  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 surface
reconstruction from noisy point clouds, 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 for reconstruction tasks such as normal filtering. We demonstrate the
effects of each application in a range of quantitative and qualitative
experiments that highlight the benefits induced by our modifications.
