research article hybrid functional-neural approach for surface reconstruction · 2019. 7. 31. ·...

Research ArticleHybrid Functional-Neural Approach for Surface Reconstruction

Andreacutes Iglesias12 and Akemi Gaacutelvez1

1 Department of Applied Mathematics and Computational Sciences ETSI Caminos Canales y Puertos University of CantabriaAvenida de los Castros sn 39005 Santander Spain

2Department of Information Science Faculty of Sciences Toho University 2-2-1 Miyama Funabashi 274-8510 Japan

Correspondence should be addressed to Andres Iglesias iglesiasunicanes

Received 28 July 2013 Accepted 8 December 2013 Published 16 January 2014

Academic Editor Yudong Zhang

Copyright copy 2014 A Iglesias and A Galvez This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper introduces a new hybrid functional-neural approach for surface reconstruction Our approach is based on thecombination of two powerful artificial intelligence paradigms on one hand we apply the popular Kohonen neural network toaddress the data parameterization problem On the other hand we introduce a new functional network called NURBS functionalnetwork whose topology is aimed at reproducing faithfully the functional structure of the NURBS surfaces These neural andfunctional networks are applied in an iterative fashion for further surface refinement The hybridization of these two networksprovides us with a powerful computational approach to obtain a NURBS fitting surface to a set of irregularly sampled noisy datapoints within a prescribed error threshold The method has been applied to two illustrative examples The experimental resultsconfirm the good performance of our approach

1 Introduction

Manufacturing industries are constantly evolving in responseto the new challenges of the globalization and the grow-ing competition in this global market Product design isplaying a central role in this process as current customersare increasingly demanding a mass customization of theproducts As a result the geometric and aesthetic propertiesof the manufactured goods (shape color and dimensions)have to be modified frequently in order to meet the newmarket demands

A major step in this process is the generation of real pro-totypes with different materials to explore and analyze theirgeometric properties and the feedback of potential customerswhen exposed to different variations of the final productPrototype generation and customization can be dramaticallyimproved by using digital technologies in which the physicalmodel is digitized stored and manipulated by computera process called reverse engineering [1 2] Typically thisprocess begins with data sampling by using 3D laser scanningand other digitizing devices This technology is intensivelyused for the construction of car bodies ship hulls airplanefuselage and other free-formobjects [2ndash7]The resulting data

points are then fitted to mathematical entities such as curvesand surfaces usually in parametric formThe output is a veryaccurate digital version of the real product which is alsosimpler and easier to store analyze and manipulate It alsosimplifies the transfer and communication processes amongdesigners manufacturers and providers making the modelavailable in just a few seconds all over the world a key aspectin our ubiquitously connected information society era

In this paper we are interested in one of the mostcritical steps of this process namely the construction ofsurfaces of the real objects from sets of digitized data pointsa field usually called surface reconstruction The preferredmathematical models for design and manufacturing are thefree-form parametric surfaces [3 8ndash16] because they are veryflexible and can be readily modified by changing a small setof parameters In this paper we use NURBS surfaces themost powerful (and most difficult to deal with) free-formparametric surfaces which have become the standard forCADCAM data representation in industrial settings and inmany other fields from digital effects for movies in computeranimation and advertisements to the design of characters incomputer graphics and video games In reverse engineeringapplications data points are usually acquired through laser

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 351648 13 pageshttpdxdoiorg1011552014351648

2 Mathematical Problems in Engineering

scanning and other digitizing devices and are thereforesubjected to measurement noise irregular sampling andother artifacts [6 7] Consequently a good fitting of datashould be generally based on approximation schemes ratherthan interpolation [1 13 14 17ndash19] Because this is the typicalcase in many real-world industrial problems in this paper wefocus on the approximation scheme to a given set of noisyirregularly sampled data points

Obtaining the best approximating surface in such cases ismuch more troublesome than it may seem at first sight Themain reasons are as follows

(i) The NURBS surfaces depend on many differentparameters (data parameters knots control pointsand weights) that are strongly interconnected witheach other leading to a strongly nonlinear continuousoptimization problem

(ii) It is also multivariate as it typically involves a largenumber of unknown variables for a large number ofdata points a case that happens very often in real-world examples

(iii) In addition it is also overdetermined because weexpect to obtain the approximating surfacewithmanyfewer parameters than the number of data points

(iv) Finally the problem is known to be multimodal thatis the least-squares objective function can exhibitmany local optima [20 21] meaning that the problemmight have several (global andor local) good solu-tions

In conclusion we have to solve a very difficult multi-modal multivariate high-dimensional continuous nonlinearoptimization problem A number of methods have beenproposed to solve this problem (see Section 2 for details)Among them those based on artificial intelligence techniqueshave received increasing attention during the last few yearsMost of such methods rely on the artificial neural networks(ANN) formalism [22] Since ANN methodology is actuallyinspired by the behavior of the human brain it is able toreproduce some of its most typical features such as the abilityto learn fromdataThis explainswhy they have been sowidelyapplied to data fitting problems

Although they are very popular the ANN are howeverlimited in many aspects A major drawback is their inabilityto reproduce mathematically the functional structure of agiven problem This limitation can be overcome with theuse of a new paradigm in artificial intelligence the so-called functional networks (FN) (see Section 4 for details)In short functional networks are a generalization of thestandard neural networks in which the scalar weights arereplaced by neural functions These neural functions canexhibit in general a multivariate character Furthermoredifferent neurons can be associated with neural functionsfrom different families of functions These FN features allowus to reproduce exactly the functional structure of theproblem by a careful choice of the functions involved whichcan hereby be associated with one or several neurons of thenetwork This procedure yields a functional structure that is

typically a replica of the underlying structure of the givenproblem

11 Aims and Structure of the Paper In this paper we proposea hybrid artificial intelligence approach to solve the surfacereconstruction problem Our approach is based on the com-bination of two powerful artificial intelligence paradigms onone hand we apply the popular Kohonen neural networkto address the data parameterization problem On the otherhand we take advantage of the remarkable properties men-tioned in previous paragraph by introducing a new functionalnetwork called NURBS functional network whose topologyis specially targeted to reproduce the functional structure ofthe NURBS surfaces These neural and functional networksare then applied iteratively for further surface refinementAs it will be shown later on the hybridization of thesetwo networks provides us with a powerful computationalapproach to solve the surface reconstruction problem Tocheck the performance of our approach it has been applied totwo illustrative examples Our experimental results show thatthe method performs very well being able to reconstruct theapproximating surface of the given set of data points with ahigh degree of accuracy

The structure of this paper is as follows in Section 2previous work regarding the surface reconstruction prob-lem is reported Then some basic concepts about NURBSsurfaces and the optimization problem to be solved aregiven in Section 3 Section 4 describes the fundamentals ofthe functional networks along with their main componentsand the differences between neural and functional networksThe proposed hybrid functional-neural method for surfacereconstruction with NURBS surfaces is described in detail inSection 5 Then some illustrative examples of its applicationare reported in Section 6 A comparison of our approachwith other ANN alternative methods is analyzed in detail inSection 7 The paper closes with the main conclusions of thiscontribution and our plans for future work in the field

2 Previous Work

Surface reconstruction has been a topic of increasing atten-tion from the scientific community during the last 20 yearswith outstanding applications in both theoretical and applieddomains Regarding the theoretical side it is a remarkablesubject in approximation theory [23 24] statistics [25]numerical analysis [26 27] geometric modeling [2 8 28]and computer-aided geometric design (CAGD) [5 29] Inaddition there is a bulk of applications in several fieldssuch as computer-aided manufacturing (CAM) [6 7] datavisualization [30] cultural heritage preservation [31] virtualreality [32] medical imaging [33] and computer animation[34] to mention just a few

In general surface reconstruction methods are classifiedin terms of the available input (2D slices isoparametriccurves clouds of points mixed information etc) Forinstance authors in [35ndash39] address the problem of obtaininga surface model from a set of given cross-sections a classicalproblem in medical science biomedical engineering and

Mathematical Problems in Engineering 3

CADCAM Other classical input data include isoparametriccurves on the surface [40] and even mixed information suchas scattered points and contours [41ndash43] or isoparametriccurves and data points [4 5 44]

In most cases however the available information aboutthe surface is typically a dense set of (usually unorganized)3D data points obtained by using some sort of digitizingdevices (see eg [14 45ndash48]) In that case the reconstructedsurface can be described using three different representationsproviding different levels of accuracy The simplest oneis given by the polygonal meshes where the data pointsare used as vertices connected by lines (edges) that worktogether to create a 3D model comprised of vertices edgesand faces Although it is the coarsest representation it isalso the most popular because of its simplicity flexibilityand excellent performance with current graphical cardsSurface reconstruction methods with polygonal meshes canbe found for instance in [30 31 47 49ndash51] and referencestherein The next level is given by the constructive solidgeometry (CSG) models where elementary geometries (suchas spheres boxes cylinders or cones) are combined in orderto produce more elaborated shapes by applying some simple(Boolean) operators union intersection and differenceThismethodology works well but presents a low level of flexibilitybeing severely limited to very simple shapes The mostsophisticated and most accurate level consists of obtainingthe real mathematical surface fitting the data points Thisissue has been analyzed from several points of view such asparametric methods [52] subdivision surfaces [53] functionreconstruction [54 55] implicit surfaces [48] and algebraicsurfaces [56] Other approaches are based on the applicationof metaheuristic techniques which have been intensivelyapplied to solve difficult optimization problems that can-not be tackled through traditional optimization algorithmsRecent schemes in this area involve particle swarm optimiza-tion [10 57 58] genetic algorithms [59ndash62] artificial immunesystems [63 64] estimation of distribution algorithms [65]firefly algorithm [66 67] and hybrid techniques [68 69]

