Orthogonal Functions and Fourier Series University of Texas at - PowerPoint PPT Presentation

Orthogonal Functions and Fourier Series University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Vector Spaces Set of vectors Closed under the following operations Vector addition: v 1 + v 2 = v 3 Scalar multiplication: s v 1 = v 2 n a v = v � Linear combinations: i i i 1 = Scalars come from some field F e.g. real or complex numbers Linear independence Basis Dimension University of Texas at Austin CS384G - Computer Graphics Fall 2008 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(b v )=(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 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: x i + y j + z k Coordinate representation: [ x y z ] a [ x y z ] b [ x y z ] [ ax bx ay by az bz ] + = + + + 1 1 1 2 2 2 1 2 1 2 1 2 University of Texas at Austin CS384G - Computer Graphics Fall 2008 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: [ x 2 x 1 ] Linear combination: a x 2 + b x + c Coordinate 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( v 1 v , ) R � 2 d is nonnegative v , v V : d( v , v ) 0 � � � i j i j d is symmetric v , v V : d( v , v ) d( v , v ) � � = i j i j j i Indiscernibles are identical v , v V : d( v , v ) 0 v v � � = � = i j i j i j The triangle inequality holds v , v , v V : d( v , v ) d( v , v ) d( v , v ) � � + � i j k i j j k i k University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Normed spaces Define the length or norm of a vector v v V : v 0 Nonnegative � � � v = 0 v 0 Positive definite � = v V , a F : a v a v Symmetric � � � = The triangle inequality holds v , v V : v v v v � � + � + i j i j i j 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 University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Norms and metrics Examples of norms: 1 p p � � D � p norm: � x � i � � p=1 manhattan norm i 1 � � = p=2 euclidean norm Metric from norm d( v , v ) v v = � 1 2 1 2 Norm from metric if d is homogeneous v , v V , a F : d( a v , a v ) a d( v , v ) � � � = i j i j i j d is translation invariant � v i ,v j , t � V :d( v i ,v j ) = d( v i + t,v j + t ) then v = d( v , 0 ) University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Inner product spaces v i v , R Define [inner, scalar, dot] product (for real spaces) s.t. � j v v , v v , v v , v + = + i j k i k j k a v , v a v , v = i j i j v , v v , v = i j j i v , v 0 � v , v = 0 v 0 � = v , a v a v , v For complex spaces: v , v v , v = = i j i j i j j i Induces a norm: v = v, v University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Some inner products Multiplication in R Dot product in Euclidean n-space D v , v v v � = 1 2 1, i 2, i i 1 = For real functions over domain [a,b] b f , g f ( x ) g ( x ) dx = � a For complex functions over domain [a,b] b f , g f ( x ) g ( x ) dx = � a Can add nonnegative weight function b f , g f ( x ) g ( x ) w ( x ) dx = � w a University of Texas at Austin CS384G - Computer Graphics Fall 2008 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 2008 Don Fussell

Orthogonality Two vectors v 1 and v 2 are orthogonal if v 1 v , 0 = 2 v 1 and v 2 are orthonormal if they are orthogonal and v , v v , v 1 = = 1 1 2 2 Orthonormal set of vectors (Kronecker delta) v , v = � i j i , j University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Examples Linear polynomials over [-1,1] (orthogonal) 1 x dx 0 B 0 ( x ) = 1, B 1 ( x ) = x � = 1 � Is x 2 orthogonal to these? 2 + Is orthogonal to them? (Legendre) 3 x 1 2 University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Fourier series � Cosine series � C i ( � ) f ( � ) = a i i = 0 C 0 ( � ) = 1, C 1 ( � ) = cos( � ), C n ( � ) = cos( n � ) 2 � � C m , C n = cos( m � )cos( n � ) d � 0 2 � 1 � (cos[( m + n ) � ] + cos[( m � n ) � ]) = 2 0 2 � � � 1 1 2( m + n ) sin[( m + n ) � ] + 2( m � n ) sin[( m � n ) � ] = 0 = � � � � 0 for m � n � 0 University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Fourier series 2 � 2 � � � � � 1 2 cos(2 n � ) + 1 1 4 n sin(2 n � ) + � � d � = for m = n � 0 = = � � � � � 2 2 � � � � 0 0 2 � 1 � 2 2cos(0) d � = 2 � for m = n = 0 = 0 � Sine series � S i ( � ) f ( � ) = b i i = 0 S 0 ( � ) = 0, S 1 ( � ) = sin( � ), S n ( � ) = sin( n � ) 2 � � S m , S n = sin( m � )sin( n � ) d � = 0 for m � n or m = n = 0 0 for m = n � 0 = � University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Fourier series � Complete series � f ( � ) = a n cos( n � ) + b n sin( n � ) n = 0 2 � � C m , S n = cos( m � )sin( n � ) d � = 0 0 Basis functions are orthogonal but not orthonormal Can obtain a n and b n by projection 2 � 2 � � � � � f , C k = f ( � )cos( k � ) d � = cos ( k � ) d � a i cos( n � ) + b i sin( n � ) 0 0 n = 0 2 � � a k cos 2 ( k � ) d � = � a k (or 2 � a k for k = 0) = 0 University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell

Fourier series 2 � a k = 1 � f ( � )cos( k � ) d � � 0 2 � a 0 = 1 � f ( � ) d � 2 � 0 Similarly for b k 2 � b k = 1 � f ( � )sin( k � ) d � � 0 University of Texas at Austin CS384G - Computer Graphics Fall 2008 Don Fussell