Uniform Convergence of a Nonlinear Energy-Based Multilevel Quantization Scheme

A popular vector quantization scheme can be constructed by centroidal Voronoi tessellations (CVTs) which also have many other applications in diverse areas of science and engineering. The development of efficient algorithms for their construction is a key to the successful applications of CVTs in practice. This paper studies the details of a new optimization-based multilevel algorithm for the numerical computation of CVTs. The rigorous proof of its uniform convergence in one space dimension and the results of computational simulations are provided. They substantiate recent claims on the significant speedup demonstrated by the new scheme in comparison with traditional methods.

[1]  Q. Du,et al.  Uniform Convergence of a Multilevel Energy-based Quantization Scheme , 2005 .

[2]  Axthonv G. Oettinger,et al.  IEEE Transactions on Information Theory , 1998 .

[3]  Sonia Martínez,et al.  Coverage control for mobile sensing networks , 2002, IEEE Transactions on Robotics and Automation.

[4]  S. P. Lloyd,et al.  Least squares quantization in PCM , 1982, IEEE Trans. Inf. Theory.

[5]  Robert M. Gray,et al.  Global convergence and empirical consistency of the generalized Lloyd algorithm , 1986, IEEE Trans. Inf. Theory.

[6]  Rex A. Dwyer.  Higher-dimensional voronoi diagrams in linear expected time , 1991, Discret. Comput. Geom..

[7]  Oliver Deussen,et al.  Beyond Stippling 
— Methods for Distributing Objects on the Plane , 2003, Comput. Graph. Forum.

[8]  B. Coull,et al.  Modelling spatial intensity for replicated inhomogeneous point patterns in brain imaging , 2004 .

[9]  Qiang Du,et al.  Convergence of the Lloyd Algorithm for Computing Centroidal Voronoi Tessellations , 2006, SIAM J. Numer. Anal..

[10]  Michele Cappellari,et al.  Adaptive spatial binning of integral-field spectroscopic data using Voronoi tessellations , 2003, astro-ph/0302262.

[11]  Jinchao Xu,et al.  Global and uniform convergence of subspace correction methods for some convex optimization problems , 2002, Math. Comput..

[12]  A. Mendes,et al.  Multi-outlet retail site location assessment , 2004 .

[13]  Franz Aurenhammer,et al.  Voronoi Diagrams , 2000, Handbook of Computational Geometry.

[14]  W. Hackbusch,et al.  Analysis of a damped nonlinear multilevel method , 1989 .

[15]  Qiang Du,et al.  Acceleration schemes for computing centroidal Voronoi tessellations , 2006, Numer. Linear Algebra Appl..

[16]  A. Brandt.  General highly accurate algebraic coarsening. , 2000 .

[17]  Q. Du,et al.  Recent progress in robust and quality Delaunay mesh generation , 2006 .

[18]  Irad Yavneh,et al.  A multigrid approach to the scalar quantization problem , 2005, IEEE Transactions on Information Theory.

[19]  William L. Briggs,et al.  A multigrid tutorial , 1987 .

[20]  Irad Yavneh,et al.  Adaptive Multiscale Redistribution for Vector Quantization , 2005, SIAM J. Sci. Comput..

[21]  Atsuyuki Okabe,et al.  Spatial Tessellations: Concepts and Applications of Voronoi Diagrams , 1992, Wiley Series in Probability and Mathematical Statistics.

[22]  J. W. Ruge,et al.  4. Algebraic Multigrid , 1987 .

[23]  Jean-Marc Chassery,et al.  Approximated Centroidal Voronoi Diagrams for Uniform Polygonal Mesh Coarsening , 2004, Comput. Graph. Forum.

[24]  Thomas A. Manteuffel,et al.  Algebraic Multigrid Based on Element Interpolation (AMGe) , 2000, SIAM J. Sci. Comput..

[25]  David L. Neuhoff,et al.  Quantization , 2022, IEEE Trans. Inf. Theory.

[26]  Tony F. Chan,et al.  Subspace correction multi-level methods for elliptic eigenvalue problems , 2002, Numer. Linear Algebra Appl..

[27]  Franz Aurenhammer,et al.  Voronoi diagrams—a survey of a fundamental geometric data structure , 1991, CSUR.

[28]  Lili Ju,et al.  Nondegeneracy and Weak Global Convergence of the Lloyd Algorithm in Rd , 2008, SIAM J. Numer. Anal..

[29]  Steven Fortune,et al.  A sweepline algorithm for Voronoi diagrams , 1986, SCG '86.

[30]  Robert M. Gray,et al.  An Algorithm for Vector Quantizer Design , 1980, IEEE Trans. Commun..

[31]  Qiang Du,et al.  Constrained Centroidal Voronoi Tessellations for Surfaces , 2002, SIAM J. Sci. Comput..

[32]  R. Kornhuber.  Adaptive monotone multigrid methods for nonlinear variational problems , 1997 .

[33]  Jinchao Xu,et al.  The method of alternating projections and the method of subspace corrections in Hilbert space , 2002 .

[34]  E. Gelman,et al.  On multilevel iterative methods for optimization problems , 1990, Math. Program..

[35]  Qiang Du,et al.  Grid generation and optimization based on centroidal Voronoi tessellations , 2002, Appl. Math. Comput..

[36]  Desheng Wang,et al.  Tetrahedral mesh generation and optimization based on centroidal Voronoi tessellations , 2003 .

[37]  Allen Gersho,et al.  Asymptotically optimal block quantization , 1979, IEEE Trans. Inf. Theory.

[38]  Qiang Du,et al.  Probabilistic methods for centroidal Voronoi tessellations and their parallel implementations , 2002, Parallel Comput..

[39]  Zhaohui Huang,et al.  Efficient Algebraic Multigrid Algorithms and Their Convergence , 2002, SIAM J. Sci. Comput..

[40]  Qiang Du,et al.  Centroidal Voronoi Tessellations: Applications and Algorithms , 1999, SIAM Rev..

[41]  Alexander V. Trushkin,et al.  On the design of an optimal quantizer , 1993, IEEE Trans. Inf. Theory.