Lecture 19: Single-view modelingAnnouncementsSlide 3Multi-view geometryProjective geometryProjective geometry—what’s it good for?Applications of projective geometryMeasurements on planesImage rectificationSolving for homographiesSlide 11Slide 12Point and line dualityIdeal points and lines3D projective geometry3D to 2D: “perspective” projectionVanishing points (1D)Vanishing points (2D)Vanishing pointsTwo point perspectiveThree point perspectiveVanishing linesSlide 23Computing vanishing pointsSlide 25Computing vanishing linesSlide 27Fun with vanishing pointsPerspective cuesSlide 30Slide 31Comparing heightsMeasuring heightComputing vanishing points (from lines)Measuring height without a rulerThe cross ratioSlide 37Slide 38Slide 393D Modeling from a photographSlide 41Slide 42Slide 43Slide 44Lecture 19: Single-view modelingCS6670: Computer VisionNoah SnavelyAnnouncements•Project 3: Eigenfaces–due tomorrow, November 11 at 11:59pm•Quiz on Thursday, first 10 minutes of classAnnouncements•Final projects–Feedback in the next few days–Midterm reports due November 24–Final presentations tentatively scheduled for the final exam period:Wed, December 16, 7:00 PM - 9:30 PMMulti-view geometry•We’ve talked about two views•And many views•What can we tell about geometry from one view?0Projective geometryReadings•Mundy, J.L. and Zisserman, A., Geometric Invariance in Computer Vision, Appendix: Projective Geometry for Machine Vision, MIT Press, Cambridge, MA, 1992, (read 23.1 - 23.5, 23.10)–available online: http://www.cs.cmu.edu/~ph/869/papers/zisser-mundy.pdfAmes RoomProjective geometry—what’s it good for?Uses of projective geometry•Drawing•Measurements•Mathematics for projection•Undistorting images•Camera pose estimation•Object recognitionPaolo UccelloApplications of projective geometry Vermeer’s Music LessonReconstructions by Criminisi et al.1 2 3 41234Measurements on planesApproach: unwarp then measureWhat kind of warp is this?Image rectificationTo unwarp (rectify) an image•solve for homography H given p and p’•solve equations of the form: wp’ = Hp–linear in unknowns: w and coefficients of H–H is defined up to an arbitrary scale factor–how many points are necessary to solve for H?pp’Solving for homographiesSolving for homographiesSolving for homographiesDefines a least squares problem:•Since is only defined up to scale, solve for unit vector•Solution: = eigenvector of with smallest eigenvalue•Works with 4 or more points2n × 992nlPoint and line duality•A line l is a homogeneous 3-vector•It is  to every point (ray) p on the line: l p=0p1p2What is the intersection of two lines l1 and l2 ?•p is  to l1 and l2  p = l1  l2Points and lines are dual in projective space•given any formula, can switch the meanings of points and lines to get another formulal1l2pWhat is the line l spanned by rays p1 and p2 ?•l is  to p1 and p2  l = p1  p2 •l can be interpreted as a plane normalIdeal points and linesIdeal point (“point at infinity”)•p  (x, y, 0) – parallel to image plane•It has infinite image coordinates(sx,sy,0)-yx-zimage planeIdeal line•l  (a, b, 0) – parallel to image plane(a,b,0)-yx-zimage plane•Corresponds to a line in the image (finite coordinates)–goes through image origin (principle point)3D projective geometryThese concepts generalize naturally to 3D•Homogeneous coordinates–Projective 3D points have four coords: P = (X,Y,Z,W)•Duality–A plane N is also represented by a 4-vector–Points and planes are dual in 3D: N P=0–Three points define a plane, three planes define a point•Projective transformations–Represented by 4x4 matrices T: P’ = TP, N’ = T-T N.3D to 2D: “perspective” projectionProjection:ΠPp 1************ZYXwwywxVanishing points (1D)Vanishing point•projection of a point at infinity•can often (but not always) project to a finite point in the imageimage planecameracenterground planevanishing pointcameracenterimage planeVanishing points (2D)image planecameracenterline on ground planevanishing pointVanishing pointsProperties•Any two parallel lines (in 3D) have the same vanishing point v•The ray from C through v is parallel to the lines•An image may have more than one vanishing point–in fact, every image point is a potential vanishing pointimage planecameracenterCline on ground planevanishing point Vline on ground planeTwo point perspectivevxvyThree point perspectiveVanishing linesMultiple Vanishing Points•Any set of parallel lines on the plane define a vanishing point•The union of all of these vanishing points is the horizon line–also called vanishing line•Note that different planes (can) define different vanishing linesv1v2Vanishing linesMultiple Vanishing Points•Any set of parallel lines on the plane define a vanishing point•The union of all of these vanishing points is the horizon line–also called vanishing line•Note that different planes (can) define different vanishing linesComputing vanishing pointsVDPP t0P0DComputing vanishing pointsProperties•P is a point at infinity, v is its projection•Depends only on line direction•Parallel lines P0 + tD, P1 + tD intersect at PVDPP t00/1///1ZYXZZYYXXZZYYXXtDDDttDtPDtPDtPtDPtDPtDPPPΠPvP0DComputing vanishing linesProperties•l is intersection of horizontal plane through C with image plane•Compute l from two sets of parallel lines on ground plane•All points at same height as C project to l–points higher than C project above l•Provides way of comparing height of objects in the sceneground planelCv1v2lFun with vanishing pointsPerspective cuesPerspective cuesPerspective cuesComparing heightsVanishingVanishingPointPointMeasuring height123455.42.83.3Camera heightHow high is the camera?q1Computing 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.txtCMeasuring height without a rulerground planeCompute Z from image measurements•Need more than vanishing