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Consistent Discretization and Minimization of the L1 Norm on Manifolds [article]

Alex Bronstein, Yoni Choukroun, Ron Kimmel, Matan Sela
2016 arXiv   pre-print
The continuous L1 norm on the manifold is often replaced by the vector l1 norm applied to sampled functions.  ...  We show that such an approach is incorrect in the sense that it does not consistently discretize the continuous norm and warn against its sensitivity to the specific sampling.  ...  Conclusion We presented a consistent discretization of the L 1 norm on manifolds as a geometrically meaningful alternative to the vector 1 norm that is frequently employed instead.  ... 
arXiv:1609.05434v1 fatcat:cnnju4u3v5hllmker5qbkgdevi

Optical flow and depth from motion for omnidirectional images using a TV-L1 variational framework on graphs

Luigi Bagnato, Pascal Frossard, Pierre Vandergheynst
2009 2009 16th IEEE International Conference on Image Processing (ICIP)  
In both cases, our graphbased algorithms provide computationally efficient solutions and significantly outperform naive implementations based on direct discretization of the operators, or on neglecting  ...  We formulate the problem in the natural spherical geometry associated with these devices and extend a recent TV-L1 variational formulation for computing the optical flow [1].  ...  The success of this algorithm depends strongly on a stable discretization of the gradient operator ∇ M on the manifold, which is not always straightforward.  ... 
doi:10.1109/icip.2009.5414552 dblp:conf/icip/BagnatoFV09 fatcat:5qq3z2tsgbcj7jtjyj72ir4qba

Some applications of compressed sensing in computational mechanics: model order reduction, manifold learning, data-driven applications and nonlinear dimensionality reduction

R. Ibañez, E. Abisset-Chavanne, E. Cueto, A. Ammar, J. -L. Duval, F. Chinesta
2019 Computational Mechanics  
Among them, maybe the most salient one is its ability of overcoming the Shannon-Nyquist sampling theorem.  ...  We consider a wide variety of applications, such as model order reduction, manifold learning, data-driven applications and nonlinear dimensionality reduction.  ...  Acknowledgements This work has been supported by the Spanish Ministry of Economy and Competitiveness through Grants Numbers DPI2017-85139-C2-1-R and DPI2015-72365-EXP and by the Regional Government of  ... 
doi:10.1007/s00466-019-01703-5 fatcat:5d7vvksvojh3te2vxlyghei2ye

Functoriality in Geometric Data (Dagstuhl Seminar 17021)

Mirela Ben-Chen, Frédéderic Chazal, Leonidas J. Guibas, Maks Ovsjanikov, Marc Herbstritt
2017 Dagstuhl Reports  
and other types of geometric data.  ...  The seminar brought together researchers interested in the fundamental questions of similarity and correspondence across geometric data sets, which include collections of GPS traces, images, 3D shapes  ...  A new manifold learning method is presented based on alternating products of diffusion operators and local kernels.  ... 
doi:10.4230/dagrep.7.1.1 dblp:journals/dagstuhl-reports/Ben-ChenCGO17 fatcat:67hkhwf73nfdljup3b4xcuwxiq

Symmetry Reduction of a Class of Hybrid Systems [chapter]

Jianghai Hu, Shankar Sastry
2002 Lecture Notes in Computer Science  
In particular, the problems of optimal collision avoidance (OCA) and optimal formation switching (OFS) of multiple agents moving on a Riemannian manifold are studied in some details.  ...  Some necessary conditions of the optimal solutions of such a system are derived based on the assumption that there is a group of symmetries acting uniformly on the domains of different discrete modes,  ...  For each (l 1 , l 2 ) ∈ E d , a subset D (l1,l2) ⊂ M l1 , called the guard associated with the discrete transition (l 1 , l 2 ), and a continuous transition relation E c (l 1 , l 2 ) ⊂ D (l1,l2) ×M l2  ... 
doi:10.1007/3-540-45873-5_22 fatcat:tmubakfd4zelbcftr56p77tcwq

Principal manifold learning by sparse grids

Christian Feuersänger, Michael Griebel
2009 Computing  
We present our sparse grid principal manifold approach, discuss its properties and report on the results of numerical experiments for one-, two-and three-dimensional model problems.  ...  For the discretization we use a sparse grid method in latent parameter space. This approach avoids, to some extent, the curse of dimension of conventional grids like in the GTM approach.  ...  (3.1) we obtain the associated discrete minimization problems (3.3).  ... 
doi:10.1007/s00607-009-0045-8 fatcat:xewaqii2t5cjzclxmczpl37jey

Human detection in images via L1-norm Minimization Learning

Ran Xu, Baochang Zhang, Qixiang Ye, Jianbin Jiao
2010 2010 IEEE International Conference on Acoustics, Speech and Signal Processing  
Experiments on two datasets validate the effectiveness and efficiency of the proposed method.  ...  In this paper we develop a novel human detection system based on L1-norm Minimization Learning (LML) method.  ...  We firstly compute blocks of HOG features on training samples and use L1-minimization to obtain weight and the sparse representation.  ... 
doi:10.1109/icassp.2010.5495930 dblp:conf/icassp/XuZYJ10 fatcat:gp5ezxme6vgcvo7bgry43kzvci

