ÿþ<html xmlns:v="urn:schemas-microsoft-com:vml" xmlns:o="urn:schemas-microsoft-com:office:office" xmlns:w="urn:schemas-microsoft-com:office:word" xmlns:m="http://schemas.microsoft.com/office/2004/12/omml" xmlns="http://www.w3.org/TR/REC-html40"> <head> <meta http-equiv=Content-Type content="text/html; charset=unicode"> <meta name=ProgId content=Word.Document> <meta name=Generator content="Microsoft Word 12"> <meta name=Originator content="Microsoft Word 12"> <link rel=File-List href="index_files/filelist.xml"> <link rel=Edit-Time-Data href="index_files/editdata.mso"> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} w\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} </style> <![endif]--> <title>L1 Homotopy</title> <!--[if gte mso 9]><xml> <o:DocumentProperties> <o:Author>Salman Asif</o:Author> <o:LastAuthor>School of ECE</o:LastAuthor> <o:Revision>102</o:Revision> <o:TotalTime>197</o:TotalTime> <o:Created>2009-03-15T08:45:00Z</o:Created> <o:LastSaved>2011-07-19T20:42:00Z</o:LastSaved> <o:Pages>2</o:Pages> <o:Words>400</o:Words> <o:Characters>2282</o:Characters> <o:Company>GT</o:Company> <o:Lines>19</o:Lines> <o:Paragraphs>5</o:Paragraphs> <o:CharactersWithSpaces>2677</o:CharactersWithSpaces> <o:Version>12.00</o:Version> </o:DocumentProperties> </xml><![endif]--> <link rel=themeData href="index_files/themedata.thmx"> <link rel=colorSchemeMapping href="index_files/colorschememapping.xml"> <!--[if gte mso 9]><xml> <w:WordDocument> <w:Zoom>90</w:Zoom> <w:DontDisplayPageBoundaries/> <w:DisplayBackgroundShape/> <w:SpellingState>Clean</w:SpellingState> <w:GrammarState>Clean</w:GrammarState> <w:TrackMoves>false</w:TrackMoves> <w:TrackFormatting/> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:DoNotPromoteQF/> <w:LidThemeOther>EN-US</w:LidThemeOther> <w:LidThemeAsian>X-NONE</w:LidThemeAsian> <w:LidThemeComplexScript>X-NONE</w:LidThemeComplexScript> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> <w:SplitPgBreakAndParaMark/> <w:DontVertAlignCellWithSp/> <w:DontBreakConstrainedForcedTables/> <w:DontVertAlignInTxbx/> <w:Word11KerningPairs/> <w:CachedColBalance/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> <m:mathPr> <m:mathFont m:val="Cambria Math"/> <m:brkBin m:val="before"/> <m:brkBinSub m:val="&#45;-"/> <m:smallFrac m:val="off"/> <m:dispDef/> <m:lMargin m:val="0"/> <m:rMargin m:val="0"/> <m:defJc m:val="centerGroup"/> <m:wrapIndent m:val="1440"/> <m:intLim m:val="subSup"/> <m:naryLim m:val="undOvr"/> </m:mathPr></w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" DefUnhideWhenUsed="false" DefSemiHidden="false" DefQFormat="false" LatentStyleCount="267"> <w:LsdException Locked="false" QFormat="true" Name="Normal"/> <w:LsdException Locked="false" QFormat="true" Name="heading 1"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 2"/> <w:LsdException Locked="false" QFormat="true" Name="heading 3"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 4"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 5"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 6"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 7"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 8"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="heading 9"/> <w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="caption"/> <w:LsdException Locked="false" QFormat="true" Name="Title"/> <w:LsdException Locked="false" Priority="1" Name="Default Paragraph Font"/> <w:LsdException Locked="false" QFormat="true" Name="Subtitle"/> <w:LsdException Locked="false" QFormat="true" Name="Strong"/> <w:LsdException Locked="false" QFormat="true" Name="Emphasis"/> <w:LsdException Locked="false" Priority="99" Name="No List"/> <w:LsdException Locked="false" Priority="99" SemiHidden="true" Name="Placeholder Text"/> <w:LsdException Locked="false" Priority="1" QFormat="true" Name="No Spacing"/> <w:LsdException Locked="false" Priority="60" Name="Light Shading"/> <w:LsdException Locked="false" Priority="61" Name="Light List"/> <w:LsdException Locked="false" Priority="62" Name="Light Grid"/> <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1"/> <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2"/> <w:LsdException Locked="false" Priority="65" Name="Medium List 1"/> <w:LsdException Locked="false" Priority="66" Name="Medium List 2"/> <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1"/> <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2"/> <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3"/> <w:LsdException Locked="false" Priority="70" Name="Dark List"/> <w:LsdException Locked="false" Priority="71" Name="Colorful Shading"/> <w:LsdException Locked="false" Priority="72" Name="Colorful List"/> <w:LsdException Locked="false" Priority="73" Name="Colorful Grid"/> <w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 1"/> <w:LsdException Locked="false" Priority="61" Name="Light List Accent 1"/> <w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 1"/> <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 1"/> <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 1"/> <w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 1"/> <w:LsdException Locked="false" Priority="99" SemiHidden="true" Name="Revision"/> <w:LsdException Locked="false" Priority="34" QFormat="true" Name="List Paragraph"/> <w:LsdException