Quaternion Fourier Transform Research Articles

Product theorem for quaternion Fourier transform

This paper presents in some detail the quaternion Fourier transform (QFT) of the product of two quaternion functions. It is shown that the proposed product theorem for the QFT is closely related to the convolution in the quaternion Fourier domain.

On Two-Dimensional Quaternion Wigner-Ville Distribution

We present the two-dimensional quaternion Wigner-Ville distribution (QWVD). The transform is constructed by substituting the Fourier transform kernel with the quaternion Fourier transform (QFT) kernel in the classical Wigner-Ville distribution definition. Based on the properties of quaternions and the QFT kernel we obtain three types of the QWVD. We discuss some useful properties of various definitions for the QWVD, which are extensions of the classical Wigner-Ville distribution properties.

Infrared small target enhancement via phase spectrum of Quaternion Fourier Transform

Small target enhancement is one of the crucial stages in infrared small target detection. In this paper, we propose a new method using phase spectrum of Quaternion Fourier Transform to enhance small targets while suppressing backgrounds for infrared images. This is inspired by the property that regularly Gaussian-like shape small targets could be considered as attractively salient signal in infrared images and the location information of such signal is implicitly contained in the phase spectrum from frequency domain. Formally, in the proposed method, we adopt the phase spectrum of Quaternion Fourier Transform instead of using traditional Fourier Transform to enhance the targets since the quaternion provides at most four data channels than only one for the latter, which could be helpful to broad types of background clutters by adding more information. For the construction of the quaternion, we present a second-order directional derivative filter via facet model to compute four second order directional derivative maps from four directions respectively as the four data channels. This filter is used to suppress noises and distinguish the targets and backgrounds into separably different textures so that it would boost the robustness of small target enhancement. In experiments, some typical infrared images with various scenes are tested to validate the effectiveness of the proposed method. The results demonstrate that our method actually has good performance and outperforms several state-of-the-art methods, which can be further used for infrared small target detection and tracking.

Different forms of Plancherel theorem for fractional quaternion Fourier transform

Fractional Fourier transform (FRFT) plays an important role in many field of optics and signal processing. The recently developed concept of two side fractional quaternion Fourier transform (FRQFT) based on quaternion algebra and FRFT has been found useful for signal and image processing. It is a generalization of the quaternion Fourier transform (QFT). Many properties of the FRQFT are already known, but an extension of the QFT's Plancherel theorem is still missing. The purpose of this paper is to introduce extensions of this theorem. Firstly, we establish scalar value Plancherel theorem for the FRQFT. Then, the concept of right side fractional quaternion Fourier transform (FRQFTr) is proposed. Furthermore, we establish quaternion value Plancherel theorem for the FRQFTr. The Plancherel theorems in QFT domain are shown to be special cases of the achieved results.

Attention selection using global topological properties based on pulse coupled neural network

Topological properties are with invariance and take priority over other features, which play an important role in cognition. This paper introduces a new attention selection model called TPA (topological properties-based attention), which adopts topological properties and quaternion. In TPA, using Unit-linking PCNN (Pulse Coupled Neural Network) hole-filter expresses an important topological property (the connectivity) in visual attention selection. Meanwhile, using the quaternion Fourier transform based phase spectrum of an image or a frame in a video obtains the spatio-temporal saliency map, which shows the result of attention selection. Adjusting the weight of a topological channel can change its influence. The experimental results show that TPA reflects the real attention selection more accurately than PQFT (Phase spectrum of Quaternion Fourier Transform).

Relationships between convolution and correlation for Fourier transform and quaternion Fourier transform

In this paper we introduce convolution theorem for the Fourier transform (FT) of two complex functions. We show that the correlation theorem for the FT can be derived using properties of convolution. We develop this idea to derive the correlation theorem for the quaternion Fourier transform (QFT) of the two quaternion functions.

Discrete quaternion Fourier transform and properties

In this paper we discuss the two-dimensional discrete quaternion Fourier transform (DQFT). We derive several properties of the DQFT which correspond to those of the (continuous) quaternion Fourier transform (QFT). Mathematical Subject Classification: 15A66, 42B10, 30G35

Quaternionic Wigner-Ville distribution of analytical signal in hyperspectral imagery

The 2D Quaternionic Fourier Transform (QFT), applied to a real 2D image, produces an invertible quaternionic spectrum. If we conserve uniquely the first quadrant of this spectrum, it is possible, after inverse transformation, to obtain, not the original image, but a 2D quaternion image, which generalize in 2D the classical notion of 1D analytical image. From this quaternion image, we compute the corresponding correlation product, then, by applying the direct QFT, we obtain the 4D Wigner-Ville distribution of this analytical signal. With reference to the shift variables ?1 , ?2 used for the computation of the correlation product, we obtain a local quaternion Wigner-Ville distribution spectrum.

Convolution Theorems for Quaternion Fourier Transform: Properties and Applications

General convolution theorems for two-dimensional quaternion Fourier transforms (QFTs) are presented. It is shown that these theorems are valid not only for real-valued functions but also for quaternion-valued functions. We describe some useful properties of generalized convolutions and compare them with the convolution theorems of the classical Fourier transform. We finally apply the obtained results to study hypoellipticity and to solve the heat equation in quaternion algebra framework.

