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Incomplete Gamma Kernels: Generalizing Locally Optimal Projection Operators

arXiv 2205.01087 ↗First posted 2 May 2022cs.CV cs.GRcited by 0

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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.

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.

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32 of 91 references.

References · 32

  1. 2016PointNet: Deep Learning on Point Sets for 3D Classification and Segmentationcited by 5,248
  2. 2017PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Spacecited by 4,401
  3. 2019Differentiable Surface Splatting for Point-based Geometry Processingcited by 239
  4. 2017A General and Adaptive Robust Loss Functioncited by 236
  5. 2018Patch-based Progressive 3D Point Set Upsamplingcited by 101
  6. 2019PointCleanNet: Learning to Denoise and Remove Outliers from Dense Point Cloudscited by 99
  7. 2018EC-Net: an Edge-aware Point set Consolidation Networkcited by 84
  8. 2021Score-Based Point Cloud Denoisingcited by 83
  9. 2019Total Denoising: Unsupervised Learning of 3D Point Cloud Cleaningcited by 58
  10. 2020Differentiable Manifold Reconstruction for Point Cloud Denoisingcited by 52
  11. 2020Pointfilter: Point Cloud Filtering via Encoder-Decoder Modelingcited by 50
  12. 20183D Point Cloud Denoising using Graph Laplacian Regularization of a Low Dimensional Manifold Modelcited by 47
20 more
  1. 2020Learning Graph-Convolutional Representations for Point Cloud Denoisingcited by 40
  2. 2017Deep Mean-Shift Priors for Image Restorationcited by 34
  3. 2019Feature Graph Learning for 3D Point Cloud Denoisingcited by 28
  4. 2021Deep Point Set Resampling via Gradient Fieldscited by 24
  5. 20193D Point Cloud Denoising via Deep Neural Network based Local Surface Estimationcited by 20
  6. 2022PD-Flow: A Point Cloud Denoising Framework with Normalizing Flowscited by 19
  7. 2021Deep Point Cloud Reconstructioncited by 15
  8. 2020Dynamic Point Cloud Denoising via Manifold-to-Manifold Distancecited by 12
  9. 2016Density-based Denoising of Point Cloudcited by 11
  10. 2021MeanShift++: Extremely Fast Mode-Seeking With Applications to Segmentation and Object Trackingcited by 9
  11. 2017Image Restoration using Autoencoding Priorscited by 8
  12. 2017On Convergence of Epanechnikov Mean Shiftcited by 6
  13. 2016Sparse Geometric Representation Through Local Shape Probingcited by 5
  14. 2017Static/Dynamic Filtering for Mesh Geometrycited by 5
  15. 2013On the Convergence and Consistency of the Blurring Mean-Shift Processcited by 4
  16. 2020Non-Local Part-Aware Point Cloud Denoisingcited by 4
  17. 2022Point Cloud Denoising via Momentum Ascent in Gradient Fieldscited by 4
  18. 2017Surface Reconstruction with Data-driven Exemplar Priorscited by 3
  19. 2019A Distributed and Approximated Nearest Neighbors Algorithm for an Efficient Large Scale Mean Shift Clusteringcited by 3
  20. 2020Surface Denoising based on Normal Filtering in a Robust Statistics Frameworkcited by 1

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