Visualization Computer Graphics II University of Missouri at Columbia Visualization •• “a“a picturepicture sayssays moremore thanthan aa thousandthousand words”words” Computer Graphics II University of Missouri at Columbia Visualization •• “a“a picturepicture sayssays moremore thanthan aa thousandthousand numbers”numbers” Computer Graphics II University of Missouri at Columbia Visualization •• VVisualizationisualization cancan facilitatefacilitate peoplepeople toto betterbetter understandunderstand thethe informationinformation embeddedembedded inin thethe givengiven dataset.dataset. •• TThehe mergemerge ofof datadata withwith thethe displaydisplay geometricgeometric objectsobjects throughthrough computercomputer graphics.graphics. Computer Graphics II University of Missouri at Columbia Data •2•2DD ddaattaasseett –– BBarar chart,chart, piepie chart,chart, graph,graph, stocks.stocks. –– IInformationnformation visualizationvisualization •3•3DD ddaattaasseett –– SScalarcalar datadata –V–Veeccttoorr ddaattaa –T–Teennssoorr ddaattaa Computer Graphics II University of Missouri at Columbia Data •2•2DD ddaattaasseett –– BBarar chart,chart, piepie chart,chart, graph,graph, stocks.stocks. –– IInformationnformation visualizationvisualization •3•3DD ddaattaasseett –– SScalarcalar datadata –V–Veeccttoorr ddaattaa –T–Teennssoorr ddaattaa Computer Graphics II University of Missouri at Columbia Data •2•2DD ddaattaasseett –– BBarar chart,chart, piepie chart,chart, graph,graph, stocks.stocks. –– IInformationnformation visualizationvisualization •3•3DD ddaattaasseett –– SScalarcalar datadata –V–Veeccttoorr ddaattaa –T–Teennssoorr ddaattaa Computer Graphics II University of Missouri at Columbia Data •2•2DD ddaattaasseett –– BBarar chart,chart, piepie chart,chart, graph,graph, stocks.stocks. –– IInformationnformation visualizationvisualization 90 80 70 60 50 East 40 West 30 North 20 10 0 1st Qtr 2nd Qtr 3rd Qtr 4th Qtr Computer Graphics II University of Missouri at Columbia Data •2•2DD ddaattaasseett –– BBarar chart,chart, piepie chart,chart, graph,graph, stocks.stocks. –– IInformationnformation visualizationvisualization Computer Graphics II University of Missouri at Columbia Volume Visualization •• DDirectirect volumevolume renderingrendering –– RRayay castingcasting –– ssplattingplatting •• IIso-surfaceso-surface extractionextraction –– MMarchingarching cubescubes Computer Graphics II University of Missouri at Columbia Iso-contour extraction f(x,y)=0f(x,y)=0 f(x,y)>0f(x,y)>0 f(x,y)<0f(x,y)<0 Computer Graphics II University of Missouri at Columbia Iso-contour extraction f(x,y)=0f(x,y)=0 f(x,y)>0f(x,y)>0 f(x,y)<0f(x,y)<0 Computer Graphics II University of Missouri at Columbia Marching Squares f(x,y)=0f(x,y)=0 f(x,y)>0f(x,y)>0 f(x,y)<0f(x,y)<0 FourFour uniqueunique casescases (after(after consideringconsidering symmetry)symmetry) Computer Graphics II University of Missouri at Columbia Marching Squares PrinciplePrinciple ofof Occam’sOccam’s razor:razor: IfIf therethere areare multiplemultiple possiblepossible explanationsexplanations ofof aa phenomenonphenomenon thatthat areare consistentconsistent withwith thethe data,data, choosechoose thethe simplestsimplest one.one. Computer Graphics II University of Missouri at Columbia Marching Squares LinearLinear interpolationinterpolation f(x,y)=0f(x,y)=0 f(x,y)>0f(x,y)>0 f(x,y)<0 f(x,y)<0 fi, j+1 > 0 fi+1, j+1 > 0 f < 0 i, j fi+1, j > 0 Computer Graphics II University of Missouri at Columbia Marching Squares fi, j = a < 0 f = b > 0 fi+∆x, j=0 i+1, j Computer Graphics II University of Missouri at Columbia Marching Squares fi, j = a < 0 f = b > 0 fi+∆x, j=0 i+1, j ∆x / h = (b-a) / (-a) ∆x = (a-b)h / a Computer Graphics II University of Missouri at Columbia Marching Squares Computer Graphics II University of Missouri at Columbia Marching Squares Computer Graphics II University of Missouri at Columbia Marching Squares-ambiguity More information are needed to resolve ambiguity Computer Graphics II University of Missouri at Columbia Iso-contour extraction f(x,y)=0f(x,y)=0 f(x,y)>0f(x,y)>0 f(x,y)<0f(x,y)<0 Computer Graphics II University of Missouri at Columbia.