Multiscale Approximation of Piecewise Smooth Two-Dimensional Function using Normal Triangulated Meshes

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dc.citation.bibtexNamearticleen_US
dc.citation.firstpage92en_US
dc.citation.issueNumber1en_US
dc.citation.journalTitleJournal of Applied and Computational Harmonic Analysisen_US
dc.citation.lastpage130en_US
dc.citation.volumeNumber19en_US
dc.contributor.authorJansen, Maartenen_US
dc.contributor.authorBaraniuk, Richard G.en_US
dc.contributor.authorLavu, Sridharen_US
dc.contributor.orgDigital Signal Processing (http://dsp.rice.edu/)en_US
dc.date.accessioned2007-10-31T00:47:23Zen_US
dc.date.available2007-10-31T00:47:23Zen_US
dc.date.issued2005-07-01en_US
dc.date.modified2006-07-19en_US
dc.date.submitted2006-07-17en_US
dc.descriptionJournal Paperen_US
dc.description.abstractMultiresolution triangulation meshes are widely used in computer graphics for representing three-dimensional(3-d) shapes. We propose to use these tools to represent 2-d piecewise smooth functions such as grayscale images,because triangles have potential to more efficiently approximate the discontinuities between the smooth pieces than other standard tools like wavelets. We show that normal mesh subdivision is an efficient triangulation, thanks to its local adaptivity to the discontinuities. Indeed, we prove that, within a certain function class, the normal mesh representation has an optimal asymptotic error decay rate as the number of terms in the representation grows. This function class is the so-called horizon class comprising constant regions separated by smooth discontinuities,where the line of discontinuity is C2 continuous. This optimal decay rate is possible because normal meshes automatically generate a polyline (piecewise linear) approximation of each discontinuity, unlike the blocky piecewise constant approximation of tensor product wavelets. In this way, the proposed nonlinear multiscale normal mesh decomposition is an anisotropic representation of the 2-d function. The same idea of anisotropic representations lies at the basis of decompositions such as wedgelet and curvelet transforms, but the proposed normal mesh approach has a unique construction.en_US
dc.description.sponsorshipTexas Instrumentsen_US
dc.description.sponsorshipDefense Advanced Research Projects Agencyen_US
dc.description.sponsorshipOffice of Naval Researchen_US
dc.description.sponsorshipNational Science Foundationen_US
dc.description.sponsorshipAir Force Office of Scientific Researchen_US
dc.identifier.citationM. Jansen, R. G. Baraniuk and S. Lavu, "Multiscale Approximation of Piecewise Smooth Two-Dimensional Function using Normal Triangulated Meshes," <i>Journal of Applied and Computational Harmonic Analysis,</i> vol. 19, no. 1, 2005.en_US
dc.identifier.doihttp://dx.doi.org/10.1016/j.acha.2005.02.006en_US
dc.identifier.urihttps://hdl.handle.net/1911/19963en_US
dc.language.isoengen_US
dc.subjectNormal offsetsen_US
dc.subjectMeshen_US
dc.subjectImageen_US
dc.subjectMultiresolutionen_US
dc.subjectWaveleten_US
dc.subjectApproximationen_US
dc.subject.keywordNormal offsetsen_US
dc.subject.keywordMeshen_US
dc.subject.keywordImageen_US
dc.subject.keywordMultiresolutionen_US
dc.subject.keywordWaveleten_US
dc.subject.keywordApproximationen_US
dc.subject.otherImage Processing and Pattern analysisen_US
dc.subject.otherMultiscale Methodsen_US
dc.subject.otherMultiscale geometry processingen_US
dc.titleMultiscale Approximation of Piecewise Smooth Two-Dimensional Function using Normal Triangulated Meshesen_US
dc.typeJournal articleen_US
dc.type.dcmiTexten_US
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