Shape-Preserving $C^1$ GP Hermite Interpolants Generated by a Subdivision Scheme

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Shape-Preserving C

¹

GP Hermite Interpolants Generated by a Subdivision Scheme

Francesca Pelosi, Paul Sablonnière

To cite this version:

Francesca Pelosi, Paul Sablonnière. Shape-Preserving C¹ GP Hermite Interpolants Generated by a Subdivision Scheme. Journal of Computational and Applied Mathematics, Elsevier, 2008, 220 (1-2), pp.686-711. �10.1016/j.cam.2007.09.013�. �hal-00071286�

Shape-Preserving C

¹

GP Hermite Interpolants Generated by a Subdivision Scheme

Francesca Pelosi

Dipartimento di Scienze Matematiche ed Informatiche, Universit`a di Siena, Pian dei Mantellini 44, Siena, Italy. E-mail: pelosi@unisi.it

Paul Sablonni` ere

INSA de Rennes, 20 avenue des Buttes de Co¨esmes, CS 14315, 35043 Rennes Cedex, France. E-mail: psablonn@insa-rennes.fr

Abstract

In this paper we present a method for the construction ofC¹ Hermite interpolants obtained from a particular family of refinable spline functions introduced by Gori &

Pitolli [16]. They constitute a one-parameter subfamily of the Hermite interpolants generated by the general Merrien’s subdivision scheme [22]. We compare this family to the other one-parameter subfamily studied by Merrien & Sablonni`ere [23] and Lyche & Merrien [19] on the solution of two-points Hermite interpolation problems with arbitrary monotonicity or convexity constraints.

Key words: refinable function, interpolation, shape-preservation, corner cutting.

1 Introduction

One of the major research items of approximation theory and of CAGD is the construction of shape-preserving smooth interpolants. In this field the litera- ture proposes a huge variety of methods, as for instanceC¹ quadratic splines [10,20,29], C¹ cubic splines [2], variable degree polynomials [7,8], parametric splines [21], parametric spline curves [14] and more general schemes [3].

On the other hand, in the last years, subdivision algorithms have been studied and used in many applications, such as wavelets and geometric design.

In order to combine these aspects, we analyze a family of refinable basis func- tions called GP B-splines, introduced by L. Gori and F. Pitolli in [16] and studied in [17,15,18,26]. They generate totally positive bases and possess many interesting properties for CAGD, such as positivity, compact support, partition

of unity, and central symmetry. We focus on the class of cubic GP B-splines, which span the spaceGP S₃ of C¹ cubic GP splines. When restricted to a sin- gle interval, say I = [0,1] for the sake of simplicity, they span the local space GP P3 of cubic GP polynomials. We then show that they can be generated by an Hermite subdivision scheme, of the form introduced by Merrien [22,23].

This scheme, that we call HS(γ), depends on a parameter γ ∈]0,2[ which plays the role of a shape parameter. Our main purpose is to study the con- struction of C¹ monotone and/or convex interpolants in GP P₃, to monotone and/or convex data, by using this Hermite subdivision scheme. We will show that, whatever be the values and the slopes of an increasing or convex func- tion f ∈GP P₃ at the endpoints of the interval [0,1], it isalways possible to construct aC¹ increasing or convex Hermite interpolant tof, by usingHS(γ).

Similar results hold for decreasing or concave functions. Moreover the scheme can be easily generalized to an arbitrary interval [a, b]. Since the construction is local, it can be extended to subintervals of an interval endowed with an arbitrary partition, with given values and slopes at the nodes . Therefore is possible to construct C¹ shape-preserving interpolants via simple algorithms.

The remaining part of the paper is organized as follows. In Section 2 we briefly recall some basic properties of general GP B-splines, while Section 3 focuses onC¹ cubic GP B-splines. Section 4 is devoted to the construction of the local Bernstein-B´ezier representation of cubic GP B-splines: in Subsection 4.1 we first construct a cubic GP Bernstein basis for a single interval, then in Subsec- tion 4.2 we give the description of the associated control net together with a Corner-Cutting Algorithm, and finally Subsection 4.3 presents the main shape- properties of the cubic GP Bernstein basis. In Section 5 we describe the main properties of quadratic GP splines and polynomials. In Section 6, we introduce the family of Hermite subdivision schemes HS(γ) depending on a parameter γ ∈]0,2[, which appears in the two formulas defining the subdivision algorithm.

This family includes cubic Hermite interpolants for γ = 1. It belongs to the family of general subdivision schemes introduced by Merrien. We then prove the C¹ convergence of the algorithm HS(γ). While in Section 7, we study a monotone Hermite interpolation problem, we give monotonicity regions and we propose an algorithm for the construction of monotone interpolants to ar- bitrary non decreasing data {y0, y⁰₀;y1, y⁰₁} by using the subdivision scheme HS(γ). In Section 8, we study a convex Hermite interpolation problem, we give convexity regions and we again propose an algorithm for convex inter- polants to arbitrary convex data {y0, y₀⁰;y1, y⁰₁} using the subdivision scheme HS(γ). In both cases, the algorithms are illustrated by some examples. In all sections, we also compare our subfamily to the subfamily defined in [23] and [19].

