THE MATHEMATICAL FOUNDATION OF IMAGE PROCESSING USES THE QUANTUM QUATERNION FOURIER INVERSION TRANSFORM

Abstract

The Fourier transform is a popular tool in many areas of practical mathematics, including data compression and signal processing. The underlying principle of the Fourier transform is that functions with the right characteristics may be expressed as a linear combination of trigonometric functions. Decomposing a signal using trigonometric functions is like taking a time-domain signal and separating its frequency-domain components; this opens up new and more efficient avenues for signal analysis and manipulation. An increasing number of people are looking at quantum image processing (QIP) as a way to enhance the efficiency of conventional methods and their applications by taking use of quantum computing's characteristics. Filtering and rectifying the frequency domain to get important picture information is the goal of both image design and interactive processing in this procedure. We demonstrate the operation of quantum operators and states in the context of quaternions and demonstrate how it may be used to expand the quantum complex Fourier transform (QCFT) to the quantum quaternion Fourier transform (QQFT). We demonstrate why convolution cannot be used to quantum image processing, and we create space-domain and quaternion-Fourier spectrum filters, including an edge detector and a quantum median filter. An enhanced method is used to provide the groundwork for the realization of picture design and interactive technology, drawing on the knowledge of FT and inverse transform.