DLT alg

1
Behaviour at infinity
Primitives pt/line/conic HZ 2.2
DLT alg HZ 4.1
Hierarchy of maps Invariants HZ 2.4
Projective transform HZ 2.3
Rectification HZ 2.7
2
Projective transform 1

• now that we have this new projective space, lets
  understand how to transform it
• necessary to understand how a camera transforms
  space
• a projective transform is a linear map in
  projective space
• projective transform projectivity homography
  collineation
• it can encode perspective projection and the
  image formation process of the pinhole camera
• Euclidean transform cannot
• Euclidean transform translation rotation
  rigid transform
• we will study projective transform and some
  special cases
• some properties are preserved (invariant) and
  others are not
• collinearity is, angle is not (and so parallel
  lines are not)
• we will study invariants of the various types of
  map

3
What is a projective transform?

• Preferred definition (matrix-based)
• A projective transform is a map
• hP2 ? P2, h(x) Hx
• where H is a nonsingular 3x3 matrix.
• Alternate definition (line preservation)
• A projective transform is an invertible map hP2
  ? P2 that preserves lines.
• i.e., x1,x2,x3 lie on same line iff h(x1), h(x2),
  h(x3) do
• invertibility means that map is one-to-one and
  onto
• first definition is algebraic, second is
  geometric
• proof of equivalence (actually just the reduction
  from algebraic to geometric)
• let h(x)Hx be a projective transform, then
• x_i lie on the line L ? Lx_i 0 ? L H-1Hx_i ?
  Hx_i lie on the line LH-1
• so h(x) preserves lines
• recall that L is a row vector
• Corollary when points transform as x ? Hx, lines
  transform as L ? LH-1
• important foreshadowing of tensor (contravariant
  vs. covariant transformation)\
• this result is used to map the vanishing line to
  the line at infinity and then map pixels
  accordingly (since pixels are points, not lines)
• HZ32-34

4
Projective transform 2

• in 1st definition, H is a homogeneous matrix,
  since scaling does not change the map
• H has 8 dof
• question how many dof do a point, line and conic
  have?
• the geometric definition motivates the term
  collineation
• homographies form a group
• can be composed, have an identity, have an
  inverse
• generalization a map from P3 to P3, or from
  P3 to P2, that preserves lines is also called a
  homography
• note that the equivalence to nonsingular matrix
  algebraic definition does not hold for maps from
  P3 to P2 (e.g., perspective projection matrix
  is singular)
• exercise why is perspective projection a
  homography?

5
Perspective projection

• perspective projection is a homography
• preserves lines given a line L in 3-space,
  consider the plane through L and C (camera
  center) a plane intersects a plane in a line
• yet persp(X,Y,Z,1) (diag(f,f,1) 0) (X,Y,Z,1) so
  matrix definition does not apply
• indeed, a special case of a homography
• interesting fact composition of two
  perspectivities is not a perspectivity
• perspectivity perspective projection between
  two planes

6
Day x (A growing list of) key facts in
projective geometry

• L1 x L2 line intersection
• P1 x P2 point join
• P.L 0 point/line incidence
• line at infinity (0,0,1)
• finding vanishing line with 7 cross products
• homography encoded by 3x3 matrix H
• if P ? HP, then L ? LH-1
• circular points I, J (1, \pm i, 0)

7
Structure from motion diagram
Camera geometry
Projective geometry
n-view geometry
Three-view geometry
Two-view geometry
Calibration
8
Behaviour at infinity
Primitives pt/line/conic HZ 2.2
DLT alg HZ 4.1
Hierarchy of maps Invariants HZ 2.4
Projective transform HZ 2.3
Rectification HZ 2.7
9
Classes of homography

• from most restrictive to most general
• Euclidean transform rigid transform
• translation and rotation only
• move the origin and rotate the coordinate frame
• similarity transform also isometrically scale
• affine transform allow different scaling in
  coordinate directions
• linear transform of Euclidean space (i.e., not
  ideal points), then Euclidean transform
• none of these transforms affect line at infinity
  ideal points remain at infinity
• in contrast, all points are created equal with
  projective transform ideal points may map to
  finite points
• projective transform linear transform of
  projective space
• HZ 2-3

10
Similarity and metric structure

• we are most interested in similarity transforms,
  since we shall define the structure of a scene up
  to a similarity
• metric structure structure defined up to a
  similarity
• it is impossible to know the absolute position or
  orientation of an imaged object, or its scale
• is this picture of Sam sitting at a desk taken in
  New York or Casablanca, on the third floor or
  tenth, leaning backward or sitting up straight?
  is this imaged Millenium Falcon in outer space
  26.7 meters long or 19.33 inches long?
• context might give an indication of scale (e.g.,
  height of door, orientation of door, known
  landmark in picture)
• therefore, the best we can expect, and our goal,
  is to retrieve the metric structure of the object
• if G is the true geometry of the object and G is
  our reconstructed geometry, then G SG for some
  similarity S

11
Invariants

• Euclidean length and area
• similarity angle, ratio of length and area
• affine parallelism
• e.g., reflection of stained glass on wall
• projective collinearity, order of contact (e.g.,
  intersection, tangency)
• initially, we shall restore structure only up to
  a projective transform (not a similarity)
• for example, a cube may be reconstructed simply
  as a polyhedron (not even a parallelepiped)
• metric reconstruction reconstruction up to a
  similarity
• projective reconstruction reconstruction up to
  a homography
• for metric reconstruction, we shall need to
  calibrate
• rectification is a 2d analog of metric
  reconstruction

12
As matrices

• projective transforms PL(3) homogeneous
  nonsingular 3x3 matrices
• Euclidean R t 0,0,1 upper left 2x2 submatrix
  is orthogonal
• motivation dont want to stretch space, leave
  3rd coordinate alone
• translation vector is 3rd column rotation or
  rotation/reflection is 2x2
• orientation-preserving Euclidean if det1 (models
  rigid motion)
• assume orientation-preserving realistic
  transforms are
• similarity sR t 0 1
• affine A 0, 0 1 3rd row is identity (0,0,1)
• A K t, no restriction on K
• motivation want ideal points to remain ideal
• Euclidean transform has 3dof (1 rotation, 2
  translation)
• similarity has 4dof
• HZ37-39

13
A similarity fixes the circular points

• What characterizes a similarity?
• Theorem A projective transform is a similarity
  iff the circular points are fixed.
• Proof
• ? Let H s cos,-s sin,tx s sin, s cos, ty 0 0
  1 be a similarity. H(1,i,0) s (cos- i sin,
  sini cos,0). Recall that c(theta)is(theta)
  ei theta. So H(1,i,0) s(ei theta, i
  ei theta,0) se-i theta (1,i,0) (1,i,0)
• ? Homework 1
• Resulting strategy we can change a projective
  transform into a similarity by finding the imaged
  circular points and moving them back to the true
  circular points (ensuring that the circular
  points are fixed)
• intuition not surprising that metric properties
  like angle can be restored once circular points
  are known
• I (1,0,0) i (0,1,0) concise package of the
  two coordinate directions

14
Goal finding circular points

• this motivates finding the images of the circular
  points I,J for metric rectification
• it is easier to find a certain dual conic (a line
  conic dual to the circular points)
• this requires an exploration of conics