Adaptive Multilinear Tensor Product Wavelets

Weiss, Kenneth; Lindstrom, Peter

doi:10.1109/TVCG.2015.2467412

Title: Adaptive Multilinear Tensor Product Wavelets

Abstract

Many foundational visualization techniques including isosurfacing, direct volume rendering and texture mapping rely on piecewise multilinear interpolation over the cells of a mesh. However, there has not been much focus within the visualization community on techniques that efficiently generate and encode globally continuous functions defined by the union of multilinear cells. Wavelets provide a rich context for analyzing and processing complicated datasets. In this paper, we exploit adaptive regular refinement as a means of representing and evaluating functions described by a subset of their nonzero wavelet coefficients. We analyze the dependencies involved in the wavelet transform and describe how to generate and represent the coarsest adaptive mesh with nodal function values such that the inverse wavelet transform is exactly reproduced via simple interpolation (subdivision) over the mesh elements. This allows for an adaptive, sparse representation of the function with on-demand evaluation at any point in the domain. In conclusion, we focus on the popular wavelets formed by tensor products of linear B-splines, resulting in an adaptive, nonconforming but crack-free quadtree (2D) or octree (3D) mesh that allows reproducing globally continuous functions via multilinear interpolation over its cells.

Authors:

Weiss, Kenneth ^[1]; Lindstrom, Peter ^[1]

Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

Publication Date:: Wed Aug 12 00:00:00 EDT 2015

Research Org.:: Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

OSTI Identifier:: 1415549

Report Number(s):: LLNL-JRNL-644249
Journal ID: ISSN 1077-2626

Grant/Contract Number:: AC52-07NA27344

Resource Type:: Accepted Manuscript

Journal Name:: IEEE Transactions on Visualization and Computer Graphics

Additional Journal Information:: Journal Volume: 22; Journal Issue: 1; Journal ID: ISSN 1077-2626

Publisher:: IEEE

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; multilinear interpolation; adaptive wavelets; multiresolution models; octrees; continuous reconstruction

Citation Formats


                    Weiss, Kenneth, and Lindstrom, Peter. Adaptive Multilinear Tensor Product Wavelets.  United States: N. p., 2015. 
Web.  doi:10.1109/TVCG.2015.2467412.

Copy to clipboard


                    Weiss, Kenneth, & Lindstrom, Peter. Adaptive Multilinear Tensor Product Wavelets.  United States.  https://doi.org/10.1109/TVCG.2015.2467412

Copy to clipboard


                    Weiss, Kenneth, and Lindstrom, Peter. Wed .  
"Adaptive Multilinear Tensor Product Wavelets".  United States.  https://doi.org/10.1109/TVCG.2015.2467412.  https://www.osti.gov/servlets/purl/1415549.

Copy to clipboard


                    
@article{osti_1415549,

  title        = {Adaptive Multilinear Tensor Product Wavelets},

  author       = {Weiss, Kenneth and Lindstrom, Peter},

  abstractNote = {Many foundational visualization techniques including isosurfacing, direct volume rendering and texture mapping rely on piecewise multilinear interpolation over the cells of a mesh. However, there has not been much focus within the visualization community on techniques that efficiently generate and encode globally continuous functions defined by the union of multilinear cells. Wavelets provide a rich context for analyzing and processing complicated datasets. In this paper, we exploit adaptive regular refinement as a means of representing and evaluating functions described by a subset of their nonzero wavelet coefficients. We analyze the dependencies involved in the wavelet transform and describe how to generate and represent the coarsest adaptive mesh with nodal function values such that the inverse wavelet transform is exactly reproduced via simple interpolation (subdivision) over the mesh elements. This allows for an adaptive, sparse representation of the function with on-demand evaluation at any point in the domain. In conclusion, we focus on the popular wavelets formed by tensor products of linear B-splines, resulting in an adaptive, nonconforming but crack-free quadtree (2D) or octree (3D) mesh that allows reproducing globally continuous functions via multilinear interpolation over its cells.},

  doi          = {10.1109/TVCG.2015.2467412},

  journal      = {IEEE Transactions on Visualization and Computer Graphics},

  number       = 1,

  volume       = 22,

  place        = {United States},

  year         = {Wed Aug 12 00:00:00 EDT 2015},

  month        = {Wed Aug 12 00:00:00 EDT 2015}

}

Copy to clipboard

Journal Article:

Free Publicly Available Full Text

Accepted Manuscript (DOE)

Publisher's Version of Record

https://doi.org/10.1109/TVCG.2015.2467412

Other availability

Search WorldCat to find libraries that may hold this journal

Citation Metrics:

Cited by: 5 works

Citation information provided by
Web of Science

Save / Share:

Export Metadata

Save to My Library

Works referencing / citing this record:

A survey on visualization of tensor field
journal, March 2019

Bi, Chongke; Yang, Lu; Duan, Yulin
Journal of Visualization, Vol. 22, Issue 3
DOI: 10.1007/s12650-019-00555-8

Data Reduction Techniques for Simulation, Visualization and Data Analysis: Survey on Scientific Data Reduction Techniques
journal, March 2018

Li, S.; Marsaglia, N.; Garth, C.
Computer Graphics Forum, Vol. 37, Issue 6
DOI: 10.1111/cgf.13336

Similar Records in DOE PAGES and OSTI.GOV collections:

Hierarchical Volume Representation with 3{radical}2 Subdivision and Trivariate B-Spline Wavelets

Conference Linsen, L ; Gray, JT ; Pascucci, V ; ...