Artificial neural networks have also been applied to thisproblem [45 70] mostly for arranging the input data incase of unorganized points After this preprocessing stepany other classical surface reconstruction method operatingon organized points is subsequently applied A work usinga combination of neural networks and partial differentialequation (PDE) techniques for the parameterization andreconstruction of surfaces from 3D scattered points can befound in [71] Two previous papers by the authors havealso addressed this problem by using functional networks[4 8] a powerful generalization of neural networks based onfunctional equations [72 73] Both works show however thatthe single application of functional networks is still unableto solve the general case The work in [4] addresses theparticular case of B-spline surface reconstruction when someadditional information (isoparametric curves) is availablein addition to the data points The fitting surfaces are ageneralization of the Gordon surfaces [40] The work in[8] combines functional networks with genetic algorithmsin order to solve the (much simpler) polynomial Beziercase The approach presented here solves the general NURBS

surface reconstruction problem based exclusively on neuraland functional networks To the best of our knowledge noprevious method is reported in the literature providing allthese features

3 Basic Concepts and Definitions

31 NURBS Surfaces Let S = 1199040 1199041 1199042 119904

119903minus1 119904119903 sub

[119886 119887] be a nondecreasing sequence of real numbers calledknots S is called the knot vector Without loss of generalitywe can assume that [119886 119887] = [0 1] The 119894th B-spline basisfunction 119873

119894119896(119906) of order 119896 (or equivalently degree 119896 minus 1) is

defined by the recurrence relations

1198731198941(119906) =

1 if 119904119894le 119906 lt 119904

119894+1

0 otherwise(1)

with 119894 = 0 119903 minus 1 and

119873119894119896(119906) =

119906 minus 119904119894

119904119894+119896minus1

minus 119904119894

119873119894119896minus1

(119906)

+

119904119894+119896

minus 119906

119904119894+119896

minus 119904119894+1

119873119894+1119896minus1

(119906)

(2)

for 119896 gt 1 Note that 119894th B-spline basis function of order1 1198731198941(119906) is a piecewise constant function with value 1 on

the interval [119904119894 119904119894+1) called the support of 119873

1198941(119906) and zero

elsewhereThis support either can be an interval or reduce toa point as knots 119904

119894and 119904119894+1

must not necessarily be different Ifnecessary the convention 00 = 0 in (2) is applied Any basisfunction of order 119896 gt 1119873

119894119896(119906) is a linear combination of two

consecutive functions of order 119896 minus 1 where the coefficientsare linear polynomials in 119906 such that its order (and henceits degree) increases by 1 Simultaneously its support is theunion of the (partially overlapping) supports of the formerbasis functions of order 119896 minus 1 and consequently it usuallyenlarges

With the same notation given a set of three-dimensionalpoints (called control points as they roughly determine theshape of the curve) P

119894119895119894=0119898119895=0119899

in a bidirectional netand two knot vectors S = 119904

0 1199041 1199042 119904

119903minus1 119904119903 and T =

1199050 1199051 119905

ℎminus1 119905ℎ a NURBS surface S(119906 V) of order (119896 119897)

(where NURBS stands for Non-Uniform Rational B-Spline)is a rational B-spline parametric surface given by

S (119906 V) =sum

119898

119894=0sum

119899

119895=0119908119894119895P119894119895119873119894119896 (

119906)119873119895119897 (V)

sum

119898

119894=0sum

119899

119895=0119908119894119895119873119894119896(119906)119873119895119897(V)

(3)

where the 119873119894119896(119906)119894and 119873

119895119897(V)119895are the B-spline basis

functions of orders 119896 and 119897 respectively defined following(1) and (2) and 119908

119894119895119894119895

are nonnegative scalar values calledweights associated with the control points P

119894119895119894119895 Without

loss of generality parameters 119906 V can be assumed to takevalues on the interval [0 1] For a proper definition of aNURBS surface in (3) the following relationships must hold(see [29]) 119903 = 119898 + 119896 ℎ = 119899 + 119897

In general a NURBS surface does not interpolate any ofits control points the interpolation only occurs for nonperi-odic knot vectors (in that case theNURBS surface does inter-polate the corner control points) [11 29] Since they are the


most common in computer graphics and industrial domainsin this work we will consider the case of nonperiodic knotvectors Note however that our method does not depend onthe kind of knot vectors used for the approximating surfaces

32 Surface Reconstruction Problem In clear contrast withmany previousmethods in this paper we focus on the generalsurface reconstruction problem which assumes that no otherinformation about the problem is available beyond the datapoints In particular our problem can be stated as followsGiven a set of (usually irregularly sampled) noisy data pointsQ assumed to lie on an unknown surface U constructto the extent possible a full mathematical representationof a surface model S that approximates U Because of itsremarkable applications in real-world engineering problemswe also demand such a mathematical representation to be aNURBS surface

Mathematically speaking we assume thatwe are providedwith a set of data points Q

120572120573120572=1119872120573=1119873

in R3 with119872119873 ⋙ 119903 ℎ Our goal is to obtain theNURBS surface S(119906 V)that fits the data points better in the discrete least-squaressense To do so we have to compute all the parameters of theapproximating surface by minimizing the least-squares error119864 defined as the sum of squares of the residuals

119864 =

119872

sum

120572=1

119873

sum

120573=1

(Q120572120573

minus

sum

119898

119894=0sum

119899

119895=0119908119894119895P119894119895119873119894119896(119906120572)119873119895119897(V120573)

sum

119898

119894=0sum

119899

119895=0119908119894119895119873119894119896(119906120572)119873119895119897(V120573)

)

2

(4)

In the case of scattered data points Q120583120583=1119877

ourmethod will work in a similar way by simply replacing theprevious expression (4) by

119864 =

119877

sum

120583=1

(Q120583minus

sum

119898

119894=0sum

119899

119895=0119908119894119895P119894119895119873119894119896(119906120583)119873119895119897(V120583)

sum

119898

119894=0sum

119899

119895=0119908119894119895119873119894119896(119906120583)119873119895119897(V120583)

)

2

(5)

The minimization of either (4) or (5) leads to the systemof equations

Qk= M sdot Pk

(6)

where the symbol (sdot)v denotes the vectorization operatorof the given matrix and M in (6) is a matrix given byM119894119895(119906120572 V120573) = [((Rv

119895119897(119906120572 V120573))

Totimes 119877119894119896(uT V

120573))

v]

T with u =

(1199061 119906

119872) 119877119894119895(119906 V) is given as

119877119894119895(119906 V) =

119908119894119895119873119894119896(119906)119873119895119897(V)

sum

119898

119894=0sum

119899

119895=0119908119894119895119873119894119896(119906)119873119895119897(V)

119894 = 0 119898 119895 = 0 119899

(7)

while otimes and (sdot)T represent the outer product operator and thetranspose of a vector or matrix respectively The indices in(4)ndash(7) vary in the ranges of values indicated throughout thesection

It is worthwhile to mention that since the lengths of Qv

and Pv are respectively 119872 times 119873 and (119898 + 1) times (119899 + 1)

the system (6) is overdetermined Premultiplication of bothsides byMT yields

MTsdotQk

= MTsdotM sdot Pk

= Δ sdot Pk (8)

where Δ = MTsdot M Note also that the tensor-product basis

functions 119877119894119895(119906 V) are generally continuous and nonlinear

so the minimization of (8) leads to the continuous nonlinearoptimization problem

minP119894119895subR3 119908119894119895ge0(119906120572 V120573)subDom(S)119904120592119905]sub[01]

10038171003817100381710038171003817

MTsdotQk

minus Δ sdot Pk10038171003817100381710038171003817

2

(9)where sdot represents the Euclidean norm

4 Functional Networks

Roughly speaking a functional network is a generalizationof the standard neural network in which the scalar weightsare replaced by neural functions Functional networks werefirstly introduced in 1998 by Castillo in [72] as a way toenhance the neural networks with new capabilities Sincethen they have been successfully applied to several problemsin science and engineering The interested reader is referredto [73 Chapter 9] (Chapter 9) for in-depth explanation aboutfunctional networks along with several illustrative examplesand applications In this section we describe the maincomponents of a functional network Differences betweenneural and functional networks are also discussed in thissection

41 Components of a Functional Network As an explanatoryexample Figure 1(a) shows the functional network of theassociative operation 119865 between two real numbers that isfunction 119865 satisfies

119865 (119865 (119909 119910) 119911) = 119865 (119909 119865 (119910 119911)) (10)It can be proved that the general solution of this equation isgiven by [73]

119865 (119909 119910) = 119891

minus1[119891 (119909) + 119891 (119910)] (11)

where119891(119909) is an arbitrary continuous and strictly monotonicfunction which can be replaced only by 119888 119891(119909) where 119888 is anarbitrary constant Such a solution can be represented by thefunctional network in Figure 1(b) Note that because of theuniqueness (except arbitrary constants) of the solution bothnetworks do actually represent the same problem In otherwords the functional network in Figure 1(b) is equivalentto (but arguably simpler than) that in Figure 1(a) FromFigure 1(b) the main components of a functional networkbecome clear

(i) Several Layers of Storing Units

(a) A Layer of Input Units This first layer contains the inputinformation In this figure this input layer consists of theunits 119909 and 119910

(b) A Set of Intermediate Layers of Storing Units They arenot neurons but units storing intermediate informationThis


F

I

I

F

F

F

x

y

z

u

(a)

f

fx

y

u

f(y)

f(x)

f(x) + f(y)

fminus1

fminus1(f(x) + f(y))

