Projective 2D geometry Appunti basati sulla parte iniziale del testo di R.Hartley e A.Zisserman “Multiple view geometry” Rielaborazione di materiale di M.Pollefeys

Projective 2D Geometry Points, lines & conics Transformations & invariants 1D projective geometry and the Cross-ratio

Homogeneous coordinates Homogeneous representation of lines 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 on if and only if The point x lies on the line l if and only if xTl=lTx=0 Homogeneous coordinates Inhomogeneous coordinates but only 2DOF

Points from lines and vice-versa Intersections of lines The intersection of two lines and is Line joining two points The line through two points and is Line joining two points: parametric equation A point on the line through two points x and x’ is y = x + q x’ Example

Ideal points and the line at infinity Intersections of parallel lines Example tangent vector normal direction Ideal points Line at infinity Note that in P2 there is no distinction between ideal points and others

A model for the projective plane exactly one line through two points exaclty one point at intersection of two lines

Duality 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

Conics Curve described by 2nd-degree equation in the plane or homogenized or in matrix form with 5DOF:

Five points define a conic For each point the conic passes through or stacking constraints yields

Pole-polar relationship The polar line l=Cx of the point x with respect to the conic C intersects the conic in two points. The two lines tangent to C at these points intersect at x

Polarity: cross ratio Cross ratio of 4 colinear points y = x + q x’ (with i=1,..,4) ratio of ratios i i Harmonic 4-tuple of colinear points: such that CR=-1

Chasles’ theorem A B C X D Conic = locus of constant cross-ratio towards 4 ref. points A B C X D

Polarity: conjugacy Given a point y and a conic C, a point z is conjugated of y wrt C if the 4-tuple (y,z,x’ ,x’’ ) is harmonic, where x’ and x’’ are the intersection points between C and line (y,z). Intersection points (y+q z) C (y +q z) = 0 is a 2-nd degree equation on q y Cy + 2 q y Cz + q z Cz = 0 --> two solutions q’ and q’’. Since y and z correspond to q = 0 and q = , harmonic -> q’+ q’’ = 0 Hence the two solutions sum up to zero: y Cz = 0, and also z Cy = 0  the point z is on the line Cy, which is called the polar line of y wrt C the polar line Cy is the locus of the points conjugate of y wrt C Conic C is the locus of points belonging to their own polar line y Cy = 0 T T T 2 T T T T

Correlations and conjugate points A correlation is an invertible mapping from points of P2 to lines of P2. It is represented by a 3x3 non-singular matrix A as l=Ax Conjugate points with respect to C (on each others polar) Conjugate lines with respect to C* (through each others pole)

Polarity and tangent lines Consider a line through a point y, tangent to a conic C: let us determine the point z conjugate of y wrt C: since the intersection points coincide, then q’ = q’’  harmonic 4-tuple: also z coincides with the tangency point  All points y on the tangent are conjugate to tangency point z wrt C  polar line of a point z on the conic C is tangent to C through z In addition: since there are two tangents to C through a point y not on C, both tangency points are conjugate to y wrt C polar line of y wrt C (=locus of the conjugate points) = line through the two tangency points of lines through y

Tangent lines to conics The line l tangent to C at point x on C is given by l=Cx l x C

Dual conics  l C C C l = 0  C = C = C A line tangent to the conic C satisfies In general (C full rank): in fact -1 Line l is the polar line of y : y = C l , but since y Cy = 0  l C C C l = 0  C = C = C T T -T -1 * -T -1 Dual conics = line conics = conic envelopes

Degenerate conics A conic is degenerate if matrix C is not of full rank e.g. two lines (rank 2) e.g. repeated line (rank 1) Degenerate line conics: 2 points (rank 2), double point (rank1) Note that for degenerate conics

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. A mapping h:P2P2 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 Theorem: Definition: Projective transformation or 8DOF projectivity=collineation=projective transformation=homography

Mapping between planes central projection may be expressed by x’=Hx (application of theorem)

Removing projective distortion select four points in a plane with known coordinates (linear in hij) (2 constraints/point, 8DOF  4 points needed) Remark: no calibration at all necessary, better ways to compute (see later)

More examples

Transformation of lines and conics For a point transformation Transformation for lines Transformation for conics Transformation for dual conics

A hierarchy of transformations Projective linear group Affine group (last row (0,0,1)) Euclidean group (upper left 2x2 orthogonal) Oriented Euclidean group (upper left 2x2 det 1) Alternative, characterize transformation in terms of elements or quantities that are preserved or invariant e.g. Euclidean transformations leave distances unchanged

Class I: Isometries orientation preserving: orientation reversing: (iso=same, metric=measure) orientation preserving: orientation reversing: 3DOF (1 rotation, 2 translation) special cases: pure rotation, pure translation Invariants: length, angle, area

Class II: Similarities (isometry + scale) 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

Class III: Affine transformations where 6DOF (2 scale, 2 rotation, 2 translation) non-isotropic scaling! (2DOF: scale ratio and orientation) Invariants: parallel lines, ratios of parallel segment lengths, ratios of areas

Action of affinities and projectivities on line at infinity Line at infinity stays at infinity, but points move along line Line at infinity becomes finite, allows to observe vanishing points, horizon,

Class VI: Projective transformations 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)

Decomposition of projective transformations decomposition unique (if chosen s>0) upper-triangular, Example:

Overview transformations Concurrency, collinearity, order of contact (intersection, tangency, inflection, etc.), cross ratio Projective 8dof Parallellism, ratio of areas, ratio of lengths on parallel lines (e.g midpoints), linear combinations of vectors (centroids). The line at infinity l∞ Affine 6dof Ratios of lengths, angles. The circular points I,J Similarity 4dof Euclidean 3dof lengths, areas.

Number of invariants? The number of functional invariants is equal to, or greater than, the number of degrees of freedom of the configuration less the number of degrees of freedom of the transformation e.g. configuration of 4 points in general position has 8 dof (2/pt) and so 4 similarity, 2 affinity and zero projective invariants

Projective geometry of 1D 3DOF (2x2-1) The cross ratio Invariant under projective transformations

Recovering metric and affine properties from images Parallelism Parallel length ratios Angles Length ratios

Note: not fixed pointwise The line at infinity The line at infinity l is a fixed line under a projective transformation H if and only if H is an affinity Note: not fixed pointwise

Affine properties from images projection rectification in fact, any point x on l’ is mapped to a point at the ∞ ∞

Affine rectification v1 l∞ v2 l1 l3 l2 l4

Distance ratios

The circular points The circular points I, J are fixed points under the projective transformation H iff H is a similarity

The circular points “circular points” l∞ Intersection points between any circle and l∞ Algebraically, encodes orthogonal directions

Conic dual to the circular points A line conic: set of lines through any of the circular points The dual conic is fixed conic under the projective transformation H iff H is a similarity Note: has 4DOF l∞ is the null vector

Angles Euclidean: Projective: in fact, e.g. (orthogonal)

Length ratios

Metric properties from images Rectifying transformation from SVD

Why ? Normally: SVD (Singular Value Decomposition) with U and V orthogonal But is symmetric  and SVD is unique  Observation : H=U orthogonal (3x3): not a P2 isometry

Metric from affine Once the image has been affinely rectified

Metric from affine

Metric from projective

Projective conic classification Diagonal Equation Conic type (1,1,1) improper conic (1,1,-1) circle (1,1,0) single real point (1,-1,0) two lines (1,0,0) single line

Affine conic classification ellipse parabola hyperbola

Fixed points and lines (eigenvectors H =fixed points) (1=2  pointwise fixed line) (eigenvectors H-T =fixed lines)