research article cuckoo search with lévy flights for ...research article cuckoo search with lévy...

Research ArticleCuckoo Search with Leacutevy Flights for Weighted Bayesian EnergyFunctional Optimization in Global-Support Curve Data Fitting

Akemi Gaacutelvez1 Andreacutes Iglesias12 and Luis Cabellos3

1 Department of Applied Mathematics and Computational Sciences ETSI Caminos Canales y PuertosUniversity of Cantabria Avenida de los Castros sn 39005 Santander Spain

2Department of Information Science Faculty of Sciences Toho University 2-2-1 Miyama Funabashi 274-8510 Japan3 Institute of Physics of Cantabria (IFCA) Avenida de los Castros sn 39005 Santander Spain

Correspondence should be addressed to Andres Iglesias iglesiasunicanes

Received 25 April 2014 Accepted 5 May 2014 Published 28 May 2014

Academic Editor Xin-She Yang

Copyright copy 2014 Akemi Galvez et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The problem of data fitting is very important in many theoretical and applied fields In this paper we consider the problem ofoptimizing a weighted Bayesian energy functional for data fitting by using global-support approximating curves By global-supportcurves we mean curves expressed as a linear combination of basis functions whose support is the whole domain of the problemas opposed to other common approaches in CADCAM and computer graphics driven by piecewise functions (such as B-splinesand NURBS) that provide local control of the shape of the curve Our method applies a powerful nature-inspired metaheuristicalgorithm called cuckoo search introduced recently to solve optimization problems A major advantage of this method is itssimplicity cuckoo search requires only two parameters many fewer than other metaheuristic approaches so the parameter tuningbecomes a very simple task The paper shows that this new approach can be successfully used to solve our optimization problemTo check the performance of our approach it has been applied to five illustrative examples of different types including open andclosed 2D and 3D curves that exhibit challenging features such as cusps and self-intersections Our results show that the methodperforms pretty well being able to solve our minimization problem in an astonishingly straightforward way

1 Introduction

The problem of data fitting is very important in manytheoretical and applied fields [1ndash4] For instance in computerdesign and manufacturing (CADCAM) data points areusually obtained from real measurements of an existinggeometric entity as it typically happens in the constructionof car bodies ship hulls airplane fuselage and other free-form objects [5ndash15] This problem also appears in the shoesindustry archeology (reconstruction of archeological assets)medicine (computed tomography) computer graphics andanimation and many other fields In all these cases theprimary goal is to convert the real data from a physical objectinto a fully usable digital model a process commonly calledreverse engineering This allows significant savings in termsof storage capacity and processing and manufacturing timeFurthermore the digital models are easier and cheaper to

modify than their real counterparts and are usually availableanytime and anywhere

Depending on the nature of these data points twodifferent approaches can be employed interpolation andapproximation In the former a parametric curve or surfaceis constrained to pass through all input data points Thisapproach is typically employed for sets of data points thatcome from smooth shapes and that are sufficiently accurateOn the contrary approximation does not require the fittingcurve or surface to pass through all input data points butjust close to them according to some prescribed distancecriteriaThe approximation scheme is particularly well suitedfor the cases of highly irregular sampling and when datapoints are not exact but subjected to measurement errorsIn real-world problems the data points are usually acquiredthrough laser scanning and other digitizing devices and aretherefore subjected to some measurement noise irregular

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 138760 11 pageshttpdxdoiorg1011552014138760

2 The Scientific World Journal

sampling and other artifacts [12 13] Consequently a goodfitting of data should be generally based on approximationschemes rather than on interpolation [16ndash20]

There are two key components for a good approximationof data points with curves a proper choice of the approx-imating function and a suitable parameter tuning Due totheir good mathematical properties regarding evaluationcontinuity and differentiability (among many others) theuse of polynomial functions (especially splines) is a classicalchoice for the approximation function [16 17 23ndash27] Ingeneral the approximating curves can be classified as global-support and local-support By global-support curves wemean curves expressed as a linear combination of basisfunctions whose support is the whole domain of the problemAs a consequence these curves exhibit a global control inthe sense that any modification of the shape of the curvein a particular location is propagated throughout the wholecurveThis is in clear contrast to the local-support approachesthat have become prevalent in CADCAM and computergraphics usually driven by piecewise functions (such as B-splines andNURBS) that provide local control of the shape ofthe curve [23 28] In this work we are particularly interestedto explore the performance of the global-support approachby using different global-support basis functions for ourapproximating curves

11 Main Contributions and Structure of the Paper In thispaper we consider the problem of optimizing a weightedBayesian energy functional for data fitting by using global-support approximating curves In particular our goal isto obtain the global-support approximating curve that fitsthe data points better while keeping the number of freeparameters of the model as low as possible To this aim weformulate this problem as a minimization problem by usinga weighted Bayesian energy functional for global-supportcurves This is one of the major contributions of this paperOur functional is comprised of two competing terms aimedat minimizing the fitting error between the original and thereconstructed data points while simultaneously minimizingthe degrees of freedom of the problem Furthermore thefunctional can be modified and extended to include variousadditional constraints such as the fairness and smoothnessconstraints typically required in many industrial operationsin computer-aided manufacturing such as CNC (computernumerically controlled) milling drilling and machining [45 12]

Unfortunately our formulation in previous paragraphleads to a nonlinear continuous optimization problem thatcannot be properly addressed by conventional mathematicaloptimization techniques To overcome this limitation in thispaper we apply a powerful nature-inspired metaheuristicalgorithm called cuckoo search introduced in 2009 by Yangand Deb to solve optimization problems [21] The algorithmis inspired by the obligate interspecific brood parasitismof some cuckoo species that lay their eggs in the nests ofhost birds of other species Since its inception the cuckoosearch (specially its variant that uses Levy flights) has beensuccessfully applied in several papers reported recently in

the literature to difficult optimization problems from differ-ent domains However to the best of our knowledge themethod has never been used so far in the context of geometricmodeling and data fitting This is also one of the majorcontributions of this paper

A critical problem when using metaheuristic approachesconcerns the parameter tuning which is well known tobe time-consuming and problem-dependent In this regarda major advantage of the cuckoo search with Levy flightsis its simplicity it only requires two parameters manyfewer than other metaheuristic approaches so the parametertuning becomes a very simple task The paper shows thatthis new approach can be successfully applied to solve ouroptimization problem To check the performance of ourapproach it has been applied to five illustrative examplesof different types including open and closed 2D and 3Dcurves that exhibit challenging features such as cusps andself-intersectionsOur results show that themethod performspretty well being able to solve our minimization problem inan astonishingly straightforward way

The structure of this paper is as follows in Section 2some previous work in the field is briefly reported ThenSection 3 introduces the basic concepts and definitions alongwith the description of the problem to be solved The fun-damentals and main features of the cuckoo search algorithmare discussed in Section 4 The proposed method for theoptimization of our weighted Bayesian energy functionalfor data fitting with global-support curves is explained inSection 5 Some other issues such as the parameter tuningand some implementation details are also reported in thatsection As the reader will see themethod requires aminimalnumber of control parameters As a consequence it is verysimple to understand easy to implement and can be appliedto a broad variety of global-support basis functions To checkthe performance of our approach it has been applied to fiveillustrative examples for the cases of open and closed 2D and3D curves exhibiting challenging features such as cusps andself-intersections as described in Section 6 The paper closeswith the main conclusions of this contribution and our plansfor future work in the field

