ATTIVITA DELL’UNIT` A DI SIENA` -...

ATTIVITA DELL’UNIT A DI SIENA

Paolo Costantini, Isabella Cravero,Paolo Costantini, Isabella Cravero,

Carla Manni, Francesca Pelosi, Maria Lucia Sampoli

Universita di Siena – Italy

COFIN 2003 – Giornate di lavoro

Roma, 16-17 Dicembre 2004

Nuovi Spazi Funzionali 1

SCHEMA DELLA PRESENTAZIONE

[1] Splines quasi-interpolanti su partizioni Powell-Sabin(C. Manni, P. Sablonniere)

[2] Studio della somma di Minkowski di due superfici parametriche(B. Juttler, M.L. Sampoli et al.)

[3] Elementi triangolari generalizzati(P. Costantini, T. Lyche, C. Manni)

[4] Superfici splines a grado variabile(P. Costantini, F. Pelosi)

[5] *****(F. Pelosi, P. Sablonniere)

BIBLIOGRAFIA 2004

P. Costantini, T. Lyche, C. Manni: On a class of weak Tchebycheff systems,preprint, submitted.

P. Costantini, C. Manni: Geometric construction of generalized cubic splines,submitted.

P. Costantini: Properties and applications of new polynomial spaces, to appearon Inter. Jour. Wavel. Mult. Infor.

P. Lamberti, C. Manni: Tensioned quasi-interpolation via geometric continuity,Adv. Comp. Math., 20 (2004), 105-127;

C. Manni, F. Pelosi: Quasi-interpolants with tension properties from and inCAGD, Computing, 72 (2004), 143-160

P. Costantini, F. Pelosi: Shape-preserving approximation of spatial data, Adv.Comp. Math. 20 (2004), 25-51

BIBLIOGRAFIA 2004

P. Costantini, C. Manni: Some applications of new spline spaces in computeraided geometric design, to appear on Rendiconti di Matematica,

P. Costantini, F. Pelosi: Data approximation using new spline spaces, preprint(2004), preprint, submitted,

I. Cravero, C. Manni: Detecting Shape of Spatial Data via Zero Moments,preprint, submitted.

R. T. Farouki, C. Y. Han, C. Manni, A. Sestini, Characterization and constructionof helical polynomial space curves, J. Comp. Appl. Math., 162 (2004), 365-392

F. Pelosi, R. T. Farouki, C. Manni, A. Sestini, Geometric Hermite interpolationby spatial Pythagorean–hodograph cubics, preprint, to appear on Adv. Comp.Math.

BIBLIOGRAFIA 2004

C. Manni, P. Sablonniere: Quasi-interpolant splines on Powell-Sabin partitions,preprint.

P. Costantini, M.L. Sampoli: A general frame for the construction of constrainedcurves, to appear on Proceedings of the Conference on Applied Mathematicsand Scientific Computing.

M.L. Sampoli, B. Juttler and and M.S. Kim, Minkowski sum of free-formsurfaces, preprint.

M. Peternell, M.L. Sampoli and B. Juttler, LN-surfaces and rational convolutionsurfaces, preprint.

QUASI-INTERPOLATION

• quasi-interpolation (q.i.) is a general approach to construct, with lowcomputational cost, efficient local approximants to a given set of data ora given function

Qf :=∑

µi(f)Bi

• µi local linear functionals (easy to compute) on f ;

• {Bi} suitable set of functions usually such that:- have compact support,- are non negative,- sum up to 1.

• Let S denote the linear space spanned by the functions {Bi}. In order tocompletely exploit the approximation power of S

Qp = p, ∀ p ∈ IPk ⊂ S.

QUASI-INTERPOLATION

• one dimension: various effective q.i. schemes based on values of f, and/orits derivatives, and/or its integrals,... ([Schoenberg, 1946], [de Boor, 1973], [Lyche and

Schumaker, 1975], [Sablonniere, 1989], [Schaback and Zongmin, 1997], [Lee, Lyche, and Schumaker, 2001],

[Gori, Pitolli, and Santi, 2002] ...)

• applications to (constrained) approximation, numerical quadrature.....([Dagnino, Demichelis and Santi, 1993], [Schaback and Zongmin, 1994], [Demichelis, 1996], [Conti, Morandi

and Rabut, 1999], [Manni and Pelosi, 2004] ... )

• two dimensions: effective q.i. schemes have been proposed for

- tensor-product spaces ([Sablonniere,... ], [Lamberti and Manni, 2004],...)

- spline spaces over special partitions ∆1 and ∆2 ([Sablonniere, 1985], [Chui, Diamond

and Raphael, 1989], [Sablonniere, 1996], [Dagnino and Lamberti, 2000], [Sablonniere, 2003], ...)

• few specific results for general triangulations.

QUASI-INTERPOLATION

•S(∆PS) := C1 quadratic splines over a PS refinement of ∆

• dim(SPS(∆)) = 3Nv (Nv:= number of vertices of ∆)

• any element of SPS(∆) is determined by its value and its gradient at thevertices of ∆;

• any element of SPS(∆) can be locally computed in each triangle of ∆providing that its values and its gradients at the three vertices are given.

QUASI-INTERPOLATION

Let us associate three functions B(j)l , j = 1, 2, 3, l = 1, . . . , Nv to any vertex

of ∆ such that

• s(x, y) =∑Nv

∑3j=1 ci,jB

• B(j)l (x, y) ≥ 0

•∑Nv

∑3j=1B

(j)l = 1

The functions B(j)l will be the Powell-Sabin B-splines.

The support for any B(j)l is the the union of all triangles containing

the vertex Vl.

QUASI-INTERPOLATION

• we are interested in q.i.s not requiring derivatives of f

Qf :=Nv∑l=1

3∑j=1

µ(j)l (f)B(j)

l , µ(j)l

(f):=∑N

k=1q(j,k)l

f(Z(j,k)l

), (D.q.i.)

• for N (j)l ≤ 3, we determine the geometric configuration of the evaluation

points Z(j,k)l , k = 1, . . . , N (j)

l in order to ensure reproduction of quadraticpolynomials;

• we give an explicit expression for all the D.q.i.s reproducing IP2 for N (j)l = 2;

• we explicitly construct a family of D.q.i.s reproducing IP2 for N (j)l = 3;

• setting Nj)l = 5 we explicitly construct a family of D.q.i.s reproducing IP2

which are projections that is

Qs = s,∀s ∈ S(∆PS);

QUASI-INTERPOLATION

• for all the presented D.q.i.s we provide upper bounds of the infinity norm ofQ in term of the position of the evaluation points Z(j,k)

l and of the family ofused PS B-splines;

• the above mentioned upper bounds ensure that the proposed D.q.i.s providethe optimal (quadratic) approximation order in the space S(∆PS);

• we characterize the used family of PS B-splines which provide D.q.i.s withlower values of the upper bounds of the infinity norm of Q;

• we discuss conditions on the used family of PS B-splines so that theevaluation points Z(j,k)

l can be taken as vertices, or belonging to the edgesof ∆: this makes the proposed D.q.i.s particularly interesting in practicalapplications.

