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Numerical reconstruction of convex polytopes from directional moments

2015
*
Advances in Computational Mathematics
*

We

doi:10.1007/s10444-014-9401-0
fatcat:7edxhfqz6rfw5od6pz2mgwqr2q
*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*. ...##
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The inverse moment problem for convex polytopes: implementation aspects
[article]

2014
*
arXiv
*
pre-print

(Discrete & Computational Geometry'12) for

arXiv:1409.3130v2
fatcat:dmy2ortj3jfb3ntdbqoirpeigu
*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 . ...##
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The Inverse Moment Problem for Convex Polytopes

2012
*
Discrete & Computational Geometry
*

The goal

doi:10.1007/s00454-012-9426-4
fatcat:2347qhktavf5rocmbmflfja2au
*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. ...##
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Reconstruction of n-dimensional convex bodies from surface tensors
[article]

2016
*
arXiv
*
pre-print

Further, we develop two algorithms that

arXiv:1606.08240v1
fatcat:ta5os467xffnbotycse6f5lofa
*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. ...##
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Reconstruction of n-dimensional convex bodies from surface tensors

2017
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Advances in Applied Mathematics
*

Further, we develop two algorithms that

doi:10.1016/j.aam.2016.09.004
fatcat:dfmlvlukbfggdpm2bvnjonbjii
*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. ...##
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Reconstruction of polytopes from the modulus of the Fourier transform with small wave length
[article]

2020
*
arXiv
*
pre-print

Let 𝒫 be an n-dimensional

arXiv:2011.06971v1
fatcat:zfazq2fh7ndzpevatpf3o2ykje
*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 - ...##
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Sparse Signal Reconstruction using Weight Point Algorithm

2018
*
Journal of ICT Research and Applications
*

By taking a large l 1 -norm value at the initial step, the intersection

doi:10.5614/itbj.ict.res.appl.2018.12.1.3
fatcat:rnhi6zdmfvh2jgf4vwv5yreswu
*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. ...##
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On the use of sparsity for recovering discrete probability distributions from their moments

2011
*
2011 IEEE Statistical Signal Processing Workshop (SSP)
*

We address the problem

doi:10.1109/ssp.2011.5967813
fatcat:3exvxndnbbavdnypdltjtumbf4
*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. ...##
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A fast direct sampling algorithm for equilateral closed polygons

2016
*
Journal of Physics A: Mathematical and Theoretical
*

We present a simple

doi:10.1088/1751-8113/49/27/275202
fatcat:tdz3rsdanfdwpn2azewciorfqy
*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). ...##
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Sparse nonnegative solution of underdetermined linear equations by linear programming

2005
*
Proceedings of the National Academy of Sciences of the United States of America
*

We explain this by the theory

doi:10.1073/pnas.0502269102
pmid:15976026
pmcid:PMC1172251
fatcat:tsll7vejtffr3jvk6gnkaav3lu
*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. ...##
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Friction Compensation in TP Model Form - AeroelasticWing as an Example System

2015
*
Acta Polytechnica Hungarica
*

The example is evaluated via

doi:10.12700/aph.12.4.2015.4.8
fatcat:p4zaofjbbnglzd6felpyr5hepe
*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. ...##
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A realizability-preserving discontinuous Galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension

2015
*
Journal of Computational Physics
*

We avoid this computation by using quadrature to generate a

doi:10.1016/j.jcp.2015.04.034
fatcat:oab75u2rhbbgdfbftizjwa4lfq
*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*. ...##
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The Convex Geometry of Linear Inverse Problems

2012
*
Foundations of Computational Mathematics
*

Thus this work extends the catalog

doi:10.1007/s10208-012-9135-7
fatcat:gurdi2wsmfdxtgl3ikctt3tt2i
*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? ...##
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Restricted isometry property of matrices with independent columns and neighborly polytopes by random sampling
[article]

2009
*
arXiv
*
pre-print

This paper considers compressed sensing matrices and neighborliness

arXiv:0904.4723v1
fatcat:3xe2z7bk3rd7jon5zfae2n5smm
*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*. ...##
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Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms

2011
*
Journal of The American Mathematical Society
*

We propose strongly consistent algorithms for

doi:10.1090/s0894-0347-2010-00683-2
fatcat:ox5lvv3usbewxl4uju6loyyqo4
*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. ...
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