Two-sided fractional quaternion Fourier transform and its application

In this paper, we introduce the two-sided fractional quaternion Fourier transform (FrQFT) and give some properties of it. The main results of this paper are divided into three parts. Firstly we give a definition of the FrQFT. Secondly based on properties of the two-sided QFT, we study the relationship between the two-sided QFT and the two-sided FrQFT, and give some differential properties of the two-sided FrQFT and the Parseval identity. Finally, we give an example to illustrate the application of the two-sided FrQFT and its inverse transform in solving partial differential equations.

Color image watermarking using multidimensional Fourier transforms

This thesis presents two vector watermarking schemes that are based on the use of complex and quaternion Fourier transforms and demonstrates, for the first time, how to embed watermarks into the coefficients consistent with our human visual systems (HVS). Watermark casting is performed by estimating the Just-Noticeable distortion (JND) of the images, to ensure watermark invisibility. The first method encodes the chromatic content of a color image as CIE a*b* chromaticity coordinates whereas the achromatic content is encoded as CIE L tristimulus value. Color watermarks (yellow and blue) are embedded in the frequency domain of the chromatic channels by using Spatio Chromatic Discrete Fourier Transform (SCDFT). It first encodes a* and b* as complex values, followed by a single discrete Fourier Transform. The most interesting characteristic of the scheme is the possibility of performing watermarking in the frequency domain of chromatic components. The second method encodes the L*a*b* components of color images and color watermarks are embedded as vectors in the frequency domain of the channels by using the Quaternion Fourier Transform (QFT). The idea is twofold: Robustness is achieved by embedding a color watermark in the coefficient with positive frequency, which spreads it to all components in the spatial domain. On the other hand, invisibility is satisfied by modifying the coefficient with negative frequency, such that the combined effects of the two are insensitive to human eyes

Color image watermarking using multidimensional Fourier transforms

This thesis presents two vector watermarking schemes that are based on the use of complex and quaternion Fourier transforms and demonstrates, for the first time, how to embed watermarks into the coefficients consistent with our human visual systems (HVS). Watermark casting is performed by estimating the Just-Noticeable distortion (JND) of the images, to ensure watermark invisibility. The first method encodes the chromatic content of a color image as CIE a*b* chromaticity coordinates whereas the achromatic content is encoded as CIE L tristimulus value. Color watermarks (yellow and blue) are embedded in the frequency domain of the chromatic channels by using Spatio Chromatic Discrete Fourier Transform (SCDFT). It first encodes a* and b* as complex values, followed by a single discrete Fourier Transform. The most interesting characteristic of the scheme is the possibility of performing watermarking in the frequency domain of chromatic components. The second method encodes the L*a*b* components of color images and color watermarks are embedded as vectors in the frequency domain of the channels by using the Quaternion Fourier Transform (QFT). The idea is twofold: Robustness is achieved by embedding a color watermark in the coefficient with positive frequency, which spreads it to all components in the spatial domain. On the other hand, invisibility is satisfied by modifying the coefficient with negative frequency, such that the combined effects of the two are insensitive to human eyes

Deep quaternion fourier transform for salient object detection

Salient object detection plays a vital role in image processing applications like image retrieval, security and surveillance in authentic-time. In recent times, advances in deep neural network gained more attention in the automatic learning system for various computer vision applications. In order to decrement the detection error for efficacious object detection, we proposed a detection classifier to detect the features of the object utilizing a deep neural network called convolutional neural network (CNN) and discrete quaternion Fourier transform (DQFT). Prior to CNN, the image is pre-processed by DQFT in order to handle all the three colors holistically to evade loss of image information, which in-turn increase the effective use of object detection. The features of the image are learned by training model of CNN, where the CNN process is done in the Fourier domain to quicken the method in productive computational time, and the image is converted to spatial domain before processing the fully connected layer. The proposed model is implemented in the HDA and INRIA benchmark datasets. The outcome shows that convolution in the quaternion Fourier domain expedite the process of evaluation with amended detection rate. The comparative study is done with CNN, discrete Fourier transforms CNN, RNN ad masked RNN.

Duality Property of Discrete Quaternion Fourier Transform

We introduce the discrete quaternionic Fourier transform (QDFT), which is generalization of discrete Fourier transform. We establish the version discrete of duality property duality related to the QDFT.