Projective Geometry and Camera model Class 2 points, lines, planes conics and quadrics transformations camera model Read tutorial chapter 2 and 3.1 http://www.cs.unc.edu/~marc/tutorial/ Chapter 1, 2 and 5 in Hartley and Zisserman

Homogeneous coordinates Homogeneous representation of 2D points and lines The point x lies on the line l if and only if Note that scale is unimportant for incidence relation equivalence class of vectors, any vector is representative Set of all equivalence classes in R3(0,0,0)T forms P2 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 Example Note: with

Ideal points and the line at infinity Intersections of parallel lines Example tangent vector normal direction Ideal points Line at infinity is a collection of all the points at infinity as x1/x2 varies, i.e. all Possible directions Line at infinity Note that in P2 there is no distinction between ideal points and others

3D points and planes Homogeneous representation of 3D points and planes The point X lies on the plane π if and only if The plane π goes through the point X if and only if

Planes from points (solve as right nullspace of )

Points from planes M is 4x3 matrix. Columns of M are null space of (solve as right nullspace of ) Representing a plane by its span M is 4x3 matrix. Columns of M are null space of

Lines Line is either joint of two points or intersection of two planes Representing a line by its span: two vectors A, B for two space points (4dof) Dual representation: P and Q are planes; line is span of row space of Example: X-axis (Alternative: Plücker representation, details see e.g. H&Z)

Points, lines and planes Plane defined by join of point X and line W is the null space of M Point X defined by the intersection of line W with plane is the null space of M

Plücker coordinates Elegant representation for 3D lines (with A and B points) (Plücker internal constraint) (two lines intersect) (for more details see e.g. H&Z)

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 Conic c is a null vector of the above matrix

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 Points lie on a point lines are tangent A line tangent to the conic C satisfies In general (C full rank): Dual conics = line conics = conic envelopes Points lie on a point lines are tangent conic to the point conic C; conic C is the envelope of lines l

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

Quadrics and dual quadrics (Q : 4x4 symmetric matrix) 9 d.o.f. in general 9 points define quadric det Q=0 ↔ degenerate quadric tangent plane 3. and thus defined by less points 4. On quadric=> tangent, outside quadric=> plane through tangent point 5. Derive X’QX=x’M’QMx=0 Dual Quadric: defines equation on planes relation to quadric (non-degenerate)

2D 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 reprented by a vector x it is true that h(x)=Hx Theorem: Definition: Projective transformation or 8DOF projectivity=collineation=projective transformation=homography

Transformation of 2D points, lines and conics For a point transformation Transformation for lines Transformation for conics Transformation for dual conics

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

Hierarchy of 2D transformations transformed squares invariants 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.

Projective geometry of 1D The cross ratio Is invariant under projective transformations in P^1. Four sets of four collinear points; each set is related to the others by a line projectivity Since cross ratio is an invariant under projectivity, the cross ratio has the same value for all the sets shown

Concurrent Lines Four concurrent lines l_i intersect the line l in the four points x_i; The cross ratio of these points is an invariant to the projective transformation of the plane Coplanar points x_i are imaged onto a line by a projection with center C. The cross ratio of the image points x_i is invariant to the position of the image line l

A hierarchy of transformations Euclidean transformations (rotation and translation) leave distances unchanged Similarity: circle imaged as circle; square as square; parallel or perpendicular lines have same relative orientation Affine: circle becomes ellipse; orthogonal world lines not imaged as orthogonoal; But, parallel lines in the square remain parallel Projective: parallel world lines imaged as converging lines; tiles closer to camera larger image than those further away. Similarity Affine projective

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 Two step process: 1.Find l the image of line at infinity in plane 2 2. Plug into H_pa Image of line at infinity in plane

Affine rectification v1 l∞ v2 l1 l3 l2 l4

The circular points Canonical coordinates of circular points Two points on l_inf which are fixed under any similarity transformation Canonical coordinates of 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∞ Circle: Line at infinity Two points on l_inf: Every circle intersects l_inf at circular points “circular points” l∞ Circle: Line at infinity Algebraically, encodes orthogonal directions

Conic dual to the circular points The dual conic is fixed conic under the projective transformation H iff H is a similarity Note: has 4DOF (3x3 ho mogeneous; symmetric, determinant is zero) l∞ is the nullvector

Angles Euclidean: Projective: (1) (l and m orthogonal) Once is identified in the projective plane, then Euclidean angles may be measured by equation (1)

Transformation of 3D points, planes and quadrics For a point transformation (cfr. 2D equivalent) Transformation for lines Transformation for conics Transformation for dual conics

Hierarchy of 3D transformations Projective 15dof Intersection and tangency Parallellism of planes, Volume ratios, centroids, The plane at infinity π∞ Affine 12dof Similarity 7dof Angles, ratios of length The absolute conic Ω∞ Euclidean 6dof Volume

The plane at infinity The plane at infinity π is a fixed plane under a projective transformation H iff H is an affinity Represents 3DOF between projective and affine canonical position contains directions two planes are parallel  line of intersection in π∞ line // line (or plane)  point of intersection in π∞

The absolute conic The absolute conic Ω∞ is a (point) conic on π. In a metric frame: or conic for directions: (with no real points) The absolute conic Ω∞ is a fixed conic under the projective transformation H iff H is a similarity Represent 5 DOF between affine and similarity Ω∞ is only fixed as a set Circle intersect Ω∞ in two circular points Spheres intersect π∞ in Ω∞

The absolute dual quadric Set of planes tangent to an ellipsoid, then squash to a pancake The absolute dual quadric Ω*∞ is a fixed conic under the projective transformation H iff H is a similarity 8 dof ( symmetric matrix, det is zero) plane at infinity π∞ is the nullvector of Ω∞ Angles:

Camera model Relation between pixels and rays in space ?

Pinhole camera

Pinhole camera model linear projection in homogeneous coordinates!

Pinhole camera model

Principal point offset

Principal point offset calibration matrix

Camera rotation and translation ~

CCD camera

General projective camera 11 dof (5+3+3) non-singular intrinsic camera parameters extrinsic camera parameters

Radial distortion Due to spherical lenses (cheap) Model: R R straight lines are not straight anymore http://foto.hut.fi/opetus/260/luennot/11/atkinson_6-11_radial_distortion_zoom_lenses.jpg

Camera model Relation between pixels and rays in space ?

Projector model Relation between pixels and rays in space (dual of camera) (main geometric difference is vertical principal point offset to reduce keystone effect) ?

Meydenbauer camera vertical lens shift to allow direct ortho-photographs

Affine cameras

Action of projective camera on points and lines projection of point forward projection of line back-projection of line

Action of projective camera on conics and quadrics back-projection to cone projection of quadric

Image of absolute conic

A simple calibration device compute H for each square (corners @ (0,0),(1,0),(0,1),(1,1)) compute the imaged circular points H(1,±i,0)T fit a conic to 6 circular points compute K from w through cholesky factorization (≈ Zhang’s calibration method)

Exercises: Camera calibration

Next class: Single View Metrology Antonio Criminisi

A hierarchy of transformations Group of invertible nxn matrices with real elements  general linear group on n dimensionsGL(n); Projective linear group: matrices related by a scalar multiplier PL(n); three subgroups: 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 (rotation and translation) leave distances unchanged Similarity Affine projective Similarity: circle imaged as circle; square as square; parallel or perpendicular lines have same relative orientation Affine: circle becomes ellipse; orthogonal world lines not imaged as orthogonoal; But, parallel lines in the square remain parallel Projective: parallel world lines imaged as converging lines; tiles closer to camera larger image than those further away.