1. Multiple View Geometry Projective Geometry & Transformations of 2D Vladimir Nedović 18-01-2008 Intelligent Systems Lab Amsterdam (ISLA) Informatics Institute, University of Amsterdam Kruislaan 403, 1098 SJ Amsterdam, The Netherlands vnedovic@science.uva.nl

2. Outline Intro to projective geometry The 2D projective plane Projective transformations Hierarchy of transformations Projective geometry of 1D Recovery of affine & metric properties from images More properties of conics

3. Intro to Projective Geometry Projective transformation: any mapping of points in the plane that preserves straight lines Projective space: an extension of a Euclidean space in which two lines always meet in a point parallel lines meet at inf. => no parallelism in proj. space parallel lines meet at inf. => no parallelism in proj. space x = x/1 y = y/1 homogeneous coordinates in P 2 (x,y,0) = (x/0,y/0,0) = (∞,∞,0) points at infinity coordinates in Euclidean R 2 (x,y) = (x,y,1) = (kx,ky,k)k ≠ 0

4. Intro to Projective Geometry (cont.) Euclidean/affine transformation of Euclidean space: points at infinity remain at infinity ≠ Projective transformation of projective space: points at infinity map to arbitrary points x’ = H x (n+1)x(n+1) non-singular matrix a point in P n, an (n+1) - vector In P 2, points at infinity form a line, in P 3 a plane, etc. e.g. an image of the real 3D world e.g. the real 3D world

5. The 2D projective plane Line l in the plane: Line l in the plane: ax + by + c = 0 – equiv. to in slope-intercept notation – thus a line could be represented by a vector (a,b,c) T A point x lies on line l iff ax + by + c = (x,y,1)(a,b,c) T = x T l = 0 Lines and points represented by homogeneous vectors (a,b,c) T = k(a,b,c) T (x,y) T = k(x,y) T k ≠ 0

6. The 2D projective plane (cont.) The intersection of two lines l and l’ is the point: x = l x l’ The line through two points x and x’ can be analogously written as l = x x x’ duality principle Set of all points at infinity (= ideal points) in P 2 (e.g. ) lies on the line at infinity l Set of all points at infinity (= ideal points) in P 2 (e.g. (x 1,x 2,0) T ) lies on the line at infinity l ∞ = (0,0,1) T P 2 = set of rays in R 3 through the origin (see Ch.1) vectors k(x 1,x 2,x 3 ) T for diff. k form a single ray (a point in P 2 ) R 3 lines in P 2 are planes in R 3

7. The 2D projective plane (cont.) Ω Λ ll’ ideal point r 1 = k(x 1,x 2,x 3 ) r 2 = k(x 1 ’,x 2 ’,x 3 ’) r1r1 r2r2 ≡ l ≡ x 1 x 2 -plane ≡ l ∞ ≡ Ω l’ є Ω l, l’, r 1, r 2 є Λ θ θ Fig 2.1 (extended) lines in P 2 are planes e.g. line l is plane e.g. line l is plane Λ x2x2 x1x1 θ x 3 = 1 points in P 2 = rays through the origin point x 1 = ray r 1 point x 1 = ray r 1

8. The 2D projective plane (cont.) Duality principle for 2D projective geometry – for every theorem there is a dual one, obtained by interchanging the roles of points and lines A curve in Euclidean space corresponds to a conic in projective space – defined using points:x T Cx = 0 C is a homog. representation, only the ratios of elements matter the ratios of elements matter – defined using (tangent) lines: l T C -1 l = 0 via the equation of a conic tangent at x: l = Cx C -1 only if C non-singular, otherwise C* if C not of full rank, the conic is degenerate

9. Projective transformations Remember slide 1? Projectivity = homography = invertible mapping in P 2 that preserves lines = invertible mapping in P 2 that preserves lines – algebraically, mapping described by the matrix H again only element ratios matter => H = homogeneous matrix – leaves all projective properties of the figure invariant x1x1 x1’x1’ Fig. 2.3 (extended) central projection preserves lines => a projectivity

10. Projective transformations (cont.) Effect of central projection (e.g. distorted shape) is described by H => inverse transformation leads back to the original (via H -1 ) H can be calculated from 4 point correspondences (i.e. 8 linear equations) between the original (e.g. the 3D world) and the projection (e.g. the image) Points transform according to H, but lines transform according to H -1 : l’ T = l T H -1 For a conic, the transformation is C’ = H -T CH -1

