1. Introduction
Shape-preserving techniques for interpolating and approximating multiscale data, that is, data with sudden large changes in magnitude and/or spacing, are important for modeling of natural and urban terrain, geophysical features, biological objects, robotic paths and many other irregular surfaces, processes and functions. Over the past decade, a new class of univariate and bivariate splines, namely,
splines, that have superior shape-preserving properties for interpolating and approximating multiscale data has arisen ([
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21]). The
-norm minimization principles on which
splines are based result in non-differentiable convex generalized geometric programs that, so far, have been more complex and more computationally expensive to solve than the programs by which other variants of splines, e.g., conventional and tension splines, T-splines,
etc., are solved, but the shape preservation provided by
splines is significantly better than the shape preservation provided by these alternative approaches.
splines have typically been calculated by minimization of global spline functionals, that is, spline functionals that extend over the whole range of the data to be interpolated. However, there have been three reports in the literature of
splines on local windows. The first such report is in [
17], where bivariate
splines were calculated by a non-iterative “domain decomposition” procedure on overlapping
windows and
subsets of these windows were pieced together to create global surfaces. With parallel computation, the domain-decomposition procedure results in sharply reduced computing time.
In 2007, a result for univariate
splines on much smaller windows arose. In [
2], Auquiert, Gibaru and Nyiri showed that, given five points on a Heaviside function with two to the left of the discontinuity and three to the right, the
spline for these five points is linear over the set of three points ([
2], Proposition 9). Even though preservation of linearity is not all of what we desire in shape preservation, it is a large part thereof. This linearity-preservation result suggests that calculation of
splines on small, 5-point windows, is geometrically meaningful. An immediate generalization of Proposition 9 of [
2] is that, if, in a set of five points, three consecutive points on one end are collinear, then the
spline through those three points is, except in the case of a V-shaped corner, linear. Such a result does not hold when the five points are embedded in a larger data set and a global
spline functional is minimized. The best result that can be achieved in the case of a global
spline functional is the following.
Theorem 1. (Theorem 2 of [
7])
If four consecutive data points , , , and lie on a straight line, then a cubic spline preserves linearity over the middle interval . If , then preserves linearity over the first interval . If , then preserves linearity over the last interval . (Here and are components of the optimal dual solution in [7].) In this case, one needs four (rather than just three) consecutive collinear points and the
spline is guaranteed to be linear only in the second interval (the interval between the second and the third of the four points). The
spline is linear in the first and third of the three cells only if additional conditions, ones that do not have clear geometric meaning, are fulfilled. Proposition 9 of [
2] is thus a significant improvement over Theorem 2 of [
7].
Proposition 9 of [
2] shows that one can preserve linearity over a larger set of points by calculating the
spline using local 5-point windows rather than globally. This has a potential strategic implication, namely, that one may be able, by replacing a global minimization problem by a set of local minimization problems, to both further improve the shape preservation capabilities of
splines and at the same time reduce the computing time because the local problems are independent of each other and can be solved in parallel. This is the opportunity that this paper wishes to investigate. The authors Auquiert, Gibaru and Nyiri of [
2] have followed up on their results of 2007 with an article [
22] containing new analytical results about preservation of linearity by windowed, rotation-invariant parametric
splines of degrees 3 and higher. In contrast, this present paper considers linearity, convexity and oscillation for 5-point-window, rotation-dependent nonparametric cubic
splines.
The precise purpose of this present paper is to provide analytical results that link linearity, convexity and oscillatory properties of the data on 5-point windows with linearity, convexity, oscillatory and uniqueness properties of the resulting
spline. In each 5-point window, a local
spline functional is used to determine the first derivative at the middle (or, near boundaries, other) point in this window. After the first derivatives at all of the data points have been determined, a
piecewise cubic interpolant, called the
spline (or “locally calculated cubic
spline”), is set up by Hermite interpolation in each interval. In
Section 2, we investigate analytical properties of the spline functional that link local geometric properties of 5-point windows of the data with geometric properties of the local
splines on these windows. Based on the analytical results for 5-point windows, we investigate in
Section 3 the properties of the
piecewise cubic interpolant that has derivatives determined by these 5-point-window
splines. In
Section 4, we summarize the results presented in the previous sections and describe potential computational implications of these results.
