Smoothness of Nonlinear and Non-Separable Subdivision Schemes

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Smoothness of Nonlinear and Non-Separable Subdivision Schemes

Basarab Matei, Sylvain Meignen, Anastasia Zakharova

To cite this version:

Basarab Matei, Sylvain Meignen, Anastasia Zakharova. Smoothness of Nonlinear and Non-Separable Subdivision Schemes. Asymptotic Analysis, IOS Press, 2011, 74 (3-4), pp.229-247. �10.3233/ASY- 2011-1052�. �hal-00452897�

Smoothness of Nonlinear and Non-Separable Subdivision Schemes

Basarab Matei^(a), Sylvain Meignen ^∗,(b) and Anastasia Zakharova^(b)

(a)

LAGA Laboratory, Paris XIII University, France, Tel:0033-1-49-40-35-71

FAX:0033-4-48-26-35-68

E-mail: [email protected]

(b)

LJK Laboratory, University of Grenoble, France Tel:0033-4-76-51-43-95

FAX:0033-4-76-63-12-63

E-mail: [email protected]

E-mail: [email protected]

Abstract

We study in this paper nonlinear subdivision schemes in a multivariate set- ting allowing arbitrary dilation matrix. We investigate the convergence of such iterative process to some limit function. Our analysis is based on some conditions on the contractivity of the associated scheme for the differences.

In particular, we show the regularity of the limit function, inL^p and Sobolev spaces.

Keywords: Nonlinear subdivision scheme, convergence of subdivision schemes, box splines

1. Introduction

Subdivision schemes have been the subject of active research in recent years. In such algorithms, discrete data are recursively generated from coarse to fine by means of local rules. When the local rules are independent of the data, the underlying refinement process is linear. This case is extensively studied in literature. The convergence of this process and the existence of the limit function was studied in [2] and [7] when the scales are dyadic. When the scales are related to a dilation matrixM, the convergence to a limit function inL^p was studied in [8] and generalized to Sobolev spaces in [5] and [13]. In the linear case, the stability is a consequence of the smoothness of the limit function.

The nonlinearity arises naturally when one needs to adapt locally the re- finement rules to the data such as in image or geometry processing. Nonlin- ear subdivision schemes based on dyadic scales were originally introduced by Harten [9][10] through the so-called essentially non-oscillatory (ENO) meth- ods. These methods have recently been adapted to image processing into essentially non-oscillatory edge adapted (ENO-EA) methods. Different ver- sions of ENO methods exist either based on polynomial interpolation as in [6][1] or in a wavelet framework [3], corresponding to interpolatory or non- interpolatory subdivision schemes respectively.

In the present paper, we study nonlinear subdivision schemes associated to dilation matrix M. After recalling the definitions on nonlinear subdivi- sion schemes in that context, we give sufficient conditions for convergence in Sobolev and L^p spaces.

2. General Setting

2.1. Notations

Before we start, let us introduce some notations that will be used through- out the paper. We denote #Q the cardinal of the set Q. For a multi-index µ = (µ1, µ2,· · · , µd) ∈ N^d and a vector x = (x1, x2,· · · , xd) ∈ R^d we define

|µ|= Pd i=1

µi,µ! = Qd i=1

µi! andx^µ = Qd i=1

xiµi.

For two multi-index m, µ∈N^d we also define



 µ m



=



 µ₁ m1



· · ·



 µ_d md



.

Let ℓ(Z^d) be the space of all sequences indexed by Z^d. The subspace of bounded sequences is denoted by ℓ^∞(Z^d) and kuk_ℓ^∞₍Z^d) is the supremum of {|u_k| : k ∈ Z^d}. We denote ℓ⁰(Z^d) the subspace of all sequences with finite support (i.e. the number of non-zero components of a sequence is finite).

As usual, let ℓ^p(Z^d) be the Banach space of sequences u on Z^d such that kuk_ℓ^p₍Z^d) <∞, where

kukℓ^p(Zd) := X

k∈Zd

|uk|^p

!¹_p

for 1≤p <∞.

As in the discrete case, we denote by L^p(R^d) the space of all measurable functions f such that kfk_L^p₍R^d)<∞, where

kfk_L^p₍R^d) :=

Z

R^d

|f(x)|^pdx ¹_p

for 1≤p <∞ and kfk_L^∞₍R^d) is the essential supremum of |f| onR^d.

Let µ ∈ N^d be a multi-index, we define ∇^µ the difference operator

∇^µ₁¹· · · ∇^µ_d^d, where ∇^µ_j^j is the µjth difference operator with respect to the

jth coordinate of the canonical basis. We define D^µ as D₁^µ1· · ·D^µ_d^d, where Dj is the differential operator with respect to thejth coordinate of the canon- ical basis. Similarly, for a vectorx∈R^dthe differential operator with respect to x is denoted by Dx.

