Multiple View Geometry Projective Geometry

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Multiple View GeometryProjective Geometry
Transformations of 2D

• Vladimir Nedovic

Intelligent Systems Lab Amsterdam
(ISLA) Informatics Institute, University of
Amsterdam Kruislaan 403, 1098 SJ Amsterdam, The
Netherlands
vnedovic_at_science.uva.nl
18-01-2008
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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

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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. gt no parallelism in
  proj. space

(x,y,0) (x/0,y/0,0) (8,8,0)
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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.

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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

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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 infinity
  l8 (0,0,1)T
• i.e. a horizon in the image

• P2 set of rays in R3 through the origin (see
  Ch.1)
• vectors k(x1,x2,x3)T for diff. k form a single
  ray (a point in P2)
• lines in P2 are planes in R3

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The 2D projective plane
(cont.)
?
l
r1 k(x1,x2,x3)
r1
r2 k(x1,x2,x3)
x1x2-plane l8 O l ? O l, l, r1, r2 ? ?
ideal point
l
r2
x1
?
x2
x3 1

• points in P2 rays through the origin
• point x1 ray r1

?
?

• lines in P2 are planes
• e.g. line l is plane ?

Fig 2.1 (extended)
O
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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

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Projective transformations

• Remember slide 1? Projectivity homography
• invertible mapping in P2 that preserves
  lines
• algebraically, mapping described by the matrix H
• again only element ratios matter gt H
  homogeneous matrix
• leaves all projective properties of the figure
  invariant

Fig. 2.3 (extended)
central projection preserves lines gt a
projectivity
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Projective transformations (cont.)

• Effect of central projection (e.g. distorted
  shape) is described by H gt 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 lT lTH-1
• For a conic, the transformation is C H-TCH-1

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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
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A transformation hierarchy Isometries

• Transformations in R2 preserving Euclidean dist.
• e is affecting orientation
• e.g. in a composition of reflection Eucl.
  trans.
• if e 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

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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

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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(?)R(-f)DR(f),
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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)

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Comparison of transformations

• Affine are between similarities and
  projectivities
• angles not preserved gt 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)

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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

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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 l8 and two
  circular points (i.e. their images)

• Affine is the most general trans. for which l8
  remains a fixed line
• but point-wise correspondence achieved only if
  the point is an eigenvector of A (i.e. if
  direction preserved)

• Once l8 is identified in the image, affine
  measurements can be made in the original
• e.g. parallel lines can be identified, length
  ratios computed, etc.

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Recovery of affine metric properties from
images (cont.)

• But identified l8 can also be transformed to l8
  (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

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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 l8 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 ratios can be measured
• direct metric rectification also possible

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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 -gt
• 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.)

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The End !