Computing the volume is difficult
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Projective clustering in high dimensions using core-sets
Proceedings of the eighteenth annual symposium on Computational geometry
Shape dimension and approximation from samples
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Shape dimension and intrinsic metric from samples of manifolds with high co-dimension
Proceedings of the nineteenth annual symposium on Computational geometry
Laplacian Eigenmaps for dimensionality reduction and data representation
Neural Computation
Curve and Surface Reconstruction: Algorithms with Mathematical Analysis (Cambridge Monographs on Applied and Computational Mathematics)
Manifold reconstruction in arbitrary dimensions using witness complexes
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Discrete laplace operator on meshed surfaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
Dimension detection via slivers
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Constructing Laplace operator from point clouds in Rd
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
A quasi-Monte Carlo method for computing areas of point-sampled surfaces
Computer-Aided Design
Towards a theoretical foundation for laplacian-based manifold methods
COLT'05 Proceedings of the 18th annual conference on Learning Theory
Approximating gradients for meshes and point clouds via diffusion metric
SGP '09 Proceedings of the Symposium on Geometry Processing
Convergence, stability, and discrete approximation of Laplace spectra
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Nonrigid Matching of Undersampled Shapes via Medial Diffusion
Computer Graphics Forum
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Integration over a domain, such as a Euclidean space or a Riemannian manifold, is a fundamental problem across scientific fields. Many times, the underlying domain is only accessible through a discrete approximation, such as a set of points sampled from it, and it is crucial to be able to estimate integral in such discrete settings. In this paper, we study the problem of estimating the integral of a function defined over a k-submanifold embedded in $d$-dimensional space, from its function values at a set of sample points. Previously, such estimation is usually obtained in a statistical setting, where input data is typically assumed to be drawn from certain probabilistic distribution. Our paper is the first to consider this important problem of estimating integral from point clouds data (PCD) under the more general non-statistical setting, and provide certain theoretical guarantees. Our approaches consider the problem from a geometric point of view. Specifically, we estimate the integral by computing a weighted sum, and propose two weighting schemes: the Voronoi and the Principle Eigenvector schemes. The running time of both methods depends mostly on the intrinsic dimension of the underlying manifold, instead of on the ambient dimensions. We show that the estimation based on the Voronoi scheme converges to the true integral under the so-called (ε, δ)-sampling condition with explicit error bound presented. This is the first result of this sort for estimating integral from general PCD. For the Principle Eigenvector scheme, although no theoretical guarantee is established, we present its connection to the Heat diffusion operator, and illustrate justifications behind its construction. Experimental results show that both new methods consistently produce more accurate integral estimations than common statistical methods under various sampling conditions.