2 Definition and properties of GP B-splines

In this section we recall some basic properties of GP B-splines, introduced in [16] and studied in [17,15,18,26]. A GP B-spline of orderm+ 1 can be defined via the following scaling (or refinement) equation:

¯

ϕ_m+1(x) =

m+1

X

k=0

¯

a_k,m+1ϕ¯_m+1(2x−k), (1)

where the coefficients depend on a parameter γ ∈]0,2[

¯

a_k,m+1 =γa_k,m+1+ (1−γ)ak−1,m−1, (2) with a_k,m = ₂m−1¹

_m

k

. The mask ¯a={¯a_k,m+1}k∈Z satisfies

X

k∈Z

¯

a_2k+1,m+1 = ^X

k∈Z

¯

a_2k,m+1 = 1.

Moreover the function ¯ϕ_m+1, solution of the refinement equation (1) is positive, compactly supported on [0, m+ 1], centrally symmetric, that is

¯

ϕ_m+1(x) = ¯ϕ_m+1(m+ 1−x), ∀x∈(0, m+ 1), and its integer translates form a partition of unity

X

k∈Z

¯

ϕ_m+1(x−k) = 1, ∀x∈R.

As in [16], it is easy to show that the symbol of the mask (2) is a Hurwitz polynomial (i.e. a polynomial with roots in the left half plane) forγ ∈]0,2[.

The symbols of the masks of B-splines of degrees m−2 and m being respec- tively pm−1(z) = 2^2−m(1 +z)^m−1 and p_m+1(z) = 2^−m(1 +z)^m+1, the symbol of the GP B-spline with mask ¯a is

¯

p_m+1(z) = 2^−m(1 +z)^m−1γ(1 +z)²+ 4(1−γ)z

= 2^−m((1 +z)^m−1γz²+ 2(2−γ)z+γ

As the discriminant of the quadratic polynomial is ∆⁰ = 4(1−γ):

1) If 0 < γ ≤1, then ∆⁰ ≥0 and its roots z =γ−2±2√

1−γ are real and negative.

2) If 1≤γ <2, then ∆⁰ <0 and its roots z =γ−2±2i√

γ−1 are complex conjugate.

As ¯p_m+1 has only real negative roots or complex roots with negative real part, it is a Hurwitz polynomial.

Thus ¯ϕ_m+1 is a ripplet, that is ¯ϕ_m+1 is totally positive, see [13] for details. As a consequence of this property, the family of integer translates{ϕ¯_m+1(.−k)}

enjoy the variation diminishing properties:

S⁻(^Xc_jϕ¯^(r)_m+1)≤S⁻(∆^rc), 0≤r≤m−2 (3)

where S⁻(b) denotes the number of strict sign changes in the sequence b = {b_j}j∈Z and (∆^rc)_j = (∆^r−1c)_j+1−(∆^r−1c)_j, with (∆⁰c)_j =c_j.

3 C¹ Cubic GP B-splines

Let us consider the case ofcubic GP B-splines(m= 3) with continuityC^m−2 = C¹ and denote by GP S3 the space of splines spanned by all combinations of translates of ¯ϕ₄(x). From the scale equation,

¯

ϕ4(x) = 1

8[γϕ¯4(2x) + 4 ¯ϕ4(2x−1) + (8−2γ) ¯ϕ4(2x−2) + 4 ¯ϕ4(2x−3) +γϕ¯4(2x−4)], we deduce that ¯ϕ₄ is entirely defined by its values at the 3 points x= 1,2,3.

Theorem 1 The values of ϕ¯₄ at x= 1,2,3 are

¯

ϕ₄(1) = ¯ϕ₄(3) = γ 2(γ+ 2) ∈

0,1 4

, ϕ¯₄(2) = 2 γ+ 2 ∈

1 2,1

. (4)

and the derivatives Dϕ¯₄ at the same points are:

Dϕ¯₄(1) = 1

2, Dϕ¯₄(2) = 0, Dϕ¯₄(3) =−1

2. (5)

Proof. From (3), we can compute successively:

8 ¯ϕ₄(1) =γϕ¯₄(2) + 4 ¯ϕ₄(1), 8 ¯ϕ₄(2) = 4 ¯ϕ₄(3) + (8−2γ) ¯ϕ₄(2) + 4 ¯ϕ₄(1).

Using the partition of unity and the symmetry w.r.t. x= 2, we also have

¯

ϕ₄(1) + ¯ϕ₄(2) + ¯ϕ₄(3) = 1.

Values (4) are solutions of these equations. With the same kind of technique it is easy to obtain the derivatives (5). 2

0 0.2 0.4 0.6 0.8 1

1 2 3 4

Fig. 1. Example of ¯ϕ₄ for γ= 0,¹₄,¹₂,³₄,1.

Moreover all values {ϕ¯4(^2k−1₂ ), k = 1,2,3,4} are obtained from the scale equation

8 ¯ϕ4

2k−1 2

!

=γϕ¯4(2k−1)+4 ¯ϕ4(2k−2)+(8−2γ) ¯ϕ4(2k−3)+4 ¯ϕ4(2k−4)+γϕ¯4(2k−5), with, of course, ¯ϕ₄(`) = 0 for`≤0 and `≥4. More generally, all values of ¯ϕ₄

at the dyadic points of its support can be computed in this way. Some plots of ¯ϕ₄, for different values of γ, are shown in Figure 1.