QUASI-INTERPOLATIONExample : f(x, y) = exp(x(1− x)(1− y)y(x− y))

−0.2 0 0.2 0.4 0.6 0.8 1 1.2−0.2

Triangulation ∆ and its Powell-Sabin refinement

0.40.6

f Qf, N(j)

0.40.6

QHf Qf : projection

0 0.2 0.4 0.6 0.8 1

f Qf, N(j)

0 0.2 0.4 0.6 0.8 1

QHf Qf : projection

0 0.2 0.4 0.6 0.8 1

Left to right: triangulations ∆(k), k=0,1,2, and their PS refinements

k Nv interp. q.i. N(j)

l=2 q.i. N

l=3 projection

0 5 0.016997 0.018356 0.016837 0.0135191 13 0.003767 0.003681 0.006568 0.0028562 41 0.000578 0.000573 0.001051 0.0004483 145 0.000078 0.000077 0.000143 0.000093

Error behaviour maxr,s=1,...,50 |f(xr, ys)−Qf(xr, ys)| of different q.i.s

SOMME DI MINKOWSKI

La somma di Minkowski di due oggetti e un’operazione geometrica di basenelle teoria degli insiemi.Negli ultimi anni si e visto come questa operazione sia utile anche inmolte applicazioni, come per esempio nella descrizione delle curve e dellesuperfici ottenute con macchine a controllo numerico o nella determinazione ditraiettorie senza collisioni nell’ ambito della robotica.

In generale dati due oggetti P e Q in IR3 la loro somma di Minkowski e datadall’insieme definito come

P ⊕Q := {p+ q, con p ∈ P, q ∈ Q}

dove p e q denotano le coordinate vettoriali di punti arbitrari in P ed in Q.

SOMME DI MINKOWSKI

Siano ora A = ∂P e B = ∂Q le superfici della frontiera di P e Qrispettivamente; il problema di determinare la superficie di frontiera dellasomma di Minkowski di P e Q, ∂(P ⊕ Q), puo essere ricondotto a quello dicalcolare la superficie di convoluzione di A e B, di solito indicata con A ? B.

Nell’ipotesi che A e B siano superfici regolari dotate ovunque di un campodi vettori normali nA ed nB rispettivamente, si puo definire la superficie diconvoluzione come

A ? B := {a+ b, con a ∈ A, b ∈ B, e nA(a)‖nB(b)}

dove nA(a) e nB(b) sono i vettori normali alle rispettive superfici in a ed in b esono tra loro paralleli.

SOMME DI MINKOWSKI

In particolare si ha che se P e Q sono oggetti convessi, la superficie ∂(P ⊕Q)della somma di Minkowski P ⊕ Q e data esattamente dalla superficie diconvoluzione A ? B.Nel caso generale invece questo non e piu vero. Infatti per due oggettinon convessi la superficie di frontiera ∂(P ⊕ Q) della somma di Minkowskie contenuta nella superficie di convoluzione A ? B.

Studiamo in dettaglio come calcolare la superficie di convoluzione A?B di duesuperfici regolari A e B.Supponiamo che le due superfici siano parametrizzate come a(u, v) e b(s, t)rispettivamente e denotiamo con nA(u, v) e nB(s, t) i relativi vettori normali.

La superficie di convoluzione A?B e data dalla somma di quei punti (vettori) ae b le cui normali nA ed nB sono parallele (ed orientate nello stesso verso).

SOMME DI MINKOWSKI

Pertanto il problema cruciale e quello di determinare una riparametrizzazionedi una delle due superfici a(u(s, t), v(s, t)) = a(s, t) in funzione dei parametridella seconda superficie b(s, t) per poter effettuare la somma di quelle parti diA e B per cui si ha che i vettori normali nA(s, t) e nB(s, t) siano paralleli.

In generale non e detto che esista una riparametrizzazione esplicita. Nelleapplicazioni e desiderabile poter descrivere la superficie di convoluzione(la somma di Minkowski) di due superfici parametriche con la stessarappresentazione usata per la descrizione delle superfici di partenza. Si puopero notare che in generale la convoluzione (somma di Minkowski) di duesuperfici parametriche razionali non e razionale.

SOMME DI MINKOWSKI

L’offset di una superficie puo essere visto come un caso particolare diconvoluzione.Infatti se A e una sfera di raggio d,e B una superficie arbitraria, allora lasuperficie di convoluzione A ? B e proprio l’offset di B a distanza d.Per gli offset si sono studiate delle classi di superfici per cui si potessedescrivere esattamente gli offset (in forma razionale), (Farouki e Sederberg’95, Patrikalakis ’87, Peternell e Pottman ’96) il problema e quindi vedere sefosse possibile generalizzare queste classi.

SOMME DI MINKOWSKI

Si e pertanto studiata una nuova classe di superfici parametriche (le superficiLN) per cui sia possibile dare una forma chiusa per la loro convoluzione(somma di Minkowski) che in particolare risulta essere razionale.

Le superfici LN sono particolari superfici (Juttler ’98, Juttler e S. ’00).Esse hanno la caratteristica di possedere un campo lineare di vettori normali.Tale proprieta, fa si che partendo da superfici polinomiali e possibiledeterminare una riparametrizzazione razionale che permette di esprimere lesuperfici offset in forma razionale.

L’idea e stata quindi quella di estendere tale proprieta al caso generale dellesuperfici di convoluzione.

In un primo lavoro si e studiato il problema della convoluzione (e quindi dellasomma di Minkowski) di due superfici LN.Attualmente si sta studiando il caso ”misto” in cui una superficie siaparametrizzabile in forma razionale mentre la seconda sia una superficie LN.

ELEMENTI TRIANGOLARI GENERALIZZATI

[1] Motivations

[2] Recent univariate results

[3] First results on bivariate triangular extensions

[4] Tensor product extensions and their practical applications

SKETCH OF THE TALK

[1] Motivations

[2] Recent univariate results

SKETCH OF THE TALK

[1] Motivations

[2] Recent univariate results (generalization of polynomial spaces)

SKETCH OF THE TALK

[1] Motivations

SKETCH OF THE TALK

[1] Motivations

[1] – Motivations

Why polynomials and polynomial splines are so popular in CAGD?

