C` alcul Num` eric d’Objectes Invariants amb Ondetes David Romero i S` anchez Joint work with Ll. Alsed` a Departament de Matem` atiques Universitat Aut` onoma de Barcelona de febrer de
Outline Motivation A Primer on Wavelets The construction of the wavelets The periodization of the wavelets Numerical Computation of Invariant Objects with Wavelets The Invariance Equation with matrices Solving the Invariance Equation by means of Haar Solving the Invariance Equation by means of Daubechies Wavelets as a possible Trojan Horse
Motivation Wavelets in Theory Wavelets in Practice In the future Motivation We are interested in approximate, via expansions of a truncated base of wavelets, complicated objects semianalitically. From such approximation, we want to predict and understand changes in the geometry or dynamical properties (among others) of such objects. We will be focused on skew products of the form � � θ n � θ n +1 = R ω ( θ n ) = θ n + ω (mod 1) , = () F σ,ε x n x n +1 = F σ,ε ( θ n , x n ) , where ε, σ, x ∈ R + , θ ∈ S 1 = R / Z , ω ∈ R \ Q . Ll. Alsed` a and D. Romero, Girona C` alcul Num` eric d’Objectes Invariants amb Ondetes //
Motivation Wavelets in Theory Wavelets in Practice In the future On the use of wavelets The complicated objects that we want to compute will be invariant objects (expressed as graphs of functions from S 1 to R ). To do this, we will assume the existence of a map ϕ : S 1 − → R such that, F σ,ε ( θ, ϕ ( θ )) = ϕ ( R ω ( θ )) , for all θ ∈ S 1 . A standard approach to compute these objects is the use of Fourier approximations: N � ϕ ∼ a 0 + ( a n cos( nθ ) + b n sin( nθ )) . n =1 Since the topology and geometry of these objects can be extremely complicate, the regularity and periodicity of the Fourier basis make this approach too costly. Ll. Alsed` a and D. Romero, Girona C` alcul Num` eric d’Objectes Invariants amb Ondetes //
Motivation Wavelets in Theory Wavelets in Practice In the future On the use of wavelets In these cases, it seems more natural to use wavelets (an orthonormal basis of L 2 ( R ) ) that adapt much better to oscillatory, irregular and highly discontinuous objects. 2 j − 1 N � � ϕ ∼ a 0 + d − j,n ψ PER − j,n ( θ ) , j =0 n =0 where ψ PER is a given wavelet. Summarizing : given a generic skew product we want to Derive properties of ϕ we need we do Massive approximations of ϕ we need Massive calculation of d − j,n and ψ PER − j,n ( θ ) Ll. Alsed` a and D. Romero, Girona C` alcul Num` eric d’Objectes Invariants amb Ondetes //
Motivation Wavelets in Theory Wavelets in Practice In the future Outline Motivation A Primer on Wavelets The construction of the wavelets The periodization of the wavelets Numerical Computation of Invariant Objects with Wavelets The Invariance Equation with matrices Solving the Invariance Equation by means of Haar Solving the Invariance Equation by means of Daubechies Wavelets as a possible Trojan Horse Ll. Alsed` a and D. Romero, Girona C` alcul Num` eric d’Objectes Invariants amb Ondetes //
Motivation Wavelets in Theory Wavelets in Practice In the future A primer on wavelets Let us start by the definition of Multi-resolution Analysis (MRA) Definition A sequence of closed subspaces of L 2 ( R ) , {V j } j ∈ Z , is a Multi-resolution Analysis if it satisfies: { 0 } ⊂ · · · ⊂ V 1 ⊂ V 0 ⊂ V − 1 ⊂ · · · ⊂ L 2 ( R ) . { 0 } = � j ∈ Z V j . �� � = L 2 ( R ) . clos j ∈ Z V j There exists a function φ ( x ) whose integer translates , φ ( x − n ) , form an orthonormal basis of V 0 . Such function is called the scaling function . For each j ∈ Z it follows that f ( x ) ∈ V j if and only if f ( x − 2 j n ) ∈ V j for each n ∈ Z . For each j ∈ Z it follows that f ( x ) ∈ V j if and only if f ( x/ 2) ∈ V j +1 . Ll. Alsed` a and D. Romero, Girona C` alcul Num` eric d’Objectes Invariants amb Ondetes //