4 Bernstein basis and control polygon

4.1 Bernstein basis

The restriction to I = [0,1] of the space of generalized cubic splines is the space of generalized cubic polynomials

GP P₃(I) ={S(x) =

3

X

j=0

a_iϕ¯₄(x+j), x∈ [0,1]},

and we can define the corresponding GP Bernstein basis {b₀, b₁, b₂, b₃}. De- noting es(x) = x^s, we can prove the following

Theorem 2 There exists a cubic GP Bernstein basis {b₀, b₁, b₂, b₃} in the space GP P₃(I) of cubic GP polynomials having the following properties:

(1) b_j(x)≥0 for x∈[0,1], with j = 0,1,2,3;

(2) ^P3_j=0b_j =e₀;

(3) b_j(0) =b_j(1) = 0, for j 6= 0,3, b₀(0) =b₃(1) = 1, and b₀(1) =b₃(0) = 0;

(4) b⁰_j(0) = 0, j = 2,3;and b⁰_j(1) = 0, j = 0,1;

(5) b_i(x) = b3−i(1−x).

The GP Bernstein basis has the following expression:

b₀(x) = 2(γ+ 2)

γ ϕ¯₄(x+ 3), b1(x) = 1

2−γ

γ²−8

γ ϕ¯4(x+ 3) + 2 ¯ϕ4(x+ 2)−γϕ¯4(x+ 1) + 2 ¯ϕ4(x)

!

, b₂(x) = 1

2−γ 2 ¯ϕ₄(x+ 3)−γϕ¯₄(x+ 2) + 2 ¯ϕ₄(x+ 1) + γ²−8 γ ϕ¯₄(x)

!

, b₃(x) = 2(γ+ 2)

γ ϕ¯₄(x).

(6)

Proof. From properties (3) and (5), we can set

b0(x) = 2(γ+ 2)

γ ϕ¯4(x+ 3), b3(x) = b0(1−x) = 2(γ+ 2) γ ϕ¯4(x), and we define

b₁(x) = πϕ¯₄(x+ 3) +ρϕ¯₄(x+ 2) +σϕ¯₄(x+ 1) +τϕ¯₄(x).

By symmetry (5), we have

b₂(x) = b₁(1−x) =πϕ¯₄(x) +ρϕ¯₄(x+ 1) +σϕ¯₄(x+ 2) +τϕ¯₄(x+ 3).

From property (2), we deduce

1 = 2(γ+ 2)

γ +π+τ

!

¯

ϕ₄(x+ 3) + (ρ+σ) ¯ϕ₄(x+ 2) + (ρ+σ) ¯ϕ4(x+ 1) + 2(γ+ 2)

γ +π+τ

!

¯ ϕ4(x), which gives the two equations

π+τ = 1− 2(γ+ 2)

γ =−γ+ 4

γ , ρ+σ = 1.

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1 0

0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

Fig. 2. Examples of cubic GP Bernstein basis in [0,1] for γ = 1 (left) and γ = ¹₄ (right).

Moreover we have b⁰₀(0) = −^γ+2_γ , and from (3)−(4), b⁰₁(0) = −b⁰₀(0) = ^γ+2_γ = πϕ¯⁰₄(3) +σϕ¯⁰₄(1) = ¹₂(σ−π) andb₁(1) = 0 =ρϕ¯⁰₄(3) +σϕ¯⁰₄(2) +τϕ¯⁰₄(1), which give the two equations:

σ−π= 2(γ+ 2)

γ , γρ+ 4σ+γτ = 0.

This system of four equations has the following solutions π =− 8−γ²

γ(2−γ), ρ= 2

2−γ, σ=− γ

2−γ, τ = 2 2−γ. Therefore we obtain the claim. 2

Figure 2 shows the cubicGP Bernstein polynomials in [0,1] for the cubic case γ = 1 and for γ = ¹₄. Reciprocally, one obtains the expression of cubic GP B-spline basis in terms of cubicGP Bernstein polynomials.

Theorem 3 For x∈[i, i+ 1], the cubic GP B-spline functions are expressed in terms of the local cubic GP Bernstein basis as follows

¯

ϕ₄(x−i+ 3) = _2(γ+2)^γ b₀(x−i), ϕ¯₄(x−i) = _2(γ+2)^γ b₃(x−i),

¯

ϕ4(x−i+ 2) = _γ+2¹ 2b0(x−i) + 2b1(x−i) +γb2(x−i) + ^γ₂b3(x−i).

¯

ϕ₄(x−i+ 1) = _γ+2^{1 γ}₂b₀(x−i) +γb₁(x−i) + 2b₂(x−i) + 2b₃(x−i),

Proof. For a general subinterval [i, i+ 1], we express the B-splines which not vanish as:

b0 b⁰₀ b1 b⁰₁ b2 b⁰₂ b3 b⁰₃ x= 0 1 −^γ+2_γ 0 ^γ+2_γ 0 0 0 0 x= 1 0 0 0 0 0 −^γ+2_γ 1 ^γ+2_γ x= ¹₂ ^γ₈ −^γ+2₄ ^4−γ₈ −^γ+2₄ ^4−γ₈ ^γ+2₄ ^γ₈ ^γ+2₄ Table 1

Values and first derivatives of the cubic GP Bernstein basis at x= 0,¹₂,1.

¯

ϕ4(x−i+ 3) =λib0(s);

¯

ϕ4(x−i+ 2) =λi+1b0(s) +µi+1b1(s) +σi+1b2(s) +υi+1b3(s);

¯

ϕ₄(x−i+ 1) =λ_i+2b₀(s) +µ_i+2b₁(s) +σ_i+2b₂(s) +υ_i+2b₃(s);

¯

ϕ₄(x−i) = λ_i+3b₃(s);

withs =x−i. Then we compute λi using ¯ϕ4(3) andλi+3 using ¯ϕ4(1). For the expression of ¯ϕ₄(x−i+ 2) and ¯ϕ₄(x−i+ 1), we use the expressions of ¯ϕ⁰₄(3) and ¯ϕ⁰₄(1). 2

We collect in Table 1 the values and the first derivatives of the cubic GP Bernstein basis atx= 0,¹₂,1, which we will use later.

