Representation and Compression of Multidimensional Piecewise Functions Using Surflets
We study the representation, approximation, and compression of functions in M dimensions that consist of constant or smooth regions separated by smooth (M -1)-dimensional discontinuities. Examples include images containing edges, video sequences of moving objects, and seismic data containing geologi...
Gespeichert in:
Veröffentlicht in: | IEEE transactions on information theory 2009-01, Vol.55 (1), p.374-400 |
---|---|
Hauptverfasser: | , , , |
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext bestellen |
Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
container_end_page | 400 |
---|---|
container_issue | 1 |
container_start_page | 374 |
container_title | IEEE transactions on information theory |
container_volume | 55 |
creator | Chandrasekaran, V. Wakin, M.B. Baron, D. Baraniuk, R.G. |
description | We study the representation, approximation, and compression of functions in M dimensions that consist of constant or smooth regions separated by smooth (M -1)-dimensional discontinuities. Examples include images containing edges, video sequences of moving objects, and seismic data containing geological horizons. For both function classes, we derive the optimal asymptotic approximation and compression rates based on Kolmogorov metric entropy. For piecewise constant functions, we develop a multiresolution predictive coder that achieves the optimal rate-distortion performance; for piecewise smooth functions, our coder has near-optimal rate-distortion performance. Our coder for piecewise constant functions employs surflets , a new multiscale geometric tiling consisting of M -dimensional piecewise constant atoms containing polynomial discontinuities. Our coder for piecewise smooth functions uses surfprints , which wed surflets to wavelets for piecewise smooth approximation. Both of these schemes achieve the optimal asymptotic approximation performance. Key features of our algorithms are that they carefully control the potential growth in surflet parameters at higher smoothness and do not require explicit estimation of the discontinuity. We also extend our results to the corresponding discrete function spaces for sampled data. We provide asymptotic performance results for both discrete function spaces and relate this asymptotic performance to the sampling rate and smoothness orders of the underlying functions and discontinuities. For approximation of discrete data, we propose a new scale-adaptive dictionary that contains few elements at coarse and fine scales, but many elements at medium scales. Simulation results on synthetic signals provide a comparison between surflet-based coders and previously studied approximation schemes based on wedgelets and wavelets. |
doi_str_mv | 10.1109/TIT.2008.2008153 |
format | Article |
fullrecord | <record><control><sourceid>proquest_RIE</sourceid><recordid>TN_cdi_proquest_journals_195938143</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><ieee_id>4729784</ieee_id><sourcerecordid>1618533331</sourcerecordid><originalsourceid>FETCH-LOGICAL-c425t-185ef9cfd6e1d89cf18b6884d0b54a2dda6dbf512f878e751584b4ca2422292c3</originalsourceid><addsrcrecordid>eNp9kcFr2zAUxsVoYWm6-2AXM-h6civJT7Z0LGFdAx0tW3IWivw0VBw707MZ_e9nJyGHHnqR9J5-38d7fIx9FvxGCG5uV8vVjeRc7w-hig9sJpSqclMqOGMzzoXODYD-yC6IXsYSlJAztv6Fu4SEbe_62LWZa-ts0W2nHk11F7KfQ9PHOm6xnTquyZ4jevwXCbP7ofWTjLI1xfZP9ntIocGeLtl5cA3hp-M9Z-v776vFQ_749GO5uHvMPUjV50IrDMaHukRR6_Eh9KbUGmq-UeBkXbuy3oRxzqArjZUSSsMGvJMgpTTSF3N2ffDdpe7vgNTbbSSPTeNa7AayRkApDehiJL-9SxZQVhKMHsGvb8CXbkjj2mSFUabQAiY3foB86ogSBrtLcevSqxXcTnHYMQ47JWGPcYySq6OvI--akFzrI510khutir31lwMXEfH0DZU0lYbiP1DVk4I</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>195938143</pqid></control><display><type>article</type><title>Representation and Compression of Multidimensional Piecewise Functions Using Surflets</title><source>IEEE Electronic Library (IEL)</source><creator>Chandrasekaran, V. ; Wakin, M.B. ; Baron, D. ; Baraniuk, R.G.</creator><creatorcontrib>Chandrasekaran, V. ; Wakin, M.B. ; Baron, D. ; Baraniuk, R.G.</creatorcontrib><description>We study the representation, approximation, and compression of functions in M dimensions that consist of constant or smooth regions separated by smooth (M -1)-dimensional discontinuities. Examples include images containing edges, video sequences of moving objects, and seismic data containing geological horizons. For both function classes, we derive the optimal asymptotic approximation and compression rates based on Kolmogorov metric entropy. For piecewise constant functions, we develop a multiresolution predictive coder that achieves the optimal rate-distortion performance; for piecewise smooth functions, our coder has near-optimal rate-distortion performance. Our coder for piecewise constant functions employs surflets , a new multiscale