Computer Vision. 2. Projective Geometry in 3D. Lars Schmidt-Thieme

Transcription

1 Computer Vision 2. Projective Geometry in 3D Lars Schmidt-Thieme Information Systems and Machine Learning Lab (ISMLL) Institute for Computer Science University of Hildesheim, Germany 1 / 26

2 Syllabus Mon (1) 0. Introduction 1. Projective Geometry in 2D: a. The Projective Plane Mon Easter Monday Mon (2) 1. Projective Geometry in 2D: b. Projective Transformations Mon Labor Day Mon (3) 2. Projective Geometry in 3D: a. Projective Space Mon (4) 2. Projective Geometry in 3D: b. Quadrics, Transformations Mon (5) 3. Estimating 2D Transformations: a. Direct Linear Transformation Mon (6) 3. Estimating 2D Transformations: b. Iterative Minimization Mon Pentecoste Day Mon (7) 4. Interest Points: a. Edges and Corners Mon (8) 4. Interest Points: b. Image Patches Mon ( 9) 5. Simulataneous Localization and Mapping: a. Camera Models Mon (10) 5. Simulataneous Localization and Mapping: b. Triangulation 1 / 26

3 Outline 1. Points, Lines, Planes in Projective Space 2. Quadrics 3. Transformations 1 / 26

4 1. Points, Lines, Planes in Projective Space Outline 1. Points, Lines, Planes in Projective Space 2. Quadrics 3. Transformations 1 / 26

5 1. Points, Lines, Planes in Projective Space Objects in 2D Revisited type repr. dim dof examples points P circular points I, J lines P line at inf. l point conics Sym(P 2 2 ) 1 5 line conics Sym(P 2 2 ) 2 5 dual conic of circ. pts. C Note: The dimensionality applies to non-degenerate cases only. 1 / 26

6 1. Points, Lines, Planes in Projective Space Homogeneous Coordinates: Points Inhomogeneous coordinates: x R 3 Homogeneous coordinates: x P 3 := (R 4 \ {(0, 0, 0, 0) T })/ x y : s R \ {0} : sx = y, x, y R 4 2 / 26

7 1. Points, Lines, Planes in Projective Space Homogeneous Coordinates: Points Inhomogeneous coordinates: Homogeneous coordinates: Example: x R 3 x P 3 := (R 4 \ {(0, 0, 0, 0) T })/ x y : s R \ {0} : sx = y, x, y R represent the same point in P3 represent a different point in P3 2 / 26

8 1. Points, Lines, Planes in Projective Space Homogeneous Coordinates: Points Inhomogeneous coordinates: Homogeneous coordinates: x R 3 x P 3 := (R 4 \ {(0, 0, 0, 0) T })/ x y : s R \ {0} : sx = y, x, y R 4 finite points: ideal points: x 1 x 2 x 3 1 x 1 x 2 x 3 0 =: ι( x 1 x 2 x 3 ) 2 / 26

9 1. Points, Lines, Planes in Projective Space Dual of Points: Planes Inhomogeneous coordinates: x 1 p R 4 :P p := { x 2 p 1 x 1 + p 2 x 2 + p 3 x 3 + p 4 = 0} x 3 Note: κ : R 4 P 3, a [a] := {a R 4 a a}. 3 / 26

10 1. Points, Lines, Planes in Projective Space Dual of Points: Planes Inhomogeneous coordinates: x 1 p R 4 :P p := { x 2 p 1 x 1 + p 2 x 2 + p 3 x 3 + p 4 = 0} x 3 Homogeneous coordinates: p P 3 : P p := {x P 3 p T x = p 1 x 1 + p 2 x 2 + p 3 x 3 + p 4 x 4 = 0} contains all finite points of p κ 1 (p): P κ(p ) ι(p p ) Note: κ : R 4 P 3, a [a] := {a R 4 a a}. 3 / 26

11 1. Points, Lines, Planes in Projective Space Intersecting Planes Zeroset / Null space: Nul(H) :={p P 3 Hp = 0} All points incident to two planes p, q (p q): PP(p, q) :={x P 3 x P p, x P q } = {x P 3 p T x = q T x = 0} Can be represented as zeroset: PP(p, q) =Nul(pq T qp T ) 4 / 26

12 1. Points, Lines, Planes in Projective Space Intersecting Planes Zeroset / Null space: Nul(H) :={p P 3 Hp = 0} All points incident to two planes p, q (p q): PP(p, q) :={x P 3 x P p, x P q } = {x P 3 p T x = q T x = 0} Can be represented as zeroset: PP(p, q) =Nul(pq T qp T ) idea: represent lines as intersection of planes 4 / 26

