Orthogonal Functions and Fourier Series University of Texas at - - PowerPoint PPT Presentation

University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Orthogonal Functions and Fourier Series

University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Vector Spaces

Set of vectors Closed under the following operations

Vector addition: v1 + v2 = v3 Scalar multiplication: s v1 = v2 Linear combinations:

Scalars come from some field F

e.g. real or complex numbers

Linear independence Basis Dimension

v v =

• =

i n i i

a

1

University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Vector Space Axioms

Vector addition is associative and commutative Vector addition has a (unique) identity element (the 0 vector) Each vector has an additive inverse

So we can define vector subtraction as adding an inverse

Scalar multiplication has an identity element (1) Scalar multiplication distributes over vector addition and field addition Multiplications are compatible (a(bv)=(ab)v)

University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Coordinate Representation

Pick a basis, order the vectors in it, then all vectors in the space can be represented as sequences of coordinates, i.e. coefficients of the basis vectors, in order. Example:

Cartesian 3-space Basis: [i j k] Linear combination: xi + yj + zk Coordinate representation: [x y z]

] [ ] [ ] [

2 1 2 1 2 1 2 2 2 1 1 1

bz az by ay bx ax z y x b z y x a + + + = +

University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Functions as vectors

Need a set of functions closed under linear combination, where

Function addition is defined Scalar multiplication is defined

Example:

Quadratic polynomials Monomial (power) basis: [x2 x 1] Linear combination: ax2 + bx + c Coordinate representation: [a b c]

University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Metric spaces

Define a (distance) metric s.t.

d is nonnegative d is symmetric Indiscernibles are identical The triangle inequality holds

R

• )

d(

2 1 v

, v

) d( ) d( :

i j j i j i

v , v v , v V v , v =

• )

d( :

• j

i j i

v , v V v , v

) d( ) d( ) d( :

k i k j j i k j i

v , v v , v v , v V v , v , v

• +
• j

i j i j i

v v v , v V v , v =

• =
• )

d( :

University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Normed spaces

Define the length or norm of a vector

Nonnegative Positive definite Symmetric The triangle inequality holds

Banach spaces – normed spaces that are complete (no holes or missing points)

Real numbers form a Banach space, but not rational numbers Euclidean n-space is Banach

v

:

• v

V v v v =

• = 0

v v V v a a F a =

• :

,

j i j i j i

v v v v V v , v +

• +
• :

University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Norms and metrics

Examples of norms:

p norm:

p=1 manhattan norm p=2 euclidean norm

Metric from norm Norm from metric if

d is homogeneous d is translation invariant then

p p D i i

x

1 1

• =

2 1 2 1

v v v , v

• =

) d(

) d( ) d( : ,

j i j i j i

v , v v , v V v , v a a a F a =

• vi,vj,t V :d(vi,vj) = d(vi + t,vj + t)

) , d( v v =

University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Inner product spaces

Define [inner, scalar, dot] product (for real spaces) s.t. For complex spaces: Induces a norm:

v v, v = R

• j

i v

, v

k j k i k j i

v , v v , v v , v v + = +

j i j i

v , v v v a a = ,

i j j i

v , v v v = ,

,

• v

v v v v =

• = 0

,

i j j i

v , v v v = ,

j i j i

v , v v v a a = ,

University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Some inner products

Multiplication in R Dot product in Euclidean n-space For real functions over domain [a,b] For complex functions over domain [a,b] Can add nonnegative weight function

• =

b a

dx x g x f g f ) ( ) ( ,

• =

b a

dx x g x f g f ) ( ) ( ,

i D i i 2, 1, 2 1

v v v , v

• =

=

1

• =

b a w

dx x w x g x f g f ) ( ) ( ) ( ,

University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Hilbert Space

An inner product space that is complete wrt the induced norm is called a Hilbert space Infinite dimensional Euclidean space Inner product defines distances and angles Subset of Banach spaces

University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Orthogonality

Two vectors v1 and v2 are orthogonal if v1 and v2 are orthonormal if they are

• rthogonal and

Orthonormal set of vectors (Kronecker delta)

=

2 1 v

, v 1 = =

2 2 1 1

v , v v , v

j i j i ,

• =

v , v

University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Examples

Linear polynomials over [-1,1] (orthogonal)

B0(x) = 1, B1(x) = x Is x2 orthogonal to these? Is orthogonal to them? (Legendre)

1 1

=

• dx

x

2 1 3

2 +

x

University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Fourier series

Cosine series

C0() =1, C1() = cos(), Cn() = cos(n)

Cm,Cn = cos(m)cos(n)d

2

• =

1 2

2

• (cos[(m + n)]+ cos[(m n)])

= 1 2(m + n) sin[(m + n)]+ 1 2(m n) sin[(m n)]

• 2

= 0 for m n 0

f () = ai

i= 0

• Ci()

University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Fourier series

= 1 2 cos(2n) + 1 2

• d =

1 4n sin(2n) + 2

• 2
• 2

= for m = n 0 = 1 2 2cos(0)d

2

• = 2

for m = n = 0

Sine series

S0() = 0, S1() = sin(), Sn() = sin(n)

Sm,Sn = sin(m)sin(n)d

2

• = 0

for m n or m = n = 0 = for m = n 0

f () = bi

i= 0

• Si()

University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Fourier series

Complete series Basis functions are orthogonal but not

• rthonormal

Can obtain an and bn by projection

f () = an

n= 0

• cos(n) + bn sin(n)

Cm,Sn = cos(m)sin(n)d

2

• = 0

f ,Ck = f ()cos(k)

2

• d =

cos

2

• (k)d

ai

n= 0

• cos(n) + bi sin(n)

= ak cos2

2

• (k)d = ak

(or 2 ak for k = 0)

University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Fourier series

ak = 1

• f ()cos(k)

2

• d

a0 = 1 2 f ()d

2

• Similarly for bk

bk = 1

• f ()sin(k)

2

• d