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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
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
Multiple multiresolution analysis
Description of expanding matrices and related
Inside filterbanks and subdivisions
Multiple multiresolution analysis
Description of expanding matrices and related
Inside filterbanks and subdivisions Remarks of their multiple counterparts
Multiple multiresolution analysis
Description of expanding matrices and related
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
Inside filterbanks and subdivisions Remarks of their multiple counterparts An efficient strategy to construct (multiple) filterbanks Case study
Multiple multiresolution analysis
Description of expanding matrices and related
Inside filterbanks and subdivisions Remarks of their multiple counterparts An efficient strategy to construct (multiple) filterbanks Case study
Multiple multiresolution analysis
Let ▼ ∈ Zs×s be an expanding matrix, i.e. all its its eigenvalues are larger than one in modulus ▼−♥ → ✵ ⇓ as ♥ increases, ▼−♥Zs → Rs ▼ ❞ ❞❡t ▼
Multiple multiresolution analysis
Let ▼ ∈ Zs×s be an expanding matrix, i.e. all its its eigenvalues are larger than one in modulus ▼−♥ → ✵ ⇓ as ♥ increases, ▼−♥Zs → Rs ▼ defines a sampling lattice ❞ = | ❞❡t(▼)| is the number of cosets
Multiple multiresolution analysis
The cosets have the form ▼Zs + ξ❥, ❥ = ✵, . . . , ❞ − ✶ where ξ❥ ∈ ▼[✵, ✶)s Zs are the coset representatives. It is well known that Zs =
❞−✶
(ξ❥ + ▼Zs)
Multiple multiresolution analysis
▼ = ✷ ✵ ✵ ✷
✶ −✶ ✶
▼ =
✶ −✶ ✶
✶ ✶ ✶ −✷
Let ❝ ∈ ℓ(Zs) be a given signal.
▼
▼
▼ ❝
❝ ▼
▼
▼
▼ ❝
❝ ▼
✶
▼
s
✵
Multiple multiresolution analysis
Let ❝ ∈ ℓ(Zs) be a given signal. Downsampling operator ↓▼ associated to ▼: ↓▼ ❝ = ❝(▼·)
▼
▼
▼ ❝
❝ ▼
✶
▼
s
✵
Multiple multiresolution analysis
Let ❝ ∈ ℓ(Zs) be a given signal. Downsampling operator ↓▼ associated to ▼: ↓▼ ❝ = ❝(▼·) Upsampling operator ↑▼ associated to ▼: ↑▼ ❝(α) =
if α ∈ ▼Zs ✵
Multiple multiresolution analysis
Filter operator ❋: ❋❝ = ❢ ∗ ❝ =
❢ (· − α)❝(α) where ❢ = ❋δ = (❢ (α) : α ∈ Zs) is the impulse response of ❋
Multiple multiresolution analysis
Critically sampled: ❞ = | ❞❡t ▼| ❋
s ❞ s
❋❝
▼ ❋❥❝
❥ ✵ ❞ ✶
s s
❥ ✵ ❞ ✶
❞ ❥ ✵
▼ ❝❥
■
Multiple multiresolution analysis
Critically sampled: ❞ = | ❞❡t ▼| Analysis filter: ❋ : ℓ(Zs) → ℓ❞(Zs) ❋❝ = [↓▼ ❋❥❝ : ❥ = ✵, . . . , ❞ − ✶] Synthesis filter:
❞
■
Multiple multiresolution analysis
Critically sampled: ❞ = | ❞❡t ▼| Analysis filter: ❋ : ℓ(Zs) → ℓ❞(Zs) ❋❝ = [↓▼ ❋❥❝ : ❥ = ✵, . . . , ❞ − ✶] Synthesis filter:
❞
Perfect reconstruction:
Multiple multiresolution analysis
By perfect reconstruction: ❝
❋
→ ❝✶
✵
❝✶
✶
. . . ❝✶
❞−✶
= ❝✶ ❞ ✶
❋✵, ●✵ − → low-pass ❋❥, ●❥, ❥ > ✵ − → high-pass Multiresolution decomposition . . .
