Radial Basis Function Method For Modelling Leaf Surface from Real

Australian Journal of Basic and Applied Sciences, 11(13) October 2017, Pages: 103-111

AUSTRALIAN JOURNAL OF BASIC AND APPLIED SCIENCES ISSN:1991-8178 EISSN: 2309-8414 Journal home page: www.ajbasweb.com

Radial Basis Function Method For Modelling Leaf Surface from Real Leaf Data 1

Moaath Oqielat,2Osama Ogilat, 3Nizar Al-Oushoush and 4A. Sami Bataineh

1,3,4 2

Albalqa Applied University, Department of Mathematics, Faculty of Science,Box. 19117, Amman. Jordan. Al-Ahliyya Amman University, Department of Mathematics, Faculty of Arts &Science,Box.19328. Amman. Jordan

Address For Correspondence: Moaath Oqielat, Albalqa Applied University, Department of Mathematics, Faculty of Science, Box. 19117, Amman. Jordan. E-mail: [email protected] ARTICLE INFO Article history: Received 28 May 2017 Accepted 22 July 2017 Available online 20 October 2017

Keywords: Interpolation, Finite elements methods, virtual leaf, Radial basis function, Clough-Tocher method.

ABSTRACT The leaves participate in the plant development and are an integral component of any plant model. Therfore accurate leaf models are essentials in many applications, such as the development of a plant model, modelling droplet movement, spraying and spreading droplet on the leaf surface as well as in studying biological system such asphotosynthesis and a canopy light nature.A smooth surface is important to assist the droplet motion that govern by a differntial equation or partial differntial equation.The leaf size, shape and position are significant in many ways. For instant the precipitation amount of pesticide or nutrients can be measured if a leaf model is manageable. This can be achieve from a surface fit to a data set measured by device such as a laser scanner or a sonic digitiser.The work presented here will structure the foundation for a theoretical revision of water droplets paths on leaves. The primary study will suppose that the droplet experiences external force includes gravity and an internal force includes viscosity and surface tension. It will be vital to create a continuous surface fit and this is certain by interpolation the data value.The problems stated in this research include: firstly, the choice of points from the set of the data. The selectionof RBF and its parameter c. We selected the Hardy’s multiquadrics in combinationwith the Rippa’s algorithm to decide the width parameter. Secondly, the use of global and local multiquadricRBF interpolates. To validate our method we apply it to two sets of scattered data points. The first set is taken from Franke (Franke (1982)) while the second set is sampled by Loch (Loch (2004)) from an Anthurium and Frangipani leaf using a laser scanner. Numerical results confirm that the proposed technique is revealed to constructs an accuarte and realistic surface of the leaves.The approach described in this paper is valid to scattered data and has the possibility for application to the numerical solution of partial differential equations.

INTRODUCTION The leaves model are imperative in the plant development and are important in any plant model.Modelling of plant has been research over the last decadesby(Davydov and Zeilfelder (2004)), while leaves model considered recently by Loch(Oqielat (2017);Lisa et al. (2016); Kempthorne et al. (2015); Oqielat et al. (2011); Oqielat et al. (2009); Oqielat et al. (2007);Dorr et al. (2014); Loch, 2004, Turner et al. (2008); Belward et al. (2008); Oqielat (2017)). Loch used laser scanner tocollect leaf data points for ear Elephant’s, Anthurium, Flameand Frangipani leaves and then applied finite element method to construct leaf model.Our goal in this paper is to use scattered data interpolation method based on radial basis function (RBF) to reconstruct the leaf surface model.The scattered data points interpolation problemis given by: Given

 distinct scattered points ( xi , yi )T , and their function values f i , i = 1,2,, P, find a function

F : D  2   that interpolates these data satisfying Open Access Journal Published BY AENSI Publication © 2017 AENSI Publisher All rights reserved

This work is licensed under the Creative Commons Attribution International License (CC BY).http://creativecommons.org/licenses/by/4.0/

ToCite ThisArticle:Moaath Oqielat, Osama Ogilat, Nizar Al-Oushoush and A. Sami Bataineh., Radial Basis Function Method For Modelling Leaf Surface from Real Leaf Data. Aust. J. Basic & Appl. Sci., 11(13): 103-111, 2017

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Moaath Oqielatet al, 2017 Australian Journal of Basic and Applied Sciences, 11(13) October 2017, Pages: 103-111

F ( xi , yi ) = f i , i = 1,, P.