A Method of Solving the Eigenproblem of the Atomic Ion Hamiltonians. I. Theory

J. Owedyk
2014 Acta Physica Polonica. A  
The aim of this paper is to present a procedure for determining the power series in N −1 Z for the approximate energy levels and eigenspaces of the nonrelativistic Hamiltonians H (N,Z) with N -electrons  ...  To this eect the theory of the best multicongurational approximations is applied.  ...  Therefore the minimizing basic manifold M minimizes the average energy of Ĥ(N ) on M ∧N .  ... 
doi:10.12693/aphyspola.125.1075 fatcat:2aoyrfy4azg7zlk2iq6cygo7xi

Fast global stereo matching via energy pyramid minimization

B. Conejo, S. Leprince, F. Ayoub, J. P. Avouac
2014 ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences  
This global discrete optimization approach guarantees that at each scale we obtain a near optimal solution, and we demonstrate its superiority over state of the art image pyramid approaches through application  ...  We efficiently address this minimization by globally optimizing a coarse to fine sequence of sparse Conditional Random Fields (CRF) directly defined on the energy.  ...  Hazard Observation and Reporting Center of Caltech, the Moore foundation through the Advanced Earth Surface Observation Project (AESOP Grant 2808), and the ANR project STEREO.  ... 
doi:10.5194/isprsannals-ii-3-41-2014 fatcat:byufvwuyibdefm3xar2na5koju

Traction microscopy to identify force modulation in subresolution adhesions

Sangyoon J Han, Youbean Oak, Alex Groisman, Gaudenz Danuser
2015 Nature Methods  
Traction reconstruction with regularization relies on minimizing the sum of the residual norm, i.e. the difference between predicted and measured deformation fields, and the solution norm of the traction  ...  The key ingredient of this method is L1-regularization, which relies assumes solution sparsity, i.e. instead of minimizing the magnitudes or derivatives of the solution the L1-norm tends to minimize the  ... 
doi:10.1038/nmeth.3430 pmid:26030446 pmcid:PMC4490115 fatcat:dvi5hnobojh3dojlnn5z7cbcxm

Variational Methods for Denoising Matrix Fields [chapter]

S. Setzer, G. Steidl, B. Popilka, B. Burgeth
2009 Mathematics and Visualization  
In this paper, we transfer successful techniques like the minimization of the Rudin-Osher-Fatemi functional and the infimal convolution to matrix fields, where our functionals couple the different matrix  ...  The restoration of scalar-valued images via minimization of an energy functional is a well-established technique in image processing.  ...  The authors like to thank J. Weickert and S. Didas for fruitful discussions.  ... 
doi:10.1007/978-3-540-88378-4_17 fatcat:yjfii3zyknbfdakj3yzlp3oib4

Harry Potter's Marauder's Map: Localizing and Tracking Multiple Persons-of-Interest by Nonnegative Discretization

Shoou-I Yu, Yi Yang, Alexander Hauptmann
2013 2013 IEEE Conference on Computer Vision and Pattern Recognition  
Nonnegative discretization is used to enforce the mutual exclusion constraint, which guarantees a person detection output to only belong to exactly one individual.  ...  Local learning approaches are used to uncover the manifold structure in the appearance space with spatio-temporal constraints.  ...  Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.  ... 
doi:10.1109/cvpr.2013.476 dblp:conf/cvpr/YuYH13 fatcat:dlyckx62ejf7zpvkpyl3n2kel4

Low-thrust Lyapunov to Lyapunov and Halo to Halo missions with L2-minimization

Maxime Chupin, Thomas Haberkorn, Emmanuel Trélat
2017 Mathematical Modelling and Numerical Analysis  
Moreover, we use continuation methods on position and on thrust, in order to gain robustness.  ...  In the Circular Restricted Three Body Problem, the knowledge of invariant manifolds helps us initialize an indirect method solving a transfer mission between periodic Lyapunov orbits.  ...  This is done by considering the minimization of the L 1 -norm of u C L1 g = min t f 0 u dt.  ... 
doi:10.1051/m2an/2016044 fatcat:nb5jfpw7wzdszkwedsanzmnc2i

Low-Thrust Lyapunov to Lyapunov and Halo to Halo with L^2-Minimization [article]

Maxime Chupin, Emmanuel Trélat
2016 arXiv   pre-print
Moreover, we use continuation methods on position and on thrust, in order to gain robustness.  ...  In the Circular Restricted Three Body Problem, the knowledge of invariant manifolds helps us initialize an indirect method solving a transfer mission between periodic Lyapunov orbits.  ...  This is done by considering the minimization of the L 1 -norm of u C L1 g = min t f 0 u dt.  ... 
arXiv:1511.02089v2 fatcat:ewbqg5bvxbg5lhufzbyxpfgn7q

Fast dual minimization of the vectorial total variation norm and applications to color image processing

Xavier Bresson, Tony Chan
2008 Inverse Problems and Imaging  
More precisely, the regularization model is based on the dual formulation of the vectorial Total Variation (VTV) norm and it may be regarded as the vectorial extension of the dual approach defined by Chambolle  ...  color wavelet shrinkage, color image decomposition, color image deblurring, and color denoising on manifolds.  ...  The authors also would like to thank the referees, Prof. Haomin Zhou, Managing Editor of Inverse Problems and Imaging and Prof.  ... 
doi:10.3934/ipi.2008.2.455 fatcat:xkjtqsurbndkrc3exe2mcppek4
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