+

(b)

Figure 1 The associativity functional network (a) original network (b) simplified network

set is optional and allows more than one neuron output tobe connected to the same unit In Figure 1(b) there are twointermediate layers of storing units which are represented bysmall circles in black

(c) A Layer of Output UnitsThis last layer contains the outputinformation In Figure 1(b) this output layer is reduced to theunit 119906 = 119891minus1(119891(119909) + 119891(119910))

(ii) One or More Layers of Neurons or Computing Units Aneuron is a computing unit which evaluates a set of inputvalues coming from the previous layer of input or interme-diate units and gives a set of output values to the next layerof intermediate or output units Neurons are represented bycircles with the name of the corresponding neural functioninside For example in Figure 1(b) we have three layers ofneurons The first one gives outputs of functions with onevariableThe second layer exhibits the sum operator of its twoinputs The last layer computes the inverse of the first layerfunction applied to output of the previous layer

(iii) A Set of Directed Links They connect the input orintermediate layers to its adjacent layer of neurons andneurons of one layer to its adjacent intermediate layers orto the output layer Connections are represented by arrowsindicating the information flow direction We remark herethat information flows in only one direction from the inputlayer to the output layer

All these elements together form the network architectureor topology of the functional network which defines thefunctional capabilities of the network

42 Differences between Functional and Neural NetworksIn next paragraphs we discuss the differences betweenfunctional and neural networks and the advantages of usingfunctional networks instead of standard neural networks

(1) In neural networks each neuron returns an output119910 = 119891(sum119908

119894119896119909119896) that depends only on the value

sum119908119894119896119909119896 where 119909

1 1199092 119909

119899are the received inputs

(see Figure 2(a)) Therefore their neural functionshave only one argument In contrast neural functionsin functional networks can have several arguments asshown in Figure 2(b)

(2) In neural networks the neural functions are univari-ate neurons can showdifferent outputs but all of themrepresent the same values In functional networks theneural functions can bemultivariate

(3) In a given functional network the neural functionscan be different (such as functions 119891

1 1198912 and 119891

3

in Figure 2(b)) while in neural networks they areidentical

(4) In neural networks there are weights which must belearned These weights do not appear in functionalnetworks where neural functions are learned instead

(5) In neural networks the neuron outputs are differentwhile in functional networks neuron outputs can becoincident This fact leads to a set of functional equa-tions which have to be solved [73 74] These func-tional equations impose strong constraints leading toa considerable reduction in the degrees of freedomof the neural functions In most cases this impliesthat neural functions can be reduced in dimension orexpressed as functions of smaller dimensions

All these features show that the functional networksexhibit more interesting possibilities than the neural net-works This implies that some problems can be solved moreefficiently by using functional networks instead of neuralnetworks

5 Our Method

In this section we describe the proposed method for solvingthe surface reconstruction problem indicated in Section 32Firstly a general overview of the method is presented Theneach step of the method is discussed in detail


x1x4

x6

x5

x2

x3

w11

w34

w35

w12

w22

w23

f(sumw1ixi)

f(sumw2ixi)

Neural network

f(sumw3ixi)

(a)

x6

x4

x5

x1

x2

x3

f1(x1 x2)

f2(x2 x3)

f3(x4 x5)

Functional network

(b)

Figure 2 Differences between (a) neural networks and (b) functional networks

51 Overview of the Method The graphical workflow inFigure 3 summarizes the main steps of our method Theinitial input consists of a set of irregularly sampled noisy3D data points assumed to lie on an unknown surfaceThe goal is to obtain the NURBS surface that approximatesthese data points optimally To this purpose we need tosolve two important subproblems data parameterization andsurface approximation To address data parameterizationwe firstly perform a principal component analysis (PCA) toobtain a parametric plane that accounts for the variabilityof data Then the data points are projected onto that para-metric plane A 2D surface parameterization is subsequentlyobtained by applying a Kohonen neural network to the setof projected 2D data points Then we apply our NURBSfunctional network to compute all other parameters of theNURBS surface approximating the data points for this initialparameterization We call that surface a base surface Thenext step consists of computing the fitting error for theapproximating surface Finally the data points are projectedonto the new base surface in order to yield a new (moreaccurate) parameterization All previous steps are repeatediteratively until a stopping criterion is reached Usual stop-ping criteria are that the fitting error becomes smaller thana given threshold value or that successive iterations of thisreconstruction pipeline no longer improve current solutions

52 Data Parameterization The parameterization step con-sists of establishing the relationships among the data pointsin the surface parametric domain This process is essentialfor a good fitting of data points Some standard proceduresare given by the uniform chord length and centripetalparameterizations However these methods are only suitablefor data points distributed in a uniform grid and tend tofail for unorganized irregularly sampled data An alterna-tive procedure is based on the idea of projecting the datapoints onto an additional surface usually called base surfacereflecting the distribution of data points and then computinga parameterization by using the projected 2D points Thesimplest case of this approach consists of using a parametricplane [11] usually orthogonal to the main viewing directionof the digitizing device A better alternative is to use asuitable 3D surface for data projection [14] usually a coarseapproximation of final fitting surface which is expected to bemodified by successive improvements of this initial surface

In our method we combine these ideas to develop arefinement process in iterative fashion At the initial stagewe project the data points onto a parametric plane reflecting

NURBS functional network

Kohonen neural

network

Fitting error computation

Data parameterization

Data projection onto base surface

Base NURBS surface fitting

Data projection onto a plane

Input 3D data points

Yes

No Output

final fitting NURBS surface

PCA

Error lt 120576(120576 threshold)

Figure 3 Graphical workflow of the proposed method

the variability of data This parametric plane is computedby using the principal component analysis (PCA) a verypopular descriptive technique in data analysis and manyother fields PCA is a powerful statistical method aimed atperforming dimensionality reduction by using the analysis ofthe correlation of data A great advantage of this method isits nonparametric nature meaning that it does not requireany parameter tuning in order to get an output Furthermorethe answer is unique and available regardless of the way thedata has been recorded or obtained Other main reasonsfor our choice are that PCA is very efficient at preservingdistances between the points and each principal componenthas the highest variance possible under the constraint that itis uncorrelated with preceding components In fact the firstprincipal component corresponds to a line passing throughthe multidimensional mean It also minimizes the sum of


squares of the distances to the points from that line Becauseof these properties the parametric plane obtained by PCA isvery adequate for the initial 2D projection

Projected 2D points are then used for an initial surfaceparameterization by applying a Kohonen neural network(also called self-organizingmap (SOM) neural network)Thisis a very popular artificial neural network for unsupervisedlearning The interested reader is referred to the nice booksin [75 76] for a comprehensive overview about the Kohonenneural network its fundamentals and mathematical basisalong with the most important variants and modificationsand several interesting applications

Algorithm 1 Kohonen neural network for 2D data parame-terization

(1) Initialize position Ψ of all neurons Φ with randomvalues within the surface boundaries

(2) Initialize the weights 120590119894119895

randomly around the cen-troid of the input space

(3) Pick a new input sample 119876120575

(4) Compute

119889119895= radic

119877

sum

119894=1

(Ψ119894(120575) minus 120590

119894119895(120575))

2

(12)

(5) Select the active neuron 120574 for which 119889120574

=

min119894=1119877

(119889119894) The winner is the closest active

neuron to the sample data 119876120575 with parametric

coordinates (119894 119895)(6) Compute the neighborhood of neuron 120574 as follows

The radius 120588 is given by

120588 (120574) =

119877

2119890

[sum120575

119894=1(110+11989410

4)]

(13)

The bubble neighborhood of the winner neuron isdefined as all neurons at positions (119894 119895) such that

10038161003816100381610038161003816

119894 minus119894

10038161003816100381610038161003816

+

10038161003816100381610038161003816

119895 minus119895

10038161003816100381610038161003816

lt 120588 (1 minus

120575

120579

) (14)

where 120579 is the run length(7) Update the weight of neuron 120574 and all neurons in its

neighborhood as

120590119894119895(120575 + 1) = 120590

119894119895(120575) + 120578 (120575) (Ψ

119894(120575) minus 120590

119894119895(120575)) (15)

with

120578 (120575) = 120577120575

1

radic2120587119890

minus12057522 (16)

where the learning rate 120577120575is given by

120577120575= 1 minus (1 minus 120577

0(1 minus

120575

120579

))

120590120575

(17)

(8) Increment 120575

(9) Repeat the steps (3)ndash(8)until120575 reaches the limit value

A very remarkable feature of theKohonen neural networkis that it incorporates a neighborhood function to preservethe topological properties of the input space [75] This prop-erty is very useful to generate a 2D grid for parameterizationwhere the neurons represent the grid nodes The neuralnetwork is trained by using the projected 2D data so thatits topology eventually reflects their shape and neighborhoodrelations To this purpose each neuron contains geometricalinformation about the coordinates of the associated node andthe topological relations with its neighbors It is importantto remark that the connections among the neurons donot change during the training stage instead the changesoccur on the geometrical values stored in the neurons Inother words the network performs a topological ordering ofthe competitive neurons such that the neighboring neuronsrepresent clusters in the two-dimensional space Eventuallythe weights will specify cluster centers whose distributionapproximates the distribution of data points and hence a suit-able 2D parameterization Because of these good propertiesthe Kohonen neural network has already been used in severalways for data parameterization in some previous works [4970 71 77]The algorithmused in this paper combines some ofthe best features of previousmethods It is briefly summarizedin Algorithm 1

