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 Title: Isotropic box-splines on the A*n lattice
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 Material Information Title: Isotropic box-splines on the A*n lattice Physical Description: Book Language: English Creator: Peters, JorgKim, Minho Publisher: Department of Computer and Information Science and Engineering, University of Florida Place of Publication: Gainesville, Fla. Publication Date: February 27, 2008 Copyright Date: 2008
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Isotropic Box-splines on the A* lattice

.In i Kim and Jorg Peters

February 27, 2008

Abstract
\\* show that the isotropic n-variate box-spline M, on the A* lat-
tice naturally generalizes bivariate box-splines on the hexagonal lattice
and trivariate box-splines on the BCC (Body-Centered Cubic) lattice and
shares the desirable properties of the box-spline i. ,I!!- Mr. Reconstruc-
tion from samples on the A* lattice with M, is information-theoretically
more t i. i for bounded frequency data than reconstruction with tensor-
product B-splines on the Cartesian lattice.

1 Introduction

As a generalization of uniform B-splines to several variables, box-splines have
been used in i ii~i areas. In most applications, shifts of one box-spline on
the Cartesian lattice are used. In particular, a family Mr of n-variate box-
splines is popular due to their linear independence and approximation proper-
ties (Section 3.2). Members of Mr are defined by r-fold convolution, in the n
directions of the Cartesian grid plus a diagonal, so that the footprint of these
box-splines is anisotropically distorted in the diagonal direction. To improve re-
construction of vector fields by isotropic convolution, convolution and shifts on
2- and 3-dimensional non-Cartesian lattices have only recently been considered
[ .1, 53, 52, 29, 28, 30, 271.
In this paper, we propose a generalization of isotropic bivariate box-splines on
the hexagonal lattice and trivariate isotropic box-splines on the BCC lattice
- to isotropic n-variate box-splines M, (Section 6.2) by leveraging a natural
connection to the A*, lattice (Section 4). This lattice is well-known in crystal-
1,-i '1,, and discrete geometry. We apply a newly derived square generator
matrix for the A* lattice in R", rather than non-square generator matrix for
embedding in ]R"+1 favored in i -I 11 _- i 1.1 Then geometric construction of
the shifts of the isotropic linear box-spline J i1 on the A* lattice simplifies to the
classical construction of box-splines by projection: The shifts of the isotropic
linear box-spline on A, are the orthogonal projection of a slab of thickness 1
decomposed into unit cubes along the diagonal direction of the cubes (1 i,.i 1).

(b) n 2

1 ,Ii.- 1: Orthogonal projection of a slab along the diagonal direction for (a)
n 1 and (b) n = 2.

By comparison, M1 has the same pre-image, but its support is distorted by its
anisotripic direction matrix. Therefore M, on the A* lattice shares the desir-
able properties of the family Mr on the Cartesian grid, and we can characterize
support, partition and quasi-interpolants for i,!! number of variables n.

2 Previous Work

Piecewise linear hat functions, the shifts of the 3-directional linear box-spline
in two variables and of the 4-directional linear box-splines in three variables are
popular basis functions for the 2D and 3D FEM (I !,,I.- 1.1. I ii Method), re-
spectively. They apply to arbitrary triangular, respectively tetrahedral meshes
in an intuitive way. Higher-degree box-splines are obtained by repeated convo-
lution along the mesh directions, hence require structured meshes. For a small
sample of the literature on the bivariate 3-directional box-spline see [20, 21,
22, 23, 39, 12, 13, 18, 19, 37, 15, 38, 25,;-., 7, 48, 3, 2, 16]. ('lLI et al. [14]
and Lai [42] gave ti !. i I evaluation of convolutions of hat functions via the
BB(Bernstein-Bezier)-form. The analogous trivariate 4-directional box-spline
evaluation has, for example, been discussed by He et al. [';it], and Casciola et
al. [8] extended the approach of ('! Ii et al. [14] to three variables. (' I! et al.
[9, 10, 11] proposed a volumetric subdivision scheme based on the 8-directional
trivariate box-spline, M2.
Interpolation on the Cartesian grid by Mr was discussed by Arge et al. [1] and
!, et al. [49] discussed the associated spline space. Neuman [47] proposed a
closed but intricate formula for evaluation of Mr. The literature refers to the
space decomposition corresponding to the polynomial pieces of M, as (n + 1)-
directional mesh.
The lattices A., and A* are well-known in I -I 11.,-i i'1~ -, discrete geometry
and related areas. A standard treatise of the I.l. I is [171. Here the lattices

