A Bayesian Approach to Manifold Topology Reconstruction

In this paper, we investigate the problem of statistical reconstruction of piecewise linear manifold topology. Given a noisy, probably undersampled point cloud from a one- or two-manifold, the algorithm reconstructs an approximated most likely mesh in a Bayesian sense from which the sample might have been taken. We incorporate statistical priors on the object geometry to improve the reconstruction quality if additional knowledge about the class of original shapes is available. The priors can be formulated analytically or learned from example geometry with known manifold tessellation. The statistical objective function is approximated by a linear programming / integer programming problem, for which a globally optimal solution is found. We apply the algorithm to a set of 2D and 3D reconstruction examples, demon-strating that a statistics-based manifold reconstruction is feasible, and still yields plausible results in situations where sampling conditions are violated

Total Denoising: Unsupervised Learning of 3D Point Cloud Cleaning

We show that denoising of 3D point clouds can be learned unsupervised, directly from noisy 3D point cloud data only. This is achieved by extending recent ideas from learning of unsupervised image denoisers to unstructured 3D point clouds. Unsupervised image denoisers operate under the assumption that a noisy pixel observation is a random realization of a distribution around a clean pixel value, which allows appropriate learning on this distribution to eventually converge to the correct value. Regrettably, this assumption is not valid for unstructured points: 3D point clouds are subject to total noise, i. e., deviations in all coordinates, with no reliable pixel grid. Thus, an observation can be the realization of an entire manifold of clean 3D points, which makes a na\"ive extension of unsupervised image denoisers to 3D point clouds impractical. Overcoming this, we introduce a spatial prior term, that steers converges to the unique closest out of the many possible modes on a manifold. Our results demonstrate unsupervised denoising performance similar to that of supervised learning with clean data when given enough training examples - whereby we do not need any pairs of noisy and clean training data.Comment: Proceedings of ICCV 201

Exploiting Structural Complexity for Robust and Rapid Hyperspectral Imaging

This paper presents several strategies for spectral de-noising of hyperspectral images and hypercube reconstruction from a limited number of tomographic measurements. In particular we show that the non-noisy spectral data, when stacked across the spectral dimension, exhibits low-rank. On the other hand, under the same representation, the spectral noise exhibits a banded structure. Motivated by this we show that the de-noised spectral data and the unknown spectral noise and the respective bands can be simultaneously estimated through the use of a low-rank and simultaneous sparse minimization operation without prior knowledge of the noisy bands. This result is novel for for hyperspectral imaging applications. In addition, we show that imaging for the Computed Tomography Imaging Systems (CTIS) can be improved under limited angle tomography by using low-rank penalization. For both of these cases we exploit the recent results in the theory of low-rank matrix completion using nuclear norm minimization

Point Cloud Structural Parts Extraction based on Segmentation Energy Minimization

In this work we consider 3D point sets, which in a typical setting represent unorganized point clouds. Segmentation of these point sets requires first to single out structural components of the unknown surface discretely approximated by the point cloud. Structural components, in turn, are surface patches approximating unknown parts of elementary geometric structures, such as planes, ellipsoids, spheres and so on. The approach used is based on level set methods computing the moving front of the surface and tracing the interfaces between different parts of it. Level set methods are widely recognized to be one of the most efficient methods to segment both 2D images and 3D medical images. Level set methods for 3D segmentation have recently received an increasing interest. We contribute by proposing a novel approach for raw point sets. Based on the motion and distance functions of the level set we introduce four energy minimization models, which are used for segmentation, by considering an equal number of distance functions specified by geometric features. Finally we evaluate the proposed algorithm on point sets simulating unorganized point clouds