C1-Interpolation for Vector Field Topology Visualizationfrey/papers... · 4 Vector Field Topology As said previously, vector field topology consists of the association of critical

C1-Interpolation for Vector Field Topology VisualizationGerik Scheuermann Xavier Tricoche Hans Hagen

University of KaiserslauternDepartment of Computer Science, Computer Graphics & CAGD

P.O. Box 3049, D-67653 KaiserslauternGermany

E-mail: scheuer|tricoche|hagen @informatik.uni-kl.de

AbstractAn application of scalar interpolation for 2D vector field topol-ogy visualization is presented. Powell-Sabin and Nielson inter-polants are considered which both make use of Nielson’s MinimumNorm Network for the precomputation of the derivatives in our im-plementation. A comparison of their results to the commonly usedlinear interpolant underlines their significant improvement of sin-gularity location and topological skeleton depiction. Evalution isbased upon the processing of polynomial vector fields with knowntopology containing higher order singularities.Keywords: vector field visualization, topology, critical point the-ory, -interpolation

1 IntroductionVector field visualization is an issue of major interest in many sci-entific and engineering areas. As a matter of fact, vector fields offera qualitative and quantitative description of numerous natural phe-nomena. In physics, they play a fundamental role in fluid dynamics,solid mechanics, electricity or magnetics, among others. They arealso impossible to circumvent for engineers that massively makeuse of them in disciplines like computational fluid dynamics (CFD),finite element analysis (FEA) and computer aided design (CAD).Typically, measurements or numerical simulations provide analystswith increasingly large vector data sets. This raw data must nextbe properly conveyed for interpretation. The aim of vector field vi-sualization is to offer a convenient way to extract this information.But to be of any interest, the display has to focus efficiently on themost meaningful aspects of the data to avoid confusing the results.

Among the existing techniques in this sphere, topology-basedmethods have proved to be very successful in enabling a good in-sight into the qualitative nature of the vector field while reducingthe size of the data. Their basic principle is to locate and classifythe critical points (i.e zeros) of the field and to draw a small numberof streamlines connecting them, separating the field into regions ofequal qualitative behavior.

As a preliminary step of the topology extraction, one has to workout the interpolation of the given discrete data. A commonly usedsolution is the computation of a linear interpolant over each cell ofthe triangulated (unstructured) point set. One problematic aspect ofthis method is that the linear interpolant is inaccurate when beingconfronted with several very close critical points or with higher or-der singularities : zeros are moved or split up and the global topol-ogy is thus likely to be altered. Furthermore, one gets piecewiseconstant differential fields (e.g. divergence, curl) that cannot bemeaningfully compared to experimental measurements or simula-tions. Consequently, consistancy is lost between the vector fieldand its associated differential fields.

This paper presents two higher order interpolation schemes ap-plied to vector field topology visualization. It is shown that the

topology is in both cases better reproduced than by piecewise linearinterpolation. Furthermore, unlike the latter, -interpolation suc-cesfully attacks the problem of additional critical points. At last,the resulting topological skeletons appear more reliable and easierto analyze.

The structure of the paper is as follows. We start with a review ofthe literature dealing with vector field topology detection and higherorder methods designed to improve the accuracy of the traditionallinear schemes. -scalar interpolation is then discussed wherewe use Nielson’s Minimum Norm Network to obtain derivative in-formation. We present two interpolation schemes, namely Powell-Sabin method and Nielson method that achieve a -continuityover the triangulation. In section 4, we repeat some basic defini-tions of vector field topology. Implementation apects are discussedin section 5. Finally, results are shown in the last part, which con-sists of a comparison of the topological skeletons obtained withboth -interpolants and the classical linear interpolation.

2 Related WorkTopology-based methods were introduced in vector field visualiza-tion by Helman and Hesselink about ten years ago (see [Hel89],[Hel91]). Their basic principle stems from critical point theory:One focuses on few features of the field, namely its critical points(where the field is zero) and the streamlines connecting them (theso-called separatrices) to get a domain decomposition into subre-gions that are all topologically equivalent to a uniform flow. Hel-man and Hesselink restricted their study to a first order approxi-mation that is, by only considering the jacobian matrix at criticalpoints to infer the local aspect of the field around them. This workgave rise to several extensions: Globus et al. ([Glo91]) developeda visualization environment called FAST in which they extract andvisualize topology of 3D vector fields; Bajaj et al. ([Baj98]) ap-plied such a topology-based method to scalar fields visualization;Nielson et al. (see [Nie97]) applied several explicit methods to thecomputation of tangential curves and topological graphs in the caseof 2D vector fields, linearly interpolated over a triangulation.