(a) n 1

are derived embedded in IR"~ Hamitouche et al. [34] recognized the need for
square generator matrices that embed the A, and A* lattices in R". Their
definition, in iterative bottom-up fashion, is more complex and the resulting
matrices are more complicated than the one to be presented below.
Frederickson [33, 31, 32] first discussed the isotropicc) bivariate splines on the
hexagonal lattice. The hexagonal lattice is known to be the optimal sampling
lattice in 2 dimensions and is equivalent to the A* lattice. Van de Ville et al.
[1, 53, 52] proposed hex-splines on the hexagonal lattice which share 11 ii!
properties with the box-splines on the hexagonal lattice. Similarly, the BCC
lattice is the optimal 3D sampling lattice [.11, 28, 27] and is equivalent to the
A* lattice [17]. Entezari et al. [29, 28, 30, 27] were the first to investigate the
isotropicc) 4- and 8-directional trivariate box-splines on the BCC lattice.

3 Notation and Background

The dimension of vectors and matrices will be determined by context. Some of
the specific vectors and matrices are:

ie the k-th unit vector,
I the 'l, iI li- matrix,
j :=(1, ,1) the diagonal vector,
J : jjT the matrix composed of Is only,
0 : (0, 0) the zero vector,
and the dot product is defined as x y := xTy E R.
Hj is the 1l i d i 1i !defined as j x = 0.
Lattices are I- I. -. in calligraphic upper case; e.g., n and An.
Following the convention in [24], an n x m matrix will be interpreted as
a set of column vectors allowing iii.LIi idl i or
a linear transformation ]RI ]R'.
Column vectors are used as either vectors or points depending on the
context.
Lattice generator matrices (Section 3.1), such as G, A* and A, are i -, l
in bold upper case.
Box-spline direction matrices (Section 3.2), such as E and T,, are I" -.
in upper case unless they are derived from generator matrices; e.g., P.
Linear ,, ... ,,., ....., ... are I I.. -. I in upper case, e.g., B but are I" -. l
in bold upper case when related to generator matrices; e.g., P.

C'(P) is the convex hull of the points in P.
A matrix B e Zmx" is unimodular [24, II- .7] if

det Z = 1, VZ C B : Z is square and rankZ = n.

If m= n then B has an integer inverse B 1 E Znn.

3.1 Lattice packing and optimal sampling

Given an m x n matrix G with m > n and rank(G) n, all integer linear com-
binations of its columns, GZ", define (the points of) an n-dimensional lattice,
, embedded in Rm:

{:= Gj e :j Z}.

G is called a generator matrix of n, and we call the columns of G a basis of
,. The choice of a generator matrix for a lattice is not unique.
Lemma 1. If B E Znxn is unimodular then G and GB generate the same
lattice points: GZ" GBZ".

-...... Since BZ" C Z" and every integer linear combination of the basis is a
lattice point, G(BZ") C GZ". Conversely, (GB)(B 1Z) GZ C GBZ"
since B 1Z" C Z". O

If one lattice can be obtained from another by a rotation, reflection and change
of scale we ,- 11. are equivalent, or similar, written [17]. Any n-dimensional
lattice , has a dual (or reciprocal, or polar) lattice given by

{ xC ." : x u E Z,Vue ,}. (1)

If G is a square generator matrix of n,, then GT is a square generator matrix
of C [171.
A Dirac comb or ~!!... function on the lattice GZ", GE Rnx", is defined as

fIIG(x) :=- 6(x Gk) (2)
kCZ-
where l is the Dirac delta function. Its Fourier transform is (see [26])
27x
IG() {UIG}() detGI L 6(uw 2 Tk) (3)

where F{} is the Fourier transform operator. Sampling a function f on the
lattice GZ" is the same as iilli .1ll1 i; f with IIIG,

(flllG)(x) = f (Gk)(x Gk).
kCZn

sampling. Although both lattices (a) and (c) can reconstruct the original signal,

(c) corresponds to a denser sampling (d).

Therefore

S{f lIG}(w) -f()* IIG(w)
2-
rdet G f(w 2G k) (4)
kCZ-

where f(w) : {f} (w). In other words, in the Fourier domain the (scaled)
Fourier transform of f, 27rf(c)/I det G is replicated on the (scaled) dual lattice
of GZ, 27G -T [26].
The sphere packing problem, "! ... ,I ..... I., we can pack identical spheres in R"?",
is one of the oldest problems in geometry [17]. The lattice packing problem is
to find the lattice that induces the densest sphere packing when the spheres are
located at the lattice points.
The lattice packing problem is closely related to the optimal sampling lattice for
multi-dimensional signal processing. Assuming the input signal has bounded
f. 1i|1. i, we can reconstruct the original signal using a sphere-shaped till I in
the frequency domain. Since the lattice in the frequency domain is the dual of
the sampling lattice, the more densely we can pack the spheres (reconstruction
t!il i -i in the frequency domain, the sparser a sampling lattice we can choose
in the space domain to reconstruct the original signal (1 !,.i!- 2). Therefore,
for input signals with bounded f!r n! the optimal n-dimensional sampling
lattice is the dual of the optimal sphere packing lattice [28, 41, 27].
A 1.. '!,, of a lattice packing is the proportion of the space occupied by the
spheres when packed. A center .1 ,. "i,, of a lattice is the number of the lattice
points per unit volume, which can be obtained by dividing its I ii-il by the
volume of the unit sphere [17]. Therefore, larger 1. !i-il (or center .1. i -i- I'
implies that its dual is a more IT, '. "- sampling lattice. Table 1 and I ,,.i!- 3
respectively show the center ,l. I.-i- and the l1. i-i of several important lat-
tices, i '1,! ii1- poorer sampling !tn! i i, of the traditional Cartesian lattice Z"
compared to other lattices.

2- 2 -/2(n+ 1) 1/2

nn/2
2"(n + 1)("1)/2

231.52-5
2-(n+2)/2 31 2-
2-(n-1)

Table 1: Center l. ii-il of several lattices. D :
[171. See Section 4 for AS, and A*k.

{j E Z : EC j(k) is even}

i i, i..- 3: D. !!-i I of the lattice packing for several lattices up to dimension 10.

3.2 The Box-splines Mr

A box-spline ME is defined by a direction matrix E. Unless mentioned specifi-
cally, we assume S E ZnX (m > n) and ranE = R". Geometrically, the value
at x E ranE of the box-spline ME is defined as the (normalized) shadow 1. i -i I
of the (m n)-dimensional volume of the intersection between the preimage of
x and the m-dimensional half-open unit cube L := [0..1)": [24, I-3] (see e.g.,
1i ,,u-. 1)
Ms(x) := vol n ( l{x} n )/I det E (5)
where E is viewed as a linear transformation E : RI R" and the preimage of
x is defined as [24, I-7]

- l{x} =T(TEE) l{x}+kerE.

Let H (E) be the collection of all the hyperplanes spanned by the columns of E.
[24, page 8] We call the shifts of all the hyperplanes in H (E) knot planes: [24,
page 16]
r(): U H+z". (7)
ffH(3)

(n 3)
(n > 3)

A, P,

ME with cE Znxm is a piecewise polynomial function on ranE, delineated
by the knot planes, of degree less than or equal to [24, page 9]

k () := m dimranS. (8)

Specifically, k (E) = m n if ranE = R".
The centered box-spline Mc of M7 is [24, I-211

ME :- M3(. + E /2). (9)

Given an invertible linear map L on R", [24, I-23]

Ms= I det LIML o L. (10)

The Fourier transform of M7 is [24, I-17]

1 exp(-i C())
ME (w): F ME } (w) [ -i ix := i V). (11)

Specifically, if M7 is centered, [24, page 11]

AMs() since( c w). (12)

By [24, page 9], the (closed) support of M7 consists of the set

suppME- E- {I t : 0 < t < 1}

where 0 := [0..1]" is the closed unit cube and t is the element of t associated
with by Et. Assuming ranE = R", let [24, page 81

B(E) : {Z C : #Z = rankZ n} (13)

In other words, B(E) is the set of all bases of E. The support of M7 is composed
of the parallelepipeds spanned by Z e B(E):
Theorem 1 (Support decomposition; [241 I-.';). Let ranE = R". I!. .. exists
points az E E{0, 1}m, Z E B(E), so that E is the essentially 1i-i ..il union of
the sets
ZO+az, Z e B(E). (14)

The cardinal spline space [24, II-1]

S := span (Ms( j))jc (15)

is the spline space spanned by the shifts of Ms on Z".

The shifts of Ms are linearly independent if and only if E is unimodular.
The spline space S, contains some polynomials. In other words, Ms can re-
produce the polynomials in [24, page 52]

HMI := H n S,
where H is the set of all the polynomials on R". Specifically,

Hm,() c LHMs
where
Ha is the set of polynomials of (total) degree up to a,
m(-) : min{#Z : Z C A()} 1 and
A(E) : {Z C : E\Z does not span}.
In other words, ME can reproduce all the polynomials up to (total) degree m(E).
The next section gives a concrete construction.
A quasi-interpolant Q7 for a box-spline ME provides a fast way of approx-
imating a function f with a spline QIf E S7 [24]. Here we focus on the
quasi-interpolant which provides the optimal approximation order m(E) + 1:
[24, page 721
(Qf) (x) := M(x j)A (f( + j)) (16)
jCZ-
where A. is a linear functional [24, III-22]

A7 (f(- + j)) := (O) (Dof) (j) (17)

and a E Z" is a multi-index with |a| := cEa(). The Appell sequence
{ga} in (17) can be computed either recursively as

{go a n ([24, III-19])
.9111 II
where
S(f) : M (j) f(-j), (18)

or from the Fourier transform M!: [24, III-34]

ga(O) ([- iD (1/M )) (0). (19)

Note that I| is the normalized a-power function

a'vi )

The Box-Spline M,. Box-splines defined by possibly repeated (n+1) distinct
convolution directions are also called box-splines on the (n + 1)-directional mesh
[1]. Given the n x (n + 1) direction matrix

T := [I I- ]

[ ii " in

-j ] E nx(n+1),

the box-splines with ...l! '"l'. !', r in each direction is defined by the n x r(n+1)
direction matrix [24, page 80]

T : T1.
j 1
As pointed out in Section 2, this family of box-splines has been widely used.
Note that TI [1 -1] in the univariate case and M. can be viewed as a
generalization of the uniform B-splines of odd degree to arbitrary dimensions.
Following the notation established in [24, page 80], we define
Mr*: MT,.

4 The Lattices An and Ai* as subsets of -_' 1

The (n-dimensional) lattice A,, embedded in ]R"+1 has points {x E Zn+1 :j x =
0} and can be generated by the (n + 1) x n matrix [17, page 109]

-1
1
Ac : .. Z(n 1)xn. (21)
-1
1

We note that j Ac 0,
(n + 1) x n matrix

i.e., all the basis elements lie in Hj C ]R"+1. We can
page 115]) that its dual A* can be generated by the

1 -n/(n + 1)
1/(n + 1)

1 1/(n + 1)
1/(n + 1)

-n/(n + 1)
j/(n + 1)
1/(n + 1)

I R(n +1)xn

Some examples of A,n and A* are:
A2 and A are equivalent to the hexagonal lattice.
A3s D is equivalent to the FCC (Face-Centered Cubic) lattice.

A DV is equivalent to the BCC (Body-Centered Cubic) lattice.
The BCC lattice and hence A*, is the optimal sampling lattice in 2 and in 3
dimensions [46, .In, 28, 29, 41, 30, 1, 27]. In dimensions higher than 3, 1 ,!..- 3
shows that An, packs spheres better than the Cartesian lattice, making A* a
better sampling lattice than Z".
The basis of A, can be taken from an n-dimensional equilateral simplex [34].
Lemma 2 (Geometric construction of An, in R"+1). Let oa be an equilateral
n-dimensional simplex one of whose vertices is located at the origin. IP. ,- the
n edges of ~Tn emanating from the origin form a basis of a lattice equivalent to

4 ....f By Lemma 1,
AcB [-I Z(n+ )xn (22)
AcB j

with
1
1 -1

B := and B-1
1
-1
also generates An. Since

I. I| _, =- Vv E AcB
-t V2 Vvj, Vk e AcB, vj Yk,

the simplex C'(0 U UveAcB v) is equilateral hence equivalent to ;!u an. O
Lemma 3 (Geometric construction of An in IR"+1). An can be generated by

P: I- 1 JER(n+l)x(n+l), (23)
n+l
the orthogonal projection of the (n+1) -dimensional Cartesian lattice Z"+1 along
the diagonal direction j.

J ...... For

B := 0 1 1 Z(n+1)x(n+1) where B-1 = ] j

we verify that

A*B= A n (n+l)I J R(n+1)x", (24)
C P ; n + 1 -j I
where A* is the matrix of the first n columns of P. The last column of P
is a linear combination of the first n columns, A*. By Lemma 1, the claim
holds. E

5 Box-splines on non-Cartesian lattices

By (10), given a square generator matrix G, a spline generated by the shifts of
the (scaled) box-spline
M := det GMGa (25)
on the (possibly non-Cartesian) lattice GZ" can be expressed as the one gener-
ated by the shifts of Ms on the Cartesian lattice Z" with change of variable:

SMH(. -j)a(j) = E (G- -k)a(Gk) (26)
jcGZ" kcZ"

where a : GZ" R is the mesh function (spline i.... h i I -, on GZ". De Boor
[21, page -.' 1] already pointed to this relationship in the bivariate case.

5.1 Spline space

Let
SG : span Ms(-- j))
\ /j'eGZ"
be the spline space spanned by the shifts of Ms on GZ". This notation becomes
consistent with (15) by omitting G I and defining

5.2 Quasi-interpolant

Lemma 4. Let D' := IoeG D-" be the composition of directional derivatives
D, := v g v(j)Dj along the columns of G and {ga} the Appell sequence of
A7 (17). 1!b. quasi-interpolant QG for SG ., ....I by the functional

A (f (. +j)) : A ((fo G) (. + G j)) (27)
= g ,(O) (Dof) (j), j GZn, (28)
a
provides the same optimal approximation power as does Q7 .. 1; .... by A7 for
S7.

i .f'... If we define

(Qf) (x): (Q, (f o G)) (G-x)
then, since f =f o Go G1,

(f Qf)() (= (( o G) Q (fo G)) (G -x) = (f- QSf) (),

for f := foG and x := G 1 ox, i.e., QG has the the same approximation power
as Q,. Since

(Q (f o G)) (G- x)

SM(G x k)A, ((fo G) ( + k))
keZ'
I detG|MG(Z j)A ((f oG)(. + G-j)),
jcGZ"

f (G +hGik)
Dk (fo G)= lim
htO h

the corresponding functional AG is for j E GZ"

A (f ( + j))

A ((foG) (+ G-lj))
Sg(0() (Do (f o G)) (G 1j)

E g,(O) (DGf)(j).
a~ ,z

6 The isotropic box-spline family M1, on A*

The box-spline family Mr and the A*, lattice have a close relationship, that is
apparent when considering the linear box-spline M1 and A*. The lattice A*
can be constructed by projecting the Cartesian lattice Z"+1 along the diagonal
direction j. By (20) and (6)

TiT(TiTiT) 1

where A* are the first n columns of P := I
Lemma 3. Therefore

1 J E R(n+1 )x(n+1) defined in

Ti 1{x} Ax + span(j), x R,j E n+1

i.e. the preimage of suppMi Ti C R" is PO C Hj C ]R+ 1 and PD is
isotropic since the directions, i.e. the columns of P := I - J, are
isometric: they have the same lengths and
isotropic: the inner product (hence the angle) between ;,i! two directions
is the same.

f(G-)

(DGikf) o G,

(by (17))

1 [ (n +l )I-J ] Ap* e R(n+l)xn
n + 1 j-J p

LL

(a) Z2 (b) AZ2 A2

1 L ,1i" 4: Geometric construction of A, in R".

However, suppMi is anisotropic, (1 ,.i.- 6d), since A, : R'" Hj C ]R"+1 is
not an orthogonal transformation:

A TA~ I J.
n+1

We can use P directly as the direction matrix of a linear box-spline

MI := Mp : i C R"'+l -- R.

Evidently M+ (and M? defined by U31 P) has an isotropic support (1 iii.- 6b
and 6e). But neither Mr nor M+ are useful since
1. M, is anisotropic and
2. the domain of M, is not R" but Hj embedded in ]R"+1.
The remedy (1 ','i.- 6c and 6f) is a square generator matrix of A, to be presented
next.

6.1 The new square generator matrices A and A

To construct square generator matrices for A,, and A* in RI, we consider a
linear map that simply scales along the diagonal j. The map transforms a point
x E R" according to
x + -(j )j
n
where c is the scaling factor.
Lemma 5 (Geometric construction of AS, in R"). AS, can be generated by

A:= I + Jc with c, : -1 / 1.
n

(b)XZ -A'

1 ,!,.i 5: Geometric construction of A* in R".

JF,..,, Any vector ij ik for i / k is parallel to Hj and hence its length remains
/2, unchanged by A and regardless of the dimension n. To show that the n-
dimensional simplex Ct({Aij : 1 < j < n} U 0) is equilateral, we verify that the
vectors ij satisfy

+ 12
Cn+1) +(n- 1)
(n /n

42. (30)

The claim follows by Lemma 2.

Note that two 11 I, !I choices of c, produce the equivalent result with respect
to Hj because I J/n projects ij on Hj.
Lemma 6 (Geometric construction of A* in ]R"). A* can be generated by

: I + c E .' with c*
n

1
v1n+-

F ..... Since

1 ( 1) ( =1) (Cn + 1) (c; + 1)

cc* + c, + c* = 0 and hence

ATrA I + (cc*n + cn + c)J

cc + c, + c + 1,

D

Under the diagonal scaling A, the length of j becomes the same as those of the
unit vectors (1 [,Li!,- 5):

IA*j IAij, V j < n.

As with A, two roots of c* result in equivalent transformations with respect to
Hj.

||Aij ||2= 2 Aij Aij

(a) Z2

(b) M+ on Hj ; R2

(e) A41 on Hj R3

(c) M* on XZ C R

(f) M* on AZ2 C R2

1i io.- 6: (top) l,11it of linear univariate box-splines and (bottom) shifts of (the
support) of linear bivariate box-splines.

(a) Mi on Z

(d) M1 on Z2

For example, for n = 2,

1 1 l//3 -1 l/v/3
2 -1 1//3 1 1//3

and for n = 3, the BCC lattice, the two choices are

1 5 -1 -1 1 -
A : -1 5 -1 or -1 1
-1 -1 5 -1 -1

6.2 A as the domain lattice of M*

We now interpret the columns of the matrix T,* := A*T, as direction vectors in
Rn.
Lemma 7. T* is isometric and isotropic.

F ......f Since (31) implies isometry, we need only verify isotropy,

1 1
(Az (-j)) (Ai,) 1 Vi and (A*i) (A*i) Vii 7 i .
n+' n+'

Therefore MT* is isotropic and AZ" can serve as a domain lattice for the box-
spline family (1 Iii,- 6c and 6f)

l1, := IdetA* MT. Mr oA*1 (32)

as in (25). Alternatively, the isotropy is verified by observing that the transfor-
mation A)A* 1 is orthogonal

(AA* )(A A* 1) I

and hence, in contrast to (29) for M1, isotropy of PO is preserved when com-
puting the preimage,

T* -{x} A* {x} + ker T*. (33)

By (-'i) a spline generated by the shifts of M* on AZ" An* can be expressed
as
S1,- ( -j)a(j) = M,( 1 -j)a(Aj). (34)
jeA*Z- jcz"
Therefore M* inherits most of the properties of Mr.
Lemma 8. M* is centered.

,,,,[.f By (9),

-JI *+ E ^ -I, (+u")

since EET,* = 0. E
Lemma 9. Mr* M T*

.,,.,,,f By (12),
F {MT } (w) J since (Tw)
CT,*
and

F {M *} (w) sin( ) since T) (- Tw) =- since (Tw)
^-T* CT* CT*

because sine is an even function. The claim holds since the Fourier transform
is invertible. E
Lemma 10 (even function). M, = M,*(-).

f,,.,' By (10) and Lemma 9,

MT, = det (-I) M T, o (-I) = MT,(--). (36)

Lemma 11 (basis functions). I ., shifts of M,* on A*Z" are linearly indepen-
dent.

Pf....f Due to (34),
S := span ( l, (.-j5))j
and ST, share the same linear independence property. Note that

B(T) = B(T) U T \( (37)

since ; n directions in T1 span R". The claim holds by Section 3.2 since

detZ -= l, VZ B(T,).

By (8), Mr, hence M*, is piecewise polynomial of (total) degree less than or
equal to (n + 1)r n.

Lemma 12 (polynomial reproduction). M,* can reproduce all the ;'.1',......'..,
of (total) degree up to 2r 1:

m(T,) 2r 1.

jF...... Due to (34), m(T*) = m(T,). For M1, we have to remove at least 2
directions so that the remaining directions in Ti no longer span R', hence

m(Ti) ((n + 1) (n 1)) 1 2 1 1.

In the same way, at most r(n 1) directions in T, span a hyperplane, therefore

m(T,) = (r(n + 1) r(n 1)) 1 = 2r 1.

Note that m(T,*) does not depend on the dimension n.
Also, since the knot planes generated by T* are linear transformed ones of F (T,),
the mesh inherits the I. 1. .1. ._- of the (n + 1)-directional mesh.
Lemma 13. 1!, .'* are n(n + 1)/2 non-parallel planes in H (T,).

'! ..... There are n planes generated by the n unit vectors in I and (, 2) ad-
ditional non-parallel planes are spanned by the diagonal direction j and n 2
additional unit vectors yielding a total of

n + n-2 n + 2n( 1)= -n(n +1)

non-parallel planes in H (T,). D

Next, we characterize the partition of R" into simplices by the knot planes in
F (TT).
Lemma 14 (Partition of (n + 1)-directional mesh = Kuhn triangulation). 1.
knot planes in H (T,) partition the unit cube 0 into n! simplices

n i
a := C(H(V,), V : U J i(J), 7E n (38)
i= j=1

where n" is the set of all the permutations of {1, n ,n}.
The partition {af }Jtrn is called Kuhn triangulation [40, Lemma 11, [43, page 140]
and [1, 5, 6].

jF.4.,.f Recall that T1 [I, -j]. All planes with normal direction ij ik, j / k,
intersect the interior of [ and are generated by Ti\ {ij, ik} i.e., as knot planes

of M1 generated by n 1 vectors including j. Unless two vertices vj, vk E V,
for some permutation 7, there exist indices a and f so that

vj(a) l, v (a) 0 and vj(,3) 0,vk(,3) =1.

and hence the knot plane with normal i, i3 separates them,

(ia ip) vj = 1 > 0 and (ia i3) vk = -1 < 0. (39)

Conversely, since knot planes excluding j are axis-aligned, neither they nor their
shifts on Z" intersect the interior of the unit cube 1. It remains to show that
no shifts of the knot planes with normal i, i3 separate vertices in of a simplex
Ca for the same fixed permutation 7. Since shifts by j E Z" within the knot
plane, i.e. j (ia i3) j(a) -j(B) 0 result in (ia i3) (x -j) = 0, we can
assume j (i, i3) j(a) j(f) > 0. Then, for all v E {0, 1}",

-j (a) +j(/ )+ < 0 v(a) = 1, v() 0
(i i) (v ) -j(a) +j() 1 < -1 v(a) =0, v() 1
-j(a) +j(() < 0 v(a) =v(B)
<0.

The case j (i, i3) < 0 corresponds to a flipped normal and yields (ia i3)
(v -j) > O

We can investigate the structure of -,q -11, by first decomposing it into paral-
lelepipeds. I ii we consider suppM*. There are two decompositions.
Lemma 15. 1! (closed) support of M* is the i .. ... '.,ll, disjoint union of the
(n + 1) parallelepipeds
{ZD: Z e B(T*)} (40)
or, .I .... i ', 1,.
{Z[+Cz : Z e B(Tf*)} (41)
where (z := T*\Z. In either decomposition, all the parallelepipeds are congruent.

F'...." See Section 7.1. O

This result is easily extended to T* since

T,* { :0 < t:0 SE T'* ETJ*

For Z E B(T*), the pair (Z, (z) is a linear transformation of the pair (I, -j).
Therefore Z is decomposed in the same way as the unit cube 0 is by Kuhn
triangulation and suppM* consists of (n + 1)! simplices. This count also agrees
with the number of modular cells in the first neighbor i, '.- I- of A* [34].
The two i- of the decomposition of suppMr in Lemma 15 can be viewed

as cubical meshes such that one is the flip of the other [4] since each cubical
mesh can be viewed as the projection of the (n + 1)-dimensional cube along two
opposite directions of its diagonal.
Here we investigate M2 that can reproduce all the cubic polynomials. (Lemma 12)
Lemma 16 (Quasi-interpolant for M*). 1., optimal quasi-interpolant of M2*
is 1. 7; ... by the functional

A (f ( + j)) : A (f(.+ )): f Df (j), j AZ". (42)

J',,,,f See Section 7.2. E

For discrete input f : A*Z -' R, we approximate the directional derivative
along C E R" by finite differences, e.g.,:

D f f( + ) + f( ) 2f. (43)
Therefore

A (f( + j)) f (j) (f( + ) + f(j 2f(j))
21

(+nf^l)f)_ f(j) (44)

where f(j) is the average of 2(n + 1) ;i1i ,1. 0 neighbor data. When specialized
to two variables, this agrees with Levin's formula [44].

7 Appendix

7.1 Proof of Lemma 15

J..,,f Due to the relation (34), we need only consider M1. Let Zj E B(T1) be
a basis of Ti and
:= -Ti\Z = .

For aj := azj in Theorem 1, there are only two choices, aj E {0, j } since no
C E Zj doubled fits into the support

Zj3+(C 9C(+( TIo, V(cZj.

Now assume

a = 0 and ak = C for Zj, Zk E B(TI),Zj / Zk.

This leads to a contradiction as we prove that the two parallelepipeds Zj 0 +aj
and Zk e +ae are not essentially 1i -ii., I but share a point p interior to both.
Let

CezTnZh
Then p E Z (0, 1)" + a, since

p1

z\ z
Zj\z ( 4 Czj + a

where aj = 0, Zj = (Zj\Z) + (Zj n Zk) and + denotes .1i-ii..i union. Also
p E Z2(0, 1)" + a~ since

1 1 1
2 Z 4 2
Cezjnz, ceT1
12 1 1 1
2 CZZ 4Z 2 2c
((- znZk CZk

(ECTI, C 0)

(3
4c

E + j z +akz
z-nZk (CZc\zj )

This establishes that there are only two alternatives. Next, we prove that all
parallelepipeds are congruent. Let's consider the decomposition

{Z*D: Z* e (T,)}.

The following lemma makes the proof easier.
Lemma 17. Let the matrices A,, Ba E Rnxn 1, 6'-, 1 as follows:

A(,k) k a
0 otherwise

and Ba(j,k)

Sj k =a
0 otherwise.

I! .. for Zj : I Aj Bj,

Zj (I+ J)Z,

I + J and Z2
i

S....f Both can be verified using the following relations.

AjJ = J, BjJ = A, j AT = J, AjB = A, BjA,

By, A2 Aj.

I -I note that A2 = I + J. For Z: := AZj and Z: : A*Zk,

Zy = (AZjZ A 1)Zk

where we can verify that

(A*ZjZ A 1)(A*ZjZkXA 1)T = ZjZA2ZTZTXA
A (I + J)A*

(A- = A)
(Lemma 17)

Therefore AZiZA* 1 is an orthonormal (rigid) transformation hence all the
parallelepipeds Z* ,Z* e B(T,) are congruent. It's quite straightforward when
one of the basis is AI. Note that ;, permutation of the columns of Z* e
B(T,) can be done with a permutation matrix, which is also an orthonormal
transformation.
The other decomposition can be verified in the same way. E

7.2 Proof of Lemma 16

We first derive the quasi-interpolant QT2 for ST2 defined by AT, (17). Then QA
for Si defined by A* can be derived by (27).

j.,.,,f Following lemma will make the computation easier.
Lemma 18. For an odd function f, T2f = 0 (18).
j.,,,,,f

T2 f = M,(-j) f (j)

(by (10))

M2(j)f(-j)

M,(-j) f(j)
j

(f -f(- ))

(change of index)

therefore pT2 f = 0.

We compute ga(0) for each degree |a|.
1. 0a =0
ga(0) = go(0) = 1 By [24, page 68].

2. a = 1
By Lemma 18, p7. || |

g 1 a1 E P-
3'a

III -(III-1111 )90 111

therefore ga(0) 0.
3. lal 2
By [24, page 11],

fT(w) :- {MT2}(w)

Yi sinc(TwC) = sinc2(~w).
CT2 sCT1

Therefore, By (19), for j / k,

1 ) (1
MT2 e\j,ij,ik sinc2 (Tw)

since2 (jTw)sinc2 (~j )sinc2 (wk)

Since sinc(0) = 1, with the help of MAPLE, we can compute

DjDk (0)
( 1 M)

(DDkyD sc2 ( )sc2 (l)sc2( ) (0)
since (ci(j)sin (ci(j)sinc (; + CI))

Also,

(D 1 ) 1
Sit ) h since2 (Tw)
~C T1e \j,ij }
Again, with the help of MAPLE, we can compute

( 2 (0)
1 M )

D s1c4 (0)
jsinC4 ,))

By (19), for j / k,

gij+i, (o) iDl'+i' 1 (0)
1T2 M

g2ij (0) (I

1
-DjDk (0)
MT2

1D ) (0)
2 M)

0 and

(DjDk

iD i 1 (0)
MT2

4. la 3
By [24, III-19],

g.= I E (T-,I
3a

III ( ( 11 II

)go + (?1 II
/31 1

) 3+ 1 (Ii=2
1/31-2

hence g(O) = 0 because
PT II II- 0 by Lemma 18,
gg = 0 for |l| 1 and

p | IIT
Summing up,

AT2 (f( +j))

0 for 131 = 2 hence Ia f3 = 1 by Lemma 18.

E g) (OD) (j)
a|

1
-=2
a-2

Df) (j)

D 2ff D U()
(SDJ)21))J^
J

Now, by (28),

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