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

Orthogonal Functions and Fourier Series

University of Texas at Austin CS384G - Computer Graphics Fall 2010 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

vv =∑=

i

n

iia

1

University of Texas at Austin CS384G - Computer Graphics Fall 2010 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 2010 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]

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

University of Texas at Austin CS384G - Computer Graphics Fall 2010 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 2010 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( 21 v,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(:

University of Texas at Austin CS384G - Computer Graphics Fall 2010 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

v0: ≥∈∀ vVv0vv =⇒= 0

vvVv aaFa =∈∈∀ :,

jijiji vvvvVv,v +≥+∈∀ :

University of Texas at Austin CS384G - Computer Graphics Fall 2010 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

ppD

iix

1

1!!"

#$$%

&∑=

2121 vvv,v −=)d(

)d()d(:, jijiji v,vv,vVv,v aaaFa =∈∈∀

€

∀vi ,vj,t ∈ V : d(vi ,vj) = d(vi + t,vj + t)

),d( 0vv =

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

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 =,

University of Texas at Austin CS384G - Computer Graphics Fall 2010 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

dxxgxfgf )()(,

∫=b

a

dxxgxfgf )()(,

i

D

ii 2,1,21 vvv,v ∑

=

=1

∫=b

aw

dxxwxgxfgf )()()(,

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

Orthogonality

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

 Orthonormal set of vectors (Kronecker delta)

0=21 v,v

1== 2211 v,vv,v

jiji ,δ=v,v

University of Texas at Austin CS384G - Computer Graphics Fall 2010 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)

01

1

=∫−

dxx

3x2 −12

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

Fourier series

Cosine series

€

C0(θ) =1, C1(θ) = cos(θ), Cn (θ) = cos(nθ)

€

Cm,Cn = cos(mθ)cos(nθ)dθ0

2π

∫

=120

2π

∫ (cos[(m + n)θ]+ cos[(m − n)θ])

=1

2(m + n)sin[(m + n)θ]+ 1

2(m − n)sin[(m − n)θ]

&

' (

)

* + 0

2π

= 0

for m ≠ n ≠ 0

€

f (θ) = aii= 0

∞

∑ Ci(θ)

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

Fourier series

€

=12cos(2nθ) +

12

#

$ %

&

' ( dθ =

14nsin(2nθ) +

θ2

#

$ %

&

' (

0

2π

∫0

2π

= π for m = n ≠ 0

€

=122cos(0)dθ

0

2π

∫ = 2π for m = n = 0

 Sine series

€

S0(θ) = 0, S1(θ) = sin(θ), Sn (θ) = sin(nθ)

€

Sm,Sn = sin(mθ)sin(nθ)dθ0

2π

∫ = 0 for m ≠ n or m = n = 0

= π for m = n ≠ 0€

f (θ) = bii= 0

∞

∑ Si(θ)

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

Fourier series  Complete series

 Basis functions are orthogonal but not orthonormal  Can obtain an and bn by projection

€

f (θ) = ann= 0

∞

∑ cos(nθ) + bn sin(nθ)

€

Cm,Sn = cos(mθ)sin(nθ)dθ0

2π

∫ = 0

€

f ,Ck = f (θ)cos(kθ)0

2π

∫ dθ = cos0

2π

∫ (kθ)dθ ain= 0

∞

∑ cos(nθ) + bi sin(nθ)

= ak cos2

0

2π

∫ (kθ)dθ = π ak (or 2π ak for k = 0)

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

Fourier series

€

ak =1π

f (θ)cos(kθ)0

2π

∫ dθ

a0 =12π

f (θ)dθ0

2π

∫

 Similarly for bk

€

bk =1π

f (θ)sin(kθ)0

2π

∫ dθ

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

Next class: Fourier Transform

  Topics: -  Derive the Fourier transform from the Fourier series -  What does it mean?