Incomplete Gamma Kernels: Generalizing Locally Optimal Projection Operators

We present incomplete gamma kernels, a generalization of Locally Optimal Projection (LOP) operators. In particular, we reveal the relation of the classical localized <inline-formula><tex-math notation="LaTeX">$ L_{1}$</tex-math><alternatives><mml:math><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="stotko-ieq1-3349967.gif"/></alternatives></inline-formula> estimator, used in the LOP operator for point cloud denoising, to the common Mean Shift framework via a novel kernel. Furthermore, we generalize this result to a whole family of kernels that are built upon the incomplete gamma function and each represents a localized <inline-formula><tex-math notation="LaTeX">$ L_{p}$</tex-math><alternatives><mml:math><mml:msub><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="stotko-ieq2-3349967.gif"/></alternatives></inline-formula> 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&#x0027;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.