Multiresolution Analysis (MRA) WTBV January 10, 2017 WTBV - PowerPoint PPT Presentation

Multiresolution Analysis (MRA) WTBV January 10, 2017 WTBV Multiresolution Analysis (MRA) January 10, 2017 1 / 43

Multiresolution (MRA) 1 Multiresolution scheme Examples Haar-MRA Shannon-MRA Piecewise-linear MRA Properties of MRA’s (I) Orthonormal systems of translates Properties of MRA’s (II) Vanishing noments, smoothness, reconstruction properties WTBV Multiresolution Analysis (MRA) January 10, 2017 2 / 43

Multiresolution (MRA) Multiresolutions analysis (MRA) was invented in 1988 by Stephane Mallat in his Ph.D. thesis Multiresolution and Wavelets (University of Pennsylvania) is an elegant theoretical framework for the study of wavelets and wavelet transforms is considered to be the central concept which integrates the many facets of wavelet transforms WTBV Multiresolution Analysis (MRA) January 10, 2017 3 / 43

Multiresolution (MRA) Multiresolution scheme Definition An MRA ( multiresolution analysis ) consists of a family { V j } j ∈ Z of subspaces of L 2 ( R ) satisfying the following properties: “nesting”: V j ⊆ V j +1 ( j ∈ Z ) 1 “density” : span { V j } j ∈ Z = L 2 ( R ) 2 “separation”: � { V j } j ∈ Z = { 0 } 3 “scaling”: 4 f ( t ) ∈ V 0 ⇔ ( D 2 j f )( t ) = 2 j / 2 f (2 j t ) ∈ V j ( f ∈ L 2 ( R ) , j ∈ Z ) “scaling function”: 5 There exists a function φ ∈ V 0 s.th. the family of its integer translates { T k φ ( t ) } k ∈ Z = { φ ( t − k ) } k ∈ Z forms a complete ON-basis of V 0 = span { T k φ } k ∈ Z (ONST) WTBV Multiresolution Analysis (MRA) January 10, 2017 4 / 43

Multiresolution (MRA) Multiresolution scheme ONST-Example: φ ( t ) = sinc( t ) = sin( π t ) Consider the function π t Are the integer translates ( T k φ )( t ) = φ ( t − k ) ( k ∈ Z ) orthogonal to each other? The answer is not obvious from looking at the graphs! How to prove orthogonality? Recipe: Go to the frequency domain! (using PP) Recall: � φ ( s ) = b ( s ) = 1 [ − 1 / 2 , 1 / 2) ( s ) (the box function) � φ | T k φ � = � � φ | � T k φ � = � b ( s ) | e − 2 π iks b ( s ) � � 1 / 2 e − 2 π iks ds = δ 0 , k = − 1 / 2 WTBV Multiresolution Analysis (MRA) January 10, 2017 5 / 43

Multiresolution (MRA) Multiresolution scheme Reminder: � φ ( s + n ) | 2 = ≡ 1 | � φ ( t ) satisfies (ONST) ⇐ ⇒ n ∈ Z � � f | T k f � = � � f | � f ( s ) � � f ( s ) e 2 π iks ds Proof : T k f � = R � n +1 � � � 2 � � � � e 2 π iks ds = f ( s ) � n n ∈ Z � 1 � � � 2 � � �� e 2 π iks ds = f ( s + n ) � 0 n ∈ Z � 1 � � � 2 � � �� e 2 π iks ds = f ( s + n ) � 0 n ∈ Z Hence in terms of Fourier series � � � � 2 � � � f | T k f � e − 2 π iks = �� f ( s + n ) � k ∈ Z n ∈ Z WTBV Multiresolution Analysis (MRA) January 10, 2017 6 / 43

Multiresolution (MRA) Multiresolution scheme Consequences The vector spaces ( V j ) j ∈ Z are ordered by inclusion 1 { 0 } ւ · · · ⊆ V − 2 ⊆ V − 1 ⊆ V 0 ⊆ V 1 ⊆ V 2 ⊆ · · · ր L 2 ( R ) For each j ∈ Z family of dilated and translated functions { φ j , k ( t ) } k ∈ Z , 2 defined by φ j , k ( t ) = 2 j / 2 φ (2 j t − k ) = ( D 2 j T k φ )( t ) , forms a complete ON-Basis (Hilbert basis) of the approximation space V j = span { φ j , k } k ∈ Z ( j ∈ Z ) From V 0 ⊆ V 1 it follows that there exists a (unique) ℓ 2 -sequence 3 h = ( h k ) k ∈ Z of complex numbers s.th. � ( S ) φ ( t ) = h k φ 1 , k ( t ) k ∈ Z This identity is the scaling identity of the MRA, the sequence h = ( h k ) k ∈ Z is the scaling filter of the MRA WTBV Multiresolution Analysis (MRA) January 10, 2017 7 / 43

Multiresolution (MRA) Multiresolution scheme Remarks Properties involving V 0 and φ ( t ) carry over to all scaling levels by using dilation, e.g., � V 0 ∋ f ( t ) = f k · ( T k φ )( t ) ⇐ ⇒ k ∈ Z � V j ∋ ( D 2 j f )( t ) = f k · ( D 2 j T k φ )( t ) k ∈ Z so each V j is a dilated copy of V 0 , and thus orthonormality is preserved � � φ j , k | φ j ,ℓ � = 2 j φ (2 j t − k ) φ (2 j t − ℓ ) dt R � = φ ( t − k ) φ ( t − ℓ ) dt = � φ 0 , k | φ 0 ,ℓ � = δ k ,ℓ R WTBV Multiresolution Analysis (MRA) January 10, 2017 8 / 43

Multiresolution (MRA) Multiresolution scheme From the scaling identity (S) and orthogonality one gets immediately � √ h k = � φ | φ 1 , k � = 2 φ ( t ) φ (2 t − k ) dt R and for all j , ℓ ∈ Z φ j ,ℓ ( t ) = 2 j / 2 φ (2 j t − ℓ ) = 2 j / 2 � h k φ 1 , k (2 j t − ℓ ) k ∈ Z = 2 ( j +1) / 2 � h k φ (2 j +1 − 2 ℓ − k ) k ∈ Z � � = h k φ j +1 , 2 ℓ + k ( t ) = h k − 2 ℓ φ j +1 , k ( t ) k ∈ Z k ∈ Z so that the scaling coefficients a j ,ℓ = � f | φ j ,ℓ � of f ∈ L 2 satisfy � � a j ,ℓ = � f | φ j ,ℓ � = h k − 2 ℓ � f | φ j +1 , k ( t ) � = h k − 2 ℓ a j +1 , k k ∈ Z k ∈ Z WTBV Multiresolution Analysis (MRA) January 10, 2017 9 / 43

Multiresolution (MRA) Multiresolution scheme The wavelet function ψ ( t ) of a MRA is defined in terms of the scaling function φ ( t ) as � g k φ 1 , k ( t ) where g k = ( − 1) k h 1 − k ( W ) ψ ( t ) = k ∈ Z The sequence g = ( g k ) k ∈ Z is the wavelet filter belonging to the MRA The wavelet functions ψ j ,ℓ ( j , ℓ ∈ Z are defined as usual The wavelet coefficients d j ,ℓ = � f | ψ j ,ℓ � of f ∈ L 2 satisfy � � d j ,ℓ = � f | ψ j ,ℓ � = g k − 2 ℓ � f | φ j +1 , k ( t ) � = g k − 2 ℓ a j +1 , k k ∈ Z k ∈ Z The Discrete Wavelet Transform (DWT) based on the functions φ ( t ) and ψ ( t ) uses these scaling and wavelet identities � � a j ,ℓ = h k − 2 ℓ a j +1 , k d j ,ℓ = g k − 2 ℓ a j +1 , k k ∈ Z k ∈ Z WTBV Multiresolution Analysis (MRA) January 10, 2017 10 / 43

Multiresolution (MRA) Multiresolution scheme Theorem For each j ∈ Z the family of wavelet functions { ψ j , k } k ∈ Z with 1 ψ j , k ( t ) = 2 j / 2 ψ (2 j t − k ) = ( D 2 j T k ψ )( t ) is a complete ON-Basis (Hilbert basis) of the wavelet (detail) space W j = span { ψ j , k } k ∈ Z For all j ∈ Z the space W j is the orthogonal complement of V j in V j +1 : 2 V j +1 = W j ⊕ V j W j ⊥ V j For every J ∈ Z one has the direct product decomposition 3 � L 2 ( R ) = V J ⊕ W j j ≥ J The family { ψ j , k } j , k ∈ Z is a complete ON-basis (Hilbert basis) of L 2 ( R ) 4 � L 2 ( R ) = W j j ∈ Z WTBV Multiresolution Analysis (MRA) January 10, 2017 11 / 43

Multiresolution (MRA) Multiresolution scheme Remarks Functions in V j and W j have resolution level ≥ 2 − j 1 Orthogonal projections on approximation and detail subspaces 2 � P j : L 2 ( R ) → V j : f �→ approximation � f | φ j , k � φ j , k k ∈ Z � Q j : L 2 ( R ) → W j : f �→ detail � f | ψ j , k � ψ j , k k ∈ Z where Q j = P j +1 − P j For all j > m one has the wavelet decomposition 3 V j +1 = V m ⊕ W m ⊕ W m +1 ⊕ · · · ⊕ W j The “density” and “separation” requirements for an MRA translate into 4 j →∞ P j f = f lim und j →−∞ P j f = 0 lim w.r.t. L 2 -convergence WTBV Multiresolution Analysis (MRA) January 10, 2017 12 / 43

Multiresolution (MRA) Examples Example (1): The Haar -MRA The scaling function is φ ( t ) = 1 [0 , 1) ( t ) For j ∈ Z the approximation space V j = span { φ j , k ( t ) } k ∈ Z ⊆ L 2 ( R ) consists of the L 2 -step functions with step width 2 − j { φ j , k ( t ) } k ∈ Z is obviously an ON-Basis of V j Density (fact about approximation by step functions): j →∞ V j = L 2 ( R ) lim Separation: an L 2 -function f ∈ � j ∈ Z V j which is constant on arbitrarily long intervals must vanish identically on R WTBV Multiresolution Analysis (MRA) January 10, 2017 13 / 43

Multiresolution (MRA) Examples Scaling filter coefficients 1 1 h 0 = √ , h 1 = √ , h k = 0 ( k � = 0 , 1) 2 2 Scaling identity 1 φ ( t ) = √ ( φ 0 , 0 ( t ) + φ 0 , 1 ( t )) = φ (2 t ) + φ (2 t − 1) 2 Wavelet filter coefficients 1 , g 1 = − 1 √ √ g 0 = , g k = 0 ( k � = 0 , 1) 2 2 Wavelet identity 1 ψ ( t ) = √ ( φ 0 , 0 ( t ) + φ 0 , 1 ( t )) = φ (2 t ) − φ (2 t − 1) 2 = 1 [0 , 1 / 2) ( t ) − 1 [1 / 2 , 1) ( t ) Fourier transforms ψ ( s ) = i · e − i π s sin( π s / 2) sinc( s / 2) � � φ ( s ) = e − i π s sinc( s ) WTBV Multiresolution Analysis (MRA) January 10, 2017 14 / 43

Multiresolution (MRA) Examples Examples (2) The Daubechies , Coiflet , and many other orthogonal filters of similar type define MRAs with filters of finite length and scaling/wavelet functions with compact support The filters are (of course!) those constructed from orthogonality and low/highpass conditions The scaling functions φ ( t ) and the wavelet functions ψ ( t ) are those functions determined by the cascade algorithm The ONST-property follows because the cascade algorithm preserves orthogonality Density and Separation do not come automatically, but have to be verified separately WTBV Multiresolution Analysis (MRA) January 10, 2017 15 / 43