# Incomplete Gamma Kernels: Generalizing Locally Optimal Projection Operators

arXiv: https://arxiv.org/abs/2205.01087 · HTML: https://mlsanity.com/paper/2205.01087/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 32 of its 91 references that are in the map.
- First posted on arXiv 2 May 2022.

## Builds on (most cited first, 30 of 32)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [Differentiable Surface Splatting for Point-based Geometry Processing](https://mlsanity.com/paper/1906.04173/) (2019)
- [A General and Adaptive Robust Loss Function](https://mlsanity.com/paper/1701.03077/) (2017)
- [Patch-based Progressive 3D Point Set Upsampling](https://mlsanity.com/paper/1811.11286/) (2018)
- [PointCleanNet: Learning to Denoise and Remove Outliers from Dense Point Clouds](https://mlsanity.com/paper/1901.01060/) (2019)
- [EC-Net: an Edge-aware Point set Consolidation Network](https://mlsanity.com/paper/1807.06010/) (2018)
- [Score-Based Point Cloud Denoising](https://mlsanity.com/paper/2107.10981/) (2021)
- [Total Denoising: Unsupervised Learning of 3D Point Cloud Cleaning](https://mlsanity.com/paper/1904.07615/) (2019)
- [Differentiable Manifold Reconstruction for Point Cloud Denoising](https://mlsanity.com/paper/2007.13551/) (2020)
- [Pointfilter: Point Cloud Filtering via Encoder-Decoder Modeling](https://mlsanity.com/paper/2002.05968/) (2020)
- [3D Point Cloud Denoising using Graph Laplacian Regularization of a Low Dimensional Manifold Model](https://mlsanity.com/paper/1803.07252/) (2018)
- [Learning Graph-Convolutional Representations for Point Cloud Denoising](https://mlsanity.com/paper/2007.02578/) (2020)
- [Deep Mean-Shift Priors for Image Restoration](https://mlsanity.com/paper/1709.03749/) (2017)
- [Feature Graph Learning for 3D Point Cloud Denoising](https://mlsanity.com/paper/1907.09138/) (2019)
- [Deep Point Set Resampling via Gradient Fields](https://mlsanity.com/paper/2111.02045/) (2021)
- [3D Point Cloud Denoising via Deep Neural Network based Local Surface Estimation](https://mlsanity.com/paper/1904.04427/) (2019)
- [PD-Flow: A Point Cloud Denoising Framework with Normalizing Flows](https://mlsanity.com/paper/2203.05940/) (2022)
- [Deep Point Cloud Reconstruction](https://mlsanity.com/paper/2111.11704/) (2021)
- [Dynamic Point Cloud Denoising via Manifold-to-Manifold Distance](https://mlsanity.com/paper/2003.08355/) (2020)
- [Density-based Denoising of Point Cloud](https://mlsanity.com/paper/1602.05312/) (2016)
- [MeanShift++: Extremely Fast Mode-Seeking With Applications to Segmentation and Object Tracking](https://mlsanity.com/paper/2104.00303/) (2021)
- [Image Restoration using Autoencoding Priors](https://mlsanity.com/paper/1703.09964/) (2017)
- [On Convergence of Epanechnikov Mean Shift](https://mlsanity.com/paper/1711.07441/) (2017)
- [Sparse Geometric Representation Through Local Shape Probing](https://mlsanity.com/paper/1612.02261/) (2016)
- [Static/Dynamic Filtering for Mesh Geometry](https://mlsanity.com/paper/1712.03574/) (2017)
- [On the Convergence and Consistency of the Blurring Mean-Shift Process](https://mlsanity.com/paper/1305.1040/) (2013)
- [Non-Local Part-Aware Point Cloud Denoising](https://mlsanity.com/paper/2003.06631/) (2020)
- [Point Cloud Denoising via Momentum Ascent in Gradient Fields](https://mlsanity.com/paper/2202.10094/) (2022)
- [Surface Reconstruction with Data-driven Exemplar Priors](https://mlsanity.com/paper/1701.03230/) (2017)

## Abstract (arXiv)

We present incomplete gamma kernels, a generalization of Locally Optimal Projection (LOP) operators. In particular, we reveal the relation of the classical localized  L1  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  Lp  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.