� Easy to compute and manipulate

� Positive, normalized bases

� Control polygon

Clssical extensions

� Rational (NURBS), exponential

Recent extensions

� Extension of polynomial spaces

�� – More flexibility on the shape

�� – “B ezier” basis and control polygon

[1] – Motivations

� Control polygon

Clssical extensions

Recent extensions

[1] – Motivations

� Control polygon

Classical extensions

Recent extensions

[1] – Motivations

� Control polygon

Classical extensions

Recent extensions

�� – “Bezier” basis and control polygon

[1] – Motivations

Few examples:

span {1, x, cos(x), sin(x)}, x ∈ [0, α], α < π ;

span{1, x, x2, cos(x), sin(x)

}, x ∈ [0, α], α < π ;

span {1, x, (1− x)m0, xm1}, x ∈ [0, 1]

(Carnicer, Mainar, Pena, Chen, Wang, Mazure, Goodman, Kvasov,Sablonniere, Lyche, Manni, Pelosi, Costantini ...)

[2] – Recent results

[P. C., T. Lyche, C. Manni: On a Class of Weak Tchebycheff Systems, (2003) submitted.]

[2] – Basic ideas

Let t ∈ [a, b] , h = b− a ; n ∈ IN , n ≥ 2

IPn = span{(

b− t

, IPn−2,

(t− a

)n}⇓ ⇓ ⇓

IPu,vn := span {u(t), IPn−2,v(t)}

where:

(1) u, v ∈ Cn+1([a, b])

(2) dim (IPu,vn ) = n+ 1

(3) any non-zero element of span{u(n−1), v(n−1)

}. cannot have two distinct zeros in [a, b]

[2] – Basic ideas

Let t ∈ [a, b] , h = b− a ; n ∈ IN , n ≥ 2

IPn = span{(

b− t

, IPn−2,

(t− a

)n}⇓ ⇓ ⇓

IPu,vn := span {u(t), IPn−2, v(t)}

where:

(1) u, v ∈ Cn+1([a, b])

(2) dim (IPu,vn ) = n+ 1

[2] – Basic ideas

Let t ∈ [a, b] , h = b− a ; n ∈ IN , n ≥ 2

IPn = span{(

b− t

, IPn−2,

(t− a

)n}⇓ ⇓ ⇓

where:

(1) u, v ∈ Cn+1([a, b])

(2) dim (IPu,vn ) = n+ 1

[2] – Basic ideas

Let t ∈ [a, b] , h = b− a ; n ∈ IN , n ≥ 2

IPn = span{(

b− t

, IPn−2,

(t− a

)n}⇓ ⇓ ⇓

where:

(1) u, v ∈ Cn+1([a, b])

(2) dim (IPu,vn ) = n+ 1

[2] – Basic ideas

Let t ∈ [a, b] , h = b− a ; n ∈ IN , n ≥ 2

IPn = span{(

b− t

, IPn−2,

(t− a

)n}⇓ ⇓ ⇓

where:

(1) u, v ∈ Cn+1([a, b])

(2) dim (IPu,vn ) = n+ 1

[2] – Basic ideas

Let t ∈ [a, b] , h = b− a ; n ∈ IN , n ≥ 2

IPn = span{(

b− t

, IPn−2,

(t− a

)n}⇓ ⇓ ⇓

where:

(1) u, v ∈ Cn+1([a, b])

(2) dim (IPu,vn ) = n+ 1

[2] – Basic results

“Bernstein” basis for the space IPu,vn

• For ` = 0, . . . , n− 1 there exists unique bu,v` ∈ IPu,v

n such thatD(i)bu,v

` (a) = 0, i = 0, · · · , `− 1, D(`)bu,v` (a) = β`,

D(i)bu,v` (b) = 0, i = 0, · · · , n− `− 1

• There exists unique bu,vn ∈ IPu,v

n such thatD(i)bu,v

n (a) = 0, i = 0, · · · , n− 1, bu,vn (b) = 1.

• The elements bu,v` ∈ IPu,v

n , ` = 0, · · · , n are linearly independent .

• It possible to select positive values β` so that• - 1 ≡

∑n`=0 b

• - bu,v` (x) > 0, x ∈ (a, b), ` = 0, · · · , n.

(positive, normalized basis)

n such thatD(i)bu,v

` (a) = 0, i = 0, · · · , `− 1, D(`)bu,v` (a) = β`,

D(i)bu,v` (b) = 0, i = 0, · · · , n− `− 1

n such thatD(i)bu,v

n (a) = 0, i = 0, · · · , n− 1, bu,vn (b) = βn.

n , ` = 0, · · · , n are linearly independent .

∑n`=0 b

• - bu,v` (x) > 0, x ∈ (a, b), ` = 0, · · · , n.

n such thatD(i)bu,v

` (a) = 0, i = 0, · · · , `− 1, D(`)bu,v` (a) = β`,

D(i)bu,v` (b) = 0, i = 0, · · · , n− `− 1

n such thatD(i)bu,v

n (a) = 0, i = 0, · · · , n− 1, bu,vn (b) = βn.

n , ` = 0, · · · , n are linearly independent.

∑n`=0 b

• - bu,v` (x) > 0, x ∈ (a, b), ` = 0, · · · , n.

n such thatD(i)bu,v

` (a) = 0, i = 0, · · · , `− 1, D(`)bu,v` (a) = β`,

D(i)bu,v` (b) = 0, i = 0, · · · , n− `− 1

n such thatD(i)bu,v

n (a) = 0, i = 0, · · · , n− 1, bu,vn (b) = βn.

n , ` = 0, · · · , n are linearly independent.

∑n`=0 b

• - bu,v` (x) > 0, x ∈ (a, b), ` = 0, · · · , n.

[2] – Basic resultsExample: “Bernstein” basis for IPu,v

3 = span {u(t), IP1, v(t)}, [a, b] = [0, 1]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u(t) = (1− t)3, v(t) = t3, u(t) =√

(1− t)9, v(t) = t3π

B-basis for the space IPu,vn

Theorem.The basis {bu,v

` ∈ IPu,vn , ` = 0, · · · , n} is strictly totally positive in (a, b) and

totally positive in [a, b].

Corollary.The basis {bu,v

` ∈ IPu,vn , ` = 0, · · · , n} is the normalized B-basis for IP u,v

Remark:all the previous results can be proved in an elementary way by usingproperties of the zeros of elements belonging to span

{u(n−1), v(n−1)

Corollary.

The basis {bu,v` ∈ IPu,v

n , ` = 0, · · · , n} is the normalized B-basis for IP u,vn .