Multiple multiresolution analysis
MRA structure... ❝ ❝✶ ❝✷ ❝✸ ❞ ✸ ❞ ✷ ❞ ✶
Multiple multiresolution analysis
Observe that
❣❥(· − ▼α) ❝(α), i.e. all reconstruction filters act as stationary subdivision
Multiple multiresolution analysis
Subdivision operator: ❙ := ❙❛,▼ : ℓ(Zs) → ℓ(Zs) defined by ❝(♥+✶) := ❙❝(♥) =
❛(· − ▼α)❝(♥)(α) where ▼ ∈ Zs×s is expanding
Multiple multiresolution analysis
Consider a set of a finite number of dilation matrices (▼❥ : ❥ ∈ Z♠) where Z♠ = {✵, . . . , ♠ − ✶} for ♠ ∈ N. ▼❥ ❛❥
s
❥
♠
❛❥ ▼❥ ♠ ❙❥ ❙❛❥ ▼❥
Multiple multiresolution analysis
Consider a set of a finite number of dilation matrices (▼❥ : ❥ ∈ Z♠) where Z♠ = {✵, . . . , ♠ − ✶} for ♠ ∈ N. Associate a mask to each ▼❥: ❛❥ ∈ ℓ (Zs) , ❥ ∈ Z♠ . ❛❥ ▼❥ ♠ ❙❥ ❙❛❥ ▼❥
Multiple multiresolution analysis
Consider a set of a finite number of dilation matrices (▼❥ : ❥ ∈ Z♠) where Z♠ = {✵, . . . , ♠ − ✶} for ♠ ∈ N. Associate a mask to each ▼❥: ❛❥ ∈ ℓ (Zs) , ❥ ∈ Z♠ . Together, ❛❥ and ▼❥ define ♠ stationary subdivision
❙❥ := ❙❛❥,▼❥
Multiple multiresolution analysis
Call ǫ = (ǫ✶, . . . , ǫ♥) ∈ Z♥
♠
a digit sequence of length ♥ =: |ǫ|.
♠ ♥ ♥ ♠ ♠
Multiple multiresolution analysis
Call ǫ = (ǫ✶, . . . , ǫ♥) ∈ Z♥
♠
a digit sequence of length ♥ =: |ǫ|. We collect all finite digit sequences in Z∗
♠ :=
Z♥
♠
and extend |ǫ| canonically to ǫ ∈ Z∗
♠.
Multiple multiresolution analysis
Consider the subdivision operator: ❙ǫ = ❙ǫ♥ · · · ❙ǫ✶.
♠
❛ ❙ ❙ ❝
s
❛ ▼ ❝ ❝
s
▼ ▼ ♥ ▼ ✶ ♥
Multiple multiresolution analysis
Consider the subdivision operator: ❙ǫ = ❙ǫ♥ · · · ❙ǫ✶. For any ǫ ∈ Z∗
♠ there exists a mask
❛ǫ = ❙ǫδ such that ❙ǫ❝ =
❛ǫ (· − ▼ǫα) ❝(α), ❝ ∈ ℓ(Zs), where ▼ǫ := ▼ǫ♥ · · · ▼ǫ✶, ♥ = |ǫ|.
Multiple multiresolution analysis
Values of ❙ǫ❝ = approximations to a function on ▼−✶
ǫ Zs.
▼
✶ s s
▼❥ ▼ ▼ ❧✐♠ ▼
✶
✵ ▼
✶ ❥
❥
♠
✶
Multiple multiresolution analysis
Values of ❙ǫ❝ = approximations to a function on ▼−✶
ǫ Zs.
In order for ▼−✶
ǫ Zs to tend to Rs:
▼❥ ▼ ▼ ❧✐♠ ▼
✶
✵ ▼
✶ ❥
❥
♠
✶
Multiple multiresolution analysis
Values of ❙ǫ❝ = approximations to a function on ▼−✶
ǫ Zs.
In order for ▼−✶
ǫ Zs to tend to Rs:
each matrix ▼❥ must be expanding, ▼ ▼ ❧✐♠ ▼
✶
✵ ▼
✶ ❥
❥
♠
✶
Multiple multiresolution analysis
Values of ❙ǫ❝ = approximations to a function on ▼−✶
ǫ Zs.
In order for ▼−✶
ǫ Zs to tend to Rs:
each matrix ▼❥ must be expanding, all the matrices ▼ǫ must be expanding ▼ ❧✐♠ ▼
✶
✵ ▼
✶ ❥
❥
♠
✶
Multiple multiresolution analysis
Values of ❙ǫ❝ = approximations to a function on ▼−✶
ǫ Zs.
In order for ▼−✶
ǫ Zs to tend to Rs:
each matrix ▼❥ must be expanding, all the matrices ▼ǫ must be expanding ⇓ The matrices ▼ǫ must all be jointly expanding i.e. ❧✐♠
|ǫ|→∞
ǫ
(1)
ρ
❥
: ❥ ∈ Z♠
(joint spectral radius condition)
Multiple multiresolution analysis
Example: adaptive subdivision/discrete shearlets Based on: ✷ ✹ ✶ ✶ ✶
Multiple multiresolution analysis
Example: adaptive subdivision/discrete shearlets Based on: parabolic scaling ✷ ✹
✶ ✶
Multiple multiresolution analysis
Example: adaptive subdivision/discrete shearlets Based on: parabolic scaling ✷ ✹
✶ ✶ ✶
Example: adaptive subdivision/discrete shearlets Based on: parabolic scaling ✷ ✹
✶ ✶ ✶
Case study . . .
Multiple multiresolution analysis
For each ❦ ∈ Z♠ Analysis filters: ❋❦ : ℓ(Zs) → ℓ❞(Zs) acting as ❋❦❝ = [↓▼❦ ❋❦,❥❝ : ❥ = ✵, . . . , ❞ − ✶]
❞ s s
❥ ✵ ❞ ✶
❞ ❥ ✵
▼❦ ❝❥
■ ❦
♠
Multiple multiresolution analysis
For each ❦ ∈ Z♠ Analysis filters: ❋❦ : ℓ(Zs) → ℓ❞(Zs) acting as ❋❦❝ = [↓▼❦ ❋❦,❥❝ : ❥ = ✵, . . . , ❞ − ✶] Synthesis filters: ●❦ : ℓ❞(Zs) → ℓ(Zs), acting as
❞
■ ❦
♠
Multiple multiresolution analysis
For each ❦ ∈ Z♠ Analysis filters: ❋❦ : ℓ(Zs) → ℓ❞(Zs) acting as ❋❦❝ = [↓▼❦ ❋❦,❥❝ : ❥ = ✵, . . . , ❞ − ✶] Synthesis filters: ●❦ : ℓ❞(Zs) → ℓ(Zs), acting as
❞
Perfect reconstruction:
❦ ∈ Z♠
Multiple multiresolution analysis
❝
▼✵
❝✶
✵ ▼✵
❝✷
✵✵
❞ ✷
✵✵ ▼✶
❝✷
✵✶
❞ ✷
✵✶
❞ ✶
✵ ▼✶
❝✶
✶ ▼✵
❝✷
✶✵
❞ ✷
✶✵ ▼✶
❝✷
✶✶
❞ ✷
✶✶
❞ ✶
✶
Multiple multiresolution analysis
Given a finitely supported ❛ Symbol: ❛♯(③) :=
❛(α)③α ❛ ❥ ③
s
❛ ▼
❥ ③
❥ ✵ ❞ ✶
Multiple multiresolution analysis
Given a finitely supported ❛ Symbol: ❛♯(③) :=
❛(α)③α Subsymbols: ❛♯
ξ❥(③) :=
❛(▼α + ξ❥)③α, ❥ = ✵, . . . , ❞ − ✶
Multiple multiresolution analysis
Start from the lowpass reconstruction filter ●✵ associated to a mask ❛.
❛ ❛ ③ ❛ ③
Multiple multiresolution analysis
Start from the lowpass reconstruction filter ●✵ associated to a mask ❛.
bank if and only if ❛ is unimodular: algebraic property involved in general simple for interpolatory schemes ❛ ③ ❛ ③
Multiple multiresolution analysis
Start from the lowpass reconstruction filter ●✵ associated to a mask ❛.
bank if and only if ❛ is unimodular: algebraic property involved in general simple for interpolatory schemes In 1D − → ❛♯(③) and ❛♯(−③) have no common zeros.