11. A hierarchy of transformations Projective transformations form a group, PL(3) – characterized by invertible 3x3 matrices In terms of increased specialization: similarityaffineprojective 1.Isometry 2.Similarity 3. Affine 4. Projective Can be described algebraically (i.e. via the transform matrix) or in terms of invariants

12. A transformation hierarchy: Isometries Transformations in R 2 preserving Euclidean dist. – ε is affecting orientation e.g. in a composition of reflection & Eucl. trans. if ε = 1, isometry = Euclidean transformation rotation matrix translation 2-vector Invariants: length, angle, area Preserves orientation if det(Z)=1 Z – Eucl. trans. model the motion of a rigid object needs 2 point correspondences

13. A transformation hierarchy: Similarity I.e. isometry + isotropic scaling – also called equi-form, since it preserves shape – in its planar form, needs 2 point correspondences If isometry does not include reflection, matrix is scaling factor Invariants: angles, parallel lines, ratio of lengths (not length itself!), ratio of areas Metric structure: something defined up to a similarity

14. A transformation hierarchy: Affine Non-singular linear transformation + translation – can be computed from 3 point correspondences – invariants: parallel lines, ratios of lengths of their segments, ratio of areas 2x2 non-singular matrix defining linear transformation Can be thought of as the composition of rotations and non-isotropic scalings – the affine matrix A is then rotation by θ scaling by λ 1 and λ 2 rotation by φ rotation back by -φ essence of affinity, separate scaling in orthog. directions A = R(θ)R(-φ)DR(φ),

15. A transformation hierarchy: Projective Most general linear trans. of homog. coords. – i.e. the one that only preserves straight lines – affine was as general, but in inhomogeneous coords. – requires 4 point correspondences – the block form of the matrix is v = (v 1,v 2 ) T (not null as with affine => non-linear effects) Invariants: cross-ratio of 4 collinear points (i.e. the ratio of ratios of line segments)

16. Comparison of transformations Affine are between similarities and projectivities: – angles not preserved => shapes skewed – but effect homogeneous over the entire plane – orientation of transformed line depends only on orientation, not on planar position of source – ideal points remain at infinity Projectivities: – area scaling varies with position – orientation of trans. line depends on both orientation & position – ideal points map to finite points (thus vanishing points modeled) Projectivity can be decomposed into a chain of more specific transformations: A = sRK + tv T, det(K) = 1

17. Projective geometry of 1D Very similar to 2D – proj. trans. of the plane implies a 1D proj. trans. of every line in the plane Proj. trans. for a line is a 2x2 homog. matrix – thus 3 point correspondences required Cross ratio is the basic projective invariant in 1D signed distance from one to another (if each is a finite point, and homog. coord. is 1) Dual to collinear points are concurrent lines, also having a P 1 geometry Dual to collinear points are concurrent lines, also having a P 1 geometry

18. Recovery of affine & metric properties from images Recover metric properties (i.e. up to a similarity) 1.by using 4 points to completely remove projective distortion 2.by identifying line at infinity l ∞ and two circular points (i.e. their images) Once l ∞ is identified in the image, affine measurements can be made in the original – e.g. parallel lines can be identified, length ratios computed, etc. Affine is the most general trans. for which l ∞ remains a fixed line – but point-wise correspondence achieved only if the point is an eigenvector of A

19. Recovery of affine & metric properties from images (cont.) But identified l ∞ can also be transformed to l ∞ = (0,0,1) T with a suitable proj. matrix – such a matrix could be – this matrix can then be applied to all points, and affine measurements made in the recovered image Figure 2.12

20. Recovery of affine & metric properties from images (cont.) Beside the line at infinity, the two circular points are fixed under similarity – i.e. a pair of complex conjugates – every circle intersects l ∞ at these Metric rectification is possible if circular points are transformed into their canonical positions – applying the transformation to the entire image results in a similarity The degenerate line conic is dual to circ. points – once it is identified, Euclidean angles and length rations can be measured – direct metric rectification also possible

21. Properties of conics Some point x and some conic C define a line l = Cx (i.e. a polar of x w.r.t. C) – the line intersects the conic at 2 points -> the tangents at these points intersect at x The conic induces a map between points & lines of P 2 – a projective invariant (involves only intersection & tangency) – called correlation, represented by a 3x3 matrix A: l = Ax For two points x and y, if x is on the polar of y, then y is on the polar of x Any conic is projectively equiv. to one with a diagonal matrix – classification based on diag. elements – hyperbola, ellipse & parabola from Eucl. geom. projectively equiv. to a circle (still valid in affine geom.)

22. The End !