Integral Transforms

This chapter presents the theory and use of the quaternion and Clifford Fourier transforms, quaternion split Fourier transform, and quaternion wavelet transforms. In addition, we will study the quaternion fractal Fourier transform, the quaternion Radon transform, and the quaternion quantum Fourier transform. We will show that using the mathematical system of geometric algebra it is possible to develop different kinds of Clifford Fourier and wavelet transforms, fractal Fourier transforms, Radon transforms, and quantum Fourier transforms very useful for image filtering, pattern recognition, feature detection, image segmentation, texture analysis, and image analysis in frequency and wavelet domains as well as together with Deep Learning in quantum computing. These techniques are fundamental for automated visual inspection, robot guidance, medical image processing, analysis of image sequences, satellite and aerial photogrammetry, as well as quantum computing, big data, and data analytics.