Multiple multiresolution analysis
Simplest filter bank − → lazy filters: translation operators τξ✐, ✐ = ✵, . . . , ❞ − ✶ In fact ■ =
❞−✶
τξ✐ ↑ ↓ τ−ξ✐, ▼
Multiple multiresolution analysis
Simplest filter bank − → lazy filters: translation operators τξ✐, ✐ = ✵, . . . , ❞ − ✶ In fact ■ =
❞−✶
τξ✐ ↑ ↓ τ−ξ✐, It: decomposes a signal modulo ▼ in the analysis recombines the components in the synthesis
Multiple multiresolution analysis
If ❛ defines an interpolatory subdivision scheme, then ●✵ can be easily completed to a perfect reconstruction filter bank. ❙❛ ▼ ❙❛❝ ▼ ❝ ❝
s
Multiple multiresolution analysis
If ❛ defines an interpolatory subdivision scheme, then ●✵ can be easily completed to a perfect reconstruction filter bank. A subdivision operator ❙❛ with dilation matrix ▼ is called interpolatory if ❙❛❝(▼·) = ❝, for any ❝ ∈ ℓ(Zs)
Multiple multiresolution analysis
The completion of an interpolatory ❛ yields the prediction–correction scheme ❋✵ ■ ❋❥
❥ ■
❙❛
▼
❥ ✶ ❞ ✶
❥
❥ ✶ ❞ ✶
Multiple multiresolution analysis
The completion of an interpolatory ❛ yields the prediction–correction scheme Analysis part: ❋✵ = ■, ❋❥ = τ−ξ❥ (■ − ❙❛ ↓▼) , ❥ = ✶, . . . , ❞ − ✶, Synthesis part:
and
❥ = ✶, . . . , ❞ − ✶.
Multiple multiresolution analysis
In terms of symbols: ❋ ♯
✵(③) = ✶,
❋ ♯
❥ (③) = ③ξ❥ − ❛♯ ξ❥(③−▼),
❥ = ✶, . . . , ❞ − ✶
✵(③) = ❛♯(③),
❋ ♯
❥ (③) = ③ξ❥,
❥ = ✶, . . . , ❞ − ✶
Multiple multiresolution analysis
Let ▼ = ΘΣΘ′ be a Smith factorization of the expanding matrix ▼, where Σ = σ✶ σ✷ ... σs and Θ, Θ′ unimodular
Multiple multiresolution analysis
1 Find s univariate interpolatory subdivision schemes
❜❥, ❥ = ✶, . . . , s with scaling factors or “arity” σ❥; ❜
s ❥ ✶
❜❥ ❜
s ❥ ✶
❜❥
❥ s
❜
Multiple multiresolution analysis
1 Find s univariate interpolatory subdivision schemes
❜❥, ❥ = ✶, . . . , s with scaling factors or “arity” σ❥;
2 Consider the tensor product
❜Σ :=
s
❜❥, ❜Σ(α) =
s
❜❥ (α❥) , α ∈ Zs, which is an interpolatory subdivision scheme for the diagonal scaling matrix Σ, i.e. ❜Σ(Σ·) = δ
Multiple multiresolution analysis
3 Set
❜▼ := ❜Σ(Θ−✶·) ❜▼ ▼ ❜▼ ③ ❜ ③
Multiple multiresolution analysis
3 Set
❜▼ := ❜Σ(Θ−✶·) Then: ❜▼ defines an interpolatory scheme for the dilation matrix ▼. ❜▼ ③ ❜ ③
Multiple multiresolution analysis
3 Set
❜▼ := ❜Σ(Θ−✶·) Then: ❜▼ defines an interpolatory scheme for the dilation matrix ▼. In terms of symbols: ❜♯
▼(③) = ❜♯ Σ
Multiple multiresolution analysis
We are considering the matrices ▼✵ := ✶ ✶ ✶ −✷
✷ −✶ ✶ −✷
where we make use of the shear matrices ❙❥ := ✶ ❥ ✵ ✶
❥ ∈ Z.
