A space-based method for the generation of a Schwartz function with infinitely many generalized vanishing moments with applications in image processing

TL;DR

This paper presents a novel space-based method to explicitly construct functions with infinitely many generalized vanishing moments, enhancing the detection of geometric features in image processing applications like the Taylorlet transform.

Contribution

It introduces a new explicit construction of functions with infinitely many vanishing moments using q-calculus, Euler function, and partition function, improving geometric feature detection.

Findings

Constructed a function with infinitely many generalized vanishing moments.

Enabled robust detection of higher-order geometric features.

Connected the construction to q-calculus and special functions.

Abstract

In this article we construct a function with infinitely many vanishing (generalized) moments. This is motivated by an application to the Taylorlet transform which is based on the continuous shearlet transform. It can detect curvature and other higher order geometric information of singularities in addition to their position and the direction. For a robust detection of these features a function with higher order vanishing moments, $\int_\mathbb{R} g(x^k)x^m dx = 0$, is needed. We show that the presented construction produces an explicit formula of a function with infinitely many vanishing moments of arbitrary order and thus allows for a robust detection of certain geometric features. The construction has an inherent connection to q-calculus, the Euler function and the partition function.