for a given application. 1. INTRODUCTION. In many applications, one wishes to interpolate or approximate univariate data by spline functions possessing ...
Mathl. Comput. Modelling Vol. 15, No. 10, pp. 97-108, 1991 Printed in Great Britain. All rights reserved
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A SURVEY ON UNIVARIATE DATA INTERPOLATION AND APPROXIMATION BY SPLINES OF GIVEN SHAPE H. GREINER Philips Forschungslabor GmbH Weisshausstrasse, 51 Aachen, Germany
(Received
April
1991)
Abstract-The construction of spline functions possessing certain shapes like monotonicity, convexity or nonnegativity is an important concern in data representation and computer-aided design. The literature on the subject is reviewed with the aim to help in the selection of the method best suited for a given application.
1. INTRODUCTION In many applications, one wishes to interpolate or approximate univariate data by spline functions possessing certain geometric properties or “shapes” like monotonicity, convexity or nonnegativity. These properties may be desirable for physical (e.g., a volume-pressure curve should have a nonnegative derivative) or practical reasons (e.g., in computer-aided design). Unfortunately, the standard procedures for constructing interpolating or approximating splines to given data often lead to functions which do not reproduce their shape, but exhibit unwanted extrema and inflection points. Our own interest in the field has been stimulated by the need to find shape preserving interpolants and approximants to thermodynamic data representing the free energy of mixing and the activities of binary mixtures as a function of composition (the free energy of mixing is necessarily convex, whereas the activities are necessarily monotone). During the last two decades, a substantial amount of research on the properties and the effective construction of shape preserving splines has been undertaken. Here, we want to give a concise review of the main results so that the interested reader can quickly sort out the literature relevant to a particular problem. In the next section, we establish the mathematical formulation and terminology. The following sections then describe the main results and algorithms available for the various problem categories. General reviews with various emphases are given in [l-6]. [S] presents many examples and FORTRAN programs. Specific applications are discussed in [7-161. 2. SPLINES
AND
THEIR
CONSTRUCTION
As is well known, a polynomial spline is a piecewise polynomial function (see e.g., [17]). More precisely, let x = (21, . . . , zn) denote a partition of the interval [zi, z,] by “knots” Zi, i = 1 , . . . , n. A real valued function s on [zi, z,] is called a spline of degree k and smoothness m if s restricted to each subinterval of the partition x is a polynomial of degree k and ifs is m times continuously differentiable. The space of all such splines will be denoted by Sp(k, m, x). In the sequel, we shall also refer to splines made up from other functions than polynomials: Rational splines are piecewise rational functions (a rational function is the quotient of two polynomials) which are continuously differentiable up to a certain order. Exponential splines are C2 functions which are defined piecewise by expressions of the form ai +
bi(r - zi) + ci exp(pi(z
- xi)) + 4 exp(-pi(+
- ti))
i = 1 ,...,n-1. Tw=t
97
by 44-W
H. GREINER
98
Here the pi are the so-called tension parameters which allow to control the shape of the spline. They were first introduced in [18,19]. Given an ordered data set (xi, yi) i = 1,. . . , n we define the slopes Ai = (pi - yi-i)/(Zi if the Ai are nonnegative and convex if the Zi-_1) i = 2,. . . , n. The data are called increasing n are nondecreasing. Ai, i=2,..., It is natural to require that the spline interpolants preserve the shape of the data and/or convex for monotone and/or convex data. Such (Zir pi) i = 1,. . . , n, i.e., are monotone splines are called monotonicity and/or convexity preserving. Clearly, the broken line interpolant is a monotonicity and convexity preserving spline interpolant, but for many applications smoother and more visually pleasing interpolants are required. If the data are not globally monotone, one wishes to construct interpolants which preserve the monotonicity locally, i.e., interpolants with a minimum number of local extrema compatible with the data. One way to ensure the preservation of local monotonicity is to require co-monotonicity of the interpolant s, i.e., s should be nondecreasing (nonincreasing) on intervals where the data are nondecreasing (nonincreasing). Similarly, for data sets which are not globally convex, one wishes to preserve local convexity, i.e., interpolants with a minimum number of inflection points co-convexity of the intercompatible with the data. This can be insured e.g., by stipulating polant s, i.e., the first derivative should be nondecreasing (nonincreasing) on intervals where the slopes Ai are nondecreasing (nonincreasing). We note that the requirement of co-monotonicity and co-convexity entrain that the local extrema and inflexion points coincide with the data points, a feature which may be considered as artificial (see [3]). Interpolation algorithms preserving local shape can be applied to any data set. Algorithms preserving global shape can only be applied to suitably segmented data sets and may encounter difficulties at the segment boundaries. Another problem is the construction of approximants of a given shape. Here, shape is defined by the application and is not necessarily implied by the data to be approximated. Basically, there are three different methods for the construction of shape preserving polynomial splines: The first method describes the spline as the solution of a variational problem. For interpolation it takes the form F(U) = subject ditions.