2 Previous Works

The problem of curve data fitting has been the subjectof research for many years First approaches in the fieldwere mostly based on numerical procedures [1 29 30]More recent approaches in this line use error bounds [31]curvature-based squared distance minimization [26] ordominant points [18] A very interesting approach to thisproblem consists in exploiting minimization of the energyof the curve [32ndash36] This leads to different functionalsexpressing the conditions of the problem such as fair-ness smoothness and mixed conditions [37ndash40] Generallyresearch in this area is based on the use of nonlinear optimiza-tion techniques that minimize an energy functional (oftenbased on the variation of curvature and other high-ordergeometric constraints) Then the problem is formulatedas a multivariate nonlinear optimization problem in which

The Scientific World Journal 3

the desired formwill be the one that satisfies various geomet-ric constraints while minimizing (or maximizing) a measureof form quality A variation of this formulation consistsin optimizing an energy functional while simultaneouslyminimizing the number of free parameters of the problemand satisfying some additional constraints on the underlyingmodel function This is the approach we follow in this paper

Unfortunately the optimization problems given by thoseenergy functionals and their constraints are very difficultand cannot be generally solved by conventionalmathematicaloptimization techniquesOn the other hand some interestingresearch carried out during the last two decades has shownthat the application of artificial intelligence techniques canachieve remarkable results regarding such optimization prob-lems [6 8 10 11 14] Most of these methods rely on somekind of neural networks such as standard neural networks [8]andKohonenrsquos SOM(self-organizingmaps) nets [10] In someother cases the neural network approach is combined withpartial differential equations [41] or other approaches [42]The generalization of these methods to functional networksis also analyzed in [6 11 14] Other approaches are based onthe application of nature-inspired metaheuristic techniqueswhich have been intensively applied to solve difficult opti-mization problems that cannot be tackled through traditionaloptimization algorithms Examples include artificial immunesystems [43] bacterial foraging [44] honey bee algorithm[45] artificial bee colony [46] firefly algorithm [47 48] andbat algorithm [49 50] A previous paper in [51] describesthe application of genetic algorithms and functional networksyielding pretty good results Genetic algorithms have alsobeen applied to this problem in both the discrete version[52 53] and the continuous version [7 54] Other meta-heuristic approaches applied to this problem include theuse of the popular particle swarm optimization technique[9 24] artificial immune systems [55 56] firefly algorithm[57 58] estimation of distribution algorithms [59] memeticalgorithms [60] and hybrid techniques [61]

3 Mathematical Preliminaries

In this paper we assume that the solution to our fittingproblem is given by a model function Φ(120585) defined on afinite interval domain []

1 ]2] Note that in this paper vectors

are denoted in bold We also assume that Φ(120585) can bemathematically represented as a linear combination of the so-called blending functions

Φ (120585) =

120575

sum120572=1

Θ120572120595120572(120585) 120585 isin []

1 ]2] (1)

In this work the family of blending functions 120595120572(120585)120572in (1)

is assumed to be linearly independent and to form a basis ofthe vector space of functions of degree le120575 minus 1 on []

1 ]2]

In this paper we consider the case in which all functions120595120572(120585)120572have their support on the whole domain []

1 ]2]

Without loss of generality we can also assume that []1 ]2]

is the unit interval [0 1] In practical terms this means thatthe blending functions provide a global control of the shape ofthe approximating curve (these functions are usually referred

to as global-support functions) as opposed to the alternativecase of local control given by the piecewise representationthat is characteristic of popular curves such as B-splines andNURBS Typical examples of global-support basis functionsare

(1) the canonical polynomial basis 120595120572(120585) = 120585120572minus1

(2) the Bernstein basis 120595120572(120585) = ( 120575minus1

120572minus1) 120585120572minus1(1 minus 120585)

120575minus120572

Other examples include the Hermite polynomial basis thetrigonometric basis the hyperbolic basis the radial basis andthe polyharmonic basis

Let us suppose now that we are given a finite set of datapoints Δ

120573120573=1120577

in a119863-dimensional space (usually119863 = 2 or119863 = 3) Our goal is to obtain a global-support approximatingcurve that best fits these data points while keeping thenumber of degrees of freedom as low as possible This leadsto a difficult minimization problem involving two different(and competing) factors the fitting error at the data pointsand the number of free parameters of the model function Inthis paper we consider the RMSE (root mean square error)as the fitting error criterion The number of free parametersis computed by following a Bayesian approach (see [62] forfurther details) This is a very effective procedure to penalizefitting models with too many parameters thus preventingdata overfitting [63] Therefore our optimization problemconsists in minimizing the following weighted Bayesianenergy functional

L =120577

2log(

120577

sum120573=1

Ω120573[Δ120573minus

120575

sum120572=1

Θ120572120595120572(120588120573)]

2

)

+120577 sdot 120574

2(2120575 minus 1

2) log (120577)

(2)

where we need a parameter value 120588120573to be associated with

each data point Δ120573 Equation (2) is comprised of two terms

the first one computes the fitting error to the data pointswhile the second one plays the role of a penalty term in orderto reduce the degrees of freedom of the model The penaltyterm also includes a real positive multiplicative factor 120574 usedto modulate how much this term will affect the whole energyfunctional

This functional L can be modified or expanded toinclude any additional constrain in our model For instanceit is very common in many engineering domains suchas computer-aided ship-hull design car-body styling andturbine-blade design to request conditions such as fairnessor smoothness In our approach these conditions can readilybe imposed by adding different energy functionals adaptedto the particular needs Suppose that instead of reducingthe degrees of freedom of our problem the smoothnessof the fitting curve is required This condition is simply


incorporated to our model by replacing the penalty term in(2) by the strain energy functional as follows

L =120577

2log(

120577

sum120573=1

Ω120573[Δ120573minus

120575

sum120572=1

Θ120572120595120572(120588120573)]

2

)

+120577 sdot 120574

2(120582int

10038171003817100381710038171003817Φ10158401015840(120585)

10038171003817100381710038171003817

2

119889120585)

(3)

Considering the vectors Ξ120572

= (120595120572(1205881) 120595

120572(120588120577))119879

with 120572 = 1 120575 where (sdot)119879 means transposition Ξ =

(Ξ1 Ξ

120575) Δ = (Δ

1 Δ

120577) Ω = (Ω

1 Ω

120577) and Θ =

(Θ1 Θ

120575)119879 (2) can be written in matricial form as

L = Ω sdot Δ119879sdot Δ minusΩ sdotΘ

119879sdot Ξ119879sdot Δ minusΩ sdot Δ

119879sdot Ξ sdotΘ

+Ω sdotΘ119879sdot Ξ119879sdot Ξ sdotΘ

(4)

Minimization of L requires differentiating (4) with respecttoΘ and equating to zero to satisfy the first-order conditionsleading to the following system of equations (called thenormal equations)

Ξ119879sdot Ξ sdotΘ = Ξ

119879sdot Δ (5)

In general the blending functions 120595120572(120585)120572are nonlinear

in 120585 leading to a strongly nonlinear optimization problemwith a high number of unknowns for large sets of data pointsa case that happens very often in practice Our strategyfor solving the problem consists in applying the cuckoosearchmethod to determine suitable parameter values for theminimization of functional L according to (2) The processis performed iteratively for a given number of iterations Sucha number is another parameter of the method that has to becalculated in order to run the algorithmuntil the convergenceof the minimization of the error is achieved

4 The Cuckoo Search Algorithm

Cuckoo search (CS) is a nature-inspired population-basedmetaheuristic algorithm originally proposed by Yang andDeb in 2009 to solve optimization problems [21] Thealgorithm is inspired by the obligate interspecific broodparasitism of some cuckoo species that lay their eggs in thenests of host birds of other species with the aim of escapingfrom the parental investment in raising their offspring Thisstrategy is also useful to minimize the risk of egg loss to otherspecies as the cuckoos can distribute their eggs amongst anumber of different nests Of course sometimes it happensthat the host birds discover the alien eggs in their nestsIn such cases the host bird can take different responsiveactions varying from throwing such eggs away to simplyleaving the nest and build a new one elsewhere Howeverthe brood parasites have at their turn developed sophisticatedstrategies (such as shorter egg incubation periods rapidnestling growth and egg coloration or pattern mimickingtheir hosts) to ensure that the host birds will care for thenestlings of their parasites

