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.

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.

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.

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

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.

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.

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.

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.

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.

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.