Projective geometry- 2D
Acknowledgements Marc Pollefeys: for allowing the use of his excellent slides on this topic http://www.cs.unc.edu/~marc/mvg/ Richard Hartley and Andrew Zisserman, "Multiple View Geometry in Computer Vision"
Homogeneous coordinates
Homogeneous representation of lines ax + by + c = 0 ()a,b,c T (ka)x + (kb)y + kc = 0,k 0 ()()a,b,c T ~ k a,b,c T equivalence class of vectors, any vector is representative Set of all equivalence classes in R3(0,0,0)T forms P2 Homogeneous representation of points T T x = ()x, y on l = ()a,b,c if and only if ax + by + c = 0 ()()()x,y,1 a,b,c T = x,y,1 l = 0 ()()x, y,1 T ~ k x, y,1 T ,k 0 The point x lies on the line l if and only if xTl=lTx=0 T Homogeneous coordinates ()x1, x2 , x3 but only 2DOF T Inhomogeneous coordinates ()x, y
Spring 2006 Projective Geometry 2D 2
1 Points from lines and vice-versa
Intersections of lines
The intersection of two lines l and l' is x = ll' Line joining two points The line through two points x and x' is l = x x'
Example
y =1
x =1 Spring 2006 Projective Geometry 2D 3
Ideal points and the line at infinity
Intersections of parallel lines
T T T l = ()a,b,c and l'= ()a,b,c' ll'= ()b,a,0
Example
x =1 x = 2
T Ideal points ()x1, x2 ,0 Note that this set lies on a single line, T Line at infinity l = ()0,0,1
2 2 Note that in P2 there is no distinction P = R l between ideal points and others
Spring 2006 Projective Geometry 2D 4
2 Summary
The set of ideal points lies on the line at infinity,
intersects the line at infinity in the ideal point
A line parallel to l also intersects in the same ideal point, irrespective of the value of c’.
In inhomogeneous notation, is a vector tangent to the line. It is orthogonal to (a, b) -- the line normal. Thus it represents the line direction. As the line’s direction varies, the ideal point varies over . --> line at infinity can be thought of as the set of directions of lines in the plane.
Spring 2006 Projective Geometry 2D 5
A model for the projective plane
Points represented by rays through origin Lines represented by planes through origin
x1x2 plane represents line at infinity
exactly one line through two points exaclty one point at intersection of two lines Spring 2006 Projective Geometry 2D 6
3 Duality
x l xTl = 0 lT x = 0 x = ll' l = x x'
Duality principle: To any theorem of 2-dimensional projective geometry there corresponds a dual theorem, which may be derived by interchanging the role of points and lines in the original theorem
Spring 2006 Projective Geometry 2D 7
Conics
Curve described by 2nd-degree equation in the plane
ax2 + bxy + cy 2 + dx + ey + f = 0
x x or homogenized x 1 , y 2 x3 x3 2 2 2 ax1 + bx1x2 + cx2 + dx1x3 + ex2 x3 + fx3 = 0
or in matrix form a b / 2 d / 2 xT C x 0 with C b / 2 c e / 2 = = d / 2 e / 2 f
5DOF: {a :b : c : d : e : f }
Spring 2006 Projective Geometry 2D 8
4 Conics …
http://ccins.camosun.bc.ca/~jbritton/jbconics.htm
Spring 2006 Projective Geometry 2D 9
Five points define a conic
For each point the conic passes through
2 2 axi + bxi yi + cyi + dxi + eyi + f = 0 or 2 2 T ()xi , xi yi , yi , xi , yi , f c = 0 c = ()a,b,c,d,e, f
stacking constraints yields
2 2 x1 x1 y1 y1 x1 y1 1 2 2 x x y y x y 1 2 2 2 2 2 2 2 2 x3 x3 y3 y3 x3 y3 1c = 0 2 2 x4 x4 y4 y4 x4 y4 1 2 2 x5 x5 y5 y5 x5 y5 1
Spring 2006 Projective Geometry 2D 10
5 Tangent lines to conics
The line l tangent to C at point x on C is given by l=Cx
l x C
Spring 2006 Projective Geometry 2D 11
Dual conics
A line tangent to the conic C satisfies lT C* l = 0
In general (C full rank): C* = C1 C* : Adjoint matrix of C. Dual conics = line conics = conic envelopes
Spring 2006 Projective Geometry 2D 12
6 Degenerate conics
A conic is degenerate if matrix C is not of full rank m e.g. two lines (rank 2) l C = lmT + mlT
e.g. repeated line (rank 1)
T C = ll l
Degenerate line conics: 2 points (rank 2), double point (rank1)
* Note that for degenerate conics ()C* C
Spring 2006 Projective Geometry 2D 13
Projective transformations
Definition: A projectivity is an invertible mapping h from P2 to itself
such that three points x1,x2,x3 lie on the same line if and only if h(x1),h(x2),h(x3) do. Theorem:
