Computer-A/ded
0010~4485(95)ooo87-9
Design. Vol. 28. No. 9, pp. 7X3-760. 1996
Copyright 0 1994 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
l3o10-44w96 $15.cQ+o.o0
ELSEVIER
Fair interpolation and
approximation
of B-splines by
energy minimization
and points
insertion
Tzvetomir
lvanov Vassilev
centripetal methods fail to give good results. Several
other algorithms for ‘optimal’ parameterization
have
been described in the literatured-6 but once the optimum
has been found, the curve is determined simply by using
the interpolation conditions or by a least squares fit. No
other shape constraints are applied.
One approach for constructing a fair curve is to apply
additional mathematical conditions to preserve its shape.
For example Carl de Boor2 proposed the ‘taut spline’
that preserves convexity of the data, and Fritsch and
Carlson’ constructed a C’ cubic interpolant
that
preserves the monotonicity. Another possibility is to
use an interactive fairing scheme8,9, where the user
decides which of the control points should be moved.
Sapidis and Farin” suggested an automation of this
process showing how to determine where a knot should
be removed and a new one inserted using the curvature
plots as a fairness criterion. Note that the resulting curve
after the fairing is always only an approximation to the
original one.
An alternative approach exploits minimization of the
energy of the curve (surface). This dates back to 1966
when Schweikert suggested the idea of a spline in
tension”. Here the standard cubic spline is enriched by
adding new exponential terms and degrees of freedom,
called tension parameters. The works of Salkauskasl2
and Foley13 use a C’ interpolant that minimizes weighted
L2-seminorms of the second, and the first and the second
derivatives, respectively. Gelniker and Gossard14 constructed a shape that naturally resists stretching and
bending. This leads to a functional expressed as an
integral of a weighted sum of the fust and second
parametric derivatives of the curve (or surface). In this
method C ’ continuous cubic Hermite polynomials are used
as a basis for curves and C’ triangular elements for
surfaces. Nowacki and Lti” went further and adopted a
fairness criterion that contains L2-seminorms of the second
and the third order parametric derivatives. Their implementation uses C2 Hermite quintic polynomial curves with
area constraints resulting in a non-linear system of
equations that is solved numerically.
An interpolation scheme, using the fairness norm14
together with Catmull-Clark surfaces, is described by
An efficient method for interpolation and approximation of
both curve and surface points using B-splines is described.
Automatic faking is presented based on minimizing an energy
functional. Additional data points, used as degrees of freedom
for the faking, are inserted only where the curve (the surface)
needs them. This reduces the number of the unknowns to a
minimum which makes the algorithm very fast and efficient
especially when a huge amount of data is concerned. Results of
applying the algorithm for about 15,000 face data points,
subject to measurement errors due to the digitization, are
presented at the end of the paper. Copyright 0 1996 Elsevier
Science Ltd
Keywords: B-splines, interpolation, smoothing
INTRODUCTION
To construct visually acceptable interpolations for sets of
data points has been a long-standing research problem.
The main difficulty arises from the fact that even when
very efficient schemes based on splines are used the
resulting curve (or surface) is often not ‘fair’ enough and
it has extraneous ‘bumps’ and ‘wiggles’ (for example, see
figures in this paper). The objective of the work,
described in this paper, is to develop a method for fair
spline interpolation and approximation for 3D curves
and surfaces that is efficient enough to be applied on
large amounts of data.
Determining the parameterization of the curve is a
fundamental problem to interpolation,
Three basic
methods have been widely used: uniform’, chord
length2 and ‘centripetal’ parameterization3.