A matrix M is called a dilation matrix if it has integer entries and if

n→∞lim M⁻ⁿ = 0. In the following, the invertible dilation matrix is always denoted by M and m stands for |det(M)|.

For a dilation matrix M and any arbitrary function Φ we put Φj,k(x) = Φ(M^jx−k).

We also recall that a compactly supported function Φ is calledL^p-stable if there exist two constants C₁, C₂ >0 satisfying

C1kck_ℓ^p₍Z^d) ≤ kX

k∈Z^d

ckΦ(x−k)k_L^p₍R^d) ≤C2kck_ℓ^p₍Z^d).

Finally, for two positive quantities A and B depending on a set of param- eters, the relation A <∼ B implies the existence of a positive constant C, independent of the parameters, such that A ≤ CB. Also A ∼ B means A <∼ B and B <∼ A.

2.2. Local, Bounded and Data Dependent Subdivision Operators, Uniform Convergence Definition

In the sequel, we will consider the general class of local, bounded and data dependent subdivision operators which are defined as follows:

Definition 1. Forv ∈ℓ^∞(Z^d), a local, bounded and data dependent subdivi- sion operator is defined by

S(v)wk =X

l∈Z^d

ak−M l(v)wl, (1)

for any w in ℓ^∞(Z^d) and where the real coefficients a_{k−M l}(v) ∈ R are such that

ak−M l(v) = 0, if kk−Mlk_ℓ^∞₍^Zd₎ > K (2) for a fixed constant K. The coefficients ak(v) are assumed to be uniformly bounded by a constant C, i.e. there is C >0 independent of v such that:

|ak(v)| ≤C.

Note that the definition of the coefficients depends on some sequencev, while S(v) acts on the sequence w.

Note also that, from (1) and (2) the new defined value S(v)wk depends only on those values l satisfying kk−Mlk_ℓ^∞₍Zd)> K. The subdivision oper- ator in this sense is local.

To simplify, in what follows a data dependent subdivision operator is an operator in the sense of Definition 1. With this definition, the associated subdivision scheme is the recursive action of the data dependent rule Sv = S(v)v on an initial set of data v⁰, according to:

v^j =Sv^j−1 =S(v^j−1)v^j−1, j ≥1. (3) 2.3. Polynomial Reproduction for Data Dependent Subdivision Operators

The study of the convergence of data dependent subdivision operators will involve the polynomial reproduction property. We recall the definition of the space P_N of polynomials of total degree N:

P_N :={P;P(x) = X

|µ|≤N

aµx^µ}.

With these notations, the polynomial reproduction properties read:

Definition 2. Let N ≥0 be a fixed integer.

1. The data dependent subdivision operatorShas the property of reproduc- tion of polynomials of total degree N if for all u∈ℓ^∞(Z^d) andP ∈P_N there exists P˜ ∈ P_N with P −P˜ ∈P_N−1 such that S(u)p= ˜p where p and p˜are defined by pk =P(k) and p˜k= ˜P(M⁻¹k).

2. The data dependent subdivision operator S has the property of exact reproduction of polynomials of total degree N if for all u ∈ ℓ^∞(Z^d) and P ∈ P_N, S(u)p = ˜p where p and p˜are defined by pk = P(k) and

˜

p_k =P(M⁻¹k).

Remark:

The case N = 0 is the so-called ”constant reproduction property”. For a data dependent subdivision operator defined as in (1), the constant repro- duction property reads P

k∈Z^d

ak−M l(v) = 1,for all v ∈ℓ^∞(Z^d).

3. Definition of Schemes for the Differences

Another ingredient for our study is the schemes for the differences asso- ciated to the data dependent subdivision operator. The existence of schemes for the differences is obtained by using the polynomial reproduction property of the data dependent subdivision operator.

Let us denote ∆^l = (∇^µ,|µ| = l) and then state the following result on the existence of schemes for the differences:

Proposition 1. Let S be a data dependent subdivision operator which re- produces polynomials up to total degree N. Then for 1 ≤ l ≤ N + 1 there

exists a data dependent subdivision rule S_l with the property that for all v,w in ℓ^∞(Z^d),

∆^lS(v)w:=S_l(v)∆^lw Proof:

Let l be an integer such that 1 ≤ l ≤N + 1. By using the definition of

∇^µ with |µ|=l, we write:

∇^µS(v)w_k=∇^µ₁¹· · · ∇^µ_d^dS(v)w_k. From the definition of S(v)w we infer that

∇^µS(v)wk=

max(µ1,···,µd)

X

m1,···,md=0

(−1)^l



 µ m



X

p∈Zd

ak−m·e−M p(v)wp,

where we have used the notationm·e=m₁e₁+· · ·+m_de_d. Straightforward computations give

∇^µS(v)wk = X

p∈Z^d

wp

max(µ1,···,µd)

