Steerability of Hermite Kernel

Steerability is a useful and important property of "kernel" functions. It enables certain complicated operations involving orientation manipulation on images to be executed with high efficiency. Thus, we focus our attention on the steerability of Hermite polynomials and their versions modulated by the Gaussian function with different powers, defined as the Hermite kernel. Certain special cases of such kernel, Hermite polynomials, Hermite functions and Gaussian derivatives are discussed in detail. Correspondingly, these cases demonstrate that the Hermite kernel is a powerful and effective tool for image processing. Furthermore, the steerability of the Hermite kernel is proved with the help of a property of Hermite polynomials revealing the rule concerning the product of two Hermite polynomials after coordination rotation. Consequently, any order of the Hermite kernel inherits steerability. Moreover, a couple sets of an explicit interpolation function and basis function can be directly obtained. We provide some examples to verify steerability of the Hermite kernel. Experimental results show the effectiveness of steerability and its potential applications in the fields of image processing and computer vision.