Uniform
knot sequence does not usually give satisfactory results
because it does not take into account the geometry of the
points. There are also cases where chord length and
Department of Informatics, TU of Rousse, 8 Studentska Street, 7017
Rousse, Bulgaria
Paper received: 10 March 1995. Revised: 28 November 1995
753
Interpolation
and approximation
of B-splines:
T I Vassilev
Halstead et ~1.‘~ for data sets with arbitrary topology. In
order to add degrees of freedom for the fairing, the
original control mesh is subdivided twice to decouple the
interpolation
constraints
from each other. This, however, dramatically
increases
the number
of control
points. As a result the method is too slow and practically
inapplicable
for large amounts of data.
This paper describes an algorithm for fair interpolation
and approximation
with C’ continuous
cubic B-splines.
The integral fairness criterion of Celniker and Gossard14
is applied for curves, but a new one is suggested for
surfaces. Its main advantage is that it preserves the twodimensionality
of the arrays
which increases
the
efficiency. Additional
data points (ADP) are used as
degrees of freedom for minimizing
the energy of the
curve (surface), unlike other methods
that use new
control points. The ADP are not distributed
evenly but
inserted only where necessary, where a lack of data is
observed. This results in a linear system of equations
with a decreased number of unknowns.
The parameterization is based on a new method for incorporating
the
data geometry and the method is presented for both
curves and surfaces. The scheme was developed to be
applicable for large amounts of data.
The rest of the paper is organized in the following way.
In the second section the construction
of a fair spline
curve is described, including
evaluation
of the energy
functional,
interpolation,
approximation
and choice of
additional data points. In the third section the method is
extended to tensor product surfaces. Finally, the results
and conclusions
are presented in the fourth and fifth
sections, respectively.
FAIR B-SPLINE
CURVES
Evaluating the energy functional
Let w(u) = [x(u), y(u), Z(U)] be a space curve parameterized by u and let V be a column vector of its B-spline
control points. Celniker and Gossardr4 suggested the
following energy functional
formula’
one can easily find that
.I
‘.
K,, =
”
7
-12
-1
7
34
-29
-12
-12
-29
34
7
1-1
-12
7
6 1
1
B,B, du = 120
.T
2
-3
0
1
-3
6
-3
0
o
-3
6
-3
1
’ h,&!du
J0
Kzs =
1
6
r
= A
6
1’
O
-3
1
(4)
21
Now. after we have expressed
the local fairness
measure
for a single segment,
the global
fairness
measure E for a B-spline curve can be found as a sum
of the local fairness measures over each of the B-spline
polynomial
segments
E = VTKV.
K = aK, + OK2
(5)
where K is a quadratic symmetric band matrix obtained
from the local stiffness matrices. Minimizing
Equation 5
for a particular choice of degrees of freedom will ensure
the fairing of the curve.
Fair B-spline interpolation with ADP
We will modify
slightly
the original
problem
of
interpolation
for the purposes of fairing. The new task
is: given a sequence of data points pi, i = 1) . ~n - 1,
some of them unknown and used as degrees of freedom,
find the vector V of B-spline control points of a curve,
which passes through the points pi and minimizes the
fairness measure in Equation 5. This is demonstrated
in
Figuw I, where the original
data set is {pI ,
, ps,
pg. . p13} and the unknown data points are (p6, p7, ps}.
Using the interpolation
constraints
with end conditions
set as degrees of freedom, the control points can be found
by solving
AV = P
E =
Ub(aiv2(u) + @ti2(U))du
s u‘l
(1)
where A is a coefficient
where W(u) and K(u) are the first and second derivatives
in respect to the parameter u. The energy of the curve is
represented
as a weighted sum of its stretching
and
bending
terms, where Q and p are freely selected
coefficients.
A single cubic B-spline segment can be written as
w,(u) = VTB,, where B, is the B-spline basis functions
vector. Evaluating
the integral of Equation
1 for the
function w, will result in the following quadratic form
Es = V;K,V,
(6)
matrix
and P is a column
vector
Y
(2)
where K, is a 4 x 4 symmetric matrix, whose entries are
calculated by evaluating the integral of Equation 1. Es is
called the ‘local fairness measure’ and K, is the ‘local
stiffness matrix’ for the segment wS(u). K, can be
represented
as a weighted sum of its stretching
and
bending terms
K, = oK~s + PK2s
For
754
a uniform
X
(3)
knot
distribution,
using
de Boor’s
Figure I
Data set for fair interpolation
with additional
data points
interpolation
and approximation
of B-splines:
T I Vassilev
cannot be performed for physical reasons. As shown by
the various experiments the author conducted, the curve
tends to have extraneous
inflection points and undulations in and around
these intervals.
