图像处理中的正交变换探讨

正交变换是一类非常重要的变换,其具有使变换前后图像能量保持不变的特性.图像的正交变换是图像处理技术的重要工具,被广泛地运用于图像特征提取、图像增强、图像复原、图像压缩和图像识别等领域.首先,论述了正交变换的定义及编码原理;其次,对正交变换中的傅立时变换和离散余弦变换的基本概念、性质、算法以及在图像处理中的应用等进行了详细的叙述;最后,利用Madab和C++编程,实现了快速离散傅立叶变换和离散余弦变换,并对两种变换结果的优劣作了全面的比较.     

一族实对称二维离散正交变换及其快速算法

本文提出了一族实对称二维离散正交变换,以二维DHART作为它的特例。文章解析了这类变换的主要性质和它跟二维DFT之间的关系,(?)二维DHART的快速算法扩展到整族变换上,并且讨论了其应用。     

基于哈尔正交函数系的抗裁剪鲁棒水印算法

提出一种基于哈尔正交函数系的抗裁剪鲁棒水印算法，该算法根据哈尔正交函数系的完备归一化正交性质，对图像进行分块哈尔正交变换，根据图像视觉系统特性和哈尔正交变换性质，提取重要的中频系数，并结合零水印嵌入技术，将水印自适应地嵌入哈尔正交变换的中频矩阵，从而提高了水印算法的有效性，对裁剪攻击具有很强的抗攻击性，实验表明：该算法具有良好的鲁棒性和有效性。     

关于Chrestenson正交变换的讨论

本文首先给出了三类新的Chrestenson正交变换；其次说明了定义在Z_p~n上的复正交函数系的实部与虚部之和能构成一个实正交函数系的充要条件，从而给出了三类新的张—哈特莱（Hartley）正交变换；最后，通过Chrestenson变换和张—哈特莱变换的关系，说明了张—哈特莱变换可用来刻划逻辑函数的某些特征。     

基于全相位反余弦双正交变换的JPEG图像压缩技术

提出了一种新型的基于全相位反余弦双正交变换(APIDCBT)的JPEG图像压缩技术,APIDCBT矩阵是一种新型的变换矩阵,给出它的具体形式,提出了双正交变换与对偶双正交基向量的概念.通过与传统的离散余弦变换(DCT)进行比较,表明了APIDCBT算法可以不用量化表,节省运算时间,与DCT可达到同样的压缩效果,并且在低码率情况下优于DCT.     

基于DTFT的正弦波频率估计方法研究

本文提出了一种利用DTFT变换获得信号的（离散时间傅里叶变换）幅频特性来进行频率估计的方法。文章论证了算法原理,介绍了关键步骤（正交变换）的实现方法,并与同类方法比较给出了若干仿真结果,说明了本算法的优势及局限。     

H.263编码中DCT在定点DSP上的实现

简要介绍了H.263编码标准及H.263编码采用的正交变换编码离散余弦变换。文中着重讨论了DCT算法的定点化,并根据TMS320C6201DSP的特点对IDCT的算法进行了改进。最后采用DSP汇编语言实现DCT快速算法。     

用矩阵的H积构造的一种新的离散正交变换

本文应用文献[1]中所定义的矩阵的H积运算,提出了一种新的离散正交变换,称之为PH变换,讨论了它的部分性质,并给出了快速算法。这种变换运算次数少(对N点变换来说,仅需5N-4(log_2N 1)次加法,乘法次数也不超过N次)且具有循环移位下,功率谱不变的性质。     

基于自适应重迭正交变换的音频信号编码

本文提出了一种基于自适应重迭正交变换的音频信号编码算法，讨论了变换系数的量化及自比特分配问题。实验结果表明该方法的编码质量优于传统的离散余弦变换，没有分块效应，而计算量则相当。     

任意长离散余弦变换的快速递归算法

该文基于Clenshaw递归公式以及离散余弦自身的对称性提出任意长离散余弦变换(DCT)的一种并行递归快速算法,给出了该算法的滤波器实现结构;与现有的其它递归算法以及基于算术傅里叶变换的余弦变换算法进行了计算复杂度的比较分析,结果表明该文算法运算量大大减少。该递归计算的滤波器结构使算法非常适合大规模集成电路(VLSI)实现。     

Complementary sets of sequences

A set of equally long finite sequences, the elements of which are either + 1 or - 1, is said to be a complementary set of sequences if the sum of autocorrelation functions of the sequences in that set is zero except for a zero-shift term. A complementary set of sequences is said to be a mate of another set if the sum of the cross-correlation functions of the corresponding sequences in these two sets is zero everywhere. Complementary sets of sequences are said to be mutually orthogonal complementary sets if any two of them are mates to each other. In this paper we discuss the properties of such complementary sets of sequences. Algorithms for synthesizing new sets from a given set are given. Recursive formulas for constructing mutually orthogonal complementary sets are presented. It is shown that matrices consisting of mutually orthogonal complementary sets of sequences can be used as operators so as to per form transformations and inverse transformations on a one- or two-dimensional array of real time or spatial functions. The similarity between such new transformations and the Hadamard transformation suggests applications of such new transformations to signal processing and image coding.     