Locked="false" Priority="29" QFormat="true" Name="Quote"/> <w:LsdException Locked="false" Priority="30" QFormat="true" Name="Intense Quote"/> <w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 1"/> <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 1"/> <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 1"/> <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 1"/> <w:LsdException Locked="false" Priority="70" Name="Dark List Accent 1"/> <w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 1"/> <w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 1"/> <w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 1"/> <w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 2"/> <w:LsdException Locked="false" Priority="61" Name="Light List Accent 2"/> <w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 2"/> <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 2"/> <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 2"/> <w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 2"/> <w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 2"/> <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 2"/> <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 2"/> <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 2"/> <w:LsdException Locked="false" Priority="70" Name="Dark List Accent 2"/> <w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 2"/> <w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 2"/> <w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 2"/> <w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 3"/> <w:LsdException Locked="false" Priority="61" Name="Light List Accent 3"/> <w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 3"/> <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 3"/> <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 3"/> <w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 3"/> <w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 3"/> <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 3"/> <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 3"/> <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 3"/> <w:LsdException Locked="false" Priority="70" Name="Dark List Accent 3"/> <w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 3"/> <w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 3"/> <w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 3"/> <w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 4"/> <w:LsdException Locked="false" Priority="61" Name="Light List Accent 4"/> <w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 4"/> <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 4"/> <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 4"/> <w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 4"/> <w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 4"/> <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 4"/> <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 4"/> <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 4"/> <w:LsdException Locked="false" Priority="70" Name="Dark List Accent 4"/> <w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 4"/> <w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 4"/> <w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 4"/> <w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 5"/> <w:LsdException Locked="false" Priority="61" Name="Light List Accent 5"/> <w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 5"/> <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 5"/> <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 5"/> <w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 5"/> <w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 5"/> <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 5"/> <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 5"/> <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 5"/> <w:LsdException Locked="false" Priority="70" Name="Dark List Accent 5"/> <w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 5"/> <w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 5"/> <w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 5"/> <w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 6"/> <w:LsdException Locked="false" Priority="61" Name="Light List Accent 6"/> <w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 6"/> <w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 6"/> <w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 6"/> <w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 6"/> <w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 6"/> <w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 6"/> <w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 6"/> <w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 6"/> <w:LsdException Locked="false" Priority="70" Name="Dark List Accent 6"/> <w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 6"/> <w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 6"/> <w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 6"/> <w:LsdException Locked="false" Priority="19" QFormat="true" Name="Subtle Emphasis"/> <w:LsdException Locked="false" Priority="21" QFormat="true" Name="Intense Emphasis"/> <w:LsdException Locked="false" Priority="31" QFormat="true" Name="Subtle Reference"/> <w:LsdException Locked="false" Priority="32" QFormat="true" Name="Intense Reference"/> <w:LsdException Locked="false" Priority="33" QFormat="true" Name="Book Title"/> <w:LsdException Locked="false" Priority="37" SemiHidden="true" UnhideWhenUsed="true" Name="Bibliography"/> <w:LsdException Locked="false" Priority="39" SemiHidden="true" UnhideWhenUsed="true" QFormat="true" Name="TOC Heading"/> </w:LatentStyles> </xml><![endif]--> <style> <!