Lecture 1.3 Basic projective geometry. Thomas Opsahl

Transcription

1 Lecture 1.3 Basic projective geometr Thomas Opsahl

2 Motivation For the pinhole camera, the correspondence between observed 3D points in the world and D points in the captured image is given b straight lines through a common point (pinhole) This correspondence can be described b a mathematical model known as the perspective camera model or the pinhole camera model This model can be used to describe the imaging geometr of man modern cameras, hence it plas a central part in computer vision

3 Motivation Before we can stud the perspective camera model in detail, we need to expand our mathematical toolbox We need to be able to mathematicall describe the position and orientation of the camera relative to the world coordinate frame Also we need to get familiar with some basic elements of projective geometr, since this will make it MUCH easier to describe and work with the perspective camera model 3

4 Introduction AA TT BB AA AA ξξ BB BB We have seen that the pose of a coordinate frame BB relative to a coordinate frame AA, denoted AA ξξ BB, can be represented as a homogeneous transformation AA TT BB in D r r t A A A 11 1 Bx A A RB t B A ξb TB = = r1 r tbx SE ( )

5 Introduction AA TT BB AA AA ξξ BB BB We have seen that the pose of a coordinate frame BB relative to a coordinate frame AA, denoted AA ξξ BB, can be represented as a homogeneous transformation AA TT BB in D and 3D A r11 r1 r13 t Bx A A A A A RB t B r1 r r3 tb ξb TB = = SE ( 3) A 0 1 r31 r3 r33 t Bz

6 Introduction And we have seen how the can transform points from one reference frame to another if we represent points in homogeneous coordinates x x p= = p 1 x x p= p = z z 1 The main reason for representing pose as homogeneous transformations, was the nice algebraic properties that came with the representation

7 Introduction Euclidean geometr AA AA AA ξξ BB RR BB, tt BB Complicated algebra Projective geometr AA AA AA ξξ BB TT BB = RR BB Simple algebra AAtt BB 00 1 p= ξ p p= R p+ t A A B A A B A B B B ( R, t ) ( R R, R t t ) ( R, R t ) ξ = ξ ξ = + A A B A A A B A B A C B C C C B C B C B ξ A A T A T A B C C C p= ξ p p = T p A A B A A B B B A A B A A B ξc = ξb ξc TC = TB TC A A 1 ξb TB In the following we will take a closer look at some basic elements of projective geometr that we will encounter when we stud the geometrical aspects of imaging Homogeneous coordinates, homogeneous transformations

8 The projective plane Points How to describe points in the plane?

9 The projective plane Points x x How to describe points in the plane? Euclidean plane R Choose a D coordinate frame Each point corresponds to a unique pair of Cartesian coordinates xx = xx, R xx = xx

10 The projective plane Points How to describe points in the plane? w x Euclidean plane R Choose a D coordinate frame Each point corresponds to a unique pair of Cartesian coordinates 1 x x x xx = xx, R xx = xx Projective plane Expand coordinate frame to 3D Each point corresponds to a triple of homogeneous coordinates xx xx = xx,, ww R xx = ww s.t. xx,, ww = λλ xx,, ww λλ R\ 0

11 The projective plane Points w x Observations 1. An point xx = xx, in the Euclidean plane has a corresponding homogeneous point xx = xx,, 1 in the projective plane x x. Homogeneous points of the form xx,, 0 does not have counterparts in the Euclidean plane The correspond to points at infinit and are called ideal points 1 x

12 The projective plane Points w x Observations 3. When we work with geometrical problems in the plane, we can swap between the Euclidean representation and the projective representation 1 x x x x x x = x = 1 x x w = = x x w w

13 Example 1. These homogeneous vectors are different numerical representations of the same point in the plane x = = 4 = The homogeneous point 1,,3 represents the same point as 1 3, 3 R

14 The projective plane Lines How to describe lines in the plane?

15 The projective plane Lines How to describe lines in the plane? Euclidean plane R 3 parameters aa, bb, cc R ll = xx, aaaa + bbbb + cc = 0 l x

16 The projective plane Lines How to describe lines in the plane? w Euclidean plane R Triple aa, bb, cc R 3 \ 00 ll = xx, aaaa + bbbb + cc = 0 l x Projective plane Homogeneous vector ll = aa, bb, cc TT ll = xx ll TT xx = 0 1 l x

17 The projective plane Lines w Observations 1. Points and lines in the projective plane have the same representation, we sa that points and lines are dual objects in l x. All lines in the Euclidean plane have a corresponding line in the projective plane 1 l x 3. The line ll = 0,0,1 TT in the projective plane does not have an Euclidean counterpart This line consists entirel of ideal points, and is know as the line at infinit