Remark:

all the previous results can be proved in an elementary way by usingproperties of the zeros of elements belonging to span

{u(n−1), v(n−1)

` ∈ IPu,vn , ` = 0, · · · , n} is the normalized B-basis for IPu,v

Remark:

all the previous results can be proved in an elementary way by usingproperties of the zeros of elements belonging to span

{u(n−1), v(n−1)

` ∈ IPu,vn , ` = 0, · · · , n} is the normalized B-basis for IPu,v

Remark:all the previous results can be proved in an elementary way by using propertiesof the zeros of elements belonging to span

{u(n−1), v(n−1)

[2] – Special caseSpecial case: IPu,v

n = IPαn

[a, b] = [0, 1] , u(x) := (1− x)α, v(x) := xα , α ∈ IR , α ≥ n− 2.

The particular and symmetric structure of the space ⇒ additionalresults:

Marsden identity : x =n∑

ξn,αi bn,α

ξn,α0 = 0, ξn,α

n = 1; ξn,αi =

i− 1n− 2

)α− 2α

, i = 1, · · · ,n− 1 .

n = IPαn

The particular and symmetric structure of the space ⇒ additionalresults:

ξn,αi bn,α

ξn,α0 = 0, ξn,α

n = 1; ξn,αi =

i− 1n− 2

)α− 2α

, i = 1, · · · ,n− 1 .

n = IPαn

The particular and symmetric structure of the space ⇒ additional results:

ξn,αi bn,α

ξn,α0 = 0, ξn,α

n = 1; ξn,αi =

i− 1n− 2

)α− 2α

, i = 1, · · · ,n− 1 .

n = IPαn

ξn,αi bn,α

ξn,α0 = 0, ξn,α

n = 1; ξn,αi =

+(i− 1n− 2

)α− 2α

, i = 1, · · · , n− 1 .

n = IPαn

ξn,αi bn,α

ξn,α0 = 0, ξn,α

n = 1; ξn,αi =

+(i− 1n− 2

)α− 2α

, i = 1, · · · , n− 1 .

0 1 1/α 1−(1/α)

ξ0 ξ

P3α , α=3π

n = IPαn

ξn,αi bn,α

ξn,α0 = 0, ξn,α

n = 1; ξn,αi =

+(i− 1n− 2

)α− 2α

, i = 1, · · · , n− 1 .

P4α , α=3π

0 1/α 1−1/α 1

ξ1 ξ

n = IPαn

ξn,αi bn,α

ξn,α0 = 0, ξn,α

n = 1; ξn,αi =

+(i− 1n− 2

)α− 2α

, i = 1, · · · , n− 1 .

P6α , α=3π

ξ0 ξ

2 ξ3 ξ

0 1/α 1−1/α 1

n = IPαn

IPαn ⊂ IPα

n+1 ⇒ degree elevation

bn,αi = θn

i bn+1,αi +

(1− θn

)bn+1,α

i+1 , i = 0,1, . . . ,n ,

whereθn0 = 1 ; θn

i =n− in− 1

, i = 1, . . . ,n ; θnn+1 = 0

n = IPαn

IPαn ⊂ IPα

bn,αi = θn

i bn+1,αi +

(1− θn

)bn+1,α

i+1 , i = 0,1, . . . ,n ,

whereθn0 = 1 ; θn

i =n− in− 1

, i = 1, . . . ,n ; θnn+1 = 0

n = IPαn

IPαn ⊂ IPα

bn,αi = θn

i bn+1,αi +

(1− θn

)bn+1,αi+1 , i = 0, 1, . . . , n ,

whereθn0 = 1 ; θn

i =n− in− 1

, i = 1, . . . ,n ; θnn+1 = 0

n = IPαn

IPαn ⊂ IPα

bn,αi = θn

i bn+1,αi +

(1− θn

)bn+1,αi+1 , i = 0, 1, . . . , n ,

whereθn0 = 1 ; θn

i =n− i

n− 1, i = 1, . . . , n ; θn

n+1 = 0

[3] – Triangular extensions

Work in progress

• no general result

• only “cubic” and “quartic” cases

[Joint work with Tom Lyche and Carla Manni]

Work in progress

[3] – NotationsStandard triangle T :

P1=(0,0,1) P

2=(1,0,0)

P3=(0,1,0)

Barycentric coordinates r, s, t ; r + s+ t = 1

[3] – Notations

space of univariate ν-degree polynomials in ξ, η:

IPν(ξ, η) := span {bνi (ξ, η), i = 0, . . . , ν} , ξ ∈ [0, 1], η = 1− ξ

Bernstein polynomials:

bνi (ξ, η) :=(ν

)ξiην−i

space of bivariate polynomials of total degree ν:

IΠν := span{Bν

i,j,k(r, s, t), i+ j + k = ν}

Bernstein polynomials on the standard triangle:

Bνi,j,k(r, s, t) =

ν!i!j!k!

risjtk ; r + s+ t = 1

[3] – Notations

space of univariate ν-degree polynomials in ξ, η:

IPν(ξ, η) := span {bνi (ξ, η), i = 0, . . . , ν} , ξ ∈ [0, 1], η = 1− ξ

Bernstein polynomials:

bνi (ξ, η) :=(ν

)ξiην−i

space of bivariate polynomials of total degree ν:

IΠν := span{Bν

i,j,k(r, s, t), i+ j + k = ν}

Bernstein polynomials on the standard triangle:

Bνi,j,k(r, s, t) =

ν!i!j!k!

risjtk ; r + s+ t = 1

[3] – Basic ideas, “quartic” case

Replace IΠ4 with IΠα4

� Extend IΠ4

- IΠ44 ≡ IΠ4

- dim (IΠα4 ) = dim (IΠ4) = 15

- IΠ4 and IΠα4 should have similar “boundary properties”

� Extend the 1D ideas of IP α4

- IP4(ξ, η) = span{η4, IP2(ξ, η), ξ4

}→ IPα

4 (ξ, η) = span {ηα, IP2(ξ, η), ξα}

- As α→∞ , IP2(ξ, η) “covers” [0, 1]

� Extend IΠ4

- IΠ44 ≡ IΠ4

- dim (IΠα4 ) = dim (IΠ4) = 15

� Extend the 1D ideas of IP α4

- IP4(ξ, η) = span{η4, IP2(ξ, η), ξ4

}→ IPα

- As α→∞ , IP2(ξ, η) “covers” [0, 1]

� Extend IΠ4

- IΠ44 ≡ IΠ4

- dim (IΠα4 ) = dim (IΠ4) = 15

� Extend the 1D ideas of IPα4

- IP4(ξ, η) = span{η4, IP2(ξ, η), ξ4

}→ IPα

- As α→∞ , IP2(ξ, η) “covers” [0, 1]