All of the quantities in this paper are real quantities. The nodes
,
, are a strictly monotonic but otherwise arbitrary partition of the finite interval
. Let
,
. At each node
, the function value
is given,
. The slope of the line segment connecting
and
is
,
. The
splines discussed in this paper are cubic polynomials in each interval
,
, and are
continuous at the nodes. The first derivative of the spline at node
,
, is denoted by
(to be determined by minimization of the
spline functional). We use
to denote the slope of the chord between neighboring points:
We use
ζ to denote the linear spline:
2. Analytical Properties of 5-Point-Window Splines
The splines that we will consider in this paper are calculated locally as described in this paragraph. For the interpolation problem under consideration in the present paper, the function values are given. In the 5-point window with middle point
,
, the derivative at
is calculated by minimizing
over the finite-dimensional spline space of
piecewise cubic polynomials
z that interpolate the data. The free parameters in the minimization of functional (
3) are the derivatives
of the spline at the five nodes. The derivative at node
that occurs at the minimum of functional (
3) is denoted by
. Whenever the minimum of functional (
3) is nonunique, we choose
to be the scalar in the optimal set (the interval
) closest to the slope
of the chord between the neighboring points, that is,
. Previously, nonuniqueness was resolved by “regularization” of the spline functional, specifically, by adding to the spline functional (
3) a sum consisting of the absolute values of various expressions involving the derivatives at the nodes times a sufficiently small number
ε (
cf. [
5,
9,
16]). The method for resolving nonuniqueness that we use in the present paper differs from the regularization approach used in previous
spline work but leads to both simpler analysis and simpler computational procedures. The derivatives at the points
and
are determined by
which is calculated by minimizing (
3) for
. Analogously, the derivatives at the points
and
are determined by
which is calculated by minimizing (
3) for
. After obtaining all of the
, a
piecewise cubic interpolant
z is set up by Hermite interpolation
for
,
(
cf. [
9]). The
piecewise cubic interpolant calculated in this manner is the
spline (locally calculated cubic
spline).
In the remainder of this section, we investigate the relation between the geometry of the 5 points in each window and the derivative at the middle point of the window. For the five points under consideration, we use the notation
,
,
,
and
. For the window with these 5 points, the objective function (
3) is
where b denotes
. Each term in the summation is a function
that is continuously differentiable and has the properties stated in the following lemma.
Lemma 2. ([
2])
is convex,and- (1)
with ,
- (2)
with ,
- (3)
with .
On the basis of Lemma 2, we have
Minimization of
is a two-level minimization problem that can be written in the form
where
and
For later use, we introduce the notation
and
where
c is a parameter.
Lemma 3. The functions and are both convex. is continuous on and differentiable except at . When , we have andWhen ,- (i)
If , then and - (ii)
If , then and - (iii)
If , then and - (iv)
If , then and - (v)
If , then and - (vi)
If , then and
When ,- (i)
If , then and - (ii)
If , then and - (iii)
If , then and - (iv)
If , then and - (v)
If , then and - (vi)
If , then and
Proof. The function
is the sum of two convex functions, so it is also convex. The convexity of
comes from the fact that it is the partial minimization of
(see [
23]).
If
, we calculate using Lemma 2
which is a nondecreasing function of
p for any fixed
q. Moreover, when
,
When
,
Analogously, if
, we calculate from Lemma 2
which is a nondecreasing function of
p for any fixed
q. When
,
When
,
When
and
,
Therefore,
and
which implies that
When
and
,
Therefore,
and
which implies that
The proofs for
and
are similar to the proof for
and are omitted. ☐
Now, we let
With this notation, we have
Remark.
Later in this paper, we will use
,
and
to classify cases of linearity, convexity and oscillation. However, for clarity of the analysis in much of the remainder of this section, we use
to denote
instead of
because
and
are defined in a symmetric manner in (
9) and (
10) and
and
are determined by
and
, which are in turn determined by
(progression outward from the middle point).
From Lemma 3,
is convex and continuous for
and is differentiable except at
and
. If
, then
If
, then
Therefore, the scalars
and
form a lower and upper bound, respectively, for
, the optimal
. Since
is convex, we could (if the lower bound is less than the upper bound) use any line search method to find
. However, simply using line search methods at this point does not reveal geometric properties of the spline and does not lead to efficient calculation of
.
The geometric properties of the set of 5 data points can be classified by looking at
,
and
. For example,
means that the the first three points lie on a straight line;
means that the first three points are convex. When
,
and
, all five points are convex. When
,
and
, the five points “oscillate.” As shown in
Table 1, there are 27 cases to consider, of which, due to symmetry, only 10 cases need be analyzed. We will analyze the location of
in these 10 cases. Recall that
is the unique optimal solution after applying the choice procedure to resolve nonuniqueness, if it occurs.
Remark. The portions of the following results related to linearity (Cases 1, 2, 4, 5, 6, 11 and 12 and cases that are equivalent to these cases) overlap with analogous linearity results in [
22]. In the present paper, however, these linearity results are presented in a wider context where not only linearity but also convexity and oscillation, measured by increases and decreases in the
, are considered.