13 1. Points, Lines, Planes in Projective Space All Planes Containing Two Points All planes containing two points x, y (x y): PP (x, y) :={p P 3 x, y P p } = {p P 3 p T x = p T y = 0} Can be represented as zeroset: PP (x, y) =Nul(xy T yx T ) this is just the dual of All points incident to two planes idea: represent lines as intersection of planes any two planes containing two points x, y will do 5 / 26

14 1. Points, Lines, Planes in Projective Space Plücker Matrix For two points x, y P 3 : Plü(x, y) :=A := xy T yx T skew symmetric: A T = A esp. zero diagonal: A i,i = 0. rank 2 (for x y) 6 / 26

15 1. Points, Lines, Planes in Projective Space Lines have 4 Degrees of Freedom 68 3 Projective Geometry and Transformations of 3D Fig A line may be specified by its points of intersection with two orthogonal planes. Each intersection point has 2 degrees of freedom, which demonstrates that a line in IP 3 has a total of 4 degrees of freedom. A direct solution for X, in terms of determinants of 3 3 submatrices, is obtained as an analogue of (3.4), though computationally a numerical solution would [HZ04, bep. obtained 68] by algorithm A5.1(p589). The two following results are direct analogues of their 2D counterparts. 7 / 26

16 1. Points, Lines, Planes in Projective Space Lines via Dual Plücker Matrices Lines can be defined easily via spans: span(x 1, x 2,..., x M ) := M Rx m := {z R M s R M : z = m=1 l(x, y) :=span(x, y) M s m x m } m=1 8 / 26

17 1. Points, Lines, Planes in Projective Space Lines via Dual Plücker Matrices Lines can be defined easily via spans: span(x 1, x 2,..., x M ) := M Rx m := {z R M s R M : z = m=1 l(x, y) :=span(x, y) M s m x m } m=1 Lines can be represented in 3D as zeroset of the dual Plücker matrix: with l(x, y) = Nul(Plü (x, y)) Plü (x, y) :=A := and Plü(x, y) :=A := xy T yx T 0 A 3,4 A 4,2 A 2,3 A 3,4 0 A 1,4 A 3,1 A 4,2 A 1,4 0 A 1,2 A 2,3 A 3,1 A 1,2 0 (Plücker-Matrix) 8 / 26

18 1. Points, Lines, Planes in Projective Space Lines via Dual Plücker Matrices Now PP(x, y) = Nul(A), A = xy T yx T l(x, y) = Nul(A ), A = pq T qp T, p, q PP(x, y) A A =(pq T qp T )(xy T yx T ) therefore for all i, j, i j: 4 0 = (A A) i,j = A i,k A j,k = =pq T xy T pq T yx T qp T xy T + qp T yx T = 0 k=1 k {i,j} A i,k A j,k as diagonals are zero i.e., A i,k 1 A j,k1 + A i,k 2 A j,k2 = 0, {1, 2, 3, 4} = {i, j, k 1, k 2 } and thus A 3,4 A = A 4,2 1,2 A = A 2,3 1,3 A = A 1,2 1,4 A = A 1,3 3,4 A = A 1,4 4,2 A 2,3 9 / 26

19 1. Points, Lines, Planes in Projective Space Operations on Points, Lines & Planes point x on plane p: p T x =0 point x on line A : A x =0 line A is on plane p: (A ) p =0 plane p joining points x, y, z: (x y z) T p =0 plane p joining point x and line A : p =A x line A joining points x, y: A =Plü (x, y) =(xy T yx T ) line A as intersection of planes p, q: A =pq T qp T point x as intersection of plane p and line A : x =(A ) p point x as intersection of planes p, q, r: (p q r) T x =0 10 / 26

20 1. Points, Lines, Planes in Projective Space Plane at Infinity p All ideal points (x 1, x 2, x 3, 0) T form a plane, the plane at infinity p := (0, 0, 0, 1) T. Two parallel planes A parallel line and plane Two parallel lines intersect in a line a point a point on p p is fixed under affine transformations. Proofs: same as for the line at infinity in P / 26

21 2. Quadrics Outline 1. Points, Lines, Planes in Projective Space 2. Quadrics 3. Transformations 12 / 26

22 2. Quadrics 3.3 Twisted cubics Quadrics Quadratic surfaces: Q Q := {x P 3 x T Qx = 0}, Q Sym(P 4 4 ) 9 degrees of freedom 9 points in general positionfig. define 3.2. Non-ruled a quadric quadrics. paraboloid. They are all projectively equivalent. The intersection of a plane p with 3.3 a quadric Twisted Q cubics is a conic A quadric Q transforms as H T QH 1 : H(Q Q ) = Q H T QH 1 This shows plots of a sphere, ellipsoi [HZ04, p. 75] Lars Schmidt-Thieme, Information Systems and Machine Fig. Learning 3.3. Ruled Lab (ISMLL), quadrics. University Twoof examples Hildesheim, of agermany hyperboloid of one sh given by equations X 2 + Y 2 = Z 2 +1and XY = Z respectively, 12 / and 26