Most methods assume that the initial scattered vector data havebeen reconstructed into a continuous field by a piecewise linear in-terpolation over a beforehand computed triangulation of the givensample points. It explains the first order restriction of formertopology-based methods. Nevertheless, the linear interpolation of avector field can yield a large number of critical points. In particular,two neighboring triangles may contain critical points of differentkinds (namely of indices +1 and -1, see section 4). Such effectsare not desirable for they artificially increase the complexity of thetopology. To address this problem, Scheuermann and Hagen (see[Sch98a]) proposed a data dependent triangulation based upon thefact that if two neighboring triangles both contain a zero, the twonew triangles obtained by swaping their common edge do not. Oneachieves in that way a significant reduction of the number of critical

points which clarifies the resulting depiction of the field.Futhermore, the linear interpolant which is clearly unable to con-

vey higher-order singularities, introduces topological artefacts suchas splitting into several simple critical points (lying in different tri-angles) of higher order zeros. To deal with this deficiency, Scheuer-mann et al. (see [Sch98b]) introduced higher order polynomials toprocess the area located around such critical points: starting with alinear interpolation over a triangulation of the points, they next lookfor neighboring triangles containing several zeros and then computeinside them a polynomial approximation of the data. The choice ofthese polynomials is motivated by Clifford analysis, mathematicalbackground of their study. Problems remain when connecting thelinear interpolated triangles with the “higher order” cells for, in thelatter, the data are not interpolated.

There has been also some work using higher order derivatives:last year, Roth and Peikert (see [Rot98]) showed the use of higherorder derivatives for finding bent vortices.

3 C1-Interpolation-interpolation over triangles is an issue that has been widely

studied for about 30 years. As a consequence, there are many ex-isting interpolants in this field. Nevertheless, in our case, we areinterested in the topology (see 4 ) extraction of the resulting in-terpolated field. That is, we have to concentrate on schemes thatare computation-efficient as well as able to result in a meaningfultopology. These considerations led us to restrict or implementationto only two methods: Nielson’s -interpolant ([Nie83]) and thePowell-Sabin scheme ([Pow77]).As a preliminary step, both methods require to be provided withderivatives information at each vertex of the scattered data. Aswe said in introduction, in a first step we have computed an (op-timal) Delaunay triangulation and are thus in a position to treat thedata globally for this goal. The derivatives are then estimated usingNielson’s Minimum Norm Network.

3.1 Derivatives Computation:Nielson’s Minimum Norm Network

Let us introduce some convenient notations: We are given a setof points . denotes the triangle with vertices

, represents the edge linking to and is a listof the indices representing the edges of the triangulation. The curvenetwork is thus defined over . We also define thefollowing directional derivative:The derivative along an edge is given by

where is the length of .Now we consider the problem of finding an interpolating curve

network which minimizes, for is the restric-tion to of some function defined on , union of all triangles :

where represents the element of arc length on the curve con-sisting of the line segment . We have then the following result:Let be the unique piecewise cubic network, with theproperties that , and

where is the edge of the triangulation with theendpoint , and

Then, among all functions , , ,the function uniquely minimizes .

Solving this linear system in and , ,one is able to build a cubic polynomial curve on each edge byHermite interpolation.

3.2 InterpolationOnce the derivatives have been estimated at each vertex of the scat-tered data, an interpolation must be processed over each trianglewhich ensures a continuity throuh the edges of the triangula-tion. We start with a brief description of the Powell-Sabin methodwhich does not fulfil the requirements of the Minimum Norm Net-work (for its restriction on the edges is not a cubic polynomial) butenables an analytic search of its roots (see 4).

3.2.1 Powell-Sabin Interpolant

This method is based on the following remark: a biquadratric poly-nomial is unable to fit both values and derivatives at each edge ofa triangle because it offers only 6 degrees of freedom and there are9 interpolation conditions to fulfil. So we need to increase the de-grees of freedom. This may be achieved thanks to a subdivision ofeach triangle into 6 subtriangles (see Figure 1). Starting with a bi-

A

B

CP

Q

R 1

2

43

5

0

O

Figure 1: Division of ABC into 6 triangles

quadratic polynomial defined over triangle OAQ, say , onethen adds a correction term each time one crosses an intern edge,moving in a clockwise direction about O. The only quadratic solu-tions for this correction term, that ensure the required continuitythrough the edge have the form:

where is the cartesian equation of the i-th crossededge and is the parameter to adjust.