X

m1,···,md=0

(−1)^l



 µ m



ak−m·e−M p(v)

= X

p∈Z^d

wpfk,p(v, µ). (4)

Let us clarify the definition offk,p(v, µ).Since the data dependent subdivision operator is local we haveak−M p(v) = 0 for any datav ∈ℓ^∞(Z^d) and any index k such that kk−Mpk_ℓ^∞₍^Zd₎ > K. Now by puttingk =ε+Mn, we get that fk,p(v, µ) is defined for pin the set

V^µ(k) :=

p:kn−p+M⁻¹(ε−m·e)k∞≤KkM⁻¹k∞, 0≤mi ≤µi∀i Then, we define V(k) := {p:kk−Mpk∞ ≤K}. Since the data dependent subdivision scheme reproduces polynomials up to total degree N, we have

for any |ν|=r≤N: X

p∈V(k)

ak−M p(v)p^ν =Pν(k) for all k ∈Z^d, (5) where Pν is a polynomial of total degree r. By tacking the differences of order |ν^′|=r+ 1 in (5) we get

X

p∈V^ν′(k)

fk,p(v, ν^′)p^ν = 0.

Note that the above equality is true for any ν such that |ν|=r. We deduce that (fk,p(v, ν^′))_k ∈Z^d is orthogonal to (p^q)_p∈Vν′(k) where |q| ≤r. Note that n(∇^νδn−β)_n∈Vν′(k),|ν|=r+ 1, β∈Z^do

spans (p^q)_p∈Vν′

(k) and we may thus write for any p∈V^ν′(k):

f_k,p(v, ν^′) = X

|ν|=r+1

X

r∈Zd

c^µ_k,r(v)∇^µδ_p−r.

Now, by using (4) we obtain for any |µ| ≤N + 1:

∇^µS(v)wk = X

p∈V^µ(k)

wp

X

|ν|=l

X

r∈Z^d

c^ν_k,r(v)∇^νδp−r

= X

p∈V^µ(k)

X

|ν|=l

c^ν_k,r(v)∇^νwp

If we now make µ vary, we obtain the desired relation.

Now that we have proved the existence of schemes for the differences, we introduce the notion of joint spectral radius for these schemes, which is a generalization of the one dimensional case which can be found in [15].

Definition 3. Let S(v) : ℓ^p(Z^d) → ℓ^p(Z^d) be a data dependent subdivision operator such that the difference operators Sl(v) : ℓ^p(Z^d)ql

→ ℓ^p(Z^d)ql

,

with q_l = #{µ,|µ| = l} exists for l ≤ N + 1. Then, to each operator S_l, l = 0,· · · , N + 1 (putting S0 =S) we can associate the joint spectral radius given by

ρp,l(S) := inf

j≥1k(Sl)^jk

1 j

ℓ^p(Zd)^ql.

In other words, ρ_p(S) is the infimum of all ρ > 0 such that for all v ∈ ℓ^p(Z^d), one has

k∆^lS^jvk_ℓ^p₍Z^d)^ql <

∼ ρ^jk∆^lvk_ℓ^p₍Z^d)^ql, (6) for all j ≥0.

Remark: Let us define a set of vectors{x¹,· · · , xⁿ}such that [x¹,· · · , xⁿ]Zⁿ = Z^d, n ≥ d (i.e. a set such that the linear combinations of its elements with coefficients in Z spans Z^d). We use the bold notation in the definition of the set so as to avoid the confusion with the coordinates of vector x. Then, consider the differences in the directionsx¹,· · · , xⁿ. One can show that there exists a scheme for that differences which we call ˜S_l for l ≤N + 1 provided the data dependent subdivision operator reproduces polynomials up to de- gree N (the proof is similar to that using the canonical directions). If we denote by ˜∆^l the difference operator of order l in the directions x¹,· · ·, xⁿ, one can see that k∆˜^lvk_ℓ^p₍Z^d)^ql˜ ∼ k∆^lvk_ℓ^p₍Z^d)^ql for all v in ℓ^p(Z^d) and where

˜

ql = #{µ,|µ|= l, µ= (µi)i=1,···,n}.Then, following (6), one can deduce that the joint spectral radius of ˜Sl is the same as that ofSl.

4. Convergence in L^p spaces

In the following, we study the convergence of data dependent subdivision schemes in L^p which corresponds to the following definition:

Definition 4. The subdivision scheme v^j = Sv^j−1 converges in L^p(R^d), if for every set of initial control points v⁰ ∈ ℓ^p(Z^d), there exists a non-trivial function v in L^p(R^d), called the limit function, such that

j→∞lim kvj −vk_L^p₍R^d) = 0.

wherevj(x) = P

k∈Z^d

v_k^jφj,k(x) with φ(x) = Q^d

i=1

max(0,1− |xi|).

