Scene Modeling for a Single ViewClasses next termBreaking out of 2Don to 3D…Camera rotations with homographiesCamera translationYes, with planar scene (or far away)So, what can we do here?Some preliminaries: projective geometrySilly Euclid: Trix are for kids!The projective planeProjective linesPoint and line dualityIdeal points and linesVanishing pointsVanishing points (2D)Vanishing pointsVanishing linesVanishing linesComputing vanishing pointsComputing vanishing linesFun with vanishing points“Tour into the Picture” (SIGGRAPH ’97)The ideaFitting the box volume2D to 3D conversion2D to 3D conversionDepth of the boxDEMOForeground ObjectsForeground ObjectsForeground DEMO (and video)Scene Modeling for a Single View15-463: Computational PhotographyAlexei Efros, CMU, Fall 2005René MAGRITTEPortrait d'Edward James…with a lot of slides stolen from Steve Seitz and David Brogan,Classes next term15-465 Animation Art and Technologythis is the learn Maya and collaborate with artists to make a movieclass. It will likely NOT be offered next year so it is important thatthe students understand that as it has been offered everyyear for a while now.http://www.cs.cmu.edu/~jkh/aat/15-869C Special Topics in Graphics: Generating Human MotionAn advanced class on techniques for generating naturalhuman motion. This is a one-time offering.15-864 Advanced Computer Graphics: Graduate Classhttp://www.cs.cmu.edu/~djames/15-864/index.htmlBreaking out of 2D…now we are ready to break out of 2DAnd enter the real world!Enough of images!We want more of the plenoptic functionWe want real 3D scenewalk-throughs:Camera rotationCamera translationCan we do it from a single photograph?on to 3D…Camera rotations with homographiesSt.Petersburgphoto by A. TikhonovVirtual camera rotationsOriginal imageCamera translationDoes it work?synthetic PPPP1PP2Yes, with planar scene (or far away)PP3 is a projection plane of both centers of projection, so we are OK!PP1PP3PP2So, what can we do here?Model the scene as a set of planes!Some preliminaries: projective geometryAmes RoomSilly Euclid: Trix are for kids!Parallel lines???(0,0,0)The projective planeWhy do we need homogeneous coordinates?• represent points at infinity, homographies, perspective projection, multi-view relationshipsWhat is the geometric intuition?• a point in the image is a ray in projective space(sx,sy,s)image plane• Each point (x,y) on the plane is represented by a ray (sx,sy,s)– all points on the ray are equivalent: (x, y, 1) ≅ (sx, sy, s)(x,y,1)yxzProjective linesWhat does a line in the image correspond to in projective space?• A line is a plane of rays through origin– all rays (x,y,z) satisfying: ax + by + cz = 0[]⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=zyxcba0 :notationvectorin• A line is also represented as a homogeneous 3-vector llplPoint and line duality• A line l is a homogeneous 3-vector• It is ⊥ to every point (ray) p on the line: lp=0p1p2What is the intersection of two lines l1and l2 ?• p is ⊥ to l1and 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 p1and p2 ?• l is ⊥ to p1and p2 ⇒ l = p1× p2 • l is the 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)yxzimage planeIdeal line•l ≅ (a, b, 0) – parallel to image plane(a,b,0)yxzimage plane• Corresponds to a line in the image (finite coordinates)Vanishing pointsVanishing point• projection of a point at infinity• Caused by ideal lineimage planecameracenterground planevanishing pointVanishing points (2D)image planecameracenterline on ground planevanishing pointVanishing pointsProperties• Any two parallel lines 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 pointimage planecameracenterCline on ground planevanishing point Vline on ground planeVanishing 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 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 define different vanishing linesComputing vanishing pointsProperties• P∞is a point at infinity, v is its projection• They depend 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 planelCFun with vanishing points“Tour into the Picture” (SIGGRAPH ’97)Create a 3D “theatre stage” of five billboardsSpecify foreground objects through bounding polygonsUse camera transformations to navigate through the sceneThe ideaMany scenes (especially paintings), can be represented as an axis-aligned box volume (i.e. a stage)Key assumptions:• All walls of volume are orthogonal• Camera view plane is parallel to back of volume• Camera up is normal to volume bottomHow many vanishing points does the box have?• Three, but two at infinity• Single-point perspectiveCan use the vanishing pointto fit the box to the particularScene!Fitting the box volumeUser controls the inner box and the vanishing point placement (# of DOF???)Q: What’s the significance of the vanishing point location?A: It’s at eye level: ray from COP to VP is perpendicular to image plane. Why?High CameraExample of user input: vanishing point and back face of view volume are definedHigh CameraExample of user input: vanishing point and back face of view volume are definedLow CameraExample of user input: vanishing point and back face of view volume are definedLow CameraExample of user input:
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