离散拟正交GaBor展开

利用一般条件的离散Ｚａｋ变换及连续Ｇｏｂｏｒ展开和离散Ｇａｂｏｒ展开间的关系。本文首次提出了在整数倍过抽样条件下，由Ｗｅｙｌ－Ｈｅｉｓｅｎｂｅｒｇ紧框架构造离散ＧＡＢＯＲ展开的方法。     

The generalized Gabor transform

The generalized Gabor transform (for image representation) is discussed. For a given function f(t), tinR, the generalized Gabor transform finds a set of coefficients a(mr) such that f(t)=Sigma(m=-infinity)(infinity)Sigma (r=-infinity)(infinity)alpha(mr )g(t-mT)exp(i2pirt/T'). The original Gabor transform proposed by D. Gabor (1946) is the special case of T=T'. The computation of the generalized Gabor transform with biorthogonal functions is discussed. The optimal biorthogonal functions are discussed. A relation between a window function and its optimal biorthogonal function is presented based on the Zak (1967) transform when T/T' is rational. The finite discrete generalized Gabor transform is also derived. Methods of computation for the biorthogonal function are discussed. The relation between a window function and its optimal biorthogonal function derived for the continuous variable generalized Gabor transform can be extended to the finite discrete case. Efficient algorithms for the optimal biorthogonal function and generalized Gabor transform for the finite discrete case are proposed.     

The undersampled discrete Gabor transform

Conventional studies on discrete Gabor transforms have generally been confined to the cases of critical sampling and oversampling in which the Gabor families span the whole signal space. In this paper, we investigate undersampled discrete Gabor transforms. For an undersampled Gabor triple (g,a,b), i.e. a·b>N, we show that the associated generalized dual Gabor window (GDGW) function is the same as the one associated with the oversampled (g,N/b,N/a), except for the constant factor (ab/N). Computations of undersampled Gabor transforms are made possible. By applying the methods (algorithms) developed in oversampled settings, the undersampled GDGW is determined. Then, we are able to obtain the best approximation of a signal x by linear combinations of vectors in the Gabor family     

离散Gabor展开及其关系

Ｇａｂｏｒ 展开是一种分析非平稳信号的工具。为了能进行数值计算,需要对连续 Ｇａｂｏｒ展开进行离散化和有限化。在过采样的一般情况下,给出了有限离散时域 Ｇａｂｏｒ 展开系数与有限离散频域 Ｇａｂｏｒ 展开系数之间的关系,并给出了一个计算实例。     

基于DCT的实值离散Gabor变换

该文提出了一种基于离散余弦变换(DCT)的实值离散Gabor变换(RDGT),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。由于这种变换仅涉及实值计算,并且可利用快速DCT,IDCT算法来加速运算,因此比传统复值离散Gabor变换在计算和实现方面更为简单,必将有效地提高非平稳信号与图像的分析、处理速度和效率。     

Discrete fourier transform on multicore

This article gives an overview on the techniques needed to implement the discrete Fourier transform (DFT) efficiently on current multicore systems. The focus is on Intel-compatible multicores, but we also discuss the IBM Cell and, briefly, graphics processing units (GPUs). The performance optimization is broken down into three key challenges: parallelization, vectorization, and memory hierarchy optimization. In each case, we use the Kronecker product formalism to formally derive the necessary algorithmic transformations based on a few hardware parameters. Further code-level optimizations are discussed. The rigorous nature of this framework enables the complete automation of the implementation task as shown by the program generator, Spiral. Finally, we show and analyze DFT benchmarks of the fastest libraries available for the considered platforms.     

基于Gabor小波的人脸表情特征提取研究

为了使计算机能更好的识别人脸表情,对基于Gabor小波变换的人脸表情识别方法进行了研究。首先对包含表情区域的静态灰度图像进行预处理,包括对确定的人脸表情区域进行尺寸和灰度归一化,然后利用二维Gabor小波变换提取脸部表情特征,使用快速PCA方法对提取的Gabor小波特征初步降维。再在低维的空间中,利用Fisher准则提取那些有利于分类的特征,最后用SVM分类器进行分类。实验结果表明,上述提出的方法比传统的方法识别速度更快,能达到实时性的要求,并且具有很好的鲁棒性,识别率高。     

连续Gabor展开离散化后的混叠问题

为能进行数值计算，需对连续Ｇａｂｏｒ展开离散化有限化。本文就时域与频域两种情况，给出了在过采样率的一般情况下，有限离散化后的Ｇａｂｏｒ展开系数与原连续Ｇａｂｏｒ展开系数之间的关系，并由此推出了离散Ｇａｂｏｒ展开系数不产生混叠的条件——连续Ｇａｂｏｒ展开离散化的采样定理。最后给出了一个计算实例     

Discrete Fourier transform of correlation functions

The Van Vleck equation relating the polarity correlation function to the direct correlation function is used to derive a new algorithm for the evaluation of the discrete Fourier transform of the direct correlation function. Application of an implementation of the algorithm to correlation-based measurement is discussed.     