4.1. Convergence in the Linear Case

When S is independent of v, the rule (1) defines a linear subdivision scheme:

Svk =X

l∈Zd

ak−M lvl.

If the linear subdivision scheme converges for anyv ∈ℓ^p(Z^d) to some function in L^p(R^d) and if there exists v⁰ such that lim

j→+∞v^j 6= 0, then {ak, k ∈ Z^d} determines a unique continuous compactly supported function Φ satisfying

Φ(x) = X

k∈Zd

a_kΦ(Mx−k) and X

k∈Zd

Φ(x−k) = 1.

Moreover, v(x) = P

k∈Z^d

v_k⁰Φ(x−k).

4.2. Convergence of Nonlinear Subdivision Schemes in L^p Spaces

In the sequel, we give a sufficient condition for the convergence of non- linear subdivision schemes in L^p(R^d). This result will be a generalization of the existing result in the linear context established in [8] and only uses the operator S1.

Theorem 1. Let S be a data dependent subdivision operator that reproduces the constants. If ρp,1(S)< m^p1, then Sv^j converges to a L^p limit function.

Proof: Let us consider

vj(x) := X

k∈Z^d

v_k^jφj,k(x), (7)

where φ(x) = Q^d

i=1

max(0,1− |x_i|) is the hat function. With this choice, one can easily check that P

k∈Zd

φ(x−k) = 1. Let ρp,1(S)< ρ < m^1p, it follows that k∆¹v^jk_(ℓ^p₍Zd))^d <

∼ ρ^jk∆¹v⁰k_(ℓ^p₍Zd))^d,

since q1 =d. We now show that the sequence vj is a Cauchy sequence inL^p: v_j+1(x)−v_j(x) = X

k∈Zd

v^j+1_k φ_j+1,k(x)−X

p∈Zd

v_p^jφ_j,p(x)

= X

k∈Zd

X

p∈Zd

(v_k^j+1−v_p^j)φj+1,k(x)φj,p(x) where we have used P

k∈Z^d

φ(· −k) = 1. Now, since the subdivision operator reproduces the constants:

vj+1(x)−vj(x) = X

p∈Z^d

X

k∈Z^d

X

l∈Z^d

ak−M l(v^j)(v^j_l −v_p^j)φj+1,k(x)φj,p(x).

Note that X

l∈Z^d

ak−M l(v^j)(v_l^j−v^j_p) = X

l∈Z^d

ak−M l(v^j)v_l^j−v^j_p =X

l∈Z^d

(ak−M l(v^j)−δp−l)v_l^j. Since P

l∈Zd

ak−M l(v^j)−δp−l = 0,

∇iδl−β, l∈ {F(k)∪ {p}}, β ∈Z^d, i= 1,· · · , d spans (a_{k−M l}−δ_p−l)_{l∈{F(k)∪{p}}}. This enables us to write:

vj+1(x)−vj(x) = X

p∈Z^d

X

k∈Z^d

X

l∈V(k)S {p}

Xd

i=1

dⁱ_k,p,l∇iv_l^jφj+1,k(x)φj,p(x).

Since | P

k∈Z^d

φ_j+1,k(x)|= 1 following the same argument as in Theorem 3.2 of [8], we may write:

kvj+1−vjk_L^p₍Rd) <

∼ m^−jp max

1≤i≤dk∇iv^jk_ℓ^p₍Zd)

<∼ m^−jpk∆¹v^jk_ℓ^p₍Zd)

∼ ( ρ

m^p1)^jk∆¹v⁰k_ℓ^p₍Z^d) (8) which proves that vj converges in L^p, since ρ < m^1p. Note that, for p =∞, we obtain that the limit function is continuous.

Furthermore, the above proof is valid for any function Φ0 satisfying the prop- erty of partition of unity when p=∞. In general, we could show, following Theorem 3.4 of [8], that the limit function in L^p is independent of the choice of a continuous and compactly supported Φ0.

4.3. Uniform Convergence of the Subdivision Schemes to C^s functions (s <

1)

We are now ready to establish a sufficient condition for theC^ssmoothness of the limit function with s <1.

Theorem 2. Let S(v) be a data dependent subdivision operator which re- produces the constants. If the scheme for the differences satisfies ρp,1(S) <

m^−s+1p, for some 0 < s < 1 then Sv^j is convergent in L^p and the limit function is C^s .