-- /* Font Definitions */ @font-face {font-family:Wingdings; panose-1:5 0 0 0 0 0 0 0 0 0; mso-font-charset:2; mso-generic-font-family:auto; mso-font-pitch:variable; mso-font-signature:0 268435456 0 0 -2147483648 0;} @font-face {font-family:"Cambria Math"; panose-1:2 4 5 3 5 4 6 3 2 4; mso-font-charset:1; mso-generic-font-family:roman; mso-font-format:other; mso-font-pitch:variable; mso-font-signature:0 0 0 0 0 0;} @font-face {font-family:Tahoma; panose-1:2 11 6 4 3 5 4 4 2 4; mso-font-charset:0; mso-generic-font-family:swiss; mso-font-pitch:variable; mso-font-signature:1627400839 -2147483648 8 0 66047 0;} @font-face {font-family:Consolas; panose-1:2 11 6 9 2 2 4 3 2 4; mso-font-charset:0; mso-generic-font-family:roman; mso-font-format:other; mso-font-pitch:auto; mso-font-signature:0 0 0 0 0 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman","serif"; mso-fareast-font-family:"Times New Roman";} h3 {mso-style-unhide:no; mso-style-qformat:yes; mso-style-link:"Heading 3 Char"; mso-style-next:Normal; margin-top:12.0pt; margin-right:0in; margin-bottom:3.0pt; margin-left:0in; mso-pagination:widow-orphan; page-break-after:avoid; mso-outline-level:3; font-size:13.0pt; font-family:"Arial","sans-serif"; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast; font-weight:bold;} a:link, span.MsoHyperlink {mso-style-unhide:no; color:blue; text-decoration:underline; text-underline:single;} a:visited, span.MsoHyperlinkFollowed {mso-style-unhide:no; color:purple; text-decoration:underline; text-underline:single;} p {mso-style-unhide:no; mso-margin-top-alt:auto; margin-right:0in; mso-margin-bottom-alt:auto; margin-left:0in; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman","serif"; mso-fareast-font-family:"Times New Roman";} pre {mso-style-unhide:no; mso-style-link:"HTML Preformatted Char"; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; font-size:10.0pt; font-family:"Courier New"; mso-fareast-font-family:"Times New Roman";} p.MsoAcetate, li.MsoAcetate, div.MsoAcetate {mso-style-unhide:no; mso-style-link:"Balloon Text Char"; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:8.0pt; font-family:"Tahoma","sans-serif"; mso-fareast-font-family:"Times New Roman";} p.MsoListParagraph, li.MsoListParagraph, div.MsoListParagraph {mso-style-priority:34; mso-style-unhide:no; mso-style-qformat:yes; margin-top:0in; margin-right:0in; margin-bottom:0in; margin-left:.5in; margin-bottom:.0001pt; mso-add-space:auto; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman","serif"; mso-fareast-font-family:"Times New Roman";} p.MsoListParagraphCxSpFirst, li.MsoListParagraphCxSpFirst, div.MsoListParagraphCxSpFirst {mso-style-priority:34; mso-style-unhide:no; mso-style-qformat:yes; mso-style-type:export-only; margin-top:0in; margin-right:0in; margin-bottom:0in; margin-left:.5in; margin-bottom:.0001pt; mso-add-space:auto; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman","serif"; mso-fareast-font-family:"Times New Roman";} p.MsoListParagraphCxSpMiddle, li.MsoListParagraphCxSpMiddle, div.MsoListParagraphCxSpMiddle {mso-style-priority:34; mso-style-unhide:no; mso-style-qformat:yes; mso-style-type:export-only; margin-top:0in; margin-right:0in; margin-bottom:0in; margin-left:.5in; margin-bottom:.0001pt; mso-add-space:auto; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman","serif"; mso-fareast-font-family:"Times New Roman";} p.MsoListParagraphCxSpLast, li.MsoListParagraphCxSpLast, div.MsoListParagraphCxSpLast {mso-style-priority:34; mso-style-unhide:no; mso-style-qformat:yes; mso-style-type:export-only; margin-top:0in; margin-right:0in; margin-bottom:0in; margin-left:.5in; margin-bottom:.0001pt; mso-add-space:auto; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman","serif"; mso-fareast-font-family:"Times New Roman";} span.Heading3Char {mso-style-name:"Heading 3 Char"; mso-style-unhide:no; mso-style-locked:yes; mso-style-link:"Heading 3"; mso-ansi-font-size:12.0pt; mso-bidi-font-size:12.0pt; font-family:"Cambria","serif"; mso-ascii-font-family:Cambria; mso-ascii-theme-font:major-latin; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:major-fareast; mso-hansi-font-family:Cambria; mso-hansi-theme-font:major-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:major-bidi; color:#4F81BD; mso-themecolor:accent1; font-weight:bold;} span.HTMLPreformattedChar {mso-style-name:"HTML Preformatted Char"; mso-style-unhide:no; mso-style-locked:yes; mso-style-link:"HTML Preformatted"; font-family:"Consolas","serif"; mso-ascii-font-family:Consolas; mso-hansi-font-family:Consolas;} span.BalloonTextChar {mso-style-name:"Balloon Text Char"; mso-style-unhide:no; mso-style-locked:yes; mso-style-link:"Balloon Text"; mso-ansi-font-size:8.0pt; mso-bidi-font-size:8.0pt; font-family:"Tahoma","sans-serif"; mso-ascii-font-family:Tahoma; mso-hansi-font-family:Tahoma; mso-bidi-font-family:Tahoma;} span.SpellE {mso-style-name:""; mso-spl-e:yes;} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; font-size:10.0pt; mso-ansi-font-size:10.0pt; mso-bidi-font-size:10.0pt;} @page WordSection1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.WordSection1 {page:WordSection1;} /* List Definitions */ @list l0 {mso-list-id:61149999; mso-list-type:hybrid; mso-list-template-ids:1549273324 67698689 67698691 67698693 67698689 67698691 67698693 67698689 67698691 67698693;} @list l0:level1 {mso-level-number-format:bullet; mso-level-text:·ð; mso-level-tab-stop:none; mso-level-number-position:left; text-indent:-.25in; font-family:Symbol;} @list l0:level2 {mso-level-number-format:bullet; mso-level-text:o; mso-level-tab-stop:none; mso-level-number-position:left; text-indent:-.25in; font-family:"Courier New";} @list l0:level3 {mso-level-number-format:bullet; mso-level-text:§ð; mso-level-tab-stop:none; mso-level-number-position:left; text-indent:-.25in; font-family:Wingdings;} @list l1 {mso-list-id:246311963; mso-list-type:hybrid; mso-list-template-ids:439901630 67698689 67698691 67698693 67698689 67698691 67698693 67698689 67698691 67698693;} @list l1:level1 {mso-level-number-format:bullet; mso-level-text:·ð; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in; font-family:Symbol;} @list l1:level2 {mso-level-tab-stop:1.0in; mso-level-number-position:left; text-indent:-.25in;} @list l1:level3 {mso-level-tab-stop:1.5in; mso-level-number-position:left; text-indent:-.25in;} @list l1:level4 {mso-level-tab-stop:2.0in; mso-level-number-position:left; text-indent:-.25in;} @list l1:level5 {mso-level-tab-stop:2.5in; mso-level-number-position:left; text-indent:-.25in;} @list l1:level6 {mso-level-tab-stop:3.0in; mso-level-number-position:left; text-indent:-.25in;} @list l1:level7 {mso-level-tab-stop:3.5in; mso-level-number-position:left; text-indent:-.25in;} @list l1:level8 {mso-level-tab-stop:4.0in; mso-level-number-position:left; text-indent:-.25in;} @list l1:level9 {mso-level-tab-stop:4.5in; mso-level-number-position:left; text-indent:-.25in;} @list l2 {mso-list-id:605118564; mso-list-type:hybrid; mso-list-template-ids:-508280962 67698689 67698691 67698693 67698689 67698691 67698693 67698689 67698691 67698693;} @list l2:level1 {mso-level-number-format:bullet; mso-level-text:·ð; mso-level-tab-stop:.25in; mso-level-number-position:left; margin-left:.25in; text-indent:-.25in; font-family:Symbol;} @list l2:level2 {mso-level-tab-stop:.75in; mso-level-number-position:left; margin-left:.75in; text-indent:-.25in;} @list l2:level3 {mso-level-tab-stop:1.25in; mso-level-number-position:left; margin-left:1.25in; text-indent:-.25in;} @list l2:level4 {mso-level-tab-stop:1.75in; mso-level-number-position:left; margin-left:1.75in; text-indent:-.25in;} @list l2:level5 {mso-level-tab-stop:2.25in; mso-level-number-position:left; margin-left:2.25in; text-indent:-.25in;} @list l2:level6 {mso-level-tab-stop:2.75in; mso-level-number-position:left; margin-left:2.75in; text-indent:-.25in;} @list l2:level7 {mso-level-tab-stop:3.25in; mso-level-number-position:left; margin-left:3.25in; text-indent:-.25in;} @list l2:level8 {mso-level-tab-stop:3.75in; mso-level-number-position:left; margin-left:3.75in; text-indent:-.25in;} @list l2:level9 {mso-level-tab-stop:4.25in; mso-level-number-position:left; margin-left:4.25in; text-indent:-.25in;} @list l3 {mso-list-id:841048228; mso-list-type:hybrid; mso-list-template-ids:-590207770 67698689 67698691 67698693 67698689 67698691 67698693 67698689 67698691 67698693;} @list l3:level1 {mso-level-number-format:bullet; mso-level-text:·ð; mso-level-tab-stop:.25in; mso-level-number-position:left; margin-left:.25in; text-indent:-.25in; font-family:Symbol;} @list l3:level2 {mso-level-tab-stop:1.0in; mso-level-number-position:left; text-indent:-.25in;} @list l3:level3 {mso-level-tab-stop:1.5in; mso-level-number-position:left; text-indent:-.25in;} @list l3:level4 {mso-level-tab-stop:2.0in; mso-level-number-position:left; text-indent:-.25in;} @list l3:level5 {mso-level-tab-stop:2.5in; mso-level-number-position:left; text-indent:-.25in;} @list l3:level6 {mso-level-tab-stop:3.0in; mso-level-number-position:left; text-indent:-.25in;} @list l3:level7 {mso-level-tab-stop:3.5in; mso-level-number-position:left; text-indent:-.25in;} @list l3:level8 {mso-level-tab-stop:4.0in; mso-level-number-position:left; text-indent:-.25in;} @list l3:level9 {mso-level-tab-stop:4.5in; mso-level-number-position:left; text-indent:-.25in;} @list l4 {mso-list-id:1357924174; mso-list-type:hybrid; mso-list-template-ids:1846683022 67698689 67698691 67698693 67698689 67698691 67698693 67698689 67698691 67698693;} @list l4:level1 {mso-level-number-format:bullet; mso-level-text:·ð; mso-level-tab-stop:.5in; mso-level-number-position:left; text-indent:-.25in; font-family:Symbol;} @list l4:level2 {mso-level-tab-stop:1.0in; mso-level-number-position:left; text-indent:-.25in;} @list l4:level3 {mso-level-tab-stop:1.5in; mso-level-number-position:left; text-indent:-.25in;} @list l4:level4 {mso-level-tab-stop:2.0in; mso-level-number-position:left; text-indent:-.25in;} @list l4:level5 {mso-level-tab-stop:2.5in; mso-level-number-position:left; text-indent:-.25in;} @list l4:level6 {mso-level-tab-stop:3.0in; mso-level-number-position:left; text-indent:-.25in;} @list l4:level7 {mso-level-tab-stop:3.5in; mso-level-number-position:left; text-indent:-.25in;} @list l4:level8 {mso-level-tab-stop:4.0in; mso-level-number-position:left; text-indent:-.25in;} @list l4:level9 {mso-level-tab-stop:4.5in; mso-level-number-position:left; text-indent:-.25in;} ol {margin-bottom:0in;} ul {margin-bottom:0in;} --> </style> <!