Quantum spectral analysis: frequency in time, with applications to signal and image processing

TL;DR

This paper introduces quantum spectral analysis (QSA), a novel method based on the Schrödinger equation, offering advantages over Fourier-based techniques for signal and image processing, including better energy handling and edge detection.

Contribution

The work presents quantum spectral analysis as a new approach, providing a complementary and advantageous alternative to traditional Fourier-based spectral methods in DSP and DIP.

Findings

QSA offers low computational complexity (O(N) for signals)

It enables edge detection, denoising, and superresolution in images

QSA has no phase uncertainties unlike Fourier methods

Abstract

A quantum time-dependent spectrum analysis, or simply, quantum spectral analysis (QSA) is presented in this work, and it is based on Schrodinger equation, which is a partial differential equation that describes how the quantum state of a non-relativistic physical system changes with time. In classic world is named frequency in time (FIT), which is presented here in opposition and as a complement of traditional spectral analysis frequency-dependent based on Fourier theory. Besides, FIT is a metric, which assesses the impact of the flanks of a signal on its frequency spectrum, which is not taken into account by Fourier theory and even less in real time. Even more, and unlike all derived tools from Fourier Theory (i.e., continuous, discrete, fast, short-time, fractional and quantum Fourier Transform, as well as, Gabor) FIT has the following advantages: a) compact support with excellent energy output treatment, b) low computational cost, O(N) for signals and O(N2) for images, c) it does not have phase uncertainties (indeterminate phase for magnitude = 0) as Discrete and Fast Fourier Transform (DFT, FFT, respectively), d) among others. In fact, FIT constitutes one side of a triangle (which from now on is closed) and it consists of the original signal in time, spectral analysis based on Fourier Theory and FIT. Thus a toolbox is completed, which it is essential for all applications of Digital Signal Processing (DSP) and Digital Image Processing (DIP); and, even, in the latter, FIT allows edge detection (which is called flank detection in case of signals), denoising, despeckling, compression, and superresolution of still images. Such applications include signals intelligence and imagery intelligence. On the other hand, we will present other DIP tools, which are also derived from the Schrodinger equation.