Convergence of uniform subdivision Amos Ron Erice, Trapani, - PowerPoint PPT Presentation

Convergence of uniform subdivision Amos Ron Erice, Trapani, Sicilia, Italia, Europa September, 2013 Cascade and subdivision defined

The challenge: provide a spectral characterization of convergence of uniform subdivision in the max-norm. Cascade and subdivision defined

The challenge: provide a spectral characterization of convergence of uniform subdivision in the max-norm. Spectral means: using the spectral structure of a finite number of linear endomorphisms, each of which with a finite rank. Cascade and subdivision defined

The challenge: provide a spectral characterization of convergence of uniform subdivision in the max-norm. Spectral means: using the spectral structure of a finite number of linear endomorphisms, each of which with a finite rank. Spectral structure means: eigenvalues and eigenvectors. Cascade and subdivision defined

The challenge: provide a spectral characterization of convergence of uniform subdivision in the max-norm. Spectral means: using the spectral structure of a finite number of linear endomorphisms, each of which with a finite rank. Spectral structure means: eigenvalues and eigenvectors. Generalizations: p < ∞ , vector subdivision, fast convergence, infinite mask, non-dyadic dilations... Cascade and subdivision defined

mask and dilation Z d / 2 is a finite mask. Considered as a discrete finite a ∈ C Z measure D is dyadic dilation: ( D f )( t ) = f ( 2 t ) . Cascade and subdivision defined The cascade operator Some necessary condition for convergence

mask and dilation Z d / 2 is a finite mask. Considered as a discrete finite a ∈ C Z measure D is dyadic dilation: ( D f )( t ) = f ( 2 t ) . Definition: the cascade operator C C : f �→ D f ∗ a . A refinable function φ is a fixed-point of C : C φ = φ. Cascade and subdivision defined The cascade operator Some necessary condition for convergence

mask and dilation Z d / 2 is a finite mask. Considered as a discrete finite a ∈ C Z measure D is dyadic dilation: ( D f )( t ) = f ( 2 t ) . Definition: the cascade operator C C : f �→ D f ∗ a . A refinable function φ is a fixed-point of C : C φ = φ. Question Given a compactly supported g , do we have � C k g − φ � ∞ → 0 ? Cascade and subdivision defined The cascade operator Some necessary condition for convergence

Necessary conditions Convergence of cascade: necessary conditions Each of the following conditions is necessary: g , φ ∈ C α , α ≥ 0. � Z d a ( j ) = 1, γ ∈ { 0 , 1 } d . j ∈ γ + 2 Z g − φ has zero mean. The PSI space S ( g ) provides approximation order 1 in the ∞ -norm, viz., for each sufficiently smooth f , as k → ∞ , dist L ∞ ( f , D k S ( g )) = O ( 2 − k ) . G 0 is the collection of compact support g that satisfy the above. Cascade and subdivision defined The cascade operator Some necessary condition for convergence

Subdivision: definition and convergence Definition: the space Q k Z d / 2 k Q k := C Z Definition: The subdivision operator S k , convergence λ �→ D k − 1 a ∗ S k − 1 λ. S k : Q 0 → Q k , Convergence: �D k g ∗ S k δ − φ � ∞ → 0 , ∀ g ∈ G 0 . Cascade and subdivision defined The cascade operator Some necessary condition for convergence

Subdivision: definition and convergence Definition: the space Q k Z d / 2 k Q k := C Z Definition: The subdivision operator S k , convergence λ �→ D k − 1 a ∗ S k − 1 λ. S k : Q 0 → Q k , Convergence: �D k g ∗ S k δ − φ � ∞ → 0 , ∀ g ∈ G 0 . C k g = D k g ∗ S k δ k k � � D j − 1 + D k = 1 + D D j − 1 . j = 1 j = 1 Cascade and subdivision defined The cascade operator Some necessary condition for convergence

The L 2 -case is spectral The Transfer operator T With f a trig. pol., and τ := | � a | 2 , � T : f �→ D − 1 ( ( τ f )( · + πγ )) . γ ∈{ 0 , 1 } d Cascade and subdivision defined The cascade operator Some necessary condition for convergence

The L 2 -case is spectral The Transfer operator T a | 2 , With f a trig. pol., and τ := | � � T : f �→ D − 1 ( ( τ f )( · + πγ )) . γ ∈{ 0 , 1 } d The transfer operator encodes L 2 -properties of a and φ , including a complete characterization of the convergence of cascade: essentially it need to have a unique dominant eigenvalue (acting on any large enough set of trig. pol.). Cascade and subdivision defined The cascade operator Some necessary condition for convergence

The L 2 -case is spectral The Transfer operator T With f a trig. pol., and τ := | � a | 2 , � T : f �→ D − 1 ( ( τ f )( · + πγ )) . γ ∈{ 0 , 1 } d The transfer operator encodes L 2 -properties of a and φ , including a complete characterization of the convergence of cascade: essentially it need to have a unique dominant eigenvalue (acting on any large enough set of trig. pol.). The transfer operator also encodes the L 2 -smoothness of φ . Cascade and subdivision defined The cascade operator Some necessary condition for convergence

The L 2 -case is spectral The Transfer operator T With f a trig. pol., and τ := | � a | 2 , � T : f �→ D − 1 ( ( τ f )( · + πγ )) . γ ∈{ 0 , 1 } d The transfer operator encodes L 2 -properties of a and φ , including a complete characterization of the convergence of cascade: essentially it need to have a unique dominant eigenvalue (acting on any large enough set of trig. pol.). The transfer operator also encodes the L 2 -smoothness of φ . The transfer cannot be used (obvious reasons) for other norms. Cascade and subdivision defined The cascade operator Some necessary condition for convergence

Characterization: joint spectral radius There are characterizations in terms of notion of joint spectral radius. Cascade and subdivision defined The cascade operator Some necessary condition for convergence

Characterization: joint spectral radius There are characterizations in terms of notion of joint spectral radius. Despite of its name, the joint spectral radius is joint but not spectral. Cascade and subdivision defined The cascade operator Some necessary condition for convergence

Convergence depends on dependence relations The space K φ K φ := { λ ∈ Q 0 : φ ∗ λ = 0 . } Convergence of cascade: (more or less) we need that S k ( K φ ) → 0 . Cascade and subdivision defined The cascade operator Some necessary condition for convergence

Convergence depends on dependence relations The space K φ K φ := { λ ∈ Q 0 : φ ∗ λ = 0 . } Convergence of cascade: (more or less) we need that S k ( K φ ) → 0 . Special case: If S k ( K φ ) = 0 for some k , then convergence. Cascade and subdivision defined The cascade operator Some necessary condition for convergence

Convergence depends on dependence relations The space K φ K φ := { λ ∈ Q 0 : φ ∗ λ = 0 . } Convergence of cascade: (more or less) we need that S k ( K φ ) → 0 . Special case: If S k ( K φ ) = 0 for some k , then convergence. Special case: If dim K φ < ∞ , then spectral. Cascade and subdivision defined The cascade operator Some necessary condition for convergence

Convergence depends on dependence relations The space K φ K φ := { λ ∈ Q 0 : φ ∗ λ = 0 . } Convergence of cascade: (more or less) we need that S k ( K φ ) → 0 . Special case: If S k ( K φ ) = 0 for some k , then convergence. Special case: If dim K φ < ∞ , then spectral. Special case: box splines, de Boor-R If φ is a box spline, then spectral. Cascade and subdivision defined The cascade operator Some necessary condition for convergence