4.2 Control Polygon and Corner-Cutting Algorithm

From Table 1, we get

b⁰₁(0) =−b⁰₀(0) = γ+ 2

γ , b⁰₃(1) =−b⁰₂(1) = γ+ 2 γ , and setting θ= _γ+2^γ , it is easy to verify that

e₁ =θb₁ + (1−θ)b₂ + b₃.

This allows to define the control polygon of the (generalized) cubic on [0,1].

Definition 4 The control polygon P of

f(x) = a₀b₀(x) +a₁b₁(x) +a₂b₂(x) +a₃b₃(x),

for x ∈ [0,1] is the polygonal line connecting the four control vertices a_j = (ξ_j, a_j), j = 0,1,2,3, where

ξ₀ = 0, ξ₁ =θ, ξ₂ = 1−θ, ξ₃ = 1.

Note that θ ∈]0,¹₂[ for γ ∈]0,2[. In particular forγ = 1, θ = ¹₃, we obtain the classical cubic control polygon. Figure 3 shows the cubic GP control polygon

Fig. 3. Control polygon forγ = ¹₂.

for γ = 1/2. Let f = ^P3_i=0a_ib_i, consider now the problem of computing the coefficients of the expansions off in the Bernstein bases of the two subintervals I₁ = [0,¹₂] and I₂ = [¹₂,1] ofI. Setting

f(t) =f₁(t) =

3

X

i=0

c_jb_j(2t) fort ∈I₁, f(t) =f₂(t) =

3

X

i=0

d_jb_j(2t−1) for t∈I₂, we can prove the following theorem.

Theorem 5 The coefficients c_j and d_j for j = 0,1,2,3 of (7) are computed from the initial coefficients {a_j, 0≤j ≤3} by the following formulae

c0 =a0, c1 = 1

2(a0+a1) a4 = 1

2(a1+a2), d2 = 1

2(a2+a3), d3 =a3, c2 = γ

2c1+ (1− γ

2)a4, d1 = (1− γ

2)a4+γ

2d2, c3 =d0 = 1

2(c2+d1).

Proof. By using the values of the GP Bernstein basis at 0,1/2,1 (see Table 1), we immediately obtain the coefficients

c₀ =f₁(0) =f(0) = a₀ and d₃ =f₂(1) =f(1) = a₃, c₃ =f₁

1 2

=d₀ =f₂

1 2

=f

1 2

= (a₀+a₃)γ

8 + (a₁+a₂)4−γ 8 ,

f⁰(0) = γ+ 2

γ (a₁−a₀) =f₁⁰(0) = 2(γ+ 2)

γ (c₁−c₀), f⁰(1) = γ+ 2

γ (a₃−a₂) = f₁⁰(1) = 2(γ+ 2)

γ (d₃−d₂), whencec₁ = ¹₂(a₀+a₁) andd₂ = ¹₂(a₂+a₃).

From the equalities f⁰(¹₂) = f₁⁰(¹₂) = f₂⁰(¹₂), we obtain the following equation 2

γ(c₃−c₂) =−1

4(a₀−a₃+a₁−a₂) = 2

γ(d₁−d₀),

which gives, after some calculations, the values ofc₂ and d₁. Finally, we have that c2 +d1 = 2c3 = 2d0, whence c3 = d0 = ¹₂(c2+d1). 2

Sincec₃ =d₀ =γ(a₀+a₃)/8 + (4−γ)(a₁+a₂)/8, the equalities of Theorem 5, become, in matrix form, the following:

[c₀, c₁, c₂, c₃ =d₀, d₁, d₂, d₃]^T = A [a₀, a₁, a₂, a₃]^T where A is defined as:

A:=











































1 0 0 0

1 2

1

2 0 0

γ 4

1 2

2−γ

4 0

γ 8

4−γ 8

4−γ 8

γ 8

0 ^2−γ₄ ¹₂ ^γ₄ 0 0 ¹₂ ¹₂

0 0 0 1











































. (7)

Note that the subdivided polygon defined by {c₀, c₁, c₂, c₃=d₀, d₁, d₂, d₃} has been obtained as convex combinations of the polygon defined by {a0, a1, a2, a3}. With the intermediate quantitya4 defined as

a₄ = 1

2(a₁+a₂),

the subdivision algorithm described in Theorem 5 can be reformulated as follows.

Definition 6 : Corner-Cutting Algorithm.

(i) starting with the polygon {a₀, a₁, a₂, a₃} we construct {c₀, c₁, a₄, d₂, d₃} by cutting the corner ina₁ with the edge(c₁, a₄)and the corner ina₂ with(a₄, d₂).

(ii) At the second step we construct the polygon{c₀, c₁, c₂, d₁, d₂, d₃}by cutting the new corner in a₄ with the edge (c₂, d₁)

c₂ = γ

2c₁+ (1−γ

2)a₄, d₁ = (1−γ

2)a₄+ γ 2d₂.

(iii) At the third step we construct the final control polygon {c₀, c₁, c₂, c₃ = d₀, d₁, d₂, d₃}by insertingc₃ =d₀ on the edgec₂, d₁. Sincec₃ =d₀ andd₃ =a₃, the subdivided polygon {c₀, c₁, c₂, c₃ =d₀, d₁, d₂, d₃}is obtained by carrying out a corner cutting scheme on {a₀, a₁, a₂, a₃}.