Proof: First, the convergence in L^p is a consequence of ρp,1(S) < m^1p. In order to prove that the limit function v be in C^s, it suffices to evaluate

|v(x)−v(y)|for kx−yk_∞ ≤1. Letj be such thatm^−j−1≤ kx−yk_∞ ≤m^−j. We then write :

|v(x)−v(y)| ≤ |v(x)−v_j(x)|+|v(y)−v_j(y)|+|v_j(x)−v_j(y)|

≤ 2kv−v_jk_L^∞₍Rd)+|v_j(x)−v_j(y)|

Note that (8) implies that kv −vjk_L^∞₍R^d) <

∼ ρ^jk∆¹v⁰k_ℓ^∞₍Z^d). Since vj is absolutely continuous, it is almost everywhere differentiable, so putting y= x+M^−jh, with h= (hi)i=1,···,d satisfying khk∞ ≤1 we get:

|v_j(x+M^−jh)−v_j(x)| ≤ |v_j(x+M^−jh)−v_j(x+M^−j(h−h_de_d))|

+|v_j(x+M^−j(h−h_de_d))−v_j(x+M^−j(h−h_de_d−h_d−1e_d−1))|

+· · ·+|v_j(x+M^−j(h₁e₁))−v_j(x)|

Then, using a Taylor expansion we remark that, there exists θ ∈]−hd, hd[ such that:

|vj(x+M^−jh)−vj(x+M^−j(h−hded))| = X

k∈Z^d

v_k^jhdDdφ(M^jx+h−k+θded) If we denote Ψd(x) = Φ(x1)· · ·Φ(xd−1)Ψ(xd), where Ψ is the characteristic function of [0,1] and Φ(x_i) = max(0,1− |x_i|), we may write:

|vj(x+M^−jh)−vj(x+M^−j(h−hded))| ∼ X

k∈Z^d

∇dv_k^jhdΨd(y) whereyi = (M^−jx+h−k+θded)i ifi < dandyd= 2(M^−jx+h+θded)d−kd

(we have used the fact that the differential of the hat function Φ is the Haar wavelet). Iterating the procedure for other differences in the sum, we get:

|vj(x+M^−jh)−vj(x)| <

∼

Xd

i=1

k∇iv^jkℓ^∞(Zd) <

∼ k∆¹v^jkℓ^∞(Zd).

Combining these results we may finally write:

|v(x)−v(y)| ≤ |v(x)−vj(x)|+|v(y)−vj(y)|+|vj(x)−vj(y)|

≤ 2kv−vjk_L^∞₍Rd)+|vj(x)−vj(y)|

<∼ ρ^jk∆¹v⁰k_ℓ^∞₍Zd)+k∆¹v^jk_ℓ^∞₍Zd)

<∼ ρ^jk∆¹v⁰k_ℓ^∞₍Z^d) <

∼ kx−yk^s_∞ with s <−log(ρ∞,1)/logm.

5. Examples of Bidimensional Subdivision Schemes

In the first part of this section, we construct an interpolatory subdivision scheme having the dilation matrix the hexagonal matrix which is:

M =



 2 1 0 −2



,

For the hexagonal dilation matrix, the coset vectors are ε₀ = (0,0)^T, ε₁ = (1,0)^T, ε2 = (1,−1)^T, ε3 = (2,−1)^T. The coset vector εi,i = 0,· · · ,3 of M defines a partition of Z² as follows:

Z² = [3

i=0

Mk+εi, k ∈Z² .

The discrete data at the level j, v^j is defined on the grid Γ^j = M^−jZ², the value v_k^j is then associated to the location M^−jk. We now define our bi- dimensional interpolatory subdivision scheme based on the data dependent subdivision operator which acts from the coarse grid Γ^j−1 to the fine grid grid Γ^j. To this end, we will compute v^j at the different coset points on the fine grid Γ^j using the existing values v^j−1 of the coarse grid Γ^j−1, as follows:

for the first coset vector ε₀ = (0,0)^T we simply put v^j_{M k+ε0} = v^j−1_k , for the coset vectors εi, i = 1,· · · ,3. the value v_{M k+ε}^j _i, i = 1,· · · ,3 is defined by affine interpolation of the values on the coarse grid. To do so, we define four different stencils on Γ^j−1 as follows:

V_k^j,1 = {M^−j+1k, M^−j+1(k+e1), M^−j+1(k+e2)}, V_k^j,2 = {M^−j+1k, M^−j+1(k+e₂), M^−j+1(k+e₁+e₂)},

W_k^j,1 = {M^−j+1(k+e1), M^−j+1(k+e2), M^−j+1(k+e1+e2)}, W_k² = {M^−j+1k, M^−j+1(k+e1), M^−j+1(k+e1+e2)}.