Therefore
the
additional
data points should be inserted
precisely
where a lack of data is observed. Figure 2 demonstrates
the insertion algorithm for the data set in Figure 1. It is as
follows.
(1) Calculate the parameter values u,, i = 1,
. II, using
the chord length or centripetal formula (Figure 2~).
(2) Compute the average length between two parameter
values Au = u,/(n - 1).
(3) For each interval
[u,, ul+,] find the number
n, = round(Au,/Au)
of the new parameter
values
(data points) and insert them (Figure 2h).
(4) Recalculate
the sequence
u, this time using the
uniform formula (Figure 2~).
We could use the parameterization
from Step 3 to
perform the actual interpolation
(or approximation).
Ln
this case, however, we will have to use a numerical
method to evaluate the integral in Equation
1, e.g.
Gaussian quadrature18. That is why the author suggests
that Step 4 should be carried out. Then the stiffness
matrix K can be computed
in advance and directly
incorporated
as a constant array into the program. This
increases the efficiency especially when a large data set
is concerned.
The final parameterization
(Step 4) is
uniform in respect to the ADP-enriched
data set but it is
non-uniform
in respect to the original
one (see the
sequence {u,.
us, uy.. , u13} in Figure 2c and the data
set {p, ~. ,ps,p9..
. pli} in Figuw I). In fact, the knot
vector (Figure 2c) is only an approximation
of the chord
length parameterization
in Figure 2h. but it is good
enough, because the smoothness of the curve is ensured
by the minimizing of the fairness measure in Equation S.
FAIR B-SPLINE
-104
4 k
Figure 4
intcrpolant
line)
Data set I. Cubic B-spline interpolant
(dashed line); fair
with ACP (dotted line); fair interpolant
with ADP (solid
V be a tzl, x n,. matrix of its control points. The following
energy functional is considered in this paper
SURFACES
Fairness criterion
Let w(u, U) be a surface parameterized
by u and IS.and let
where the suffixes mean partial derivatives in respect to
the parameters
u and ?I. A tensor product
B-spline
surface can be expressed as
wju, 11)= BIfVB,.
where B,, and
functions.
(17)
B,. are
vectors
of the
B-spline
basis
Interpolation and approximation with ADP
The interpolation
(approximation)
the following system:
constraints
result in
A,,VA,, = D or V = A, ‘DA,. ’
(18)
Suppose we have inserted additional data points only in
the direction
of the parameter
u (see Figure 3), then
Table 1
Figure 3
parameter
Surface data set with new data points inscrted 111direction
11
01
.\
)
Data set
0.0
2.0
0.0
0.5
I, Figure4
3.0
6.5
4.0
7.6
5.0
6.5
6.0
1.2
7.0
0.62
8.0
0.6
9.0
0.6
Interpolation
and approximation
of B-splines:
T I Vassilev
Y
i
x
7-e54-32I--
X
a
b
2
Figure 6 Face data set: (a) least-squares
squares approximation
with ADP
IO-
approximation;
(b) fair least-
16.
freedom need to be slightly modified for tensor product
surfaces. New parameter values (data points) for the
sequence ui will appear in each one of the n, isoparametric curves (see Figure 3), which may be computed as
follows:
lo-6-
X
0
Figure 5 Data set 2. Cubic B-spline interpolant
interpolant
(I = 1, /3 = 0.2 (dotted line); ADP
p = 1 (solid line)
(dashed line); ADP
interpolant
a = 0.2,
Equation 18 becomes:
V = A;‘(D,X,
+ Ds)A,*
(19)
where X, is a r, x n, matrix of the unknowns and D, is a
II, x n, matrix of the known data points.