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman","serif";} </style> <![endif]--><!--[if gte mso 9]><xml> <o:shapedefaults v:ext="edit" spidmax="23554"/> </xml><![endif]--><!--[if gte mso 9]><xml> <o:shapelayout v:ext="edit"> <o:idmap v:ext="edit" data="1"/> </o:shapelayout></xml><![endif]--> </head> <body lang=EN-US link=blue vlink=purple style='tab-interval:.5in'> <div class=WordSection1> <div align=center> <table class=MsoNormalTable border=1 cellspacing=0 cellpadding=0 style='margin-left:22.45pt;border-collapse:collapse;border:none;mso-border-alt: solid windowtext .5pt;mso-yfti-tbllook:1184;mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-border-insideh:.5pt solid windowtext;mso-border-insidev:.5pt solid windowtext'> <tr style='mso-yfti-irow:0;mso-yfti-firstrow:yes;height:51.7pt'> <td width=883 colspan=2 style='width:662.2pt;border:solid windowtext 1.0pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:51.7pt'> <p class=MsoNormal align=center style='text-align:center'><span style='font-size:18.0pt;mso-bidi-font-size:16.0pt;font-family:"Arial","sans-serif"; color:#000082'>L1 Homotopy: A MATLAB Toolbox for Homotopy Algorithms in L1 Norm Minimization Problems <o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:1;height:206.5pt'> <td width=884 colspan=2 valign=top style='width:663.05pt;border:solid white 1.0pt; border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid white .5pt; mso-border-top-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt; height:206.5pt'> <p class=MsoNormal><span style='font-size:14.0pt;font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span style='font-size:15.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'>Introduction: <o:p></o:p></span></b></p> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span style='font-size:14.0pt;font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></b></p> <p class=MsoNormal><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt; font-family:"Arial","sans-serif"'>This package is a collection of MATLAB routines for solving some L1 norm minimization problems using homotopy techniques. These problems are usually encountered in the recovery of sparse signals from linear incoherent measurements. <o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt; font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt; font-family:"Arial","sans-serif"'>This package contains scripts for solving the following optimization problems:<o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt; font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l1 level1 lfo2; tab-stops:list .5in'><![if !supportLists]><span style='font-size:13.0pt; mso-bidi-font-size:14.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size: 14.0pt;font-family:"Arial","sans-serif"'>Basis pursuit <span class=SpellE>denoising</span> (BPDN) / LASSO<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l1 level1 lfo2; tab-stops:list .5in'><![if !supportLists]><span style='font-size:13.0pt; mso-bidi-font-size:14.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span class=SpellE><span style='font-size: 13.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'>Dantzig</span></span><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'> selector<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l1 level1 lfo2; tab-stops:list .5in'><![if !supportLists]><span style='font-size:13.0pt; mso-bidi-font-size:14.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size: 14.0pt;font-family:"Arial","sans-serif"'>L1 decoding<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l1 level1 lfo2; tab-stops:list .5in'><![if !supportLists]><span style='font-size:13.0pt; mso-bidi-font-size:14.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size: 14.0pt;font-family:"Arial","sans-serif"'>Robust L1 decoding<o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt; font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt; font-family:"Arial","sans-serif"'>In addition to solving these problems for any given set of parameters, we have some dynamic algorithms to update their solution when <o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt; font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l4 level1 lfo4; tab-stops:list .5in'><![if !supportLists]><span style='font-size:13.0pt; mso-bidi-font-size:14.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size: 14.0pt;font-family:"Arial","sans-serif"'>New measurements are sequentially added to the system<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;text-indent:-.25in;mso-list:l4 level1 lfo4; tab-stops:list .5in'><![if !supportLists]><span style='font-size:13.0pt; mso-bidi-font-size:14.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size: 14.0pt;font-family:"Arial","sans-serif"'>The unknown signal varies over time and we get a new measurement vector<o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:14.0pt;font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span style='font-size:15.