23 2. Quadrics Quadrics / Signature Q = USU T SVD: S diagonal, UU T = I = HS H T S diagonal with S i,i {+1, 1, 0} signature of quadric Q: σ(q) := {i {1, 2, 3, 4} S i,i = +1} {i {1, 2, 3, 4} S i,i = 1} 13 / 26

24 2. Quadrics Quadrics / Types rank σ diagonal equation point set 4 4 (1, 1, 1, 1) x 2 + y 2 + z = 0 no real points 2 (1, 1, 1, -1) x 2 + y 2 + z 2 1 = 0 sphere 0 (1, 1, -1, -1) x 2 + y 2 z 2 1 = 0 hyperboloid of one sheet 3 3 (1, 1, 1, 0) x 2 + y 2 + z 2 = 0 one point (0, 0, 0, 1) T 1 (1, 1, -1, 0) x 2 + y 2 z 2 = 0 cone at origin 2 2 (1, 1, 0, 0) x 2 + y 2 = 0 single line (z-axis) 0 (1, -1, 0, 0) x 2 y 2 = 0 two planes x = ±y 1 1 (1, 0, 0, 0) x 2 = 0 one plane x = 0 14 / 26

25 2. Quadrics 3.3 Twisted cubics 75 Quadrics / Types a) rank = 4, σ = 2 : sphere / ellipsoid 3.3 Twisted cubics 75 Fig Non-ruled quadrics. This shows plots of a sphere, ellipsoid, hyperboloid of two sheets and paraboloid. They are all projectively equivalent. b) rank = 4, σ = 0 : hyperboloid Fig Non-ruled quadrics. This shows plots of a sphere, ellipsoid, hyperboloid of two sheets and paraboloid. They are all projectively equivalent. [HZ04, p. 75] 15 / 26

26 2. Quadrics Quadrics / Types (2/2) c) rank = 3, σ = 1 : cone d) rank = 2, σ = 0 : two planes 76 3 Projective Geometry and Transformations of 3D Fig Degenerate quadrics. The two most important degenerate quadrics are shown, the cone and two planes. Both these quadrics are ruled. The matrix representing the cone has rank 3, and the nullvector represents the nodal point of the cone. The matrix representing the two (non-coincident) planes has rank 2, and the two generators of the rank 2 null-space are two points on the intersection line of the planes. [HZ04, p. 76] A conic in the 2-dimensional projective plane may be described as a parametrized 16 / 26

27 2. Quadrics Absolute Dual Quadric Q Plane/dual quadrics: Q Q := {p P3 p T Q p = 0}, Q Sym(P 4 4 ) Absolute dual quadric: Q := ( I 0 0 T 0 ) = / 26

28 2. Quadrics Absolute Dual Quadric Q Invariant under Similarity The absolute dual quadric Q is invariant under projectivity H H is a similarity. proof: ( A t H = v T v 4 ( HQ H T A t = v T v 4 ), ) ( I 0 0 T 0 ) ( A T v t T v 4 ) 18 / 26

29 2. Quadrics Absolute Dual Quadric Q Invariant under Similarity The absolute dual quadric Q is invariant under projectivity H H is a similarity. proof: ( ) A t H = v T, v 4 ( ) ( HQ H T A t A = T ) v v T v 4 0 T 0 ( ) AA T Av! = v T A T v T = Q v v = 0, AA T = I, i.e., H is a similarity 18 / 26

30 2. Quadrics Absolute Dual Quadric Q p is the nullvector of Q. The angle between two planes is given by cos θ(p, q) := p T Q q p T Q p q T Q q esp. two planes p, q are orthogonal iff p T Q q = 0. proofs: as in P / 26

31 2. Quadrics Absolute Conic Ω C Ω :=Q Q P p ={x P 3 x x x 2 3 = 0, x 4 = 0} H is a similarity transform Ω is invariant under H 20 / 26

32 2. Quadrics Objects in 3D type repr. dim dof examples points P lines Skew(P 4 4 ) 1 5 planes P plane at inf. p point quadrics Sym(P 4 4 ) 2 9 plane quadrics Sym(P 4 4 ) 3 9 absolute dual quadric Q conic p Q 1 8 absolute conic Ω Note: The dimensionality applies to non-degenerate cases only. 21 / 26

33 3. Transformations Outline 1. Points, Lines, Planes in Projective Space 2. Quadrics 3. Transformations 22 / 26