Substituting Equations 17 and 19 in Equation 16 and
finding its minimum leads to a system
D;rKAUD,X, + D;rKAUDO= 0
(20)
where KAu and D, are the same as in Equation 11 but
computed for the knot sequence Ui. Finding the solution
of Equation 20 requires solving rz, linear systems with r,
unknowns. Note that the system matrix DzKAUD, is
common and it can be computed and decomposed in
advance which speeds up the method. A detailed proof of
Equation 20 is given in the Appendix.
If we insert additional data points in the direction of w
we will obtain nearly the same system as Equation 20 but
with matrices computed for the sequence vi. Once all the
data points have been found the control points are
calculated from Equation 18.
Parameterization and choice of degrees of freedom
The parameterization
and the choice of degrees of
Table 2
I, 2 and 5
x
Y
0.0
0.0
Data set 2, Figures
1.0
0.1
2.0
0.2
3.0
0.4
4.0
1.0
5.0
6.6
6.0
7.2
7.0
7.4
8.0
7.5
9.0
7.6
(1) For each interval [ui, Ui+,], using Steps l-3 from the
previous section for each curve, find the number
= lElX(nij,~ = l,...,n,).
%max
(2) For each interval [Ui,Ui+1]insert nimaxnew parameter
values.
(3) Recalculate the sequence Ui, using the uniform
formula Ui = i, thus making the parameterization
common for all the isoparametric curves, which is
necessary for tensor product surfaces.
Controlling the error of approximation can be done in
the same way as for curves but this time the values L, and
L, for both parameter directions should be varied.
RESULTS
The algorithms were implemented on a Silicon Graphics
Indy workstation using the open GL library to render the
surfaces. The results are shown in Figures 4-8. The
original data points are marked with crosses and the new
ones (inserted by the algorithm) with squares. The
curvature and slope distributions are given below every
curve as indicators for the fairness. All the experiments
(except Figure 5) were conducted with coefficient values
inEquation 1 (~=/3= 1.
An alternative method for fair interpolation has been
implemented that follows from the idea for inserting
Table 3
curves
Performance
of the different
Method
Fair interpolation
Fair interpolation
Approximation
Fair approximation
methods
Figure
with ACP
with ADP
with ADP
Figure 6a
Figure 6b
for approximating
Number of
polynomial
segments L
Time (s)
354
125
70
70
71.3
4.9
0.05
1.2
757
Interpolation
and approximation
of B-splines:
T I Vassilev
(4
(b)
Figure 7 (a) B-spline
interpolation
surface
interpolation:
(b) fair B-spline
surface
enough additional control vertices so that the interpolation constraints
are decoupled from each other16. This
method uses the same energy functional
but additional
control points (ACP) as degrees of freedom. Figure 4
compares the quality of the schemes with the data set
given in Table 1. Type 1 end conditions
(given tangent
vectors) have been imposed
for the cubic B-spline
interpolation.
The
method
with
ACP
obviously
improves the smoothness
of the curve but it neither
cures the violated monotonicity
in interval
one nor
removes the extraneous inflection points in intervals 6, 7.
Figure 5 demonstrates
the results with another data set
(Table 2). It also shows the influence of the parameters
N
and p, which can be used as shape controls. If N > fl the
seminorm of the first derivative takes a bigger part in the
758
Figure 8 (a) Least-squares
surface approximation
for face data set; (b)
fair least-squares
surface approximation
for face data set
fairness measure and the interpolant
(dotted line) is more
stretched (as if somebody pulls it at the ends). This effect
can be best observed for the limit case (/I = 0) but then
the interpolant
might not be smooth enough. When
(I < ,!3the second derivative seminorm takes the bigger
part and the resulting curve (solid line) is more bent.
