快速超复数傅氏变换和超复数互相关的新算法及应用

超复数把彩色图像作为一个矢量整体进行处理,与传统算法相比,它能更好地描述图像的色彩关联,超复数互相关已广泛应用到彩色图像处理的各个领域.本文首先分析和介绍了目前的实现二维超复数傅氏变换和超复数互相关的快速方法,然后通过把超复数按实部和各个虚部展开,分别进行传统的快速傅氏变换,再把对应的单位虚向量还原,从而为超复数傅氏变换和超复数互相关提出了一种新的快速算法.分析表明:本文提出的方法比目前现有的方法更简单易行,且计算量更小.最后,本文介绍了我们把超复数互相关技术应用到彩色目标跟踪上获得的一些新结果.     

彩色图像超复数空间的自适应水印算法

 针对彩色载体图像,本文提出了一种超复数频域的自适应水印算法,它首先对彩色载体图像进行快速超复数傅氏变换,在超复数频谱实部选择合适频段的基础上,再利用人类视觉系统对彩色载体图像的纹理、边缘和亮度的掩蔽特性,对选择的频段赋予不同的掩蔽强度而嵌入水印,从而在超复数频域内实现了一种彩色载体图像自适应的水印算法.实验结果表明,通过彩色图像的自适应掩蔽,大大提高了超复数频域水印算法的不易感知性和鲁棒性,且抗攻击性能也优于无自适应掩蔽的超复数频域水印算法;和现有文献的比较结果验证了本算法的这些优越性.     

Estimation of Motions in Color Image Sequences Using Hypercomplex Fourier Transforms

Although the motion estimation problem has been extensively studied, most of the proposed estimation approaches deal mainly with monochrome videos. The most usual way to apply them also in color image sequences is to process each color channel separately. A different, more sophisticated approach is to process the color channels in a “holistic” manner using quaternions, as proposed by Ell and Sangwine. In this paper, we extend standard spatiotemporal Fourier-based approaches to handle color image sequences, using the hypercomplex Fourier transform. We show that translational motions are manifested as energy concentration along planes in the hypercomplex 3-D Fourier domain and we describe a methodology to estimate the motions, based on this property. Furthermore, we compare the three-channels-separately approach with our approach and we show that the computational effort can be reduced by a factor of 1/3, using the hypercomplex Fourier transform. Also, we propose a simple, accompanying method to extract the moving objects in the hypercomplex Fourier domain. Our experimental results on synthetic and natural images verify our arguments throughout the paper.      

Fractional Fourier transforms of hypercomplex signals

An overview is given to a new approach for obtaining generalized Fourier transforms in the context of hypercomplex analysis (or Clifford analysis). These transforms are applicable to higher-dimensional signals with several components and are different from the classical Fourier transform in that they mix the components of the signal. Subsequently, attention is focused on the special case of the so-called Clifford-Fourier transform where recently a lot of progress has been made. A fractional version of this transform is introduced and a series expansion for its integral kernel is obtained. For the case of dimension 2, also an explicit expression for the kernel is given.     

Robust multiple color images encryption using discrete Fourier transforms and chaotic map

Based on discrete Fourier transforms and logistic-exponent-sine map, this paper investigates an encryption algorithm for multiple color images. In the encryption process, each color image represented in trinion matrix is performed by block-wise discrete trinion Fourier transforms. Then the first real matrix is constructed by splicing real and imaginary parts of transformed results. Followed by two-dimensional discrete Fourier transform, the second real matrix is synthesized only using half of the spectrum on the basis of the conjugate symmetry property. In order to further enhance the randomness of interim result, the random multi-resolution singular value decomposition is exploited. Afterwards, a sharing process is utilized to get final cipherimages. Numerical simulations performed on 300 color images have shown that quality of correctly decrypted images is much better, where the PSNR value is up to 305.03 dB. The number of changing pixel rate and unified average changing intensity are respectively greater than 99.99% and 33.33%, indicating good sensitivity. The comparison with other methods under noise and cropping attacks validates the reliability of the proposed algorithm.     