Multiple multiresolution analysis
It is easily verified that ❞❡t ▼✵ = ❞❡t ▼✶ = −✸ ▼✵
✶ ✷ ✶
✶✸ ▼✶ ✸ ▼✵ ▼✶
Multiple multiresolution analysis
It is easily verified that ❞❡t ▼✵ = ❞❡t ▼✶ = −✸ ▼✵ is anisotropic (eigenvalues:
✶ ✷
√ ✶✸
✸ ▼✵ ▼✶
Multiple multiresolution analysis
It is easily verified that ❞❡t ▼✵ = ❞❡t ▼✶ = −✸ ▼✵ is anisotropic (eigenvalues:
✶ ✷
√ ✶✸
√ ✸) ▼✵ ▼✶
Multiple multiresolution analysis
It is easily verified that ❞❡t ▼✵ = ❞❡t ▼✶ = −✸ ▼✵ is anisotropic (eigenvalues:
✶ ✷
√ ✶✸
√ ✸) ▼✵ and ▼✶ are jointly expanding so they define a reasonable subdivision scheme.
Multiple multiresolution analysis
Multiple multiresolution analysis
Multiple multiresolution analysis
Initial data M0 M0 M0 M0 M0 M0 M0 M0 M0 M0 M0 M0 M0 M0 M0
Sequence 0 0 0 0 0 0
M0 M0 M0 M0 M0 M0
Multiple multiresolution analysis
Initial data M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1 M1
Sequence 1 1 1 1 1 1
M1 M1 M1 M1 M1 M1
Multiple multiresolution analysis
Initial data M1 M0 M1 M1 M0 M1 M0 M1 M0 M1 M1 M0 M1 M0 M1
Sequence 0 1 0 1 0 1
M0 M1 M0 M1 M0 M1
Multiple multiresolution analysis
Initial data M0 M1 M0 M0 M1 M0 M1 M0 M1 M0 M0 M1 M0 M1 M0
Sequence 1 0 1 0 1 0
M1 M0 M1 M0 M1 M0
Multiple multiresolution analysis
In ”Multiple MRA” one considers functions of the form φη(▼ǫ · −α), α ∈ Zs. ▼
Multiple multiresolution analysis
In ”Multiple MRA” one considers functions of the form φη(▼ǫ · −α), α ∈ Zs. φη : limit function of subdivision ▼
Multiple multiresolution analysis
In ”Multiple MRA” one considers functions of the form φη(▼ǫ · −α), α ∈ Zs. φη : limit function of subdivision Role of ▼ǫ: scale & rotate
Multiple multiresolution analysis
In ”Multiple MRA” one considers functions of the form φη(▼ǫ · −α), α ∈ Zs. φη : limit function of subdivision Role of ▼ǫ: scale & rotate Can we get ”all rotations” by appropriate ǫ?
Multiple multiresolution analysis
In ”Multiple MRA” one considers functions of the form φη(▼ǫ · −α), α ∈ Zs. φη : limit function of subdivision Role of ▼ǫ: scale & rotate Can we get ”all rotations” by appropriate ǫ? → Slope resolution
Multiple multiresolution analysis
Action of: ▼✶▼✶ (blue), ▼✵▼✶ (red), ▼✶▼✵ (green), ▼✵▼✵ (cyan)
−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1
−5 −4 −3 −2 −1 1 2 3 4 5 −6 −4 −2 2 4 6
Multiple multiresolution analysis
Action of:
▼✶▼✶▼✶▼✶▼✶▼✶ (blue), ▼✵▼✶▼✵▼✶▼✵▼✶ (red), ▼✶▼✵▼✶▼✵▼✶▼✵ (green), ▼✵▼✵▼✵▼✵▼✵▼✵ (cyan)
−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1
−150 −100 −50 50 100 150 −150 −100 −50 50 100 150
Multiple multiresolution analysis
Can all directions, i.e., all lines through the origin, be generated by applying an appropriate ▼ǫ to a given reference line?
Multiple multiresolution analysis
Given the reference line ▲① := R ①, ① ∈ R✷ and a target line ▲② := R ②, ② ∈ R✷ we ask whether there exists ǫ ∈ Z∗
♠ such that
▲② ∼ ▼ǫ▲①.
