Danping lecture04-projective geometry SVD

Lecture 04- Projective Geometry

EE382-Visual localization & PerceptionDanping Zou @Shanghai Jiao Tong

University

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Projective geometry

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• Key concept - Homogenous coordinates– Represent an -dimensional vector by a

dimensional coordinate

– Can represent infinite points or lines

Danping Zou @Shanghai Jiao Tong University

Homogenous coordinate Cartesian coordinate

August Ferdinand Möbius 1790-1868

2D projective geometry• Point representation:

– A point can be represented by a homogenous coordinate:

Danping Zou @Shanghai Jiao Tong University 3

Homogeneous coordinates:

Inhomogeneous coordinates:

2D projective geometry• Line representation:

– A line is represented by a line equation:

Hence a line can be naturally represented by a homogeneous coordinate:

It has the same scale equivalence relationship:


2D projective geometry• A point lie on a line is simply described as:

where and .


A point or a line has only two degree of freedom (DoF, 自由度).

2D projective geometry• Intersection of lines：

– The intersection of two lines is computed by the cross production of the two homogenous coordinates of the two lines:


Intersection of lines

Here, ‘ ‘ represents cross production

2D projective geometry• Example of intersection of two lines: Let the two

lines be

• The intersection point is computed as :

• The inhomogeneous coordinate is


2D projective geometry• Line across two points:

• The line across the two points is obtained by the cross production of the two homogenous coordinates of the two points:


2D projective geometry• Idea point (point at infinity):

– Intersection of parallel lines– Let two parallel lines bewhere . Their intersection is computed as


Its inhomogeneous coordinates are meaningless:

The point with the last coordinate x3 =0 is called idea points

2D projective geometry• Line at infinity : all idea points lies on


A geometric interpolation

• Points and lines in the 2D plane are represented by 3D rays and planes respectively.

• The idea points lies in -plane and - plane represents

2D projective geometry• Projective transformation:

– A projective transformation is an invertible mapping such that three points ଵ ଶ ଷ lie on the same line if

and only if ଵ ଶ ଷ do.– A projective transformation is also called as

• A collineation• A homography

– A homography is a linear transformation on homogeneous 3-vectors represented by a non-singular 3 × 3 matrix:


2D projective geometry• Given a transformation of points ,

the transformation of lines is given by .


2D projective geometry• An example of projective transformation


2D projective geometry• A hierarchy of transformations


Isometries

Similarity transformation

Affine transformation

Projective transformation

2D projective geometry• Isometrics : Euclidean transformation [+ Reflection]


Euclidean transformation

Reflection + Euclidean transformation

Rotation

Translation

A isometric transformation has 3 degree of freedom, whose invariants include length, angle and area.

3DoF

2D projective geometry• Similarity transformation


A isometric transformation has 4 degree of freedom, with one more degree of freedom on the scaling.

It preserves the ‘shape’, angle.

2D projective geometry• An Affine transformation is a non-singular linear

transformation which has the following form:


6DoF

2D projective geometry• Projective transformation

• Can be decomposed into a chain of essential transformations:


Similarity Pure Affine Pure projective

8DoF

Scale

Summary


* Page 44 Multiple-view Geometry for Computer Vision

Summary• Homogenous/inhomogenous coordinates• 2D points vs.. 2D lines• Points / lines at infinity• Projective transformation, Homography• Hierarchy of transformations:

– Isometrics (Euclidean) ->similarity->affine->projective


3D projective geometry• The homogeneous a 3D point is represented by

• Its inhomogeneous coordinates is

• The projective transformation in 3D space is a linear transformation on homogeneous 4-vectors, represented by a non-singular matrix:


3D projective geometry• A plane in 3D space is written as

• Let the point on the plane be ，we have :


Plane normal

3D projective geometry• Transformation of planes :


3D projective geometry• Joint and incidence relations

– A plane is defined uniquely by three distinct points– Three planes intersect in a unique point– Two distinct planes intersect in a unique line


3D projective geometry• Three points define a plane :

• The rank of is 3, how to solve this equation?– Method #1. Compute the basis of the null space of– Method #2. Let ସ , solve the following equation:


About the null space & SVD• Fundamentals

– Range （column space） : range(A) is the space spanned by the columns of the matrix A.

