3D photography Marc Pollefeys Fall

3D photography course schedule (tentative) LectureExercise Sept 26Introduction- Oct. 3Geometry & Camera modelCamera calibration Oct. 10Single View MetrologyMeasuring in images Oct. 17Feature Tracking/matching (Friedrich Fraundorfer) Correspondence computation Oct. 24Epipolar GeometryF-matrix computation Oct. 31Shape-from-Silhouettes (Li Guan) Visual-hull computation Nov. 7Stereo matchingProject proposals Nov. 14Structured light and active range sensing Papers Nov. 21Structure from motionPapers Nov. 28Multi-view geometry and self-calibration Papers Dec. 5Shape-from-XPapers Dec. 123D modeling and registrationPapers Dec. 19Appearance modeling and image-based rendering Final project presentations

Projective Geometry and Camera model Class 2 points, lines, planes conics and quadrics transformations camera model Read tutorial chapter 2 and 3.1

Homogeneous coordinates Homogeneous representation of 2D points and lines equivalence class of vectors, any vector is representative Set of all equivalence classes in R 3  (0,0,0) T forms P 2 The point x lies on the line l if and only if Homogeneous coordinates Inhomogeneous coordinates but only 2DOF Note that scale is unimportant for incidence relation

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 Ideal points Line at infinity tangent vector normal direction Note that in P 2 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 (solve as right nullspace of ) Representing a plane by its span

Lines Example: X -axis (4dof) Representing a line by its span Dual representation (Alternative: Plücker representation, details see e.g. H&Z)

Points, lines and planes

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

Conics Curve described by 2 nd -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

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 A line tangent to the conic C satisfies Dual conics = line conics = conic envelopes In general ( C full rank):

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 relation to quadric (non-degenerate)

2D projective transformations A projectivity is an invertible mapping h from P 2 to itself such that three points x 1,x 2,x 3 lie on the same line if and only if h(x 1 ),h(x 2 ),h(x 3 ) do. Definition: A mapping h : P 2  P 2 is a projectivity if and only if there exist a non-singular 3x3 matrix H such that for any point in P 2 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 Transformation for lines Transformation for conics Transformation for dual conics For a point transformation

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

Hierarchy of 2D transformations Projective 8dof Affine 6dof Similarity 4dof Euclidean 3dof Concurrency, collinearity, order of contact (intersection, tangency, inflection, etc.), cross ratio Parallellism, ratio of areas, ratio of lengths on parallel lines (e.g midpoints), linear combinations of vectors (centroids). The line at infinity l ∞ Ratios of lengths, angles. The circular points I,J lengths, areas. invariants transformed squares

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

Affine rectification v1v1 v2v2 l1l1 l2l2 l4l4 l3l3 l∞l∞

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∞l∞ 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 l ∞ is the nullvector l∞l∞

Angles Euclidean: Projective: (orthogonal)

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

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

The plane at infinity The plane at infinity π  is a fixed plane under a projective transformation H iff H is an affinity 1.canonical position 2.contains directions 3.two planes are parallel  line of intersection in π ∞ 4.line // line (or plane)  point of intersection in π ∞

The absolute conic The absolute conic Ω ∞ is a fixed conic under the projective transformation H iff H is a similarity The absolute conic Ω ∞ is a (point) conic on π . In a metric frame: or conic for directions: (with no real points) 1.Ω ∞ is only fixed as a set 2.Circle intersect Ω ∞ in two circular points 3.Spheres intersect π ∞ in Ω ∞

The absolute dual quadric The absolute dual quadric Ω * ∞ is a fixed conic under the projective transformation H iff H is a similarity 1.8 dof 2.plane at infinity π ∞ is the nullvector of Ω ∞ 3.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

Principal point offset calibration matrix

Camera rotation and translation ~

CCD camera

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

Radial distortion Due to spherical lenses (cheap) Model: R R straight lines are not straight anymore

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 forward projection of line back-projection of line projection of point

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

Image of absolute conic

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

Exercises: Camera calibration

Next class: Single View Metrology Antonio Criminisi