Ensuring the interpolation conditions for values and derivatives andforcing

one gets a non singular linear system with 12 (not independent)variables. By solving it, one obtains the desired piecewise bi-quadratic interpolant. (see [Pow77]).

3.2.2 Nielson’s Blending MethodThe second -continuous method used is Nielson’s Side-Vertex blending method (see [Nie83]). This scheme profits morefrom the Minimum Norm Network we introduced previously for itrespects the cubic curves built on the edges on the triangulation.However, since it consists of a rational function, its zeros may notbe found analytically (see 5.2.1).So to extend the scalar values defined on the edges to the wholedomain, Nielson proposes the following formula:For any point with barycentric coordinates in atriangle with vertices , one sets:

=

where

,,

, ,

and

This defines a 9-parameter, interpolant.

4 Vector Field TopologyAs said previously, vector field topology consists of the associationof critical points with some particular streamlines. In this work, weadopted the concept of topological skeleton proposed by Helmanand Hesselink (see [Hel91]).Let us recall that we consider the eigenvalues of the jacobian matrix(restricting our analysis to a linear approximation):

Depending on their sign and on their imaginary part, one gets 6possible configurations for the vector field around a singular point(see Figure 2). To describe the qualitative nature of a critical point,one can also use its index:

Let z be an isolated zero of the vector field .Then there is a neighborhood of containing only one criticalpoint. Let and be a closed disc aroundz of radius . Let be the boundary curve of

. We define the index of the critical point of the vector fieldas:

ind d

where is the angle cordinate of the vector field, namely

d d arctan d d

The index measures the number of rotations of the flow around acritical point.

Notice that for the special case of singularities of first order, thepossible values of the index are +1 and -1.

Attracting Node R1,R2<0 I1,I2=0

R1=R2=0I1=-I2<>0

CenterRepelling Node R1,R2>0 I1,I2=0

Saddle PointR1<0,R2>0I1=I2=0

Attracting FocusR1,R2<0

I1=-I2<>0

Repelling focusR1,R2>0

I1=-I2<>0

Figure 2: possible configurations of 1st-order singular pointsdenote the real parts of the eigenvalues and their

imaginary parts

5 Locating Critical PointsThe topology extraction of a vector field starts with the location ofits singular points. In this section we explain how we have detectedthem for both interpolants.

5.1 Powell-Sabin CaseIn the presentation of this method, we underlined the fact that it ispossible to determine algebraically the zeros of a biquadratic poly-nomial. The method is as follows.Let

be two quadratic polynomials which common roots have to befound. We may consider each bivariate polynomial as a polyno-mial in the variable (i.e. becomes a parameter):

We next introduce the so-called resultant of the system, defined by:

with the property that if and only if andhave a root in common. Now is a 4th-order polyno-mial in the variable which roots may be found thanks to classicalmethods. The found values must now be replaced in either theequation of or to get a quadratic polynomial whichroots are the zeros of the system.In our special case, we obviously have to check if the roots lie in-side the subtriangle where the considered biquadratic polynomialdescribes the vector field.

5.2 Nielson Case

To compute the position of the zeros for Nielson bi-rational inter-polant, one has to solve a system of two equations of fifth order.Since this is a very difficult algebraic task, we use numerical algo-rithms. Unfortunately, the existing algorithms need to be providedwith a “good” initial guess to start their search and we can not infereasily, a priori, an approximate location of the singular points inthe interpolated field. These remarks led us to adopt the followingheuristic: as a first step, we find out which triangles may potentiallycontain one or more zeros (actually, one could find up to 25 zerosfor such a polynomial system, even if this is practically very un-likely to occur); then we divide each so-called “candidate triangle”in 25 subtriangles, in which we process the same analysis; in thelast step we use the barycenter of each candidate subtriangle as firstguess for a numerical search, assuming that only one zero, at themost, is in a subtriangle. Let us detail these topics.

