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Sampling and Reconstruction of Surfaces and Higher Dimensional Manifolds

Emil Saucan, Eli Appleboim, Yehoshua Y. Zeevi
2007 Journal of Mathematical Imaging and Vision  
We present new sampling theorems for surfaces and higher dimensional manifolds. The core of the proofs resides in triangulation results for manifolds with boundary, not necessarily bounded.  ...  The method is based upon geometric considerations that are further augmented for 2-dimensional manifolds (i.e surfaces).  ...  Lorentz for his careful reading of the manuscript and his many helpful comments and suggestions.  ... 
doi:10.1007/s10851-007-0048-z fatcat:lns62pnkinakfmafnuyfyief5a

Sampling and Reconstruction of Surfaces and Higher Dimensional Manifolds

Emil Saucan, Eli Appleboim, Yehoshua Y. Zeevi
2008 Journal of Mathematical Imaging and Vision  
We present new sampling theorems for surfaces and higher dimensional manifolds. The core of the proofs resides in triangulation results for manifolds with boundary, not necessarily bounded.  ...  The method is based upon geometric considerations that are further augmented for 2-dimensional manifolds (i.e surfaces).  ...  Lorentz for his careful reading of the manuscript and his many helpful comments and suggestions.  ... 
doi:10.1007/s10851-008-0103-4 fatcat:7syqgml3b5hpxgpyehbsiiimju

Generalized Ricci curvature based sampling and reconstruction of images

A. Shu Lin, B. Zhongxuan Luo, C. Jielin Zhang, D. Emil Saucan
2015 2015 23rd European Signal Processing Conference (EUSIPCO)  
We introduce a novel method of image sampling based on viewing grayscale images as manifolds with density, and sampling them according to the generalized Ricci curvature introduced by Bakry, Emery and  ...  A variation of this approach, due to Morgan and his students is also considered. This new paradigm generalizes ideas and results that are by now common in Imaging and Graphics.  ...  (This is a generalization of the similar sampling criterion using Gaussian/sectional curvature for surfaces [5] and higher dimensional manifolds [14] .)  ... 
doi:10.1109/eusipco.2015.7362454 dblp:conf/eusipco/LinLZS15 fatcat:unvw2b7fhbbxnd56m24b2h6hou

Geodesic-HOF: 3D Reconstruction Without Cutting Corners [article]

Ziyun Wang, Eric A. Mitchell, Volkan Isler, Daniel D. Lee
2020 arXiv   pre-print
Our results show that taking advantage of these learned lifted coordinates yields better performance for estimating surface normals and generating surfaces than using point cloud reconstructions alone.  ...  To address this issue, we propose learning an image-conditioned mapping function from a canonical sampling domain to a high dimensional space where the Euclidean distance is equal to the geodesic distance  ...  The key insight of our approach is to embed points sampled from the surface of a 3-dimensional object into a higher-dimensional space such that geodesic distances on the surface of the object can be computed  ... 
arXiv:2006.07981v1 fatcat:msyxgquzkjbx7fv77vawjkvaey

Curvature Based Clustering for DNA Microarray Data Analysis [chapter]

Emil Saucan, Eli Appleboim
2005 Lecture Notes in Computer Science  
Zeevi, Journal of Fourier Analysis and Applications, Special Issue -"Analysis on the Sphere II" 13(6), 2007, 711-727. (10) Sampling and Reconstruction of Surfaces and Higher Dimensional Manifolds, with  ...  ( 1 ) 1 Sampling and Reconstruction of Surfaces and Higher Dimensional Manifolds, with Eli Appleboim and Yehoshua Y. Zeevi, Technion CCIT Report #591, June 2006 (EE PUB #1543 June 2006).  ... 
doi:10.1007/11492542_50 fatcat:lney3vpjbzggza4cgcnqfbqheu

Dimensionality reduction and principal surfaces via Kernel Map Manifolds

Samuel Gerber, Tolga Tasdizen, Ross Whitaker
2009 2009 IEEE 12th International Conference on Computer Vision  
The manifold is represented as a parametrized surface represented by a set of parameters that are defined on the input samples.  ...  The representation also provides a natural mapping from high to low dimensional space, and a concatenation of these two mappings induces a projection operator onto the manifold.  ...  This work was supported by the NIH/NCBC grant U54-EB005149 and the NSF grant CCF-073222.  ... 
doi:10.1109/iccv.2009.5459193 dblp:conf/iccv/GerberTW09 fatcat:a7xsk7wjjbh3xlzmnzrfw7m5oa

Jeffrey's prior sampling of deep sigmoidal networks [article]

Lorien X. Hayden, Alexander A. Alemi, Paul H. Ginsparg, James P. Sethna
2017 arXiv   pre-print
In connection with the results we present, we discuss problems of sampling high-dimensional manifolds as well as recent work [M. Transtrum, G. Hart, and P.  ...  Neural networks have been shown to have a remarkable ability to uncover low dimensional structure in data: the space of possible reconstructed images form a reduced model manifold in image space.  ...  For the DBN and SdA, the reconstructed manifold is 30 dimensional and the factor is a meager 1.03.  ... 
arXiv:1705.10589v1 fatcat:pzsnvtcxincshnbm4vq7dx6pk4

Shape Dimension and Approximation from Samples

Tamal K. Dey, Joachim Giesen, Samrat Goswami, Wulue Zhao
2003 Discrete & Computational Geometry  
There are many scientific and engineering applications where an automatic detection of shape dimension from sample data is necessary.  ...  We present a Voronoi based dimension detection algorithm that assigns a dimension to a sample point which is the topological dimension of the manifold it belongs to.  ...  The first author thanks Marshall Bern for suggesting the problem of dimension detection from sample points and for his insightful comments.  ... 
doi:10.1007/s00454-002-2838-9 fatcat:dhbbswabszeb5eoraerot2ebjy

