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.
Highlights
Orthogonal transformations have very useful properties in solving science and engineering problems
The Discrete Tchebichef Transform (DTT) provides another transformation method using the Chebyshev moments [13], [14], which has as good energy compression properties as the Discrete Cosine Transform (DCT) and works better for a certain class of images [15]
By jumping to the construction directly, we sacrifice some mathematical insights and certainties, we provide a way to significantly broaden the base of discrete orthogonal matrices for engineering analyses
Summary
Orthogonal transformations have very useful properties in solving science and engineering problems. The Discrete Tchebichef Transform (DTT) provides another transformation method using the Chebyshev moments [13], [14], which has as good energy compression properties as the DCT and works better for a certain class of images [15] Both of the above example transformations are defined upon orthogonal polynomials. It starts with the definition of discrete orthogonality, makes use of even and odd functions, inspired by the DCT and DTT, to simplify the problems, constructs the linear equation system for deriving the coefficients of the polynomials, proves that a unique solution exists, and inductively obtains the solution.
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