1	Announcements Project 3 questions Final project out today
2	Projective geometry Readings 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.pdf
3	Projective geometry—what’s it good for? Uses of projective geometry Drawing Measurements Mathematics for projection Undistorting images Focus of expansion Camera pose estimation, match move Object recognition
4	Applications of projective geometry
5	Measurements on planes
6	Image rectification
7	Solving for homographies
8	Solving for homographies
9	The projective plane Why do we need homogeneous coordinates? represent points at infinity, homographies, perspective projection, multi-view relationships What is the geometric intuition? a point in the image is a ray in projective space
10	Projective lines What does a line in the image correspond to in projective space?
11	Point and line duality A line l is a homogeneous 3-vector It is ^ to every point (ray) p on the line: l p=0
12	Ideal points and lines Ideal point (“point at infinity”) p @ (x, y, 0) – parallel to image plane It has infinite image coordinates
13	Homographies of points and lines Computed by 3x3 matrix multiplication To transform a point: p’ = Hp To transform a line: lp=0 ® l’p’=0
14	3D projective geometry These 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 Projective transformations Represented by 4x4 matrices T: P’ = TP, N’ = N T^-1
15	3D to 2D: “perspective” projection Matrix Projection:
16	Vanishing points Vanishing point projection of a point at infinity
17	Vanishing points (2D)
18	Vanishing points Properties 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 point in fact every pixel is a potential vanishing point
19	Vanishing lines Multiple 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 lines
20	Vanishing lines Multiple 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 lines
21	Computing vanishing points
22	Computing vanishing points Properties 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_¥
23	Computing vanishing lines Properties 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 scene
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25	Fun with vanishing points
26	Perspective cues
27	Perspective cues
28	Perspective cues
29	Comparing heights
30	Measuring height
31	Computing vanishing points (from lines) Intersect p₁q₁ with p₂q₂
32	Measuring height without a ruler
33	The cross ratio A Projective Invariant Something that does not change under projective transformations (including perspective projection)
34	Measuring height
35	Measuring height
36	Measuring height
37	Computing (X,Y,Z) coordinates Okay, we know how to compute height (Z coords) how can we compute X, Y?
38	3D Modeling from a photograph
39	Camera calibration Goal: estimate the camera parameters Version 1: solve for projection matrix
40	Vanishing points and projection matrix
41	Calibration using a reference object Place a known object in the scene identify correspondence between image and scene compute mapping from scene to image
42	Chromaglyphs
43	Estimating the projection matrix Place a known object in the scene identify correspondence between image and scene compute mapping from scene to image
44	Direct linear calibration
45	Direct linear calibration
46	Direct linear calibration Advantage: Very simple to formulate and solve Disadvantages: Doesn’t tell you the camera parameters Doesn’t model radial distortion Hard to impose constraints (e.g., known focal length) Doesn’t minimize the right error function
47	Alternative: multi-plane calibration
48	Some Related Techniques Image-Based Modeling and Photo Editing Mok et al., SIGGRAPH 2001 http://graphics.csail.mit.edu/ibedit/ Single View Modeling of Free-Form Scenes Zhang et al., CVPR 2001 http://grail.cs.washington.edu/projects/svm/ Tour Into The Picture Anjyo et al., SIGGRAPH 1997 http://koigakubo.hitachi.co.jp/little/DL_TipE.html