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'>Download: </span></b><span style='font-size:15.0pt;mso-bidi-font-size:14.0pt; font-family:"Arial","sans-serif"'><o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:14.0pt;font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt; font-family:"Arial","sans-serif"'>Homotopy package (.<a href="L1_homotopy.zip">zip</a>)<o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt; font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></p> <p class=MsoListParagraphCxSpFirst style='text-indent:-.25in;mso-list:l0 level1 lfo9'><![if !supportLists]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:Symbol; mso-fareast-font-family:Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size: 14.0pt;font-family:"Arial","sans-serif"'>BPDN/LASSO <o:p></o:p></span></p> <p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space: auto;text-indent:-.25in;mso-list:l0 level2 lfo9'><![if !supportLists]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New"'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'>BPDN <span class=SpellE>homotopy</span> with positivity constraint<o:p></o:p></span></p> <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo9'><![if !supportLists]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:Symbol; mso-fareast-font-family:Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span class=SpellE><span style='font-size: 13.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'>Dantzig</span></span><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'> selector (PD-Pursuit)<o:p></o:p></span></p> <p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space: auto;text-indent:-.25in;mso-list:l0 level2 lfo9'><![if !supportLists]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New"'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'>DS <span class=SpellE>homotopy</span> with positivity constraints<o:p></o:p></span></p> <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo9'><![if !supportLists]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:Symbol; mso-fareast-font-family:Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size: 14.0pt;font-family:"Arial","sans-serif"'>Dynamic update <span class=SpellE>homotopy</span><o:p></o:p></span></p> <p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space: auto;text-indent:-.25in;mso-list:l0 level2 lfo9'><![if !supportLists]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New"'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'>Sequential measurements<o:p></o:p></span></p> <p class=MsoListParagraphCxSpMiddle style='margin-left:1.5in;mso-add-space: auto;text-indent:-.25in;mso-list:l0 level3 lfo9'><![if !supportLists]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:Wingdings; mso-fareast-font-family:Wingdings;mso-bidi-font-family:Wingdings'><span style='mso-list:Ignore'>§<span style='font:7.0pt "Times New Roman"'>&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'>One measurement at a time<o:p></o:p></span></p> <p class=MsoListParagraphCxSpMiddle style='margin-left:1.5in;mso-add-space: auto;text-indent:-.25in;mso-list:l0 level3 lfo9'><![if !supportLists]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:Wingdings; mso-fareast-font-family:Wingdings;mso-bidi-font-family:Wingdings'><span style='mso-list:Ignore'>§<span style='font:7.0pt "Times New Roman"'>&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'>Multiple measurements<o:p></o:p></span></p> <p class=MsoListParagraphCxSpMiddle style='margin-left:1.0in;mso-add-space: auto;text-indent:-.25in;mso-list:l0 level2 lfo9'><![if !supportLists]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:"Courier New"; mso-fareast-font-family:"Courier New"'><span style='mso-list:Ignore'>o<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'>Time-varying signal<o:p></o:p></span></p> <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo9'><![if !supportLists]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:Symbol; mso-fareast-font-family:Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size: 14.0pt;font-family:"Arial","sans-serif"'>Dictionary update<o:p></o:p></span></p> <p class=MsoListParagraphCxSpMiddle style='text-indent:-.25in;mso-list:l0 level1 lfo9'><![if !supportLists]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:Symbol; mso-fareast-font-family:Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size: 14.0pt;font-family:"Arial","sans-serif"'>Multiple measurements<o:p></o:p></span></p> <p class=MsoListParagraphCxSpLast style='text-indent:-.25in;mso-list:l0 level1 lfo9'><![if !supportLists]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:Symbol; mso-fareast-font-family:Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size: 14.0pt;font-family:"Arial","sans-serif"'>Re-weighted L1<o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt; font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt; font-family:"Arial","sans-serif"'>User s guide (.