Cavoretto, Roberto

De Rossi, Alessandra

Dell&apos

Di Tommaso, F.

English

DE ROSSI, Alessandra

Institutional Research Information System University of Turin

Proceedings of the 17th International Conferenceon Computational and Mathematical Methodsin Science and Engineering, CMMSE 20174–8 July, 2017.An efficient technique for the interpolation on compacttriangulationsRoberto Cavoretto1, Alessandra De Rossi1, Francesco Dell’Accio2 andFilomena Di Tommaso21 Department of Mathematics “Giuseppe Peano”, University of Torino2 Department of Mathematics and Computer Science, University of Calabriaemails: roberto.cavoretto@unito.it, alessandra.derossi@unito.it,francesco.dellaccio@unical.it, ditommaso@mat.unical.itAbstractIn this paper we present an efficient scheme for the computation of triangular Shep-ard method. More precisely, it is well known that the triangular Shepard methodreaches an approximation order better than the Shepard one [4], but it needs to identifyuseful general triangulation of the node set. Here we propose a searching techniqueused to detect and select the nearest neighbor points in the interpolation scheme [2, 3].It consists in determining the closest points belonging to the different neighborhoodsand consequently applies to the triangulation-based approach. Numerical results showefficiency of the interpolation procedure.Key words: scattered data interpolation, triangular Shepard method, fast computa-tion, approximation algorithmsMSC 2000: 65D05, 65D15, 41A051 IntroductionScattered data consists of a set of points Xn = {x1, . . . , xn} and corresponding functionalvalues f1, . . . , fn, where the points have no structure or order between their relative loca-tions. Among the various approaches to interpolating scattered data, the Shepard method[8] is one of the earliest techniques. It defines an interpolating function as a convex com-bination of the functional values, that is a linear combination of them with non negativecoefficients (or weight functions or basis functions) which are inverse distances to the scat-tered points and form a partition of unity. The main drawback of the Shepard method isc©CMMSE ISBN: 978-84-617-8694-7Page 516 of  2288An efficient technique for the interpolationits low polynomial precision (only constants) that badly affects the reconstructed surface.Several variants of the Shepard method have been considered to overcome this drawback.Among them, the triangular Shepard method [7] is a convex combination of local linear in-terpolants with triangle-based weight functions, which are the product of inverse distancesfrom the vertices of triangles and form a partition of unity. The main feature of the tri-angular Shepard method is its linear precision without using derivative data, although, forachieving a good accuracy of approximation, it needs to identify useful general triangulationof the node set.An efficient organization of the scattered data, when local interpolation is used, turnsout to be crucial. To this aim, in literature, techniques known as kd-trees, which are notspecifically implemented for a specific interpolation scheme, have already been designed,[1, 6]. In this paper, we propose the use of a versatile partitioning structure, called theblock-based partitioning structure, given in [3], which is suitably adapted to triangularShepard interpolation.The paper is organized as follows. In Section 2 we consider interpolation using thetriangular Shepard method and the selection of the compact triangulation. Section 3 isdevoted to present the searching technique used to detect and select the nearest neighborpoints in our interpolation scheme. Finally, Section 4 shows some numerical results.2 Interpolation on compact triangulations2.1 Triangular Shepard methodIn 1983 Little [7] introduced a variant of the Shepard method, called triangular Shepard,which is a Shepard-like convex combination of the linear interpolants of a set of triangles.More precisely, if we denote by Xn = {x1, x2, . . . , xn} a set of nodes of R2 with associatedfunction data f1, . . . , fn and by Tm = {t1, t2, . . . , tm} a set of triangles which vertices arepoints of Xn, the triangular Shepard operator is defined byKµ[f ](x) =m∑j=1Bµ,j(x)Lj(x), µ > 0, (1)where Lj(x) is the linear interpolant on the vertices