1 Basic geometric concepts to understand Affine, Euclidean geometries (inhomogeneous coordinates) projective geometry (homogeneous coordinates) plane at infinity: affine geometry

2 Naturally everything starts from the known vector space Intuitive introduction

3 Vector space to affine: isomorph, one-to-one vector to Euclidean as an enrichment: scalar prod. affine to projective as an extension: add ideal elements Pts, lines, parallelism Angle, distances, circles Pts at infinity

4 P2 and R2 Relation between Pn (homo) and Rn (in-homo): Rn --> Pn, extension, embedded in Pn --> Rn, restriction,

5 Examples of projective spaces Projective plane P2 Projective line P1 Projective space P3

6 Pts are elements of P2 Projective plane 4 pts determine a projective basis 3 ref. Pts + 1 unit pt to fix the scales for ref. pts Relation with R2, (x,y,0), line at inf., (0,0,0) is not a pt Pts at infinity: (x,y,0), the line at infinity Space of homogeneous coordinates (x,y,t) Pts are elements of P2

7 Line equation: Lines: Linear combination of two algebraically independent pts Operator + is ‘span’ or ‘join’

8 Point/line duality: Point coordinate, column vector A line is a set of linearly dependent points Two points define a line Line coordinate, row vector A point is a set of linearly dependent lines Two lines define a point What is the line equation of two given points? ‘line’ (a,b,c) has been always ‘homogeneous’ since high school!

9 Given 2 points x1 and x2 (in homogeneous coordinates), the line connecting x1 and x2 is given by Given 2 lines l1 and l2, the intersection point x is given by NB: ‘cross-product’ is purely a notational device here.

10 Conics: a curve described by a second-degree equation 3*3 symmetric matrix 5 d.o.f 5 pts determine a conic Conics

11 Projective line Finite pts: Infinite pts: how many? A basis by 3 pts Fundamental inv: cross-ratio Homogeneous pair (x1,x2)

12 Pts, elements of P3 Relation with R3, plane at inf. planes: linear comb of 3 pts Basis by 4 (ref pts) +1 pts (unit) Projective space P3

13 planes In practice, take SVD

14 Key points Homo. Coordinates are not unique 0 represents no projective pt finite points embedded in proj. Space (relation between R and P) pts at inf. (x,0) missing pts, directions hyper-plane (co-dim 1): dualily between u and x,

15 2D general Euclidean transformation: 2D general affine transformation: 2D general projective transformation: Introduction to transformation Colinearity Cross-ratio

16 Projective transformation = collineation = homography Consider all functions All linear transformations are represented by matrices A Note: linear but in homogeneous coordinates!

17 How to compute transformatins and canonical projective coordinates?

18 Geometric modeling of a camera u v X u O X’ u’ P3 P2 How to relate a 3D point X (in oxyz) to a 2D point in pixels (u,v)?

19 X Y Z x y u v X x O f Camera coordinate frame

20 x o y uv X Y Z x y u v X x O f Image coordinate frame

21 Focal length in horizontal/vertical pixels (2) (or focal length in pixels + aspect ratio) the principal point (2) the skew (1) 5 intrinsic parameters one rough example: 135 film In practice, for most of CCD cameras: alpha u = alpha v i.e. aspect ratio=1 alpha = 90 i.e. skew s=0 (u0,v0) the middle of the image only focal length in pixels?

22 Xw Yw Zw XwXw X Y Z x y u v X x O f World (object) coordinate frame

23 World coordinate frame: extrinsic parameters Relation between the abstract algebraic and geometric models is in the intrinsic/extrinsic parameters! 6 extrinsic parameters

24 Finally, we have a map from a space pt (X,Y,Z) to a pixel (u,v) by

25 It turns the camera into an angular/direction sensor! Direction vector: What does the calibration give us? Normalised coordinates:

26 Camera calibration Given Estimate C decompose C into intrinsic/extrinsic from image processing or by hand

27 Decomposition analytical by equating K(R,t)=P

28 Pose estimation = calibration of only extrinsic parameters Given Estimate R and t

29 3-point algebraic method First convert pixels u into normalized points x by knowing the intrinsic parameters Write down the fundamental equation: Solve this algebraic system to get the point distances first Compute a 3D transformation 3 reference points == 3 beacons

30 given 3 corresponding 3D points: 3D transformation estimation Compute the centroids as the origin Compute the scale (compute the rotation by quaternion) Compute the rotation axis Compute the rotation angle

31 Linear pose estimation from 4 coplanar points Vector based (or affine geometry) method O A B C D x_a x_d

32 Midterm statistics Total Q1: Q2: Q3: Q4: Q5: Q6: ~ ~ ~ ~ ~