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University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell
Orthogonal Functions andFourier Series
University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell
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
University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell
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)
University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell
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 +++=+
University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell
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]
University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell
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(:
University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell
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 +!+"# :
University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell
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 =
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: 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 2008 Don Fussell
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 )()()(,
University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell
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
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 areorthogonal and
Orthonormal set of vectors (Kronecker delta)
0=21v,v
1==2211v,vv,v
jiji ,!=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)
0
1
1
=!"
dxx
2
132+x
University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell
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(")
University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell
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(")
University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell
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)
University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell
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#
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