1. Projective geometry

Transcription

1 1. Projective geometry Homogeneous representation of points and lines in D space D projective space Points at infinity and the line at infinity Conics and dual conics Projective transformation Hierarchy of D projective transformation

2 Notation Scalars: x, y,... Vectors: x, y,... Transpose: x >, y >,... Matrices: A, B, P,... Transpose: A >, B >, P >,...

3 Homogeneous representation of lines in D space A line on a xy plane: ax + by + c =0 Any line can be represented by a -vector l =[a, b, c] > The same line is given by k(a, b, c) > =(ka, kb, kc) > kax + kby + kc = k(ax + by + c) =0 In this sense, (a, b, c) > is equivalent to k(a, b, c) > Such vectors are called homogeneous vectors This equivalence will be represented by Two vectors on both sides are parallel (a, b, c) > / k(a, b, c) > /

4 Homogeneous representation of points in D space A point (x, y) is on a line l =(a, b, c) >, l > x = a b c x 4y5 = ax + by + c =0 1 Homogeneous representation of a point: x =(x, y, 1) > The same point is given by: k(x, y, 1) > =(kx, ky, k) > (x, y, 1) > / k(x, y, 1) > To recover the original (inhomogeneous) coordinates from homogeneous coordinates, simply do x =(x 1,x,x ) > x1, x x x

5 D projective space Sets of equivalent vectors in this sense form a special space called the projective space x x x x x x D Euclidean space D projective space

6 Intersection of lines Q) What is the crossing point of lines? l = a 4b5 l 0 = c a 0 4b 0 5 c 0 A) It is given by their cross product: x l l 0 l l 0 Proof: l > x = l > (l l 0 )=0 (Scalar triple product) x =1 y =1 E.g. Crossing point of and? l =[ 1, 0, 1] > l 0 =[0, 1, 1] > x = l l 0 = = (1/1, 1/1) = (1, 1) y l y = 1 l 0 (1,1) x = 1 x

7 Points at infinity l =(a, b, c) > l 0 =(a, b, c 0 ) > Consider the intersection of and l l 0 =(bc 0 bc, ca ac 0,ab ab) > =(c 0 c)(b, a, 0) > Conversion to inhomogeneous coordinates (b/0, a/0) = (1, 1) (x, y, 0) > A point with homogeneous coordinates is called a point at infinity A point (x 1,x,x ) > gives a finite point if x 6=0and gives a point at infinity if x =0

8 Points at infinity In inhomogeneous domain, parallel lines do not have an intersection In homogenous domain, parallel lines have an intersection at a point at infinity Points at infinity and finite points can be treated equally ß an advantage of using homogeneous representation Note that there are an infinite number of ʻpoints at infinityʼ y (a, b, 0) > x

9 The line at infinity l =(0, 0, 1) > The special line is called the line at infinity The name comes from the fact that every point at infinity will lie on this line x 1 x 0 5 =0 This line is unique, and will be denoted by l 1 =(0, 0, 1) >

10 Duality of points and lines There is symmetry between points and lines l > x = x > l =0 Exchange of points and lines in a proposition wonʼt change its correctness E.g. Intersection of lines is given by, x = l l 0 A line passing through two points is given by l = x x 0 Prove this directly without using the duality (1 st assignment)

11 Conics A conic = a curve obtained as the intersection of a cone with a plane Ellipse Hyperbola Parabola Any conic on a xy plane is given by ax + bxy + cy + dx + ey + f =0 x x 1 /x and y x /x ax 1 + bx 1 x + cx + dx 1 x + ex x + fx =0

12 Conics Or, can also be written as C = x > Cx =0 a b/ d/ 4b/ c e/5 d/ e/ f Any symmetric matrix represents a conic Matrices scaled by constants give the same conic Specifying five points determines a conic (passing these points)

13 Tangents to a conic l The line tangent to a conic at a point is given by C l = Cx x x l x > l = x > Cx =0 Proof: lies on, since Assume that there exists another point and. l y lying on C y Then, for any, it should hold that (x + y) > C(x + y) =0 x l This means that any point x + y should lie on C. This is possible only if C is a line or there exists no such point y.