[3] – Basic ideas, “quartic” caseB

0404 =s4

B0044 =t4 B

4004 =r4 B

1034 =4rt3 B

2024 =6r2t2 B

3014 =4r3t

B0134 =4t3s

B0224 =6t2s2

B0314 =4ts3

B3104 =4sr3

B2204 =6s2r2

B1304 =4s3r

B2114 =12r2st

B1214 =12rs2t

B1124 =12rst2

IΠ4 = span{B4

0,0,4, B41,0,3, B

42,0,2, B

43,0,1, B

44,0,0, B

43,1,0, B

42,2,0, B

41,3,0,

B40,4,0, B

40,3,1, B

40,2,2, B

40,1,3, B

41,1,2, B

42,1,1, B

41,2,1

B0404 =s4

B0044 =t4 B

4004 =r4

First step: “vertex” functions

0404 =s4

B0044 =t4 B

4004 =r4 B

1034 =4rt3 B

2024 =6r2t2 B301

4 =4r3t

B0134 =4t3s

B0224 =6t2s2

B0314 =4ts3

B3104 =4sr3

B2204 =6s2r2

B1304 =4s3r

Second step: “boundary” functions

Take, e.g., the edge e2 : r + s = 1 ↔ t = 0

(1) B4,αi,j,0(r, s,0) ∈ IP4(s, r) ≡ span

{r4, IP2(s, r), s4

∑i+j=4 B4,α

i,j,0(r, s, t) ≡ (r + s)4 (“4− decay′′)

(2) B4,αi,j,0(r, s, t) ∈ span

{r4, (r + s)2IP2, s4

(1) B4i,j,0(r, s, 0) ∈ IP4(s, r) ≡ span

{r4, IP2(s, r), s4

∑i+j=4 B4

i,j,0(r, s, t) ≡ (r + s)4 (“4− decay′′)

(2) B4i,j,0(r, s, t) ∈ span

{r4, (r + s)2IP2, s4

(1) B4i,j,0(r, s, 0) ∈ IP4(s, r) ≡ span

{r4, IP2(s, r), s4

∑i+j=4B

4i,j,0(r, s, t) ≡ (r + s)4 (“4− decay′′)

(2) W4i,j,0(r, s, t) ∈ span

{r4, (r + s)2IP2, s4

(1) B4i,j,0(r, s, 0) ∈ IP4(s, r) ≡ span

{r4, IP2(s, r), s4

∑i+j=4B

4i,j,0(r, s, t) ≡ (r + s)4 (“4− decay′′)

(3) B4i,j,0(r, s, t) ∈ span

{r4, (r + s)2IP2(s, r), s4

0404 =s4

B0044 =t4 B

4004 =r4

(s+r)2P2(s,r) (t+s)2P

2(t,s)

(r+t)2P2(r,t)

Second step: Reformulation of the “boundary” functions

0404 =s4

B0044 =t4 B

4004 =r4 B

1034 =4rt3 B

2024 =6r2t2 B

3014 =4r3t

B0134 =4t3s

B0224 =6t2s2

B0314 =4ts3

B3104 =4sr3

B2204 =6s2r2

B1304 =4s3r

Third step: “internal” functions

0404 =s4

B0044 =t4 B

4004 =r4 B

1034 =4rt3 B

2024 =6r2t2 B

3014 =4r3t

B0134 =4t3s

B0224 =6t2s2

B0314 =4ts3

B3104 =4sr3

B2204 =6s2r2

B1304 =4s3r

Π1=span{t,r,s}

Requirement for the first step:- More flexibility in the vertex functions

B0404,α =sα

B0044,α =tα B

4004,α =rα

Requirement for the first step:- More flexibility in the vertex functions

Take, e.g., the edge e2 : r + s = 1 ↔ t = 0.

Requirements for the boundary functions

(1) B4,αi,j,0(r, s,0) ∈ IPα

4 (s, r) ≡ span {rα, IP2(s, r), sα}

(2)∑

i+j=4 B4,αi,j,0(r, s, t) ≡ (r + s)α (“α− decay′′)

(2) B4,αi,j,0(r, s, t) ∈ span

{r4, (r + s)(α− 2)IP2, s4

(1) B4,αi,j,0(r, s, 0) ∈ IPα

(2)∑

i+j=4 B4,αi,j,0(r, s, t) ≡ (r + s)4 (“4− decay′′)

(2) B4,αi,j,0(r, s, t) ∈ span

{r4, (r + s)2IP2, s4

(1) B4,αi,j,0(r, s, 0) ∈ IPα

(2)∑

i+j=4B4,αi,j,0(r, s, t) ≡ (s+ r)α (“α− decay′′)

(2) W4,αi,j,0(r, s, t) ∈ span

{r4, (s + r)2IP2, s4

(1) B4,αi,j,0(r, s, 0) ∈ IPα

(2)∑

i+j=4B4,αi,j,0(r, s, t) ≡ (s+ r)α (“α− decay′′)

(3) B4,αi,j,0(r, s, t) ∈ span

{rα, (s+ r)α−2IP2(s, r), sα

B0404,α =sα

(s+r)α−2P2(s,r) (t+s)α−2P

2(t,s)

(r+t)α−2P2(r,t) B

4004,α =rα B

0044,α =tα

Second step: “boundary” functions

B0404,α =sα

(s+r)α−2P2(s,r) (t+s)α−2P

2(t,s)

(r+t)α−2P2(r,t) B

4004,α =rα B

0044,α =tα

Π1=span{t,r,s}

[3] – Triangular extension, “quartic” case

Definition

IΠα4 := span

{tα, (r + t)α−2IP2(r, t), rα, (s+ r)α−2IP2(s, r),

sα, (t+ s)α−2IP2(t, s), IΠ1

}that is

IΠα4 := span

{tα, (t+ r)α−2t2, (t+ r)α−2tr, (t+ r)α−2r2,rα, (r + s)α−2r2, (r + s)α−2rs, (r + s)α−2s2,sα, (s+ t)α−2s2, (s+ t)α−2st, (s+ t)α−2t2,IΠ1}

Theorem

The above functions are linearly independent

[3] – Some Hermite problems

Problem H(P1): find ψ ∈ IΠα4 s.t.

ψ(P1) = f1 ; D`(−e1)`ψ(P2) = 0 , ` ≤ 3

D`+m(−e2)`(e3)mψ(P3) = 0 , `+m ≤ 3 .

Problem He1(P1): find ψ ∈ IΠα4 s.t.

ψ(P1) = f1 ; De1ψ(P1) = fe11 ; D`

(−e1)`ψ(P2) = 0 , ` ≤ 2

D`+m(−e2)`(e3)mψ(P3) = 0 , `+m ≤ 3 .