(1)

To evaluate the accuracy of the RBF method, we apply itto two sets of a real data points sampled from aFrangipani and Anthurium leaf surface using laser scanner. The research present in this paper is comprises of four main sections. In § 2, an overview of the radial basis function (RBF) is presented. In § 3, The RBF technique is evaluated using six test functions and data points chosen from Franke (Franke (1982)). The quality of the approximation of the methods is measured numerically using the maximum error and the root mean square error (RMS). In § 4, Frangipani and Anthurium leaf surface is

§ 5.

constructed using the RBF method. Finally, conclusion and future work is presented in Radial Basis Functions Method The approximation radial basis function F to the function N

F ( x) = ai ri ,

f (x) , is given by:

x  R2

(3)

i =1

wher ri = x  xi

with . denoting the Euclidean norm. The centres of the approximationRBF are

xi , i = 1,2,, N . If the coefficients of the RBF ai , i = 1,, N satisfies the system

a = F



with  ij =  x j  xi



i, j = 1, , N

(4)

and

F = ( f1 ,, f N )T . then F (x) is called interpolation function of f at x1 ,, xN . The advantage of using RBF methodcan be seen in proposing a smooth surface depictionto approximate the function values at points. This method has many application in fieldsfor instant,medical imaging (Carr et al. (1997)),software to drive laser scanners (Carr et al. (2003)), and the solution of parabolic, elliptic and hyperbolic partial differential equationsby Hardy (Hardy (1990)). Powell (Powell (1991)) review the theory of RBF. The computational costs in assessing the RBF to large sets of data points can become time consuming because a large compact matrix system of size N  N has to be solved in order to calculate the RBF coefficients

ai , i = 1,2,.., N in equation (4),this system can become ill-conditioned with very small in magnitude singular values(Franke (1982)). Frankecompared around 30 interpolation schemes and found thatthe most accurate schemes were based on fitting RBFsand the application of global methods should be limited to sets of up to 100-200 data points.Beatson (Beatson et al. (2001))used fast evaluation techniques to reduce the cost of computing the radial basis function. Well known examples of radial basis function methods include thin plate splines, Gaussian RBF andHardy’s multiquadric (Hardy (1990)) which is adopted in this paper. The multiquadric RBF is given by:

 x  xi  =

x  xi  c 2 .

(5)

The accuracy of RBF interpolat depends strongly on the parameter c where this parameter is specified by the user, see for example (Niceno (2003)). For some values of c the problem may become ill-conditioned. Franke

D where D is the diameter of the minimal circle n enclosing all data points. Hardy (Hardy (1990)) suggested a value of c = 0.815d where and d is the distance (Franke (1982))and Foley (Foley (1987))used c = 1.25

between a data point and its closest neighbour. Franke (Franke (1982)) studied the accuracy of the inverse multiquadric and the multiquadric interpolant and found that the choice of the parameter c has great impact on the accuracy of the RBF. Carlson(Carlson (1991))recurrent the computation of the RMS error with different choices of c and stated the optimal value of c that minimizes the RMS. Rippa (Rippa (1999))performed experimentson the influence that the parameter c has on theapproximation quality achieved using Gaussian, inverse multiquadric and multiquadricinterpolan. Rippaproved that the value of c has great impact on the approximation quality of these RBFs. Rippa considered two sets of data points and nine different test functions defined on the unit square. Nine test functions and two sets of data are considered by Rippa. He constructed the data vector F =  f1 , f 2 ,  , f N 

T

function over the set of data points so that

by computing each test

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Moaath Oqielatet al, 2017 Australian Journal of Basic and Applied Sciences, 11(13) October 2017, Pages: 103-111

S ( xi ) = fi , i = 1,2,, N.

(6)

An algorithm is proposed by Rippa for the choice of a good value for the parameter c that minimizes the RMS error between the RBF interpolant and the unknown function from which the data vector F was sampled. The value of c is selected by minimizing the cost function. Themnbrak and brent routines from Numerical Recipes(Press (1992))were used to do the minimization. The mnbrak routine is given some tolerance and two initial values c1 and c2 . It returns three numbers b1 , b2 and b3 that bracket the minimum. After bracketing the minimum, thethree numbers and the tolerance parameter are passed into thefunction brent that uses Brent's method to minimize thecost function. We refer to the minimum value of the cost function asthe "good value" of c .The cost function is given by:Let E be the vector

H = H 1 ,  , H N 

T

(7)

with

H s = f s  F s ( xs ), s = 1,, N , where 

s

(8)

is the interpolant to the data set with the point

( xs , f s ) removed, so that:

 a  x  x . N

F s ( x) =

s i

i =1,i  s

(9)

i

Rippa showed that

Es =

as , a ss

where

(10)

as is as defined in equation (6) and a s is the solution of

a s = e s , where

(11)

e

s

is the

s

th

column of the N  N identity matrix.Finally, the cost function

C (c)

C (c ) = H (c ) 1 ,

is given by: (12)

and

copt = arg min H (c) 1 .