Recall that from equation (
1), we have
Case 1. In this case,
and
. From Lemma 3,
The unique optimal solution is therefore .
Case 2. In this case,
,
and
. From Lemma 3,
Table 1.
27 cases in the 5-point window method.
Table 1.
27 cases in the 5-point window method.
Case | Sign of | Same as Case | Linearity | Convexity | Oscillation |
---|
| | | / Concavity |
---|
1 | 0 | 0 | 0 | | Yes | Yes | No |
2 | 0 | 0 | + | | Yes | Yes | No |
3 | 0 | 0 | − | 2 | Yes | Yes | No |
4 | 0 | + | 0 | | Yes | Yes | No |
5 | 0 | + | + | | Yes | Yes | No |
6 | 0 | + | − | | Yes | No | No |
7 | 0 | − | 0 | 4 | Yes | Yes | No |
8 | 0 | − | + | 6 | Yes | No | No |
9 | 0 | − | − | 5 | Yes | Yes | No |
10 | + | 0 | 0 | 2 | Yes | Yes | No |
11 | + | 0 | + | | Yes | Yes | No |
12 | + | 0 | − | | Yes | No | No |
13 | + | + | 0 | 5 | Yes | Yes | No |
14 | + | + | + | | No | Yes | No |
15 | + | + | − | | No | No | No |
16 | + | − | 0 | 6 | Yes | No | No |
17 | + | − | + | | No | No | Yes |
18 | + | − | − | 15 | No | No | No |
19 | − | 0 | 0 | 2 | Yes | Yes | No |
20 | − | 0 | + | 12 | Yes | No | No |
21 | − | 0 | − | 11 | Yes | Yes | No |
22 | − | + | 0 | 6 | Yes | No | No |
23 | − | + | + | 15 | No | No | No |
24 | − | + | − | 17 | No | No | Yes |
25 | − | − | 0 | 5 | Yes | Yes | No |
26 | − | − | + | 15 | No | No | No |
27 | − | − | − | 14 | No | Yes | No |
The unique optimal solution is .
Case 4. In this case,
,
and
. From Lemma 3,
Any solution in
is optimal. Since
, the unique solution (the
in the optimal interval closest to
) is
.
Case 5 and 6. In Case 5,
,
and
. In Case 6,
,
,
and
. In both cases, from Lemma 3,
The unique optimal solution in both cases is
.
Case 11. In this case,
,
and
. From Lemma 3,
The unique optimal solution is
.
Case 12. In this case,
,
,
and
. From Lemma 3,
Any solution that lies in
is optimal. Since
, the unique solution is
.
Case 14. In this case,
,
and
. From Lemma 3,
Therefore, the optimal
lies in
. This case is divided into 4 subcases as follows.
Subcase 14-1. If
, then, from Lemma 3,
From the condition
, the interval
is not empty, and any
in this interval is optimal. The unique solution is
Subcase 14-2. If
, then, from Lemma 3,
From the condition
, the interval
is not empty. Since
is strictly increasing on this interval, there exists exactly one
in this interval such that
.
Subcase 14-3. If
, then, from Lemma 3,
From the condition
, the interval
is not empty and any
in this interval is optimal. The unique solution is
Subcase 14-4. If
, then, from Lemma 3,
From the condition
, the interval
is not empty. Since
is strictly increasing on this interval, there exists exactly one
in this interval such that
.
Case 15. In this case,
,
,
and
. Then, from Lemma 3,
Therefore,
lies in
This case is divided into 3 subcases as follows.
Subcase 15-1. If
, then, from Lemma 3,
when
lies in the interval
Therefore, any
in this interval is optimal. The unique solution is
.
Subcase 15-2. If
, then, from Lemma 3,
The unique optimal solution is
.
Subcase 15-3. If
, then
is in the interval
Since
is strictly increasing on this interval, there exists exactly one
in this interval such that
.
Case 17. In this case,
,
,
,
and
. Then, from Lemma 3,
Therefore,
lies in
. This case is divided into 2 subcases as follows.
Subcase 17-1. If
, then, from Lemma 3,
From the condition
, the interval
is not empty. Since
is strictly increasing on this interval, there exists exactly one
in this interval such that
.
Subcase 17-2. If
, then, from Lemma 3,
From the condition
, the interval
is not empty and any
in this interval is optimal. The unique solution is
3. Linkage of Geometric Properties of Data Points and Spline
In this section, based on the analytic results for the solution at the middle point in each 5-point window, we present two theorems that link the local linearity, convexity and oscillatory properties of the original data set with the local linearity, convexity and oscillatory properties of the locally calculated spline. In particular, we show that the locally calculated spline does not “over-oscillate”.
