A class of anisotropic multiple multiresolution analysis Mariantonia - PowerPoint PPT Presentation

A class of anisotropic multiple multiresolution analysis Mariantonia Cotronei University of Reggio Calabria, Italy MAIA 2013, Erice, September 2013 Jointly with: Mira Bozzini, Milvia Rossini, Tomas Sauer Multiple multiresolution analysis �

Description of expanding matrices and related objects Multiple multiresolution analysis �

Description of expanding matrices and related objects Inside filterbanks and subdivisions Multiple multiresolution analysis �

Description of expanding matrices and related objects Inside filterbanks and subdivisions Remarks of their multiple counterparts Multiple multiresolution analysis �

Description of expanding matrices and related objects Inside filterbanks and subdivisions Remarks of their multiple counterparts An efficient strategy to construct (multiple) filterbanks Multiple multiresolution analysis �

Description of expanding matrices and related objects Inside filterbanks and subdivisions Remarks of their multiple counterparts An efficient strategy to construct (multiple) filterbanks Case study Multiple multiresolution analysis �

D escription of expanding matrices and related objects I nside filterbanks and subdivisions R emarks of their multiple counterparts A n efficient strategy to construct (multiple) filterbanks C ase study Multiple multiresolution analysis �

▼ ❞ ❞❡t ▼ Expanding matrices Let ▼ ∈ Z s × s be an expanding matrix, i.e. all its its eigenvalues are larger than one in modulus � ▼ − ♥ � → ✵ ⇓ as ♥ increases, ▼ − ♥ Z s → R s Multiple multiresolution analysis �

Expanding matrices Let ▼ ∈ Z s × s be an expanding matrix, i.e. all its its eigenvalues are larger than one in modulus � ▼ − ♥ � → ✵ ⇓ as ♥ increases, ▼ − ♥ Z s → R s ▼ defines a sampling lattice ❞ = | ❞❡t ( ▼ ) | is the number of cosets Multiple multiresolution analysis �

The cosets have the form ▼ Z s + ξ ❥ , ❥ = ✵ , . . . , ❞ − ✶ where ξ ❥ ∈ ▼ [ ✵ , ✶ ) s � Z s are the coset representatives. It is well known that ❞ − ✶ Z s = � ( ξ ❥ + ▼ Z s ) ❥ = ✵ Multiple multiresolution analysis �

Separable/Nonseparable � ✷ � � � ✵ ✶ ✶ ▼ = ▼ = − ✶ ✵ ✷ ✶ Multiple multiresolution analysis �

Isotropy/Anisotropy � ✶ � � � ✶ ✶ ✶ ▼ = ▼ = − ✶ − ✷ ✶ ✶ Multiple multiresolution analysis �

▼ ▼ ▼ ❝ ❝ ▼ ▼ ▼ ✶ s ❝ ▼ ▼ ▼ ❝ ✵ Down/upsampling Let ❝ ∈ ℓ ( Z s ) be a given signal. Multiple multiresolution analysis �

▼ ▼ ✶ s ❝ ▼ ▼ ▼ ❝ ✵ Down/upsampling Let ❝ ∈ ℓ ( Z s ) be a given signal. Downsampling operator ↓ ▼ associated to ▼ : ↓ ▼ ❝ = ❝ ( ▼ · ) Multiple multiresolution analysis �

Down/upsampling Let ❝ ∈ ℓ ( Z s ) be a given signal. Downsampling operator ↓ ▼ associated to ▼ : ↓ ▼ ❝ = ❝ ( ▼ · ) Upsampling operator ↑ ▼ associated to ▼ : � ❝ ( ▼ − ✶ α ) if α ∈ ▼ Z s ↑ ▼ ❝ ( α ) = ✵ otherwise Multiple multiresolution analysis �

Filtering Filter operator ❋ : � ❋❝ = ❢ ∗ ❝ = ❢ ( · − α ) ❝ ( α ) α ∈ Z s where ❢ = ❋ δ = ( ❢ ( α ) : α ∈ Z s ) is the impulse response of ❋ Multiple multiresolution analysis �

s ❞ s ❋ ❋❝ ▼ ❋ ❥ ❝ ❥ ✵ ❞ ✶ ❞ s s ● ❞ ● ❝ ❥ ❥ ✵ ❞ ✶ ● ❥ ▼ ❝ ❥ ❥ ✵ ●❋ ■ ❞ -channel filter bank Critically sampled: ❞ = | ❞❡t ▼ | Multiple multiresolution analysis �

●❋ ■ ❞ -channel filter bank Critically sampled: ❞ = | ❞❡t ▼ | Analysis filter: ❋ : ℓ ( Z s ) → ℓ ❞ ( Z s ) ❋❝ = [ ↓ ▼ ❋ ❥ ❝ : ❥ = ✵ , . . . , ❞ − ✶ ] Synthesis filter: ● : ℓ ❞ ( Z s ) → ℓ ( Z s ) ❞ � ● [ ❝ ❥ : ❥ = ✵ , . . . , ❞ − ✶ ] = ● ❥ ↑ ▼ ❝ ❥ , ❥ = ✵ Multiple multiresolution analysis �

❞ -channel filter bank Critically sampled: ❞ = | ❞❡t ▼ | Analysis filter: ❋ : ℓ ( Z s ) → ℓ ❞ ( Z s ) ❋❝ = [ ↓ ▼ ❋ ❥ ❝ : ❥ = ✵ , . . . , ❞ − ✶ ] Synthesis filter: ● : ℓ ❞ ( Z s ) → ℓ ( Z s ) ❞ � ● [ ❝ ❥ : ❥ = ✵ , . . . , ❞ − ✶ ] = ● ❥ ↑ ▼ ❝ ❥ , ❥ = ✵ Perfect reconstruction: ●❋ = ■ Multiple multiresolution analysis �

❞ -channel filter bank By perfect reconstruction:   ❝ ✶ ✵ � ❝ ✶ ❝ ✶ �   ❋ ✶ ● ❝ →  = → ❝   . ❞ ✶ .   .  ❝ ✶ ❞ − ✶ ❋ ✵ , ● ✵ − → low-pass ❋ ❥ , ● ❥ , ❥ > ✵ − → high-pass Multiresolution decomposition . . . Multiple multiresolution analysis �

Iterated filter bank MRA structure... ❝ ❞ ✶ ❝ ✶ ❞ ✷ ❝ ✷ ❞ ✸ ❝ ✸ Multiple multiresolution analysis �

Observe that � ● ❥ ↑ ❝ = ❣ ❥ ∗ ↑ ▼ ❝ = ❣ ❥ ( · − ▼ α ) ❝ ( α ) , α ∈ Z s i.e. all reconstruction filters act as stationary subdivision operators with dilation matrix ▼ . Multiple multiresolution analysis �

Stationary subdivision Subdivision operator: ❙ := ❙ ❛ , ▼ : ℓ ( Z s ) → ℓ ( Z s ) defined by ❝ ( ♥ + ✶ ) := ❙❝ ( ♥ ) = � ❛ ( · − ▼ α ) ❝ ( ♥ ) ( α ) α ∈ Z s where ▼ ∈ Z s × s is expanding Multiple multiresolution analysis �

▼ ❥ s ❛ ❥ ❥ ♠ ❛ ❥ ▼ ❥ ♠ ❙ ❥ ❙ ❛ ❥ ▼ ❥ Multiple subdivision Consider a set of a finite number of dilation matrices ( ▼ ❥ : ❥ ∈ Z ♠ ) where Z ♠ = { ✵ , . . . , ♠ − ✶ } for ♠ ∈ N . Multiple multiresolution analysis �

❛ ❥ ▼ ❥ ♠ ❙ ❥ ❙ ❛ ❥ ▼ ❥ Multiple subdivision Consider a set of a finite number of dilation matrices ( ▼ ❥ : ❥ ∈ Z ♠ ) where Z ♠ = { ✵ , . . . , ♠ − ✶ } for ♠ ∈ N . Associate a mask to each ▼ ❥ : ❛ ❥ ∈ ℓ ( Z s ) , ❥ ∈ Z ♠ . Multiple multiresolution analysis �

Multiple subdivision Consider a set of a finite number of dilation matrices ( ▼ ❥ : ❥ ∈ Z ♠ ) where Z ♠ = { ✵ , . . . , ♠ − ✶ } for ♠ ∈ N . Associate a mask to each ▼ ❥ : ❛ ❥ ∈ ℓ ( Z s ) , ❥ ∈ Z ♠ . Together, ❛ ❥ and ▼ ❥ define ♠ stationary subdivision operators ❙ ❥ := ❙ ❛ ❥ , ▼ ❥ Multiple multiresolution analysis �

♥ ♠ ♠ ♥ ♠ Multiple subdivision Call ǫ = ( ǫ ✶ , . . . , ǫ ♥ ) ∈ Z ♥ ♠ a digit sequence of length ♥ =: | ǫ | . Multiple multiresolution analysis �

Multiple subdivision Call ǫ = ( ǫ ✶ , . . . , ǫ ♥ ) ∈ Z ♥ ♠ a digit sequence of length ♥ =: | ǫ | . We collect all finite digit sequences in � Z ∗ Z ♥ ♠ := ♠ ♥ ∈ N and extend | ǫ | canonically to ǫ ∈ Z ∗ ♠ . Multiple multiresolution analysis �

♠ ❛ ❙ s ❙ ❝ ❛ ▼ ❝ ❝ s ▼ ▼ ♥ ▼ ✶ ♥ Multiple subdivision Consider the subdivision operator: ❙ ǫ = ❙ ǫ ♥ · · · ❙ ǫ ✶ . Multiple multiresolution analysis �

Multiple subdivision Consider the subdivision operator: ❙ ǫ = ❙ ǫ ♥ · · · ❙ ǫ ✶ . For any ǫ ∈ Z ∗ ♠ there exists a mask ❛ ǫ = ❙ ǫ δ such that � ❝ ∈ ℓ ( Z s ) , ❙ ǫ ❝ = ❛ ǫ ( · − ▼ ǫ α ) ❝ ( α ) , α ∈ Z s where ▼ ǫ := ▼ ǫ ♥ · · · ▼ ǫ ✶ , ♥ = | ǫ | . Multiple multiresolution analysis �

▼ ❥ ▼ ✶ s s ▼ ▼ ✶ ❧✐♠ ▼ ✵ ✶ ▼ ❥ ✶ ♠ ❥ Multiple subdivision Values of ❙ ǫ ❝ = approximations to a function on ▼ − ✶ ǫ Z s . Multiple multiresolution analysis �

▼ ❥ ▼ ▼ ✶ ❧✐♠ ▼ ✵ ✶ ▼ ❥ ✶ ♠ ❥ Multiple subdivision Values of ❙ ǫ ❝ = approximations to a function on ▼ − ✶ ǫ Z s . ǫ Z s to tend to R s : In order for ▼ − ✶ Multiple multiresolution analysis �

▼ ▼ ✶ ❧✐♠ ▼ ✵ ✶ ▼ ❥ ✶ ♠ ❥ Multiple subdivision Values of ❙ ǫ ❝ = approximations to a function on ▼ − ✶ ǫ Z s . ǫ Z s to tend to R s : In order for ▼ − ✶ each matrix ▼ ❥ must be expanding, Multiple multiresolution analysis �

▼ ✶ ❧✐♠ ▼ ✵ ✶ ▼ ❥ ✶ ♠ ❥ Multiple subdivision Values of ❙ ǫ ❝ = approximations to a function on ▼ − ✶ ǫ Z s . ǫ Z s to tend to R s : In order for ▼ − ✶ each matrix ▼ ❥ must be expanding, all the matrices ▼ ǫ must be expanding Multiple multiresolution analysis �

Multiple subdivision Values of ❙ ǫ ❝ = approximations to a function on ▼ − ✶ ǫ Z s . ǫ Z s to tend to R s : In order for ▼ − ✶ each matrix ▼ ❥ must be expanding, all the matrices ▼ ǫ must be expanding ⇓ The matrices ▼ ǫ must all be jointly expanding i.e. � = ✵ , � ▼ − ✶ � � ❧✐♠ (1) ǫ | ǫ |→∞ or, equivalently, ▼ − ✶ � � : ❥ ∈ Z ♠ ρ < ✶ ❥ (joint spectral radius condition) Multiple multiresolution analysis �

✷ ✹ ✶ ✶ ✶ Multiple subdivision Example: adaptive subdivision/discrete shearlets Based on: Multiple multiresolution analysis �