University of Texas at Austin CS384G - Computer Graphics Spring 2010 Don FussellOrthogonal Functions andFourier SeriesUniversity of Texas at Austin CS384G - Computer Graphics Spring 2010 Don FussellVector SpacesSet of vectorsClosed under the following operationsVector addition: v1 + v2 = v3Scalar multiplication: s v1 = v2Linear combinations:Scalars come from some field Fe.g. real or complex numbersLinear independenceBasisDimensionvv =!=iniia1University of Texas at Austin CS384G - Computer Graphics Spring 2010 Don FussellVector Space AxiomsVector addition is associative and commutativeVector addition has a (unique) identity element(the 0 vector)Each vector has an additive inverseSo we can define vector subtraction as adding aninverseScalar 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 Spring 2010 Don FussellCoordinate RepresentationPick 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]][][][212121222111bzazbyaybxaxzyxbzyxa +++=+University of Texas at Austin CS384G - Computer Graphics Spring 2010 Don FussellFunctions as vectorsNeed a set of functions closed under linearcombination, whereFunction addition is definedScalar multiplication is definedExample:Quadratic polynomialsMonomial (power) basis: [x2 x 1]Linear combination: ax2 + bx + cCoordinate representation: [a b c]University of Texas at Austin CS384G - Computer Graphics Spring 2010 Don FussellMetric spacesDefine a (distance) metric s.t.d is nonnegatived is symmetricIndiscernibles are identicalThe triangle inequality holdsR!)d(21v,v)d()d(:ijjijiv,vv,vVv,v =!"0)d(: !"#jijiv,vVv,v)d()d()d(:kikjjikjiv,vv,vv,vVv,v,v !+"#jijijivvv,vVv,v =!="# 0)d(:University of Texas at Austin CS384G - Computer Graphics Spring 2010 Don FussellNormed spacesDefine the length or norm of a vectorNonnegativePositive definiteSymmetricThe triangle inequality holdsBanach spaces – normed spaces that are complete(no holes or missing points)Real numbers form a Banach space, but not rationalnumbers Euclidean n-space is Banachv0: !"# vVv0vv =!= 0vvVv aaFa =!!" :,jijijivvvvVv,v +!+"# :University of Texas at Austin CS384G - Computer Graphics Spring 2010 Don FussellNorms and metricsExamples of norms:p norm:p=1 manhattan normp=2 euclidean normMetric from normNorm from metric ifd is homogeneousd is translation invariantthenppDiix11!!"#$$%&'=2121vvv,v !=)d()d()d(:,jijijiv,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 Spring 2010 Don FussellInner product spacesDefine [inner, scalar, dot] product (for real spaces) s.t.For complex spaces:Induces a norm:vv,v =R!jiv,vkjkikjiv,vv,vv,vv +=+jijiv,vvv aa =,ijjiv,vvv =,0, !vv0vvv =!= 0,ijjiv,vvv =,jijiv,vvv aa =,University of Texas at Austin CS384G - Computer Graphics Spring 2010 Don FussellSome inner productsMultiplication in RDot product in Euclidean n-spaceFor real functions over domain [a,b]For complex functions over domain [a,b]Can add nonnegative weight function!=badxxgxfgf )()(,!=badxxgxfgf )()(,iDii 2,1,21vvv,v!==1!=bawdxxwxgxfgf )()()(,University of Texas at Austin CS384G - Computer Graphics Spring 2010 Don FussellHilbert SpaceAn inner product space that is complete wrtthe induced norm is called a Hilbert spaceInfinite dimensional Euclidean spaceInner product defines distances and anglesSubset of Banach spacesUniversity of Texas at Austin CS384G - Computer Graphics Spring 2010 Don FussellOrthogonalityTwo vectors v1 and v2 are orthogonal ifv1 and v2 are orthonormal if they areorthogonal andOrthonormal set of vectors (Kronecker delta)0=21v,v1==2211v,vv,vjiji ,!=v,vUniversity of Texas at Austin CS384G - Computer Graphics Spring 2010 Don FussellExamplesLinear polynomials over [-1,1] (orthogonal)B0(x) = 1, B1(x) = xIs x2 orthogonal to these?Is orthogonal to them? (Legendre)011=!"dxx2132+xUniversity of Texas at Austin CS384G - Computer Graphics Spring 2010 Don FussellFourier seriesCosine series! C0(") = 1, C1(") = cos("), Cn(") = cos(n")! Cm,Cn= cos(m")cos(n")d"02#$=1202#$(cos[(m + n)"] + cos[(m % n)"])=12(m + n)sin[(m + n)"] +12(m % n)sin[(m % n)"]& ' ( ) * + 02#= 0for m , n , 0! f (") = aii= 0#$Ci(")University of Texas at Austin CS384G - Computer Graphics Spring 2010 Don FussellFourier series! =12cos(2n") +12# $ % & ' ( d"=14nsin(2n") +"2# $ % & ' ( 02)*02)=)for m = n + 0! =122cos(0)d"02#$= 2#for m = n = 0Sine series! S0(") = 0, S1(") = sin("), Sn(") = sin(n")! Sm,Sn= sin(m")sin(n")d"02#$= 0 for m % n or m = n = 0=#for m = n % 0! f (") = bii= 0#$Si(")University of Texas at Austin CS384G - Computer Graphics Spring 2010 Don FussellFourier seriesComplete seriesBasis functions are orthogonal but notorthonormalCan obtain an and bn by projection! f (") = ann= 0#$cos(n") + bnsin(n")! Cm,Sn= cos(m")sin(n")d"02#$= 0! f ,Ck= f (")cos(k")02#$d"= cos02#$(k") d"ain= 0%&cos(n") + bisin(n")= akcos202#$(k") d"=#ak(or 2#akfor k = 0)University of Texas at Austin CS384G - Computer Graphics Spring 2010 Don FussellFourier series! ak=1"f (#)cos(k#)02"$d#a0=12"f (#) d#02"$Similarly for bk! bk=1"f
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