Multiresolution methods provide a means for representing data at multiple levels of detail. They are typically based on a hierarchical data organization scheme and update rules needed for data value computation. We use a data organization that is based on what we call n{radical}2 subdivision. The main advantage of subdivision, compared to quadtree (n = 2) or octree (n = 3) organizations, is that the number of vertices is only doubled in each subdivision step instead of multiplied by a factor of four or eight, respectively. To update data values we use n-variate B-spline wavelets, which yields better approximations formore »« less
Full Text Available
AMM: Adaptive Multilinear Meshes

Journal Article Bhatia, Harsh ; Hoang, Duong ; Morrical, Nate ; ... - IEEE Transactions on Visualization and Computer Graphics

Adaptive representations are increasingly indispensable for reducing the in-memory and on-disk footprints of large-scale data. Usual solutions are designed broadly along two themes: reducing data precision, e.g., through compression, or adapting data resolution, e.g., using spatial hierarchies. Additionally, recent research suggests that combining the two approaches, i.e., adapting both resolution and precision simultaneously, can offer significant gains over using them individually. However, there currently exist no practical solutions to creating and evaluating such representations at scale. In this work, we present a new resolution-precision-adaptive representation to support hybrid data reduction schemes and offer an interface to existing tools and algorithms.more »« less
https://doi.org/10.1109/tvcg.2022.3165392

Full Text Available
Adaptive mesh refinement for steady flows in Nek5000

Journal Article Offermans, N. ; Peplinski, A. ; Marin, O. ; ... - Computers and Fluids

Adaptive mesh refinement is performed in the framework of the spectral element method augmented by approaches to error estimation and control. The h-refinement technique is used for adapting the mesh, where selected grid elements are split by a quadtree (2D) or octree (3D) structure. Continuity between parent-child elements is enforced by high-order interpolation of the solution across the common faces. Parallel mesh partitioning and grid management respectively, are taken care of by the external libraries ParMETIS and p4est. Two methods are considered for estimating and controlling the error of the solution. The first error estimate is local and based onmore »« less
Cited by 10
https://doi.org/10.1016/j.compfluid.2019.104352

Full Text Available
Large Scale Isosurface Bicubic Subdivision-Surface Wavelets for Representation and Visualization

Conference Bertram, M ; Duchaineau, M A ; Hamann, B ; ...

We introduce a new subdivision-surface wavelet transform for arbitrary two-manifolds with boundary that is the first to use simple lifting-style filtering operations with bicubic precision. We also describe a conversion process for re-mapping large-scale isosurfaces to have subdivision connectivity and fair parameterizations so that the new wavelet transform can be used for compression and visualization. The main idea enabling our wavelet transform is the circular symmetrization of the filters in irregular neighborhoods, which replaces the traditional separation of filters into two 1-D passes. Our wavelet transform uses polygonal base meshes to represent surface topology, from which a Catmull-Clark-style subdivision hierarchymore »« less
https://doi.org/10.1109/VISUAL.2000.885720

Full Text Available
Generalized b-spline subdivision-surface wavelets and lossless compression

Conference Bertram, M ; Duchaineau, M A ; Hamann, B ; ...

We present a new construction of wavelets on arbitrary two-manifold topology for geometry compression. The constructed wavelets generalize symmetric tensor product wavelets with associated B-spline scaling functions to irregular polygonal base mesh domains. The wavelets and scaling functions are tensor products almost everywhere, except in the neighborhoods of some extraordinary points (points of valence unequal four) in the base mesh that defines the topology. The compression of arbitrary polygonal meshes representing isosurfaces of scalar-valued trivariate functions is a primary application. The main contribution of this paper is the generalization of lifted symmetric tensor product B-spline wavelets to two-manifold geometries. Surfacesmore »« less
Full Text Available

Similar Records

Title: Adaptive Multilinear Tensor Product Wavelets

Abstract

Citation Formats

A survey on visualization of tensor field journal, March 2019

Data Reduction Techniques for Simulation, Visualization and Data Analysis: Survey on Scientific Data Reduction Techniques journal, March 2018

A survey on visualization of tensor field
journal, March 2019

Data Reduction Techniques for Simulation, Visualization and Data Analysis: Survey on Scientific Data Reduction Techniques
journal, March 2018