Multichannel transforms for signal/image processing

This paper presents a novel approach to the Fourier analysis of multichannel time series. Orthogonal matrix functions are introduced and are used in the definition of multichannel Fourier series of continuous-time periodic multichannel functions. Orthogonal transforms are proposed for discrete-time multichannel signals as well. It is proven that the orthogonal matrix functions are related to unitary transforms (e.g., discrete Hartley transform (DHT), Walsh-Hadamard transform), which are used for single-channel signal transformations. The discrete-time one-dimensional multichannel transforms proposed in this paper are related to two-dimensional single-channel transforms, notably to the discrete Fourier transform (DFT) and to the DHT. Therefore, fast algorithms for their computation can be easily constructed. Simulations on the use of discrete multichannel transforms on color image compression have also been performed.     

Closed-Form Orthogonal DFT Eigenvectors Generated by Complete Generalized Legendre Sequence

In this paper, we propose a new method for deriving the closed-form orthogonal discrete Fourier transform (DFT) eigenvectors of arbitrary length using the complete generalized Legendre sequence (CGLS). From the eigenvectors, we then develop a novel method for computing the DFT. By taking a specific eigendecomposition to the DFT matrix, after proper arrangement, we can derive a new fast DFT algorithm with systematic construction of an arbitrary length that reduces the number of multiplications needed as compared with the existing fast algorithm. Moreover, we can also use the proposed CGLS-like DFT eigenvectors to define a new type of the discrete fractional Fourier transform, which is efficient in implementation and effective for encryption and OFDM.      

Quadratic optimization for simultaneous matrix diagonalization

Simultaneous diagonalization of a set of matrices is a technique that has numerous applications in statistical signal processing and multivariate statistics. Although objective functions in a least-squares sense can be easily formulated, their minimization is not trivial, because constraints and fourth-order terms are usually involved. Most known optimization algorithms are, therefore, subject to certain restrictions on the class of problems: orthogonal transformations, sets of symmetric, Hermitian or positive definite matrices, to name a few. In this paper, we present a new algorithm called QDIAG that splits the overall optimization problem into a sequence of simpler second order subproblems. There are no restrictions imposed on the transformation matrix, which may be nonorthogonal, indefinite, or even rectangular, and there are no restrictions regarding the symmetry and definiteness of the matrices to be diagonalized, except for one of them. We apply the new method to second-order blind source separation and show that the algorithm converges fast and reliably. It allows for an implementation with a complexity independent of the number of matrices and, therefore, is particularly suitable for problems dealing with large sets of matrices.     

A two-dimensional fast lattice recursive least squares algorithm

This paper is mainly devoted to the derivation of a new two-dimensional fast lattice recursive least squares (2D FLRLS) algorithm. This algorithm updates the filter coefficients in growing-order form with linear computational complexity. After appropriately defining the “order” of 2D data and exploiting the relation with 1D multichannel, “order” recursion relations and shift invariance property are derived. The geometrical approaches of the vector space and the orthogonal projection then can be used for solving this 2D prediction problem. We examine the performances of this new algorithm in comparison with other fast algorithms     

Iterative maximum entropy power spectrum estimation for multirate systems

In multirate systems, observations are generally insufficient to determine the power spectrum of the input signal. In this paper, we reformulate the problem using a novel matrix notation and the discrete entropy function. Then we present an iterative maximum entropy power spectrum estimation algorithm for the solution of this problem. Contrary to the existing solutions, the new algorithm is computationally efficient since it is based on fast Fourier transform (FFT) and simple matrix calculations. Furthermore, simulation results show that the new algorithm converges to the maximum entropy solution and can be successfully used in multirate statistical data estimation.     

平行多视点算法实时求取深度信息

引入了平行多视点的概念,从平行多视点获取的多幅图象中可以求得深度信息。通过利用图象的FFT变换,将时域中很难解决的问题变换到频域中来解决,避免了复杂的匹配运算,极大地简化了求取图象点、面深度信息的问题,并且提高了深度求取的实时性、全面性和准确性。介绍了算法原理及特点,并给出了实验结果及分析。     

Geometric transformations on the hexagonal grid

The hexagonal grid has long been known to be superior to the more traditional rectangular grid system in many aspects in image processing and machine vision related fields. However, systematic developments of the mathematical backgrounds for the hexagonal grid are conspicuously lacking. The purpose of this paper is to study geometric transformations on the hexagonal grid. Formulations of the transformation matrices are carried out in a symmetrical hexagonal coordinate frame. A trio of new trigonometric functions are defined in this paper to facilitate the rotation transformations. A fast algorithm for rounding an arbitrary point to the nearest hexagonal grid point is also presented.     

Fast Hartley transform algorithm

The discrete Hartley transform is a new tool for the analysis, design and implementation of digital signal processing algorithms and systems. It is strictly symmetrical concerning the transformation and its inverse. A new fast Hartley transform algorithm has been developed. Applied to real signals, it is faster than a real fast Fourier transform, especially in the case of the inverse transformation. The speed of operation for a fast convolution can thus be increased.     

Fast 2-D distance transformations

The performance of a number of image processing methods depends on the output quality of a distance transformation (DT) process. Most of the fast DT methodologies are not accurate, whereas other error-free DT algorithms are not very fast. In this paper, a novel, fast, simple, and error-free DT algorithm is presented. By recording the relative x- and y-coordinates of the examined image pixels, an optimal algorithm can be developed to achieve the DT of an image correctly and efficiently in constant time without any iteration. Furthermore, the proposed method is general since it can be used by any kind of distance function, leading to accurate image DTs.