to the interpolating For approximation
In u”(z)* J Xl
conditions u(xi) = yi, it is formulated as
F(u) =
=, u”(z)*dx J +1
+ (Y2
dx = Min, i = 1,. . . , n and appropriate
Wi(u(xi)
boundary
con-
- yi)* = Min,
i=l
with weights wi, i = 1,. . . , n. The “smoothing” parameter cr controls the closeness of the “smoothing” spline to the data. In both cases, the shape of the spline is enforced by defining various constraints like nonnegativity (U 1 0) , monotonicity (u’ 3 0) or convexity (u” 1 0) . In all cases, it can be shown that the solution of the constrained variational problem is a cubic C* spline whose knots need not coincide with the original data points xi. If the problem is formulated in a Hilbert space setting, it is readily shown that F is a strictly convex functional, and that the constraint set is closed and convex. Hence, the problem has a unique solution. This functional analytic approach is developed in [20,21]. The second method is based on Hermite interpolation. One considers splines of Hermite polynomials (in practice quadratic, cubic and sometimes quintic polynomials are used) which are determined by their function and derivative values at the end points of the knot intervals. To construct an interpolation scheme for a given shape, one first has to identify a set of derivative values at the data knots admissible for the given shape. In a second step, an algorithm for selecting admissible derivative values has to be formulated. One way to accomplish this is to start
99
Univariate data interpolation
with some interpolating spline and to modify or “constrain” its derivatives such that they lie in the admissible set. The third method for the construction of shape preserving splines proceeds by the construction with of suitably defined Bezier curves (cf. [17]): a shape preserving piecewise linear interpolant breakpoints in the knot intervals IQ, zi+l[ is constructed. The restriction of the spline on the interval [pi, zi+J is then obtained as the Bernstein polynomial of the piecewise interpolant on that interval. Here, the Bernstein polynomial of degree n of the function f defined on the interval [a, b] is defined by n-k
.
By the well known shape preserving properties of the Bernstein polynomials (cf. [17]), these splines have the same shape as the linear interpolants from which they are obtained. Furthermore, by locating the breakpoints of the linear interpolants properly, one can ensure the differentiability of the splines. Exponential splines are obtained by solving systems of linear equations for their coefficients [19]. As the exponential spline interpolants converge with increasing tension parameters pi to the braken line interpolant, they can always be rendered shape preserving. The problem is how large the tension parameters have to be chosen in order to obtain an appealing curve. [22] describes a whole class of smooth interpolating splines depending on a parameter such that with incressing parameter the spline converges to the broken line interpolant. These generalized splines in “tension” include the exponential spline, a rational spline and several cubic splines. The construction of shape preserving rational splines is complicated by the fact that their coefficients have to be obtained from systems of nonlinear equations, for which existence and convergence proofs may be difficult to establish. A further concern are the approximation properties of the interpolating and approximating splines, i.e., the question how fast they converge to a given function. It can be shown that for polynomial splines the convergence rates are the same as those obtained for the unconstrained case. The interested reader should consult [23-301. The convergence properties of exponential splines are investigated in [31]. 3. CONVEXITY Given convex the minimization
subject
PRESERVING
SPLINE
INTERPOLATION
data (zi, yi) i = 1,. . . , n, a convex interpolant problem III U”(Z)’ dz = Min, J Xl
can be found
as the solution
of
to u(zi)=yi,
i=l,...,n
U”(Z) 1 0
for z1 5 2 5 zn.
