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Projective Geometry and Geometric Invariance in Computer Vision. Babak N. Araabi Electrical and Computer Eng. Dept. University of Tehran. Workshop on image and video processing Mordad 13, 1382. Do parallel lines intersect?. Perhaps!. Fifth Euclid's postulate.

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projective geometry and geometric invariance in computer vision

Projective Geometry and Geometric Invariance in Computer Vision

Babak N. Araabi

Electrical and Computer Eng. Dept.

University of Tehran

Workshop on image and video processing

Mordad 13, 1382

fifth euclid s postulate
Fifth Euclid's postulate
  • For any line L and a point P not on L, there exists a unique line that is parallel to L (never meets L) and passes through P.
  • At first glance it would seem that the parallel postulate ought to be a theorem deducible from the other more basic postulates.
  • For centuries mathematicians tried to prove it.
  • Eventually it was discovered that the parallel postulate is logically independent of the other postulates, and you get a perfectly consistent system even if you assume that parallel postulate is false.
alternative postulates
Alternative postulates
  • Non-Euclidean geometry; though not a typical one
  • The projective axiom: Any two lines intersect (in exactly one point).
  • More intuitive approach:Take each line of ordinary Euclidean geometry and add to it one extra object called a point at infinity. In addition, the collection of all the extra objects together is also called a line in projective plane (called the line at infinity).
why projective geometry
Why Projective Geometry?
  • A more appropriate framework when dealing with projection related issues.
  • Perspective projection: Photophraphy Human vision
  • Perspective projection is a non-linear mapping with Euclidean coordinates, but a linear mapping with homogeneous coordinates
  • 2D Projective Geometry
  • 3D Projective Geometry
  • Application: Invariant matching
  • From 2D images to 3D space
homogeneous coordinates

Homogeneous representation of points


if and only if

Homogeneous coordinates

but only 2DOF

Inhomogeneous coordinates

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

The point x lies on the line l if and only if xTl=lTx=0

points from lines and vice versa

Line joining two points

The line through two points and is

Points from lines and vice-versa

Intersections of lines

The intersection of two lines and is


ideal points and the line at infinity

tangent vector

normal direction


Ideal points

Line at infinity

Ideal points and the line at infinity

Intersections of parallel lines

Note that in P2 there is no distinction

between ideal points and others

a model for the projective plane
A model for the projective plane

exactly one line through two points

exaclty one point at intersection of two lines


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.

For instance, the basic axiom that "for any two points, there is a unique line that intersects both those points", when turned around, becomes "for any two lines, there is a unique point that intersects (i.e., lies on) both those lines"


In projective geometry, points and lines are completely interchangeable!


or homogenized

or in matrix form



Curve described by 2nd-degree equation in the plane


6 parameters

five points define a conic


stacking constraints yields

Five points define a conic

For each point the conic passes through

projective transformations


A mapping h:P2P2is a projectivity if and only if there exist a non-singular 3x3 matrixH such that for any point in P2 reprented by a vector x it is true that h(x)=Hx

Definition: Projective transformation



Projective transformations


A projectivity is an invertible mapping h from P2 to itself such that three points x1,x2,x3lie on the same line if and only if h(x1),h(x2),h(x3) do.

projectivity=collineation=projective transformation=homography

mapping between planes
Mapping between planes

central projection may be expressed by x’=Hx

(application of theorem)

removing projective distortion
Removing projective distortion

select four points in a plane with know coordinates

(linear in hij)

(2 constraints/point, 8DOF  4 points needed)

Remark: no calibration at all necessary,

better ways to compute

transformation of lines and conics

Transformation for conics

Transformation of lines and conics

For a point transformation

Transformation for lines

a hierarchy of transformations
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:

Class I: Isometries

(iso=same, metric=measure)

3DOF (1 rotation, 2 translation)

special cases: pure rotation, pure translation

Invariants: length, angle, area

class ii similarities
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
Class III: Affine transformations

6DOF (2 scale, 2 rotation, 2 translation)

non-isotropic scaling! (2DOF: scale ratio and orientation)

Invariants: parallel lines, ratios of parallel lengths,

ratios of areas

class vi projective transformations
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 of area

action of affinities and projectivities on line at infinity
Action of affinities and projectivitieson 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,

decomposition of projective transformations
Decomposition of projective transformations

decomposition unique (if chosen s>0)



overview transformations
Overview transformations

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.

number of invariants
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
Projective geometry of 1D

3DOF (2x2-1)

The cross ratio

Invariant under projective transformations

projective 3d geometry
Projective 3D Geometry
  • Points, lines, planes and quadrics
  • Transformations
  • П∞, ω∞and Ω ∞
3d points
3D points

3D point

in R3

in P3

projective transformation

(4x4-1=15 dof)



Euclidean representation


3D plane

Dual: points ↔ planes, lines ↔ lines

planes from points

(solve as right nullspace of )

Planes from points

Or implicitly from coplanarity condition

points from planes

(solve as right nullspace of )

Points from planes

Representing a plane by its span



Example: X-axis

hierarchy of transformations
Hierarchy of transformations



Intersection and tangency

Parallellism of planes,

Volume ratios, centroids,

The plane at infinity π∞





The absolute conic Ω∞




the plane at infinity
The plane at infinity

The plane at infinity π is a fixed plane under a projective transformation H iff H is an affinity

  • canonical position
  • contains directions
  • two planes are parallel  line of intersection in π∞
  • line // line (or plane)  point of intersection in π∞
how to define an invariant 2d identifier patch on an image
How to define an invariant 2D identifier patch on an image?
  • Consider a multilateral identifier patch.
  • Lines are invariant under the perspective projection.
  • 3 (4) reference points are required to obtain an affine (projective) invariant representation of the multilateral identifier patch.
  • Weak-perspective/perspective camera models.
affine invariant representation of a planar point






Affine invariant representation of a planar point
  • Three reference points.
  • Ratio of areas are invariant.
  • X at intersection of R1R2 and R3A.
  • (r1,r2) invariants defined by:
projective invariant representation of a planar point







Projective invariant representation of a planar point
  • Four reference points.
  • Ratio of ratio of areas are invariant.
  • X at intersection of R1R4 and R3A.
  • (r1,r2) invariants defined by:
affine invariant parallelogram three reference points





Affine invariant parallelogram:Three reference points

Select A to make a parallelogram with three reference points R1, R2, and R3

perspective projection










Focal point




Image plane

Perspective projection
  • Perspective projection of 6 (feature) points from 3D world (dorsal fin) to 2D image:

pi=(xi,yi,zi) qi=(ui,vi) i=1,…,6

3d and 2d projective invariants
3D and 2D Projective Invariants
  • I1, I2, and I3 are 3D projective invariants (under PGL(4)).
  • i1, i2, i3, and i4 are 2D projective invariants (under PGL(3)).
  • There is no general case view-invariant.
  • 2D invariants impose constraints (in the form of polynomial relations) on 3D invariants.

* PGL(n): Projective General Linear group of order n.

invariants of point sets algebraic invariants













Invariants of point sets:Algebraic invariants

Transfer images to a reference frame.

A unique projective/affine transformation between any two sets of 4/3 corresponding points.

Coordinate of any other point is invariant in the reference frame.

Reference frame


invariant relations
Invariant Relations
  • Constraints (here for 6 points) typically have the following form
  • This is a fundamental object-image relation for six arbitrary 3D points and their 2D perspective projections.
  • Therefore having at least 6 points in correspondence over a number of images, 3D invariants (Ik’s) can be estimated and invariant relations can be used as geometric invariant models.

n 6 points3n-15 Ik’s2n-8 ik’s2n-11 relations