1 A camera is modeled as a map from a space pt (X,Y,Z) to a pixel (u,v) by ‘homogeneous coordinates’ have been used to ‘treat’ translations ‘multiplicatively’ in matrices (translation was ‘additive’ in vectors), but this is not the full story … Motivation This will be re-used in computer graphics, computer vision and robotics!

2 Basic geometric concepts to understand Affine, Euclidean geometries (inhomogeneous coordinates) projective geometry (homogeneous coordinates) plane at infinity: affine geometry (absolute conic: Euclidean geometry) NB: some of the content in this part is only optional!!!, don’t worry

3 Introduction to projective geometry Intuitive ideas from projective geometry (Formal definition of projective spaces)

4 Naturally everything starts from the known vector space add two vectors multiply any vector by any scalar zero vector – origin finite basis Intuitive introduction

5 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

6 Algebraic extension to pts at infinity: introduction of homogeneous coordiantes Points at infinity: Rq: the homogeneous coordinates are not unique, up to a scale.

7 The direction d is a pt at infinity: On a plane, Can we see the pts at infinity?

8 a projective space is an affine space + some pts at infinity a projective space is a space of ‘homogeneous coordinates’ or Provisional summary

9 ((Formal) definition of projective geometry) Given K=R or C, can be defined as the nonzero equivalent classes determined by the relation ~ on If there is non-zero real number such that Any element of the equivalent class will be called the homogeneous coordinates of the point.

10 Definition: a pt x is said to be linearly dependent on a set of pts if A projective space is nothing but a quotient space (space of equivalent classes): A space of homogeneous coordinates Basic structure: linear dependence of points

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

12 One example of construction of projective line by quotient space

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

14 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

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

16 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!

17 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.

18 Compute the intersection point of two lines, each defined by two points

19 Conics: a curve described by a second-degree equation 3*3 symmetric matrix 5 d.o.f 5 pts determine a conic affine classification with pts at inf the line tangent to a conic at a pt dual conic pole and polar one numerical example Conics

20 Tangent to a conic at a pt x on C is given by l=Cx Dual conic (in line coordinates) is given by l^T C^{-1} l = 0 Polar of a pt x is l = C x and is also a tangent on C from x

21 Line at infinity

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

23 Euclidean coordinate: the distance Affine coordinate: the ratio of the distances (x-a/a-o) Projective coordinate: the ratio of the ratio of the distances (cross-ratio, double ratio) ((x-a)/(a-o)) / ((x-b)(b-o))

24 Pts, elements of P3 Relation with R3, plane at inf. lines: linear comb of 2 pts, but 3*4 matrix, complicated …back later planes: linear comb of 3 pts Basis by 4 (ref pts) +1 pts (unit) quadrics: two classes---ruled and unruled (topology of P3) Plane equation:... Line equation? Projective space P3

25 planes In practice, take SVD Homework: compute plane normal vector?

26 How many d.o.f??? 6 2*2 minors, Two lines intersect in space iff (Plucker coordinates of lines in P3)

27 (Quadric surfaces) Ruled: hyperboloid of one sheet, 1,1,-1,-1---topo torus Unruled: sphere, ellipsoid, hyperboloid and paraboloid: 1,1,1, topo sphere

28 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,

29 2D general Euclidean transformation: 2D general affine transformation: 2D general projective transformation: Introduction to transformation

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

31 (n+1)*(n+1) -1 d.o.f. all projective properties are left invariant by A all transformations form a group GL(n,R) Check the most important one: linear dependency, i.e. lines into lines as line is just a span Starting pt for new investigation: Klein’s Erlangen program Inversely, we may also prove that any 1-1 transf. Preserving lines is a linear trans in homogeneous coord. Properties N+2 pts to determine a trans. = a proj. basis

32 on pts, lines and conics: Transforms contravariantly Co-variantly to preserve incidence Co-variantly NB: co-,contra-variance is w.r.t. the basis trans. Transpose is of no importance, il accommodates row/column vectors Some numerical examples of transformation on P2 (Some examples of transformations)

33 How to compute (canonical or standard) coordinates?--- affine case by definition, vector(x4-x1) = a vector(x2-x1) + b vector(x3-x1) by canonical transformation, x1->(0,0), x2->(1,0), x3->(0,1), get transfromation A, then Ax4 Given 4 pts, x1, x2, x3, x4, find the affine coord of x4 w.r.t. x1, x2 and x3: How to solve Ax=b?

34 Canonical projective coordinates? Given 5 pts, x1, x2, x3, x4, x5find the affine coord of x5 w.r.t. x1, x2, x3, x4: By canonical transformation: How to solve Ax=0?

35 A transformation between 2 spaces?

36 Exercise Compute the transformation from (0,0,1), (1,0,1), (0,1,1) and (1,1,1) into (0,0,1), (1,1/4,1),(0,1,1) and (1,3/4,1)

37 (Geometry as an invariant theory of transformation groups) projective geom. GL(n,R) cross-ratio affine geom. Subgroup A(n,R) ratio Euclidean geom. Subgroup E(n,R) distance Hierarchy of geometry: All proj. Transformations nicely form a group! Each geometry is associated with a (sub)group!

38 Affine transformation is a projective one which leaves the line at inf. invariant: x3=x3’=0 Example of dim 2 From projective to affine:

39 Similarity transformation is an affine one which leaves the circular pts I and J invariant What are the circular points? From affine to euclidean

40 Intuitive introduction of circular pts The pair of circular points The line at infinity of a usual plane Circular points

41 Affine transformation leaves the plane at inf. invariant Similarity (euclidean) leaves the absolute conic (globally, not point-wise) invariant What is the absolute conic? Example of transformation in P3

42 (Absolute conic) A space conic on the plane at infinity: In point coordinates: In plane coordinates: –rank 3 space quadric=absolute quadric Euclidean structure in projective space by the absolute conic

43 The plane at infinity A usual plane in 3D The absolute conic The pair of circular points The line at infinity of a usual plane

44 Key message from projective geometry for vision ‘abstract camera’ is a projective transformation from P3 to P2, so 3*4 matrix the intrinsic parameters of the camera are the image of the absolute conic!

45 Summary transformation and geometry group of transformation affine group: hyper-plane at inf. euclidean group: absolute pts