Integration with stochastic point processes

© 2016 ACM. We present a novel comprehensive approach for studying error in integral estimation with point distributions based on point process statistics. We derive exact formulae for bias and variance of integral estimates in terms of the spatial or spectral characteristics of integrands and first- And-second order product density measures of general point patterns. The formulae allow us to study and design sampling schemes adapted to different classes of integrands by analyzing the effect of sampling density, weighting, and correlations among point locations separately. We then focus on non- Adaptive correlated stratified sampling patterns and specialize the formulae to derive closed-form and easy- To- Analyze expressions of bias and variance for various stratified sampling strategies. Based on these expressions, we perform a theoretical error analysis for integrands involving the discontinuous visibility function.We show that significant reductions in error can be obtained by considering alternative sampling strategies instead of the commonly used random jittering or low discrepancy patterns. Our theoretical results agree with and extend various previous results, provide a unified analytic treatment of point patterns, and lead to novel insights. We validate the results with extensive experiments on benchmark integrands as well as real scenes with soft shadows

Perceptually based downscaling of images

We propose a perceptually based method for downscaling images that provides a better apparent depiction of the input image. We formulate image downscaling as an optimization problem where the difference between the input and output images is measured using a widely adopted perceptual image quality metric. The downscaled images retain perceptually important features and details, resulting in an accurate and spatio-temporally consistent representation of the high resolution input. We derive the solution of the optimization problem in closed-form, which leads to a simple, efficient and parallelizable implementation with sums and convolutions. The algorithm has running times similar to linear filtering and is orders of magnitude faster than the state-of-the-art for image downscaling. We validate the effectiveness of the technique with extensive tests on many images, video, and by performing a user study, which indicates a clear preference for the results of the new algorithm.</jats:p

Robust Poisson Surface Reconstruction

Abstract. We propose a method to reconstruct surfaces from oriented point clouds with non-uniform sampling and noise by formulating the problem as a convex minimization that reconstructs the indicator func-tion of the surface’s interior. Compared to previous models, our recon-struction is robust to noise and outliers because it substitutes the least-squares fidelity term by a robust Huber penalty; this allows to recover sharp corners and avoids the shrinking bias of least squares. We choose an implicit parametrization to reconstruct surfaces of unknown topology and close large gaps in the point cloud. For an efficient representation, we approximate the implicit function by a hierarchy of locally supported basis elements adapted to the geometry of the surface. Unlike ad-hoc bases over an octree, our hierarchical B-splines from isogeometric analysis locally adapt the mesh and degree of the splines during reconstruction. The hi-erarchical structure of the basis speeds-up the minimization and efficiently represents clustered data. We also advocate for convex optimization, in-stead isogeometric finite-element techniques, to efficiently solve the min-imization and allow for non-differentiable functionals. Experiments show state-of-the-art performance within a more flexible framework.