<span class=SpellE>pdf</span>)<span style='mso-spacerun:yes'>  </span></span><span style='mso-bidi-font-size: 14.0pt;font-family:"Arial","sans-serif"'>(<i style='mso-bidi-font-style:normal'><a href="http://www.phdcomics.com/comics/archive.php?comicid=971">still under preparation</a></i>)<o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt; font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span style='font-size:15.0pt;mso-bidi-font-size:16.0pt;font-family:"Arial","sans-serif"'>Related papers:</span></b><b style='mso-bidi-font-weight:normal'><span style='font-size:13.0pt;mso-bidi-font-size:16.0pt;font-family:"Arial","sans-serif"'><o:p></o:p></span></b></p> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span style='font-size:14.0pt;mso-bidi-font-size:16.0pt;font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></b></p> <p class=MsoNormal style='margin-left:.25in;text-indent:-.25in;mso-list:l3 level1 lfo6; tab-stops:list .25in'><![if !supportLists]><span style='font-size:14.0pt; mso-bidi-font-size:16.0pt;font-family:Symbol;mso-fareast-font-family:Symbol; mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size: 12.0pt;font-family:"Arial","sans-serif"'>M. Salman Asif and Justin Romberg, <i><a href="http://arxiv.org/abs/0903.1443"><span style='color:windowtext; text-decoration:none;text-underline:none'>Dynamic updating for L1 minimization</span></a></i>, submitted to IEEE Journal of selected topics in signal processing, special issue on compressive sensing.</span><span style='font-size:14.0pt;mso-bidi-font-size:16.0pt;font-family:"Arial","sans-serif"'><o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:14.0pt;mso-bidi-font-size:16.0pt; font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span style='font-size:15.0pt;mso-bidi-font-size:16.0pt;font-family:"Arial","sans-serif"'>Related <span class=SpellE>softwares</span> and links:</span></b><b style='mso-bidi-font-weight: normal'><span style='font-size:16.0pt;font-family:"Arial","sans-serif"'><o:p></o:p></span></b></p> <p class=MsoNormal><span style='font-size:14.0pt;font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.25in;text-indent:-.25in;line-height: 150%;mso-list:l2 level1 lfo8;tab-stops:list .25in'><![if !supportLists]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;line-height:150%; font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size: 14.0pt;line-height:150%;font-family:"Arial","sans-serif"'>l1_magic [<a href="http://www.acm.caltech.edu/l1magic/">link</a>]<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.25in;text-indent:-.25in;line-height: 150%;mso-list:l2 level1 lfo8;tab-stops:list .25in'><![if !supportLists]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;line-height:150%; font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size: 14.0pt;line-height:150%;font-family:"Arial","sans-serif"'>CVX [<a href="http://www.stanford.edu/~boyd/cvx/">link</a>]<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.25in;text-indent:-.25in;line-height: 150%;mso-list:l2 level1 lfo8;tab-stops:list .25in'><![if !supportLists]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;line-height:150%; font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span class=SpellE><span style='font-size: 13.0pt;mso-bidi-font-size:14.0pt;line-height:150%;font-family:"Arial","sans-serif"'>WaveLab</span></span><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;line-height:150%; font-family:"Arial","sans-serif"'> [<a href="http://www-stat.stanford.edu/~wavelab/">link</a>]<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.25in;text-indent:-.25in;line-height: 150%;mso-list:l2 level1 lfo8;tab-stops:list .25in'><![if !supportLists]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;line-height:150%; font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size: 14.0pt;line-height:150%;font-family:"Arial","sans-serif"'>GPSR [<a href="http://www.lx.it.pt/%7Emtf/GPSR">link</a>]<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.25in;text-indent:-.25in;line-height: 150%;mso-list:l2 level1 lfo8;tab-stops:list .25in'><![if !supportLists]><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;line-height:150%; font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:Symbol'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:13.0pt;mso-bidi-font-size: 14.0pt;line-height:150%;font-family:"Arial","sans-serif"'>FPC_AS [<a href="http://www.caam.rice.edu/~optimization/L1/FPC_AS/">link</a>]<o:p></o:p></span></p> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span style='font-size:15.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></b></p> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span style='font-size:15.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'>License:</span></b><span style='font-size:15.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'><o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:14.0pt;font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt; font-family:"Arial","sans-serif"'>L1 Homotopy © 2009<span style='mso-spacerun:yes'>  </span><a href="http://users.ece.gatech.edu/~sasif">Salman Asif</a>, <a href="http://users.ece.gatech.edu/~justin">Justin Romberg</a><o:p></o:p></span></p> <p><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'>This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.<o:p></o:p></span></p> <p><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'>This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. &lt;<a href="http://www.gnu.org/licenses/">http://www.gnu.org/licenses/</a>&gt;.