We determine to which stencils each point of Γ^j belongs to, and we then define the prediction as its barycentric coordinates. Since we use an affine interpolant we have:

v_{M k}^j =v^j−1_k and v_{M k+ε}^j ₁ = 1

2v_k^j−1+1

2v_k+e^j−1₁. (9) To compute the rules for the coset point ε₂, V_k¹ or V_k² can be used leading respectively to:

v_{M k+ε}^j,1 ₂ = 1

4v_k+e^j−1₁ +1

2v_k+e^j−1₂ + 1 4v^j−1_k v_{M k+ε}^j,2 ₂ = 1

2v_k^j−1+1

4v_k+e^j−1₂ +1

4v_k+e^j−1_1+e2. (10) When one considers the rules for the coset point ε3, W_k¹ or W_k² can be used leading respectively to:

v_{M k+ε}^j,1 ₃ = 1

4v^j−1_k+e2 + 1

4v_k+e^j−1_1+e2 +1 2v_k+e^j−1₁ v_{M k+ε}^j,2 ₃ = 1

4v^j−1_k + 1

4v^j−1_k+e1+ 1

2v_k+e^j−1_1+e2. (11) This nonlinear scheme is converging in L^∞ since we have the following result:

Proposition 2. The prediction defined by (9), (10), (11) satisfies:

k∆¹v_M.+ε^j _ik_ℓ^∞₍Z^d) ≤ 3

4k∆¹v^j−1k_ℓ^∞₍Z^d)

We do not detail the proof here but the result is obtained by computing the differences in the canonical directions at each coset points. If we then use Theorem 2 we can find the regularity of the corresponding limit function is C^s with s <−^log(3/4)_log(4) ≈0.207.

In the second pat of the section, we build an example of bidimensional subdivision scheme based on the same philosophy but this time using as dilation matrix the quincunx matrix defined by:

M =



 −1 1 1 1



,

whose coset vectors are ε0 = (0,0)^T and ε1 = (0,1)^T. Note that a0,0 = 1 and since the nonlinear subdivision operator reproduces the constants we have P

i

aM i+ε= 1 for all coset vectors ε. To build the subdivision operator, we consider the subdivision rules based on interpolation by of first degree polynomials on the grid Γ^j−1. v_{M k+ǫ}^j ₁ corresponds to a point inside the cell delimited by M^−j+1{k, k+e₁, k+e₂, k+e₁ +e₂}. There are four potential stencils, leading in this case only to two subdivision rules:

ˆ

v^j,1_{M k+ǫ1} = ¹₂(v_k^j−1+v^j−1_k+e1+e2) (12)

ˆ

v^j,2_{M k+ǫ1} = ¹₂(v_k+e^j−1₁+v^j−1_k+e2) (13) Note also, that since the scheme is interpolatory we have the relation: v_{M k}^j = v^j−1_k . Let us now prove a contraction property for the above scheme.

Proposition 3. The nonlinear subdivision scheme defined by (12) and (13) satisfies the following property:

1. when k =Mk^′:

kv_M.+ε^j,1 ₁ −v_M.^j k_ℓ^∞₍Z^d) ≤ 1

2k∆¹v^j−2_. k_ℓ^∞₍Z^d)

kv_M.+ε^j,2 ₁ −v_M.^j k_ℓ^∞₍Z^d) ≤ k∆¹v_.^j−2k_ℓ^∞₍Z^d)

2. when k =Mk^′+ε1, we can show that:

kv_M.+ε^j,2 ₁ −v_M.^j k_ℓ^∞₍Z^d) ≤ 1

2k∆¹v^j−2_. k_ℓ^∞₍Z^d)

kv_M.+ε^j,1 ₁ −v_M.^j k_ℓ^∞₍Z^d) ≤ k∆¹v_.^j−2k_ℓ^∞₍Z^d)

The proof of this theorem is obtained computing all the potential differ- ences. This theorem shows that the nonlinear subdivision scheme converges in L^∞ since ρ1,∞(S)<1.

6. Convergence in Sobolev Spaces

In this section, we extend the result established in [13] on the convergence of linear subdivision scheme to our nonlinear setting. We will first recall the notion of convergence in Sobolev spaces in the linear case. Following [12]

Theorem 4.2, when Φ0(x) = P

k∈Z^d

akΦ0(Mx−k) is L^p-stable, the so-called

”moment condition of order k + 1 for a” is equivalent to the polynomial reproduction property of polynomial of total degree k for the subdivision scheme associated to a. In what follows, we will say that Φ0 reproduces polynomial of total degree k. When the subdivision associated to a exactly reproduces polynomials, we will say that Φ0 exactly reproduces polynomi- als. We then have the following definition for the convergence of subdivision schemes in Sobolev spaces in the linear case [13]:

Definition 5. We say thatv^j =Sv^j−1 converges in the Sobolev spaceW_N^k(R^d) if there exists a function v in W_N^k(R^d) satisfying:

j→+∞lim kvj−vk_W^k

N(R^d) = 0 where v is in W_N^k(R^d), and vj = P

k∈Zd

v^j_kΦ0(M^jx−k) for any Φ0 reproducing polynomials of total degree k.

We are going to see that in the nonlinear case, to ensure the convergence we are obliged to make a restriction on the choice of Φ0. We will first give some results when the matrix M is an isotropic dilation matrix, we will also emphasize a particular class of isotropic matrices, very useful in image processing.

6.1. Definitions and Preliminary Results

Definition 6. We say that a matrix M is isotropic if it is similar to the diagonal matrix diag(σ1, . . . , σd), i.e. there exists an invertible matrixΛ such that

M = Λ⁻¹diag(σ1, . . . , σd)Λ, with |σ1|=. . .=|σd| being the eigenvalues of matrix M.

Evidently, for an isotropic matrix holds |σ1| = . . . = |σd| = σ = m^1d. Moreover, for any given norm in R^d, any integer n and any v ∈R^d we have

σⁿkuk <

∼ kMⁿuk <

∼ σⁿkuk.

A particular class of isotropic matrices is when there exists a set ˜e1,e˜2,· · · ,e˜q

such that:

M˜ei =λie˜γ(i) (14)

whereγ is a permutation of{1,· · · , q}. Such matrices are particular cases of isotropic matrices since M^q = λI where I is the identity matrix and where λ= Q^d

i=1

λ_i. For instance, when d= 2, the quincunx (resp. hexagonal) matrix satisfies M² = 2I (resp. M² = 4I).

We establish the following property on joint spectral radii that will be useful when dealing with the convergence in Sobolev spaces.

Proposition 4. Assume thatS reproduces polynomials up to total degreeN. Then,

ρ_p,n+1(S)≥ 1 kMk∞

ρ_p,n(S), for all n = 0, . . . , N.

Remark: If M is an isotropic matrix and S reproduces polynomials up to total degree N, then

ρp,n+1(S)≥σ⁻¹ρp,n(S), for all n = 0, . . . , N.

Proof: It is enough to prove

ρp,1(S)≥ 1

kMk_∞ρp(S).

According to the definition of spectral radius there exists ρ > ρp,1(S) such that for any u⁰

kS₁(u^j−1). . . S₁(u⁰)∇uk_ℓ^p₍Zd) <

∼ ρ^jk∇uk_(ℓ^p₍Zd).

Using the notation ω^j :=S(u^j−1)·. . .·S(u⁰)u we obtain k∇ω^jk_ℓ^p₍Zd) <

∼ ρ^jk∇uk_ℓ^p₍Zd).

Since

ω_l^j =X

n

A^j_l,nun,

where

A^j_l,n = X

l1,...,lj−1

al−M lj−1(u^j−1)alj−1−M lj−2 ·. . .·al1−M n(u⁰).

We can write down the ℓ^p-norm as follows:

kω^jk^p_ℓp(Z^d) = X

k∈Z^d m^j

X

i=1

|ω^j

M^jk+ε^j_i|^p,

where {ε^j_i}^m_i=1^j are the representatives of cosets of M^j. First note that:

kk−nk∞ ≤ kk−n+M^−jε^j_ik∞+kM^−jε^j_ik∞.

Note that M^−jε^j_i belongs to the unit square so that kM^−jε^j_ik∞≤K1. When A^j

M^jk+ε^j_i,n 6= 0, one can prove that there exists K2 >0 such that kk−n+M^−jε^j_ik∞ ≤K2,

the proof being similar to that of Lemma 2 in [11]. From these inequalities it follows that if A^j

M^jk+ε^j_i,n 6= 0 there existsK₃ >0 such that kk−nk∞≤K3,

that is, for a fixed k, the values of ω_l^j for l ∈ {M^jk+ε^j_i}^m_i=1^j depend only on un with n:{kk−nk∞≤K3}.

Let us now fix k and define ˜u such that

˜ ul =







ul, if kk−lk∞≤K3; 0, otherwise.

Let ˜ω^j :=S(u^j−1)·. . .·S(u⁰)˜u, then

˜ ω_l^j =







ω_l^j, if l∈ {M^jk+ε^j_i}^m_i=1^j; 0, if kk−M^−jlk∞≥K4, since if A^j_l,n 6= 0, then

kk−M^−jlk_∞≤ kk−nk_∞+kn−M^−jlk_∞≤K₃+K₂ :=K₄.

Moreover, fromkk−M^−jlk∞ ≤K4, it follows thatkM^jk−lk∞≤K4kM^jk∞. Taking all this into account, we get

X

k∈Zd

X

l∈{M^jk+ε^j_i}

|ω_l^j|^p = X

k∈Zd

X

l∈{M^jk+ε^j_i}

|˜ω^j_l|^p ≤ X

k∈Zd

X

kM^jk−lk∞≤K4kM^jk∞

|˜ω_l^j|^p

<∼ kMk^j_∞k∆¹ω˜_l^jk_ℓ^p₍Zd) <

∼ (kMk∞ρ)^jk∆¹uk˜ _ℓ^p₍Zd). That is, kω^jk_ℓ^p₍Z^d) <

∼ (kMk∞ρ)^jkuk_ℓ^p₍Z^d), consequently ρp(S) <

∼ kMk∞ρ.