This interesting and surprising breeding behavioral pat-tern is the metaphor of the cuckoo search metaheuristicapproach for solving optimization problems In the cuckoosearch algorithm the eggs in the nest are interpreted as a poolof candidate solutions of an optimization problem while thecuckoo egg represents a new coming solution The ultimategoal of themethod is to use these new (and potentially better)solutions associated with the parasitic cuckoo eggs to replacethe current solution associated with the eggs in the nest Thisreplacement carried out iteratively will eventually lead to avery good solution of the problem

In addition to this representation scheme the CS algo-rithm is also based on three idealized rules [21 22]

(1) Each cuckoo lays one egg at a time and dumps it in arandomly chosen nest

(2) The best nests with high quality of eggs (solutions)will be carried over to the next generations

(3) The number of available host nests is fixed and a hostcan discover an alien egg with a probability 119901

119886isin

[0 1] In this case the host bird can either throwthe egg away or abandon the nest so as to build acompletely new nest in a new location

For simplicity the third assumption can be approximatedby a fraction 119901

119886of the 119899 nests being replaced by new

nests (with new random solutions at new locations) For amaximization problem the quality or fitness of a solution cansimply be proportional to the objective function Howeverother (more sophisticated) expressions for the fitness func-tion can also be defined

Based on these three rules the basic steps of the CSalgorithm can be summarized as shown in the pseudocodereported in Algorithm 1 Basically the CS algorithm startswith an initial population of 119899 host nests and it is performediteratively In the original proposal the initial values of the 119895thcomponent of the 119894th nest are determined by the expression119909119895

119894(0) = rand sdot (up119895

119894minus low119895

119894) + low119895

119894 where up119895

119894and low119895

119894

represent the upper and lower bounds of that 119895th componentrespectively and rand represents a standard uniform randomnumber on the open interval (0 1) Note that this choiceensures that the initial values of the variables are within thesearch space domain These boundary conditions are alsocontrolled in each iteration step

For each iteration 119892 a cuckoo egg 119894 is selected randomlyand new solutions x

119894(119892 + 1) are generated by using the

Levy flight a kind of random walk in which the steps aredefined in terms of the step lengths which have a certainprobability distribution with the directions of the steps beingisotropic and random According to the original creators ofthemethod the strategy of using Levy flights is preferred overother simple random walks because it leads to better overallperformance of the CS The general equation for the Levyflight is given by

x119894(119892 + 1) = x

119894(119892) + 120572 oplus levy (120582) (6)

where 119892 indicates the number of the current generation and120572 gt 0 indicates the step size which should be related to


beginObjective function 119891(x) x = (119909

1 119909

119863)119879

Generate initial population of 119899 host nests x119894(119894 = 1 2 119899)

While (119905 lt MaxGeneration) or (stop criterion)Get a cuckoo (say 119894) randomly by Levy flightsEvaluate its fitness 119865

119894

Choose a nest among 119899 (say 119895) randomlyif (119865119894gt 119865119895)

Replace 119895 by the new solutionendA fraction (119901

119886) of worse nests are abandoned and new ones are built via Levy flights

Keep the best solutions (or nests with quality solutions)Rank the solutions and find the current best

end whilePostprocess results and visualization

end

Algorithm 1 Cuckoo search algorithm via Levy flights as originally proposed by Yang and Deb in [21 22]

the scale of the particular problem under study The symboloplus is used in (6) to indicate the entrywise multiplication Notethat (6) is essentially a Markov chain since next locationat generation 119892 + 1 only depends on the current locationat generation 119892 and a transition probability given by thefirst and second terms of (6) respectively This transitionprobability is modulated by the Levy distribution as

levy (120582) sim 119892minus120582 (1 lt 120582 le 3) (7)

which has an infinite variance with an infinite mean Herethe steps essentially form a random walk process with apower-law step-length distribution with a heavy tail Fromthe computational standpoint the generation of randomnumbers with Levy flights is comprised of two steps firstly arandom direction according to a uniform distribution is cho-sen then the generation of steps following the chosen Levydistribution is carried out The authors suggested using theso-called Mantegnarsquos algorithm for symmetric distributionswhere ldquosymmetricrdquo means that both positive and negativesteps are considered (see [64] for details) Their approachcomputes the factor

120601 = (Γ (1 + 120573) sdot sin ((120587 sdot 120573) 2)

Γ (((1 + 120573) 2) sdot 120573 sdot 2(120573minus1)2))

1120573

(8)

where Γ denotes the Gamma function and 120573 = 32 in theoriginal implementation by Yang and Deb [22] This factor isused in Mantegnarsquos algorithm to compute the step length 119904 as

120589 =119906

|V|1120573 (9)

where 119906 and V follow the normal distribution of zero meanand deviation 1205902

119906and 1205902V respectively where 120590119906 obeys the

Levy distribution given by (8) and 120590V = 1 Then the stepsize120578 is computed as

120578 = 001120589 (x minus xbest) (10)

where 120589 is computed according to (9) Finally x is modifiedas x larr x+120578 sdotΥ whereΥ is a random vector of the dimensionof the solution x and that follows the normal distribution119873(0 1)

The CS method then evaluates the fitness of the newsolution and compares it with the current one In case the newsolution brings better fitness it replaces the current one Onthe other hand a fraction of the worse nests (according to thefitness) are abandoned and replaced by new solutions so as toincrease the exploration of the search space looking for morepromising solutions The rate of replacement is given by theprobability 119901

119886 a parameter of the model that has to be tuned

for better performance Moreover for each iteration step allcurrent solutions are ranked according to their fitness and thebest solution reached so far is stored as the vector xbest (usedeg in (10))

This algorithm is applied in an iterative fashion until astopping criterion is met Common terminating criteria arethat a solution is found that satisfies a lower threshold valuethat a fixed number of generations have been reached or thatsuccessive iterations no longer produce better results

5 The Method

We have applied the cuckoo search algorithm discussed inprevious section to our optimization problem described inSection 3 The problem consists in minimizing the weightedBayesian energy functional given by (2) for a given familyof global-support blending functions To this aim we firstlyneed a suitable representation of the variables of the problemWe consider an initial population of 119899 nests representing thepotential solutions of the problem Each solution consists ofa real-valued vector of dimension 119863 sdot 120575 + 3120577 + 2 containingthe parameters 120588

120573 vector coefficients Θ

120572 and weights Ω

120573 120575

and 120574 The structure of this vector is also highly constrainedOn one hand the set of parameters 120588

120573120573is constrained to

lie within the unit interval [0 1] In computational termsthis means that different controls are to be set up in order


to check for this condition to hold On the other hand theordered structure of data points means that those parametersmust also be sorted Finally weights are assumed to be strictlypositive real numbers

Regarding the fitness function it is given by eitherthe weighted Bayesian energy functional in (2) or by theweighted strain energy functional in (3) where the formerpenalizes any unnecessarily large number of free parametersfor the model while the latter imposes additional constraintsregarding the smoothness of the fitting curve Note also thatthe strength of the functionals can be modulated by theparameter 120574 to satisfy additional constraints

51 Parameter Tuning A critical issue when working withmetaheuristic approaches concerns the choice of suitableparameter values for the method This issue is of paramountimportance since the proper choice of such values willlargely determine the good performance of the methodUnfortunately it is also a very difficult task On one handthe field still lacks sufficient theoretical results to answerthis question on a general basis On the other hand thechoice of parameter values is strongly problem-dependentmeaning that good parameter values for a particular problemmight be completely useless (even counterproductive) forany other problem These facts explain why the choiceof adequate parameter values is so troublesome and veryoften a bottleneck in the development and application ofmetaheuristic techniques

The previous limitations have been traditionally over-come by following different strategies Perhaps the mostcommon one is to obtain good parameter values empiricallyIn this approach several runs or executions of the methodare carried out for different parameter values and a statisticalanalysis is performed to derive the values leading to thebest performance However this approach is very time-consuming especially when different parameters influenceeach other This problem is aggravated when the meta-heuristic depends on many different parameters leading toan exponential growth in the number of executions Thecuckoo search method is particularly adequate in this regardbecause of its simplicity In contrast to other methods thattypically require a large number of parameters the CS onlyrequires two parameters namely the population size 119899 andthe probability 119901

119886 This makes the parameter tuning much

easier for CS than for other metaheuristic approachesSome previous works have addressed the issue of param-

eter tuning for CS They showed that the method isrelatively robust to the variation of parameters For instanceauthors in [21] tried different values for 119899 = 5 10 15 20 50

100 150 250 and 500 and 119901119886= 0 001 005 01 015 02

025 04 and 05 They obtained that the convergence rateof the method is not very sensitive to the parameters usedimplying that no fine adjustment is needed for the methodto perform well Our experimental results are in goodagreement with these empirical observations We performedseveral trials for the parameter values indicated above andfound that our results do not differ significantly in any caseWe noticed however that some parameter values are more

adequate in terms of the number of iterations required toreach convergence In this paper we set the parameters 119899 and119901119886to 100 and 025 respectively

52 Implementation Issues Regarding the implementationall computations in this paper have been performed on a26GHz Intel Core i7 processor with 8GB RAM The sourcecode has been implemented by the authors in the nativeprogramming language of the popular scientific programMATLAB version 2012a We remark that an implementationof the CS method has been described in [21] Similarlya vectorized implementation of CS in MATLAB is freelyavailable in [65] Our implementation is strongly based(although not exactly identical) on that efficient open-sourceversion of the CS

6 Experimental Results

We have applied the CS method described in previoussections to different examples of curve data fitting To keepthe paper in manageable size in this section we describe onlyfive of them corresponding to different families of global-support basis functions and also to open and closed 2Dand 3D curves In order to replicate the conditions of real-world applications we assume that our data are irregularlysampled and subjected to noise Consequently we consider anonuniform sampling of data in all our examples Data pointsare also perturbed by an additive Gaussian white noise ofsmall intensity given by a SNR (signal-to-noise ratio) of 60in all reported examples

First example corresponds to a set of 100noisy data pointsobtained by nonuniform sampling from the Agnesi curveThe curve is obtained by drawing a line 119874119861 from the originthrough the circle of radius 119903 and center (0 119903) and thenpicking the point with the 119910 coordinate of the intersectionwith the circle and the 119909 coordinate of the intersection ofthe extension of line 119874119861 with the line 119910 = 2119903 Then theyare fitted by using the Bernstein basis functions Our resultsare depicted in Figure 1(a) where the original data points aredisplayed as red emptied circles whereas the reconstructedpoints appear as blue plus symbols Note the good matchingbetween the original and the reconstructed data points Infact we got a fitness value of 198646 times 10

minus3 indicatingthat the reconstructed curve fits the noisy data points withhigh accuracy The average CPU time for this example is301563 seconds We also computed the absolute mean valueof the difference between the original and the reconstructeddata points for each coordinate and obtained good results(9569738 times 10

minus4 1776091 times 10minus3) This good performanceis also reflected in Figure 1(b) where the original data pointsand the reconstructed Bezier fitting curve are displayed asblack plus symbols and a blue solid line respectively

Second example corresponds to the Archimedean spiralcurve (also known as the arithmetic spiral curve) This curveis the locus of points corresponding to the locations overtime of a point moving away from a fixed point with aconstant speed along a line which rotates with constantangular velocity In this example we consider a set of 100


04

06

08

1

12

14

16

18

2

minus4 minus3 minus2 minus1 0 1 2 3 4

(a)

04

06

08

1

12

14

16

18

2


(b)

Figure 1 Application of the cuckoo search algorithm to the Agnesi curve (a) original data points (red emptied circles) and reconstructedpoints (in blue plus symbol) (b) data points (black plus symbol) and fitting curve (solid blue line)

minus5

minus4

minus3

minus2

minus1

0

1

2

minus3 minus2 minus1 0 1 2 3 4 5 6

(a)

2

minus5

minus4

minus3

minus2

minus1

0

1


(b)

Figure 2 Application of the cuckoo search algorithm to the Archimedean spiral curve (a) original data points (red emptied circles) andreconstructed points (in blue plus symbol) (b) data points (black plus symbol) and fitting curve (solid blue line)

noisy data points from such a curve that are subsequentlyfitted by using the canonical polynomial basis functions Ourresults for this example are depicted in Figure 2 We omitthe interpretation of this figure because it is similar to theprevious one Once again note the good matching betweenthe original and the reconstructed data points In this case weobtained a fitness value of 112398times10minus2 for these data pointswhile the absolute mean value of the difference between theoriginal and the reconstructed data points for each coordinateis (1137795 times 10minus2 6429596 times 10minus3) The average CPU timefor this example is 468752 seconds We conclude that the CSmethod is able to obtain a global-support curve that fits thedata points pretty well

Third example corresponds to a hypocycloid curve Thiscurve belongs to a set of a much larger family of curvescalled the roulettes Roughly speaking a roulette is a curvegenerated by tracing the path of a point attached to a curve

that is rolling upon another fixed curve without slippage Inprinciple they can be any two curves The particular caseof a hypocycloid corresponds to a roulette traced by a pointattached to a circle of radius 119903 rolling around the inside of afixed circle of radius 119877 where it is assumed that 119877 = 119896 sdot 119903If 119896 = 119877119903 is a rational number then the curve eventuallycloses on itself and has 119877 cusps (ie sharp corners wherethe curve is not differentiable) In this example we consider aset of 100 noisy data points from the hypocycloid curve with5 cusps They are subsequently fitted by using the Bernsteinbasis functions Figure 3 shows our results graphically In thiscase the best fitness value is 200706times10minus3 while the absolutemean value of the difference between the original and thereconstructed data points for each coordinate is (1661867 times10minus3 1521872times10minus3)The averageCPU time for this example

is 982813 seconds In this case the complex geometry ofthe curve involving several cusps and self-intersections leads


minus8

minus6

minus4

minus2

0

2

4

6

8

minus10 minus5 0 5 10

(a)

minus8

minus6

minus4

minus2

0

2

4

6

8


(b)

Figure 3 Application of the cuckoo search algorithm to the hypocycloid curve example (a) original data points (red emptied circles) andreconstructed points (in blue plus symbol) (b) data points (black plus symbol) and fitting curve (solid blue line)

to this relatively large CPU time in comparison with theprevious (much simpler) examples In fact this example isvery illustrative about the ability of the method to performwell even in case of nonsmooth self-intersecting curves

Fourth example corresponds to the so-called piriformcurve which can be defined procedurally in a rather complexway Once again we consider a set of 100 noisy data pointsfitted by using the Bernstein basis functions Our resultsare shown in Figure 4 The best fitness value in this case is117915times10

minus3 while the absolutemean value of the differencebetween the original and the reconstructed data points foreach coordinate is (864616 times 10minus4 5873391 times 10minus4) Theaverage CPU time for this example is 3276563 seconds Notethat this curve has a cusp in the leftmost part moreover thedata points tend to concentrate around the cuspmeaning thatthe data parameterization is far from uniform However themethod is still able to recover the shape of the curvewith greatdetail

