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Projective Geometry and Geometric Invariance in Computer Vision

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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

- 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

- 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?

- 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

Overview

- 2D Projective Geometry
- 3D Projective Geometry
- Application: Invariant matching
- From 2D images to 3D space

Homogeneous representation of points

on

if and only if

Homogeneous coordinates

but only 2DOF

Inhomogeneous coordinates

Homogeneous coordinatesHomogeneous 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

The line through two points and is

Points from lines and vice-versaIntersections of lines

The intersection of two lines and is

Example

normal direction

Example

Ideal points

Line at infinity

Ideal points and the line at infinityIntersections of parallel lines

Note that in P2 there is no distinction

between ideal points and others

A model for the projective plane

exactly one line through two points

exaclty one point at intersection of two lines

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"

DualityIn projective geometry, points and lines are completely interchangeable!

A mapping h:P2P2is 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

or

8DOF

Projective transformationsDefinition:

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

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

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

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

(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

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

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 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 unique (if chosen s>0)

upper-triangular,

Example:

Overview transformations

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.

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 3D Geometry

- Points, lines, planes and quadrics
- Transformations
- П∞, ω∞and Ω ∞

Hierarchy of transformations

Projective

15dof

Intersection and tangency

Parallellism of planes,

Volume ratios, centroids,

The plane at infinity π∞

Affine

12dof

Similarity

7dof

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

- 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?

- 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.

R2

R1

X

A

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:

R2

R1

X

R4

A

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:

A

R2

R1

Affine invariant parallelogram:Three reference pointsSelect A to make a parallelogram with three reference points R1, R2, and R3

p3

p2

q1

p4

q2

q3

q4

q5

Focal point

p5

q6

p6

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

- 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.

P1

P2

P4

P3

P3

P1

P2

P1

P3

P2

P4

Invariants of point sets:Algebraic invariantsTransfer 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

Globalinvariant

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

or

Questions?

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