Tensor transform-based quaternion fourier transform algorithm

One-Dimensional Quaternion Fourier Transform with Application to Probability Theory

The Fourier transform occupies a central place in applied mathematics, statistics, computer sciences, and engineering. In this work, we introduce the one-dimensional quaternion Fourier transform, which is a generalization of the Fourier transform. We derive the conjugate symmetry of the one-dimensional quaternion Fourier transform for a real signal. We also collect other properties, such as the derivative and Parseval’s formula. We finally study the application of this transformation in probability theory.

A Variation on Inequality for Quaternion Fourier Transform, Modified Convolution and Correlation Theorems for General Quaternion Linear Canonical Transform

The quaternion linear canonical transform is an important tool in applied mathematics and it is closely related to the quaternion Fourier transform. In this work, using a symmetric form of the two-sided quaternion Fourier transform (QFT), we first derive a variation on the Heisenberg-type uncertainty principle related to this transformation. We then consider the general two-sided quaternion linear canonical transform. It may be considered as an extension of the two-sided quaternion linear canonical transform. Based on an orthogonal plane split, we develop the convolution theorem that associated with the general two-sided quaternion linear canonical transform and then derive its correlation theorem. We finally discuss how to apply general two-sided quaternion linear canonical transform to study the generalized swept-frequency filters.

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 and masked RNN.

Algorithm Development for the Non-Destructive Testing of Structural Damage

Monitoring of structures to identify types of damages that occur under loading is essential in practical applications of civil infrastructure. In this paper, we detect and visualize damage based on several non-destructive testing (NDT) methods. A machine learning (ML) approach based on the Support Vector Machine (SVM) method is developed to prevent misdirection of the event interpretation of what is happening in the material. The objective is to identify cracks in the early stages, to reduce the risk of failure in structures. Theoretical and experimental analyses are derived by computing the performance indicators on the smart aggregate (SA)-based sensor data for concrete and reinforced-concrete (RC) beams. Validity assessment of the proposed indices was addressed through a comparative analysis with traditional SVM. The developed ML algorithms are shown to recognize cracks with a higher accuracy than the traditional SVM. Additionally, we propose different algorithms for microwave- or millimeter-wave imaging of steel plates, composite materials, and metal plates, to identify and visualize cracks. The proposed algorithm for steel plates is based on the gradient magnitude in four directions of an image, and is followed by the edge detection technique. Three algorithms were proposed for each of composite materials and metal plates, and are based on 2D fast Fourier transform (FFT) and hybrid fuzzy c-mean techniques, respectively. The proposed algorithms were able to recognize and visualize the cracking incurred in the structure more efficiently than the traditional techniques. The reported results are expected to be beneficial for NDT-based applications, particularly in civil engineering.