Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization

TL;DR

This paper explores the properties of continuous Fourier extensions of the discrete cosine transform (DCT), highlighting differences from the discrete Fourier transform (DFT) and demonstrating applications in interpolation and image compression.

Contribution

It introduces the continuous extension of DCT (CEDCT), analyzing its approximation, convergence, and locality properties, and compares it with the continuous extension of DFT.

Findings

CEDCT closely approximates the original function between grid points.

The derivative of CEDCT converges to the derivative of the original function as N increases.

CEDCT demonstrates potential for interpolation and data compression in 2D images.

Abstract

A versatile method is described for the practical computation of the discrete Fourier transforms (DFT) of a continuous function $g(t)$ given by its values $g_{j}$ at the points of a uniform grid $F_{N}$ generated by conjugacy classes of elements of finite adjoint order $N$ in the fundamental region $F$ of compact semisimple Lie groups. The present implementation of the method is for the groups SU(2), when $F$ is reduced to a one-dimensional segment, and for $SU(2)\times ... \times SU(2)$ in multidimensional cases. This simplest case turns out to result in a transform known as discrete cosine transform (DCT), which is often considered to be simply a specific type of the standard DFT. Here we show that the DCT is very different from the standard DFT when the properties of the continuous extensions of these two discrete transforms from the discrete grid points $t_j; j=0,1, ... N$ to all points $t \in F$ are considered. (A) Unlike the continuous extension of the DFT, the continuous extension of (the inverse) DCT, called CEDCT, closely approximates $g(t)$ between the grid points $t_j$. (B) For increasing $N$, the derivative of CEDCT converges to the derivative of $g(t)$. And (C), for CEDCT the principle of locality is valid. Finally, we use the continuous extension of 2-dimensional DCT to illustrate its potential for interpolation, as well as for the data compression of 2D images.