This scheme can be formulated in matrix form:

[c₀, c₁, c₂, c₃ =d₀, d₁, d₂, d₃]^T = S [a₀, a₁, a₂, a₃]^T where S is defined as:

S:=











































1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 ¹₂ ¹₂ 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1













































































1 0 0 0 0 0 1 0 0 0 0 ^γ₂ 1−^γ₂ 0 0 0 0 1−^γ₂ ^γ₂ 0 0 0 0 1 0 0 0 0 0 1































































1 0 0 0

1 2

1 2 0 0 0 ¹₂ ¹₂ 0 0 0 ¹₂ ¹₂ 0 0 0 1





























. (8)

Theorem 7 Assume 0 ≤ γ ≤ 2, then the matrix S defined in (8) is totally positive.

Proof. The matrix S is the product of 3 matrices which are bidiagonal and whose entries are nonnegative for 0 ≤ γ ≤ 2, thus they are totally positive.

Since the product of totally positive matrices is totally positive, we can con- clude that also S is totally positive. 2

4.3 Shape Properties of the cubic GP Bernstein basis

Theorem 8 For γ ∈]0,2[, the GP P3 Bernstein basis is totally positive.

The proof follows straightforward the steps of proof for Theorem 11 in [19].

Fig. 4. One step of corner cutting scheme.

Since theGP P₃ Bernstein basis is totally positive, any functionf ∈GP P₃(I) has the same properties (positivity, convexity and monotonicity) as those of the corresponding control polygon. In particular

Corollary 9 For γ ∈]0,2[, the Bernstein basis {b₀, b₁, b₂, b₃}of GP P₃ has the following properties:

(1) b0 is nonnegative, decreasing and convex on [0,1].

(2) b₁ is nonnegative and concave on[0,1/2]and nonnegative, decreasing and convex on [1/2,1].

(3) b2 is nonnegative, increasing and convex on[1,1/2] and nonnegative and concave on [1/2,1].

(4) b₃ is nonnegative, increasing and convex on [0,1].

Proof. From Theorem 5 and Definition 4 it is immediate to see that the ordinates of the control polygon of the polynomialbjis thej−thunit vectorEj

forj = 0,1,2,3.Thus the nonnegativity ofb_j follows from the nonnegativity of its control polygon and for the same reason the monotonicity and the convexity properties of b0 and b3 hold. For the properties of b1 and b2 it suffices to do one subdivision step and to use the relative control polygons: then the proof is similar. 2

5 Quadratic GP splines and polynomials

A similar study can be done for quadratic GP splines and GP polynomials.

We denote their spaces respectively by GP S₂ and GP P₂. As we use the same techniques as in previous sections, we only give the main results in the present one without entering into details. The standard quadratic GP B-spline ¯ϕ₃ has support [0,3] and its values are obtained from its refinement equation

¯

ϕ₃(x) = 1

4(γ( ¯ϕ₃(2x) + ¯ϕ₃(2x−3)) + (4−γ)( ¯ϕ₃(2x−1) + ¯ϕ₃(2x−2)). Moreover, we have ¯ϕ⁰₄(x) = ¯ϕ₃(x)−ϕ¯₃(x−1), therefore iff =^P3_i=0α_iϕ₄(x+3−

i), forx∈I, then we deduce thatf⁰ =^P2_j=0∆αjϕ¯3(x+2−j). The symbol ¯p3(z) of ¯ϕ₃ being a Hurwitz polynomial, the integer translates of this B-spline form a totally positive system having the classical variation diminishing properties.

From the above scale equation, we also derive the values of ¯ϕ3 at knots and midpoints of its support :

¯

ϕ3(1) = ¯ϕ3(2) = 1/2, ϕ¯3(1/2) = ¯ϕ3(5/2) = γ

8, ϕ¯3(3/2) = 1− γ 4. The Bernstein basis of the local space GP P₂ of quadratic GP polynomials on I is defined by

β₀(x) = 2 ¯ϕ₃(x+2), β₁(x) =−ϕ¯₃(x+2)+ ¯ϕ₃(x+1)−ϕ¯₃(x), β₂(x) = 2 ¯ϕ₃(x).

Conversely, one has

¯

ϕ₃(x+ 2) = 1

2β₀(x), ϕ¯₃(x+ 1) = 1

2β₀(x) +β₁(x) +1

2β₂(x), ϕ¯₃(x) = 1 2β₂(x).

Therefore, ifg =^P2_i=0c_iβ_i =^P2_j=0γ_jϕ₃(x+ 2−j), one has c0 = 1

2(γ0+γ1), c1 =γ1, c2 = 1

2(γ1+γ2).

and we derive in particular

∆c₀ = 1

2(∆γ₀), ∆c₁ = 1 2∆γ₁.

This implies that the monotonicity of the sequence of coefficients in the Bern- stein basis is equivalent to its monotonicity in the B-spline basis. In particular, when it is nondecreasing, then g is also nondecreasing because of the variation diminishing property of the latter.

One can also express derivatives of the Bernstein basis of GP P₃ in terms of

the Bernstein basis of GP P₂ as follows (we recall that θ= _γ+2^γ ) : b⁰₀(x) = −1

θβ₀(x), b⁰₁(x) = 1

θβ₀(x) + 1

1−2θβ₁(x), b⁰₂(x) = 1

1−2θβ₁(x)− 1

θβ₂(x), b⁰₃(x) = 1 θβ₂(x).