of tj , j = 1, . . . ,m, and the weightfunctions Bµ,j(x) are defined byBµ,j(x) =3∏ℓ=11‖x− xjℓ‖µm∑k=13∏ℓ=11‖x− xkℓ‖µ, j = 1, . . . ,m. (2)c©CMMSE ISBN: 978-84-617-8694-7Page 517 of  2288R. Cavoretto, A. De Rossi, F. Dell’Accio, F. Di TommasoAs noticed by Little himself, the triangular Shepard operator (1) exceeds the Shepardmethod both in polynomial precision and esthetic behaviour. In fact, it has linear precisionwhile Shepard method achieves only constant precision. The better polynomial precision ofthe triangular Shepard method reflects on a higher order of approximation. As shown in [4]the triangular Shepard method reaches quadratic approximation order while the Shepardmethod achieves at most linear approximation order [5]. We remark that the definition ofthe triangular Shepard operator (1) requires an appropriate list of triangles to be identified.However, these triangles can realize only a general triangulation of the node set Xn, that isa triangulation in which some triangles may overlap or be disjoint.2.2 Selection of the compact triangulationIn order to identify useful general triangulation of the node set Xn we take into accounttheoretical results achieved in [4], which link the bound for the remainder term of the linearinterpolant Lj(x) with the bound for the remainder of the triangular Shepard operator(1). In particular, all triangles should have a rather regular form (as near as possible toequilateral triangles) and two quantities, denoted by h′ and h′′, play a key role in the choiceof these triangles: the first one is the fill distance in the maximum norm and controlsthe uniformity of the triangle distribution, the second one excludes the presence of largetriangles.For each node xi of the node set Xn we choose, among the 15 triangles with a vertex inxi and other 2 vertices among its 6 nearest neighbors in Xn, the one which locally reducesthe bound2‖x− xj1‖2 + 4hjCj‖x− xj1‖ (3)for the error of the local linear interpolant, where xj1 denotes the first vertex of tj , hjdenotes the maximum edge length of tj and Cj is a constant which depends only on theshape of tj . We omit duplicate triangles and we get a triangulation Tm of the nodes withm ≤ n triangles, in which some of the triangles may overlap or be disjoint.3 Localizing searching techniqueIn this section we present the searching technique used to detect and select the nearestneighbor points in our interpolation scheme [2, 3]. Though this procedure can be appliedon a generic domain R ⊆ R2 and in higher dimensions, for our purpose we here pursue thisdescription focusing on the unit square, i.e. R = [0, 1]× [0, 1].Firstly, we define a circular neighborhood of radiusδ =2d, (4)c©CMMSE ISBN: 978-84-617-8694-7Page 518 of  2288An efficient technique for the interpolationwithd =⌊√n2⌋, (5)where each neighborhood is centred at a data point belonging to R. To localize points inour method, we set n/d2 = 4. Note that the larger (smaller) the value of d is, the more(less) localizing the scheme is.In order to determine the closest points belonging to the different neighborhoods andconsequently apply our triangulation-based approach, we propose a new structure thatpartitions the domain in blocks of square shape. Such technique results in an effectivesearching procedure which is quite efficient from a computational viewpoint. For this scopewe partition the domain R with b2 square blocks, b being the number of blocks along oneside of the unit square defined asb =⌈1δ⌉. (6)It follows that the side of each square block turns out to be equal to the neighborhoodradius. Hence, such a choice (seemingly trivial) allows us to examine in the searching processonly a small number of blocks, significantly reducing the computational effort compared tostandard or more advanced searching procedures such kd-trees [1, 9]. In fact, our searchingroutine is performed in a constant time, independently from the initial number of nodesconsidered.In our partitioning technique square blocks are numbered from 1 to b2, following thelexicographic order “bottom to top, left to right”. By a repeated use of a quicksort routinethe set Xn is thus partitioned by the block-based partitioning structure into b2 subsets Xnk ,k = 1, . . . , b2, where Xnk are the points belonging to nine blocks: the k-th block and