Fair B-spline least squares approximation
(Figure 6)
was applied on 119 points of a face profile obtained
through digitization and subject to substantial noise. The
effect of fairing is obvious in the areas of the forehead
and the mouth.
The performance
of the different
methods
for this data set is given in Table 3. The
efficiency of the scheme with ADP improves by a factor
of 14 compared with that of ACP.
Figure 7 shows B-spline surface interpolation
of the
Interpolation
Table 4
Performance
and approximation
of B-splines:
T I Vassilev
of the different methods for approximating
surfaces
Method
Fair interpolation
Approximation
Fair approximation
Figure
Figure 8a
Figure 8b
Number of
segments
Farin, G Curves and Surfaces for Computer Aided Geometric
Design Academic Press (1993)
de Boor, C A Practical Guide to Splines Springer-Verlag, New
Time (s)
L”
L”
130
130
21.1
56
56
70
70
3.2
6.8
6
data points in Figure 3. The unwanted bumps and
wiggles (Figure 7~2)were removed by the fairing (Figure
7b). Least squares B-spline fair approximation has been
applied on a ‘noisy’ data set, of about 15,000 points. The
extraneous undulations (Figure 8~) were smoothed by the
algorithm (Figure 8b), automatically inserting new data
points where necessary. The performance of the different
methods for this data set is given in Table 4. It is obvious
that the fairing is not too expensive compared to the pure
approximation, the quality, however, is much better.
10
11
12
CONCLUSIONS
13
14
A method for fair interpolation and approximation of
both curve and surface points using B-splines has been
described. The suggested energy functional for surfaces
(Equation 16) preserves the two-dimensionality of the
arrays of control and data points. Solving this kind of
system, as Farin’ shows, leads to 0(n4) computations,
which are normally 0(n6) if the points are arranged in a
vector. Although the algorithms were implemented for the
cubic case, they could be easily extended for splines of any
order k. Additional data points (ADP) are inserted as
degrees of freedom for the fairing only where necessary.
This approach proved to be very suitable for the case of
measured (or digitized) data when the points have been
originally planned to be evenly distributed, but for some
practical reason measurements cannot be performed in all
of the intervals. An important feature is that the
algorithms for ADP insertion and for the actual fairing
are independent. So, if for some practical applications
another insertion scheme proves to be more appropriate,
it can be used without any limitations with the described
fairing procedure. The idea of ADP can be extended to
other methods for global fair interpolation (approximation). It will improve the performance, especially when a
larger amount of data is concerned.
ACKNOWLEDGEMENTS
The author would like to thank the British Council and
Microsoft who sponsored his work. Figures 6 and 8 are
produced from data obtained from a digitization of a
Spitting Image puppet of Margaret Thatcher. Thanks to
Spitting Image production for providing the puppet and
to the UK Advisory Committee on Computer Graphics
for funding the digitization. The work described in this
paper was done at Queen Mary and Westfield College,
University of London. The author is very grateful to his
supervisor Mel Slater and to the referees for their help
and useful comments.