53 Surface Fitting Amajor property of functional networksis their ability to reproduce the functional structure of theunderlying mathematical function of the data points In thispaper we introduce a new functional network (depicted inFigure 4) especially designed to reproduce the functionalstructure of NURBS surfaces the NURBS functional net-work Its workflow can be traced graphically by proceedingupwardly in Figure 4 given the surface parameter values 119906and V and two orders 119896 and 119897 what this functional networkessentially does is to compute the values of the basis functions119873119894119896at 119906 and 119873

119895119897at V and then the bivariate tensor-product

basis functions 119873119894119896(119906)119873119895119897(V) (119894 = 0 119898 119895 = 0 119899)

Each bivariate basis function is then multiplied twice firstlyby the scalar weight 119908

119894119895and then by such a weight and its

associated vector weight P119894119895 Summation on indices 119894 and

119895 is applied to both expressions to obtain the numeratorand denominator of (3) to which the quotient operator issubsequently applied Note that two different (scalar andvector) time operators are considered to account for themultiplication by 119908

119894119895and P

119894119895 respectively Note also that

those vectors P119894119895

play the role of weights of the neuralfunctions as well with themeaning that a functional networkwith119889-dimensional vectors asweights can be understood as119889parallel functional networks with scalar weights Note finallythat the first three layers of this functional network applyfunctions to either the same or independent arguments of theprevious layer so NURBS functional networks are well suitedfor partial bottom-up parallelization

Now we use the surface parameterization obtained inprevious steps to compute an approximating surface to


times times

times times times times times times times times times times times times times times times times


times times times times times times times times times times times times times times

w00 w01 w02 w0n

u v

w10 w11 w20 w21 w22 w2n wm0 wm1 wm2 wmnw1nw12

middot middot middotmiddot middot middot middot middot middotmiddot middot middot

middot middot middot

middot middot middotmiddot middot middotmiddot middot middotmiddot middot middot









middot middot middotmiddot middot middot

N0k N1k N2k Nmk N0l N1l N2l Nnl

P00 P01 P10 P11 P12 P1n P20 P21 P22 P2n Pm0 Pm1 Pm2 PmnP02 P0n

S(u )

+ +

Figure 4 Graphical representation of the NURBS functional network

the data points This functional network is trained by super-vised learning in which couples of input-output values D =

(119868119894 119874119894) | 119894 = 1 119877 corresponding to the surface

parameters (119906120583 V120583)120583and their associated data points Q

120583

are presented to the network Unlike ANN where the neuronfunctions are assumed to be fixed and known and only theweights are learned in functional networks the functions arelearned during the structural learning (which obtains thesimplified network and functional structures) and estimatedduring the parametric learning (which consists of obtainingthe optimal neuron function from a given family) In ourcase the structural learning corresponds to the topology ofthe NURBS functional network where the input data arethe order (119896 119897) and the length (119903 ℎ) of the knot vectors of

the approximating surface These values determine the num-ber of neurons in each layer of the functional network

The parametric learning concerns the estimation of theneuron functions It is usually accomplished by consideringlinear combinations of given functional families and estimat-ing the associated parameters from the available data Forour choice of B-spline basis functions during the structurallearning the parametric learning is required to determine thescalar and vector weights of our NURBS functional networkVector weights are learned by using expression (8) where Δis a symmetric square matrix This system is solved by usingthe singular value decomposition (SVD) which provides thebest numerical answer for the minimization problem in (9)in those cases in which the exact solution is not possible


Table 1 Number of data points and input parameters of theapproximating NURBS surfaces for the two examples discussed inthis paper

Example Number of data points Order (119903 ℎ) IterationsMask 11830 (4 4) (17 14) 2Umbrella 8926 (3 3) (6 27) 4

(see [78] for details) Finally the scalar weights are obtainedby least-squares minimization of expression (9) where allother relevant parameters of this minimization problem havealready been obtained as described in previous steps

54 Surface Refinement Once the approximating surface isobtained the initial parametric plane used for data param-eterization is replaced by this approximating surface whichbecomes the new base surface Data points are then projectedonto this base surface by computing the nearest point on thesurface to each 3D data point by following the procedureindicated in [79] A new data parameterization and surfacefitting steps are computed according to Sections 52 and 53respectively The resulting fitting surface becomes the newbase surface and so on In general this procedure yieldsa more refined (ie more accurate) fitting surface to datapoints

This process is repeated iteratively until a stopping cri-terion is reached Usual stopping criteria are that the fittingerror becomes smaller than a given threshold value or thatsuccessive iterations of this reconstruction pipeline no longerimprove current solutions In the former case we assumethat an error threshold value is provided as an input of theproblem (as it usually happens in many industrial problems)and we compute the fitting error according to either (4) or(5) In the latter case we compare the fitting error betweensuccessive iterations and the process is stopped when nofurther improvement is achieved

6 Illustrative Examples

Our method has been tested with several examples of differ-ent clouds of data points To keep the paper at manageablesize we discuss here two of them Examples in this paperare shown in Figures 5-6 For each example two differentpictures are displayed at (a) we show the original cloud ofinput data points represented as small red points at (b) thebest approximating NURBS surface is shown as obtainedwith our functional-neural approach Our input consists ofsets of irregularly sampled data points (this fact can readilybe seen from simple visual inspection of the point cloudsat (a)) which are also affected by measurement noise of asignal-to-noise ratio of 15 1 in all examples In this paper afitting error threshold value 120598 = 10minus3 is consideredThe otherrelevant parameters of the approximating NURBS surfacesare reported in Table 1 where the examples are arranged inrows For each example the following data are arranged incolumns number of data points order of the approximatingNURBS surface length of knot vectors and number of

iterations required to obtain the fitting surface with a fittingerror below the given threshold

A simple visual inspection of the figures clearly shows thatour method yields a very good approximating surface to datapoints in all cases The low number of iterations required toobtain the fitting surface for the given threshold also confirmsthe good behavior of the method From these examples andmany others not reported here for the sake of brevity weconclude that the presented method performs very well withremarkable capability to provide a satisfactory solution to thegeneral reconstruction problem with NURBS surfaces

Regarding the implementation issues all computationsin this paper have been performed on a 29GHz Intel Corei7 processor with 8GB of RAM The source code has beenimplemented by the authors in the native programminglanguage of the popular scientific program Matlab version2010b

7 Comparison with Other Approaches

In this section we compare the presented method with otheralternative approaches for surface reconstruction based onneural networks A careful revision of the literature in thefield gives six previous contributions in the field the worksin [4 8 49 70 71 77]This small number is a clear indicationof the difficulty and originality of the present work

Comparative results of these methods are summarizedin Table 2 The different methods are arranged in rows andsorted by year of publication For each reported methodthe columns give a brief description about its main featuresColumns 2 4 and 6 give a binary answer to three differentquestions whether or not the indicated method providesalgorithms for the subproblems of data parameterizationsurface fitting and support for NURBS surfaces respectivelyAnswer true is marked with a check () otherwise withsymbol (times) Wherever a positive answer is found a shortdescription about the specific techniques incorporated in thatmethod for the reported subproblem is given in columns3 and 5 respectively Note that all methods except [4 8]address the data parameterization subproblem with a Koho-nen neural network and all methods except [70 77] providea procedure to compute the approximating surface either asa Bezier surface [8] a B-spline surface [4 71] or a polygonalmesh [49] The works in [70 77] assume that any traditionalsurface fitting technique will be used for this particular sub-problem

The most important difference of this method withrespect to alternative approaches is that previous methodsdo not provide support for NURBS surfaces (see column6 in Table 2) On the contrary NURBS surfaces are fullysupported in our method In fact the examples shown in thepaper are more difficult to reconstruct through simple poly-nomial surfaces and require a larger number of parametersIn clear contrast ourmethod requires only a small number ofparameters Furthermore the solution is reachedwith a smallnumber of iterations To the best of our knowledge this isthe first neural-based approach providing full support for all


(a) (b)

Figure 5 Mask example (a) original data points (b) fitting surface

(a) (b)

Figure 6 Umbrella example (a) original data points (b) fitting surface

steps of the general surface reconstruction problem by usingNURBS surfaces

8 Conclusions and Future Work

This paper proposes a new hybrid functional-neuralapproach to solve the surface reconstruction problem In ourapproach we combine two powerful artificial intelligenceparadigms the popular Kohonen neural network to addressthe data parameterization problem and a new functionalnetwork called NURBS functional network to reproducethe functional structure of the NURBS surfaces Theseneural and functional networks are then applied iterativelyfor further surface refinement The hybridization of thesetwo networks provides us with a powerful computational

approach to solve the surface reconstruction problemThe performance of our approach has been tested by itsapplication to two illustrative examples Our results showthat themethod performs very well being able to reconstructthe approximating surface of the given set of data points witha high degree of accuracy and a low number of iterations

The main limitation of this approach is that it requiressome initial input such as the order of the approximatingNURBS surface and the length of knot vectors which arestrongly dependent on the particular set of data points Con-sequently their optimal valuesmight be difficult to choose forend users thus preventing themethod for automatic human-independent reconstruction This limitation opens the doorfor future research in the area in order to develop efficientalgorithms for automatic determination of those optimal


Table 2 Comparison of the proposed method with other alternative methods based on neural networks for the surface reconstructionproblem

Author year andreference

Dataparameterization Method used Surface

fitting Method used NURBSsupported

Hoffmann and Varady(1998) [70] Kohonen neural network times times

Yu (1999) [49] Kohonen neural network Polygonal mesh (edge swap) times

Hoffmann (1999) [77]

Modified Kohonen neuralnetwork times times

Barhak and Fischer(2001) [71]