itseight neighboring blocks, see Figure 1. In such framework, we are able to get an optimalprocedure to find the interpolation nodes closest to each of points.4 Numerical resultsIn our numerical experiments we report results obtained by using the triangular Shepardinterpolant (1). All the experiments are carried out by means of a grid of ne = 21 × 21evaluation points on R = [0, 1]× [0, 1] by using several sets of Halton points [6]. The usedtest function is the Franke’s one,f(x, y) = 0.75 exp(−(9x− 2)2 + (9y − 2)24)+ 0.50 exp(−(9x− 7)2 + (9y − 3)24)+0.75 exp(−(9x+ 1)249− (9y + 1)4)− 0.20 exp (−(9x− 4)2 − (9y − 7)2) .c©CMMSE ISBN: 978-84-617-8694-7Page 519 of  2288R. Cavoretto, A. De Rossi, F. Dell’Accio, F. Di Tommaso0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.91xy  1  b  2bb+1 b2b2−b+1Figure 1: Example of 2D block-based partitioning structure with a data point set: in redthe k-th block and its eight neighboring blocks, in blue a set of scattered points containedin the unit square R.n RMSE told tnew10000 2.87e-4 27.3056 7.078420000 1.53e-4 242.5568 18.196240000 7.51e-5 – 49.393880000 3.59e-5 – 245.2871Table 1: CPU times (in seconds) and RMSEs on a grid of ne = 21 × 21 evaluation pointsfor the Franke’s test function using different sets of n Halton points.In Table 1 we show the CPU times computed in seconds and the root mean square error(RMSE)RMSE =√∑nei=1 ei2newithei = |f(Pi)−K2[f ](Pi)|, (7)Pi being an evaluation point in R. In particular, we compare performance of the procedurewhich computes all the distances between the scattered points (told) with the one using theblock-based searching technique (tnew).c©CMMSE ISBN: 978-84-617-8694-7Page 520 of  2288An efficient technique for the interpolation00.20.40.60.8110.500.20.40.60.81.21000.20.40.60.8110.5×10-4406820Figure 2: Approximation of Franke’s test function (left) and the absolute errors (right) ona grid of ne = 21× 21 evaluation points using n = 40000 Halton points.Figure 2 shows the reconstruction of Franke’s test function and the absolute values eion n = 40000 Halton points.AcknowledgementsThis research has been accomplished within the RITA “Research ITalian network on Ap-proximation”. This work was partially supported by the projects “Metodi e modelli nu-merici per le scienze applicate” and “Approssimazione multivariata e algoritmi efficienti conapplicazioni a problemi algebrici, differenziali e integrali” of the Department of Mathemat-ics of the University of Torino. Moreover, this research was supported by GNCS–INdAM2016-2017 project and by a research fellow of the Centro Universitario Cattolico.References[1] S. Arya, D.M. Mount, N.S. Netanyahu, R. Silverman, A.Y. Wu, An optimalalgorithm for approximate nearest neighbor searching in fixed dimensions, J. ACM 45(1998) 891–923.[2] R. Cavoretto, A. De Rossi, A meshless interpolation algorithm using a cell-basedsearching procedure, Comput. Math. Appl. 67 (2014) 1024–1038.[3] R. Cavoretto, A. De Rossi, E. Perracchione, Efficient computation of partitionof unity interpolants through a block-based searching technique, Comput. Math. Appl.71 (2016) 2568–2584.c©CMMSE ISBN: 978-84-617-8694-7Page 521 of  2288R. Cavoretto, A. De Rossi, F. Dell’Accio, F. Di Tommaso[4] F. Dell’Accio, F. Di Tommaso, K. Hormann, On the approximation order of thetriangular Shepard interpolation, IMA J. Numer. Anal. 36 (2016) 359–379.[5] R. Farwig, Rate of convergence of Shepard’s global interpolation formula, Math. Comp46 (1986) 577–590.[6] G.E. Fasshauer, Meshfree approximation methods with MATLAB, World Scientific,Singapore, 2007.[7] F. Little, Convex combination surfaces, in R.E. Barnhill, W. Boehm (eds.),Surfaces in Computer Aided Geometric Design, North-Holland, Amsterdam, 1983.[8] D. Shepard, A two-dimensional interpolation function for irregularly-spaced data., inProceedings of the 23rd ACM National Conference, New York: ACM Press, pp. 517524.[9] H. Wendland, Scattered data approximation, Cambridge University Press, Cam-bridge, 2005.c©CMMSE ISBN: 978-84-617-8694-7Page 522 of  2288

An efficient technique for the interpolation on compact triangulations

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An efficient technique for the interpolation on compact triangulations

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