  In this paper, we propose a novel formulation to extend CNNs to
two-dimensional (2D) manifolds using orthogonal basis functions, called Zernike
polynomials. In many areas, geometric features play a key role in understanding
scientific phenomena. Thus, an ability to codify geometric features into a
mathematical quantity can be critical. Recently, convolutional neural networks
(CNNs) have demonstrated the promising capability of extracting and codifying
features from visual information. However, the progress has been concentrated
in computer vision applications where there exists an inherent grid-like
structure. In contrast, many geometry processing problems are defined on curved
surfaces, and the generalization of CNNs is not quite trivial. The difficulties
are rooted in the lack of key ingredients such as the canonical grid-like
representation, the notion of consistent orientation, and a compatible local
topology across the domain. In this paper, we prove that the convolution of two
functions can be represented as a simple dot product between Zernike polynomial
coefficients; and the rotation of a convolution kernel is essentially a set of
2-by-2 rotation matrices applied to the coefficients. As such, the key
contribution of this work resides in a concise but rigorous mathematical
generalization of the CNN building blocks.