geometric tiling consisting of M -dimensional piecewise constant atoms containing polynomial discontinuities. Our coder for piecewise smooth functions uses surfprints , which wed surflets to wavelets for piecewise smooth approximation. Both of these schemes achieve the optimal asymptotic approximation performance. Key features of our algorithms are that they carefully control the potential growth in surflet parameters at higher smoothness and do not require explicit estimation of the discontinuity. We also extend our results to the corresponding discrete function spaces for sampled data. We provide asymptotic performance results for both discrete function spaces and relate this asymptotic performance to the sampling rate and smoothness orders of the underlying functions and discontinuities. For approximation of discrete data, we propose a new scale-adaptive dictionary that contains few elements at coarse and fine scales, but many elements at medium scales. Simulation results on synthetic signals provide a comparison between surflet-based coders and previously studied approximation schemes based on wedgelets and wavelets.</description><identifier>ISSN: 0018-9448</identifier><identifier>EISSN: 1557-9654</identifier><identifier>DOI: 10.1109/TIT.2008.2008153</identifier><identifier>CODEN: IETTAW</identifier><language>eng</language><publisher>New York, NY: IEEE</publisher><subject>Algorithms ; Applied sciences ; Approximation ; Asymptotic properties ; Coders ; Coding, codes ; Compressing ; Compression ; Data compression ; Dictionaries ; discontinuities ; Discontinuity ; Entropy ; Exact sciences and technology ; Geology ; Image coding ; Information theory ; Information, signal and communications theory ; Instruments ; Mathematical analysis ; metric entropy ; multidimensional signals ; Multidimensional systems ; multiscale representations ; nonlinear approximation ; Optimization ; rate-distortion ; Sampling methods ; Signal and communications theory ; Signal resolution ; Simulation ; sparse representations ; Studies ; surflets ; Telecommunications and information theory ; Two dimensional displays ; Video sequences ; wavelets</subject><ispartof>IEEE transactions on information theory, 2009-01, Vol.55 (1), p.374-400</ispartof><rights>2009 INIST-CNRS</rights><rights>Copyright Institute of Electrical and Electronics Engineers, Inc. (IEEE) Jan 2009</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c425t-185ef9cfd6e1d89cf18b6884d0b54a2dda6dbf512f878e751584b4ca2422292c3</citedby><cites>FETCH-LOGICAL-c425t-185ef9cfd6e1d89cf18b6884d0b54a2dda6dbf512f878e751584b4ca2422292c3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://ieeexplore.ieee.org/document/4729784$$EHTML$$P50$$Gieee$$H</linktohtml><link.rule.ids>314,780,784,796,4024,27923,27924,27925,54758</link.rule.ids><linktorsrc>$$Uhttps://ieeexplore.ieee.org/document/4729784$$EView_record_in_IEEE$$FView_record_in_$$GIEEE</linktorsrc><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=20985343$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Chandrasekaran, V.</creatorcontrib><creatorcontrib>Wakin, M.B.</creatorcontrib><creatorcontrib>Baron, D.</creatorcontrib><creatorcontrib>Baraniuk, R.G.</creatorcontrib><title>Representation and Compression of Multidimensional Piecewise Functions Using Surflets</title><title>IEEE transactions on information theory</title><addtitle>TIT</addtitle><description>We study the representation, approximation, and compression of functions in M dimensions that consist of constant or smooth regions separated by smooth (M -1)-dimensional discontinuities. Examples include images containing edges, video sequences of moving objects, and seismic data containing geological horizons. For both function classes, we derive the optimal asymptotic approximation and compression rates based on Kolmogorov metric entropy. For piecewise constant functions, we develop a multiresolution predictive coder that achieves the optimal rate-distortion performance; for piecewise smooth functions, our coder has near-optimal rate-distortion performance. Our coder for piecewise constant functions employs surflets , a new multiscale geometric tiling consisting of M -dimensional piecewise constant atoms containing polynomial discontinuities. Our coder for piecewise smooth functions uses surfprints , which wed surflets to wavelets for piecewise smooth approximation. Both of these schemes achieve the optimal asymptotic approximation performance. Key features of our algorithms are that they carefully control the potential growth in surflet parameters at higher smoothness and do not require explicit estimation of the discontinuity. We also extend our results to the corresponding discrete function spaces for sampled data. We provide asymptotic performance results for both discrete function spaces and relate this asymptotic performance to the sampling rate and smoothness orders of the underlying functions and discontinuities. For approximation of discrete data, we propose a new scale-adaptive dictionary that contains few elements at coarse and fine scales, but many elements at medium scales. Simulation results on synthetic signals provide a comparison between surflet-based coders and previously studied approximation schemes based on wedgelets and wavelets.