14 Dual conics C A conic gives a set of points. A dual conic is another type of conics, which gives a set of lines Also known as line conics l > C l =0 Consider, where C C 1 This gives a set of tangent lines to conic ---- (*) C C l > C l =0 x > Cx =0 Show the above (*) is true ( nd assignment).

15 D projective transformation Projective transformation (different names: projectivity, homography, colineation) Definition: An invertible mapping from/to D projective spaces that satisfies that if three points lie on a line, then the mapped points lie on a line. Intuitively, any mapping from a plane to a plane that maps a line to a line Such a transformation ( x! x 0 ) is given by the following equation: x 0 / Hx or x 0 / h 11 h 1 h 1 4h 1 h h 5 x h 1 h h

16 An example: central projection Central projection: a point on a plane is mapped onto a point on another plane as shown below x x π π Obviously, central projection maps a line to a line, and thus it is a d projective transformation

17 Calculating a projective transformation If you want to represent the points before/after the mapping in inhomogeneous coordinates x 0 / h 11 h 1 h 1 4h 1 h h 5 x h 1 h h 4 x 0 y / h 11 h 1 h 1 x 4h 1 h h 5 4y5 = h 1 h h 1 h 11 x + h 1 y + h 1 4h 1 x + h y + h 5 h 1 x + h y + h x 0 = h 11x + h 1 y + h 1 h 1 x + h y + h y 0 = h 1x + h y + h h 1 x + h y + h

18 Transformation of lines and conics A D projective transformation mapping a point as x 0 / Hx maps a line as proof: l 0 / H > l x 0> l 0 =(x > H > )l 0 =(x > H > )(H > l)=x > l =0 and maps a conic as C 0 / H > CH 1 proof: x 0> C 0 x 0 = x > H > C 0 Hx = x > H > H > CH 1 Hx = x > Cx =0

19 Transformation of lines and conics l π l π C C π π

20 Hierarchy of D projective transformation Projective transformation can be classified into the following four types in the order of increasing degrees of freedom: 1. Euclidean transformation / Isometry. Similarity transformation. Affine transformation 4. Projective transformation (full-projective ---) x 0 / h 11 h 1 h 1 4h 1 h h 5 x h 1 h h

21 Euclidean transformation D projective transformations given as follows: " cos sin t x x 0 / 4" sin cos t y 5 x Called Euclidean trans. if ε=1; isometry if ε=-1 or 1. Combination of D rotation and translation Length and area are preserved (invariant)

22 Similarity transformation D projective transformations given as follows: s cos s sin t x x 0 / 4s sin s cos t y 5 x Combination of D rotation, translation, and scaling Shape and angle are preserved (invariant)

23 Affine transformation D projective transformations given as follows: a 11 a 1 t x x 0 / 4a 1 a t y 5 x Parallelism is preserved Points at infinity are mapped to points at infinity x 0 / = a 11 a 1 t x x 1 4a 1 a t y 5 4x a 11 x 1 + a 1 x 4a 1 x 1 + a x 5 0 5

24 Projective transformation The most general one: x 0 / h 11 h 1 h 1 4h 1 h h 5 x h 1 h h Colinearity and cross-ratio are preserved Points at infinity can be mapped to finite points x 0 / 5 = h 11 h 1 h 1 x 1 4h 1 h h 5 4x 5 h 1 h h 0 h 11 x 1 + h 1 x 4h 1 x 1 + h x 5 h 1 x 1 + h x

25 Transformations and images Images of a plane created by different transformations [Hartley-Zisserman0] similarity affine projective

26 Image rectification Given a projective transform of a plane, we want to find a transformation that maps it onto a similarity transform of the same plane x x 1 x 0 x 0 1 x x 4 x 0 x 0 4 [Hartley-Zisserman0] Any projective trans. is determined by four point pairs