Color image registration based on quaternion Fourier transformation

The traditional Fourier Mellin transform is applied to quaternion algebra in order to investigate quaternion Fourier transformation properties useful for color image registration in frequency domain. Combining with the quaternion phase correlation, we propose a method for color image registration based on the quaternion Fourier transform. The registration method, which processes color image in a holistic manner, is convenient to realign color images differing in translation, rotation, and scaling. Experimental results on different types of color images indicate that the proposed method not only obtains high accuracy in similarity transform in the image plane but also is computationally efficient.

The Balian-Low Theorem for the Windowed Quaternionic Fourier Transform

In this paper we present the Balian-Low theorem for the twosided windowed quaternionic Fourier transform (WQFT), a theorem which expresses the fact that time-frequency concentrations are incompatible with non-redundancy whenever Gabor systems form orthonormal bases or frames. Since uncertainty principles are closely connected with representations of the kernel of the Fourier transform under consideration, we construct a suitable representation for the kernel of our two-sided WQFT which in turn provides suitable Gabor systems. We proceed by deriving several important properties of the WQFT, such as shift and modulation operators, a reconstruction formula, orthogonality relations and a Heisenberg uncertainty principle for the WQFT. Finally, we establish the Balian-Low theorem for Gabor orthonormal bases associated with discrete versions of the kernels of the WQFT and of the right-sided WQFT.

Bottom-Up Saliency Detection Model Based on Human Visual Sensitivity and Amplitude Spectrum

With the wide applications of saliency information in visual signal processing, many saliency detection methods have been proposed. However, some key characteristics of the human visual system (HVS) are still neglected in building these saliency detection models. In this paper, we propose a new saliency detection model based on the human visual sensitivity and the amplitude spectrum of quaternion Fourier transform (QFT). We use the amplitude spectrum of QFT to represent the color, intensity, and orientation distributions for image patches. The saliency value for each image patch is calculated by not only the differences between the QFT amplitude spectrum of this patch and other patches in the whole image, but also the visual impacts for these differences determined by the human visual sensitivity. The experiment results show that the proposed saliency detection model outperforms the state-of-the-art detection models. In addition, we apply our proposed model in the application of image retargeting and achieve better performance over the conventional algorithms.

Double random-phase encryption based on discrete quaternion fourier-transforms

In this paper, a new method for color-image encryption using Discrete Quaternion Fourier-Transforms (DQFT) combined with double random-phase encryption is reported, by which color images can be processed as single channel architecture, instead of as separated color components in three channels, so that the complexity of the encryption system can be effectively reduced without any reduction of its security. The principle of both encryption and decryption is detailed and the noise robustness of the system has been examined experimentally.

Steganographic Technique for High Volume Secrete Data Transmission through Colour Image (STHVSDTCI)

This paper presents a novel steganographic technique which demonstrates the secrete data transmission technique through colour image in frequency domain based on the Discrete Fourier Transformation (DFT) which pertain to colour image authentication. Secrete transmission is done by hiding secrete message/secrete image into the transformed frequency component of source image. The DFT is applied on sub-image block called mask of size 2 x 2 in row major order. Three secrete message/image bits are fabricated within the transformed real frequency component of each source image byte except the LSB of first frequency component of each mask. After embedding, a delicate readjust phase is incorporated in all the frequency component of each mask, to keep the quantum value positive and non fractional in spatial domain. Robustness is achieved by hiding an authenticating or secretes message/image in the frequency component with both positive and negatives quantum values and invisibility is satisfied in spatial domain using delicate re-adjust phase. Inverse DFT (IDFT) is performed as final steps after embedding to transform embedded image in frequency domain to spatial domain. Experimental results conformed that the proposed algorithm performs better than discrete cosine transformation (DCT), Quaternion Fourier Transformation (QFT) based scheme and Spatio Chromatic DFT (SCDFT). Keywords: QFT, DFT, IDFT, DCT, MD and SCDFT

Windowed Fourier transform of two-dimensional quaternionic signals

In this paper, we generalize the classical windowed Fourier transform (WFT) to quaternion-valued signals, called the quaternionic windowed Fourier transform (QWFT). Using the spectral representation of the quaternionic Fourier transform (QFT), we derive several important properties such as reconstruction formula, reproducing kernel, isometry, and orthogonality relation. Taking the Gaussian function as window function we obtain quaternionic Gabor filters which play the role of coefficient functions when decomposing the signal in the quaternionic Gabor basis. We apply the QWFT properties and the (right-sided) QFT to establish a Heisenberg type uncertainty principle for the QWFT. Finally, we briefly introduce an application of the QWFT to a linear time-varying system.

Directional Uncertainty Principle for Quaternion Fourier Transform

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.

An uncertainty principle for quaternion Fourier transform

We review the quaternionic Fourier transform (QFT). Using the properties of the QFT we establish an uncertainty principle for the right-sided QFT. This uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion-valued signals in the spatial and frequency domains. It is shown that only a Gaussian quaternion signal minimizes the uncertainty.

Relationships Among Various 2-D Quaternion Fourier Transforms

The recently developed quaternion Fourier transform (QFT) based on quaternion algebra has been found useful for color image processing and signal analysis. However, due to the noncommutative property of quaternion algebra, there exist various definitions for the QFT. The purpose of this letter is to establish and present an in-depth discussion on the relationships among the various QFTs.

Doubled random-phase encryption based on discrete quaternion fourier-transforms

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.

Quaternion Fourier Transform on Quaternion Fields and Generalizations

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.