Therefore, iff =^P3_i=0a_ib_i, then we obtain f⁰ = ∆a₀

θ β₀+ ∆a₁

1−2θβ₀+ ∆a₂ θ β₂

6 Cubic Hermite Subdivision Scheme

In this section we restrict our attention to the interval I = [0,1] and we consider a cubic GP-polynomial f(x), defined as

f(x) =

3

X

i=0

c_iω_i(x) (9)

where ω_i(x) = ¯ϕ₄(x−i+ 3), and satisfying the Hermite interpolation condi- tions:

f(0) =y₀, f(1) =y₁, f⁰(0) =y₀⁰, f⁰(1) =y⁰₁. (10) The coefficients c_i are obtained by imposing the above conditions,

c0= 1 4−γ²

h(γ+ 2)(2y1−γy0) + (γ²−8)y₀⁰ −2γy₁⁰ⁱ, c₁= 1

4−γ² [(γ + 2)(2y₀−γy₁) +γ(2y₀⁰ +γy⁰₁)], c2= 1

4−γ² [(γ + 2)(2y1−γy0)−γ(2y⁰₁+γy₀⁰)], c₃= 1

4−γ²

h(γ+ 2)(2y₀−γy₁)−(γ²−8)y⁰₁+ 2γy₀⁰ⁱ.

Let us compute the values of f and f⁰ at x = 1/2. By using the values and the first derivatives of the functions ω_i, (4)-(5), we obtain

f

1 2

=y₀+y₁

2 − γ(4−γ)

8(2 +γ)(y⁰₁−y₀⁰),

f⁰

1 2

=γ+ 2

2 (y₁−y₀)−γ

4(y⁰₁+y₀⁰).

Therefore we have obtained the initial step of a Merrien subdivision algorithm [23], for the Hermite data (10). Now we consider the general formulation.

Suppose that a function f and its first derivative p are given at 0 and 1 and take the values {y₀, y₀⁰, y₁, y₁⁰}; f and p are built by induction on the set of dyadic points D = ∪_nD_n where D_n = {x = jh, j = 0, . . . ,2ⁿ−1}. At step n, setting h = 2⁻ⁿ, then for two consecutive points a =jh and b = (j + 1)h of D_n, f and p are evaluated at the midpoint m = (a+b)/2 of [a, b] by the formulas:

f(m) =f(a) +f(b)

2 +αh(p(b)−p(a)),

(11) p(m) = (1−β)f(b)−f(a)

h +βp(b) +p(a)

2 ,

where

α=−γ(4−γ)

8(2 +γ) and β =−γ

2. (12)

The construction produces an Hermite subdivision schemeHS(γ), which de- pends on the parameter γ ∈]0,2[ : reiterating the process, we definef and p on the set of dyadic numbersD=∪_nD_n, which is dense in [0,1].

Remark 10 The above construction shows that an Hermite interpolant in GP P3([0,1]) can be constructed using Merrien’s algorithm with the particular choice (12) of the parameters(α, β). We call ECS (Extended Cubic Splines), the one parameter family of curves obtained with such values of (α, β), since it contains the Hermite interpolating cubic polynomials on [0,1], for γ = 1, (α, β) = (−¹₈,−¹₂).

Remark 11 The class of subdivision schemes introduced above belongs to a more generalC¹ interpolating subdivision scheme recently studied in [19]. Thus all general results hold for this particular case.

As in [19], the algorithm can be formulated in a general way. Starting with Hermite data f₀, p₀, f₁, p₁ at the endpoints of a finite interval [0,1], we set f₀⁰ = f₀, p⁰₀ = p₀, f₁⁰ = f₁, p⁰₁ = p₁, then for n = 0,1,2, . . . ,2⁻ⁿ and k = 0,1, . . . ,2ⁿ−1

f_2iⁿ⁺¹:=f_iⁿ, f_2i+1ⁿ⁺¹ := f_i+1ⁿ +f_iⁿ

2 + α

2ⁿ(pⁿ_i+1−pⁿ_i), (13)

pⁿ⁺¹_2i :=pⁿ_i, pⁿ⁺¹_2i+1 := (1−β)f_i+1ⁿ −f_iⁿ

2ⁿ +βpⁿ_i+1+pⁿ_i

2 . (14)

Following the same steps of [19], formula (13) can be formulated so that only values of f are involved. Similarly (14) can be formulated only in terms of values of p. We use the notation µ=−^β₂(2 +β) and ν =−^β₂(1 +β):



















































f_8iⁿ⁺¹ f_8i+1ⁿ⁺¹ f_8i+2ⁿ⁺¹ f_8i+3ⁿ⁺¹ f_8i+4ⁿ⁺¹ f_8i+5ⁿ⁺¹ f_8i+6ⁿ⁺¹ f_8i+7ⁿ⁺¹



















































= 1 4



















































4 0 0 0 0

1 +µ 2(2−µ) µ+ν−1 −2ν ν

0 4 0 0 0

−µ 2(1 +µ) 2−µ−ν 2ν −ν

0 0 4 0 0

−ν 2ν 2−µ−ν 2(1 +µ) −µ

0 0 0 4 0

ν −2ν µ+ν−1 2(2−µ) 1 +µ















































































f_4iⁿ f_4i+1ⁿ f_4i+2ⁿ f_4i+3ⁿ f_4i+4ⁿ





























and





















pⁿ⁺¹_4i pⁿ⁺¹_4i+1 pⁿ⁺¹_4i+2 pⁿ⁺¹_4i+3





















=





















1 0 0

µ 1 + ^β₂ −ν

0 1 0

−ν 1 + ^β₂ µ



































pⁿ_2i pⁿ_2i+1 pⁿ_2i+2















.