Multiple multiresolution analysis
We represent lines by means of slopes, setting ▲(s) := R ✶ s
s ∈ R ∪ {±∞}, where s = ±∞ corresponds to (the same) vertical line. s ✵
✶ ✷
s ✵
♠
s s ▲ s ▼ ▲s ▼✵✶ ▼✵▼✶ ▼✵✶ ▼✶▼✵
Multiple multiresolution analysis
We represent lines by means of slopes, setting ▲(s) := R ✶ s
s ∈ R ∪ {±∞}, where s = ±∞ corresponds to (the same) vertical line. Theorem For each s ∈ (✵, ✶
✷), any s′ ∈ R and any δ > ✵ there exists
ǫ ∈ Z∗
♠ such that
|s′ − sǫ| < δ, ▲(sǫ) = ▼ǫ▲s. ▼✵✶ ▼✵▼✶ ▼✵✶ ▼✶▼✵
Multiple multiresolution analysis
We represent lines by means of slopes, setting ▲(s) := R ✶ s
s ∈ R ∪ {±∞}, where s = ±∞ corresponds to (the same) vertical line. Theorem For each s ∈ (✵, ✶
✷), any s′ ∈ R and any δ > ✵ there exists
ǫ ∈ Z∗
♠ such that
|s′ − sǫ| < δ, ▲(sǫ) = ▼ǫ▲s. ▼✵✶ ▼✵▼✶ ▼✵✶ ▼✶▼✵
Multiple multiresolution analysis
We represent lines by means of slopes, setting ▲(s) := R ✶ s
s ∈ R ∪ {±∞}, where s = ±∞ corresponds to (the same) vertical line. Theorem For each s ∈ (✵, ✶
✷), any s′ ∈ R and any δ > ✵ there exists
ǫ ∈ Z∗
♠ such that
|s′ − sǫ| < δ, ▲(sǫ) = ▼ǫ▲s. Indeed even combinations of ▼✵✶ = ▼✵▼✶ and ▼✵✶ = ▼✶▼✵ are sufficient to satisfy the claim of the theorem.
Multiple multiresolution analysis
Smith factorizations of ▼✵, ▼✶: ▼✵ = ✹ ✶ ✶ ✵ ✶ ✸ ✶ −✷ −✶ ✸
▼✶ = ✺ ✶ ✶ ✵ ✶ ✸ ✶ −✷ −✶ ✸
Multiple multiresolution analysis
Possible choices for the ternary interpolatory schemes piecewise linear interpolant: ❜✷ = ✶ ✸ (. . . , ✵, ✶, ✷, ✸, ✷, ✶, ✵, . . . ) ❜✷ ✶ ✽✶ ✵ ✹ ✺ ✵ ✸✵ ✻✵ ✽✶ ✻✵ ✸✵ ✵ ✺ ✹ ✵
Multiple multiresolution analysis
Possible choices for the ternary interpolatory schemes piecewise linear interpolant: ❜✷ = ✶ ✸ (. . . , ✵, ✶, ✷, ✸, ✷, ✶, ✵, . . . ) four point scheme based on local cubic interpolation ❜✷ = ✶ ✽✶ (. . . , ✵, −✹, −✺, ✵, ✸✵, ✻✵, ✽✶, ✻✵, ✸✵, ✵, −✺, −✹, ✵, . . . )
Multiple multiresolution analysis
The schemes are obtained from ❜♯
▼(③) = ❜♯ Σ
which result in the following two symbols ❆♯
✶(③✶, ③✷) = ③−✷ ✶
✸
✶
✷ , ❆♯
✷(③✶, ③✷) = −③−✺ ✶
✽✶
✶
✹ ✹③✷
✶ − ✶✶③✶ + ✹
Multiple multiresolution analysis
Theorem Suppose: ❜❥, ❥ = ✶, . . . , s define univariate subdivision schemes with scaling factors σ❥ ≥ ✶ ❙❜❥✶ = ✶. ❜▼ ▼ ❙❇ ❉
✶❙❜