York (1978)
Lee, E ‘Choosing nodes in parametric curve interpolation’
Comput.-Aided Des. Vol21 No 6 (1989) pp 363-370
Hartley, P and Judd, C ‘Parameterization and shape of B-spline
curves’ Comput.-Aided Des. Vol 12 No 5 (1980) pp 235-238
Nielson, G and Foley, T ‘A survey of applications of an affine
invariant norm’ in Lyche, T. and Shumaker, L. (Eds.) Mathematical Methods in CAGD Academic Press (1989) pp 445-468
Marin, S P ‘An approach to data parameterization in parametric
cubic spline interpolation problems’ Approx. Theory Vol41 (1984)
pp 64-86
Fritsch, F N and Carlson, R E ‘Monotone piecewise cubic internolation’ SIAM J. Numer. Anal. Vol 17 No 2 (1980) pp 238-246
Kjellander, J A P ‘Smoothing of cubic parametric splines’
Comput.-Aided Des. Vol 15 No 5 (1983) pp 175-179
Rogers. D F. Satterfield. S G and Rodrieuez, F A ‘Shiphull,
B-&ink surfaces and CAD/CAM' in Computer Graphics-Theory
and Applications Springer-Verlag (1983)
Sapidis, N and Farin, G ‘Automatic fairing algorithm for B-spline
curves’ Comput.-Aided Des. Vo122 No 2 (1990) pp 121-129
Schweikert, D G ‘An interpolation curve using a spline in tension’
J. Math. Phys. Vol45 (1966) pp 312-317
Salkauskas, K ‘C’ splines for interpolation of rapidly varying
data’ Rocky Mountain J. Math. Vol 14 (1984) pp 239-250
Foley, T A ‘A shape preserving interpolant with tension controls’
Comma. Aided Geom. Des. Vol 5 (1988) DD 105-l 18
Celniker, G and Gossard, D ‘Deformable curve and surface finite
elements for free-form shape design’ Proc. SIGGRAPHVol25
No
4 (1991) pp 257-266
15
16
17
18
Nowacki, H and Lii, X ‘Fairing composite polynomial curves with
constraints’ Comput. Aided Geom. Des. Vol 11 (1994) pp 1-15
Halstead, M, Kass, M and DeRose, T ‘Efficient, fair interpolation
using Catmull-Clark surfaces’ Proc. SIGGRAPH, Annual series
(1993) pp 35-44
Forsythe, G and Moler, C Computer Solution of Linear Algebraic
Systems Prentice-Hall (1967) pp 114-l 19
Press, W H, Flannery, B P, Teukolsky, S A and Vetterling, W T
Numerical Recipes, The Art of Scientt$c Computations Cambridge
University Press, New York (1989) pp 121-126
APPENDIX
Equation 16 is a sum of four similar terms and here the
solution to the first one is given. From Equation 17
w:v = BTVTB
2)
uBTVB
11 ‘v
(Al)
where the dots mean first derivative. We have to compute
d
cl w;,dudv = cl 8%
d
= Clax,,
f f
B;A;~(D,x,
B=V=i B=VB dudv
Cl
u I4 21
+ Do)=A;=IiUti;fA,’
(D,X, + Ds)A,%,dudv
= 2c,
D~A,=i3,li~A,‘(D,X,
+ D,,)
A,‘&,iS;A,=dudv
= 2c,D;A;=
(ji.B:du)A;l(D,,Xu
= 2c,D;fA,TK1uA;1(D,X,
+ D,,)A,’
+ Do)A,‘KI,A,=
(-42)
759
Interpolation
and approximation
of B-splines:
T I Vassilev
Results
similar
to Equation
A2 will be obtained
evaluating
the remaining
three terms of Equation
16.
Then the system of equations reads (++ means if and only
4
D:A&,KI,
+
LZKd A,’ (DuX,, + Do)A, ’
((Y,K,~ + ,/&K2,)A,T = 0 #
D;A,TK,A,‘(D,X,
D:K,,(D,X,
+ Do)A,‘K,,A,.T
+ Do)KA,. = 0
= 0 ++
(A3)
Here cl = (Y,,cY,.,cz = ,&,a,,, c3 = c@,,, c4 = /L&. The
matrix KAt, is symmetric,
band and positive definite
and its inverse KA:, exists. Multiplying
both sides of
760
Equation
A3 with K,&! results in
D,TKA~D,X, + D,TKAuDo= 0
(Ad)
Tzvetomir Ivanov Vussilev gruduuted
jiom
the Technicrrl b’niversi!,~ qf
Roussr, Bulgaria in 1990. He i.5 currently a lecturer at the Department of
Informatics at the same university. His
research interests include computer-aided
geometric d~@yn, data processing, and
computer graphics