(i) Kohonen neuralnetwork

(ii) Partial differentialequations (PDE)

(i) Gradient descent algorithm(GDA)

(ii) Random surface errorcorrection (RSEC)

times

Iglesias et al (2004) [4] times

Tensor-product functionalnetwork times

Galvez et al (2007) [8] Genetic algorithms Functional network times

This method (2013) Kohonen neural network NURBS functional network

values Another important limitation of the method occurswhen the data points cannot be unambiguously projectedonto the base surface This problem can arise with closedsurfaces often represented in implicit form In those casesthe PCA method might fail to extract the real tendency ofdataWe are currently working on new strategies to overcomethis problem Future work also includes the extension ofthis approach to other kinds of approximating surfaces aswell as the possible application of this methodology to someinteresting industrial problems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper Any commercialidentity mentioned in this paper is cited solely for scientificpurposes

Acknowledgments

This research has been kindly supported by the ComputerScience National Program of the Spanish Ministry of Econ-omy and Competitiveness Project ref no TIN2012-30768Toho University (Funabashi Japan) and the University ofCantabria (Santander Spain) The authors are particularlygrateful to the Department of Information Science of TohoUniversity for all the facilities given to carry out this workSpecial thanks are owed to the editor and the four anonymousreviewers for their useful comments and suggestions thatallowed us to improve the final version of this paper

References

[1] T Varady R R Martin and J Cox ldquoReverse engineering ofgeometric modelsmdashan introductionrdquo CAD Computer AidedDesign vol 29 no 4 pp 255ndash268 1997

[2] T Varady and R Martin ldquoReverse engineeringrdquo in Handbookof Computer Aided Geometric Design G Farin J Hoschek andM Kim Eds pp 651ndash681 Elsevier Science 2002

[3] R E BarnhillGeometric Processing for Design andManufactur-ing SIAM Philadelphia Pa USA 1992

[4] A Iglesias G Echevarrıa and A Galvez ldquoFunctional networksfor B-spline surface reconstructionrdquo Future Generation Com-puter Systems vol 20 no 8 pp 1337ndash1353 2004

[5] A Iglesias andAGalvez ldquoA new artificial intelligence paradigmfor computer aided geometric designrdquo in Artificial Intelligenceand Symbolic Computation vol 1930 of Lectures Notes inArtificial Intelligence pp 200ndash213 2001

[6] N M Patrikalakis and T Maekawa Shape Interrogation forComputer Aided Design and Manufacturing Springer Heidel-berg Germany 2002

[7] H Pottmann S Leopoldseder M Hofer T Steiner and WWang ldquoIndustrial geometry recent advances and applicationsin CADrdquo CAD Computer Aided Design vol 37 no 7 pp 751ndash766 2005

[8] A Galvez A Iglesias A Cobo J Puig-Pey and J EspinolaldquoBezier curve and surface fitting of 3D point cloudsthrough genetic algorithms functional networks and least-squares approximationrdquo in Computational Science and ItsApplicationsmdashICCSA 2007 vol 4706 of Lectures Notes inComputer Science pp 680ndash693 2007

[9] G E Farin Curves and Surfaces for Computer-Aided GeometricDesign Morgan Kaufmann San Francisco Calif USA 5thedition 2001

[10] AGalvez andA Iglesias ldquoEfficient particle swarmoptimizationapproach for data fitting with free knot B-splinesrdquo CAD Com-puter Aided Design vol 43 no 12 pp 1683ndash1692 2011

[11] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters 1993

[12] J Ling and S Li ldquoFitting B-spline curves by least squaressupport vector machinesrdquo in Proceedings of the InternationalConference on Neural Networks amp Brain Proceedings (ICNNBrsquo05) pp 905ndash909 IEEE Press Beijing China October 2005

[13] T C M Lee ldquoOn algorithms for ordinary least squares regres-sion spline fitting a comparative studyrdquo Journal of StatisticalComputation and Simulation vol 72 no 8 pp 647ndash663 2002

[14] W Y Ma and J P Kruth ldquoParameterization of randomlymeasured points for least squares fitting of B-spline curves andsurfacesrdquo Computer-Aided Design vol 27 no 9 pp 663ndash6751995


[15] W P Wang H Pottmann and Y Liu ldquoFitting B-spline curvesto point clouds by curvature-based squared distance minimiza-tionrdquoACMTransactions on Graphics vol 25 no 2 pp 214ndash2382006

[16] H P Yang W P Wang and J G Sun ldquoControl point adjust-ment for B-spline curve approximationrdquo CAD Computer AidedDesign vol 36 no 7 pp 639ndash652 2004

[17] P Dierckx Curve and Surface Fitting with Splines OxfordUniversity Press Oxford UK 1993

[18] H Park and J-H Lee ldquoB-spline curve fitting based on adaptivecurve refinement using dominant pointsrdquoCADComputerAidedDesign vol 39 no 6 pp 439ndash451 2007

[19] L A Piegl andW Tiller ldquoLeast-squares B-spline curve approx-imation with arbitrary end derivativesrdquo Engineering with Com-puters vol 16 no 2 pp 109ndash116 2000

[20] D L B Jupp ldquoApproximation to data by splines with free knotsrdquoSIAM Journal on Numerical Analysis vol 15 no 2 pp 328ndash3431978

[21] J R Rice The Approximation of Functions vol 2 Addison-Wesley Reading Mass USA 1969

[22] J Hertz A Krogh and R G Palmer Introduction to the Theoryof Neural Computation Addison-Wesley Reading Mass USA1991

[23] M G Cox ldquoAlgorithms for spline curves and surfacesrdquo in Fun-damental Developments of Computer-Aided Geometric DesignL Piegl Ed pp 51ndash76 Academic Press London UK 1993

[24] R H Franke and L L Schumaker ldquoA bibliography ofmultivariate approximationrdquo in Topics in Multivariate Approx-imation C K Chui L L Schumaker and F I Utreras EdsAcademic Press New York NY USA 1986

[25] N R Draper and H Smith Applied Regression Analysis Wiley-Interscience 3rd edition 1998

[26] D R Forsey and R H Bartels ldquoSurface fitting with hierarchicalsplinesrdquo ACM Transactions on Graphics vol 14 pp 134ndash1611995

[27] H Pottmann S Leopoldseder and M Hofer ldquoApproximationwith active bspline curves and surfacesrdquo in Proceedings of the10th Pacific Conference on Computer Graphics and Applicationspp 8ndash25 IEEE Computer Society Press 2002

[28] M Eck and H Hoppe ldquoAutomatic reconstruction of B-splinesurfaces of arbitrary topological typerdquo in Proceedings of theComputer Graphics Conference (SIGGRAPH rsquo96) pp 325ndash334August 1996

[29] L Piegl and W Tiller The NURBS Book Springer BerlinGermany 1997

[30] M Prasad A Zisserman and A Fitzgibbon ldquoSingle viewreconstruction of curved surfacesrdquo in Proceedings of the IEEEComputer Society Conference on Computer Vision and PatternRecognition (CVPR rsquo06) vol 2 pp 1345ndash1354 IEEE ComputerSociety Press Los Alamitos Calif USA June 2006

[31] M Levoy K Pulli B Curless et al ldquoThe digital Michelangeloproject 3D scanning of large statuesrdquo in Proceedings of the27th Annual Conference on Computer Graphics and InteractiveTechniques (SIGGRAPH rsquo00) pp 131ndash144 New Orleans LaUSA July 2000

[32] M C Leu X Peng and W Zhang ldquoSurface reconstruction forinteractive modeling of freeform solids by virtual sculptingrdquoCIRP AnnalsmdashManufacturing Technology vol 54 no 1 pp 131ndash134 2005

[33] C V Alvino and A J Yezzi Jr ldquoTomographic reconstruction ofpiecewise smooth imagesrdquo in Proceedings of the IEEE Computer

Society Conference on Computer Vision and Pattern Recognition(CVPR rsquo04) vol 1 pp I576ndashI581 Los Alamitos Calif USA July2004

[34] J Rossignac and B Borrel ldquoMulti-resolution 3D approxima-tions for rendering complex scenesrdquo in Geometric Modeling inComputer Graphics pp 455ndash465 Springer 1993

[35] C L Bajaj F Bernardini and G Xu ldquoAutomatic reconstructionof surfaces and scalar fields from 3D scansrdquo in Proceedings ofthe 22nd Annual ACM Conference on Computer Graphics andInteractive Techniques pp 109ndash118 August 1995

[36] C L Bajaj E J Coyle and K-N Lin ldquoArbitrary topology shapereconstruction from planar cross sectionsrdquo Graphical Modelsand Image Processing vol 58 no 6 pp 524ndash543 1996

[37] M Jones and M Chen ldquoA new approach to the construction ofsurfaces from contour datardquo Computer Graphics Forum vol 13no 3 pp 75ndash84 1994

[38] D Meyers S Skinnwer and K Sloan ldquoSurfaces from contoursrdquoACM Transactions on Graphics vol 11 no 3 pp 228ndash258 1992

[39] H Park and K Kim ldquoSmooth surface approximation to serialcross-sectionsrdquoCADComputer Aided Design vol 28 no 12 pp995ndash1005 1996

[40] W J Gordon ldquoSpline-blended surface interpolation throughcurve networksrdquo Journal of Mathematics andMechanics vol 18no 10 pp 931ndash952 1969

[41] H Fuchs Z M Kedem and S P Uselton ldquoOptimal surfacereconstruction from planar contoursrdquo Communications of theACM vol 20 no 10 pp 693ndash702 1977

[42] T Maekawa and K H Ko ldquoSurface construction by fittingunorganized curvesrdquo Graphical Models vol 64 no 5 pp 316ndash332 2002

[43] V Savchenko A Pasko O Okunev and T Kunii ldquoFunctionrepresentation of solids reconstructed from scattered surfacepoints and contoursrdquo Computer Graphics Forum vol 14 no 4pp 181ndash188 1995

[44] G Echevarrıa A Iglesias andAGalvez ldquoExtending neural net-works for B-spline surface reconstructionrdquo in ComputationalSciencemdashICCS vol 2330 of Lectures Notes in Computer Sciencepp 305ndash314 2002

[45] P Gu and X Yan ldquoNeural network approach to the reconstruc-tion of freeform surfaces for reverse engineeringrdquo Computer-Aided Design vol 27 no 1 pp 59ndash64 1995

[46] B Guo ldquoSurface reconstruction from points to splinesrdquo CADComputer Aided Design vol 29 no 4 pp 269ndash277 1997

[47] H Hoppe T DeRose T Duchamp J McDonald and WStuetzle ldquoSurface reconstruction from unorganized pointsrdquoComputer Graphics (ACM) vol 26 no 2 pp 71ndash78 1992

[48] C T Lim G M Turkiyyah M A Ganter and D W StortildquoImplicit reconstruction of solids from cloud point setsrdquo inProceedings of the 3rd Symposium on Solid Modeling andApplications pp 393ndash402 May 1995

[49] Y Yu ldquoSurface reconstruction from unorganized points usingself-organizing neural networksrdquo in Proceedings of IEEE Visual-ization rsquo99 pp 61ndash64 1999

[50] M S Floater ldquoParametrization and smooth approximation ofsurface triangulationsrdquo Computer Aided Geometric Design vol14 no 3 pp 231ndash250 1997

[51] C Oblonsek and N Guid ldquoA fast surface-based procedurefor object reconstruction from 3D scattered pointsrdquo ComputerVision and ImageUnderstanding vol 69 no 2 pp 185ndash195 1998

[52] R M Bolle and B C Vemuri ldquoOn three-dimensional surfacereconstructionmethodsrdquo IEEETransactions on PatternAnalysisand Machine Intelligence vol 13 no 1 pp 1ndash13 1991


[53] F J M Schmitt B A Barsky and W-H Du ldquoAn adaptivesubdivision method for surface fitting from sampled datardquoComputer Graphics (ACM) vol 20 no 4 pp 179ndash188 1986

[54] T A Foley ldquoInterpolation of scattered data on a sphericaldomainrdquo in Algorithms for Approximation II J C Mason andM G Cox Eds pp 303ndash310 Chapman and Hall London UK1990

[55] S Sclaroff and A Pentland ldquoGeneralized implicit functions forcomputer graphicsrdquo ACM Siggraph Computer Graphics vol 25no 4 pp 247ndash250 1991

[56] V Pratt ldquoDirect least-squares fitting of algebraic surfacesrdquoComputer Graphics (ACM) vol 21 no 4 pp 145ndash152 1987

[57] A Galvez A Cobo J Puig-Pey and A Iglesias ldquoParticleswarm optimization for Bezier surface reconstructionrdquo inCom-putational SciencemdashICCS 2008 vol 5102 of Lectures Notes inComputer Science pp 116ndash125 2008

[58] A Galvez andA Iglesias ldquoParticle swarmoptimization for non-uniform rational B-spline surface reconstruction from clouds of3D data pointsrdquo Information Sciences vol 192 pp 174ndash192 2012

[59] A Galvez A Iglesias and J Puig-Pey ldquoIterative two-stepgenetic-algorithm-based method for efficient polynomial B-spline surface reconstructionrdquo Information Sciences vol 182 no1 pp 56ndash76 2012

[60] M Sarfraz and S A Raza ldquoCapturing outline of fonts usinggenetic algorithms and splinesrdquo in Proceedings of the 5thInternational Conference on Information Visualization (IVrsquo 01)pp 738ndash743 IEEE Computer Society Press 2001

[61] F Yoshimoto M Moriyama and T Harada ldquoAutomatic knotadjustment by a genetic algorithm for data fitting with asplinerdquo in Proceedings of the International Conference on ShapeModeling pp 162ndash169 IEEE Computer Society Press 1999

[62] F Yoshimoto T Harada and Y Yoshimoto ldquoData fitting witha spline using a real-coded genetic algorithmrdquo CAD ComputerAided Design vol 35 no 8 pp 751ndash760 2003

[63] A Galvez A Iglesias and A Andreina ldquoDiscrete Bezier curvefitting with artificial immune systemsrdquo in Intelligent ComputerGraphics 2012 vol 441 of Studies in Computational Intelligencepp 59ndash75 2013

[64] E Ulker and A Arslan ldquoAutomatic knot adjustment using anartificial immune system for B-spline curve approximationrdquoInformation Sciences vol 179 no 10 pp 1483ndash1494 2009

[65] X Zhao C Zhang B Yang and P Li ldquoAdaptive knot placementusing a GMM-based continuous optimization algorithm in B-spline curve approximationrdquo CAD Computer Aided Design vol43 no 6 pp 598ndash604 2011

[66] A Galvez and A Iglesias ldquoFirey algorithm for polynomialBezier surface parameterizationrdquo Journal of Applied Mathemat-ics vol 2013 Article ID 237984 9 pages 2013

[67] A Galvez and A Iglesias ldquoFirey algorithm for explicit B-spline curve fitting to data pointsrdquo Mathematical Problems inEngineering vol 2013 Article ID 528215 12 pages 2013

[68] A Galvez and A Iglesias ldquoA new iterative mutually-coupledhybrid GA-PSO approach for curve fitting in manufacturingrdquoApplied Soft Computing vol 13 no 3 pp 1491ndash1504 2013

[69] A Galvez and A Iglesias ldquoFrom nonlinear optimization toconvex optimization through firey algorithm and indirectapproach with applications to CADCAMrdquoThe ScientificWorldJournal vol 2013 Article ID 283919 10 pages 2013

[70] M Hoffmann and L Varady ldquoFree-form surfaces for scattereddata by neural networksrdquo Journal for Geometry and Graphicsvol 2 no 1 pp 1ndash6 1998

[71] J Barhak and A Fischer ldquoParameterization and reconstructionfrom 3D scattered points based on neural network and PDEtechniquesrdquo IEEE Transactions on Visualization and ComputerGraphics vol 7 no 1 pp 1ndash16 2001

[72] E Castillo ldquoFunctional networksrdquoNeural Processing Letters vol7 no 3 pp 151ndash159 1998

[73] E Castillo A Iglesias and R Ruiz-Cobo Functional Equationsin Applied Sciences Elsevier Science Amsterdam The Nether-lands 2005

[74] E Castillo and A Iglesias ldquoSome characterizations of familiesof surfaces using functional equationsrdquo ACM Transactions onGraphics vol 16 no 3 pp 296ndash318 1997

[75] T Kohonen Self-Organization and Associative Memory vol 8of Springer Series in Information Sciences Springer New YorkNY USA 3rd edition 1989

[76] T Kohonen Self-Organizing Maps vol 30 of Springer Series inInformation Sciences Springer Berlin Germany 3rd edition2001

[77] M Hoffmann ldquoModified Kohonen neural network for surfacereconstructionrdquo Publicationes Mathematicae vol 54 pp 857ndash864 1999

[78] WH Press B P Flannery S A Teukolsky andW T VetterlingNumerical Recipes Cambridge University Press CambridgeUK 2nd edition 1986

[79] J Puig-Pey A Galvez A Iglesias J Rodrıguez P Corcuera andF Gutierrez ldquoSome applications of scalar and vector fields togeometric processing of surfacesrdquo Computers amp Graphics vol29 no 5 pp 723ndash729 2005

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of


scanning and other digitizing devices and are thereforesubjected to measurement noise irregular sampling andother artifacts [6 7] Consequently a good fitting of datashould be generally based on approximation schemes ratherthan interpolation [1 13 14 17ndash19] Because this is the typicalcase in many real-world industrial problems in this paper wefocus on the approximation scheme to a given set of noisyirregularly sampled data points

Obtaining the best approximating surface in such cases ismuch more troublesome than it may seem at first sight Themain reasons are as follows

(i) The NURBS surfaces depend on many differentparameters (data parameters knots control pointsand weights) that are strongly interconnected witheach other leading to a strongly nonlinear continuousoptimization problem

(ii) It is also multivariate as it typically involves a largenumber of unknown variables for a large number ofdata points a case that happens very often in real-world examples

(iii) In addition it is also overdetermined because weexpect to obtain the approximating surfacewithmanyfewer parameters than the number of data points

(iv) Finally the problem is known to be multimodal thatis the least-squares objective function can exhibitmany local optima [20 21] meaning that the problemmight have several (global andor local) good solu-tions

In conclusion we have to solve a very difficult multi-modal multivariate high-dimensional continuous nonlinearoptimization problem A number of methods have beenproposed to solve this problem (see Section 2 for details)Among them those based on artificial intelligence techniqueshave received increasing attention during the last few yearsMost of such methods rely on the artificial neural networks(ANN) formalism [22] Since ANN methodology is actuallyinspired by the behavior of the human brain it is able toreproduce some of its most typical features such as the abilityto learn fromdataThis explainswhy they have been sowidelyapplied to data fitting problems

Although they are very popular the ANN are howeverlimited in many aspects A major drawback is their inabilityto reproduce mathematically the functional structure of agiven problem This limitation can be overcome with theuse of a new paradigm in artificial intelligence the so-called functional networks (FN) (see Section 4 for details)In short functional networks are a generalization of thestandard neural networks in which the scalar weights arereplaced by neural functions These neural functions canexhibit in general a multivariate character Furthermoredifferent neurons can be associated with neural functionsfrom different families of functions These FN features allowus to reproduce exactly the functional structure of theproblem by a careful choice of the functions involved whichcan hereby be associated with one or several neurons of thenetwork This procedure yields a functional structure that is

typically a replica of the underlying structure of the givenproblem

11 Aims and Structure of the Paper In this paper we proposea hybrid artificial intelligence approach to solve the surfacereconstruction problem Our approach is based on the com-bination of two powerful artificial intelligence paradigms onone hand we apply the popular Kohonen neural networkto address the data parameterization problem On the otherhand we take advantage of the remarkable properties men-tioned in previous paragraph by introducing a new functionalnetwork called NURBS functional network whose topologyis specially targeted to reproduce the functional structure ofthe NURBS surfaces These neural and functional networksare then applied iteratively for further surface refinementAs it will be shown later on the hybridization of thesetwo networks provides us with a powerful computationalapproach to solve the surface reconstruction problem Tocheck the performance of our approach it has been applied totwo illustrative examples Our experimental results show thatthe method performs very well being able to reconstruct theapproximating surface of the given set of data points with ahigh degree of accuracy

The structure of this paper is as follows in Section 2previous work regarding the surface reconstruction prob-lem is reported Then some basic concepts about NURBSsurfaces and the optimization problem to be solved aregiven in Section 3 Section 4 describes the fundamentals ofthe functional networks along with their main componentsand the differences between neural and functional networksThe proposed hybrid functional-neural method for surfacereconstruction with NURBS surfaces is described in detail inSection 5 Then some illustrative examples of its applicationare reported in Section 6 A comparison of our approachwith other ANN alternative methods is analyzed in detail inSection 7 The paper closes with the main conclusions of thiscontribution and our plans for future work in the field

2 Previous Work

Surface reconstruction has been a topic of increasing atten-tion from the scientific community during the last 20 yearswith outstanding applications in both theoretical and applieddomains Regarding the theoretical side it is a remarkablesubject in approximation theory [23 24] statistics [25]numerical analysis [26 27] geometric modeling [2 8 28]and computer-aided geometric design (CAGD) [5 29] Inaddition there is a bulk of applications in several fieldssuch as computer-aided manufacturing (CAM) [6 7] datavisualization [30] cultural heritage preservation [31] virtualreality [32] medical imaging [33] and computer animation[34] to mention just a few

In general surface reconstruction methods are classifiedin terms of the available input (2D slices isoparametriccurves clouds of points mixed information etc) Forinstance authors in [35ndash39] address the problem of obtaininga surface model from a set of given cross-sections a classicalproblem in medical science biomedical engineering and








119903minus1 119904119903 sub




1198731198941(119906) =

1 if 119904119894le 119906 lt 119904

119894+1

0 otherwise(1)

with 119894 = 0 119903 minus 1 and

119873119894119896(119906) =

119906 minus 119904119894

119904119894+119896minus1

minus 119904119894

119873119894119896minus1

(119906)

+

119904119894+119896

minus 119906

119904119894+119896

minus 119904119894+1

119873119894+1119896minus1

(119906)

(2)



1198941(119906) and zero


119894and 119904119894+1





119894119895119894=0119898119895=0119899


0 1199041 1199042 119904

119903minus1 119904119903 and T =

1199050 1199051 119905



S (119906 V) =sum

119898

119894=0sum

119899

119895=0119908119894119895P119894119895119873119894119896 (

119906)119873119895119897 (V)

sum

119898

119894=0sum

119899

119895=0119908119894119895119873119894119896(119906)119873119895119897(V)

(3)

where the 119873119894119896(119906)119894and 119873



119894119895119894119895


119894119895119894119895 Without







120572120573120572=1119872120573=1119873


119864 =

119872

sum

120572=1

119873

sum

120573=1

(Q120572120573

minus

sum

119898

119894=0sum

119899

119895=0119908119894119895P119894119895119873119894119896(119906120572)119873119895119897(V120573)

sum

119898

119894=0sum

119899

119895=0119908119894119895119873119894119896(119906120572)119873119895119897(V120573)

)

2

(4)



119864 =

119877

sum

120583=1

(Q120583minus

sum

119898

119894=0sum

119899

119895=0119908119894119895P119894119895119873119894119896(119906120583)119873119895119897(V120583)

sum

119898

119894=0sum

119899

119895=0119908119894119895119873119894119896(119906120583)119873119895119897(V120583)

)

2

(5)


Qk= M sdot Pk

(6)


119895119897(119906120572 V120573))

Totimes 119877119894119896(uT V

120573))

v]

T with u =

(1199061 119906

119872) 119877119894119895(119906 V) is given as

119877119894119895(119906 V) =

119908119894119895119873119894119896(119906)119873119895119897(V)

sum

119898

119894=0sum

119899

119895=0119908119894119895119873119894119896(119906)119873119895119897(V)

119894 = 0 119898 119895 = 0 119899

(7)





MTsdotQk

= MTsdotM sdot Pk

= Δ sdot Pk (8)





10038171003817100381710038171003817

MTsdotQk

minus Δ sdot Pk10038171003817100381710038171003817

2






119865 (119909 119910) = 119891

minus1[119891 (119909) + 119891 (119910)] (11)






F

I

I

F

F

F

x

y

z

u

(a)

f

fx

y

u

f(y)

f(x)

f(x) + f(y)

fminus1


+

(b)










sum119908119894119896119909119896 where 119909

1 1199092 119909





1 1198912 and 119891

3





5 Our Method



x1x4

x6

x5

x2

x3

w11

w34

w35

w12

w22

w23

f(sumw1ixi)

f(sumw2ixi)

Neural network

f(sumw3ixi)

(a)

x6

x4

x5

x1

x2

x3

f1(x1 x2)

f2(x2 x3)

f3(x4 x5)

Functional network

(b)






Kohonen neural

network







Yes

No Output


PCA












(4) Compute

119889119895= radic

119877

sum

119894=1

(Ψ119894(120575) minus 120590

119894119895(120575))

2

(12)


=

min119894=1119877





120588 (120574) =

119877

2119890

[sum120575

119894=1(110+11989410

4)]

(13)


10038161003816100381610038161003816

119894 minus119894

10038161003816100381610038161003816

+

10038161003816100381610038161003816

119895 minus119895

10038161003816100381610038161003816

lt 120588 (1 minus

120575

120579

) (14)


neighborhood as

120590119894119895(120575 + 1) = 120590

119894119895(120575) + 120578 (120575) (Ψ

119894(120575) minus 120590

119894119895(120575)) (15)

with

120578 (120575) = 120577120575

1

radic2120587119890

minus12057522 (16)


120577120575= 1 minus (1 minus 120577

0(1 minus

120575

120579

))

120590120575

(17)











119894119895and P






times times




w00 w01 w02 w0n

u v
















S(u )

+ +





120583




















(a) (b)


(a) (b)





















times








Acknowledgments


References


























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119903minus1 119904119903 sub




1198731198941(119906) =

1 if 119904119894le 119906 lt 119904

119894+1

0 otherwise(1)

with 119894 = 0 119903 minus 1 and

119873119894119896(119906) =

119906 minus 119904119894

119904119894+119896minus1

minus 119904119894

119873119894119896minus1

(119906)

+

119904119894+119896

minus 119906

119904119894+119896

minus 119904119894+1

119873119894+1119896minus1

(119906)

(2)



1198941(119906) and zero


119894and 119904119894+1





119894119895119894=0119898119895=0119899


0 1199041 1199042 119904

119903minus1 119904119903 and T =

1199050 1199051 119905



S (119906 V) =sum

119898

119894=0sum

119899

119895=0119908119894119895P119894119895119873119894119896 (

119906)119873119895119897 (V)

sum

119898

119894=0sum

119899

119895=0119908119894119895119873119894119896(119906)119873119895119897(V)

(3)

where the 119873119894119896(119906)119894and 119873



119894119895119894119895


119894119895119894119895 Without







120572120573120572=1119872120573=1119873


119864 =

119872

sum

120572=1

119873

sum

120573=1

(Q120572120573

minus

sum

119898

119894=0sum

119899

119895=0119908119894119895P119894119895119873119894119896(119906120572)119873119895119897(V120573)

sum

119898

119894=0sum

119899

119895=0119908119894119895119873119894119896(119906120572)119873119895119897(V120573)

)

2

(4)



119864 =

119877

sum

120583=1

(Q120583minus

sum

119898

119894=0sum

119899

119895=0119908119894119895P119894119895119873119894119896(119906120583)119873119895119897(V120583)

sum

119898

119894=0sum

119899

119895=0119908119894119895119873119894119896(119906120583)119873119895119897(V120583)

)

2

(5)


Qk= M sdot Pk

(6)


119895119897(119906120572 V120573))

Totimes 119877119894119896(uT V

120573))

v]

T with u =

(1199061 119906

119872) 119877119894119895(119906 V) is given as

119877119894119895(119906 V) =

119908119894119895119873119894119896(119906)119873119895119897(V)

sum

119898

119894=0sum

119899

119895=0119908119894119895119873119894119896(119906)119873119895119897(V)

119894 = 0 119898 119895 = 0 119899

(7)





MTsdotQk

= MTsdotM sdot Pk

= Δ sdot Pk (8)





10038171003817100381710038171003817

MTsdotQk

minus Δ sdot Pk10038171003817100381710038171003817

2






119865 (119909 119910) = 119891

minus1[119891 (119909) + 119891 (119910)] (11)






F

I

I

F

F

F

x

y

z

u

(a)

f

fx

y

u

f(y)

f(x)

f(x) + f(y)

fminus1


+

(b)










sum119908119894119896119909119896 where 119909

1 1199092 119909





1 1198912 and 119891

3





5 Our Method



x1x4

x6

x5

x2

x3

w11

w34

w35

w12

w22

w23

f(sumw1ixi)

f(sumw2ixi)

Neural network

f(sumw3ixi)

(a)

x6

x4

x5

x1

x2

x3

f1(x1 x2)

f2(x2 x3)

f3(x4 x5)

Functional network

(b)






Kohonen neural

network







Yes

No Output


PCA












(4) Compute

119889119895= radic

119877

sum

119894=1

(Ψ119894(120575) minus 120590

119894119895(120575))

2

(12)


=

min119894=1119877





120588 (120574) =

119877

2119890

[sum120575

119894=1(110+11989410

4)]

(13)


10038161003816100381610038161003816

119894 minus119894

10038161003816100381610038161003816

+

10038161003816100381610038161003816

119895 minus119895

10038161003816100381610038161003816

lt 120588 (1 minus

120575

120579

) (14)


neighborhood as

120590119894119895(120575 + 1) = 120590

119894119895(120575) + 120578 (120575) (Ψ

119894(120575) minus 120590

119894119895(120575)) (15)

with

120578 (120575) = 120577120575

1

radic2120587119890

minus12057522 (16)


120577120575= 1 minus (1 minus 120577

0(1 minus

120575

120579

))

120590120575

(17)











119894119895and P






times times




w00 w01 w02 w0n

u v
















S(u )

+ +





120583




















(a) (b)


(a) (b)





















times








Acknowledgments


References


























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













120572120573120572=1119872120573=1119873


119864 =

119872

sum

120572=1

119873

sum

120573=1

(Q120572120573

minus

sum

119898

119894=0sum

119899

119895=0119908119894119895P119894119895119873119894119896(119906120572)119873119895119897(V120573)

sum

119898

119894=0sum

119899

119895=0119908119894119895119873119894119896(119906120572)119873119895119897(V120573)

)

2

(4)



119864 =

119877

sum

120583=1

(Q120583minus

sum

119898

119894=0sum

119899

119895=0119908119894119895P119894119895119873119894119896(119906120583)119873119895119897(V120583)

sum

119898

119894=0sum

119899

119895=0119908119894119895119873119894119896(119906120583)119873119895119897(V120583)

)

2

(5)


Qk= M sdot Pk

(6)


119895119897(119906120572 V120573))

Totimes 119877119894119896(uT V

120573))

v]

T with u =

(1199061 119906

119872) 119877119894119895(119906 V) is given as

119877119894119895(119906 V) =

119908119894119895119873119894119896(119906)119873119895119897(V)

sum

119898

119894=0sum

119899

119895=0119908119894119895119873119894119896(119906)119873119895119897(V)

119894 = 0 119898 119895 = 0 119899

(7)





MTsdotQk

= MTsdotM sdot Pk

= Δ sdot Pk (8)





10038171003817100381710038171003817

MTsdotQk

minus Δ sdot Pk10038171003817100381710038171003817

2






119865 (119909 119910) = 119891

minus1[119891 (119909) + 119891 (119910)] (11)






F

I

I

F

F

F

x

y

z

u

(a)

f

fx

y

u

f(y)

f(x)

f(x) + f(y)

fminus1


+

(b)










sum119908119894119896119909119896 where 119909

1 1199092 119909





1 1198912 and 119891

3





5 Our Method



x1x4

x6

x5

x2

x3

w11

w34

w35

w12

w22

w23

f(sumw1ixi)

f(sumw2ixi)

Neural network

f(sumw3ixi)

(a)

x6

x4

x5

x1

x2

x3

f1(x1 x2)

f2(x2 x3)

f3(x4 x5)

Functional network

(b)






Kohonen neural

network







Yes

No Output


PCA












(4) Compute

119889119895= radic

119877

sum

119894=1

(Ψ119894(120575) minus 120590

119894119895(120575))

2

(12)


=

min119894=1119877





120588 (120574) =

119877

2119890

[sum120575

119894=1(110+11989410

4)]

(13)


10038161003816100381610038161003816

119894 minus119894

10038161003816100381610038161003816

+

10038161003816100381610038161003816

119895 minus119895

10038161003816100381610038161003816

lt 120588 (1 minus

120575

120579

) (14)


neighborhood as

120590119894119895(120575 + 1) = 120590

119894119895(120575) + 120578 (120575) (Ψ

119894(120575) minus 120590

119894119895(120575)) (15)

with

120578 (120575) = 120577120575

1

radic2120587119890

minus12057522 (16)


120577120575= 1 minus (1 minus 120577

0(1 minus

120575

120579

))

120590120575

(17)











119894119895and P






times times




w00 w01 w02 w0n

u v
















S(u )

+ +





120583




















(a) (b)


(a) (b)





















times








Acknowledgments


References


























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










F

I

I

F

F

F

x

y

z

u

(a)

f

fx

y

u

f(y)

f(x)

f(x) + f(y)

fminus1


+

(b)










sum119908119894119896119909119896 where 119909

1 1199092 119909





1 1198912 and 119891

3





5 Our Method



x1x4

x6

x5

x2

x3

w11

w34

w35

w12

w22

w23

f(sumw1ixi)

f(sumw2ixi)

Neural network

f(sumw3ixi)

(a)

x6

x4

x5

x1

x2

x3

f1(x1 x2)

f2(x2 x3)

f3(x4 x5)

Functional network

(b)






Kohonen neural

network







Yes

No Output


PCA












(4) Compute

119889119895= radic

119877

sum

119894=1

(Ψ119894(120575) minus 120590

119894119895(120575))

2

(12)


=

min119894=1119877





120588 (120574) =

119877

2119890

[sum120575

119894=1(110+11989410

4)]

(13)


10038161003816100381610038161003816

119894 minus119894

10038161003816100381610038161003816

+

10038161003816100381610038161003816

119895 minus119895

10038161003816100381610038161003816

lt 120588 (1 minus

120575

120579

) (14)


neighborhood as

120590119894119895(120575 + 1) = 120590

119894119895(120575) + 120578 (120575) (Ψ

119894(120575) minus 120590

119894119895(120575)) (15)

with

120578 (120575) = 120577120575

1

radic2120587119890

minus12057522 (16)


120577120575= 1 minus (1 minus 120577

0(1 minus

120575

120579

))

120590120575

(17)











119894119895and P






times times




w00 w01 w02 w0n

u v
















S(u )

+ +





120583




















(a) (b)


(a) (b)





















times








Acknowledgments


References


























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x1x4

x6

x5

x2

x3

w11

w34

w35

w12

w22

w23

f(sumw1ixi)

f(sumw2ixi)

Neural network

f(sumw3ixi)

(a)

x6

x4

x5

x1

x2

x3

f1(x1 x2)

f2(x2 x3)

f3(x4 x5)

Functional network

(b)






Kohonen neural

network







Yes

No Output


PCA












(4) Compute

119889119895= radic

119877

sum

119894=1

(Ψ119894(120575) minus 120590

119894119895(120575))

2

(12)


=

min119894=1119877





120588 (120574) =

119877

2119890

[sum120575

119894=1(110+11989410

4)]

(13)


10038161003816100381610038161003816

119894 minus119894

10038161003816100381610038161003816

+

10038161003816100381610038161003816

119895 minus119895

10038161003816100381610038161003816

lt 120588 (1 minus

120575

120579

) (14)


neighborhood as

120590119894119895(120575 + 1) = 120590

119894119895(120575) + 120578 (120575) (Ψ

119894(120575) minus 120590

119894119895(120575)) (15)

with

120578 (120575) = 120577120575

1

radic2120587119890

minus12057522 (16)


120577120575= 1 minus (1 minus 120577

0(1 minus

120575

120579

))

120590120575

(17)











119894119895and P






times times




w00 w01 w02 w0n

u v
















S(u )

+ +





120583




















(a) (b)


(a) (b)





















times








Acknowledgments


References


























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















(4) Compute

119889119895= radic

119877

sum

119894=1

(Ψ119894(120575) minus 120590

119894119895(120575))

2

(12)


=

min119894=1119877





120588 (120574) =

119877

2119890

[sum120575

119894=1(110+11989410

4)]

(13)


10038161003816100381610038161003816

119894 minus119894

10038161003816100381610038161003816

+

10038161003816100381610038161003816

119895 minus119895

10038161003816100381610038161003816

lt 120588 (1 minus

120575

120579

) (14)


neighborhood as

120590119894119895(120575 + 1) = 120590

119894119895(120575) + 120578 (120575) (Ψ

119894(120575) minus 120590

119894119895(120575)) (15)

with

120578 (120575) = 120577120575

1

radic2120587119890

minus12057522 (16)


120577120575= 1 minus (1 minus 120577

0(1 minus

120575

120579

))

120590120575

(17)











119894119895and P






times times




w00 w01 w02 w0n

u v
















S(u )

+ +





120583




















(a) (b)


(a) (b)





















times








Acknowledgments


References


























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










times times




w00 w01 w02 w0n

u v
















S(u )

+ +





120583




















(a) (b)


(a) (b)





















times








Acknowledgments


References


























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























(a) (b)


(a) (b)





















times








Acknowledgments


References


























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























times








Acknowledgments


References


























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article hybrid functional-neural approach for surface reconstruction · 2019. 7. 31. ·...

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