DGT与DCT在图像编码中的性能比较

介绍了二维实值离散Gabor变换（RDGT）的快速算法,并着重探讨了二维实值离散Gabor变换与二维离散余弦变换在图像编码中的性能及差异。     

Novel DCT-Based Real-Valued Discrete Gabor Transform and Its Fast Algorithms

The oversampled Gabor transform is more effective than the critically sampled one in many applications. The biorthogonality relationship between the analysis window and the synthesis window of the Gabor transform represents the completeness condition. However, the traditional discrete cosine transform (DCT)-based real-valued discrete Gabor transform (RGDT) is available only in the critically sampled case and its biorthogonality relationship for the transform has not been unveiled. To bridge these important gaps, this paper proposes a novel DCT-based RDGT, which can be applied in both the critically sampled case and the oversampled case, and their biorthogonality relationships can be derived. The proposed DCT-based RDGT involves only real operations and can utilize fast DCT algorithms for computation, which facilitates computation and implementation by hardware or software as compared to that of the traditional complex-valued discrete Gabor transform. This paper also develops block time-recursive algorithms for the efficient and fast computation of the RDGT and its inverse transform. Unified parallel lattice structures for the implementation of these algorithms are presented. Computational complexity analysis and comparisons have shown that the proposed algorithms provide a more efficient and faster approach for discrete Gabor transforms as compared to those of the existing discrete Gabor transform algorithms. In addition, an application in the noise reduction of the nuclear magnetic resonance free induction decay signals is presented to show the efficiency of the proposed RDGT for time-frequency analysis.      

Multirate-based fast parallel algorithms for DCT-kernel-based real-valued discrete Gabor transform

Fast parallel algorithms for the DCT-kernel-based real-valued discrete Gabor transform (RDGT) and its inverse transform are presented based on multirate signal processing. An analysis convolver bank is designed for the RDGT and a synthesis convolver bank is designed for its inverse transform. The parallel channels in each of the two convolver banks have a unified structure and can apply the fast DCT algorithms to reduce computation. The computational complexity of each parallel channel is low and depends mainly on the length of the discrete input signal and the number of the Gabor frequency sampling points. Every parallel channel corresponds to one RDGT coefficient, and all the RDGT coefficients are computed in parallel during the analysis process and are finally reconstructed in parallel as pieces of the original signal during the synthesis process. The computational complexity related to the computational time of each RDGT coefficient or each piece of the reconstructed signal in the proposed parallel algorithms is analyzed and compared with those in the existing major parallel algorithms for the RDGT and its inverse transform. The results indicate that the proposed multirate-based fast parallel algorithms for the RDGT are attractive for real-time signal processing.     

Daugman's Gabor transform as a simple generative back propagation network

Much work has been performed on learning mechanisms for neural networks. A particular area of interest has been the use of neural networks for image processing problems. Two important pieces of work in this area are unified. An architecture and learning scheme for neural networks called generative back propagation has been previously developed and a system for image compression and filtering based on 2-D Gabor transformations which used a neural network type architecture described. Daugman's procedure is exactly replicated, a procedure which used a four layer neural network as a two-layer generative back propagation network with half of the units. The GBP update rule is shown to perform the same change as Daugman's rule, but more efficiently.<>     

On characterization of the optimal biorthogonal window functionsfor Gabor transform

Gabor transforms have been recognized as useful tools in signal analysis. It is known that the solutions for the biorthogonal analysis window function γ(t) given a synthesis window function h(t) in Gabor transforms are not unique in general. Among these solutions, the minimum norm solution has already been given by Wexler and Raz (1990) in the discrete-time case and has been studied by Janssen, Ron and Shen, and Daubechies et al. (1995), in the continuous-time case. The minimum norm solution in the discrete-time case was also proved to be equal to the most orthogonal-like solution by Qian and Chen (1993). In this note, we consider a general optimal-solution problem, where the minimum norm and the most orthogonal-like solutions are two special cases. We prove that these optimal solutions in many cases are equal. We also prove that it remains true in the continuous-time case     

Discrete cosine transform in error control coding

The authors define a new class of real-number linear block codes using the discrete cosine transform (DCT). They also show that a subclass with a BCH-like structure can be defined and, therefore, encoding/decoding algorithms for BCH codes can be applied, A (16,10) DCT code is given as an example     

离散小波变换在医学图像中的应用

本文以三维医学体数据作为研究对象,充分利用离散小波变换,对其进行多分辨率分解,把三维医学体数据分解成三个不同方向上的子医学体数据。实验结果表明,三维医学体数据经过离散小波变换后被分割成八个子频带,其主要能量和特征集中在低频子带系数中,可以用来提取特征向量构造零水印。     

Discrete Hartley transform in error control coding

A new class of real-valued linear code obtained by using the discrete Hartley transform (DHT) is defined. The authors have derived the limitation on the choice of parity frequencies so as to define DHT codes with the cyclic-shift property. Then, by introducing the well-established encoding/decoding algorithm for cyclic codes in error control coding, they have constructed the encoder/decoder for the DHT cyclic codes