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Numerical reconstruction of convex polytopes from directional moments

Mathieu Collowald, Annie Cuyt, Evelyne Hubert, Wen-Shin Lee, Oliver Salazar Celis
2015 Advances in Computational Mathematics  
We reconstruct an n-dimensional convex polytope from the knowledge of its directional moments up to a certain order.  ...  Numerical illustrations are given for the reconstruction of 2-d and 3-d convex polytopes.  ...  approach for the reconstruction of polytopes from their directional moments.  ... 
doi:10.1007/s10444-014-9401-0 fatcat:7edxhfqz6rfw5od6pz2mgwqr2q

The inverse moment problem for convex polytopes: implementation aspects [article]

Nick Gravin, Danny Nguyen, Dmitrii Pasechnik, Sinai Robins
2014 arXiv   pre-print
(Discrete & Computational Geometry'12) for reconstructing an N-vertex convex polytope P in R^d from the knowledge of O(Nd) of its moments.  ...  We give a detailed technical report on the implementation of the algorithm presented in Gravin et al.  ...  From harmonic moments only, one may reconstruct vertices of a convex polytope P .  ... 
arXiv:1409.3130v2 fatcat:dmy2ortj3jfb3ntdbqoirpeigu

The Inverse Moment Problem for Convex Polytopes

Nick Gravin, Jean Lasserre, Dmitrii V. Pasechnik, Sinai Robins
2012 Discrete & Computational Geometry  
The goal of this paper is to present a general and novel approach for the reconstruction of any convex d-dimensional polytope P, from knowledge of its moments.  ...  In particular, we show that the vertices of an N-vertex polytope in R^d can be reconstructed from the knowledge of O(DN) axial moments (w.r.t. to an unknown polynomial measure od degree D) in d+1 distinct  ...  Robins were supported by Singapore Ministry of Education ARF Tier 2 Grant MOE2011-T2-1-090.  ... 
doi:10.1007/s00454-012-9426-4 fatcat:2347qhktavf5rocmbmflfja2au

Reconstruction of n-dimensional convex bodies from surface tensors [article]

Astrid Kousholt
2016 arXiv   pre-print
Further, we develop two algorithms that reconstruct shape of n-dimensional convex bodies.  ...  One algorithm requires knowledge of a finite number of surface tensors, whereas the other algorithm is based on noisy measurements of a finite number of harmonic intrinsic volumes.  ...  This research was supported by Centre for Stochastic Geometry and Advanced Bioimaging, funded by a grant from the Villum foundation.  ... 
arXiv:1606.08240v1 fatcat:ta5os467xffnbotycse6f5lofa

Reconstruction of n-dimensional convex bodies from surface tensors

Astrid Kousholt
2017 Advances in Applied Mathematics  
Further, we develop two algorithms that reconstruct shape of n-dimensional convex bodies.  ...  One algorithm requires knowledge of a finite number of surface tensors, whereas the other algorithm is based on noisy measurements of a finite number of harmonic intrinsic volumes.  ...  This research was supported by Centre for Stochastic Geometry and Advanced Bioimaging, funded by a grant from the Villum foundation.  ... 
doi:10.1016/j.aam.2016.09.004 fatcat:dfmlvlukbfggdpm2bvnjonbjii

Reconstruction of polytopes from the modulus of the Fourier transform with small wave length [article]

Konrad Engel, Bastian Laasch
2020 arXiv   pre-print
Let 𝒫 be an n-dimensional convex polytope and 𝒮 be a hypersurface in ℝ^n.  ...  This paper investigates potentials to reconstruct 𝒫 or at least to compute significant properties of 𝒫 if the modulus of the Fourier transform of 𝒫 on 𝒮 with wave length λ, i.e., |∫_𝒫 e^-i1/λ𝐬·𝐱  ...  Acknowledgement This work was partly supported by the European Social Fund (ESF) and the Ministry of Education, Science and Culture of Mecklenburg-Western Pomerania (Germany) within the project NEISS -  ... 
arXiv:2011.06971v1 fatcat:zfazq2fh7ndzpevatpf3o2ykje

Sparse Signal Reconstruction using Weight Point Algorithm

Koredianto Usman, Hendra Gunawan, Andriyan B. Suksmono
2018 Journal of ICT Research and Applications  
By taking a large l 1 -norm value at the initial step, the intersection of l 1 -norm and the constraint curves forms a convex polytope and by exploiting the fact that any convex combination of the polytope's  ...  In this paper we propose a new approach of the compressive sensing (CS) reconstruction problem based on a geometrical interpretation of l 1 -norm minimization.  ...  At the moment there are two main groups of CS reconstruction algorithms, i.e. convex optimization and the greedy algorithm.  ... 
doi:10.5614/itbj.ict.res.appl.2018.12.1.3 fatcat:rnhi6zdmfvh2jgf4vwv5yreswu

On the use of sparsity for recovering discrete probability distributions from their moments