Problem H(e1)2(P1): find ψ ∈ IΠα

4 s.t.

ψ(P1) = f1 ; De1ψ(P1) = fe11 ; D2

(e1)2ψ(P1) = f

1 ; D`(−e1)`ψ(P2) = 0 , ` ≤ 1 ;

D`+m(−e2)`(e3)mψ(P3) = 0 , `+m ≤ 3 .

Problem H(e1)3(P1): find ψ ∈ IΠα

4 s.t.

ψ(P1) = fν ; De1ψ(P1) = fe11 ; D2

(e1)2ψ(P1) = f

1 ; D3(e1)3

ψ(P1) = fe31

ψ(P2) = 0 ; D`+m(−e2)`(e3)mψ(P3) = 0 , `+m ≤ 3 .

Problem He1(−e2)(P1): find ψ ∈ IΠα4 s.t.

ψ(P1) = f1 ; D`+r(e1)`(−e2)rψ(P1) = f

(e1)`(−e2)

1 ; `+ r = 1, 2 ;D`

(e2)`ψ(P2) = D`(−e2)`ψ(P3) = 0, ` = 0, 1, 2;D2

(−e1)e2ψ(P2) = D2

e3(−e2)ψ(P3) = 0 .

Similar Hermite problems can be “centered” in P2 and P3.

TheoremThere exists an unique solution to each of the above Hermite problems.

Similar Hermite problems can be “centered” in P2 and P3.

TheoremThere exists an unique solution to each of the above Hermite problems.

[3] – B ezier basis

Theorem

It is possible to find values f∗∗ such that the solutions of the properinterpolation problems produce linearly independent functions

B4,αi,j,k ; i+ j + k = 4

such that:(1) B4,4

i,j,k = B4i,j,k ;

(2) B4,αi,j,k(r, s, t) ≥ 0 ⇔ B4

i,j,k(r, s, t) ≥ 0 ;

(3)∑

i+j+k=4B4,αi,j,k ≡ 1

Theorem

B4,αi,j,k ; i+ j + k = 4

such that:(1) B4,4

i,j,k = B4i,j,k ;

(2) B4,αi,j,k(r, s, t) ≥ 0 ⇔ B4

i,j,k(r, s, t) ≥ 0 ;

(3)∑

i+j+k=4 B4,αi,j,k ≡ 1

Theorem

B4,αi,j,k ; i+ j + k = 4

such that:(1) B4,4

i,j,k = B4i,j,k ;

(2) B4,αi,j,k(r, s, t) ≥ 0 ⇔ B4

i,j,k(r, s, t) ≥ 0 ;

(3) sumi+j+k=4B4,αi,j,k ≡ 1

Theorem

B4,αi,j,k ; i+ j + k = 4

such that:(1) B4,4

i,j,k = B4i,j,k ;

(2) B4,αi,j,k(r, s, t) ≥ 0 ⇔ B4

i,j,k(r, s, t) ≥ 0 ;

(3)∑

i+j+k=4B4,αi,j,k ≡ 1

Theorem

B4,αi,j,0(r, s, 0) = b4,α

i (s, r)

(On the edge e2 the basis functions of IΠα4 and IPα

4 (r, s) coincide)

Theorem

B4,αi,j,0(r, s, 0) = b4,α

i (s, r)

(On the edge e2 the basis functions of IΠα4 and IPα

4 (r, s) coincide)

B4,α0,0,4 : H(P1) with f1 = 1.

B4,α1,0,3: He1(P1) with f1 = 0, fe1

1 = −De1B4,α0,0,4(P1) .

B4,α1,0,3 example: α = 4

B4,α1,0,3 example: α = 4 + π

B4,α1,0,3 example: α = 8 + e

B4,α2,0,2 : H(e1)2

(P1) with f1 = 0 ;

1 = −D`(e1)`(B

4,α0,0,4(P1) +B4,α

1,0,3(P1)) ; ` = 1, 2.

B4,α1,1,2 : He1(−e3)(P1) with f1 = 0 ;

fe`1(−e3)

1 = −D`+m

e`1(−e3)m

∑2k=4B

4,αi,j,k(P1) ; 1 ≤ `+m ≤ 2

B4,α1,1,2 example: α = 4

B4,α1,1,2 example: α = 4 + π

B4,α1,1,2 example: α = 8 + e

[3] – Control polygon

t ∈ IΠα4 ⇔ t =

∑i+j+k=4

ξ(t)i,j,kB

4,αi,j,k(r, s, t) .

Theoremξ(t)0,0,4 = 1 ; ξ(t)1,0,3 = ξ

(t)0,1,3 = 1− 1

α; ξ(t)1,1,2 = 1− 2

ξ(t)2,0,2 = ξ

(t)0,2,2 =

; ξ(t)3,0,1 = ξ(t)2,1,1 = ξ

(t)1,2,1 = ξ

(t)0,3,1 =

ξ(t)4,0,0 = ξ

(t)3,1,0 = ξ

(t)2,2,0 = ξ

(t)1,3,0 = ξ

(t)0,4,0 = 0 .

[3] – Control polygon

t ∈ IΠα4 ⇔ t =

∑i+j+k=4

ξ(t)i,j,kB

4,αi,j,k(r, s, t) .

Theoremξ(t)0,0,4 = 1 ; ξ(t)1,0,3 = ξ

(t)0,1,3 = 1− 1

α; ξ(t)1,1,2 = 1− 2

ξ(t)2,0,2 = ξ

(t)0,2,2 =

; ξ(t)3,0,1 = ξ(t)2,1,1 = ξ

(t)1,2,1 = ξ

(t)0,3,1 =

ξ(t)4,0,0 = ξ

(t)3,1,0 = ξ

(t)2,2,0 = ξ

(t)1,3,0 = ξ

(t)0,4,0 = 0 .

[3] – Control polygonSimilar representations hold for r ∈ IΠα

4 and s ∈ IΠα4

ψ =∑

i+j+k=4

βi,j,kB4,αi,j,k

the piecewise linear function L3,α connecting the coefficients βi,j,k at thepoints Qi,j,k given by

Q0,0,4 = (0, 0, 1), Q1,0,3 =

α, 0, 1−

), Q2,0,2 =

), Q3,0,1 =

α, 0,

Q4,0,0 = (1, 0, 0), Q3,1,0 =

), Q2,2,0 =

), Q1,3,0 =

α, 1−

Q0,4,0 = (0, 1, 0), Q0,3,1 =

(0, 1−

), Q0,2,2 =

), Q0,1,3 =

α, 1−

Q1,1,2 =

α, 1−

), Q2,1,1 =

), Q1,2,1 =

α, 1−

)is the control polygon with the usual end tangent properties.

