and StirMark attacks with a low rate of false alarm.     

最优多进Haar小波及在图像压缩中的应用

我们提出了最优多进Haar小波的概念,证明了其存在性和唯一性,给出了最优多进Haar小波构造的通用方法,并证明了最优多进Haar小波具有线性相位,在消失矩意义下,我们所得到的最优多进Haar小波优于离散余弦变换.同时,我们用图像缩编码的方法验证了最优多进Haar小波的性能优于离散余弦变换的,新的变换可以化为精确的小整数运算,能非常廉价地用集成电路实现,该变换的实用意义在于给图像和视频压缩提供了一个更好的选择.     

随机激励下结构的非线性分析

针对随机时间序列载荷激励下的非线性系统,提出一种基于Z变换的递归方法．对于所获得的响应时间序列的识别,建议了一种离散小波变换（DWT）的技术．     

A FAST SINE TRANSFORM ALGORITHM FOR TOEPLITZ MATRICES AND ITS APPLICATIONS

In this paper, a fast algorithm for the discrete sine transform(DST) of a Toeplitz matrix of order N is derived. Only O(N log N) O(M) time is needed for the computation of M elements. The auxiliary storage requirement is O(N). An application of the new fast algorithm is also discussed.     

The expected profile of digital search trees

A digital search tree (DST) is a fundamental data structure on words that finds various applications from the popular Lempel-Ziv?78 data compression scheme to distributed hash tables. The profile of a DST measures the number of nodes at the same distance from the root; it depends on the number of stored strings and the distance from the root. Most parameters of DST (e.g., depth, height, fillup) can be expressed in terms of the profile. We study here asymptotics of the average profile in a DST built from sequences generated independently by a memoryless source. After representing the average profile by a recurrence, we solve it using a wide range of analytic tools. This analysis is surprisingly demanding but once it is carried out it reveals an unusually intriguing and interesting behavior. The average profile undergoes phase transitions when moving from the root to the longest path: at first it resembles a full tree until it abruptly starts growing polynomially and oscillating in this range. These results are derived by methods of analytic combinatorics such as generating functions, Mellin transform, poissonization and depoissonization, the saddle point method, singularity analysis and uniform asymptotic analysis.     

On the solutions of generalized discrete Poisson equation

The set of common numerical and analytical problems is introduced in the form of the generalized multidimensional discrete Poisson equation. It is shown that its solutions with square-summable discrete derivatives are unique up to a constant. The proof uses the Fourier transform as the main tool. The necessary condition for the existence of the solution is provided.     

On the construction of wavelets on a bounded interval

This paper presents a general approach to a multiresolution analysis and wavelet spaces on the interval [–1, 1]. Our method is based on the Chebyshev transform, corresponding shifts and the discrete cosine transformation (DCT). For the wavelet analysis of given functions, efficient decomposition and reconstruction algorithms are proposed using fast DCT-algorithms. As examples for scaling functions and wavelets, polynomials and transformed splines are considered.     

On Segal�CBargmann analysis for finite Coxeter groups and its heat kernel

We prove identities involving the integral kernels of three versions (two being introduced here) of the Segal?CBargmann transform associated to a finite Coxeter group acting on a finite dimensional, real Euclidean space (the first version essentially having been introduced around the same time by Ben Sa?d and ?rsted and independently by Soltani) and the Dunkl heat kernel, due to R?sler, of the Dunkl Laplacian associated with the same Coxeter group. All but one of our relations are originally due to Hall in the context of standard Segal?CBargmann analysis on Euclidean space. Hall??s results (trivial Dunkl structure and arbitrary finite dimension) as well as our own results in???-deformed quantum mechanics (non-trivial Dunkl structure, dimension one) are particular cases of the results proved here. So we can understand all of these versions of the Segal?CBargmann transform associated to a Coxeter group as Hall type transforms. In particular, we define an analogue of Hall??s Version C generalized Segal?CBargmann transform which is then shown to be Dunkl convolution with the Dunkl heat kernel followed by analytic continuation. In the context of Version C we also introduce a new Segal?CBargmann space and a new transform associated to the Dunkl theory. Also we have what appears to be a new relation in this context between the Segal?CBargmann kernels for Versions A and C.     

A fast wavelet block Jacobi method

In this paper, we develop a fast block Jacobi method for linear systems based on discrete wavelet transform (DWT). Traditional wavelet-based methods for linear systems do not fully utilize the sparsity and the multi-level block structure of the transformed matrix after DWT. For the sake of numerical efficiency, we truncate the transformed matrix to be a sparse matrix by letting the small values be zero. To combine the advantages of the direct method and the iterative method, we solve the sub-systems appropriately based on the multi-level block structure of the transformed matrix after DWT. Numerical examples show that the proposed method is very numerically effective.     

On the properties of a new tensor product of matrices

Previously, the author introduced a new tensor product of matrices according to which the matrix of the discrete Walsh-Paley transform can be represented as a power of the second-order discrete Walsh transform matrix H with respect to this product. This power is an analogue of the representation of the Sylvester-Hadamard matrix in the form of a Kronecker power of H. The properties of the new tensor product of matrices are examined and compared with those of the Kronecker product. An algebraic structure with the matrix H used as a generator element and with these two tensor products of matrices is constructed and analyzed. It is shown that the new tensor product operation proposed can be treated as a convenient mathematical language for describing the foundations of discrete Fourier analysis.     

Gaussian DCT Coefficient Models

It has been known that the distribution of the discrete cosine transform (DCT) coefficients of most natural images follow a Laplace distribution. However, recent work has shown that the Laplace distribution may not be a good fit for certain type of images and that the Gaussian distribution will be a realistic model in such cases. Assuming this alternative model, we derive a comprehensive collection of formulas for the distribution of the actual DCT coefficient. The corresponding estimation procedures are derived by the method of moments and the method of maximum likelihood. Finally, the superior performance of the derived distributions over the Gaussian model is illustrated. It is expected that this work could serve as a useful reference and lead to improved modeling with respect to image analysis and image coding.