Anna Cohen, Arie Yeredor
2011 2011 IEEE Statistical Signal Processing Workshop (SSP)  
We address the problem of determining the probability distribution of a discrete random variable from its moments, using a sparsity-based approach.  ...  If the random variable can take at most K different values from a potential set of M K values, then its moments can be represented as linear measurements of a K-sparse probabilities vector, where the measurement  ...  A convex hull of a set of vertices is the set of all points which are convex combinations of these vertices. A convex polytope is the convex hull of all of its vertices.  ... 
doi:10.1109/ssp.2011.5967813 fatcat:3exvxndnbbavdnypdltjtumbf4

A fast direct sampling algorithm for equilateral closed polygons

Jason Cantarella, Bertrand Duplantier, Clayton Shonkwiler, Erica Uehara
2016 Journal of Physics A: Mathematical and Theoretical  
We present a simple direct sampler which is fast and numerically stable, and analyze its runtime using a new formula for the volume of equilateral polygon space as a Dirichlet-type integral.  ...  direct samplers (which require extended-precision arithmetic to evaluate numerically unstable polynomials).  ...  This work was supported by grants from the Simons Foundation (#354225, Clayton Shonkwiler, and #284066, Jason Cantarella).  ... 
doi:10.1088/1751-8113/49/27/275202 fatcat:tdz3rsdanfdwpn2azewciorfqy

Sparse nonnegative solution of underdetermined linear equations by linear programming

D. L. Donoho, J. Tanner
2005 Proceedings of the National Academy of Sciences of the United States of America  
We explain this by the theory of convex polytopes. Let aj denote the jth column of A, 1 < j Յ n, let a0 ‫؍‬ 0 and P denote the convex hull of the aj.  ...  polytopes ͉ cyclic polytopes ͉ combinatorial optimization ͉ convex hull of Gaussian samples ͉ positivity constraints in ill-posed problems Neighborly Polytopes.  ...  They also follow from recent work by Fuchs (9) , who gave a direct proof of uniqueness.  ... 
doi:10.1073/pnas.0502269102 pmid:15976026 pmcid:PMC1172251 fatcat:tsll7vejtffr3jvk6gnkaav3lu

Friction Compensation in TP Model Form - AeroelasticWing as an Example System

2015 Acta Polytechnica Hungarica  
The example is evaluated via numerical simulations.  ...  The aim of this paper is to fit the friction compensation problem in the field of modern polytopic and Linear Matrix Inequality (LMI) based control design methodologies.  ...  One direction of LMI-based theories needs the system to be formulated in convex polytopic form.  ... 
doi:10.12700/aph.12.4.2015.4.8 fatcat:p4zaofjbbnglzd6felpyr5hepe

A realizability-preserving discontinuous Galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension

Graham Alldredge, Florian Schneider
2015 Journal of Computational Physics  
We avoid this computation by using quadrature to generate a convex polytope which approximates this set.  ...  a linear scaling limiter when the numerical solution leaves the set of realizable moments (that is, those moments associated with a positive underlying distribution).  ...  The convex polytope R Q p | u0=1 ⊂ R N is known as the cyclic polytope and plays a special role in the study of convex polytopes.  ... 
doi:10.1016/j.jcp.2015.04.034 fatcat:oab75u2rhbbgdfbftizjwa4lfq

The Convex Geometry of Linear Inverse Problems

Venkat Chandrasekaran, Benjamin Recht, Pablo A. Parrilo, Alan S. Willsky
2012 Foundations of Computational Mathematics  
Thus this work extends the catalog of simple models that can be recovered from limited linear information via tractable convex programming.  ...  of the number of generic measurements required for exact and robust recovery of models from partial information.  ...  Suppose one is given access to a linear combination of moments of an atomically supported measure. How can we reconstruct the support of the measure?  ... 
doi:10.1007/s10208-012-9135-7 fatcat:gurdi2wsmfdxtgl3ikctt3tt2i

Restricted isometry property of matrices with independent columns and neighborly polytopes by random sampling [article]

Radosław Adamczak, Alexander E. Litvak, Alain Pajor, Nicole Tomczak-Jaegermann
2009 arXiv   pre-print
This paper considers compressed sensing matrices and neighborliness of a centrally symmetric convex polytope generated by vectors ± X_1,...,± X_N∈^n, (N> n).  ...  We show that such "sensing" matrices are valid for the exact reconstruction process of m-sparse vectors via ℓ_1 minimization with m< Cn/^2 (cN/n).  ...  It is well known that the ψ r -norm of a random variable may be estimated from the growth of the moments.  ... 
arXiv:0904.4723v1 fatcat:3xe2z7bk3rd7jon5zfae2n5smm

Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms

Gabriele Bianchi, Richard J. Gardner, Markus Kiderlen
2011 Journal of The American Mathematical Society  
We propose strongly consistent algorithms for reconstructing the characteristic function 1_K of an unknown convex body K in R^n from possibly noisy measurements of the modulus of its Fourier transform  ...  The approach is via the closely related problem of reconstructing K from noisy measurements of its covariogram, the function giving the volume of the intersection of K with its translates.  ...  of K 0 in the direction u.  ... 
doi:10.1090/s0894-0347-2010-00683-2 fatcat:ox5lvv3usbewxl4uju6loyyqo4
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