[32,33] show that the solution is a convex C2 cubic spline with at most [3n/2]+ 1 knots which need not coincide with the original knots pi. In [34,35] it is shown how this spline can be computed and a number of instructive examples are presented. The general case with &-norm, 1 5 p < CO, is treated in [36]. [37] demonstrates that the convex interpolation problem can be solved by a quadratic Cl-spline if one introduces at most one additional knot per interval. The solution is characterized by a minimum of ]u”], . [17, Chapter 161 shows that “extraneous” inflection points in natural C2 cubic spline interpolants can be removed by the introduction of at most one additional knot per data interval. Furthermore, the user can control the ‘%autness” of the spline by a “tension” parameter. A Fortran program is available. For given data (zi, yi) i = 1, . . . , n, [38-401 elucidate necessary and sufficient conditions for the existence of a convex interpolating spline:
H. GREINER
100
Let k and 0 < m < k - 1 and 1 = min(k - m - 1, m) be given. Then interpolating spline s E Sp(k, I, x) if and only if the system of inequalities
(k - m) Pi + mpi k
0 prescribed is a C2 cubic spline with knots ti i = 1,. . . , n and at most [+I + 1 additional knots. Algorithms for the solution of this problem are presented in [75-771. [78] considers the more general constraints 4(z) < u’(z) 5 $(z) where 4 and $J are measureable functions. An effective algorithm for 4 and $J continuous and piecewise linear functions is stated. [3] shows that local monotonicity preserving C’ cubic splines can be obtained as the solution of the variational problem V(u)
= g
w;
J*‘+lu”(2)’
dz + 2
+I
i=l
Diu’(zi)’
= Min,
i=l
where u ranges over all cubic C1 spline subject to the interpolating and appropriate end conditions. The general solution of this problem is the so called Y-spline. The interval weights wi control the tightness of the spline in the ith interval, whereas the point tension parameters vi control the magnitude of the derivatives u’(zi). The requirements of monotonicity and convexity then become linear inequalities for the spline coefficients. Computationally, we are thus dealing with a positive definite quadratic minimization problem subject to linear inequality constraints which can be treated by standard numerical software. The approach produces nice comonotone interpolants, but already simple examples demonstrate that they are not necessarily convexity preserving. In contrast to the convex case, monotonicity preserving Hermite interpolation by C’ cubic splines is always possible (but not by C2 cubic splines, cf. [79]): for given monotonic data (Zi,W) i = l,..., n and m a monotonic spline s ES(X, 2m + 1, m) satisfying sj(zi) = 0 for j = l,... , m can be constructed [80]. But unfortunately, its visual appearance is very often is determined unpleasing (see the example given in [Sl]). In [82] it is shown that this interpolant by the requirements of monotonicity preservation and linearity of the interpolation operator and has poor approximation properties. Better interpolants can be obtained by nonlinear cubic Hermite spline interpolation. [81] derives necessary and sufficient conditions for a cubic polynomial to be monotone on an interval in terms of function and derivative values at the interval ends. A two-pass algorithm for monotonicity preserving cubits is formulated: In a first pass over the data, numerical approximations to the derivatives are calculated. In a second pass, the derivative values are then modified (where necessary) to ensure monotonicity preservation. The procedure is nonlocal and cannot be applied to nonmonotonic data. The method given in [83] is free of these drawbacks: A function G such that the slopes di = G(Ai-1, Ai) generate co-monotone cubic splines is indicated. [84] contains a local algorithm for co-monotone interpolating splines from Sp(x, n, R) with k < n - L. The splines are obtained as Bernstein polynomials of suitable linear interpolants whose slopes in the knots are defined by a function di = G(Ai_1, Ai). the derivative In [79] the algorithm of [81] is analysed and a simple procedure for modifying values is introduced. The method is local and can deal with nonmonotonic data. In [45] the end slopes of monotonic polynomials of arbitrary degree on an interval are identified. [l] states algorithms for the construction of cubic and quintic Hermite splines preserving local monotonicity. Monotinicity preserving rational splines are considered in [15,58,59,85-871. 6. APPROXIMATION [78] considers