</description><subject>Algorithms</subject><subject>Applied sciences</subject><subject>Approximation</subject><subject>Asymptotic properties</subject><subject>Coders</subject><subject>Coding, codes</subject><subject>Compressing</subject><subject>Compression</subject><subject>Data compression</subject><subject>Dictionaries</subject><subject>discontinuities</subject><subject>Discontinuity</subject><subject>Entropy</subject><subject>Exact sciences and technology</subject><subject>Geology</subject><subject>Image coding</subject><subject>Information theory</subject><subject>Information, signal and communications theory</subject><subject>Instruments</subject><subject>Mathematical analysis</subject><subject>metric entropy</subject><subject>multidimensional signals</subject><subject>Multidimensional systems</subject><subject>multiscale representations</subject><subject>nonlinear approximation</subject><subject>Optimization</subject><subject>rate-distortion</subject><subject>Sampling methods</subject><subject>Signal and communications theory</subject><subject>Signal resolution</subject><subject>Simulation</subject><subject>sparse representations</subject><subject>Studies</subject><subject>surflets</subject><subject>Telecommunications and information theory</subject><subject>Two dimensional displays</subject><subject>Video sequences</subject><subject>wavelets</subject><issn>0018-9448</issn><issn>1557-9654</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2009</creationdate><recordtype>article</recordtype><sourceid>RIE</sourceid><recordid>eNp9kcFr2zAUxsVoYWm6-2AXM-h6civJT7Z0LGFdAx0tW3IWivw0VBw707MZ_e9nJyGHHnqR9J5-38d7fIx9FvxGCG5uV8vVjeRc7w-hig9sJpSqclMqOGMzzoXODYD-yC6IXsYSlJAztv6Fu4SEbe_62LWZa-ts0W2nHk11F7KfQ9PHOm6xnTquyZ4jevwXCbP7ofWTjLI1xfZP9ntIocGeLtl5cA3hp-M9Z-v776vFQ_749GO5uHvMPUjV50IrDMaHukRR6_Eh9KbUGmq-UeBkXbuy3oRxzqArjZUSSsMGvJMgpTTSF3N2ffDdpe7vgNTbbSSPTeNa7AayRkApDehiJL-9SxZQVhKMHsGvb8CXbkjj2mSFUabQAiY3foB86ogSBrtLcevSqxXcTnHYMQ47JWGPcYySq6OvI--akFzrI510khutir31lwMXEfH0DZU0lYbiP1DVk4I</recordid><startdate>200901</startdate><enddate>200901</enddate><creator>Chandrasekaran, V.</creator><creator>Wakin, M.B.</creator><creator>Baron, D.</creator><creator>Baraniuk, R.G.</creator><general>IEEE</general><general>Institute of Electrical and Electronics Engineers</general><general>The Institute of Electrical and Electronics Engineers, Inc. (IEEE)</general><scope>97E</scope><scope>RIA</scope><scope>RIE</scope><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7SP</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>F28</scope><scope>FR3</scope></search><sort><creationdate>200901</creationdate><title>Representation and Compression of Multidimensional Piecewise Functions Using Surflets</title><author>Chandrasekaran, V. ; Wakin, M.B. ; Baron, D. ; Baraniuk, R.G.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c425t-185ef9cfd6e1d89cf18b6884d0b54a2dda6dbf512f878e751584b4ca2422292c3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2009</creationdate><topic>Algorithms</topic><topic>Applied sciences</topic><topic>Approximation</topic><topic>Asymptotic properties</topic><topic>Coders</topic><topic>Coding, codes</topic><topic>Compressing</topic><topic>Compression</topic><topic>Data compression</topic><topic>Dictionaries</topic><topic>discontinuities</topic><topic>Discontinuity</topic><topic>Entropy</topic><topic>Exact sciences and technology</topic><topic>Geology</topic><topic>Image coding</topic><topic>Information theory</topic><topic>Information, signal and communications theory</topic><topic>Instruments</topic><topic>Mathematical analysis</topic><topic>metric entropy</topic><topic>multidimensional signals</topic><topic>Multidimensional systems</topic><topic>multiscale representations</topic><topic>nonlinear approximation</topic><topic>Optimization</topic><topic>rate-distortion</topic><topic>Sampling methods</topic><topic>Signal and communications theory</topic><topic>Signal resolution</topic><topic>Simulation</topic><topic>sparse representations</topic><topic>Studies</topic><topic>surflets</topic><topic>Telecommunications and information theory</topic><topic>Two dimensional displays</topic><topic>Video sequences</topic><topic>wavelets</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Chandrasekaran, V.</creatorcontrib><creatorcontrib>Wakin, M.B.</creatorcontrib><creatorcontrib>Baron, D.</creatorcontrib><creatorcontrib>Baraniuk, R.G.