18 The projective plane Lines w l x Properties of lines in the projective plane 1. In the projective plane, all lines intersect, parallel lines intersect at infinit Two lines ll 1 and ll intersect in the point xx = ll 1 ll. The line passing through points xx 1 and xx is given b ll = xx 1 xx 1 l x

19 Example Determine the line passing through the two points, 44 and 55, 1111 Homogeneous representation of points Homogeneous representation of line Equation of the line x 5 = 4 x = l = x1 x = [ x1] x = = 3 = x+ + = 0 = 3x Matrix representation of the cross product uu vv uu vv where 0 u3 u def [ u] = u3 0 u 1 u u1 0

20 Example A point at infinit 1

21 The projective plane Transformations Some important transformations like the action of a pose ξξ on points in the plane happen to be linear in the projective plane and non-linear in the Euclidean plane The most general invertible transformations of the projective plane are known as homographies or projective transformations / linear projective transformations / projectivities / collineations Definition A homograph of is a linear transformation on homogeneous 3-vectors represented b a homogeneous, non-singular 3 3 matrix HH xx ww = h 11 h 1 h 13 h 1 h h 3 h 31 h 3 h 33 So HH is unique up to scale, i.e. HH = λλλλ λλ R\ 0 xx ww

22 The projective plane Transformations One characteristic of homographies is that the preserve lines, in fact an invertible transformation of that preserves lines is a homograph Examples Central projection from one plane to another is a homograph Hence if we take an image with a perspective camera of a flat surface from an angle, we can remove the perspective distortion with a homograph xx xxx Perspective distortion Without distortion Images from

23 The projective plane Transformations One characteristic of homographies is that the preserve lines, in fact an invertible transformation of that preserves lines is a homograph Examples Central projection from one plane to another is a homograph Two images, captured b perspective cameras, of the same planar scene is related b a homograph Image 1 Image

24 The projective plane Transformations One characteristic of homographies is that the preserve lines, in fact an invertible transformation of that preserves lines is a homograph Examples Central projection from one plane to another is a homograph Two images, captured b perspective cameras, of the same planar scene is related b a homograph One can show that the product of two homographies also must be a homograph We sa that the homographies constitute a group the projective linear group PPPP 3 Within this group there are several more specialized subgroups

25 Transformations of the projective plane Transformation of Matrix #DoF Preserves Visualization Translation II tt 00 TT 1 Euclidean RR tt 00 TT 1 Similarit ssrr tt 00 TT 1 Affine aa 11 aa 1 aa 13 aa 1 aa aa Homograph /projective h 11 h 1 h 13 h 1 h h 3 h 31 h 3 h 33 Orientation + all below 3 Lengths + all below 4 Angles + all below 6 Parallelism, line at infinit + all below 8 Straight lines 6

26 The projective space The relationship between the Euclidean space R 3 and the projective space 3 is much like the relationship between R and In the projective space We represent points in homogeneous coordinates xx λλxx λλ xx = = λλ R\ 0 zz λλzz ww λλww Points at infinit have last homogeneous coordinate equal to zero Planes and points are dual objects Π = xx 3 ππ TT xx = 0 x x x = x = z z 1 x x w x = x = z w w z w The plane at infinit are made up of all points at infinit 7

27 Transformations of the projective space Transformation of 33 Matrix #DoF Preserves Translation II tt 00 TT 1 aa aa Euclidean RR tt 00 TT 1 Similarit ssrr tt 00 TT 1 Affine aa 11 aa 1 aa 1 aa aa 31 aa aa 3 aa 4 aa 33 aa Orientation + all below 6 Volumes, volume ratios, lengths + all below 7 Angles + all below 1 Parallelism of planes, The plane at infinit + all below Homograph /projective h 11 h 1 h 13 h 14 h 1 h h 3 h 4 h 31 h 3 h 33 h 34 h 41 h 4 h 43 h Intersection and tangenc of surfaces in contact, straight lines

28 Summar The projective plane Homogeneous coordinates Line at infinit Points & lines are dual Additional reading Szeliski:.1.,.1.3, The projective space 3 Homogeneous coordinates Plane at infinit Points & planes are dual Linear transformations of and 3 Represented b homogeneous matrices Homographies Affine Similarities Euclidean Translations 9

29 Summar The projective plane Homogeneous coordinates Line at infinit Points & lines are dual The projective space 3 Homogeneous coordinates Plane at infinit Points & planes are dual Linear transformations of and 3 Represented b homogeneous matrices Homographies Affine Similarities Euclidean Translations Additional reading Szeliski:.1.,.1.3, MATLAB WARNING When we work with linear transformations, we represent them as matrices that act on points b right multiplication T : n n x = M x Matlab seem to prefer left multiplication instead T : n n x = x T T T So if ou use built in matlab functions when ou work with transformations, be careful!!! R M L 30