4 and s ∈ IΠα4

ψ =∑

i+j+k=4

βi,j,kB4,αi,j,k

Q0,0,4 = (0, 0, 1), Q1,0,3 =

α, 0, 1−

), Q2,0,2 =

), Q3,0,1 =

α, 0,

Q4,0,0 = (1, 0, 0), Q3,1,0 =

), Q2,2,0 =

), Q1,3,0 =

α, 1−

Q0,4,0 = (0, 1, 0), Q0,3,1 =

(0, 1−

), Q0,2,2 =

), Q0,1,3 =

α, 1−

Q1,1,2 =

α, 1−

), Q2,1,1 =

), Q1,2,1 =

α, 1−

4 and s ∈ IΠα4

ψ =∑

i+j+k=4

βi,j,kB4,αi,j,k

Q0,0,4 = (0, 0, 1), Q1,0,3 =

α, 0, 1−

), Q2,0,2 =

), Q3,0,1 =

α, 0,

Q4,0,0 = (1, 0, 0), Q3,1,0 =

), Q2,2,0 =

), Q1,3,0 =

α, 1−

Q0,4,0 = (0, 1, 0), Q0,3,1 =

(0, 1−

), Q0,2,2 =

), Q0,1,3 =

α, 1−

Q1,1,2 =

α, 1−

), Q2,1,1 =

), Q1,2,1 =

α, 1−

[3] – Control polygon: example

α = 4

α = 5

α = 7.1416

α = 10.7183

[3] – Triangular extension, examples

α = 4

α = 4 + π

α = 8 + e

[3] – Triangular extension, “cubic” case

Similar construction

tα rα

(r+t)α−1P1(r,t)

(s+r)α−1P1(s,r)

(t+s)α−1P1(t+s)

Π0=span{1}

IΠ3 → IΠα3

[3] – Triangular extension, comments

� Extension of cubic and quartic bivariate polynomials� IΠα

3 → IΠ0 , IΠα4 → IΠ1

� Further extensions ?� • General functions: tα → u(t), rα → v(r), sα → w(s)� • (tα1, rα2, sα3 for possible applications of IΠα

� • in practical tension schemes)� • IΠα

3 → IΠ0 , IΠα4 → IΠ1

tα rα

(r+t)βPn−2

(s+r)βPn−2

(s,r) (t+s)βPn−2

Πn−3

β=α−(n−2)

[4] – Tensor-product extensions

[Based on joint works with Francesca Pelosi (2000-2004)]

[4] – Basic univariate results

Particular case: IP u,vn = IPk0,k1

n , n = 2, 3, 4 ; k0, k1 ∈ IN , k0, k1 ≥ n− 2.

u(x) := (1− x)k0 , v(x) := xk1

Case n = 2IPk0,k1

2 := span{(1− x)k0, IP0, x

x ∈ [a, b] ⊂ [0, 1] ; p ∈ IP0 ; ψ ∈ IPk0,k12 , ψ(x) = a0(1− x)k0 + p(x) + a1x

limk0,k1→∞

ψ(x) = p(x)(

IPk0,k12 → IP0

n , n = 2, 3, 4 ; k0, k1 ∈ IN , k0, k1 ≥ n− 2.

u(x) := (1− x)k0 , v(x) := xk1

Case n = 2IPk0,k1

2 := span{(1− x)k0, IP0, x

limk0,k1→∞

ψ(x) = p(x)(

IPk0,k12 → IP0

n , n = 2, 3, 4 ; k0, k1 ∈ IN , k0, k1 ≥ n− 2.

u(x) := (1− x)k0 , v(x) := xk1

Case n = 2IPk0,k1

2 := span{(1− x)k0, IP0, x

limk0,k1→∞

ψ(x) = p(x)(

IPk0,k12 → IP0

n , n = 2, 3, 4 ; k0, k1 ∈ IN , k0, k1 ≥ n− 2.

u(x) := (1− x)k0 , v(x) := xk1

Case n = 2IPk0,k1

2 := span{(1− x)k0, IP0, x

limk0,k1→∞

ψ(x) = p(x)(

IPk0,k12 → IP0

n , n = 2, 3, 4 ; k0, k1 ∈ IN , k0, k1 ≥ n− 2.

u(x) := (1− x)k0 , v(x) := xk1

Case n = 3IPk0,k1

3 := span{(1− x)k0, IP1, x

limk0,k1→∞

ψ(x) = p(x)(

IPk0,k13 → IP0

n , n = 2, 3, 4 ; k0, k1 ∈ IN , k0, k1 ≥ n− 2.

u(x) := (1− x)k0 , v(x) := xk1

Case n = 3IPk0,k1

3 := span{(1− x)k0, IP1, x

limk0,k1→∞

ψ(x) = p(x)(

IPk0,k13 → IP1

n , n = 2, 3, 4 ; k0, k1 ∈ IN , k0, k1 ≥ n− 2.

u(x) := (1− x)k0 , v(x) := xk1

Case n = 4IPk0,k1

4 := span{(1− x)k0, IP2, x

limk0,k1→∞

ψ(x) = p(x)(

IPk0,k14 → IP2

n , n = 2, 3, 4 ; k0, k1 ∈ IN , k0, k1 ≥ n− 2.

u(x) := (1− x)k0 , v(x) := xk1

Case n = 4IPk0,k1

4 := span{(1− x)k0, IP2, x

limk0,k1→∞

ψ(x) = p(x)(

IPk0,k14 → IP2

[4] – Basic univariate resultsSPLINE SPACES

[possibly extended] knot sequence

[· · · < r−2 < r−1 <]r0 < r1 < · · · < rN [< rN+1 < rN+2 < · · ·]

[possibly extended] degree sequence

[· · · < k−2 < k−1 <]k0 < k1 < · · · < kN [< kN+1 < kN+2 < · · ·]

IPki,ki+1n,i := span

{(1− ρ)ki, IPn−2, ρ

ki+1}, ρ =

r − riri+1 − ri

GENERAL SPLINE SPACES

VSkn,r =

{s ∈ Cm[r0, rN ] : s|[ri,ri+1] ∈ IPki,ki+1

[4] – “Quartic” splines

n = 4 ,m = 2 (the case m = 3 is also possible)