– Null space : null(A) is a space where all the vectors satisfy

– Rank : The rank is the dimension of the column space


About the null space & SVD• Use SVD to compute the null space of a matrix.• The Singular Value Decomposition (SVD) is a basic

tool to analysis a matrix. • Many problems of linear algebra can be better

understood if we first ask the question：what if we take the SVD ?


SVD• The SVD is motivated by the following geometric

fact:– The image of the unit sphere under any matrix

is a hyper-ellipse (超椭球）


Orthogonal bases of the original space

Orthogonal bases of the target space

SVD• From the geometric interpretation, we have the

following equations: . • By collecting all those equations together, we have

or compactly, . Since has orthogonal columns, we get


SVD• Back to the 2D case, we have


Left singular vectors Singular values Right singular vectors

SVD• Step by step interpretation

– Step 1. map (rotation + with/without reflection) the bases to the canonical frame.



– Step 2. Stretch the axes



– Step 3. rotate the canonical axes


SVD• In the step 2, what if we set ?


• We have :is the null space of A

Null space from SVD• So the null space of a matrix can be obtained by

SVD decomposition.• The bases of the null space are the right singular

vectors that corresponds to the zero singular values.


3D projective geometry• Back to the plane problem: Three points define a

plane :

• Its null space can be solved by SVD


3D projective geometry• Three planes define a point:

• Its null space can be solved by SVD


Summary• 3D points vs. 3D Planes • Transformations of 3D points and 3D lines• Three planes define a point / Three points define a

plane• How to solve a homogenous equation• Null space of a matrix• Geometric interpretation of SVD


3D projective geometry• Hierarchy of 3D transformation


• Euclidean transformation (Rigid body transformation)

• A mapping is a rigid body transformation if it satisfies 1) Length is preserved :

2) The cross product is preserved:

3D projective geometry


3D projective geometry• Homogenous form vs. inhomogenous form

• About the rotation matrix


Homogenous:

Inhomogenous:

3D projective geometry• 3D Rotation matrix transforms three

orthogonal axes into r1, r2, r3 in the new coordinate system.


r1

r2

r3

More about 3D rotation will be discussed in the later lectures.

3D projective geometry• Exercise:

1. Write a small program to compute the harries corner detector. Show all the intermediate results as shown in the slides. Requirements:1) source codes + executable (any language, but with details how to run it)2) result analysis (world <= 2 pages)

2. Write a small program to rectify a distorted planar image to a normal one as shown (Page 13)Requirements - The same as the previous one.


Deadline : Oct 27th

3D Lines• A line is defined by join of two points or intersection

of two planes.

• Lines have 4 degrees of freedom in 3D-space


3D Lines• Plücker matrix: A line can be defined from two

points and

• Plücker matrix is a skew-symmetric matrix


3D Lines• The rank of a Plücker matrix is 2. Its null space is the

pencil of planes with the line as the axis.

• Under the point transformation , the matrix transforms as :


3D Lines• Determinant of a Plücker matrix


3D Lines• Independent of the two selected points

• Let and be two different point on the same line, we have


Equivalent up to scale

3D Lines• Dual Plücker matrix :

– Let two planes be

• has similar properties to Plücker matrix


3D Lines• Join and incidence properties

(1) Plane from the join of a point and a line

if and only if , is on


3D Lines• Join and incidence properties

(2) Point from the intersection of a line and a plane

if and only if , is on


3D Lines• Plücker line : A compact and meaningful

representation from the Plücker matrix

• Correspond to the Plücker matrix as :


3D Lines• Plücker constrain is satisfied:


3D Lines• Plücker line representation


1. n is a vector normal to the plane containing the line and the origin

2. v is a direction vector of line, oriented from a to b

3. is the orthogonal distance from the line to the origin

Summary• Plücker matrix• Dual Plücker matrix• Plücker lines


Danping lecture04-projective geometry SVD

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