(13)

cR

The solution of the linear system a = F : The solution of equation (4) is a unique if and only if the matrix  in (6) is invertible.However, the approximation solution of the linear systemis computed by applying the truncated singular value decomposition (TSVD) of  (Tony (1990)). N

 = UV T = ui i viT ,

(14)

i =1

ui and vi are the columns of the matrices U and V , respectively, and  i are the singular values of  . where the left and right singular vectors

We applied TSVD (Moroney (2006))to castoff the small singular values regarding to the benchmark where the singular values that are equal to, or less than, the product of the largest singular value with a chosen target  (machine epsilon) are ignored. Thus, if

 i  1

we ignore

 i , i = 2,, N .

A new matrix

t is then

t defined by:

formed with rank t

 t = ui i viT ,

t  rank( )

(15)

i =1

and the solution to (6) is then approximated by: t

a =  †t F =  i =1

uiT F

i

where the matrix

vi ,

 †t is the pseudoinverse of the matrix  t .

(16)

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The LocalandGlobalRBFApproximations: Inthispaperwehaveappliedinterpolation basedon multiquadricradialbasis functionsfortwo types of scattered data. Inparticular, twovariantsareinvestigated whichwe refertoast h e local multiquadric RBF interpolation 𝐹𝑚 (𝑥) and theglobal multiquadricR B Finterpolation 𝐹𝑛 (𝑥) . Theglobal RBF methodusesallpoints(N) onthe surfaceto constructthe RBFinterpolant,whilethelocalR B F me t h o d usesonlyalocalsetofpoints(m)onthesurfaceforthis purpose. We nowelaborate oneachofthesevariants. Global Method: GlobalmultiquadricRBFinterpolant.

T x1 , x1 ,... xN (our caseN=100)andadatavector f =  f1 , f 2 ,  , f N  , determine theinterpolatingRBF 𝐹𝑁 𝑥 that interpolatethedatavector f .

Given Ndatapoints

Local Method: LocalmultiquadricRBFinterpolant. GivenNdata

x1 , x1 ,...xN

points

f =  f1 , f 2 ,  , f N 

T

,andadatavector

onlyasubsetofmdatavalues f =  f1 , f 2 ,  , f m  anddata points T

,choose

x1 , x1 ,... xm , (typically m =20and m

=40)such that these m points representthe closestm points toeachpointofinterest onthe surfacearound whichthe RBFinterpolantistobeconstructed. Wenowoutlinethisprocedureinthefollowing Algorithm. Algorithm 1: Construction Surfaces using the RBF method: INPUT: N data points ( xi , f i ), i = 1,, N and corresponding data vector.





Step 1:Select a subset of n  N data points to built the surface triangulation. Step 2:Compuate the RBF linear system (4)Using eitherthe local method OR the global method Step 3: Estimatethe parameter c either locally OR globally. Step 4:Solve the linear system using the TSVD method. Step 5:Apply eitherthe global methodORthe local methodto construct the surface value. Numerical Experimentation for the Franke Data Set: The numerical experiments for the surface fitting technique showed in section § 2 is presented in this section.A data set taken from Franke (Franke (1982)) are used to evaluate the accuracy of our method either globally or locally. Franke data sets consists of two subsets (100 points and 33 points) and six test functions (see oqielatet al. (2007); oqielat et al. (2009)).The global RBF interpolant is constructed using all N = 100 data points and thenusedtoestimatethesurfacevaluesforthe33points. While, the local RBF interpolant is constructed by choosingtheclosest20or40pointstoeachofthe33points. The parameter c in the two cases was estimated either globally using the n = 100 data points, or locally using a selection of m = 20 or m = 40 neighbouring data points for each surface value. To assess the quality of the methodwecomputedthe root mean square error (RMS) given by: q

[ F (a , b )  f (a , b )]

2

i

RMS = where

i

i

i =1

q

i

,

(19)

f (ai , bi ) is the exact value of the function and F (ai , bi ) is the estimate value at the equivalent

points. Table 1:The RMS error comparison using the global multiquadric RBF (

m = 20

or

n = 100

points.

