Incomplete Gamma Kernels: Generalizing Locally Optimal Projection Operators

TL;DR

In conclusion, 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 are illustrated.

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.

Figures (12)

• Figure 1: Relation between LOP and Mean Shift in the example of the 2D Fish model. Minimizing the localized$L_1$ attraction energy with the Gaussian kernel $K_{\mathrm{Gaussian}}$ (left) results in the same trajectory $\bm{q}^{(t)}$ as applying Mean Shift on a global kernel density estimate with the kernel $K_{\mathrm{LOP}}$ (right).
• Figure 2: Interpolation between 2D incomplete gamma kernels $K_{\Gamma}$ with varying ${ p \in [1, 2] }$ and fixed ${ \sigma^2 = 1 / 32 }$. Each kernel corresponds to a localized attraction energy minimization with the respective $p$-norm.
• Figure 3: Comparison of LOP and Gaussian kernels in spatial and frequency domain. Filtering with the LOP kernel $K_{\mathrm{LOP}}$ better preserves higher frequency information.
• Figure 4: Exemplary point cloud denoising of the Elephant model (302458.0 points) with 30.0 WLOP huang2009consolidation iterations (${ h = 6 }$, ${ \mu = 0.4 }$) for different incomplete gamma kernels $K_{\Gamma}$ using varying ${ p \in (0, 2] }$ and fixed ${ \sigma^2 = 1 / 32 }$. The model has been corrupted with ${ \sigma_{\mathrm{noise}} = 0.3 }$ ($80%$ points) and ${ \sigma_{\mathrm{outlier}} = 1.5 }$ Gaussian noise ($20%$ points) respectively to account for both typical sensor noise and heavy outliers. Higher $p$-norms result in more regular but oversmoothed point distributions whereas lower values better preserve features. Unit of $h, \sigma_{\mathrm{noise}}, \sigma_{\mathrm{outlier}}$: $[\% \text{ BB diagonal}]$.
• Figure 5: Comparison of LOP and Gaussian M-estimators for ${ \sigma^2 = 1 / 2 }$. Due to the close relation to Mean Shift, these robust loss functions $\rho$ do not only share the shape of the corresponding kernels $K$ but also have similar properties and form localized versions of the common global $L_2$ and $L_1$ loss functions.