More Single View GeometryFinal ProjectsPop Quiz: which is 1,2,3-point perspectiveAutomatic Photo Pop-upResultsSlide 6Slide 7Slide 8Slide 9How can we model this scene?Finding world coordinates (X,Y,Z)Measurements on planesUnwarp ground planeSlide 14Comparing heightsPerspective cuesSlide 17Slide 18Measuring heightComputing vanishing points (from lines)Criminisi ’99Measuring height without a rulerSlide 23Slide 24What if vz is not infinity?The cross ratioSlide 28Slide 29Measuring heights of peopleForensic Science: measuring heights of suspectsAssessing geometric accuracySlide 33Criminisi et al., ICCV 99Interactive silhouette cut-outOcclusion fillingComplete 3D reconstructionReconstruction from single photographsA virtual museum @ MicrosoftMore Single View Geometry15-463: Computational PhotographyAlexei Efros, CMU, Fall 2006…with a lot of slides stolen from Steve SeitzCyclops Odilon Redon 1904Final ProjectsAre coming up fast!Undergrads can work in pairs, but project must be bigger. Sample Topics:•Full 360 panorama construction (spherical or cylindrical)•Render in synthetic object into real scene•Automatic Tour into the Picture (can use Pop-up labeling code)•Build a virtual CMU campus environment•Implement a paper discussed in class (e.g. Video Textures)•Come up with art project that uses Comp. Photography•Etc.Project proposals due next Tuesday!Pop Quiz: which is 1,2,3-point perspectiveImage AImage BImage CAutomatic Photo Pop-upOriginal ImageGeometric LabelsFit Segments Cut and Fold Novel ViewResultsInput ImageAutomatic Photo Pop-upCut and FoldResultsAutomatic Photo Pop-upInput ImageCut and FoldResultsAutomatic Photo Pop-upInput ImageResultsAutomatic Photo Pop-upInput ImagesResultsAutomatic Photo Pop-upInput ImageHow can we model this scene?1. Find world coordinates (X,Y,Z) for a few points2. Connect the points with planes to model geometry•Texture map the planesFinding world coordinates (X,Y,Z)1. Define the ground plane (Z=0)2. Compute points (X,Y,0) on that plane3. Compute the heights Z of all other points1 2 3 41234Measurements on planesApproach: unwarp, then measureWhat kind of warp is this?Unwarp ground planeOur old friend – the homographyNeed 4 reference points with world coordinates p = (x,y)p’ = (X,Y,0)pp’xFinding world coordinates (X,Y,Z)1. Define the ground plane (Z=0)2. Compute points (X,Y,0) on that plane3. Compute the heights Z of all other pointsComparing heightsPerspective cuesPerspective cuesComparing heightsVanishingVanishingPointPointMeasuring height123455.42.83.3Camera heightq1Computing vanishing points (from lines)Intersect p1q1 with p2q2 vp1p2q2Least squares version•Better to use more than two lines and compute the “closest” point of intersection•See notes by Bob Collins for one good way of doing this:–http://www-2.cs.cmu.edu/~ph/869/www/notes/vanishing.txtVanishing pointVanishing lineVanishing point Vertical vanishing point(at infinity)Criminisi ’99CMeasuring height without a rulerground planeCompute Z from image measurements•Need more than vanishing points to do thisZMeasuring heightRHvzrbtHb0t0vvxvyvanishing line (horizon)Measuring heightvzrbt0vxvyvanishing line (horizon)vt0m0What if the point on the ground plane b0 is not known?•Here the guy is standing on the box•Use one side of the box to help find b0 as shown aboveb0t1b1What if vz is not infinity?The cross ratioA Projective Invariant•Something that does not change under projective transformations (including perspective projection)P1P2P3P414232413PPPPPPPPThe cross-ratio of 4 collinear pointsCan permute the point ordering•4! = 24 different orders (but only 6 distinct values)This is the fundamental invariant of projective geometry1iiiiZYXP34212431PPPPPPPPvZ rtbtvbrrvbtZZimage cross ratioMeasuring heightB (bottom of object)T (top of object)R (reference point)ground planeHCTBRRBTscene cross ratio1ZYXP1yxpscene points represented asimage points asRHRHRMeasuring heightRHvzrbtRHZZtvbrrvbtimage cross ratioHb0t0vvxvyvanishing line (horizon)Measuring heights of peopleHere we go !reference185.3 cmForensic Science: measuring heights of suspectsVanishing lineReference heightReference heightAssessing geometric accuracyFlagellation, Piero della FrancescaEstimated relative heightsAre the heights of the 2 groups of people consistent with each other?Assessing geometric accuracyEstimated relative heightsThe Marriage of the Virgin, RaphaelCriminisi et al., ICCV 99Complete approach•Load in an image•Click on lines parallel to X axis–repeat for Y, Z axes•Compute vanishing points•Specify 3D and 2D positions of 4 points on reference plane•Compute homography H•Specify a reference height •Compute 3D positions of several points•Create a 3D model from these points•Extract texture maps–Cut out objects–Fill in holes•Output a VRML modelInteractive silhouette cut-outOcclusion fillingGeometric filling by exploiting:•symmetries•repeated regular patternsTexture synthesis•repeated stochastic patternsComplete 3D reconstructionPlanar measurementsHeight measurementsAutomatic vanishing point/line computationInteractive segmentationOcclusion fillingObject placement in 3D modelSingleimage3DmodelSingleViewalgorithmsReconstruction from single photographsReconstruction of the garden Hut from a single imageA virtual museum @ A virtual museum @ MicrosoftMicrosoftA.Criminisi http://research.microsoft.com/~antcrim/A.Criminisi
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