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Numerical reconstruction of convex polytopes from directional moments
2015
Advances in Computational Mathematics
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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]
2014
arXiv
pre-print
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(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
2012
Discrete & Computational Geometry
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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]
2016
arXiv
pre-print
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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
2017
Advances in Applied Mathematics
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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]
2020
arXiv
pre-print
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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
2018
Journal of ICT Research and Applications
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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
2011
2011 IEEE Statistical Signal Processing Workshop (SSP)
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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
2016
Journal of Physics A: Mathematical and Theoretical
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2015 pre-print arXiv:1510.02466v1 fatcat:mlvbylvdznhcll23w5gpj6saueOther Versions
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
2005
Proceedings of the National Academy of Sciences of the United States of America
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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
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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
2015
Journal of Computational Physics
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2015 pre-print arXiv:1501.03660v2 fatcat:5gyxi6e2qbfifpojcrl3s6mdkuOther Versions
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
2012
Foundations of Computational Mathematics
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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]
2009
arXiv
pre-print
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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
2011
Journal of The American Mathematical Society
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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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