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Color image steganography using discrete quaternion Fourier transforms
基于离散四元数傅里叶变换的彩色图像信息隐藏技术

GAI Qi,QIAO Zhao-liang,
盖琦
,乔兆亮

计算机应用研究 , 2011,
Abstract: This paper proposed a new method for color image steganography using discrete quaternion Fourier transforms (DQFT).The host color image could be described as a quaternion matrix and should be transformed to the DQFT domain,the secret image should be encrypted first and then the encrypted color image could be embedded into the DQFT domain of the host image. The principle of embedding was detailed and the robustness of the system had been examined experimentally.Experimental results show that this method is robust against Gaussian noise,salt & pepper noise,cropping and image scaling attack.
Doubled random-phase encryption based on discrete quaternion fourier-transforms
基于离散四元数傅里叶变换的双随机相位加密技术貂

Gai Qi,Wang Ming-Wei,Li Zhi-Lei,Zhai Hong-Chen,
盖琦
,王明伟,李智磊,翟宏琛

物理学报 , 2008,
Abstract: In this paper, a new method for color-image encryption using discrete quaternion fourier-transforms (DQFT) combined with doubled random-phase encryption as used in the optical implementation is proposed, by which color images can be processed as a whole, rather than as separated color components in three channels, so that the complexity of the encryption system can be effectively reduced without any reduction in its security. The principle of both encryption and decryption is detailed and the robustness of the system has been examined experimentally.
Fast complexified quaternion Fourier transform  [PDF]
Salem Said,Nicolas Le Bihan,Stephen J. Sangwine
Mathematics , 2006, DOI: 10.1109/TSP.2007.910477
Abstract: A discrete complexified quaternion Fourier transform is introduced. This is a generalization of the discrete quaternion Fourier transform to the case where either or both of the signal/image and the transform kernel are complex quaternion-valued. It is shown how to compute the transform using four standard complex Fourier transforms and the properties of the transform are briefly discussed.
OPS-QFTs: A new type of quaternion Fourier transforms based on the orthogonal planes split with one or two general pure quaternions  [PDF]
Eckhard Hitzer
Computer Science , 2013, DOI: 10.1063/1.3636721
Abstract: We explain the orthogonal planes split (OPS) of quaternions based on the arbitrary choice of one or two linearly independent pure unit quaternions $f,g$. Next we systematically generalize the quaternionic Fourier transform (QFT) applied to quaternion fields to conform with the OPS determined by $f,g$, or by only one pure unit quaternion $f$, comment on their geometric meaning, and establish inverse transformations. Keywords: Clifford geometric algebra, quaternion geometry, quaternion Fourier transform, inverse Fourier transform, orthogonal planes split
Convolution products for hypercomplex Fourier transforms  [PDF]
Roxana Bujack,Hendrik De Bie,Nele De Schepper,Gerik Scheuermann
Mathematics , 2013,
Abstract: Hypercomplex Fourier transforms are increasingly used in signal processing for the analysis of higher-dimensional signals such as color images. A main stumbling block for further applications, in particular concerning filter design in the Fourier domain, is the lack of a proper convolution theorem. The present paper develops and studies two conceptually new ways to define convolution products for such transforms. As a by-product, convolution theorems are obtained that will enable the development and fast implementation of new filters for quaternionic signals and systems, as well as for their higher dimensional counterparts.
Connecting spatial and frequency domains for the quaternion Fourier transform  [PDF]
Hendrik De Bie,Nele De Schepper,Todd A. Ell,Klaus Rubrecht,Stephen J. Sangwine
Mathematics , 2015,
Abstract: The quaternion Fourier transform (qFT) is an important tool in multi-dimensional data analysis, in particular for the study of color images. An important problem when applying the qFT is the mismatch between the spatial and frequency domains: the convolution of two quaternion signals does not map to the pointwise product of their qFT images. The recently defined `Mustard' convolution behaves nicely in the frequency domain, but complicates the corresponding spatial domain analysis. The present paper analyses in detail the correspondence between classical convolution and the new Mustard convolution. In particular, an expression is derived that allows one to write classical convolution as a finite linear combination of suitable Mustard convolutions. This result is expected to play a major role in the further development of quaternion image processing, as it yields a formula for the qFT spectrum of the classical convolution.
Quaternion Fourier Transform on Quaternion Fields and Generalizations  [PDF]
Eckhard Hitzer
Computer Science , 2013, DOI: 10.1007/s00006-007-0037-8
Abstract: We treat the quaternionic Fourier transform (QFT) applied to quaternion fields and investigate QFT properties useful for applications. Different forms of the QFT lead us to different Plancherel theorems. We relate the QFT computation for quaternion fields to the QFT of real signals. We research the general linear ($GL$) transformation behavior of the QFT with matrices, Clifford geometric algebra and with examples. We finally arrive at wide-ranging non-commutative multivector FT generalizations of the QFT. Examples given are new volume-time and spacetime algebra Fourier transformations.
Spectrum of Signals on the Quaternion Fourier Transform Domain  [PDF]
Guicheng Yang, Yingxiong Fu
Journal of Applied Mathematics and Physics (JAMP) , 2013, DOI: 10.4236/jamp.2013.14007
Abstract:

The quaternion Fourier transform plays a vital role in the representation of two-dimensional signals. This paper characterizes spectrum of quaternion-valued signals on the quaternion Fourier transform domain by the partial derivative.

Directional Uncertainty Principle for Quaternion Fourier Transform  [PDF]
Eckhard Hitzer
Mathematics , 2013, DOI: 10.1007/s00006-009-0175-2
Abstract: This paper derives a new directional uncertainty principle for quaternion valued functions subject to the quaternion Fourier transformation. This can be generalized to establish directional uncertainty principles in Clifford geometric algebras with quaternion subalgebras. We demonstrate this with the example of a directional spacetime algebra function uncertainty principle related to multivector wave packets.
On the positivity of Fourier transforms  [PDF]
Bertrand G. Giraud,Robi Peschanski
Physics , 2014,
Abstract: Characterizing in a constructive way the set of real functions whose Fourier transforms are positive appears to be yet an open problem. Some sufficient conditions are known but they are far from being exhaustive. We propose two constructive sets of necessary conditions for positivity of the Fourier transforms and test their ability of constraining the positivity domain. One uses analytic continuation and Jensen inequalities and the other deals with Toeplitz determinants and the Bochner theorem. Applications are discussed, including the extension to the two-dimensional Fourier-Bessel transform and the problem of positive reciprocity, i.e. positive functions with positive transforms.
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