34 3. Transformations 78 3 Projective Geometry and Transformations of 3D Hierarchy of Transformations Group Matrix Distortion Invariant properties Projective 15 dof [ ] A t Intersection and tangency of surfaces in contact. Sign of Gaussian v T v curvature. Affine 12 dof [ ] A t Parallelism of planes, volume ratios, centroids. The plane at infin- 0 T 1 ity, π, (see section 3.5). Similarity 7 dof [ sr ] t 0 T 1 The absolute conic, Ω, (see section 3.6). Euclidean 6 dof [ R ] t 0 T 1 Volume. [HZ04, p. 78] 22 / 26

35 3. Transformations Rotations in 3D Rotations in 3D can be described by a rotation axis and a rotation angle. 23 / 26

36 3. Transformations Rotations in 3D Rotations in 3D can be described by a rotation axis and a rotation angle. Pure rotations (rotations along an axis through the origin) can be described by 1. a rotation axis direction (an axis through the origin) and a rotation angle, or Pure rotations have 3 dof. 23 / 26

37 3. Transformations Rotations in 3D Rotations in 3D can be described by a rotation axis and a rotation angle. Pure rotations (rotations along an axis through the origin) can be described by 1. a rotation axis direction (an axis through the origin) and a rotation angle, or 2. Euler-Tait-Bryan angles: R x (α) := 0 cos α sin α, R y (β) := 0 sin α cos α R = R z (γ)r y (β)r x (α), cos β 0 sin β sin β 0 cos β, R z (γ) := cos γ sin γ 0 sin γ cos γ Pure rotations have 3 dof. 23 / 26

38 3. Transformations Rotations in 3D Rotations in 3D can be described by a rotation axis and a rotation angle. Pure rotations (rotations along an axis through the origin) can be described by 1. a rotation axis direction (an axis through the origin) and a rotation angle, or 2. Euler-Tait-Bryan angles: R x (α) := 0 cos α sin α, R y (β) := 0 sin α cos α 3. a proper orthogonal matrix: R = R z (γ)r y (β)r x (α), cos β 0 sin β sin β 0 cos β, R z (γ) := R R 3 3 : RR T = R T R = I, det R = 1 cos γ sin γ 0 sin γ cos γ Pure rotations have 3 dof. 23 / 26

39 3. Transformations The Screw Decomposition Any Euclidean transformation, i.e., a 3D rotation R followed by a translation t, can be represented as a rotation followed by a translation along the same axis (called skrew axis) [HZ04, p. 79] 24 / 26

40 3. Transformations The Screw Decomposition Any Euclidean transformation, i.e., a 3D rotation R followed by a translation t, can be represented as a rotation followed by a translation along the same axis (called skrew axis) Proof: 1. if t is orthogonal to the rotation axis of R: planar transformation. 3.5 The plane at infinity 79 / x / x R( θ ) y / O / y / O / t S θ y y O a x O b x Fig D Euclidean motion and a screw axis. (a) The frame {x, y} undergoes a translation t and a rotation by θ to reach the frame {x,y }. The motion is in the plane orthogonal to the rotation axis. (b) This motion is equivalent a single rotation about the screw axis S. The screw axis lies on the 2. generally: decompose perpendicular bisector of ttheinto line joining t corresponding points, such that the angle between the lines joining S to the corresponding points is orthogonal and t θ. In the figure the corresponding parallel. points are the two frame origins and θ has the value 90. a a [HZ04, p. 79] 24 / 26

41 3. Transformations Summary (1/2) The projective space P 3 is an extension of the Euclidean space R 3 with ideal points. Points and planes in P 3 are parametrized by homogenuous coordinates, planes by homogeneous skew-symmetric matrices. Each two parallel lines intersect in an ideal point, each two parallel planes intersect in a line of ideal points, all ideal points form the plane at infinity p. Quadrics are surfaces of order 2 (hyperboloid, paraboloid, ellipsoid), parametrized by a symmetric matrix Q containing all points x with x T Qx = / 26

42 3. Transformations Summary (2/2) Projectivities H are invertibles mappings of P 3 onto P 3 that preserve lines. Lines a transform via H T a, quadrics Q via H T QH 1. There exist several subgroups of the group of projectivities: Isometries rotate and translate figures. preserving lengths Similarities additionally (isotropic) scale figures. preserving ratio of lengths, angles, the plane at infinity p Affine transforms additionally non-isotropic scale figures. preserving ratio of lengths on parallel lines, parallel lines, the absolute conic Ω Projectivities additionally move the plane at infinity. preserving cross ratios Any projectivity can be decomposed into a chain of an pure projective, a pure affine transform and a similarity. 26 / 26

43 Further Readings [HZ04, ch. 3]. for the derivation of the dual Plücker coordinates [Cox98, p. 88f] 27 / 26

44 References Harold Scott Macdonald Coxeter. Non-euclidean geometry. Cambridge University Press, Richard Hartley and Andrew Zisserman. Multiple view geometry in computer vision. Cambridge university press, / 26