Envelope detection using generalized analytic signal in 2D QLCT domains

The hypercomplex 2D analytic signal has been proposed by several authors with applications in color image processing. The analytic signal enables to extract local features from images. It has the fundamental property of splitting the identity, meaning that it separates qualitative and quantitative information of an image in form of the local phase and the local amplitude. The extension of analytic signal of linear canonical transform domain from 1D to 2D, covering also intrinsic 2D structures, has been proposed. We use this improved concept on envelope detector. The quaternion Fourier transform plays a vital role in the representation of multidimensional signals. The quaternion linear canonical transform (QLCT) is a well-known generalization of the quaternion Fourier transform. Some valuable properties of the two-sided QLCT are studied. Different approaches to the 2D quaternion Hilbert transforms are proposed that allow the calculation of the associated analytic signals, which can suppress the negative frequency components in the QLCT domains. As an application, examples of envelope detection demonstrate the effectiveness of our approach.     

结合Fourier变换对称性和随机多分辨率奇异值分解的彩色图像加密

提出一种基于Fourier变换对称性和随机多分辨率奇异值分解(R-MRSVD)的彩色图像加密算法。首先计算归一化明文图像的平均值作为logistic-exponent-sine映射的初值,并生成随机矩阵和位置索引;然后对每个颜色通道分别进行二维离散Fourier变换,根据共轭对称性仅保留一半的频谱系数,并提取实部分量和虚部分量构建实数矩阵;最后对实数矩阵进行R-MRSVD和Josephus置乱操作,得到密文图像。将明文图像的像素特征作为混沌序列的初值,保证算法具有高敏感性和高安全性,同时实值的密文便于存储和传输。对算法的解密图像质量、统计特性、密钥敏感性、抗选择明文攻击、鲁棒性等性能进行测试,仿真结果表明,所提加密算法具有可行性和安全性。     

Vector phase correlation

Phase correlation has been applied for over 25 years to applications such as registration of images, but there has never been a simple generalisation to vector images. Here, for the first time, a vector approach to phase correlation, based on hypercomplex Fourier transforms, is proposed and demonstrated     

斜坡类信号频谱分析时应注意的一个问题

斜坡类信号属功率无限信号,针对常见于各种复变函数与积分变换类教科书中此类信号的傅里叶变换结果提出了质疑,认为以傅里叶变换所体现的此类信号的频谱特性不具有实际意义或应用背景。通过对傅里叶变换基本定义、运算性质、物理意义以及斜坡信号频谱特性的分析,表明在信号频谱分析过程中,傅里叶变换这一频谱分析工具不一定适用于功率无限信号。     

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.     

Novel adaptive color space transform and application to image compression

Various contemporary standards by Joint Picture Expert Group, which is used for compression, exploited the correlation among the color components using a component color space transform before the subband transform stage. The transforms used to de-correlate the colors are primarily the fixed kernel transforms, which are not suitable for large class of images. In this paper an image dependent color space transform (ID-CCT), exploiting the inter-channel redundancy optimally and which is very much suitable for compression has been proposed. Also the comparative performance has been evaluated and a significant improvement has been observed, objectively as well as subjectively over other quantifiable methods.     

Linear, Worst-Case Estimators for Denoising Quantization Noise in Transform Coded Images

Transform-coded images exhibit distortions that fall outside of the assumptions of traditional denoising techniques. In this paper, we use tools from robust signal processing to construct linear, worst-case estimators for the denoising of transform compressed images. We show that while standard denoising is fundamentally determined by statistical models for images alone, the distortions induced by transform coding are heavily dependent on the structure of the transform used. Our method, thus, uses simple models for the image and for the quantization error, with the latter capturing the transform dependency. Based on these models, we derive optimal, linear estimators of the original image that are optimal in the mean-squared error sense for the worst-case cross correlation between the original and the quantization error. Our construction is transform agnostic and is applicable to transforms from block discrete cosine transforms to wavelets. Furthermore, our approach is applicable to different types of image statistics and can also serve as an optimization tool for the design of transforms/quantizers. Through the interaction of the source and quantizer models, our work provides useful insights and is instrumental in identifying and removing quantization artifacts from general signals coded with general transforms. As we decouple the modeling and processing steps, we allow for the construction of many different types of estimators depending on the desired sophistication and available computational complexity. In the low end of this spectrum, our lookup table based estimator, which can be deployed in low complexity environments, provides competitive PSNR values with some of the best results in the literature.     

2-D and 1-D multipaired transforms: frequency-time type wavelets

A concept of multipaired unitary transforms is introduced. These kinds of transforms reveal the mathematical structure of Fourier transforms and can be considered intermediate unitary transforms when transferring processed data from the original real space of signals to the complex or frequency space of their images. Considering paired transforms, we analyze simultaneously the splitting of the multidimensional Fourier transform as well as the presentation of the processed multidimensional signal in the form of the short one-dimensional (1-D) “signals”, that determine such splitting. The main properties of the orthogonal system of paired functions are described, and the matrix decompositions of the Fourier and Hadamard transforms via the paired transforms are given. The multiplicative complexity of the two-dimensional (2-D) 2^r×2^r-point discrete Fourier transform by the paired transforms is 4^r/2(r-7/3)+8/3-12 (r>3), which shows the maximum splitting of the 5-D Fourier transform into the number of the short 1-D Fourier transforms. The 2-D paired transforms are not separable and represent themselves as frequency-time type wavelets for which two parameters are united: frequency and time. The decomposition of the signal is performed in a way that is different from the traditional Haar system of functions     

A new numerical Fourier transform in d-dimensions

The classical method of numerically computing Fourier transforms of digitized functions in one or in d-dimensions is the so-called discrete Fourier transform (DFT) efficiently implemented as fast Fourier transform (FFT) algorithms. In many cases, the DFT is not an adequate approximation to the continuous Fourier transform, and because the DFT is periodical, spectrum aliasing may occur. The method presented in this contribution provides accurate approximations of the continuous Fourier transform with similar time complexity. The assumption of signal periodicity is no longer posed and allows the computation of numerical Fourier transforms in a broader domain of frequency than the usual half-period of the DFT. In addition, this method yields accurate numerical derivatives of any order and polynomial splines of any odd degree. The numerical error on results is easily estimated. The method is developed in one and in d dimensions, and numerical examples are presented.     

一种基于DWT-DCT变换域的全息水印技术

在研究傅里叶全息技术基础上,提出了一种结合离 散小波变换和离散波变换(DWT-DCT)变换的彩色图像数字水印算法,使得水印 拥有强鲁棒性和不可见性。首先,对水印信息进行傅里叶变换,并生成全息水印图; 其次,将载体图 像在RGB空间内的蓝色B分量进行小波分解,对其低频部分进行DCT;最后, 将水印全息图嵌入其高频系数 中。实验结果发现,以k=0.05的强度嵌入水印后,含水印图像的峰值 信噪比(PSNR)达到47,能有效抵抗 信号处理和裁切攻击,同时提取出的水印和原始水印的相似性(NC值)能保持在0. 75以上,说明本文水印 算法有较强的鲁棒性和不可见性。本文算法能够抵抗裁切、滤波、噪声和JPEG压缩攻击,可 用于数字图像版权保护方面。     

Content protection in grayscale and color images based on robust digital watermarking

The concept of digital rights management architecture for multimedia will be presented in this paper, with focus on robust digital image watermarking, which is able to undergo geometrical attacks performed against the embedded watermarks. Geometrical attacks are still an open problem for many digital watermarking algorithms used in present time. Most of geometrical attacks can be described by using affine transforms. A new approach to improve robustness against geometrical attacks is presented. The discrete Fourier transform and log-polar mapping are used for watermark embedding and for watermark detection. Some attacks against the embedded watermarks are performed and the results are given. Also two modifications, usage of the proposed method as a synchronization template and implementation for the color images, are presented and experimental results are given too.     

Discrete fractional Fourier transform based on orthogonalprojections

The continuous fractional Fourier transform (FRFT) performs a spectrum rotation of signal in the time-frequency plane, and it becomes an important tool for time-varying signal analysis. A discrete fractional Fourier transform has been developed by Santhanam and McClellan (see ibid., vol.42, p.994-98, 1996) but its results do not match those of the corresponding continuous fractional Fourier transforms. We propose a new discrete fractional Fourier transform (DFRFT). The new DFRFT has DFT Hermite eigenvectors and retains the eigenvalue-eigenfunction relation as a continous FRFT. To obtain DFT Hermite eigenvectors, two orthogonal projection methods are introduced. Thus, the new DFRFT will provide similar transform and rotational properties as those of continuous fractional Fourier transforms. Moreover, the relationship between FRFT and the proposed DFRFT has been established in the same way as the conventional DFT-to-continuous-Fourier transform     

单透镜成象系统的分数维傅立叶变换分析

在菲涅耳衍射与分数维傅立叶变换关系的理论基础上 ,给出了这一关系的另一表示方式 ,以此理论和透镜的分数维傅立叶变换特性分析了放大率大于 1和小于 1的单透镜成象系统。结果表明 ,作为菲涅耳衍射的分数维傅立叶变换可以与透镜的分数维傅立叶变换组成分数维傅立叶变换群 ,两次分数维傅立叶变换完成了一次成象过程     

Parallel vector processing of multidimensional orthogonal transforms for digital signal processing applications

This paper presents vector and parallel algorithms and implementations of one- and two-dimensional orthogonal transforms. The speed performances are evaluated on Cray X-MP/48 vector computer. The sinusoidal orthogonal transforms are computed using fast real Fourier transform (FFT) kernel. The non-sinusoidal orthogonal transform algorithms are derived by using direct factorizations of transform matrices. Concurrent processing is achieved by using the multitasking capability of Cray X-MP/48 to transform long data vectors and two-dimensional data vectors. The discrete orthogonal transforms discussed in this paper include: Fourier transform (DFT), cosine transform (DCT), sine transform (DST), Hartley transform (DHT), Walsh transform (DWHT) and Hadamard transform (DHDT). The factors affecting the speedup of vector and parallel processing of these transforms are considered. The vectorization techniques are illustrated by an FFT example.This work is supported in part by the National Science Foundation, Pittsburgh Supercomputing Center (grant number ECS-880012P) and by the PEW Science Education Program.     

A General Description of Linear Time-Frequency Transforms and Formulation of a Fast, Invertible Transform That Samples the Continuous S-Transform Spectrum Nonredundantly

Examining the frequency content of signals is critical in many applications, from neuroscience to astronomy. Many techniques have been proposed to accomplish this. One of these, the S-transform, provides simultaneous time and frequency information similar to the wavelet transform, but uses sinusoidal basis functions to produce frequency and globally referenced phase measurements. It has shown promise in many medical imaging applications but has high computational requirements. This paper presents a general transform that describes Fourier-family transforms, including the Fourier, short-time Fourier, and S- transforms. A discrete, nonredundant formulation of this transform, as well as algorithms for calculating the forward and inverse transforms are also developed. These utilize efficient sampling of the time-frequency plane and have the same computational complexity as the fast Fourier transform. When configured appropriately, this new algorithm samples the continuous S-transform spectrum efficiently and nonredundantly, allowing signals to be transformed in milliseconds rather than days, as compared to the original S-transform algorithm. The new and efficient algorithms make practical many existing signal and image processing techniques, both in biomedical and other applications.