VSk4,r =

{s ∈ C2[r0, rN] : s|[ri,ri+1] ∈ IPki,ki+1

• For limit values of the degrees: k0, . . . ,kN →∞

VSk4,r → S0

• For technical reasons in data approximation

IPk0,k14,0 = span

{IP2, rk1

}, IPkN−1,kN

4,N−1 = span{(1− r)kN−1, IP2

}so that

dim(VSk

)= dim S0

VSk4,r =

{s ∈ C2[r0, rN] : s|[ri,ri+1] ∈ IPki,ki+1

VSk4,r → S0

IPk0,k14,0 = span

{IP2, rk1

}, IPkN−1,kN

4,N−1 = span{(1− r)kN−1, IP2

}so that

dim(VSk

)= dim S0

VSk4,r =

{s ∈ C2[r0, rN ] : s|[ri,ri+1] ∈ IPki,ki+1

VSk4,r → S0

IPk0,k14,0 = span

{IP2, rk1

}, IPkN−1,kN

4,N−1 = span{(1− r)kN−1, IP2

}so that

dim(VSk

)= dim S0

VSk4,r =

• For limit values of the degrees: k0, . . . , kN →∞

VSk4,r → S0

IPk0,k14,0 = span

{IP2, rk1

}, IPkN−1,kN

4,N−1 = span{(1− r)kN−1, IP2

}so that

dim(VSk

)= dim S0

VSk4,r =

• For limit values of the degrees: k0, . . . , kN →∞

VSk4,r → S0

IPk0,k14,0 = span

{IP2, r

k1}, IPkN−1,kN

4,N−1 = span{(1− r)kN−1, IP2

}• so that

dim(VSk

)= dim S0

[4] – “Quartic” splines: geometric construction

GEOMETRIC CONSTRUCTION: the polynomial control points areobtained via convex combinations of the spline control points

Di−1

hi−1

Classical example: C2 cubic splines

FUNDAMENTAL RESULT: The geometric construction holds for VSk4,r

REMARK: approximately a quadratic computational cost

Di−1

Si−1

Example: ki = ki+1 = 6

Si−1

Di−1

i+1−

fi−1+

fi+2−

Si−1

Di−1

i+1−

fi−1+

fi+2−

Si−1

Di−1

i+1−

fi−1+

fi+2−

Si−1

Di−1

i+1−

fi−1+

fi+2−

Si−1

Di−1

i+1−

fi−1+

fi+2−

REMARK: approximately the computational cost of quadratic splines .

WHY GEOMETRIC CONSTRUCTION IS SO IMPORTANT ?

⇒ the spline shape can be controlled using the de Boor control polygon

⇒ the asymptotic properties of the spline can be deduced from those ofthe de Boor control polygon

⇒ tool for the construction of a B-spline basis

→ applications to least squares approximation→ extensions to tensor-product and Boolean sum parametric surfaces

[4] – “Quartic” splines: B-splines

s ∈ VSk4,r, s =

∑N1 DiN

∑N0 SiN

s ∈ VSk4,r, s =

∑N1 DiN

∑N0 SiN

k = {4, 4, 4, 4, 4, 4, 4, 4}

s ∈ VSk4,r, s =

∑N1 DiN

∑N0 SiN

k = {4, 4, 4, 10, 10, 4, 4, 4}

s ∈ VSk4,r, s =

∑N1 DiN

∑N0 SiN

k = {10, 10, 10, 10, 10, 10, 10, 10}

s ∈ VSk4,r, s =

∑N1 DiN

∑N0 SiN

k = {40, 40, 40, 40, 40, 40, 40, 40}

s ∈ VSk4,r, s =

∑N1 DiN

∑N0 SiN

C0 Quadratic B-spline Basis

[4] – “Quartic” splines: tensor-product

• r = {r0, . . . , rN}, kr = {kr0, . . . , k

• s = {s0, . . . , sM}, ks = {ks0, . . . , k

4,r⊗

• Classical geometric construction

• Surface with the same shape of the De Boor control polygon

[4] – “Quartic” splines: tensor-product

• B–spline basis

Dj : kr

i = ksj = 4 kr

2 = kr3 = 40 kr

2 = kr3 = ks

2 = ks3 = 40

Sj : kr

i = ksj = 4 kr

2 = kr3 = 40 kr

2 = kr3 = ks

2 = ks3 = 40

[4] – “Quartic” splines: surface approximation

• Data:{(ui, vj, Ii,j); i = 1, . . . , Nu ; j = 1, . . . ,Mv},

BASIC ASSUMPTIONS

• The object can be approximated by a tensor product surface

• The shape of the object is “simple”, weak noise ⇒ it can be describedby biquadratic patches

• The parameter values (ui, vj) are given

(1) Extract significant knot sequences from the parameter valuesr = {r0, . . . , rN} ⊂ {u0, . . . , uNu} , s = {s0, . . . , sM} ⊂ {v0, . . . , vMv}

(2) Compute σ∗ ∈ S02

2 s.t.

Nu∑i=1

Mv∑j=1

‖σ∗(ui, vj)− Ii,j‖2 <Nu∑i=1

Mv∑j=1

‖σ(ui, vj)− Ii,j‖2, ∀σ

(3) Define the shape of data as the shape of σ∗

(4) Compute s∗ ∈ VSkr

4,r⊗

4,s s.t.

Nu∑i=1

Mv∑j=1

‖s∗(ui, vj)− Ii,j‖2 <Nu∑i=1

Mv∑j=1

‖s(ui, vj)− Ii,j‖2, ∀s

(5) Convergence Result:

- If kri , k

sj →∞ for all indices,

=⇒ ‖s∗ − σ∗‖∞ −→ 0

- For sufficient large degree sequences kr and ks,

=⇒ s∗ ∈ VSkr

4,r⊗

4,s has the same shape of σ∗

(6) Find kr and ks

• s∗ reproduces the convexity of σ∗ (“zero moment” approach)• Check on the fairness properties

[4] – “Quartic” splines: example

Data points Grid lines selection

C0 Biquadratic Surface kr2 = 10, kr

i = 4, ksj = 10

⇒ Local shape control using weighted least squares

C0 Biquadratic Surface kr2 = 10, kr

i = 4, ksj = 10

⇒ Local shape control using weighted least squares

[4] – “Quadratic” splinesn = 2,m = 1 VSk

2,r ={s ∈ C1[r0, rN ] : s|[ri,ri+1] ∈ IPki,ki+1

}B-spline Basis, tensor-product extension

Useful for the reproduction of density functions from histograms• area/volume matching conditions• co-monotonicity

0 5 10 15−20

BRONCHUSSTOMACH

COLONOVARY

BREAST

SURVIVAL TIME(days)

−500

BRONCHUSSTOMACH

COLONOVARY

BREAST

SURVIVAL TIME(days)

−500

[4] – “Cubic” splines

n = 3,m = 2 VSk3,r =

All the properties and applications of cubic splines• interpolation, approximation, free-form design• NURBS

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