Now, if ρ→ρ_p,1(S) we get ρ_p(S)≤ kMk_∞ρ_p,1(S).

6.2. Convergence in Sobolev Spaces When M is Isotropic

First, Let us recall that the Sobolev norm onW_N^p(R^d) is defined by:

kfk_W^p

N(Rd)=kfk_L^p₍Rd)+ X

|µ|≤N

kD^µfk_L^p₍Rd). (15) If one considers a setx¹,· · · , xⁿsuch that [x¹,· · · , xⁿ]Zⁿ =Z^d, an equivalent norm is given by:

kfk_W^p

N(R^d)=kfk_L^p₍^Rd₎+ X

|µ|≤N

kD˜^µfk_L^p₍^Rd₎. (16) where ˜D^µ =D^µ_x1¹· · ·D_x^µnⁿ.

We then enounce a convergence theorem for general isotropic matrix M:

Theorem 3. Let S be a data dependent nonlinear subdivision scheme which exactly reproduces polynomials up to total degree N−1, then the subdivision scheme Sv^j converges in W_N^p(R^d), provided Φ₀ is compactly supported and exactly reproduces polynomials up to total degree N−1 and

ρp,N(S)< m^1p−ds for some s > N. (17) Proof: Note that because of Proposition 4, the hypotheses of Theorem 3 imply that ρp,k(S)< m^1dρp,k+1(S)< m^1p−s−d1,which means that (17) is also true for k < N. Let us now show that vj is a Cauchy sequence in L^p. To do so, let us define

qj(x) = Xd

l=1

λj,lxl,

where Λ = (λj,l) is defined in (6). For a multi-index µ = (µ1, . . . , µd) ∈ Z^d let

qµ(x) =q^µ₁¹(x). . . q^µ_d^d(x).

Since Λ is invertible, the set {qµ :|µ|=N} forms a basis of the space of all polynomials of exact degree N, which proves that

kD^µ(vj+1−vj)k_L^p₍R^d) ∼ kqµ(D)(vj+1−vj)k_L^p₍R^d)

Now, we use the fact that, sinceM is isotropic,q_µ(D)(f(M^jx)) =σ^jµ(q_µ(D)f)(M^jx) where ˜σ^µ=

Qd i=1

σ_i^µi ([5]). We can thus write:

qµ(D)(vj+1−vj) =qµ(D) X

l∈Z^d

v_l^j+1Φ0(M^j+1x−l)−X

l∈Z^d

v_l^jΦ0(M^jx−l)

! .

We use now the scaling equation of Φ₀ to get qµ(D)(vj+1−vj) = qµ(D) X

l∈Zd

v_l^j+1Φ0(M^j+1x−l)−X

l∈Zd

X

r∈Zd

v_r^jgl−M rΦ0(M^j+1x−l)

!

= X

l∈Z^d

(v_l^j+1− X

r∈Z^d

v^j_rgk−M r)qµ(D) Φ0(M^j+1x−l)

= X

l∈Z^d

X

r∈Z^d

(al−M r(v^j)−gl−M r)v^j_rσ˜^µ(j+1)(qµ(D)Φ0)(M^j+1x−l).

Since S and Φ₀ exactly reproduce polynomials up to total degree N −1, we have for |µ| ≤N −1:

X

r∈Z^d

(al−M r(v^j)−gl−M r)r^µ= 0.

Remark that g_{l−M r} = 0 for kl−Mrk>K˜ since Φ₀ is compactly supported.

Sincen

∇^νδl−β,|ν|=N, r∈F(l) =n

kl−Mrk ≤max(K,K˜)o

, β ∈Z^do spans (a_{l−M r}(v^j)−g_{l−M r})_r∈F(l), we deduce:

q_µ(D)(v_j+1−v_j) = X

l∈Zd

X

r∈F(l)

X

|ν|=N

c^ν_r(v^j)∇^νv_r^jσ˜^µ(j+1)(q_µ(D)Φ₀)(M^j+1x−l), Consequently,

kq_µ(D)(v_j+1−v_j)k_L^p₍Rd) <

∼ σ^(j+1)Nm^−(j+1)/p(ρ_p,N(S))^jk∆^Nv⁰k_(ℓ^p₍Zd))^qN

Since ρ_p,N(S)< m^1/p−s/d, with s > N we obtain kqµ(D)(vj+1−vj)k_L^p₍^Rd₎ <

∼ σ^j(N−s)k∆^Nv⁰k_(ℓ^p₍^Zd₎₎qN.

From this we deduce that kqµ(D)(vj+1−vj)kL^p(Rd) tends to 0 withj. Making µ vary, we deduce the convergence inW_N^p(R^d)

We now show that when the matrixM satisfies (14) and when Φ0 is a box spline satisfying certain properties, the limit function is in W_N^p(R^d). Before