❙❇ ✶ ❙❇ ❧✐♠
♥
s✉♣
❝ ✶
❙♥
❇
❝
✶ ♥
❉ ❉ ❝ ❝ ❝ ❝
❥
❝ ❥ ✶ s
Theorem Suppose: ❜❥, ❥ = ✶, . . . , s define univariate subdivision schemes with scaling factors σ❥ ≥ ✶ ❙❜❥✶ = ✶. Then ❜▼ is a convergent subdivision scheme with dilation matrix ▼ iff the vector scheme ❙❇Σ defined by ∇❉(Θ′Θ)−✶❙❜Σ = ❙❇Σ∇ satisfies ✶ > ρ∞ (❙❇Σ | ∇) := ❧✐♠
♥→∞ s✉♣ ∇❝≤✶
❇Σ∇❝
where ❉Λ is the dilation operator ❉Λ❝ = ❝(Λ·) ∇ is the forward difference operator ∇❝ = [❝(· + ǫ❥) − ❝ : ❥ = ✶, . . . , s]
❆♯
✶(③✶, ③✷) = ③−✷
✶
✸ (✶ + ③✶ + ③✷ ✶) ✷, ▼✵ =
✶ ✶ ✶ −✷
2 −2 2 1
Multiple multiresolution analysis
❆♯
✶(③✶, ③✷) = ③−✷
✶
✸ (✶ + ③✶ + ③✷ ✶) ✷, ▼✶ =
✷ −✶ ✶ −✷
2 −2 2 1
Multiple multiresolution analysis
❆♯
✷(③✶, ③✷) = − ③−✺
✶
✽✶ (✶ + ③✶ + ③✷ ✶) ✹ (✹③✷ ✶ − ✶✶③✶ + ✹),
▼✵ = ✶ ✶ ✶ −✷
5 −5 5 1
Multiple multiresolution analysis
❆♯
✷(③✶, ③✷) = − ③−✺
✶
✽✶ (✶ + ③✶ + ③✷ ✶) ✹ (✹③✷ ✶ − ✶✶③✶ + ✹),
▼✶ = ✷ −✶ ✶ −✷
5 −5 5 1
Multiple multiresolution analysis
❆♯
✶(③✶, ③✷) = ③−✷
✶
✸ (✶ + ③✶ + ③✷ ✶) ✷ and ▼✵
Analysis
❋✵ ❋✶ ❋✷ ✵ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ − ✷
✸
✶ ✵ − ✶
✸
✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ − ✶
✸
✵ ✶ − ✷
✸
✵
Synthesis
✵ ✵ ✵ ✵ ✵
✶ ✸ ✷ ✸
✶
✷ ✸ ✶ ✸
✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✶ ✵
Multiple multiresolution analysis
❆♯
✶(③✶, ③✷) = ③−✷
✶
✸ (✶ + ③✶ + ③✷ ✶) ✷ and ▼✶
Analysis
❋✵ ❋✶ ❋✷ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ − ✶
✸
✵ ✶ − ✷
✸
✵ ✵ ✵ ✵ ✵ ✵ ✵ − ✷
✸
✶ ✵ − ✶
✸
✵ ✵ ✵ ✵ ✵
Synthesis
✵ ✵ ✵ ✵ ✵
✶ ✸ ✷ ✸
✶
✷ ✸ ✶ ✸
✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✵
Multiple multiresolution analysis
c
M0c1
M0 c2 00 d2 00 M1c2
01d2
01d1
M1c1
1 M0c2
10d2
10 M1c2
11d2
11d1
1c
M0c1
M0c2
00d2
00 M1 c2 01 d2 01d1
M1c1
1 M0c2
10d2
10 M1c2
11d2
11d1
1c
M0c1
M0c2
00d2
00 M1c2
01d2
01d1
M1c1
1 M0 c2 10 d2 10 M1c2
11d2
11d1
1c
M0c1
M0c2
00d2
00 M1c2
01d2
01d1
M1c1
1 M0c2
10d2
10 M1 c2 11 d2 11d1
1 c
M0c1
M0 c2 00 d2 00 M1c2
01d2
01d1
M1c1
1 M0c2
10d2
10 M1c2
11d2
11d1
1c
M0c1
M0c2
00d2
00 M1 c2 01 d2 01d1
M1c1
1 M0c2
10d2
10 M1c2
11d2
11d1
1c
M0c1
M0c2
00d2
00 M1c2
01d2
01d1
M1c1
1 M0 c2 10 d2 10 M1c2
11d2
11d1
1c
M0c1
M0c2
00d2
00 M1c2
01d2
01d1
M1c1
1 M0c2
10d2
10 M1 c2 11 d2 11d1
1 Multiple multiresolution analysis