Linear Algebra & Geometry why is linear algebra useful in computer vision?

Some of the slides in this lecture are courtesy to Prof. Octavia I. Camps, Penn State University

References:-Any book on linear algebra!-[HZ] – chapters 2, 4

Why is linear algebra useful in computer vision?

• Representation– 3D points in the scene– 2D points in the image

• Coordinates will be used to– Perform geometrical transformations– Associate 3D with 2D points

• Images are matrices of numbers– Find properties of these numbers

Agenda

1. Basics definitions and properties2. Geometrical transformations3. SVD and its applications

P = [x,y,z]

Vectors (i.e., 2D or 3D vectors)

Image

3D worldp = [x,y]

Vectors (i.e., 2D vectors)

P

x1

x2θ

v

Magnitude:

Orientation:

Is a unit vector

If , Is a UNIT vector

Vector Addition

vw

v+w

Vector Subtraction

vw

v-w

Scalar Product

vav

Inner (dot) Product

vw

α

The inner product is a SCALAR!

Orthonormal BasisP

x1

x2θ

v

ij

Magnitude:kuk = kv ⇥ wk = kvkkwk sin↵

Vector (cross) Product

The cross product is a VECTOR!

wv

αu

Orientation:

i = j⇥ k

j = k⇥ i

k = i⇥ j

Vector Product Computation

)1,0,0()0,1,0()0,0,1(

=

=

=

kji

1||||1||||1||||

=

=

=

kji

kji )()()( 122131132332 yxyxyxyxyxyx −+−+−=

Matrices

Sum:

Example:

A and B must have the same dimensions!

Pixel’s intensity value

An⇥m =

2

6664

a11 a12 . . . a1ma21 a22 . . . a2m...

......

...an1 an2 . . . anm

3

7775

Matrices

Product:

A and B must have compatible dimensions!

An⇥m =

2

6664

a11 a12 . . . a1ma21 a22 . . . a2m...

......

...an1 an2 . . . anm

3

7775

Matrix Transpose

Definition:

Cm×n = ATn×m

cij = aji

Identities:

(A + B)T = AT + BT

(AB)T = BTAT

If A = AT , then A is symmetric

Matrix Determinant

Useful value computed from the elements of a square matrix A

det[a11

]= a11

det

[a11 a12

a21 a22

]= a11a22 − a12a21

det

a11 a12 a13

a21 a22 a23

a31 a32 a33

= a11a22a33 + a12a23a31 + a13a21a32

− a13a22a31 − a23a32a11 − a33a12a21

Matrix Inverse

Does not exist for all matrices, necessary (but not sufficient) thatthe matrix is square

AA−1 = A−1A = I

A−1 =

[a11 a12

a21 a22

]−1

=1

det A

[a22 −a12

−a21 a11

], det A 6= 0

If det A = 0, A does not have an inverse.

2D Geometrical Transformations

2D Translation

t

P

P’

2D Translation Equation

P

x

y

tx

tyP’

t

2D Translation using Matrices

P

x

y

tx

tyP’

t

Homogeneous Coordinates

• Multiply the coordinates by a non-zero scalar and add an extra coordinate equal to that scalar. For example,

Back to Cartesian Coordinates:

• Divide by the last coordinate and eliminate it. For example,

• NOTE: in our example the scalar was 1

2D Translation using Homogeneous Coordinates

P

x

y

tx

tyP’

tt

P

Scaling

P

P’

Scaling Equation

P

x

y

sx x

P’sy y

P

P’=S·PP’’=T·P’

P’’=T · P’=T ·(S · P)=(T · S)·P = A · P

Scaling & Translating

P’’

Scaling & Translating

A

Translating & Scaling = Scaling & Translating ?

Rotation

P

P’

Rotation Equations

Counter-clockwise rotation by an angle θ

P

x

y’P’

θ

x’y

Degrees of Freedom

R is 2x2 4 elements

Note: R belongs to the category of normal matrices and satisfies many interesting properties:

Rotation+ Scaling +TranslationP’= (T R S) P

If sx=sy, this is asimilarity transformation!

Transformation in 2D

-Isometries

-Similarities

-Affinity

-Projective

Transformation in 2D

Isometries:

- Preserve distance (areas)- 3 DOF- Regulate motion of rigid object

[Euclideans]

Transformation in 2D

Similarities:

- Preserve- ratio of lengths- angles

-4 DOF

Transformation in 2D

Affinities:

Transformation in 2D

Affinities:

-Preserve:- Parallel lines- Ratio of areas- Ratio of lengths on collinear lines- others…

- 6 DOF

Transformation in 2D

Projective:

- 8 DOF- Preserve:

- cross ratio of 4 collinear points- collinearity - and a few others…

Eigenvalues and Eigenvectors

A eigenvalue λ and eigenvector u satisfies

Au = λu

where A is a square matrix.

I Multiplying u by A scales u by λ

Eigenvalues and Eigenvectors

Rearranging the previous equation gives the system

Au− λu = (A− λI)u = 0

which has a solution if and only if det(A− λI) = 0.

I The eigenvalues are the roots of this determinant which ispolynomial in λ.

I Substitute the resulting eigenvalues back into Au = λu andsolve to obtain the corresponding eigenvector.

UΣVT = A• Where U and V are orthogonal matrices,

and Σ is a diagonal matrix. For example:

Singular Value Decomposition

Singular Value decomposition

that

• Look at how the multiplication works out, left to right:

• Column 1 of U gets scaled by the first value from Σ.

• The resulting vector gets scaled by row 1 of VT to produce a contribution to the columns of A

An Numerical Example

• Each product of (column i of U)·(value i from Σ)·(row i of VT) produces a

+=

An Numerical Example

We can look at Σ to see that the first column has a large effect

while the second column has a much smaller effect in this example

An Numerical Example

• For this image, using only the first 10 of 300 singular values produces a recognizable reconstruction

• So, SVD can be used for image compression

SVD Applications

• Remember, columns of U are the Principal Components of the data: the major patterns that can be added to produce the columns of the original matrix

• One use of this is to construct a matrix where each column is a separate data sample

• Run SVD on that matrix, and look at the first few columns of U to see patterns that are common among the columns

• This is called Principal Component Analysis (or PCA) of the data samples

Principal Component Analysis

• Often, raw data samples have a lot of redundancy and patterns

• PCA can allow you to represent data samples as weights on the principal components, rather than using the original raw form of the data

• By representing each sample as just those weights, you can represent just the “meat” of what’s different between samples.

• This minimal representation makes machine learning and other algorithms much more efficient

Principal Component Analysis