Until a few years ago all the work in the area of univariate subdivision was limited to consider just binary (Chaikin [
Since subdivision schemes propose efficient iterative algorithms to produce the smooth curves and surfaces from a discrete set of control points by subdividing them according to some refining rules, recursively. These refining rules are very helpful and useful for the creation of smooth curves and surfaces in computational geometry and geometric designing due to their wide range of applications in many areas like engineering, medical science and image processing etc.
In this article an algorithm has been introduced to produce the quaternary 
point (for any integer
) approximating subdivision schemes. This algorithm has been developed using the Cox-de Boor recursion formula, in the form of uniform B-spline blending functions to produce piecewise polynomials of order 
over the interval 
(for detail, see Section 2).
The quaternary subdivision scheme can be defined in terms of a mask consisting of a finite set of non-zero coefficients 
as follows
The formal definitions and the notion for the convergence analysis of the quaternary subdivision scheme are as follows:
The quaternary convergent subdivision scheme 
with the corresponding mask 
necessarily satisfies
It follows that the symbol of a convergent subdivision scheme satisfies the conditions 
and
for
.
Introducing a symbol called the Laurent polynomial
of a mask 
with finite support. In view of Dyn [
A subdivision scheme 
is uniform convergent if and only if there is an integer
, such that
subdivision 
with symbol 
is related to S with symbol
, where 
and satisfying the property
where 
and
The norm 
of a subdivision scheme 
with a mask 
is defined by
and
where
where
and
The paper is organized as follows, in Section 2 the algorithm to construct 
-point (for any integer
) quaternary schemes has been introduced. Three examples are considered to produce the masks of 2-point (corner cutting), 3-point and 4-point schemes in the same section. In Section 3, the polynomial reproduction property has been discussed. The conclusion is drawn in Section 4.
-point approximating subdivision schemes using the uniform B-spline basis functions and the Cox-de Boor recursion relation. The Cox-de Boor recursion relation, in view of Buss [
(or of order
, i.e.,
). For this, consider 
to be a set of 
non-decreasing real numbers in such a way that
. The values
’s, not necessarily uniformly spaced, are called knots of non-uniform spline and the set 
is called knot vector. The uniform B-splines are just the special case of non-uniform B-splines in which the knots are equally spaced such that 
is a constant for 
(i.e.,
) . Note that the blending functions 
of order 
depend only on the knot positions and are defined by induction on 
as follows.
let
. Setting
, 
is defined by the Cox-de Boor formula as,
are piecewise polynomials of degree 
and that the breaks between pieces occur at the knots
.
of order 
over the interval
, together with the properties [
.
.
for all t.
have continuous 
derivatives, that is, they are 
-continuous.
of the proposed quaternary 
-point scheme can be calculated using the following recurrence relation
is a uniform B-spline basis function of degree
. In the following, some examples are considered to produce the masks of 2-point, 3-point and 4-point quaternary approximating schemes after setting 
and 4, respectively, in recurrence relation (3).
in above relation (3). It may be noted that the linear uniform B-spline basis function 
produces the mask of 2-point quarternary scheme (which is also called corner cutting scheme). Thus 2-point scheme (after adjusting the mask) to refine the control polygon is defined as follows:
.
corresponding to the symbol 
is
. So, the Laurent polynomial 
for the mask of the scheme can be written as
. Taking
and
, it can be written as
is
is contractive, by Theorem 3, and so 
, consider 
and
; it can be written as
is
is contractive. Consequently, 
is convergent and
.
in recursion relation (3). The quadratic uniform B-spline basis functions 
are obtained. The mask 
of the proposed quaternary 3-point scheme can be calculated from these basis functions. The 3-point scheme (after adjusting the mask), to refine the control polygon, is defined as:
.
in relation (3) the cubic B-spline basis functions 
can be calculated. Thus, 4-point scheme is defined as follows
.
implies that the scheme has 
approximation order. For this, polynomial reproduction can be made from initial date which has been sampled from some polynomial function. In view of [
and definitions in the following manner.
the corresponding parametric shift and attach the data 
for 
to the parameter values
if it is convergent and its continuous limit function (for any polynomial
) is equal to 
and initial data
with respect to the parametrization defined in (8) if and only if
can be performed to prove this theorem following [
and
. Then a subdivision symbol 
satisfies
satisfies
derivative of the symbol) which in turn is equivalent to require that 
for some c(z).
. Then the smoothed scheme 
with the symbol
satisfies the conditions
, but it has only linear reproduction.
with
, we have
. Thus, the 
derivative of 
is
is
derivative of 
is
.
.
point (for any integer
) approximating subdivision scheme has been developed which generates the smooth limiting curves. The construction of the quaternary scheme is associated with an algorithm of uniform B-spline basis functions developed from the Cox-de Boor recursion formula. The objective is to introduce the quaternary subdivision schemes, which have smaller support and higher smoothness, comparing to binary and ternary schemes. Moreover the polynomial reproduction property has also been discussed.