The last example corresponds to a 3D closed curve calledEight Knot curve Two images of the curve from differentviewpoints are shown in Figure 5 The CS method is appliedto a set of 100 noisy data points for the Bernstein basisfunctions Our results are shown in Figure 6 The best fitnessvalue in this case is 3193634 times 10

minus2 while the absolutemean value of the difference between the original and thereconstructed data points for each coordinate is (27699870times10minus2 2863125 times 10minus2 13710703 times 10minus2) The average CPU

time for this example is 875938 seconds

7 Conclusions and Future Work

This paper addresses the problem of approximating a set ofdata points by using global-support curves while simulta-neously minimizing the degrees of freedom of the modelfunction and satisfying other additional constraints Thisproblem is formulated in terms of a weighted Bayesian

energy functional that encapsulates all these constraints intoa single mathematical expression In this way the originalproblem is converted into a continuous nonlinear multi-variate optimization problem which is solved by using ametaheuristic approach Our method is based on the cuckoosearch a powerful nature-inspired metaheuristic algorithmintroduced recently to solve optimization problems Cuckoosearch (especially its variant that uses Levy flights) has beensuccessfully applied to difficult optimization problems indifferent fields However to the best of our knowledge thisis the first paper applying the cuckoo search methodology inthe context of geometric modeling and data fitting

Our approach based on the cuckoo search method hasbeen tested on five illustrative examples of different typesincluding open and closed 2D and 3D curves Some examplesalso exhibit challenging features such as cusps and self-intersections They have been fitted by using two differentfamilies of global-support functions (Bernstein basis func-tions and the canonical polynomial basis) with satisfactoryresults in all cases The experimental results show that themethod performs pretty well being able to solve our difficultminimization problem in an astonishingly straightforwardway We conclude that this new approach can be successfullyapplied to solve our optimization problem

Amajor advantage of thismethod is its simplicity cuckoosearch requires only two parameters many fewer than othermetaheuristic approaches so the parameter tuning becomesa very simple taskThis simplicity is also reflected in the CPUruntime of our examples Even though we are dealing with aconstrained continuous multivariate nonlinear optimizationproblem andwith curves exhibiting challenging features suchas cusps and self-intersections a typical single executiontakes less than 10 seconds of CPU time for all the examplesreported in this paper In addition the method is simpleto understand easy to implement and does not require anyfurther pre-postprocessing


minus1

minus05

0

05

1

minus05 0 05 1 15 2 25

(a)

minus1

minus05

0

05

1

minus05 0 05 1 15 2 25

(b)

Figure 4 Application of the cuckoo search algorithm to the piriform curve example (a) original data points (red emptied circles) andreconstructed points (in blue plus symbol) (b) data points (black plus symbol) and fitting curve (solid blue line)

minus5

0

5

minus15minus10 minus5

0 510 15

20

minus10

0

10

(a)

010

20

05

10

0

5

minus5

minus5

minus10minus10

(b)

Figure 5 Two different viewpoints of the 3D Eight Knot curve

0 5 10 15 200

10

0

5

minus5

minus10minus15

minus10minus5

(a)

0 5 10 15 20 minus10minus5

0510

0

5

minus5

minus15minus10

minus5

(b)

Figure 6 Application of the cuckoo search algorithm to the 3D Eight Knot curve example (a) original data points (red emptied circles) andreconstructed points (in blue plus symbol) (b) data points (black plus symbol) and fitting curve (solid blue line)


In spite of these encouraging results further researchis still needed to determine the advantages and limitationsof the present method at full extent On the other handsome modifications of the original cuckoo search have beenclaimed to outperform the initial method on some bench-marks Our implementation has been designed according tothe specifications of the original method and we did not testany of its subsequent modifications yet We are currentlyinterested in exploring these issues as part of our future workThe hybridization of this approach with other competitivemethods for better performance is also part of our futurework

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 work

References

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

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

[3] J R RiceNumerical Methods Software and Analysis AcademicPress New York NY USA 2nd edition 1993

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

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

[6] G Echevarrıa A Iglesias andAGalvez ldquoExtending neural net-works for B-spline surface reconstructionrdquo in ComputationalSciencemdashICCS 2002 vol 2330 of Lectures Notes in ComputerScience pp 305ndash314 Springer Berlin Germany 2002

[7] A Galvez A Iglesias and J Puig-Pey ldquoIterative two-stepgenetic-algorithm-based method for efficient polynomialB-spline surface reconstructionrdquo Information Sciences vol 182no 1 pp 56ndash76 2012

[8] P Gu and X Yan ldquoNeural network approach to thereconstruction of freeform surfaces for reverse engineeringrdquoComputer-Aided Design vol 27 no 1 pp 59ndash64 1995

[9] 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

[10] M Hoffmann ldquoNumerical control of Kohonen neural networkfor scattered data approximationrdquo Numerical Algorithms vol39 no 1ndash3 pp 175ndash186 2005

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

[12] N M Patrikalakis and T Maekawa Shape Interrogationfor Computer Aided Design and Manufacturing SpringerHeidelberg Germany 2002

[13] H Pottmann S Leopoldseder M Hofer T Steiner and WWang ldquoIndustrial geometry recent advances and applicationsin CADrdquo Computer Aided Design vol 37 no 7 pp 751ndash7662005

[14] A Iglesias andAGalvez ldquoA new artificial intelligence paradigmfor computer aided geometric designrdquo in Artificial Intelligenceand Symbolic Computation vol 1930 of Lectures Notes in Artifi-cial Intelligence pp 200ndash213 Springer Berlin Germany 2001

[15] T Varady and R Martin ldquoReverse engineeringrdquo in Handbookof Computer Aided Geometric Design G Farin J Hoschek andM Kim Eds Elsevier Science Amsterdam The Netherlands2002

[16] 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

[17] W Y Ma and J P Kruth ldquoParameterization of randomly mea-sured points for least squares fitting of B-spline curves and sur-facesrdquo Computer-Aided Design vol 27 no 9 pp 663ndash675 1995

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

[19] L A Piegl and W Tiller ldquoLeast-squares B-spline curveapproximation with arbitrary end derivativesrdquo Engineeringwith Computers vol 16 no 2 pp 109ndash116 2000

[20] T Varady R R Martin and J Cox ldquoReverse engineering ofgeometric modelsmdashan introductionrdquo Computer Aided Designvol 29 no 4 pp 255ndash268 1997

[21] X Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NABIC rsquo09) pp 210ndash214 IEEECoimbatore India December 2009

[22] X S Yang and S Deb ldquoEngineering optimisation by cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 4 pp 330ndash343 2010

[23] G Farin Curves and Surfaces for CAGD Morgan KaufmannSan Francisco Calif USA 5th edition 2002

[24] AGalvez andA Iglesias ldquoEfficient particle swarmoptimizationapproach for data fitting with free knot B-splinesrdquo ComputerAided Design vol 43 no 12 pp 1683ndash1692 2011

[25] L Jing and L Sun ldquoFitting B-spline curves by least squaressupport vector machinesrdquo in Proceedings of the InternationalConference on Neural Networks and Brain (ICNNB rsquo05) vol 2pp 905ndash909 IEEE Press Beijing China October 2005

[26] W P Wang H Pottmann and Y Liu ldquoFitting B-splinecurves to point clouds by curvature-based squared distanceminimizationrdquo ACM Transactions on Graphics vol 25 no 2pp 214ndash238 2006

[27] H P YangW PWang and J G Sun ldquoControl point adjustmentfor B-spline curve approximationrdquo Computer Aided Designvol 36 no 7 pp 639ndash652 2004

[28] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Wellesley Mass USA 1993

[29] D L B Jupp ldquoApproximation to data by splines with freeknotsrdquo SIAM Journal of Numerical Analysis vol 15 no 2 pp328ndash343 1978


[30] M J D Powell ldquoCurve fitting by splines in one variablerdquo inNumerical Approximation to Functions and Data J G HayesEd Athlone Press London UK 1970

[31] H Park ldquoAn error-bounded approximate method for repre-senting planar curves in B-splinesrdquo Computer Aided GeometricDesign vol 21 no 5 pp 479ndash497 2004

[32] Y J Ahn C Hoffmann and P Rosen ldquoGeometric constraintson quadratic Bezier curves using minimal length and energyrdquoJournal of Computational and Applied Mathematics vol 255pp 887ndash897 2014

[33] L Fang and D C Gossard ldquoMultidimensional curve fittingto unorganized data points by nonlinear minimizationrdquoComputer-Aided Design vol 27 no 1 pp 48ndash58 1995

[34] H Martinsson F Gaspard A Bartoli and J M LavestldquoEnergy-based reconstruction of 3D curves for quality controlrdquoin Energy Minimization Methods in Computer Vision andPattern Recognition vol 4679 of Lecture Notes in ComputerScience pp 414ndash428 Springer Berlin Germany 2007

[35] T I Vassilev ldquoFair interpolation and approximation of B-splinesby energy minimization and points insertionrdquo Computer-AidedDesign vol 28 no 9 pp 753ndash760 1996

[36] C Zhang P Zhang and F Cheng ldquoFairing spline curves andsurfaces by minimizing energyrdquo Computer Aided Design vol33 no 13 pp 913ndash923 2001

[37] G Brunnett and J Kiefer ldquoInterpolation with minimal-energysplinesrdquo Computer-Aided Design vol 26 no 2 pp 137ndash1441994

[38] H P Moreton and C H Sequin ldquoFunctional optimizationfor fair surface designrdquo Computer Graphics vol 26 no 2 pp167ndash176 1992

[39] E Sarioz ldquoAn optimization approach for fairing of ship hullformsrdquo Ocean Engineering vol 33 no 16 pp 2105ndash2118 2006

[40] R C Veltkamp and W Wesselink ldquoModeling 3D curves ofminimal energyrdquo Computer Graphics Forum vol 14 no 3 pp97ndash110 1995

[41] 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

[42] A Iglesias and A Galvez ldquoHybrid functional-neural approachfor surface reconstructionrdquo Mathematical Problems inEngineering vol 2014 Article ID 351648 13 pages 2014

[43] J D Farmer N H Packard and A S Perelson ldquoThe immunesystem adaptation and machine learningrdquo Physica DNonlinear Phenomena vol 22 no 1ndash3 pp 187ndash204 1986

[44] KM Passino ldquoBiomimicry of bacterial foraging for distributedoptimization and controlrdquo IEEE Control Systems Magazine vol22 no 3 pp 52ndash67 2002

[45] S Nakrani and C Tovey ldquoOn honey bees and dynamic serverallocation in internet hosting centersrdquo Adaptive Behavior vol12 no 3-4 pp 223ndash240 2004

[46] D Karaboga and B Basturk ldquoA powerful and efficientalgorithm for numerical function optimization artificial beecolony (ABC) algorithmrdquo Journal of Global Optimization vol39 no 3 pp 459ndash471 2007

[47] X S Yang ldquoFirefly algorithms for multimodal optimizationrdquo inStochastic Algorithms Foundations and Applications vol 5792of Lecture Notes in Computer Science pp 169ndash178 SpringerBerlin Germany 2009

[48] X S Yang ldquoFirefly algorithm stochastic test functions anddesign optimizationrdquo International Journal of Bio-InspiredComputation vol 2 no 2 pp 78ndash84 2010

[49] X S Yang ldquoA new metaheuristic Bat-inspired Algorithmrdquo inNature Inspired Cooperative Strategies for Optimization (NICSO2010) vol 284 of Studies in Computational Intelligence pp65ndash74 Springer Berlin Germany 2010

[50] X S Yang ldquoBat algorithm for multi-objective optimisationrdquoInternational Journal of Bio-Inspired Computation vol 3 no 5pp 267ndash274 2011

[51] A Galvez A Iglesias A Cobo J Puig-Pey and J EspinolaldquoBezier curve and surface fitting of 3D point clouds throughgenetic algorithms functional networks and least-squaresapproximationrdquo in Computational Science and Its Applica-tionsmdashICCSA 2007 vol 4706 of Lecture Notes in ComputerScience pp 680ndash693 Springer Berlin Germany 2007

[52] M Sarfraz and S A Raza ldquoCapturing outline of fonts usinggenetic algorithms and splinesrdquo in Proceedings of the 5thInternational Conference on Information Visualisation (IV rsquo01)pp 738ndash743 IEEE Computer Society Press London UK 2001

[53] F Yoshimoto M Moriyama and T Harada ldquoAutomaticknot adjustment by a genetic algorithm for data fitting witha splinerdquo in Proceedings of the International Conference onShape Modeling and Applications pp 162ndash169 IEEE ComputerSociety Press 1999

[54] F Yoshimoto T Harada and Y Yoshimoto ldquoData fitting witha spline using a real-coded genetic algorithmrdquo Computer AidedDesign vol 35 no 8 pp 751ndash760 2003

[55] A Galvez and A Iglesias ldquoFirefly algorithm for polynomialBezier surface parameterizationrdquo Journal of AppliedMathematics vol 2013 Article ID 237984 9 pages 2013

[56] 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

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

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

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

[60] A Galvez and A Iglesias ldquoNew memetic self-adaptive fireflyalgorithm for continuous optimizationrdquo International Journalof Bio-Inspired Computation In press

[61] A Galvez and A Iglesias ldquoA new iterative mutually coupledhybrid GA-PSO approach for curve fitting in manufacturingrdquoApplied Soft Computing Journal vol 13 no 3 pp 1491ndash15042013

[62] G E Schwarz ldquoEstimating the dimension of a modelrdquo Annalsof Statistics vol 6 no 2 pp 461ndash464 1978

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

[64] X S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress Frome UK 2nd edition 2010

[65] ldquoVectorized cuckoo search implementation in Matlab freelyrdquohttpwwwmathworkscommatlabcentralfileexchange29809-cuckoo-search-csalgorithm

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



sampling and other artifacts [12 13] Consequently a goodfitting of data should be generally based on approximationschemes rather than on interpolation [16ndash20]

There are two key components for a good approximationof data points with curves a proper choice of the approx-imating function and a suitable parameter tuning Due totheir good mathematical properties regarding evaluationcontinuity and differentiability (among many others) theuse of polynomial functions (especially splines) is a classicalchoice for the approximation function [16 17 23ndash27] Ingeneral the approximating curves can be classified as global-support and local-support By global-support curves wemean curves expressed as a linear combination of basisfunctions whose support is the whole domain of the problemAs a consequence these curves exhibit a global control inthe sense that any modification of the shape of the curvein a particular location is propagated throughout the wholecurveThis is in clear contrast to the local-support approachesthat have become prevalent in CADCAM and computergraphics usually driven by piecewise functions (such as B-splines andNURBS) that provide local control of the shape ofthe curve [23 28] In this work we are particularly interestedto explore the performance of the global-support approachby using different global-support basis functions for ourapproximating curves

11 Main Contributions and Structure of the Paper In thispaper we consider the problem of optimizing a weightedBayesian energy functional for data fitting by using global-support approximating curves In particular our goal isto obtain the global-support approximating curve that fitsthe data points better while keeping the number of freeparameters of the model as low as possible To this aim weformulate this problem as a minimization problem by usinga weighted Bayesian energy functional for global-supportcurves This is one of the major contributions of this paperOur functional is comprised of two competing terms aimedat minimizing the fitting error between the original and thereconstructed data points while simultaneously minimizingthe degrees of freedom of the problem Furthermore thefunctional can be modified and extended to include variousadditional constraints such as the fairness and smoothnessconstraints typically required in many industrial operationsin computer-aided manufacturing such as CNC (computernumerically controlled) milling drilling and machining [45 12]