The capability of the 5-point local window method to preserve linearity is described in the following theorem.
Theorem 4. (Proposition 3 of [22]) If any three consecutive points in a five-point window are collinear with slope , then except in Cases 4 and 7. Proof. See Cases 1, 2, 4, 5, 6, 11 and 12 in
Section 2. ☐
Theorem 4 indicates that local linearity of the data is preserved in the 5-point-window spline with the “reasonable” exception of when two lines intersect at the point . continuity of the spline prevents linearity from being preserved in both intervals bordering on a corner .
Convexity is not as simple as linearity. To study the convexity of the spline, we need to consider not just a node but the whole interval . In this interval, the spline is determined by and , which are calculated using the six data points , in the two overlapping 5-point windows for and . In the remainder of this section, we focus on the spline in the interval and assume that these six data points (and, therefore, also their linear spline interpolant) are convex on . The analysis in the rest of this section will reveal that the spline in is not always convex, but, when not, the oscillation is not large.
Lemma 5. The following statements are equivalent:- (i)
The cubic spline function is convex on the interval ;
- (ii)
;
- (iii)
.
Remark. Condition (iii) in Lemma 5 is equivalent to Proposition 3.1 in [
24].
Proof. Recall from the definition in
Section 1 that
,
.
The second derivative of the cubic spline function on
is
Let
,
, then
Hence the cubic spline function is convex on the interval
if and only if
☐
Every contiguous set of six points comes from two 5-point windows. Let Case denote Case/Subcase α for the left window (27 cases and 9 subcases) and Case/Subcase β for the right window (also 27 cases and 9 subcases). After applying Lemma 5 to all convex situations and eliminating equivalent cases, we can identify that the spline is convex on in Cases 1↔1, 1↔2, 2↔5, 10↔2, 11↔5, 5↔14-3, is not convex in Cases 2↔4, 4↔11, 5↔14-1, 5↔14-2, 5↔14-4, 14-1↔14-4, 14-2↔14-4, and is not determined in Cases 5↔13, 14-1↔14-1, 14-1↔14-2, 14-1↔14-3, 14-2↔14-3. However, the spline does not have extraneous oscillation on as is shown in the remainder of this section.
Lemma 6. , where and .
Proof. The proof comes directly from the analysis of the 27 cases. ☐
Remark.
Lemma 6 does not hold for global splines. Consider, for example, the 11 data points , , , , , , , , , and . By the 5-point-window method, (the derivative at ) is 0. In contrast, the of the global spline is 0.37304.
Lemma 7. If , then the cubic spline is bounded above by the linear spline on the interval .
Proof. Given
, the cubic
spline on
can be written as
Let
,
, then
☐
Theorem 8 If the linear spline is convex on the interval , that is, , then the cubic spline is bounded above by the linear spline on the interval .
Proof. The proof comes from Lemmas 6 and 7.
☐
The results in this section indicate that the splines produced by the 5-point-window method with the proposed choice procedure for resolving nonuniqueness preserve linearity and convexity in many cases and do not oscillate excessively. From Lemma 6, we see that the calculated by this method is always bounded by and . This property is a prime factor in restricting oscillation of splines and may in the future lead to additional theoretic results about the properties of splines for non-over-oscillating interpolation of oscillatory data.
4. Conclusions
In summary, the results presented in this paper indicate that a new class of univariate
interpolating splines calculated using 5-point windows as suggested by [
2] has superior geometric shape preservation properties—better than those of
splines calculated using global functionals. Lemma 6 ensures that the optimal solution
(the first derivative at node
) of 5-point-window
splines is bounded by
and
. This property does not hold for globally calculated
splines and is not known to hold for locally calculated
splines with uniqueness being enforced by adding a regularization term to the spline functional as was done in [
7,
9]. Theorems analogous to Theorem 8 that will assist in understanding how local convexity and oscillation in the data set translate into local convexity and oscillation of the
spline are an excellent topic for future research. The results presented here for univariate interpolation are a basis for development of locally calculated univariate
approximating splines and locally calculated bivariate
interpolating and approximating splines.
The algorithmic implications of the analytical results of the present paper are large. In the past, there have been a few published reports and more unpublished reports about deficiencies of the primal affine, primal-dual and active-set algorithms that have been used to minimize
splines. The convergence of these algorithms for medium to large data sets is often unsatisfactory. In addition, the discretization required by the primal affine and primal-dual algorithms is not desirable. The results of the present paper are a basis on which an efficient algorithm that minimizes the original continuum spline functional (not a discretization thereof) can be constructed. In a companion [
25] article, we present such an algorithm and provide computational results for it.