A General Method for Generating Discrete Orthogonal Matrices

TL;DR

This paper introduces a versatile, direct method for generating discrete orthogonal matrices using polynomial construction, eliminating reliance on continuous functions and enabling efficient creation of known and novel transforms for applications like coding and cryptography.

Contribution

The paper presents a general, constructive approach for generating discrete orthogonal matrices directly from polynomial construction, bypassing the need for continuous orthogonal functions.

Findings

Method simplifies the generation of discrete orthogonal matrices.

Demonstrates how to derive Discrete Cosine and Tchebichef Transforms using the method.

Shows how to create new orthogonal matrices from arbitrary values.

Abstract

Discrete orthogonal matrices have several applications in information technology, such as in coding and cryptography. It is often challenging to generate discrete orthogonal matrices. A common approach widely in use is to discretize continuous orthogonal functions that have been discovered. The need of certain continuous functions is restrictive. To simplify the process while improving the efficiency and flexibility, we present a general method for generating orthogonal matrices directly through the construction of certain even and odd polynomials from a set of distinct positive values, bypassing the need of continuous orthogonal functions. We provide a constructive proof by induction that not only asserts the existence of such polynomials, but also tells how to iteratively construct them. Besides the derivation of the method as simple as a few nested loops, we discuss two well-known discrete transforms, the Discrete Cosine Transform and the Discrete Tchebichef Transform. How they can be achieved using our method with the specific values, and show how to embed them into the transform module of video coding. By the same token, we also show some examples of how to generate new orthogonal matrices from arbitrarily chosen values.