High-veracity functional imaging in scanning probe microscopy via Graph-Bootstrapping

Xin Li, Liam Collins, Keisuke Miyazawa, Takeshi Fukuma, Stephen Jesse, Sergei V. Kalinin
2018 Nature Communications  
To meet this challenge, we present a data-driven approach, Graph-Bootstrapping, based on low-dimensional manifold learning of the full SPM spectra and demonstrate its successes for high-veracity mechanical  ...  The key objective of scanning probe microscopy (SPM) techniques is the optimal representation of the nanoscale surface structure and functionality inferred from the dynamics of the cantilever.  ...  Department of Energy, Office of Science User Facility at Oak Ridge National Laboratory.  ... 
doi:10.1038/s41467-018-04887-1 pmid:29930246 pmcid:PMC6013493 fatcat:irlqbqp53zbgjfn5h6xhoierna

Spectral sampling of manifolds

A. Cengiz Öztireli, Marc Alexa, Markus Gross
2010 ACM SIGGRAPH Asia 2010 papers on - SIGGRAPH ASIA '10  
We propose a new method to solve this problem based on spectral analysis of manifolds which results in faithful reconstructions and high quality isotropic samplings, is efficient, out-of-core, feature  ...  Although our main focus is on sampling surfaces, the analysis and algorithms are general and can be applied for simplifying and resampling point clouds lying near a manifold of arbitrary dimension.  ...  The models used in the paper are courtesy of AIM@SHAPE Shape Repository, Stanford University Computer Graphics Laboratory, Cyberware, SensAble, MIT CSAIL, INRIA, IMATI and Clemson University.  ... 
doi:10.1145/1882262.1866190 fatcat:e5jkdty2avgw5mryo6h6w53p2i

Spectral sampling of manifolds

A. Cengiz Öztireli, Marc Alexa, Markus Gross
2010 ACM Transactions on Graphics  
We propose a new method to solve this problem based on spectral analysis of manifolds which results in faithful reconstructions and high quality isotropic samplings, is efficient, out-of-core, feature  ...  Although our main focus is on sampling surfaces, the analysis and algorithms are general and can be applied for simplifying and resampling point clouds lying near a manifold of arbitrary dimension.  ...  The models used in the paper are courtesy of AIM@SHAPE Shape Repository, Stanford University Computer Graphics Laboratory, Cyberware, SensAble, MIT CSAIL, INRIA, IMATI and Clemson University.  ... 
doi:10.1145/1882261.1866190 fatcat:cuziwmjmejhhzplgs3wojklv6e

Spectral sampling of manifolds

A. Cengiz Öztireli, Marc Alexa, Markus Gross
2010 ACM SIGGRAPH Asia 2010 papers on - SIGGRAPH ASIA '10  
We propose a new method to solve this problem based on spectral analysis of manifolds which results in faithful reconstructions and high quality isotropic samplings, is efficient, out-of-core, feature  ...  Although our main focus is on sampling surfaces, the analysis and algorithms are general and can be applied for simplifying and resampling point clouds lying near a manifold of arbitrary dimension.  ...  The models used in the paper are courtesy of AIM@SHAPE Shape Repository, Stanford University Computer Graphics Laboratory, Cyberware, SensAble, MIT CSAIL, INRIA, IMATI and Clemson University.  ... 
doi:10.1145/1866158.1866190 fatcat:wf6jzidycva5tapfumn3vqbv2y

Prediction of high-dimensional states subject to respiratory motion: a manifold learning approach

Wenyang Liu, Amit Sawant, Dan Ruan
2016 Physics in Medicine and Biology  
We evaluated and compared the proposed method with PCA-based approach on level-set surfaces reconstructed from point clouds captured by a 3D photogrammetry system.  ...  In this study, we extend such rationale to a more general manifold and propose a framework for high-dimensional motion prediction with manifold learning, which allows to learn more descriptive features  ...  We also acknowledge the support from UCLA dissertation year fellowship and Vision RT.  ... 
doi:10.1088/0031-9155/61/13/4989 pmid:27299958 pmcid:PMC4975535 fatcat:aayz2e6fkrflhjiw3vjarkyqkq

Geometric Understanding of Deep Learning [article]

Na Lei, Zhongxuan Luo, Shing-Tung Yau, David Xianfeng Gu
2018 arXiv   pre-print
data concentrates close to a low-dimensional manifold, deep learning learns the manifold and the probability distribution on it.  ...  We further introduce the concepts of rectified linear complexity for deep neural network measuring its learning capability, rectified linear complexity of an embedding manifold describing the difficulty  ...  We uniformly sample the surface, there are 235, 771 samples in total. The number of cells in the cell decomposition induced by the reconstruction map is 230051.  ... 
arXiv:1805.10451v2 fatcat:d2dbdlkqqnavxg5ajqtxalxa6q

Manifold bootstrapping for SVBRDF capture

Yue Dong, Jiaping Wang, Xin Tong, John Snyder, Yanxiang Lan, Moshe Ben-Ezra, Baining Guo
2010 ACM Transactions on Graphics  
The first acquires representatives of high angular dimension but sampled sparsely over the surface, while the second acquires keys of low angular dimension but sampled densely over the surface.  ...  Abstract Manifold bootstrapping is a new method for data-driven modeling of real-world, spatially-varying reflectance, based on the idea that reflectance over a given material sample forms a low-dimensional  ...  Wang et al. [2006] reconstruct the BRDF manifold from samples over the surface and from these predict reflectance variation over time.  ... 
doi:10.1145/1778765.1778835 fatcat:vqmiumhxbrhtji6lxmecdxxbwy
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