论文标题
不完整的伽马内核：概括本地最佳投影操作员
Incomplete Gamma Kernels: Generalizing Locally Optimal Projection Operators
论文作者
论文摘要
我们提出了不完整的伽马内核，这是局部最佳投影（LOP）操作员的概括。特别是，我们揭示了在LOP操作员用于点云Deoising中使用的经典本地化$ L_1 $估算器的关系，并通过新颖的内核与通用的平均移位框架的关系。此外，我们将此结果推广到建立在不完整的伽马功能上的整个内核中，每个核心都代表了本地化的$ L_P $估算器。通过得出核心家族的各种特性，涉及分布，均值转移以及其他方面，例如严格的积极确定性，我们可以更深入地了解操作员的投射行为。从这些理论见解中，我们说明了几种应用，从改进的加权LOP（WLOP）密度加权方案和更准确的连续LOP（CLOP）内核近似值到一组新型稳健损耗函数的定义。这些不完整的伽马损失包括特殊情况，包括高斯和LOP损失，可以应用于包括正常过滤在内的各种任务。此外，我们表明，新颖的内核可以作为神经网络的先验。我们在一系列定量和定性实验中演示了每个应用程序的效果，这些实验突出了我们的修改所产生的好处。
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.