m = 40

Function

c

F1 F2

0.2506

F3

N  100

points) and local multiquadric RBF interpolants (

c

was computed globally by Rippa method using the

points) for the six test functions. The parameter

Global Method 2.3e-003

Local Method m=40 2.6e-003

m=20 2.6e-003

0.1560

5.3e-003

5.3e-003

5.3e-003

0.5907

7.4e-005

1.1e-004

2.9e-004

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Moaath Oqielatet al, 2017 Australian Journal of Basic and Applied Sciences, 11(13) October 2017, Pages: 103-111

F4

1.1974

2.2e-007

3.3e-006

5.1e-005

F5

0.4909

1.6e-005

2.6e-004

3.9e-004

F6

0.8901

1.3e-003

5.6e-004

2.8e-004

Table 2: The RMS error comparison using the local multiquadric RBF interpolant ( m =

20

m = 40 points) for the six test 20 or m = 40 ) points.

or

functions. The parameter c was computed locally by Rippa method using the same ( m = Local Method (m=40) Local Method (m=20) Function

[cmin cmax ]

RMS

[cmin cmax ]

RMS

F1 F2

[0.1565 3.4001]

2.8e-003

[0.0518 10]

4.9e-003

[0.1009 2.1613]

5.2e-003

[0.0282 10.462]

5.1e-003

F3 F4 F5

[0.4645 3.8262]

4.2e-004

[0.3533 10.344]

1.3e-003

[1.1355 2.6431]

5.0e-006

[0.7738 10.844]

1.8e-004

[0.3086 0.5959]

1.7e-004

[0.2358 8.0684]

7.7e-004

F6

[1.4729 4.3317]

1.9e-004

[2.0418 10.951]

2.0e-004

RESULTS AND DISCUSSION Tables 1 and 2 show the RMS errors for the six test functions using the global and local multiquadric RBF method where the parameter c computed in both cases using Rippa method.We observe thatthe RMS errors obtained using the global RBF method is more accurate than the RMS produced using the local RBF method. Moreover, the RMS error produced using the local RBF method constructed with m = 40 points was always found to be more accurate than the RMS produced for the surface representation constructed from m = 20 points (forboth cases whether c is approximated globally or locally). Furthermore, estimating the c value locally (table 2) helps to produce a more accurate RMS for some of the functions than using c computed globally (Table 1). However, the parameter c computed using global methodis less computationally costly than using c locally because a dense matrix system (6) of size either 20×20 or 40×40 needs to be solved at each time the local R B F is estimated. Table 3 shows as expected the global values of c were always contained in the local ranges of c given for each of the functions. Insummary, usingtheglobalRBF ismoreaccurateandlesscostlythanusingthe localRBF.Now, we investigate the suitability of the global RBF method for a real leaf data set in the following section. Application of the multiquadric RBF technique to a real Leaf data Set: A set of data points sampled from real leaf surface using laser scanner are requires to reconstruct the surface of a leaf. In this paper we used a real data points of Frangipani and Anthurium leaves sampled using a laser scanner by Loch (Loch (2004)). The multiquadric RBF are then applied to estimate the surface values for the Anthurium and Frangipani leaves. The Anthurium leaf data set consists of 4,688 leaf surface points and 79 boundary points, see figure 1. While Frangipani leaf data set comprises of 3,388 leaf surface points and 17 boundary points, see figure 3.

Fig. 1:The data points for the Anthurium Leaf. There are 79 boundary points (represented by the red dots) and 4,555 surface points (represented by the green dots)

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Moaath Oqielatet al, 2017 Australian Journal of Basic and Applied Sciences, 11(13) October 2017, Pages: 103-111

. Fig. 2: The model of the Anthurium leaf surfacegenerated from the points (shown in figure 1) using the global RBF method. Now to apply the multiquadric RBFmethod to the Frangipani and Anthurium leaf data a new reference plane for the data are required, see oqielat (oqielat et al. (2009)). The reference plane of the laser scanner leaf data points may not essentialcorrespond with the xy -plane in the data point coordinate system. To solve this issue a least squares fit to the leaf data points is then used as reference plane and then rotate the coordinate system to obtaine the xy -plane as the new reference plane. This rotations can be succeed by rotating the normal vector of the reference plane about the y -axis into the yz -plane and then rotating about the x -axis into the xz -plane, for more details (see oqielat et al. (2007); oqielat et al. (2009)). Finally, After we generated the new reference plane of the leaves data sets, The global multiquadric RBF method was applied to reconstruct the surface of those two leaves (shown in figure 2 and figure 4). For the Anthurium leaf, The global RBF approach based on using 212 pointsto generate one global RBF and then use it to estimate the surface values for all leaf data points. While, For the Frangipani leaf, The global RBF approach based on using 141 points to generate one global RBF and then use it to estimate the surface values for all leaf data points. The selection process for the212 points and 141 points are explained in (see oqielat et al. (2007); oqielat et al. (2009)).In both cases we used Rippa method(Rippa (1999))to estimate the parameter c globally using the same points that we used to built the global RBF. The results obtained for this surface fitting method are shown in Table 3 and Table 4.

Fig. 3:The data points for the Frangipani Leaf. There are 17 boundary points (represented by the Red dots) and 3,388 surface points (represented by the green dots).

Fig.4: The model of the Frangipani leaf surface generated from the points (shown in figure 3) using the global RBF method. Numerical Experiments for the Leaf Surface: In this section we show the outcome of applying themultiquadricRBFtechnique to the Frangipani and Anthurium leaf data points. As stated before a subset of the leaf surface data are used to construct one global RBF,

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Moaath Oqielatet al, 2017 Australian Journal of Basic and Applied Sciences, 11(13) October 2017, Pages: 103-111

the remaining data points of the leaf data (say m ) were used to evaluate the method quality using two error metrics. The first error metric is the relative root mean square error RMS (see equation 19), which is given by:

RelativeRMS =

RMS , i = 1,2,, m max ( f i )  min( f i )

The second error metric is the maximum error associated with the surface fit in relation to the maximum variation in z as

MaximumError 

max  F (ai , bi )  f i



, max ( f i )  min( f i ) where F (ai , bi ), i = 1,2,, m are the multiquadric RBF estimated values at the data points ( m ) and f i

are the given function values at the same data points. Table3:RMS and maximum error computed using the global multiquadric RBF for Anthurium leaf. Global Method Maximum error 0.042 Relative RMS 0.006 Number of point tested 4555 Table4:RMS and maximum error computed using the global multiquadric RBF for Frangipani leaf. Global Method Maximum error 0.097 Relative RMS 0.014 Number of point tested 3264

RESULTS AND DISCUSSION Table 3 and 4 shows the maximum errors and the relative RMS using the global RBF method for the Anthurium and Frangipani leaves shown in Figures 2 and 4.For the Anthurium and Frangipani leaf there were a total of 4555 and 3264 data points respectively used to evaluate the accuracy of the surface. In conclusion, from the results given in the tables3 and 4it appears that the global multiquadric RBF method produces an accuarte leaf surface representation and this is confirm our finding for Franke data set given in section 3. Conclusions and Future Research: In this research we presents a surface fitting method, based on Radial basis function, for modelling leaf surfacefrom three-dimensional scanned data points.We showed that the method produces an accurat leaf surface representation. The surface can be used to determine the path of pesticide or water droplet on a leaf surface. This model will help in the assessment of different pesticide formulations and then in the effectiveness of treatment. REFERENCES Anderson, W.K., 1994. A Grid Generation and Flow Solution Method for the Euler Equations on Unstructured grids. J. Comput. Phys., 110:23-38. Belward, J., I.Turner and M.Oqielat, 2008. Numerical Investigation of Linear Least Square Methods forDerivativesEstimation. CTAC 08 Computational Techniques and applications conference, Australia, July2008. Beatson, R.K., J.B. Cherrie and C.T. Mouat, 1999. Fast fitting of c basis functions: Methods based on preconditioned GMRES iteration. Advances in Computational Mathematics, 11:253-270. Beatson, R.K., J.B. Cherrie and D.L. Ragozin, 2001. Fast evaluation of radial basis functions: Methods for four-dimensional polyharmonic splines. SIAM Journal on Mathematical Analysis, 32(6):1272-1310. Borga,M.and A. Vizzaccaro, 1997. On the interpolation of hydrologic variables: Formal equivalence of multiquadric surface fitting and kriging. Journal of Hydrology, 195:160-171. Breslin, M., 2001. Spatial interpolation and fractal analysis applied to rainfall data. Ph.D. Thesis. Department of Mathematics, University of Queensland, Australia. Carlson,R.E.and T.A. Foley, 1991. The parameter R2 in multiquadric interpolation. Comput. Math. Appl, 21:29-42. Clough,R.W.and J.L. Tocher, 1965. Finite element stiffness matrices for analysis of plate bending. In Proceedings of the Conference on Matrix Methods in Structural Mechanics. Wright-Patterson A.F.B., Ohio, pp; 515-545. Carr, J.C., R.K. Beatson, J.B. Cherrie, T.J. Mitchell, W.R. Fright, B.C. McCallum and T.J. Evans, 2001. Reconstruction and representation of 3D objects with radial basis functions. ACM SIGGRAPH, 12-17 August 2001, Los Angeles, CA, pp; 67-76.

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