<o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt; font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal><span style='font-size:14.0pt;font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:2;height:242.95pt'> <td width=459 valign=top style='width:344.1pt;border-top:none;border-left: solid white 1.0pt;border-bottom:solid windowtext 1.0pt;border-right:solid white 1.0pt; mso-border-top-alt:solid white .5pt;mso-border-alt:solid white .5pt; mso-border-bottom-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt; height:242.95pt'> <p class=MsoNormal align=center style='text-align:center'><a name="_top"></a><a name="_About_Me:"></a><span style='font-size:14.0pt;font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal align=center style='text-align:center'><span style='font-size:14.0pt;font-family:"Arial","sans-serif";mso-no-proof:yes'><!--[if gte vml 1]><v:shapetype id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter"/> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0"/> <v:f eqn="sum @0 1 0"/> <v:f eqn="sum 0 0 @1"/> <v:f eqn="prod @2 1 2"/> <v:f eqn="prod @3 21600 pixelWidth"/> <v:f eqn="prod @3 21600 pixelHeight"/> <v:f eqn="sum @0 0 1"/> <v:f eqn="prod @6 1 2"/> <v:f eqn="prod @7 21600 pixelWidth"/> <v:f eqn="sum @8 21600 0"/> <v:f eqn="prod @7 21600 pixelHeight"/> <v:f eqn="sum @10 21600 0"/> </v:formulas> <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/> <o:lock v:ext="edit" aspectratio="t"/> </v:shapetype><v:shape id="Picture_x0020_1" o:spid="_x0000_i1026" type="#_x0000_t75" alt="image001" style='width:259.5pt;height:171pt;visibility:visible; mso-wrap-style:square'> <v:imagedata src="index_files/image001.gif" o:title="image001"/> </v:shape><![endif]--><![if !vml]><img border=0 width=346 height=228 src="index_files/image001.gif" alt=image001 v:shapes="Picture_x0020_1"><![endif]></span><span style='font-size:14.0pt;font-family:"Arial","sans-serif"'><o:p></o:p></span></p> <p class=MsoNormal align=center style='text-align:center'><span style='font-size:14.0pt;font-family:"Arial","sans-serif"'><span style='mso-spacerun:yes'>          </span><o:p></o:p></span></p> <p class=MsoNormal align=center style='text-align:center'><u><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'>BPDN homotopy path</span></u><span style='font-size:14.0pt;font-family:"Arial","sans-serif"'><o:p></o:p></span></p> </td> <td width=424 valign=top style='width:318.1pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid white 1.0pt; mso-border-left-alt:solid white .5pt;mso-border-left-alt:solid white .5pt; mso-border-bottom-alt:solid windowtext .5pt;mso-border-right-alt:solid white .5pt; padding:0in 5.4pt 0in 5.4pt;height:242.95pt'> <p class=MsoNormal><span style='font-size:14.0pt;font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal><span style='font-size:14.0pt;font-family:"Arial","sans-serif"'><span style='mso-spacerun:yes'>    </span><span style='mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_2" o:spid="_x0000_i1025" type="#_x0000_t75" alt="image003" style='width:259.5pt;height:171pt;visibility:visible;mso-wrap-style:square'> <v:imagedata src="index_files/image002.gif" o:title="image003"/> </v:shape><![endif]--><![if !vml]><img border=0 width=346 height=228 src="index_files/image002.gif" alt=image003 v:shapes="Picture_x0020_2"><![endif]></span><o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:14.0pt;font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal align=center style='text-align:center'><span class=SpellE><u><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'>Dantzig</span></u></span><u><span style='font-size:13.0pt;mso-bidi-font-size:14.0pt;font-family:"Arial","sans-serif"'> selector <span class=SpellE>homotopy</span> path</span></u><u><span style='font-size:14.0pt;font-family:"Arial","sans-serif"'><o:p></o:p></span></u></p> </td> </tr> <tr style='mso-yfti-irow:3;mso-yfti-lastrow:yes;height:53.5pt'> <td width=459 valign=top style='width:344.1pt;border-top:none;border-left: solid white 1.0pt;border-bottom:solid windowtext 1.0pt;border-right:solid white 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-top-alt:windowtext; mso-border-left-alt:white;mso-border-bottom-alt:windowtext;mso-border-right-alt: white;mso-border-style-alt:solid;mso-border-width-alt:.5pt;padding:0in 5.4pt 0in 5.4pt; height:53.5pt'> <p class=MsoNormal><i style='mso-bidi-font-style:normal'><span style='font-size:14.0pt;font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></i></p> <p class=MsoNormal><i style='mso-bidi-font-style:normal'><span style='font-size:14.0pt;font-family:"Arial","sans-serif"'>Last updated: 06/26/09<o:p></o:p></span></i></p> </td> <td width=424 valign=top style='width:318.1pt;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid white 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid white .5pt; mso-border-top-alt:windowtext;mso-border-left-alt:white;mso-border-bottom-alt: windowtext;mso-border-right-alt:white;mso-border-style-alt:solid;mso-border-width-alt: .5pt;padding:0in 5.4pt 0in 5.4pt;height:53.5pt'> <p class=MsoNormal align=center style='text-align:center'><span style='font-size:14.0pt;font-family:"Arial","sans-serif"'><o:p>&nbsp;</o:p></span></p> </td> </tr> </table> </div> <p class=MsoNormal><a name="_Research:"></a><a name="_Courses:_1"></a><a name="_Links:"></a><a name="_Courses:"></a><o:p>&nbsp;</o:p></p> </div> </body> </html>