Incomplete Gamma Kernels: Generalizing Locally Optimal Projection Operators

TL;DR

This paper introduces incomplete gamma kernels, a new family of localized estimators generalizing LOP operators, with theoretical analysis and diverse applications including denoising, robust losses, and neural network priors.

Contribution

It presents a novel kernel family based on incomplete gamma functions, linking LOP and Mean Shift, with theoretical properties and practical applications.

Findings

Improved point cloud denoising with new kernels

Development of robust loss functions including Gaussian and LOP as special cases

Enhanced neural network priors with kernel-based regularization

Abstract

We present incomplete gamma kernels, a generalization of Locally Optimal Projection (LOP) operators. In particular, we reveal the relation of the classical localized $ L_1 $ estimator, used in the LOP operator for point cloud denoising, to the common Mean Shift framework via a novel kernel. Furthermore, we generalize this result to a whole family of kernels that are built upon the incomplete gamma function and each represents a localized $ L_p $ estimator. By deriving various properties of the kernel family concerning distributional, Mean Shift induced, and other aspects such as strict positive definiteness, we obtain a deeper understanding of the operator's projection behavior. From these theoretical insights, we illustrate several applications ranging from an improved Weighted LOP (WLOP) density weighting scheme and a more accurate Continuous LOP (CLOP) kernel approximation to the definition of a novel set of robust loss functions. These incomplete gamma losses include the Gaussian and LOP loss as special cases and can be applied to various tasks including normal filtering. Furthermore, we show that the novel kernels can be included as priors into neural networks. We demonstrate the effects of each application in a range of quantitative and qualitative experiments that highlight the benefits induced by our modifications.