A mapping h:P2P2 is a projectivity if and only if there exist a non-singular 3x3 matrix H such that for any point in P2 represented by a vector x it is true that h(x)=Hx
Definition: Projective transformation x' h h h x 1 11 12 13 1 x' h h h x 2 = 21 22 23 2 or x'= H x x'3 h31 h32 h33 x3 8DOF projectivity=collineation=projective transformation=homography
Spring 2006 Projective Geometry 2D 14
7 Mapping between planes
central projection may be expressed by x’=Hx (application of theorem)
Spring 2006 Projective Geometry 2D 15
Removing projective distortion
select four points in a plane with know coordinates x' h x + h y + h x' h x + h y + h x'= 1 = 11 12 13 y'= 2 = 21 22 23 x'3 h31x + h32 y + h33 x'3 h31x + h32 y + h33
x'()h31x + h32 y + h33 = h11x + h12 y + h13 (linear in hij) y'()h31x + h32 y + h33 = h21x + h22 y + h23
(2 constraints/point, 8DOF 4 points needed) Remark: no calibration at all necessary, better ways to compute (see later) Spring 2006 Projective Geometry 2D 16
8 Transformation of lines and conics
For a point transformation x'= H x Transformation for lines l'= H-T l
Transformation for conics C'= H-TCH-1
Transformation for dual conics C'* = HC*HT
Spring 2006 Projective Geometry 2D 17
Distortions under center projection
Similarity: squares imaged as squares. Affine: parallel lines remain parallel; circles become ellipses. Projective: Parallel lines converge.
Spring 2006 Projective Geometry 2D 18
9 Class I: Isometries
(iso=same, metric=measure)
x' cos sin t x x y' sin cos t y = y = ±1 1 0 0 1 1
orientation preserving: =1 orientation reversing: = 1 R t x' H x x T = E = T R R = I 0 1 3DOF (1 rotation, 2 translation) special cases: pure rotation, pure translation
Invariants: length, angle, area Spring 2006 Projective Geometry 2D 19
Class II: Similarities
x' s cos ssin t x (isometry + scale) x y' ssin s cos t y = y 1 0 0 1 1
sR t x' H x x T = S = T R R = I 0 1
4DOF (1 scale, 1 rotation, 2 translation)
also know as equi-form (shape preserving) metric structure = structure up to similarity (in literature) Invariants: ratios of length, angle, ratios of areas, parallel lines
Spring 2006 Projective Geometry 2D 20
10 Class III: Affine transformations
x' a a t x 11 12 x y' a a t y = 21 22 y 1 0 0 1 1
A t x' H x x = A = T 0 1
1 0 A = R()( R )DR () D = 0 2 6DOF (2 scale, 2 rotation, 2 translation) non-isotropic scaling! (2DOF: scale ratio and orientation)
Invariants: parallel lines, ratios of parallel lengths, ratios of areas
Spring 2006 Projective Geometry 2D 21
Class VI: Projective transformations
A t T x' H x x v = v ,v = P = T ()1 2 v v
8DOF (2 scale, 2 rotation, 2 translation, 2 line at infinity)
Action non-homogeneous over the plane
Invariants: cross-ratio of four points on a line (ratio of ratio)
Spring 2006 Projective Geometry 2D 22
11 Action of affinities and projectivities on line at infinity
x1 x A t A 1 x = T 2 x2 0 v 0 0 Line at infinity stays at infinity, but points move along line
x1 x A t A 1 x = T 2 x2 v v 0 v1x1 + v2 x2
Line at infinity becomes finite, allows to observe vanishing points, horizon.
Spring 2006 Projective Geometry 2D 23
Decomposition of projective transformations
sR tK 0 I 0 A t H H H H = S A P = T T T = T 0 10 1v v v v
decomposition unique (if chosen s>0) A = sRK + tvT
K upper-triangular, det K =1 Example: 1.707 0.586 1.0 H 2.707 8.242 2.0 = 1.0 2.0 1.0
2cos 45 2sin 45 1.00.5 1 01 0 0 H 2sin 45 2cos 45 2.0 0 2 00 1 0 = 0 0 1 0 0 11 2 1
Spring 2006 Projective Geometry 2D 24
12 Overview transformations
h11 h12 h13 Projective Concurrency, collinearity, h21 h22 h23 order of contact (intersection, 8dof tangency, inflection, etc.), h31 h32 h33 cross ratio
a11 a12 tx Parallellism, ratio of areas, a a t ratio of lengths on parallel Affine 21 22 y lines (e.g midpoints), linear 6dof 0 0 1 combinations of vectors (centroids).
The line at infinity l sr11 sr12 tx Similarity sr sr t 21 22 y 4dof Ratios of lengths, angles. 0 0 1 The circular points I,J
r11 r12 tx r r t Euclidean 21 22 y lengths, areas. 3dof 0 0 1 Spring 2006 Projective Geometry 2D 25
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