</creatorcontrib><collection>IEEE All-Society Periodicals Package (ASPP) 2005-present</collection><collection>IEEE All-Society Periodicals Package (ASPP) 1998-Present</collection><collection>IEEE Electronic Library (IEL)</collection><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Electronics & Communications Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>ANTE: Abstracts in New Technology & Engineering</collection><collection>Engineering Research Database</collection><jtitle>IEEE transactions on information theory</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Chandrasekaran, V.</au><au>Wakin, M.B.</au><au>Baron, D.</au><au>Baraniuk, R.G.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Representation and Compression of Multidimensional Piecewise Functions Using Surflets</atitle><jtitle>IEEE transactions on information theory</jtitle><stitle>TIT</stitle><date>2009-01</date><risdate>2009</risdate><volume>55</volume><issue>1</issue><spage>374</spage><epage>400</epage><pages>374-400</pages><issn>0018-9448</issn><eissn>1557-9654</eissn><coden>IETTAW</coden><abstract>We study the representation, approximation, and compression of functions in M dimensions that consist of constant or smooth regions separated by smooth (M -1)-dimensional discontinuities. Examples include images containing edges, video sequences of moving objects, and seismic data containing geological horizons. For both function classes, we derive the optimal asymptotic approximation and compression rates based on Kolmogorov metric entropy. For piecewise constant functions, we develop a multiresolution predictive coder that achieves the optimal rate-distortion performance; for piecewise smooth functions, our coder has near-optimal rate-distortion performance. Our coder for piecewise constant functions employs surflets , a new multiscale geometric tiling consisting of M -dimensional piecewise constant atoms containing polynomial discontinuities. Our coder for piecewise smooth functions uses surfprints , which wed surflets to wavelets for piecewise smooth approximation. Both of these schemes achieve the optimal asymptotic approximation performance. Key features of our algorithms are that they carefully control the potential growth in surflet parameters at higher smoothness and do not require explicit estimation of the discontinuity. We also extend our results to the corresponding discrete function spaces for sampled data. We provide asymptotic performance results for both discrete function spaces and relate this asymptotic performance to the sampling rate and smoothness orders of the underlying functions and discontinuities. For approximation of discrete data, we propose a new scale-adaptive dictionary that contains few elements at coarse and fine scales, but many elements at medium scales. Simulation results on synthetic signals provide a comparison between surflet-based coders and previously studied approximation schemes based on wedgelets and wavelets.</abstract><cop>New York, NY</cop><pub>IEEE</pub><doi>10.1109/TIT.2008.2008153</doi><tpages>27</tpages><oa>free_for_read</oa></addata></record> |
fulltext | fulltext_linktorsrc |
identifier | ISSN: 0018-9448 |
ispartof | IEEE transactions on information theory, 2009-01, Vol.55 (1), p.374-400 |
issn | 0018-9448 1557-9654 |
language | eng |
recordid | cdi_proquest_journals_195938143 |
source | IEEE Electronic Library (IEL) |
subjects | Algorithms Applied sciences Approximation Asymptotic properties Coders Coding, codes Compressing Compression Data compression Dictionaries discontinuities Discontinuity Entropy Exact sciences and technology Geology Image coding Information theory Information, signal and communications theory Instruments Mathematical analysis metric entropy multidimensional signals Multidimensional systems multiscale representations nonlinear approximation Optimization rate-distortion Sampling methods Signal and communications theory Signal resolution Simulation sparse representations Studies surflets Telecommunications and information theory Two dimensional displays Video sequences wavelets |
title | Representation and Compression of Multidimensional Piecewise Functions Using Surflets |
url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2024-12-25T02%3A05%3A31IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_RIE&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Representation%20and%20Compression%20of%20Multidimensional%20Piecewise%20Functions%20Using%20Surflets&rft.jtitle=IEEE%20transactions%20on%20information%20theory&rft.au=Chandrasekaran,%20V.&rft.date=2009-01&rft.volume=55&rft.issue=1&rft.spage=374&rft.epage=400&rft.pages=374-400&rft.issn=0018-9448&rft.eissn=1557-9654&rft.coden=IETTAW&rft_id=info:doi/10.1109/TIT.2008.2008153&rft_dat=%3Cproquest_RIE%3E1618533331%3C/proquest_RIE%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=195938143&rft_id=info:pmid/&rft_ieee_id=4729784&rfr_iscdi=true |