5.2.1 Finding Candidate Triangles

The aim of this procedure is to avoid numerical searchs in vain. Tokeep efficiency in our processing, we have to take away the trianglesthat can not contain any critical point. But to be of any practical use,this dichotomy has to be fast, so we focus on the control polygonsof the cubic polynomial defined along the edges of each triangle.The reason is that when we build Nielson’s interpolant over eachtriangle, we compute a blending of the splines on the edges so thata kind of energy criterion is minimized (see 3.2.2). Consequently, ifno spline on the border crosses the X-Y plane, we assume that alsothe interpolant over the triangle does not which has been confirmedby our numerical tests. So we have to check for each dimension,if a spline has a root. To speed up that process, we approximatethe behaviour of the spline by its control polygon, easily defined byboth value and derivative of the field at both vertices of the edge.Five1 generic configurations may occur (see Figure 3), from whichfour may lead to a zero (namely, in case 1, one has no zero, whereasin cases 2 and 3, there is exactly one zero, and in cases 4 and 5 onehas either 2 or no zeros). If we get such “zero”-configurations forboth dimensions then the triangle is marked as “candidate” and willbe processed further.

This kind of sign test is similar to the scheme proposed by Asi-mov et al. to find candidate cells in the case of a bilinear interpolant([Tut92]).

1a sixth configuration is theoretically possible which has 3 roots but thissituation does not occur in our case for the splines on the edges minimizethe pseudonorm introduced in 3.1

0 10

Case 4

0 10

Case 3100

0 10

0 10

Case 1

Case 2

Case 5

Figure 3: generic configurations of the Bezier control polygonfor a cubic polynomial

5.2.2 Processing of Subtriangles

As we said, before each candidate triangle is divided into 25 sub-triangles, to avoid finding several zeros in the same cell. This ismotivated by the fact that our birational interpolant may have up to25 zeros on the one hand and that 2 zeros should not be too close to-gether on the other hand, for this would mean an oscillation of theinterpolant, quite incompatible with its pseudo-energy minimiza-tion property. Then we compute the value of the index (see 4) ofeach subtriangle: a value +1 or -1 shows the presence of a criticalpoint (see Figure 4). Notice that even if higher order singularities- e.g. with index +2, -3, ... - may theoretically occur, they do notoccur here in practical cases.

Saddle:index -1

no singularity:index 0

Figure 4: index of subtriangles

Remark that the index method was not used for the “big” trian-gles because one may get several critical points in the same triangle,which can lead to a 0 index computed on its border, while it actu-ally contains singularities (for example, the problem occurs when asaddle and an attracting focus lie in the same triangle: the sum oftheir indices is and one misses two critical points as inFigure 5).

Saddle:index -1

Attracting Node:index +1

Figure 5: triangle with index 0 containing a saddle and an at-tracting focus

5.2.3 Numerical SearchThe former steps intended to provide a “good” first guess for a nu-merical search. We have eliminated all the triangles that do notcontain a critical point and determined (small) subtriangles that ac-tually contain a single zero. We can now take the barycenter ofeach selected subtriangle as first guess. For the numerical search,the Newton-Raphson algorithm is applied, which works satisfacto-rily for our needs.

6 Results6.1 Test DatasetsThe test of our interpolation schemes requires a precise descriptionof the topology of the underlying vector field. A suitable solutionis the use of vector fields given by a single formula with knowntopology. Furthermore, to prove the accuracy of our algorithm, wemust be able to design topological aspects of the field, like closecritical points, higher order critical points,...

The only vector fields that are usually known topologically arelinear fields or some special cases with strong restrictions. In aprevious paper, we proved a theorem that enables the design ofpolynomial vector fields with higher order singularities. We bringback here the main results (see [Sch97]).

Let be the canonical basis of and let

(where and ) be a linear vector field. Forit has a unique zero at . Forhas one saddle with index -1. For it has one

critical point with index 1. The special types in this case can be gotfrom the following list:

1. circle at .

2. node at .

3. spiral at .

4. focus at .

In cases 2)-4) one has a sink for and a source for. For one gets a whole line of zeros.

For our needs, we use the following theorem:

Let be the vector field

with

and let be the unique zero of . Then haszeros at , , and the index of at is the sum ofthe indices of at .

(That is, we only make use of linear factors).

Remember that a critical point with index -1 is a saddle point,whereas a critical point with index +1 may be a circle, a node, aspiral or a focus. Practically, it means that when we design ourvector fields we are able to locate the saddle points and the criticalpoints of index +1 (the precise nature of which is unknown) as wellas to define critical points of higher order by givinga multiplicity higher than 1 in the expression of .

6.2 ExamplesThe presentation of our results is based upon the comparison of the

-methods to a piecewise linear interpolation of the same data.For each case, the exact topology is used as reference.

6.2.1 First Example

In the first example, we use several simple critical points and oneof higher order. The definition of this field is:Let

(An overview of the topology is proposed in Figure 6).

Starting with a sample of 500 vectors, one gets the topologicalskeleton of Figure 7, in the linear case. The resulting topology ishere erroneous: singularities are missed which entails the deforma-tion or disappearence of separatrices. Globally, this depiction of thefield should be considered as totally unsatisfying.Nielson -interpolant produces the result shown in Figure 8.

In this case, the global aspect of the topology has been respected.The only topology deformation occurs at the expected location ofthe higher order singularity: it has been split up in attracting andrepelling foci.With the same points sample one gets the topology of Figure 9,when applying Powell-Sabin’s method. No significant differenceappears here, compared to Nielson’s method.

By doubling the number of sample points, one gets for all theinterpolants a globally satisfying depiction of the topological skele-ton. Nevertheless, the area locating around the higher order sin-gularity remains problematic in the linear case as shown in theenlargement proposed in Figure 10. That is, although one could

expect an improvement of the topology approximation with morepoints, as far as the higher order singularity is concerned the re-sults are worse: the whole aspect of the field in this area has beendeformed and the presence of an higher order singularity is impos-sible to guess.Nielson’s method offers in this case the same kind of result as for500 points. However the two foci have become closer which rep-resents an improvement of the higher order singularity approxima-tion.Otherwise, in this case, Powell-Sabin interpolant confuses thetopology depiction by introducing two additional singularities thathave no meaningful impact on the global aspect of the topologicalskeleton(see Figure 11).

6.2.2 Second Example

The second example is defined by the following expression:Let

that is, one has to face two topological difficulties(see Figure 12):two very close saddle points on the one hand and a singularity ofhigher order (namely of index + 2) on the other hand. Starting witha sample of 500 vectors, one gets for the linear case the pictureproposed in Figure 13. The most problematic aspect of this fieldapproximation takes place at the expected second order singularity:it is replaced by 4 first order singularities (namely one saddle and 3foci) and consequently the original topology is totally lost locally.Furthemore the two close saddle points have been taken away fromanother.Considering Nielson’s scheme, the results is as shown in Figure 14.

Because the two saddles have been maintained very close fromanother and the two foci resulting from the higher order singularitysplit lie in the same triangle, this topology approximation should beregarded as very satisfying and does not require more data points.The same remark applies to Powell-Sabin interpolant as one can seein Figure 15. If one tries to improve the topology depiction in thelinear case by increasing the number of sample points one has toface on the contrary a deterioration of the interpolant performanceas already observed with the former example. Figure 16 demon-strates this effect (note that the coarse aspect of the streamlines isdue to the enlargement for we have to focus on an area that is about4000 smaller as the original one). Some artefacts also occur in thearea located around the saddles as shown in Figure 17.

7 ConclusionA piecewise linear interpolation of scattered vector data is not an ac-curate way of reproducing the inner structure of the original vectorfield in difficult cases. Furthermore, when the topology is unknowna priori, it introduces additional singularities that confuse the resultsand inconvenience the interpretation.

By using two interpolation methods, we have arrived at aclearly improved approximation of the original field with the sameinitial information. Close singularities as well as critical points ofhigher order are that way better conveyed by both schemes that al-ways keep close to the expected topological skeleton.

Nevertheless we should underline that Nielson’s interpolant,thanks to its respect of the precomputed minimum norm network,produces thoughtlessly better results. So in our opinion, it comesout to be the right choice for -interpolation for the purpose ofvector field topology visualization.

AcknowledgementsThis research was partly made possible by financial support by theConseil General de l’Isere (France). The second author received a“bourse de projet a l’etranger” during his stay at the university ofKaiserslautern from March 98 to June 98.

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Figure 6: ex.1: topology of the original vector field

Figure 7: ex.1: linear interpolated vector field (500 vectors)

Figure 8: ex.1: Nielson C1-interpolated vector field (500 vectors)

Figure 9: ex.1: Powell-Sabin C1-interpolated vector field (500vectors)

Figure 10: ex.1: linear interpolated vector field around thehigher order singularity (1000 vectors)

Figure 11: ex.1: Powell-Sabin C1-interpolated vector fieldaround the higher order singularity (1000 vectors)

Figure 12: ex.2: topology of the original vector field

Figure 13: ex.2: linear interpolated vector field (500 vectors)

Figure 14: ex.2: Nielson C1-interpolated vector field (500 vec-tors)

Figure 15: ex.2: Powell-Sabin C1-interpolated vector field (500vectors)

Figure 16: ex.2: linear interpolated vector field around thehigher order singularity (5000 vectors)

Figure 17: ex.2: linear interpolated vector field around the twosaddles (5000 vectors)