A Space-Based Method for the Generation of a Schwartz Function with Infinitely Many Vanishing Moments of Higher Order with Applications in Image Processing

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, $$\int _\mathbb {R}g(x^k)x^mdx=0$$ , is needed. We show that the presented construction produces an explicit formula of a function with $$\infty $$ 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.