Introduction to Multiresolution Analysis of Wavelets

Transcription

1 Introuction to Multiresolution Analysis o Wavelets Aso Ray Proessor o Mechanical Engineering The Pennsylvania State University University Par PA 68 Tel: (84) Eail: axr@psu.eu

2 Organization o the Presentation What is Multiresolution Analysis (MRA)? Scaling Functions an Average Spaces Wavelet Functions an Detail Spaces An Exaple o Scaling an Wavelet Functions Ipleentation o MRA as Filter Bans Subban Filtering Schee Signal Analysis an Synthesis Using QMFs Exaples o MRA Using Matlab Deo

3 Let { } What is Multiresolution Analysis? be a sequence o saple signals at the intervals t τ > i.e. scale τ. With no loss o generality we set τ. Split into: average at coarse scale τ an etails at ine scale τ + The process goes on recursively to yiel a progressively coarser sequences with etails This yiels: L reove at every scale t τ in + + L + + N N or soe N Z an +. L An orthonoral base is create by having the scale s in the recursive extraction o etails. The tie translation an ilation operators T D : L( R) L ( R) are eine respectively by: T ( ( t ) ( T )( ( t ) Z D t t ( ) ( ) ( D )( ( Both T an D are unitary operators i.e. T * T an D * D. It ollows that DT T D an D T T / D

4 Scaling Functions an Average Spaces Deinition : Multiresolution analysis (MRA) o signals belonging to L ( R) is eine as a sequence o close subspaces Z with the ollowing properties: V V V + h ( V h( V+ h( V h( t + ) V Closure V L ( R ) a scaling unction ϕ( ) V such that: (Explicit expressions or ( ) U an IV { } n n n n t ϕ( i.e. ϕ ˆ () ϕ ( ( D T ϕ)( ( T ϕ)( ϕ( t ) R the collection { ϕ t ) Z} ( is an orthonoral basis or V ϕ ay not be available) Rear: a sequence { h } l ( Z) such that the scaling unction satisies: ϕ( t ) h ϕ(t ) h i.e. ˆ h () an ( D ϕ)( ht ϕ( Z Z Z The set j j ϕ j ϕ j ( ϕ( t ); Z is an orthonoral basis or V j

5 Wavelet Functions an Detail Spaces Deinition : We construct the etail space W j { V L R): g g V } W j as orthogonal copleent o j ( j V j W j V j The space L ( R) can be partitione as: L ( R) l Theore : For any o integer l let g( T ) T h* ( T ) polynoials can be uniquely expresse as: an V K W j u ( h( v( T ) + g( T ) w( T ) or soe v w P { } W closure o g( w( T )ϕ: w P V j in V j i.e. K j L ( R) W j j. Then every u P the space o Deinition 3: The other wavelet ψ W D W is eine as: ψ D g( T ) ϕ i.e. ψ( t ) g ϕ(t ) so that ψ n Z n D T ψ W closure o { w( T )ψ : w P} i.e. ψ n( ψ( t n) Theore : The wavelets { ψ : n Z} that span W n or an orthonoral basis or ( R) L

6 AN EXAMPLE OF SCALING AND WAVELET FUNCTIONS t [) Haar Scaling Function: ϕ( t ) χ[ )( ϕ ξ πξ sin( πξ) ˆ ( ) e i t [) πξ The projection o a signal L ( R) to V is: ( P )( ) * t ϕ n ϕ n ( n χ ( [ n ( n+ )) ( n+ ) where ϕ* n n θ ( θ) average o in [ n ( n + )) n The ilation equation is: Dϕ χ ( χ + χ ) ( I + T )ϕ [) l Haar Wavelet Function: g( T ) T h* ( T ) Hence ψ g( ϕ Setting [) [) h ( ( I + T ) l l ( T T ) D D ( χ χ ) χ [ ) χ [ ) [ l l) [ l l+ ) l l l l+ l yiels g ( I T ) ( an ψ [ ) [ χ χ ) or any o integer l (by Theore ) Theore 3 (Jannsen): Let ϕ be a scaling unction or an orthogonal MRA. Except or the Haar wavelet ϕ cannot be a probability ensity. Rear: It is possible to construct biorthogonal bases whose scaling unctions are ensities..

7 Ipleentation o MRA as Filter Bans Binary Downsapling: ( ){ L L} { L L} Binary Upsapling: ( ){ L L} { L L} Theore 3: The binary ownsapling an upsapling operators have the ollowing properties: * I V on V ( ) ( u( ) ϕ u( T ) ϕ u n ϕ n ( v ( T )) ϕ vn ϕ n an ( T v( ) ϕ vn + ϕ n ( v ( T ) ) ϕ vn ϕ n an ( T v( ) ϕ vn+ ϕ n + T T IV Signal ecoposition into requency bans: low (averaging) an high (etails) Low-pass ilter H * : V V eine as: H * h( T ) H h * ( ; an H * V DV V High-pass ilter G * : W V eine as: G * w( ψ g( ϕ Gu( ϕ g * ( u( T ψ; ) * W an G W DW

8 Subban Filtering Schee Low-Pass Ban-liiting Filter Operator H h * (. average + g * ( High-Pass Ban-liiting Filter Operator G + h(t ) etails + Low-Pass Interpolating Filter Operator H* + g( High-Pass Interpolating Filter Operator G* Analysis Filter Ban Synthesis Filter Ban

9 Signal Analysis an Synthesis Using Quarature Mirror Filters (QMF) H H H 3 L Wavelet Decoposition with QMFs G G G G L 3 L H* H* H* 3 L Wavelet Coposition with QMFs G* G* G* G* L 3 L