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

**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

**homogeneous** coordinatesin P2 x = x/1 y = y/1 coordinates in Euclidean R2 (x,y) = (x,y,1) = (kx,ky,k) k ≠ 0 points at infinity 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 (x,y,0) = (x/0,y/0,0) = (∞,∞,0)

**(n+1)x(n+1)** non-singular matrix a point in Pn, an (n+1) - vector e.g. the real 3D world e.g. an image of the real 3D world 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 • In P2, points at infinity form a line, in P3 a plane, etc.

**Lines and points represented by homogeneous vectors** (a,b,c)T = k(a,b,c)T k ≠ 0 (x,y)T = k(x,y)T • A point xlies on line l iff ax + by + c = (x,y,1)(a,b,c)T = xTl = 0 The 2D projective 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

**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 duality principle l = x x x’ • Set of all points at infinity (= ideal points) in P2 (e.g. (x1,x2,0)T) lies on the line at infinityl∞ = (0,0,1)T • P2= set of rays in R3 through the origin (see Ch.1) • vectors k(x1,x2,x3)Tfor diff. k form a single ray (a point in P2) • lines in P2 are planes in R3

**The 2D projective plane (cont.)** θ l’ l ideal point r1 = k(x1,x2,x3) r1 r2 = k(x1’,x2’,x3’) x1x2-plane ≡ l∞ ≡ Ω l’ єΩ l, l’, r1, r2єΛ r2 x1 θ x2 x3 = 1 • points in P2 = rays through the origin • point x1= ray r1 Λ θ • lines in P2 are planes • e.g. line l is plane Λ Fig 2.1 (extended) Ω

**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: xTCx = 0 • C is a homog. representation, only the ratios of elements matter • defined using (tangent) lines: lTC-1l = 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

**x1** x1’ Projective transformations • Remember slide 1? Projectivity = homography = invertible mapping in P2that 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 Fig. 2.3 (extended) central projection preserves lines => a projectivity

**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= lTH-1 • For a conic, the transformation is C’ = H-TCH-1

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

**rotation matrix** translation 2-vector A transformation hierarchy: Isometries • Transformations in R2 preserving Euclidean dist. • εis affecting orientation • e.g. in a composition of reflection & Eucl. trans. • if ε = 1, isometry = Euclidean transformation • Eucl. trans. model the motion of a rigid object • needs 2 point correspondences Z • Invariants: length, angle, area • Preserves orientation if det(Z)=1

**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

**essence of affinity,** separate scaling in orthog. directions rotation by φ rotation by θ scaling by λ1 and λ2 rotation back by -φ 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 A = R(θ)R(-φ)DR(φ),

**v = (v1,v2)T** (not null as with affine => non-linear effects) 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 • Invariants: cross-ratio of 4 collinear points (i.e. the ratio of ratios of line segments)

**Projectivity can be decomposed into a chain of more specific** transformations: A = sRK + tvT, det(K) = 1 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)

**signed distance ** from one to another (if each is a finite point, and homog. coord. is 1) 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 Dual to collinear points are concurrent lines, also having a P1 geometry

**Recovery of affine & metric properties from images** • Recover metric properties (i.e. up to a similarity) • by using 4 points to completely remove projective distortion • by identifying line at infinity l∞and two circular points (i.e. their images) • 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 • 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.

**Figure 2.12** 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

**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

**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 P2 • 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.)

**The End !**