Unfortunately our formulation in previous paragraphleads to a nonlinear continuous optimization problem thatcannot be properly addressed by conventional mathematicaloptimization techniques To overcome this limitation in thispaper we apply a powerful nature-inspired metaheuristicalgorithm called cuckoo search introduced in 2009 by Yangand Deb to solve optimization problems [21] The algorithmis inspired by the obligate interspecific brood parasitismof some cuckoo species that lay their eggs in the nests ofhost birds of other species Since its inception the cuckoosearch (specially its variant that uses Levy flights) has beensuccessfully applied in several papers reported recently in

the literature to difficult optimization problems from differ-ent domains However to the best of our knowledge themethod has never been used so far in the context of geometricmodeling and data fitting This is also one of the majorcontributions of this paper

A critical problem when using metaheuristic approachesconcerns the parameter tuning which is well known tobe time-consuming and problem-dependent In this regarda major advantage of the cuckoo search with Levy flightsis its simplicity it only requires two parameters manyfewer than other metaheuristic approaches so the parametertuning becomes a very simple task The paper shows thatthis new approach can be successfully applied to solve ouroptimization problem To check the performance of ourapproach it has been applied to five illustrative examplesof different types including open and closed 2D and 3Dcurves that exhibit challenging features such as cusps andself-intersectionsOur results show that themethod performspretty well being able to solve our minimization problem inan astonishingly straightforward way

The structure of this paper is as follows in Section 2some previous work in the field is briefly reported ThenSection 3 introduces the basic concepts and definitions alongwith the description of the problem to be solved The fun-damentals and main features of the cuckoo search algorithmare discussed in Section 4 The proposed method for theoptimization of our weighted Bayesian energy functionalfor data fitting with global-support curves is explained inSection 5 Some other issues such as the parameter tuningand some implementation details are also reported in thatsection As the reader will see themethod requires aminimalnumber of control parameters As a consequence it is verysimple to understand easy to implement and can be appliedto a broad variety of global-support basis functions To checkthe performance of our approach it has been applied to fiveillustrative examples for the cases of open and closed 2D and3D curves exhibiting challenging features such as cusps andself-intersections as described in Section 6 The paper closeswith the main conclusions of this contribution and our plansfor future work in the field

2 Previous Works

The problem of curve data fitting has been the subjectof research for many years First approaches in the fieldwere mostly based on numerical procedures [1 29 30]More recent approaches in this line use error bounds [31]curvature-based squared distance minimization [26] ordominant points [18] A very interesting approach to thisproblem consists in exploiting minimization of the energyof the curve [32ndash36] This leads to different functionalsexpressing the conditions of the problem such as fair-ness smoothness and mixed conditions [37ndash40] Generallyresearch in this area is based on the use of nonlinear optimiza-tion techniques that minimize an energy functional (oftenbased on the variation of curvature and other high-ordergeometric constraints) Then the problem is formulatedas a multivariate nonlinear optimization problem in which








Φ (120585) =

120575

sum120572=1

Θ120572120595120572(120585) 120585 isin []

1 ]2] (1)



1 ]2]


1 ]2]







120575minus120572



120573120573=1120577


L =120577

2log(

120577

sum120573=1

Ω120573[Δ120573minus

120575

sum120572=1

Θ120572120595120572(120588120573)]

2

)

+120577 sdot 120574

2(2120575 minus 1

2) log (120577)

(2)







L =120577

2log(

120577

sum120573=1

Ω120573[Δ120573minus

120575

sum120572=1

Θ120572120595120572(120588120573)]

2

)

+120577 sdot 120574

2(120582int

10038171003817100381710038171003817Φ10158401015840(120585)

10038171003817100381710038171003817

2

119889120585)

(3)


= (120595120572(1205881) 120595

120572(120588120577))119879


(Ξ1 Ξ

120575) Δ = (Δ

1 Δ

120577) Ω = (Ω

1 Ω

120577) and Θ =

(Θ1 Θ






(4)



119879sdot Δ (5)










119886isin






119894(0) = rand sdot (up119895

119894minus low119895

119894) + low119895

119894 where up119895

119894and low119895

119894





x119894(119892 + 1) = x

119894(119892) + 120572 oplus levy (120582) (6)




1 119909

119863)119879



119894






end





120601 = (Γ (1 + 120573) sdot sin ((120587 sdot 120573) 2)

Γ (((1 + 120573) 2) sdot 120573 sdot 2(120573minus1)2))

1120573

(8)


120589 =119906

|V|1120573 (9)




120578 = 001120589 (x minus xbest) (10)






5 The Method




120573 120575












100 150 250 and 500 and 119901119886= 0 001 005 01 015 02











04

06

08

1

12

14

16

18

2


(a)

04

06

08

1

12

14

16

18

2


(b)


minus5

minus4

minus3

minus2

minus1

0

1

2


(a)

2

minus5

minus4

minus3

minus2

minus1

0

1


(b)







minus8

minus6

minus4

minus2

0

2

4

6

8


(a)

minus8

minus6

minus4

minus2

0

2

4

6

8


(b)














minus1

minus05

0

05

1

minus05 0 05 1 15 2 25

(a)

minus1

minus05

0

05

1

minus05 0 05 1 15 2 25

(b)


minus5

0

5


0 510 15

20

minus10

0

10

(a)

010

20

05

10

0

5

minus5

minus5

minus10minus10

(b)


0 5 10 15 200

10

0

5

minus5

minus10minus15

minus10minus5

(a)

0 5 10 15 20 minus10minus5

0510

0

5

minus5

minus15minus10

minus5

(b)






Acknowledgments


References










































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Φ (120585) =

120575

sum120572=1

Θ120572120595120572(120585) 120585 isin []

1 ]2] (1)



1 ]2]


1 ]2]







120575minus120572



120573120573=1120577


L =120577

2log(

120577

sum120573=1

Ω120573[Δ120573minus

120575

sum120572=1

Θ120572120595120572(120588120573)]

2

)

+120577 sdot 120574

2(2120575 minus 1

2) log (120577)

(2)







L =120577

2log(

120577

sum120573=1

Ω120573[Δ120573minus

120575

sum120572=1

Θ120572120595120572(120588120573)]

2

)

+120577 sdot 120574

2(120582int

10038171003817100381710038171003817Φ10158401015840(120585)

10038171003817100381710038171003817

2

119889120585)

(3)


= (120595120572(1205881) 120595

120572(120588120577))119879


(Ξ1 Ξ

120575) Δ = (Δ

1 Δ

120577) Ω = (Ω

1 Ω

120577) and Θ =

(Θ1 Θ






(4)



119879sdot Δ (5)










119886isin






119894(0) = rand sdot (up119895

119894minus low119895

119894) + low119895

119894 where up119895

119894and low119895

119894





x119894(119892 + 1) = x

119894(119892) + 120572 oplus levy (120582) (6)




1 119909

119863)119879



119894






end





120601 = (Γ (1 + 120573) sdot sin ((120587 sdot 120573) 2)

Γ (((1 + 120573) 2) sdot 120573 sdot 2(120573minus1)2))

1120573

(8)


120589 =119906

|V|1120573 (9)




120578 = 001120589 (x minus xbest) (10)






5 The Method




120573 120575












100 150 250 and 500 and 119901119886= 0 001 005 01 015 02











04

06

08

1

12

14

16

18

2


(a)

04

06

08

1

12

14

16

18

2


(b)


minus5

minus4

minus3

minus2

minus1

0

1

2


(a)

