Orthogonal Functions and Fourier Seriesfussell/courses/cs384g-fall...University of Texas at Austin CS384G - Computer Graphics Spring 2010 Don Fussell Functions as vectors Need a set

University of Texas at Austin CS384G - Computer Graphics Spring 2010 Don Fussell

Orthogonal Functions andFourier Series


Vector SpacesSet of vectorsClosed under the following operations

Vector addition: v1 + v2 = v3

Scalar multiplication: s v1 = v2

Linear combinations:

Scalars come from some field Fe.g. real or complex numbers

Linear independenceBasisDimension

vv =!=

i

n

i

ia

1


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

So we can define vector subtraction as adding aninverse

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


Coordinate Representation

Pick a basis, order the vectors in it, then allvectors in the space can be represented assequences of coordinates, i.e. coefficients ofthe basis vectors, in order.Example:

Cartesian 3-spaceBasis: [i j k]Linear combination: xi + yj + zkCoordinate representation: [x y z]

][][][ 212121222111 bzazbyaybxaxzyxbzyxa +++=+


Functions as vectors

Need a set of functions closed under linearcombination, where

Function addition is definedScalar multiplication is defined

Example:Quadratic polynomialsMonomial (power) basis: [x2 x 1]Linear combination: ax2 + bx + cCoordinate representation: [a b c]


Metric spaces

Define a (distance) metric s.t.d is nonnegatived is symmetricIndiscernibles are identical

The triangle inequality holds

R!)d(21v,v

)d()d(: ijjiji v,vv,vVv,v =!"

0)d(: !"# jiji v,vVv,v

)d()d()d(: kikjjikji v,vv,vv,vVv,v,v !+"#

jijiji vvv,vVv,v =!="# 0)d(:


Normed spaces

Define the length or norm of a vectorNonnegativePositive definiteSymmetricThe triangle inequality holds

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

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

v

0: !"# vVv

0vv =!= 0

vvVv aaFa =!!" :,

jijiji vvvvVv,v +!+"# :


Norms and metrics

Examples of norms:p norm:

p=1 manhattan normp=2 euclidean norm

Metric from normNorm from metric if

d is homogeneous

d is translation invariant

then

ppD

i

ix

1

1!!

"

#

$$

%

&'=

2121vvv,v !=)d(

)d()d(:, jijiji v,vv,vVv,v aaaFa =!!"

!

"vi ,vj,t # V : d(vi ,vj) = d(vi + t,vj + t)

),d( 0vv =


Inner product spaces

Define [inner, scalar, dot] product (for real spaces) s.t.

For complex spaces:

Induces a norm: vv,v =

R!ji v,v

kjkikji v,vv,vv,vv +=+

jiji v,vvv aa =,

ijji v,vvv =,

0, !vv

0vvv =!= 0,

ijji v,vvv =, jiji v,vvv aa =,


Some inner products

Multiplication in RDot 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

dxxgxfgf )()(,

!=b

a

dxxgxfgf )()(,

i

D

i

i 2,1,21vvv,v !

=

=1

!=b

a

wdxxwxgxfgf )()()(,


Hilbert Space

An inner product space that is complete wrtthe induced norm is called a Hilbert spaceInfinite dimensional Euclidean spaceInner product defines distances and anglesSubset of Banach spaces


Orthogonality

Two vectors v1 and v2 are orthogonal if

v1 and v2 are orthonormal if they areorthogonal and

Orthonormal set of vectors (Kronecker delta)

0=21v,v

1==2211v,vv,v

jiji ,!=v,v


Examples

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

B0(x) = 1, B1(x) = x

Is x2 orthogonal to these?Is orthogonal to them? (Legendre)

0

1

1

=!"

dxx

2

132+x


Fourier series

Cosine series

!

C0(") =1, C

1(") = cos("), C

n(") = cos(n")

!

Cm,C

n= cos(m")cos(n")d"

0

2#

$

=1

20

2#

$ (cos[(m + n)"]+ cos[(m % n)"])

=1

2(m + n)sin[(m + n)"]+

1

2(m % n)sin[(m % n)"]

&

' (

)

* + 0

2#

= 0

for m , n , 0

!

f (") = aii= 0

#

$ Ci(")


Fourier series

!

=1

2cos(2n") +

1

2

#

$ %

&

' ( d" =

1

4nsin(2n") +

"

2

#

$ %

&

' (

0

2)

*0

2)

= ) for m = n + 0

!

=1

22cos(0)d"

0

2#

$ = 2# for m = n = 0

Sine series

!

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

!

Sm,S

n= sin(m")sin(n")d"

0

2#

$ = 0 for m % n or m = n = 0

= # for m = n % 0

!

f (") = bii= 0

#

$ Si(")


Fourier seriesComplete series

Basis functions are orthogonal but notorthonormalCan obtain an and bn by projection

!

f (") = ann= 0

#

$ cos(n") + bn sin(n")

!

Cm,S

n= cos(m")sin(n")d"

0

2#

$ = 0

!

f ,Ck = f (")cos(k")0

2#

$ d" = cos0

2#

$ (k")d" ain= 0

%

& cos(n") + bi sin(n")

= ak cos2

0

2#

$ (k")d" = # ak (or 2# ak for k = 0)


Fourier series

!

ak =1

"f (#)cos(k#)

0

2"

$ d#

a0

=1

2"f (#)d#

0

2"

$

Similarly for bk

!

bk =1

"f (#)sin(k#)

0

2"

$ d#

Documents

Orthogonal Functions and Fourier Seriesfussell/courses/cs384g-fall...University of Texas at Austin CS384G - Computer Graphics Spring 2010 Don Fussell Functions as vectors Need a set