Now that f and p have been defined on D = ∪^∞_n=0D_n, we shall study their continuity on D and also on [0,1], again the result follows from the general case (see Proposition 4 [19]).

Definition 12 (f, p) is a C¹ interpolant on a set A if f is continuous and admits a first derivative f⁰ with f⁰ =p.

Theorem 13 For γ ∈]0,2[, i.e. β ∈]−1,0[andα∈[¹₂(√

3−2),0[, then(f, p) is a C¹ interpolant on [0,1].

The minimum value ¯α of α = ^β(2+β)_4(1−β) is attained at ¯β = −1−√

3, and it is equal to ¯α=α( ¯β) = ¹₂(√

3−2). Consider now the matrices U_nⁱ ∈R², defined as

U_nⁱ =















p((i+ 1)2⁻ⁿ)−p(i2⁻ⁿ)

f((i+1)2⁻ⁿ)−f(i2⁻ⁿ)

2⁻ⁿ − ^p((i+1)2−n₂^{)−p(i2−n)}















,

for n∈N andi= 0, . . . ,2ⁿ−1. There exist two matrices Λ₁ and Λ₋₁ ofR^2×2 such that

U_n+1²ⁱ = Λ₁U_nⁱ, U_n+1²ⁱ⁺¹ = Λ₋₁U_nⁱ, where

Λ_ε=















1

2 ε(1−β) ε^8α+1₄ ^1+β₂















, ε =±1. (15)

In [23] it is shown that if the generalized spectral radius of the set Σ = {Λ₁,Λ−1}satisfiesρ(Σ)<1, then the functionf⁰ =pis H¨older with exponent

−log₂(ρ). An equivalent condition is the existence of a matrix norm|| · || such that ||Λ_ε||<1, ε=±1. In that case, we have ρ(Σ) = maxε=±1(||Λ_ε||).

Sinceα= ^β₄ ^2+β_1−β, withβ ∈]−1,0[, the matrices (15) can be rewritten in the following form

Λ_ε =















1

2 ε(1−β)

ε(1+β)(1+2β) 4(1−β)

1+β 2















, ε=±1. (16)

Forγ = 1, i.e. β=−¹₂ andα=−¹₈ (cubic splines), we haveρ(Λ_ε) = ¹₂. For the general case, we first compute det(Λ_ε) =−¹₂β(1 +β), then the characteristic polynomial P(λ) = det(Λε−λI) is given by

P(λ) =λ²−1

2(2 +β)λ− 1

2β(1 +β), (17)

the roots of which are respectively

λ₁ = 1 +β, λ₂ =−1 2β.

Forβ ∈ [−¹₂,0[, we have λ₁ ≥λ₂, hence ρ({Λ₁,Λ−1})≥1 +β, Now we prove that ρ({Λ₁,Λ−1})≤ 1 +β. Consider, for a positive real number θ, the norm defined in R² by ||(x, y)||θ = |x|+θ|y| and the associated matrix norm in R^2×2 defined by ||M||_θ = max{|m₁₁|+θ|m₂₁|,|m₁₂|/θ+|m₂₂|}. By choosing θ = ^2(1−β)_β+1 , we get ||Λ₋₁||_θ =||Λ₁||_θ = 1 +β, thusρ({Λ₁,Λ₋₁})≤1 +β.

For β ∈]−1,−¹₂[, we have λ₂ > λ₁, then we get ||Λ−1||_θ =−β and ||Λ₁||_θ = 1 +β, thus ρ({Λ₁,Λ−1}) =−β. Both results prove the following

Proposition 14 For γ ∈]0,2[, i.e. β ∈] −1,0[ , then the function p = f⁰ is H¨older with exponent ω(β) = −log₂(1 +β) for β ∈ [−¹₂,0[ and ω(β) =

−log₂(−β) for β ∈]−1,−¹₂]. Therefore

|f⁰(x)−f⁰(y)| ≤C|x−y|^ω(β), x, y ∈[0,1].

Now we can express a functionf ∈GP P₃(I), defined from the Hermite inter- polation scheme (12), in terms of the GP Bernstein basis,f(x) =^P3_i=0aibi(x).

Let P be the polygonal line connecting the four control points (ξ_i, a_i), as in Definition 4. As a consequence of Theorem 9 in [19], we have the following convergence result.

Theorem 15 Let P_n the control polygon obtained from the control polygon P of a function f ∈GP P₃(I)aftern steps of repeated corner cutting scheme, as described in Theorem 5. Then

n→∞lim P_n =f.

7 Monotone Interpolants

The aim of this section is to construct monotone Hermite C¹ interpolants by subdivision. Using a classical model problem stated in [2], with the data {y₀, y₀⁰, y₁, y₁⁰}={0, x,1, y}where (x, y)∈R²+, we are looking for a parameter γ ensuring the C¹-convergence of the algorithm (11) to functions f, p = f⁰ such that p≥0 on [0,1].

Definition 16 For (α, β) = (−^γ(4−γ)_8(2+γ),−^γ₂) we define the monotonicity region M(α, β) ={(x, y)∈R²+:p≥0}.

