1	Announcements Panorama signups Panorama project issues? Nalwa handout
2	Projective geometry Readings Mundy, J.L. and Zisserman, A., Geometric Invariance in Computer Vision, Chapter 23: Appendix: Projective Geometry for Machine Vision, MIT Press, Cambridge, MA, 1992, pp. 463-534 (for this week, 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 is a line 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 The ray from C through v point is parallel to the lines An image may have more than one 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 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_¥
22	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 Provides way of comparing height of objects in the scene
23	Fun with vanishing points
24	Perspective cues
25	Perspective cues
26	Perspective cues
27	Comparing heights
28	Measuring height
29	Computing vanishing points (from lines) Intersect p₁q₁ with p₂q₂
30	Measuring height without a ruler
31	The cross ratio A Projective Invariant Something that does not change under projective transformations (including perspective projection)
32	Measuring height
33	Measuring height
34	Measuring height
35	Measurements within reference plane Solve for homography H relating reference plane to image plane H maps reference plane (X,Y) coords to image plane (x,y) coords Fully determined from 4 known points on ground plane Option A: physically measure 4 points on ground Option B: find a square, guess the dimensions Option C: Note H = columns 1,2,4 projection matrix derive on board Given (x, y), can find (X,Y) by H^-1
36	Criminisi et al., ICCV 99 Complete 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 Output a VRML model
37	Vanishing points and projection matrix
38	3D Modeling from a photograph
39	3D Modeling from a photograph
40	Camera calibration Goal: estimate the camera parameters Version 1: solve for projection matrix
41	Calibration: Basic Idea 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 Advantages: Very simple to formulate and solve Once you know the projection matrix, can compute intrinsics and extrinsics using matrix factorizations
47	Alternative: Multi-plane calibration
48	Summary Things to take home from this lecture Homogeneous coordinates and their geometric intuition Homographies Points and lines in projective space projective operations: line intersection, line containing two points ideal points and lines (at infinity) Vanishing points and lines and how to compute them Single view measurement within a reference plane height Cross ratio Camera calibration using vanishing points direct linear method