the variational V(u) =
BY MONOTONE
SPLINES
problem
Jzn
u”(x)’ dz +
Xl
2 wi(u(xi) i=l
- yi)' = Min
103
Univariate data interpolation
subject to the restrictions 4(z) _< u’(z) 5 $(z) . It is shown that for continuous and piecewise linear functions 4 and II, the solution is given by a C2 cubic spline, which can be efficiently computed by solving a set of nonlinear equations by Newton’s method. [72] studies the approximation of data by monotone functions defined as linear combinations of user defined basis functions in the L1 and L, norm by applying methods of linear programming. In [73] the basis functions are taken as B-splines and algorithms for obtaining monotone approximants in the L2 norm (2) are given. 7. NONNEGATIVE
SPLINES
Given nonnegative data (zi, yi) with yi > 0, i = 1,. . . , n, one wishes to construct splines with ~(2) 2 0 for tl 5 z 5 c,,. [75] shows that the solution of the problem
interpolating
In
J
~“(2)~ dz = Min,
z1
subject to the interpolating conditions and u(x) > 0 for xl 5 x _< x,, is a C2 cubic spline and indicate an effective algorithm. The same problem is also addressed in [76] by methods of optimal control. [l] states necessary conditions on the derivatives of cubic and quintic Hermite polynomials ensuring nonnegativity. A simple constraining algorithm for the derivatives is given. Necessary and sufficient conditions for the derivatives of cubic Hermite splines for nonnegativity are formulated in [88]. Nonnegative interpolation with rational quadratic splines is analyzed in [89,90]. Nonnegative exponential splines are considered in [91]. [73] studies the least-square approximation of data by nonnegative B-splines. Yet the formulation of a general algorithm for nonnegative smoothing splines seems to be an open problem. 8. CONCLUSION We have tried to offer a review of the available methods for shape preserving interpolation and approximation by polynomial, exponential and rational splines. In our opinion, quadratic and cubic splines are the most versatile in that for nearly all cases they can be computed effectively. constructive existence proofs have been given. Furthermore, Compared to polynomial splines, exponential splines are costly to evaluate. The use of rational splines seems to be restricted to interpolation. They are quite powerful for certain applications, but their computation can be quite involved (see [S]). REFERENCES 1. R.L. Dougherty, A. Edelman and J.M. Hyman, Nonnegativity-, monotonicity-, or convexity-preserving cubic and quintic Hermite interpolation, Mathemalics of Compulation 52, 471-494 (1989). 2. D.R. Ferguson, Construction of curves and surfaces using numerical optimization techniques, Computer-Aided Design 18, 15-21 (1986). 3. T.A. Foley, A shape preserving interpolant with tension controls, Compater Aided Geomeiric Design 5, 105-118 (1988). 4. J.A. Roulier, Constrained interpolation, SIAM J. Sci. Slat. Compul. 1, 333-344 (1980). 5. J.W. Schmidt and H. Spiith, Results and problems in shape preserving interpolation and approximation with polynomial splines, In Splines in Numerical Analysis, (Edited by J.W. Schmidt), pp. 159-171, Akademie Verlag, Berlin, (1989). 6. H. Spiith, Eindimensionale Spline-Interpolations-Algorithmen, R. Oldenbourg, Miinchen, (1990). 7. H.M. Anthony, An automatic algorithm for immunoassay curve calibration using controlled quadratic splines, IMA J. Math. Appl. Med. Biol. 6, 91-110 (1989). 8. L.E. Deimel, C.L. Doss, R.J. Fornaro, D.F. McAllister and J.A. Roulier, Application of shape preserving spline interpolation to interactive editing of photogrammetric data, Conf. Proc. Siggraph 12, 93-99 (1978). 9. B. Dimsdale, Convex cubic splines, IBM J. Res. Develop. 22, 168-178 (1978). 10. A. Lahtinen, On the construction of monotony preserving taper curves, Acla For. Fennica 203, l-34 (1988). 11. S.P. Marin, An approach to data parametrization in parametric cubic spline interpolation problems, Journal of Approximation Theory 41, 64-86 (1984).
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