Comparative study regarding the methods of interpolation

Transcription

1 Recent Avances in Geoesy an Geomatics Engineering Comparative stuy regaring the methos of interpolation PAUL DANIEL DUMITRU, MARIN PLOPEANU, DRAGOS BADEA Department of Geoesy an Photogrammetry Faculty of Geoesy Technical University for Civil Engineering Bucharest -4, Bv. Lacul Tei, Sector, Bucharest ROMANIA Abstract: - An important problem between specialists is the interpolation of the surfaces etermine by classics or moern techniques. Choosing the proper technique is practically a rigorous stuy regaring all the interpolation methos an using that metho accoring to the nees of the project. The etermination of the Digital terrain moel, the igital surface moel from GNSS, LiDAR or classic measurements, the etermination of the surface of the quasi geoi surface with the geometric metho from GNSS an leveling observations are one of the projects that use the interpolation techniques. In this paper are shortly presente the main methos for interpolating the surfaces as: polynomial interpolation, Delaunay triangulation, nearest neighbor, natural neighbor, Kriging, inverse istance weighting (IDW) an spline functions. The stuy ens with a comparative stuy regaring the characteristics of each metho presente. Key-Wors: -interpolation, surface, Delaunay triangulation, Kriging, Inverse istance weighting (IDW) Introuction Surface moeling is the process of a natural or artificial surface etermination by using one or more mathematical equations. An universal algorithm for surface moeling is not available for all applications, each metho of generating surfaces have a number of avantages an isavantages that must be taken into account at its iscretion. Moeling the -imensional surface in space involves fining a function z = f (x, y) that represents the entire surface of the values z = f (x, y) associate with the point P (x, y) arrange irregularly. In aition, this function can preict the values z = f (x, y) an for other positions regularly arrange. Such a feature is known as interpolation function. There are two types of interpolation functions, exact an approximate. In a known point of a certain value, to etermine the exact interpolation of the same value is the problem propose. In fact, a metho is exact only when it is known beforehan the expression of the function z = f (x, y), if any. There are some accurate methos that can be use with a smoothing factor an in this case it can go from exact methos to approximate methos. Interpolating methos For interpolating scatter or regular istribute ata there are approximate or exactly methos. They can be use for multiple purposes an epens of the type of the works. Below are shortly presente the main interpolating methos an base on this analysis coul be chosen the proper one. Some of the methos coul be use for interpolating ata in a regular gri an other in irregular gri. They coul be use to generate surfaces for D analyses or for etermine other values for unknown points. Knowing the D position that has attache a value for the thir coorinate the surfaces coul be moel for any type of project. Some of the methos are implemente in powerful applications that can be use to generate surfaces as Digital Terrain Moel, Digital Terrain Surface, (quasi)geoi conversion surface, etc..:polynomial interpolation, Delaunay triangulation with linear or spline interpolation, nearest neighbor, natural neighbor, inverse istance weighte, Kriging metho, raial basis functions, polyharmonic functions, moifie Shepar, local polynomial, minimum curvature, etc.. ISBN:

2 Recent Avances in Geoesy an Geomatics Engineering.. Polynomial interpolation For the polynomial interpolation metho it is necessary to etermine a polynomial that has the property to go through some ata points by using ifferent methos. This metho is use to etermine the general tren of the values of a polynomial function z = f (x, y) for a certain area. Polynomials can vary for the egrees number, representing ifferent geometric surfaces: a plane, a bilinear surface (etc.) quarant area, a cubic surface, another appropriately efine area. In aition to the variables (x, y), the maximum power of the polynomial equation of these variables may represent other parameters. The general equation () of a spatial surface efine by a polynomial of orer n + m is: f(x,y) =a 00 + a 0 x + a 0 y + a xy + a 0 x + a 0 y + a 0 x + a x y+ a xy + a 0 y +. + a mn x m y n () where a mn - coefficients of the variables (x, y) m - the egree of x the function f (x, y) n - the egree of the variable y of the function f (x, y) The relation () is a polynomial of orer m + n which escribes the tenency of a surface etermine by a set of m + n + points... Multivariate interpolation The multivariate interpolation or spatial interpolation is represente by a function that consists of more than one variable. The interpolation function is known in the set-point given by coorinates (x i, y i, H i ), an the problem is to fin the values of interpolation functions for an arbitrary point (x, y, H). Multivariate interpolation is performe using several methos epening on the arrangement of points in the ata set. Thus, one can speak of interpolation in a regular gri (with a preefine spacing an not necessarily uniform) or the interpolation in an irregular network for a ranom arrangement of the ata set. Regular network is a spatial representation by rectangles. Each cell of the network can be represente by its inex (i, j) in the case of twoimensional spatial representation (Fig.) an the (i, j, k) in the case of three-imensional representations, each noe of the network of cells having the coorinates (i x, y j) in twoimensional space an (i x, y j, k z) in threeimensional space, x, y an z spacing representing the network or network step in any irection. Fig. Regular gri interpolation [0] An irregular network is a spatial representation by simple shapes like triangles or tetraheral, with an irregular shape shown in Fig.. Irregular gris may be use in the finite element analysis where the input ata are the areas that have an irregular shape. Compare to regular networks, irregular networks require connectivity list or rules on the representation of the ata set in irregular network. In aition for the representation by triangles or tetraheral generating another form of surface elements is quarilateral (4 knots) or hexaheron (8 noes). Fig. Irregular network ISBN:

3 Recent Avances in Geoesy an Geomatics Engineering On such surfaces form equation generate by the above forms or algorithms use to generate such networks nothing can be conclue, each interpolate or generate surface following its typical characteristics. In the case of interpolation in a regular gri are several methos known in the function use. Thus we have: bilinear interpolation Bicubic Interpolation Interpolation metho nearest neighbor. Interpolation in irregular networks can be achieve using the following methos: nearest neighbor metho natural neighbor metho Delaunay triangulation with linear interpolation metho the inverse istance weighte metho Kriging metho raial basis function metho polyharmonic spline functions metho... Bilinear interpolation Bilinear interpolation or interpolation of st orer is use for rectangular networks where noes know its coorinates, aiming to fin the value within the cell (Fig. ) nees to be etermine epening on the values of the altitues of noes by a bilinear interpolation. Then the shape of the equation use for the interpolation of H p value is (): H P = (m - x)(n - y)h + (m - x)yh + x(n - y)h + xyh 4 () X P X YP Y where x= şi y= () X X Y Y.4. Bicubic spline interpolation Bicubic interpolation is use to interpolate the regular gri points using a two-imensional space. Unlike surfaces obtaine by bilinear interpolation or nearest neighbor interpolate surface the metho obtain smoother surfaces. For this metho it is be consier a cell in the network of 6 noes an the interpolate point P (Fig.4): Fig.4 Bicubic spline interpolation [] Fig. Bilinear interpolation in regular gri For a point P belonging to the cell forme by the four points of the network an which has known planimetric coorinates x p an y p, the altitue H p The values of f = p (x, y) as well as the partial erivatives of the function f x, f y an f xy are known in the corners of the unit square (h =, up) efine by the coorinate points 6,7,,0 6 (0.0) 7 (.0) 0 (0.), an (,). Bicubic spline interpolation is performe using the relation (4): i j f ( x, y) = p( x, y) = a x y (4) i= 0 j= 0 ij ISBN:

4 Recent Avances in Geoesy an Geomatics Engineering.5. Delaunay triangulation with linear interpolation Triangulation with linear interpolation metho is a metho well known, being one of the first methos use before the evelopment of the intensive computing. It is base on iviing the omain D in the R n space into triangles. Fig. 5 Delaunay Triangulation Then each triangle efine, thru it corners, creates a plane surface from linear parts []. In analytic geometry, Delaunay triangulation for a set P of points in a plane is a triangulation characterize by none of the points from P is in the circle circumscribing of the triangle (Fig.5)..6. Nearest-neighbor interpolation metho Nearest neighbor metho was use for the first time by JG Skell an then by PJ Clark an FC Evans who introuce a statistical test to etermine the significance of nearest neighbor in orer to calculate the eviation from the general tren. This metho is a subclass of Delaunay triangulation methos type. Interpolation nearest neighbor metho (also known as the maximum proximity interpolation metho) is a simple variant of the metho of multivariate interpolation in one or more imensions []. Nearest neighbor algorithm selects the nearest point value an oes not take into account the values of other neighboring points, proucing a constant interpolation. Nearest neighbor metho is base on a comparison of the istribution of istances between a point an the nearest neighboring points of a set of ranomly istribute ata (5): ( ( ) ) / x i y ( x, y ) x y = ( x y )( x y ) = (5) = i i.7. Inverse istance weighting (IDW) interpolation metho The inverse istance weighte metho is a multivariate interpolation metho, ie a process of assigning values to unknown points by using values of ranomly istribute points. The metho of inverse istance weighte interpolation means that the result is influence by the nearby points an ignores unknown istant points []. Interpolation result is the weighte average of the values of the set of ranom points, the weight of each ranom point is reuce as the istance from the point interpolate ranom point increase. The best known metho of interpolation using the inverse istance weighte metho is the metho of Shepar, the simplest form of interpolation []. The form of an interpolation equation for Shepar metho is represente by the following relation (6): n F ( x, y ) = i= w i f i (6) where n is the number of ranom points in the ata set, f i the value function at points interpolate, an w i is the weight attache to each ranom point. Classical formula of the calculation of the weight is given by (7): w p hi i = n p h j j= (7) where p is a positive real number chosen ranomly an calle the power parameter (typically p = ) an h i is the istance from point to point interpolate from the original ata set. This can be etermine by the relationship (8): h i = ( x xi ) + ( y yi ) (8) The pair (x, y) represents the coorinates of the interpolate an (x i, y i ) coorinates of the original ata set. The calculation of the weight varies from unit value ranomly chosen point to a value close to zero as the istance between ranom points increases. The calculation of the weight is normalize so that their sum is close to the value of the unit. The effect of the weighting function is that the surface interpolates each point of the ata set being influence by the istance between points an points from the interpolate ata set. Relation (7) is the general form for calculating the weights in ISBN:

5 Recent Avances in Geoesy an Geomatics Engineering practice using the following relationship (9): w i = R Rh n R h j = Rhj j h i i (9) where h i is the istance from the interpolate point to point i of the set of ranom ata R is the istance from the interpolate point to the farthest point of the set of ata an the number of ranom points in the ata set. Equation (9) has been evelope to rive superior results compare to its classical form (Franke & Nielson, 980) [6]. Through the global metho, each of the ranom point in the ata set is searche for each point is from the interpolate for etermining point closest to the point of interpolation. This metho is quick sets of ata with a relatively small number of points istribute ranomly, but can be slow ata sets with a larger number of points. The local metho, ranom points are temporarily in the form of triangles arrange to form an irregular triangular gri before the interpolation begins. To calculate the closest points, etermine the triangle that contains the point of interpolating by Delaunay triangulation, aiming to fin a systematic way to the nearest points. Local metho is much faster than the global large sets of ranom points..8. Kriging interpolation metho Kriging is a metho name after a South African mining engineer name DG Krige, who evelope this metho in an attempt to preict more accurately the surface of mineral reserves [7][8]. In recent ecaes, Kriging metho has become a funamental tool in geostatistics. Kriging metho is a group of geostatic techniques to interpret the value of a ranom fiel such altitue etermine by geographical location. Kriging metho is base on calculating the variogram. Variogram analysis is to calculate the experimental variogram (Fig.6) of the available ata an to compare it with the variogram moel. Variogram moel is chosen from a set of mathematical functions that escribe the spatial relationship of the point ata set. They can be linear functions, exponential or spherical chosen in such a way that the graphical representation of curves to fit experimental variogram of a mathematical function. Fig.6 Experimental an moel variogram for Kriging metho [] Experimental variogram is calculate by averaging the ifferences in values of a function f (x, y) calculate on the basis of irections an istances between pairs of points (x, y). Variogram represents the egree of spatial epenence an is enote by γ (h). The expression use to represent the variogram is given by the following (0): γ ( h) = [ f ( x) f ( x+ h) ] n where: h - the istance between two points γ (h) - is semi-varoigram n - number of points in the ata set (0) Once the experimental variogram is calculate, the next step is to efine a moel of it. Variogram moel is represente by a simple mathematical function that moels the experimental variogram trens. This function can be linear, cubic, spherical, exponential, logarithmic or any other function that escribes how the tren nearest experimental variogram. As can be seen in Fig. 6, inicating small ifferences form variogram moel an experiment, proving that the variance function variogram moel is small[7][8]. Once the variogram moel is built it will be use to calculate weights equations for Kriging interpolation metho. Basic equation for Kriging metho, given that it is base on etermining the istances between points of the ata set is represente by (6). This equation is just as important as the equation use in the metho of inverse istance weighte interpolation except that in this case the weights are not associate with ISBN:

6 Recent Avances in Geoesy an Geomatics Engineering ranom functions which are reference functions to the creation of the variogram moel. To interpolate a point P which is surroune by points P, P an P must first etermine the weights w, w an w. They are etermine by solving the following system of equations: w S ) + w S( ) + w S( ) = S( ) ( p S( ) + ws( ) + ws( ) S( p S( ) ws( ) + ws( ) S( p w = w + = ) () where S ( ij ) is variogram moel measure at a istance equal to the istance between points i an j. For example, S ( p ) variogram moel is measure at a istance equal to the istance between the points P an P. Since it is necessary that the sum of weights is equal to the unit will enter the fourth equation: w + w + w =.0 () Because now there are four equations an three unknowns, it also introuces another variable to the final set of equations. The unknown is the Lagrange multiplier λ an aims to minimize or maximize the function of the variogram moel efine in the sense of introucing constraints in the moel. In this case the unknowns etermine will efine a function that will pass through the original ata set points. What the sum of weights of the unknowns is equal to unity, the equation following the introuction of the conition λ becomes: w + w + w + 0=.0 () This equation is ae to the relations (): w S ) + w S( ) + w S( ) + λ = S( ) ( p S( ) + ws( ) + ws( ) + λ S( p S( ) + ws( ) + ws( ) + λ S( p + w + w + 0=.0 w = w = w (4) The linear system of equations will be solve to etermine the three values of the weights w, w an w. The value of the interpolation function fp of a point P is calculate by the formula: F( x = w f (5) p, y p ) w f+ w f + Using the metho of least squares to solve the system of equations (6) for etermining the unknowns the error is minimize. There are two types of Kriging interpolation metho known as Kriging punctual an block Kriging. Both methos generate an interpolate surface form a rectangular network. Punctual Kriging metho estimates the values of points in the rectangular lattice. Block Kriging metho estimates the average block centere rectangular lattice an generates ) ) ) smooth surfaces. The blocks have the size an shape of a rectangular gri of cells. Since the block Kriging interpolation metho oes not estimate the value of a point, this is not a solution for the generation of interpolate surfaces containing the points on which the surface moel was generate. Even if the values that are intene to be interpolate are in a noe of a rectangular network, block Kriging metho estimates for that noe a value close to the original [7][8]. The Kriging metho involves the following techniques for interpolation approach: orinary Kriging metho simple Kriging metho universal Kriging metho.9. Natural neighbor interpolation metho The metho of natural neighbor interpolation is base on Voronoi mosaic. Voronoi mosaic may be efine as the partition of a level of n points in each convex polygon so that contains exactly one point an each point in a given polygon is closer to the center than to any other point. Using natural neighbor interpolation metho has many beneficial features. It can be use both for interpolation an for extrapolation an works well in conjunction with a set of ranom locations in ifferent cell types. The metho has been use for the first time by Sibson (98). A more etaile escription of natural neighbor interpolation metho using the multi-imensional space has been performe by Owen (99) [5]. The basic equation use in natural neighbor interpolation metho is ientical to that use in the interpolation using the inverse istance weighte metho. Conclusion After analyzing the known methos of multivariate interpolation they can be presente their main characteristics, the goal being to choose the best metho of interpolation of the available ata set. Delaunay triangulation with linear interpolation metho is a proven metho of surfaces interpolation base on iviing a space omain into triangles. Each triangle is efine by the three vertices forming a plane surface resulting from the combine interpolate flat triangles. The metho presents a fast interpolation algorithm but has the isavantage that the interpolation function is limite to the area boune by convex ranom set of ata points. The resulting surface is not smoothe an isolines are represente by line segments []. Division into ISBN:

7 Recent Avances in Geoesy an Geomatics Engineering triangles can be ambiguous given the simple example below by choosing ifferent triangulation of the same points it can get ifferent shapes (valley or hill) even if Delaunay criteria were met (Fig.7): Fig. 7 The ambiguity of Delaunay triangulation [] Natural neighbor can be unerstoo intuitively as all points ajacent to a given point P an forming a mosaic type Voronoi incluing point P. The area create using the natural neighbor interpolate surface is continuous at any point except the points that were the basis for interpolation. It presents a fast interpolation algorithm an the resulting surface is smoothe except the points in which it was mae. It has the isavantages that the interpolation function is limite to the area boune by convex ranom set of ata points an the resulte surface cannot be accepte in many fiels such as geology or hyrology. Inverse istance weighte metho was among the first computer implemente metho of interpolation, being easily programme. an the metho coul be implemente in all the applications. The isavantages are: the calculation metho is time consuming when the number of points in the ata set is large. It increases the number of istances that have to be calculate. It generates so-calle "bull's eyes aroun a point in the interpolate fiel. For this reason, the resulting function is not supporte for the vast majority of applications (Fig.8). Kriging interpolation metho, base on a statistical formulation of the best linear estimation, is the most use metho to solve the problem of interpolation an approximation of the surfaces. An important concept erive from this metho is the empirical or experimental evaluation using the variogram. Variogram is approximate using empirical or theoretical variogram moel. In orer to generate it the linear variogram moel, Gaussian or exponential moel coul be use. The interpolation algorithm is base on a statistical linear formulation for the best estimation, which means that there is no other better interpolation methos in statistical terms but equations weights are calculate for each noe of the network. If there is a large number of points than the number of equations to be solve will be large. For this reason the metho is use to solve problems in small areas. The metho prouces grooves (pits) or circular isolines into the interpolate moel. As a final conclusion the table below (Table ) presents the avantages an isavantage of all interpolation methos presente in the current paper. Table. Interpolation methos [9][0][][] Fig. 8 Bull s eyes effect for IDW metho [0] The resulting surface is obtaine by interpolation point that is locate on a circle whose raius has previously been etermine. The avantages are: simple evelopment an implementation of the algorithm easy programming Metho Inverse istance weighting (IDW) Natural neighbor Type of Avantages/Disavantages metho Exact Tens to generate the Smooth - effect of "bull's eye". when the It is a simple metho. smoothing Does not extrapolate. parameter is All values are known interpolate from points use to generate the moel. In this case is possible that it is not approximating fair the lanforms (valleys or hills) Exact Interpolation is performe using the calculation of weights for neighboring points ISBN:

8 Recent Avances in Geoesy an Geomatics Engineering Metho Nearestneighbor Kriging - Geostatistical moels (stochastic) Raial basis function Moifie Shepar Triangulation with linear interpolation Triangulation with bicubic spline interpolation Spline Functions Type of metho Exact Exact Avantages/Disavantages after the concept of Voronoi polygons. It is a goo way for large ata sets. A typical implementation is not extrapolate. Mainly use for filling atasets (networks with missing values) High flexibility base on variogram moelling. Can extrapolate an estimate errors. Computing spee is affecte by the number of points in the ata set an the size of the interpolate surface. Exact Similar to Kriging Smooth - metho an using the when the variogram moels. smoothing Flexible but not give parameter is information on the known statistical properties of items in the ataset. Exact Subclass metho using Smooth - an inverse istance when the weighte. smoothing It coul not generate the parameter is correct forms of relief. known Does extrapolation. Exact Exact Accurately as possible an smoothe It is a metho base on Delaunay triangulation. Requires a meium or large number of ata to generate acceptable results. It is a metho base on Delaunay triangulation. Use cubic spline functions instea of linear functions. Generates very smooth surfaces. It is available as a separate proceure an is incorporate in most interpolation methos Biharmonic bicubic spline functions or commonly use Metho Polynomial regression References: Type of metho Smooth Avantages/Disavantages Prouce an estimate surface that fits the points in the ataset. The most use are the linear (grae I) an the quarics (grae II). Approximation ege can be a problem if there are insufficient ata. [] Lee DT, Schachter BJ. Two Algorithms for Constructing a Delaunay Triangulation, International Journal of Computer an Information Sciences [J].980,9():9. [] Franke, R.: Scattere Data Interpolation: Test of Some Methos, Mathematics of Computations, 98, (57):8. [] Watson, D. F., Philip, G. M: A refinement of inverse istance weighte interpolation. Geo- Processing : 5-7, 985. [4] Ch. Yang, S. Kao, F. Lee, P. Hung, Twelve Different Interpolation methos: A Case Stuy of Surfer 8.0, Geo-Imagery Briging Continents, XXth ISPRS Congress, - July 004 Istanbul, 006 [5] Steven J. Owen, An Implementation of Natural Neighbor Interpolation in Three Dimensions, Brigham Young University. Department of Engineering., 99 [6] Franke, R. an Nielson, G., Smooth interpolation of large sets of scattere ata. Int. J. Numer. Meth. Engng.,980, 5: oi: 0.00/nme.6050 [7] Royle, A. G., F. L. Clausen, & P. Freeriksen, 98, "Practical universal kriging an automatic contouring," Geo-Processing, Vol., No. 4, pp [8] Deutsch, C.V., & A.G. Journel, 99, GSLIB: Geostatistical Software Library an User's Guie, Oxfor University Press, New York, 40 p. [9] (visit at ) [0] (visit at ) [] (visit at ) [] Mgr. Ph.D. Miroslav Dressler, Art of Surface Interpolation, Kunstat, 009, [] (visit at ) ISBN: