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• 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

x2

v

Magnitude:

Orientation:

Is a unit vector

If , Is a UNIT vector

vw

v+w

• Vector Subtraction

vw

v-w

• Scalar Product

vav

• Inner (dot) Product

vw

The inner product is a SCALAR!

• Orthonormal BasisP

x1

x2

v

ij

• Magnitude:kuk = kv wk = kvkkwk sin

Vector (cross) Product

The cross product is a VECTOR!

wv

u

Orientation:

• i = j kj = k ik = 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!

Pixels intensity value

Anm =

2

6664

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

......

...an1 an2 . . . anm

3

7775

• Matrices

Product:

A and B must have compatible dimensions!

Anm =

2

6664

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

......

...an1 an2 . . . anm

3

7775

• Matrix Transpose

Definition:

Cmn = ATnm

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 a12a21 a22

]= a11a22 a12a21

det

a11 a12 a13a21 a22 a23a31 a32 a33

= a11a22a33 + a12a23a31 + a13a21a32 a13a22a31 a23a32a11 a33a12a21

• Matrix Inverse

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

AA1 = A1A = I

A1 =

[a11 a12a21 a22

]1=

1

det A

[a22 a12a21 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

Psy y

• P

P=SPP=TP

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

P

x

yP

xy

• 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 and eigenvector u satisfies

Au = u

where A is a square matrix.

I Multiplying u by A scales u by

• Eigenvalues and Eigenvectors

Rearranging the previous equation gives the system

Au u = (A I)u = 0

which has a solution if and only if det(A I) = 0.I The eigenvalues are the roots of this determinant which is

polynomial in .

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

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

and 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 .

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 )(row i of VT) produces a

+=

An Numerical Example

• We can look at 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 whats different between samples.

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

Principal Component Analysis

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