2

minus5

minus4

minus3

minus2

minus1

0

1


(b)







minus8

minus6

minus4

minus2

0

2

4

6

8


(a)

minus8

minus6

minus4

minus2

0

2

4

6

8


(b)














minus1

minus05

0

05

1

minus05 0 05 1 15 2 25

(a)

minus1

minus05

0

05

1

minus05 0 05 1 15 2 25

(b)


minus5

0

5


0 510 15

20

minus10

0

10

(a)

010

20

05

10

0

5

minus5

minus5

minus10minus10

(b)


0 5 10 15 200

10

0

5

minus5

minus10minus15

minus10minus5

(a)

0 5 10 15 20 minus10minus5

0510

0

5

minus5

minus15minus10

minus5

(b)






Acknowledgments


References










































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





L =120577

2log(

120577

sum120573=1

Ω120573[Δ120573minus

120575

sum120572=1

Θ120572120595120572(120588120573)]

2

)

+120577 sdot 120574

2(120582int

10038171003817100381710038171003817Φ10158401015840(120585)

10038171003817100381710038171003817

2

119889120585)

(3)


= (120595120572(1205881) 120595

120572(120588120577))119879


(Ξ1 Ξ

120575) Δ = (Δ

1 Δ

120577) Ω = (Ω

1 Ω

120577) and Θ =

(Θ1 Θ






(4)



119879sdot Δ (5)










119886isin






119894(0) = rand sdot (up119895

119894minus low119895

119894) + low119895

119894 where up119895

119894and low119895

119894





x119894(119892 + 1) = x

119894(119892) + 120572 oplus levy (120582) (6)




1 119909

119863)119879



119894






end





120601 = (Γ (1 + 120573) sdot sin ((120587 sdot 120573) 2)

Γ (((1 + 120573) 2) sdot 120573 sdot 2(120573minus1)2))

1120573

(8)


120589 =119906

|V|1120573 (9)




120578 = 001120589 (x minus xbest) (10)






5 The Method




120573 120575












100 150 250 and 500 and 119901119886= 0 001 005 01 015 02











04

06

08

1

12

14

16

18

2


(a)

04

06

08

1

12

14

16

18

2


(b)


minus5

minus4

minus3

minus2

minus1

0

1

2


(a)

2

minus5

minus4

minus3

minus2

minus1

0

1


(b)







minus8

minus6

minus4

minus2

0

2

4

6

8


(a)

minus8

minus6

minus4

minus2

0

2

4

6

8


(b)














minus1

minus05

0

05

1

minus05 0 05 1 15 2 25

(a)

minus1

minus05

0

05

1

minus05 0 05 1 15 2 25

(b)


minus5

0

5


0 510 15

20

minus10

0

10

(a)

010

20

05

10

0

5

minus5

minus5

minus10minus10

(b)


0 5 10 15 200

10

0

5

minus5

minus10minus15

minus10minus5

(a)

0 5 10 15 20 minus10minus5

0510

0

5

minus5

minus15minus10

minus5

(b)






Acknowledgments


References










































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





1 119909

119863)119879



119894






end





120601 = (Γ (1 + 120573) sdot sin ((120587 sdot 120573) 2)

Γ (((1 + 120573) 2) sdot 120573 sdot 2(120573minus1)2))

1120573

(8)


120589 =119906

|V|1120573 (9)




120578 = 001120589 (x minus xbest) (10)






5 The Method




120573 120575












100 150 250 and 500 and 119901119886= 0 001 005 01 015 02











04

06

08

1

12

14

16

18

2


(a)

04

06

08

1

12

14

16

18

2


(b)


minus5

minus4

minus3

minus2

minus1

0

1

2


(a)

2

minus5

minus4

minus3

minus2

minus1

0

1


(b)







minus8

minus6

minus4

minus2

0

2

4

6

8


(a)

minus8

minus6

minus4

minus2

0

2

4

6

8


(b)














minus1

minus05

0

05

1

minus05 0 05 1 15 2 25

(a)

minus1

minus05

0

05

1

minus05 0 05 1 15 2 25

(b)


minus5

0

5


0 510 15

20

minus10

0

10

(a)

010

20

05

10

0

5

minus5

minus5

minus10minus10

(b)


0 5 10 15 200

10

0

5

minus5

minus10minus15

minus10minus5

(a)

0 5 10 15 20 minus10minus5

0510

0

5

minus5

minus15minus10

minus5

(b)






Acknowledgments


References










































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in











100 150 250 and 500 and 119901119886= 0 001 005 01 015 02











04

06

08

1

12

14

16

18

2


(a)

04

06

08

1

12

14

16

18

2


(b)


minus5

minus4

minus3

minus2

minus1

0

1

2


(a)

2

minus5

minus4

minus3

minus2

minus1

0

1


(b)







minus8

minus6

minus4

minus2

0

2

4

6

8


(a)

minus8

minus6

minus4

minus2

0

2

4

6

8


(b)














minus1

minus05

0

05

1

minus05 0 05 1 15 2 25

(a)

minus1

minus05

0

05

1

minus05 0 05 1 15 2 25

(b)


minus5

0

5


0 510 15

20

minus10

0

10

(a)

010

20

05

10

0

5

minus5

minus5

minus10minus10

(b)


0 5 10 15 200

10

0

5

minus5

minus10minus15

minus10minus5

(a)

0 5 10 15 20 minus10minus5

0510

0

5

minus5

minus15minus10

minus5

(b)






Acknowledgments


References










































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




04

06

08

1

12

14

16

18

2


(a)

04

06

08

1

12

14

16

18

2


(b)


minus5

minus4

minus3

minus2

minus1

0

1

2


(a)

2

minus5

minus4

minus3

minus2

minus1

0

1


(b)







minus8

minus6

minus4

minus2

0

2

4

6

8


(a)

minus8

minus6

minus4

minus2

0

2

4

6

8


(b)














minus1

minus05

0

05

1

minus05 0 05 1 15 2 25

(a)

minus1

minus05

0

05

1

minus05 0 05 1 15 2 25

(b)


minus5

0

5


0 510 15

20

minus10

0

10

(a)

010

20

05

10

0

5

minus5

minus5

minus10minus10

(b)


0 5 10 15 200

10

0

5

minus5

minus10minus15

minus10minus5

(a)

0 5 10 15 20 minus10minus5

0510

0

5

minus5

minus15minus10

minus5

(b)






Acknowledgments


References










































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




minus8

minus6

minus4

minus2

0

2

4

6

8


(a)

minus8

minus6

minus4

minus2

0

2

4

6

8


(b)














minus1

minus05

0

05

1

minus05 0 05 1 15 2 25

(a)

minus1

minus05

0

05

1

minus05 0 05 1 15 2 25

(b)


minus5

0

5


0 510 15

20

minus10

0

10

(a)

010

20

05

10

0

5

minus5

minus5

minus10minus10

(b)


0 5 10 15 200

10

0

5

minus5

minus10minus15

minus10minus5

(a)

0 5 10 15 20 minus10minus5

0510

0

5

minus5

minus15minus10

minus5

(b)






Acknowledgments


References










































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




minus1

minus05

0

05

1

minus05 0 05 1 15 2 25

(a)

minus1

minus05

0

05

1

minus05 0 05 1 15 2 25

(b)


minus5

0

5


0 510 15

20

minus10

0

10

(a)

010

20

05

10

0

5

minus5

minus5

minus10minus10

(b)


0 5 10 15 200

10

0

5

minus5

minus10minus15

minus10minus5

(a)

0 5 10 15 20 minus10minus5

0510

0

5

minus5

minus15minus10

minus5

(b)






Acknowledgments


References










































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







Acknowledgments


References










































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



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