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University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Orthogonal Functions and Fourier Series
![Page 2: Orthogonal Functions and Fourier Seriesfussell/courses/cs384g-spring... · 2017-03-30 · University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Functions](https://reader035.vdocument.in/reader035/viewer/2022062303/5fb749e189fb3536c035832e/html5/thumbnails/2.jpg)
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
![Page 3: Orthogonal Functions and Fourier Seriesfussell/courses/cs384g-spring... · 2017-03-30 · University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Functions](https://reader035.vdocument.in/reader035/viewer/2022062303/5fb749e189fb3536c035832e/html5/thumbnails/3.jpg)
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)
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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 +++=+
![Page 5: Orthogonal Functions and Fourier Seriesfussell/courses/cs384g-spring... · 2017-03-30 · University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Functions](https://reader035.vdocument.in/reader035/viewer/2022062303/5fb749e189fb3536c035832e/html5/thumbnails/5.jpg)
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]
![Page 6: Orthogonal Functions and Fourier Seriesfussell/courses/cs384g-spring... · 2017-03-30 · University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Functions](https://reader035.vdocument.in/reader035/viewer/2022062303/5fb749e189fb3536c035832e/html5/thumbnails/6.jpg)
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(:
![Page 7: Orthogonal Functions and Fourier Seriesfussell/courses/cs384g-spring... · 2017-03-30 · University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Functions](https://reader035.vdocument.in/reader035/viewer/2022062303/5fb749e189fb3536c035832e/html5/thumbnails/7.jpg)
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 +≥+∈∀ :
![Page 8: Orthogonal Functions and Fourier Seriesfussell/courses/cs384g-spring... · 2017-03-30 · University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Functions](https://reader035.vdocument.in/reader035/viewer/2022062303/5fb749e189fb3536c035832e/html5/thumbnails/8.jpg)
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 =
![Page 9: Orthogonal Functions and Fourier Seriesfussell/courses/cs384g-spring... · 2017-03-30 · University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Functions](https://reader035.vdocument.in/reader035/viewer/2022062303/5fb749e189fb3536c035832e/html5/thumbnails/9.jpg)
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 =,
![Page 10: Orthogonal Functions and Fourier Seriesfussell/courses/cs384g-spring... · 2017-03-30 · University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Functions](https://reader035.vdocument.in/reader035/viewer/2022062303/5fb749e189fb3536c035832e/html5/thumbnails/10.jpg)
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 )()()(,
![Page 11: Orthogonal Functions and Fourier Seriesfussell/courses/cs384g-spring... · 2017-03-30 · University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Functions](https://reader035.vdocument.in/reader035/viewer/2022062303/5fb749e189fb3536c035832e/html5/thumbnails/11.jpg)
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
![Page 12: Orthogonal Functions and Fourier Seriesfussell/courses/cs384g-spring... · 2017-03-30 · University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Functions](https://reader035.vdocument.in/reader035/viewer/2022062303/5fb749e189fb3536c035832e/html5/thumbnails/12.jpg)
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
![Page 13: Orthogonal Functions and Fourier Seriesfussell/courses/cs384g-spring... · 2017-03-30 · University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Functions](https://reader035.vdocument.in/reader035/viewer/2022062303/5fb749e189fb3536c035832e/html5/thumbnails/13.jpg)
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
![Page 14: Orthogonal Functions and Fourier Seriesfussell/courses/cs384g-spring... · 2017-03-30 · University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Functions](https://reader035.vdocument.in/reader035/viewer/2022062303/5fb749e189fb3536c035832e/html5/thumbnails/14.jpg)
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)
![Page 15: Orthogonal Functions and Fourier Seriesfussell/courses/cs384g-spring... · 2017-03-30 · University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Functions](https://reader035.vdocument.in/reader035/viewer/2022062303/5fb749e189fb3536c035832e/html5/thumbnails/15.jpg)
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)
![Page 16: Orthogonal Functions and Fourier Seriesfussell/courses/cs384g-spring... · 2017-03-30 · University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Functions](https://reader035.vdocument.in/reader035/viewer/2022062303/5fb749e189fb3536c035832e/html5/thumbnails/16.jpg)
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)
![Page 17: Orthogonal Functions and Fourier Seriesfussell/courses/cs384g-spring... · 2017-03-30 · University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Functions](https://reader035.vdocument.in/reader035/viewer/2022062303/5fb749e189fb3536c035832e/html5/thumbnails/17.jpg)
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
![Page 18: Orthogonal Functions and Fourier Seriesfussell/courses/cs384g-spring... · 2017-03-30 · University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Functions](https://reader035.vdocument.in/reader035/viewer/2022062303/5fb749e189fb3536c035832e/html5/thumbnails/18.jpg)
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?