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.

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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


Do parallel lines intersect

Do parallel lines intersect?

Perhaps!


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


Overview

Overview

  • 2D Projective Geometry

  • 3D Projective Geometry

  • Application: Invariant matching

  • From 2D images to 3D space


Projective geometry and geometric invariance in computer vision

Projective 2D Geometry


Homogeneous coordinates

Homogeneous representation of points

on

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

Example


Ideal points and the line at infinity

tangent vector

normal direction

Example

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

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"

Duality

In projective geometry, points and lines are completely interchangeable!


Conics

or homogenized

or in matrix form

with

Conics

Curve described by 2nd-degree equation in the plane

5DOF:

6 parameters


Five points define a conic

or

stacking constraints yields

Five points define a conic

For each point the conic passes through


Projective transformations

Theorem:

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

or

8DOF

Projective transformations

Definition:

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


More examples

More examples


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)

upper-triangular,

Example:


Overview transformations

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

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 geometry and geometric invariance in computer vision

Projective 3D Geometry


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)


Planes

Transformation

Euclidean representation

Planes

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


Lines

Lines

(4dof)

Example: X-axis


Points lines and planes

Points, lines and planes


Hierarchy of transformations

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

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∞


Projective geometry and geometric invariance in computer vision

Application inInvariant Matching


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

R3

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:


Projective invariant representation of a planar point

R3

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:


Affine invariant parallelogram three reference points

R3

A

R2

R1

Affine invariant parallelogram:Three reference points

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


Affine invariant parallelogram construction of identifier patch 1

R3

A

B2

B1

R2

R1

Affine invariant parallelogram:Construction of identifier patch (1)

Select B1 on R2R3


Affine invariant parallelogram construction of identifier patch 2

R3

B2

C2

C1

B1

R1

Affine invariant parallelogram:Construction of identifier patch (2)

Select C1 on B1R1


Affine invariant parallelogram construction of identifier patch 3

R3

D2

C2

C1

D1

R1

Affine invariant parallelogram:Construction of identifier patch (3)

Select D1 on C1R1


Projective geometry and geometric invariance in computer vision

From 2D to 3D


Perspective projection

p1

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

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

P4

P1

P2

P4

P3

P3

P1

P2

P1

P3

P2

P4

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

Globalinvariant


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

or


Where are we going

Where are we going?


Projective geometry and geometric invariance in computer vision

Thanks for yourAttention!

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