Motivation Wavelets in Theory Wavelets in Practice In the future A primer on wavelets If we fix an MRA, we know that V j ⊂ V j − 1 . Then, following [Mal], define the subspace W j as the orthogonal complement of V j on V j − 1 . That is V j − 1 = W j ⊕ V j . We are looking for an orthonormal basis of W j which is given by the formula � x − 2 j n � 1 √ ψ j,n ( x ) = 2 j ψ := wavelets . 2 j Such basis is obtained from a function called the mother wavelet ψ ( x ) which ψ ( x ) verifies a relation with φ ( x ) . [Mal] Mallat, St´ ephane, A wavelet tour of signal processing , Academic Press Inc., San Diego, CA, , xxiv+. Ll. Alsed` a and D. Romero, Girona C` alcul Num` eric d’Objectes Invariants amb Ondetes //
Motivation Wavelets in Theory Wavelets in Practice In the future A primer on wavelets The integer translates , ψ ( x − n ) , of ψ ( x ) form an orthonormal basis of W 0 . Moreover, from [Mal]: Mallat and Meyer Theorem For every j ∈ Z the family { ψ j,n } n ∈ Z is an orthonormal basis of each W j , The wavelets { ψ j,n } ( j,n ) ∈ Z × Z are an orthonormal basis of L 2 ( R ) for all j, n ∈ Z . Ll. Alsed` a and D. Romero, Girona C` alcul Num` eric d’Objectes Invariants amb Ondetes //
Motivation Wavelets in Theory Wavelets in Practice In the future Summarizing � � W j := V j − 1 \V j L 2 ( R ) = clos L 2 ( R ) = clos V j W j j ∈ Z j ∈ Z V j = span { φ j,n ( x ) } n ∈ Z W j = span { ψ j,n ( x ) } n ∈ Z V 0 = span { φ ( x − n ) } n ∈ Z W 0 = span { ψ ( x − n ) } n ∈ Z � 1 2 e − iω � h ∗ ( ω + π ) � ψ ( ω ) := φ ( ω ) √ φ ( x ) ψ ( x ) ∞ h (2 − p ω ) � � � φ ( ω ) = √ 2 p =1 � � h [ n ] e − inω � � h ( ω ) = 1 2 φ ( x 1 2 ψ ( x √ 2 ) = h [ n ] φ ( x − n ) √ 2 ) = g [ n ] φ ( x − n ) n ∈ Z n ∈ Z n ∈ Z h [ n ] g [ n ] g [ n ] := ( − 1) 1 − n h [1 − n ] Ll. Alsed` a and D. Romero, Girona C` alcul Num` eric d’Objectes Invariants amb Ondetes //
Motivation Wavelets in Theory Wavelets in Practice In the future Examples of mother wavelets Shannon wavelet (no compact support) Daubechies wavelet (compact support) No closed formula ψ ( x ) = sin(2 π ( x − 1 / 2)) − sin( π ( x − 1 / 2)) 2 π ( x − 1 / 2) π ( x − 1 / 2) 0 . 48296291314 . . . if n = 0 , √ 0 . 83651630373 . . . if n = 1 , 2 if n = 0 , 2 h [ n ] = 0 . 22414386804 . . . if n = 2 , √ 2 − 1 ( n − 1) / 2 h [ n ] = if n odd, − 0 . 12940952255 . . . if n = 3 , πn 0 otherwise. 0 otherwise. Ll. Alsed` a and D. Romero, Girona C` alcul Num` eric d’Objectes Invariants amb Ondetes //
Motivation Wavelets in Theory Wavelets in Practice In the future Examples of mother wavelets 1 ψ ( x ) := 1 [0 , 1 2 ) ( x ) − 1 [ 1 2 , 1) ( x ) 0 � 1 if x ∈ [ a, b ) , where 1 [ a,b ) ( x ) = 0 otherwise. − 1 � 1 0 0 . 5 1 if n = 0 , 1 , √ h [ n ] = 2 0 otherwise. Haar wavelet (compact support) It is the unique Daubechies wavelet with an explicit formula. Ll. Alsed` a and D. Romero, Girona C` alcul Num` eric d’Objectes Invariants amb Ondetes //
Motivation Wavelets in Theory Wavelets in Practice In the future Fixing and translating the wavelet We will be focused on the Daubechies wavelets family. Each Daubechies wavelet minimize its support, [1 − p, p ] , constrained to the maximal number of vanishing moments, p : � p x k ψ ( x ) dx = 0 for 0 ≤ k < p. 1 − p Since our framework is S 1 = R / Z , we transform a R -function into a S 1 -function by setting ψ PER j,n as follows: x x ∈ R : frac( x )= θ ���� � ( � θ + ι ) − 2 j n � �� � � ( θ + ι ) = 2 − j/ 2 � ψ PER j,n ( θ ) = ψ j,n ψ . 2 j ι ∈ Z ι ∈ Z j,n are 1 -periodic functions belonging to L 1 ( S 1 ) . ψ PER Ll. Alsed` a and D. Romero, Girona C` alcul Num` eric d’Objectes Invariants amb Ondetes //
Motivation Wavelets in Theory Wavelets in Practice In the future Fixing and translating the wavelet It is known that an orthonormal basis of L 2 ( S 1 ) is given by − j,n with j ≥ 0 and n = 0 , 1 , . . . , 2 j − 1 } provided that ψ ( x ) { 1 , ψ PER is an orthonormal wavelet from a R -MRA (see [HeWe]). Hence, once ψ is given, we are (almost) ready to compute 2 j − 1 N � � d − j,n ψ PER ϕ ∼ a 0 + − j,n ( θ ) . j =0 n =0 [HeWe] Hern´ andez, Eugenio and Weiss, Guido, A first course on wavelets , CRC Press, Boca Raton, FL, , xx+. Ll. Alsed` a and D. Romero, Girona C` alcul Num` eric d’Objectes Invariants amb Ondetes //
Recommend
More recommend