For σ >0, we define the triangular domain

T(σ) ={(x, y)∈R²+ :x+y≤σ}.

For η >0, we define the square domain Q(η) = [0, η]². For δ >0, we define the strip

B(δ) = {(x, y)∈R²+ :−1/2δ≤x−y≤1/2δ}.

Proposition 17 Let P be the control polygon (Definition 4) of a function f ∈GP P₃ interpolating the data {0, x,1, y}. Then P is increasing if and only if the pair (x, y) lies in T ^β−1_β =T ¹_θ.

Proof. Setting f =^P3_i=0a_ib_i, and imposing the interpolation conditions, we get a₀ =f(0) = 0 and a₃ =f(1) = 1. Since f⁰(0) =x, f⁰(1) =y, we also get a₁ =θxanda₂ = 1−θy. Thus the control polygonP is increasing if and only a1 ≤a2, i.e. θx≤1−θy, which is equivalent to x+y≤ ¹_θ, i.e. (x, y)∈T ¹_θ, which completes the proof. 2

Theorem 18 For γ ∈]0,2[, β = −^γ₂ ∈]−1,0[, and α =−^γ(4−γ)_8(2+γ) ∈ [¹₂(√ 3− 2),0[, the square region Q¹_θ = Q^β−1_β is included in the monotonicity region M(α, β).

Proof. with the Hermite data{0, x; 1, y}, the ordinates of the control polygon P are

a₀ = 0, a₁ =θx, a₂ = 1−θy, a₃ = 1,

with θ = _γ+2^γ = −_1−β^β . After one step of the Corner-Cutting Algorithm, we obtain

c₀ = 0, c₁ = 1

2θx, c₂ = 1

4((2−γ) + 2θx−(2−γ)θy) d₁ = 1

4((2 +γ) + (2−γ)θx−2θy), d₂ = 1− 1

2θy, d₃ = 1, c₃ =d₀ = 1

8(4 + (4−γ)θ(x−y)).

By hypothesis, we have 0 ≤ θx ≤ 1 and 0 ≤ θy ≤ 1 since ^β−1_β = ¹_θ. Let us now compute the normalized Hermite data {0, X₀; 1, Y₀} (respectively {0, X1; 1, Y1}) at the ends of the left (resp. right) subintervalI0 = [0,¹₂] (resp.

I₁ = [¹₂,1]). First we observe that c₁, d₂ and a₄ ∈ [0,1], therefore also c₂, d₁ and finally c₃ =d₀ ∈ [0,1]. On I₀, as f(¹₂) = c₃ > 0, we divide all the coeffi- cients by c3 in order to get the right normalization. Then, the vertices of the corresponding normalized polygon P₀ on [0,1] are the following

{(0,0), (θ,c₁

c₃), (1−θ,c₁

c₃), (1,1)}.

Therefore the end point derivatives are respectively X0 = c₁

θc₃ ≥0, Y0 = c₃−c₂ θc₃ , and we obtain

θ(X₀+Y₀) = c−1−c2+c3

c₃ .

As 4(c₂ −c₁) = (2−γ)(1 −θy) ≥ 0, we have θ(X₀ +Y₀) ≤ 1. Second, as c1 ≤c2, we haveY0 ≥0, therefore θ(X0+Y0)≥0. We can conclude that the polygon P₀ is non decreasing.

On the right subintervalI₁, we substractd₀ >0 from the ordinates in order to

get the zero value at the left endpoint. Then, we divide them by d₃−d₁ >0 in order to get the value 1 at the right endpoint. Indeed 8(d₃−d₀) = 4−(4− γ)θ(x−y)>0 sinceθ(x−y)< _4−γ⁴ for γ ∈]0,2[.

Thus, the vertices of the right normalized polygonP₁ on [0,1] are the following {(0,0), θ,d1−d0

d₃−d₀

!

, 1−θ,d2−d0

d₃−d₀

!

, (1,1)}.

Therefore the end point derivatives are respectively X₁ = d₁−d₀

θ(d3−d0) ≥0, Y₁ = d₃−d₂ θ(d3−d0), and we obtain

θ(X1+Y1) = d3−d2+d1−d0

d₃−d₀ .

First we have 8(d₁−d₀) = γ(2−θ(x+y))>0 andd₃−d₂ = ¹₂θy ≥0, therefore bothθX₁ and θY₁ are non negative. Second, as we have

1−θ(X1+Y1) = d₂−d₁ d₃−d₀,

and 4(d₂−d₁) = (2−γ)(1−θx)≥0, we conclude that 0≤ θ(X₁+Y₁) ≤1, which proves thatP₁ is also non decreasing.

By repeating the corner cutting process, we obtain a sequence of non decreas- ing control polygons which uniformly converges to the function f. Therefore f is also non decreasing. 2

Figure 5 shows some examples of monotonicity regions for β = −3/8,−1/2,

−1/4,−1/8.

The above theorem defines a square contained in M(α, β), in order to give a better localization of the latter, we shall define two regions M^f(α, β) and M(α, β) such that:

M(α, β)⊆M(α, β)⊆M(α, β),^f

both containing the square Q(^β−1_β ). Suppose f is monotone increasing, then f⁰(1/2)≥0, which gives:

x+y≤ 2(β−1)

β , (18)

thusM(α, β)⊆T(2(β−1)/β), whereT(2(β−1)/β) is a triangular domain. If we impose the nonnegativity of the derivatives at 1/4,3/4,3/8,5/8, we obtain the inequalities: