fuzzy b-spline surface modeling

Research ArticleFuzzy B-Spline Surface Modeling

Rozaimi Zakaria Abd Fatah Wahab and R U Gobithaasan

School of Informatics amp Applied Mathematics Universiti Malaysia Terengganu (UMT) 21030 Kuala TerengganuTerengganu Malaysia

Correspondence should be addressed to Rozaimi Zakaria rozaimi zyahoocom

Received 23 October 2013 Revised 10 January 2014 Accepted 18 February 2014 Published 6 July 2014

Academic Editor Jong Hae Kim

Copyright copy 2014 Rozaimi Zakaria et al 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 discusses the construction of a fuzzy B-spline surface modelThe construction of this model is based on fuzzy set theorywhich is based on fuzzy number and fuzzy relation concepts The proposed theories and concepts define the uncertainty data setswhich represent fuzzy datacontrol points allowing the uncertainties data points modeling which can be visualized and analyzedThe fuzzification and defuzzification processes were also defined in detail in order to obtain the fuzzy B-spline surface crisp modelFinal section shows an application of fuzzy B-spline surface modeling for terrain modeling which shows its usability in handlinguncertain data

1 Introduction

Data points are collected from physical objects to captureits geometric entity and representation in a digital systemthat is CAD systems This data is collected by using specificdevices such as scanning tools However the recorded data donot necessarily represent error-free data It is due to the factthat the errors are produced by the limitations of the toolsenvironmental factors human errors and so forth Usuallythese kinds of data which have uncertainty characteristicscannot be used directly to produce digitized models Hencedesigners use certain type of digital filter to remove andamend the errors involved which is a painstaking process [1]One may not truly capture the digital model of a scannedmodel due to the reasons stated above regardless of executingthe time consuming digital filter

In order to make the uncertain data useable for analysisand model building these data have to be defined in adifferent approachwhichwill incorporate uncertainties of themeasurements In this paper we propose fuzzy set theorywhichwas introduced by Zadeh in 1965 [2] It has beenwidelyused in dealing with uncertain matters for wise decisionmaking processes Readers are referred to [3ndash5] for detailedexplanation regarding the subject matter

Natural spline Bezier and B-spline functions are exampleof functions which can be used to create CAD models with

data points [6ndash8] These curves and surfaces created withthe stated functions are the standard approach to representa set of collected data points These curves and surfacesare used to visualize and analyze the CAD models B-splinefunctions can be used to design curves and surface usingeither approximation or interpolation methods to model thereal data points [9ndash15] The reason why B-spline functionswith its weight known as nonuniform rational B-spline(NURBS) being used inmany applications is that the designercan easily tweak the control points to obtain a desired shapeeasily

This paper discusses modeling of interpolating B-splinesurface using fuzzy set theory and it is organized as followsSection 2 reviews the previous work on modeling uncertaindata via B-spline function in the form of curves and surfaceSection 3 discusses the representation of data points usingfuzzy set theory fuzzy number and fuzzy relation conceptsSection 4 discusses blending of fuzzy datacontrol point withB-spline curve and surface function where the end resultsin fuzzy B-spline curve and surface model This section alsodefines fuzzification and defuzzification processes To showthe application of fuzzy B-spline surface model we apply theproposed method to model lakebed from a set of uncertaindata in Section 5 This section also compares the result ofstatistical analysis to show the effectiveness of proposed fuzzyB-spline model

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 285045 8 pageshttpdxdoiorg1011552014285045

2 Journal of Applied Mathematics

2 Previous Works

The requirement of fuzzy set theory is essential in handlingambiguous data in order to create a model using B-splinecurve and surface function The designers are unable tochoose the appropriate control points which are exposed toerrors and uncertainties due to the reasons stated aboveTherefore this is the reason why we define the uncertain databy using fuzzy set theory and model them through B-splinecurve and surface function

There are a number of methods that have been developedin dealing with uncertain data Examples include modelingsurface of Mount Etna which was proposed by Gallo et al[16 17] In this paper they developed a fuzzy B-spline modelwhich was conceptualized from fuzzy numbers in the formof intervals They also proposed alpha level within (0 1] Thedata of Mount Etna was represented in the form of intervalfuzzy surface based on alpha value Anile et al [18] furtherenhanced this method for modeling uncertain and sparsedata

Both of themethods discussed above do not have defuzzi-fication process to obtain a crisp fuzzy surface (defuzzifiedsurface) Although both methods employed fuzzificationphase it is still in interval surface form which is not in singlesurface form Furthermore the fuzzification and defuzzifica-tion process of B-spline fuzzy system [19] were not elaboratedin detail Therefore this paper elucidates defuzzificationprocess upon the application of fuzzification process

3 The Process of Defining Fuzzy Data

This section defines uncertain data based on the concept offuzzy number [3ndash5 20] based on the interval of fuzzy number

Definition 1 Let119877 be a universal set in which119877 is a real num-ber and 119860 is subset of 119877 A Fuzzy setlarrrarr119860 in 119877 is called a fuzzynumber and expressed using the 120572-level with various 120572-cutthat is if for every 120572 isin (0 1] there exist setlarrrarr119860

120572in 119877 where

larrrarr119860120572= 119909 isin 119877 120583

119860120572

(119909) gt 120572 andlarrrarr119860120572= 119909 isin 119877 120583

119860120572

(119909) ge 120572

[21]Definition 1 provides the basis to define uncertain data

in which these data are in the form of real numbers We usetriangular fuzzy number for defining the uncertain data in theform of interval Therefore the triangular fuzzy number canbe defined in Definition 2 as follows [21]

Definition 2 If triangular fuzzy number is represented aslarrrarr119860 = (119886 119889 119888) andlarrrarr119860

120572is a 120572-cut operation of triangular fuzzy

number then crisp interval by 120572-cut operation is obtained aslarrrarr119860120572= [119886120572 119888120572] = [(119889 minus 119886)120572 + 119886 minus(119888 minus 119889)120572 + 119888] with 120572 isin (0 1]

where the membership function 120583larrrarr119860

(119909) is given by

120583larrrarr119860

(119909) =

0 for 119909 lt 119886119909 minus 119886

119889 minus 119886for 119886 le 119909 le 119889

119888 minus 119909

119888 minus 119889for 119889 le 119909 le 119888

0 for 119909 gt 119888

(1)

1

0

120572

120583

a a120572 c120572d c

x

b

Figure 1 Triangular fuzzy numberlarrrarr119860 = (119886 119887 119888)

where 119886 and 119888 are left and right fuzzy number which formthe interval fuzzy number and 119889 is the crisp point in theinterval The 120572 symbol means the 120572-level values of triangularfuzzy 120572-cut (see Figure 1)

In Figure 1 the 119887 value is the crisp fuzzy number whichhas full membership function that is it is equal to 1

Before we define the uncertain data points we mustdefine the fuzzy relation which is used as a converter fromthe definition of fuzzy number to the definition of fuzzy datapoints in real numbers Then the definition of fuzzy relationand also the definition of the relation between two fuzzypoints can be given in Definition 3 till Definition 5

Definition 3 Let119883119884 sube 119877 be universal sets then

larrrarr119877 = ((119909 119910) 120583larrrarr

119877

(119909 119910)) | (119909 119910) sube 119883 times 119884 (2)

is called a fuzzy relation on119883 times 119884 [4 21]

Definition 4 Let 119883119884 sube 119877 and larrrarr119860 = 119909 120583larrrarr119860

(119909) | 119909 isin 119883

andlarrrarr119861 = 119910 120583larrrarr119861

(119910) | 119910 isin 119884 are two fuzzy sets Thenlarrrarr119877 =[(119909 119910) 120583larrrarr

119877

(119909 119910)] (119909 119910) isin 119883 times 119884 is a fuzzy relation onlarrrarr119860

andlarrrarr119861 if 120583larrrarr119877

(119909 119910) le 120583larrrarr119860

(119909) forall(119909 119910) isin 119883times119884 and 120583larrrarr119877

(119909 119910) le


(119910) forall(119909 119910) isin 119883 times 119884 [4]

Definition 5 Let 119883119884 sube 119877 with larrrarr119872 = 119909 120583larrrarr119872

(119909) | 119909 isin 119883


(119910) | 119910 isin 119884 represent two fuzzy data Then

the fuzzy relation between both fuzzy data is given bylarrrarr119875 =[(119909 119910) 120583larrrarr

119875

(119909 119910)] (119909 119910) isin 119883 times 119884The uncertain data points can be defined after fuzzy

number and fuzzy relation had been defined This uncertaindata point becomes fuzzy data point (FDP) as shown inDefinition 6

Journal of Applied Mathematics 3

Crisp data point

Ordinary data point Fuzzy data point0 1 2 3 4 5 6

0

02

04

06

08

1

0 1 2 3 4 5 60

02

04

06

08

1

0 1 2 3 4 5 60

02

04

06

08

1

x

x x120583A(x)

120583A(x)

120583A(x)

Figure 2 The process of defining FDPs

Definition 6 Let 119863 = (119909 119910) 119909 isin 119883 119910 isin 119884 | 119909 and 119910 arefuzzy data andlarrrarr119863 = 119875

119894| 119875 is data point is the set of FDPs

which is 119863119894isin 119863 sub 119883 times 119884 sube 119877 with 119877 is the universal set

and 120583119875(119863119894) 119863 rarr [0 1] is membership function defined as

120583119875(119863119894) = 1 in whichlarrrarr119863 = (119863

119894 120583119863(119863119894)) | 119863

119894isin 119877 Therefore

120583119875(119863119894) =

0 if 119863119894notin 119877

119888 isin (0 1) if 119863119894

larrrarrisin 119877

1 if 119863119894isin 119877

(3)

with 120583119863(119863119894) = ⟨120583

119875(119863larr119894) 120583119875(119863119894) 120583119875(119863rarr119894)⟩ where 120583

119863(119863larr119894)

and 120583119863(119863rarr119894) are left-grade and right-grade membership

values respectively This can be written as

larrrarr119863 =

larrrarr119863119894= (119909119894 119910119894) | 119894 = 0 1 119899 (4)

for all 119894larrrarr119863119894= ⟨larrrarr119863larr

119894 119863119894larrrarr119863rarr

119894⟩withlarrrarr119863

larr

119894119863119894 andlarrrarr119863

rarr

119894being

left FDP crisp data point and right FDP respectively [20]The procedure in defining FDP is illustrated in Figure 2

Figure 2 shows the approach to transform ordinary datapoint to Fuzzy data point (FDP) The membership grades ofFDPs in the form of (119909 119910) are illustrated in Figure 3

Figure 3 shows the formation of FDP by using the defini-tions of fuzzy relation and fuzzy numberThe construction ofFDPs is in 119909- and 119910-axis

0 1 2 3 4 5 6 7 8

02

46

0

1

y

x

120583(xy)

08

06

04

02

Figure 3 FDP form after being defined by fuzzy relation

Definition 6 gives us the definition of FDP in 2D formandfor FDP in 3D form similar concept is applicable given by (5)(based on (4)) Consider

larrrarr119863 =

larrrarr119863119894= (119909119894 119910119894 119911119894) | 119894 = 0 1 119899 (5)

4 The Proposed Method

This section discusses blending of FDPs into B-spline func-tion to produce fuzzy B-spline curves and surfaces Thefuzzification and defuzzification process towards fuzzy B-spline model in the form of either curves or surface are alsodiscussed


harrP0

harrP1

harrP2

harrP3

harrP4

(a)

harrP(00)

harrP(10)

harrP(20)

harrP(12)

harrP(22)

harrP(33)

harrP(32)harr

P(31)harrP(30)

(b)

Figure 4 The fuzzy B-spline models in the form of (a) curves and (b) surface

Based on B-spline function [6ndash8] we can define the fuzzyB-spline model [22 23] as follows

Definition 7 A fuzzy B-spline curve is a functionlarrrarr119861119904(119905)whichrepresents a curve to the set of real fuzzy numbers and it isdefined as

larrrarr119861119904 (119905) =

119896+ℎminus1

sum119894=1

larrrarr119875119894119861119894ℎ(119905) (6)

where larrrarr119875119894are fuzzy control points which are also known as

fuzzy data point and 119861119894ℎ(119905) are B-spline basic function with

crisp knot sequences 1199051 1199052 119905

119898=119889+119899+1where 119889 represents

the degree of B-spline function and 119899 represents the numbersof control points

Definition 8 A fuzzy B-spline surface is defined by the fol-lowing equation

larrrarr119861119904119878 (119904 119905) =

119898

sum119894=0

119899

sum119895=0

larrrarr119875(119894119895)119873119894119901(119904)119873119895119902(119905) (7)

where (i) 119873119894119901(119904) and 119873

119895119902(119905) are B-spline basic function of

degrees 119901 and 119902 with crisp parameters of 119904 and 119905 in [0 1] (ii)each vector knot must satisfy the conditions 119903 = 119898 + 119901 + 1and 119904 = 119899 + 119902 + 1 (iii)larrrarr119875

(119894119895)= ⟨larrrarr119875larr

(119894119895) 119875(119894119895)larrrarr119875rarr

(119894119895)⟩ are fuzzy

control point in 119894th row and 119895th columnTherefore both of Definitions 7 and 8 can be illustrated

in the form of numerical examples with Figure 4For fuzzy B-spline curve model the resultant curve inter-

polates the first and last fuzzy control pointsThis fuzzy curvewas designed based on five fuzzy controldata points (119894 =0 1 4)The same concept is applied to surface which usesdiagonal fuzzy controldata points having 16 fuzzy controldata points

For fuzzy B-spline curve model the fuzzy controldatapoints are defined for 119909-element and for fuzzy B-splinesurface model and the fuzzy controldata points are defined

at 119911-element Therefore defining uncertain data can be doneby combining either the tuple axis or one of the axes whilemaintaining other axes as crisp values [21]

Upon defining the fuzzy B-spline model we use thealpha-cut operation of triangular fuzzy number to do thefuzzification process based on Definition 2 Therefore thefuzzification process of fuzzy control points is given by thefollowing definition which is intact with the B-spline surfacefunction

Definition 9 Let larrrarr119875(119894119895)

be the set of fuzzy control pointswhere 119894 = 0 1 119898 and 119895 = 0 1 119899 Thenlarrrarr119875

(119894119895)120572119896

is thealpha-cut operation of fuzzy control point which is given asthe following equation where 120572

119896isin (0 1] with 119896 = 1 2 119897

larrrarr119875(119894119895)120572119896

= ⟨larrrarr119875larr

(119894119895)120572119896

119875(119894119895)larrrarr119875rarr

(119894119895)120572119896

⟩

= ⟨[(119875(119894119895)minuslarrrarr119875larr

(119894119895)) 120572119896 +larrrarr119875larr

(119894119895)] 119875(119894119895)

[minus (larrrarr119875rarr

(119894119895) minus 119875(119894119895))120572119896 +larrrarr119875rarr

(119894119895)]⟩

(8)

Upon fuzzification the next procedure is the defuzzifi-cation process Defuzzification process is applied to obtain afuzzy solution in a single value The result of defuzzificationprocess is also known as fuzzy crisp solution Therefore thedefuzzification process is defined as in Definition 10

Definition 10 The defuzzification of larrrarr119875(119894119895)120572119896

and 119875(119894119895)120572119896

canbe given as

119875120572119896

= 119875(119894119895)120572119896

where 119875(119894119895)120572119896

=1

3sum119894=0119895=0

⟨119875larr

(119894119895)120572119896

119875(119894119895) 119875rarr

(119894119895)120572119896

⟩

(9)


(a) (b)

Figure 5 Fuzzy B-spline surface after fuzzification and defuzzification processes

The illustration of fuzzification and defuzzification pro-cesses based onDefinitions 9 and 10 respectively is illustratedin Figure 5 with the alpha value being 05

Figure 5 shows the fuzzy B-spline surface after fuzzifi-cation (Figure 5(a)) and defuzzification of B-spline surface(Figure 5(b)) FromFigure 5(b) the fuzzification process wasapplied by means of alpha-cut operation with the value ofalpha as 05 Finally the defuzzification of B-spline surfaceis equal to crisp B-spline surface because the left and rightinterval of fuzzy control points is equal

The constructed fuzzy B-spline surface alongwith defuzz-ification process has its merits The advantage includes theeffectiveness model which can be used to model either thecrisp data (exact data) or fuzzy data (with various 120572-cuts)compared to the crisp model which can be used to model thecrisp data only but not for fuzzy data

5 Lakebed Modeling

This section illustrates an application example of fuzzy B-spline surface model proposed in this paper The proposedmodel is used to generate the lakebed by using collected datapoints which is exposed to various kinds of errors

These errors which occurred during data point retrievalinclude the wavy water surface condition which gives theuncertain data reading Therefore every set of data pointsin modeling underwater ground surface has the error ofaccuracy to a certain extent Figure 6 shows the scenario

Figure 6 shows the process of getting uncertain depthlake data which has been taken by echo sounder We canclearly comprehend the reason for modeling the uncertaindata indicating the depth of lake with fuzzy B-spline surfacemodel This uncertainty in the data exists for 119911-elements(depth)

Fuzzy number concept and fuzzy relation definition areutilized to define this uncertain data We represent the fol-lowing algorithm to illustrate the steps to be modeled withfuzzy B-spline from fuzzification to defuzzification processes

Algorithm 11Step 1 Define the uncertain data of lakebed by usingDefinition 6

Ship with echo sounderWavy water surface

harrz

Figure 6 The illustration of collecting the depth of lake

Step 2 Blend the FDPs of lakebed together with B-splinesurface function which is given as

larr997888997888rarr119861119904119878119871 (119904 119905) =

119898

sum119894=07

119899

sum119895=07

119871larrrarr119863(119894119895)1198731198943(119904)1198731198953(119905) (10)

which is in the form of bicubic surfaceNote that the set of FDPs of lakebed of Kenyir Lake has

64 data points where 119894 times 119895 with 119894 119895 = 0 1 7

Step 3 Apply alpha-cut operation as fuzzification process(Definition 9) towards (10) with the alpha value being 05Consider

larr997888997888rarr11986111990411987811987112057205

(119904 119905) =

119898

sum119894=07

119899

sum119895=07

119871larrrarr119863(119894119895)12057205

1198731198943(119904)1198731198953(119905) (11)

where 119871larrrarr119863(119894119895)12057205

are the fuzzified data points of lakebed afteralpha-cut operation was applied with

119871larrrarr119863(119894119895)12057205

= ⟨119871larrrarr119863larr

(119894119895)12057205

119871

119863(119894119895)119871larrrarr119863rarr

(119894119895)12057205

⟩

= ⟨[(119871

119863(119894119895)minus119871larrrarr119863larr

(119894119895)) 05 +119871larrrarr119863larr

(119894119895)] 119871

119863(119894119895)

[minus (119871larrrarr119863rarr

(119894119895) minus119871

119863(119894119895)) 05 +

119871larrrarr119863rarr

(119894119895)]⟩

(12)


Defuzzificationprocess

Fuzzification process(alpha-cut operation 120572 = 05)

Figure 7 The process of modeling fuzzy data of Kenyir Lake lakebed

Figure 8 The defuzzification process of fuzzy data of a lakebed forKenyir Lake with 120572 = 02

Step 4 Use Definition 10 to defuzzify B-spline surface modelof lakebed which is

11986111990411987811987112057205

(119904 119905) =

119898

sum119894=07

119899

sum119895=07

119871

119863(119894119895)12057205

1198731198943(119904)1198731198953(119905) (13)

where 119871119863(119894119895)12057205

are defuzzified data points of lakebed with

119871

119863(119894119895)12057205

=1

3sum

119894=07119895=07

⟨119871larrrarr119863larr

(119894119895)12057205

119863(119894119895)119871larrrarr119863rarr

(119894119895)12057205

⟩

(14)

The result of Algorithm 11 can be illustrated in Figure 7

Figure 7 shows the processes of defining the uncertaindata of lakebed which is then modelled by using fuzzy B-spline surfaceThe fuzzy data points of lakebed can be definedas the fuzzy control points because we used the approxima-tion method to create the fuzzy B-spline surface which usedfuzzy data points as fuzzy control points After achievingfuzzy B-spline surface modeling we then apply the fuzzi-fication process which utilized the alpha-cut operation of


triangular fuzzy number with the alpha value as 05 Thenwe defuzzify the fuzzy lake surface based on the definition ofdefuzzification By setting the new alpha values that are 02and 09 we may obtain the result of the defuzzified lakebedmodel as illustrated in Figures 8 and 9 respectively

In order to investigate the effectiveness of the output oflakebed we find the errors between the defuzzification datapoints and crisp data points of lakebed which can be giventhrough (15) Therefore the error between those data can beillustrated by Figures 10 11 and 12 for different alpha valuesas follows

sum119899

119896=0119899

119871119863119896119890

sum119896=0119899

119899 ( 119871119863119896119890

)where 119871119863

119896119890

=

119871119863119896minus 119871119863119896

119871119863119896

with 119896 = 0 1 2 119899 = 64

(15)

Figure 10 until Figure 12 shows the errors betweendefuzzified and crisp data points of lakebed in order to seethe effectiveness of proposed model in modeling uncertaindata of lakebed The average percentage of the errors is


0 10 20 30 40 50 60

004

006

008

010

012

014Er

ror (

)

Number of data points (m)

Figure 10 The error between defuzzified and crisp data points oflakebed with 120572 = 05

0 10 20 30 40 50 60

005

010

015

020


Erro

r (

)


0 10 20 30 40 50 600005

0010

0015

0020

0025


Erro

r (

)


00413623m 00661796m and 000827245m respectivelyThese errors are acceptable for terrain modeling

6 Conclusion

In this paper we proposed a new paradigm in modelingthe uncertain data by using the hybrid method betweenfuzzy set theory and B-spline function surface function Thismodel has an upper hand in dealing with uncertain problem

compared to the existing model which only can be used indealing with and modeling crisp data

The fuzzification and defuzzification processes were alsoelucidated exclusively for fuzzy B-spline surface model Forfuzzification process the alpha-cut operation was appliedwhich is in the formof triangular functionsThis fuzzificationprocess was applied to obtain the fuzzy interval of fuzzy datapoints where the crisp fuzzy solution is in this interval It isthen followed by the defuzzification process to find crisp B-spline surface which focused on the defuzzification of fuzzydata points

Finally to identify the effectiveness of fuzzy B-splinesurface model this model was applied to modeling of theuncertain data of lakebed The process of defining fuzzifi-cation and defuzzification of uncertain data of lakebed canbe represented by an algorithmwhich is applicable in dealingwith various fuzzy data The errors produced in the case oflakebed modeling indicate that it can be used for terrainmodeling

This work can expand further to solve 1198662 Hermite dataproblems which occur in designing aesthetic splines [24]One may create a system to optimize and propose a suitablealpha-cut which satisfies given 1198662 Hermite data which mayfacilitate the designers in creating aesthetic shapes efficiently

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge Research Manage-ment and Innovation Centre (RMIC) of Universiti MalaysiaTerengganu and Ministry of Higher Education (MOHE)Malaysia for their funding (FRGS 59244) and providing thefacilities to conduct this research They further acknowledgeanonymous referees for their constructive comments whichimproved the readability of this paper

References

[1] K T Miura R Shirahata S Agari S Usuki and R U Gob-ithaasan ldquoVariational formulation of the log aesthetic surfaceand development of discrete surface filtersrdquo Computer AidedDesign and Application vol 9 no 6 pp 901ndash914 2012

[2] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[3] D Dubois and H Prade Fuzzy Sets and Systems Theory andApplications vol 144 ofMathematics in Science and EngineeringAcademic Press New York NY USA 1980

[4] H J Zimmermann Fuzzy Set Theory and Its ApplicationsKluwer Academic New York NY USA 1985

[5] G J Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplication Prentice Hall New York NY USA 1995

[6] D F Rogers An Introduction to NURBS With Historical Per-spective Academic Press New York NY USA 2001

[7] G FarinCurves and Surfaces for CAGD A Practical Guide Aca-demic Press New York NY USA 5th edition 2002


[8] F YamaguchiCurves and Surfaces inComputerAidedGeometricDesign Springer Berlin Germany 1988

[9] B Aggarwal B-spline finite elements for plane elasticity problems[MS thesis] Texas AampM University 2006

[10] C-C Leung P C-K Kwok K-Y Zee and F H-Y Chan ldquoB-spline interpolation for bend intra-oral radiographsrdquo Comput-ers in Biology and Medicine vol 37 no 11 pp 1565ndash1571 2007

[11] H-B Jung ldquoAn interpolation method of b-spline surface forhull form designrdquo International Journal of Naval Architectureand Ocean Engineering vol 2 no 4 pp 195ndash199 2010

[12] T Maekawa T Noda S Tamura T Ozaki and K-I MachidaldquoCurvature continuous path generation for autonomous vehicleusingB-spline curvesrdquoCADComputerAidedDesign vol 42 no4 pp 350ndash359 2010

[13] S Okaniwa A Nasri L Hongwei A Abbas Y Kineri and TMaekawa ldquoUniform B-spline curve interpolation with pre-scribed tangent and curvature vectorsrdquo IEEE Transactions onVisualization and Computer Graphics vol 18 no 9 pp 1474ndash1487 2012

[14] H Pungotra G K Knopf and R Canas ldquoAn alternative meth-odology to represent B-spline surface for applications in virtualreality environmentrdquo Computer-Aided Design and Applicationsvol 10 no 4 pp 711ndash726 2013

[15] AANada ldquoUse of B-spline surface tomodel large-deformationcontinuum plates procedure and applicationsrdquo NonlinearDynamics vol 72 no 1-2 pp 243ndash263 2013

[16] G Gallo M Spagnuolo and S Spinello ldquoFuzzy B-splines a sur-face model encapsulating uncertaintyrdquo Graphical Models vol62 no 1 pp 40ndash55 2000

[17] G Gallo M Spagnuolo and S Spinello ldquoRainfall estimationfrom sparse data with fuzzy B-splinesrdquo Journal of GeographicInformation and Decision Analysis vol 2 pp 194ndash203 1998

[18] A M Anile B Falcidieno G Gallo M Spagnuolo and SSpinello ldquoModeling uncertain data with fuzzy B-splinesrdquo FuzzySets and Systems vol 113 no 3 pp 397ndash410 2000

[19] T Yanhua and L Hongxing ldquoFaired MISO B-spline fuzzysystems and its applicationsrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 870595 9 pages 2013

[20] G J Klir U S Clair and B Yuan Fuzzy SetTheory Foundationand Application Prentice Hall Upper Saddle River NJ USA1997

[21] R Zakaria andA BWahab ldquoFuzzy B-splinemodeling of uncer-tainty datardquo Applied Mathematical Sciences vol 6 pp 6971ndash6991 2012

[22] A F Wahab J M Ali A A Majid and A O M Tap ldquoFuzzy setin geometricmodelingrdquo in Proceedings of the International Con-ference onComputer Graphics Imaging andVisualization (CGIVrsquo04) pp 227ndash232 July 2004

[23] N A A Karim A FWahab R U Gobithaasan and R ZakarialdquoModel of fuzzy B-spline interpolation for fuzzy datardquo Far EastJournal of Mathematical Sciences vol 72 pp 269ndash280 2013

[24] K T Miura D Shibuya R U Gobithaasan and S UsukildquoDesigning log-aesthetic splines with G2 continuityrdquoComputer-Aided Design and Applications vol 10 no 6 pp 1021ndash1032 2013

Submit your manuscripts athttpwwwhindawicom

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2 Previous Works

The requirement of fuzzy set theory is essential in handlingambiguous data in order to create a model using B-splinecurve and surface function The designers are unable tochoose the appropriate control points which are exposed toerrors and uncertainties due to the reasons stated aboveTherefore this is the reason why we define the uncertain databy using fuzzy set theory and model them through B-splinecurve and surface function

There are a number of methods that have been developedin dealing with uncertain data Examples include modelingsurface of Mount Etna which was proposed by Gallo et al[16 17] In this paper they developed a fuzzy B-spline modelwhich was conceptualized from fuzzy numbers in the formof intervals They also proposed alpha level within (0 1] Thedata of Mount Etna was represented in the form of intervalfuzzy surface based on alpha value Anile et al [18] furtherenhanced this method for modeling uncertain and sparsedata

Both of themethods discussed above do not have defuzzi-fication process to obtain a crisp fuzzy surface (defuzzifiedsurface) Although both methods employed fuzzificationphase it is still in interval surface form which is not in singlesurface form Furthermore the fuzzification and defuzzifica-tion process of B-spline fuzzy system [19] were not elaboratedin detail Therefore this paper elucidates defuzzificationprocess upon the application of fuzzification process

3 The Process of Defining Fuzzy Data

This section defines uncertain data based on the concept offuzzy number [3ndash5 20] based on the interval of fuzzy number

Definition 1 Let119877 be a universal set in which119877 is a real num-ber and 119860 is subset of 119877 A Fuzzy setlarrrarr119860 in 119877 is called a fuzzynumber and expressed using the 120572-level with various 120572-cutthat is if for every 120572 isin (0 1] there exist setlarrrarr119860

120572in 119877 where

larrrarr119860120572= 119909 isin 119877 120583

119860120572

(119909) gt 120572 andlarrrarr119860120572= 119909 isin 119877 120583

119860120572

(119909) ge 120572

[21]Definition 1 provides the basis to define uncertain data

in which these data are in the form of real numbers We usetriangular fuzzy number for defining the uncertain data in theform of interval Therefore the triangular fuzzy number canbe defined in Definition 2 as follows [21]

Definition 2 If triangular fuzzy number is represented aslarrrarr119860 = (119886 119889 119888) andlarrrarr119860

120572is a 120572-cut operation of triangular fuzzy

number then crisp interval by 120572-cut operation is obtained aslarrrarr119860120572= [119886120572 119888120572] = [(119889 minus 119886)120572 + 119886 minus(119888 minus 119889)120572 + 119888] with 120572 isin (0 1]

where the membership function 120583larrrarr119860

(119909) is given by


(119909) =

0 for 119909 lt 119886119909 minus 119886

119889 minus 119886for 119886 le 119909 le 119889

119888 minus 119909

119888 minus 119889for 119889 le 119909 le 119888

0 for 119909 gt 119888

(1)

1

0

120572

120583

a a120572 c120572d c

x

b

Figure 1 Triangular fuzzy numberlarrrarr119860 = (119886 119887 119888)

where 119886 and 119888 are left and right fuzzy number which formthe interval fuzzy number and 119889 is the crisp point in theinterval The 120572 symbol means the 120572-level values of triangularfuzzy 120572-cut (see Figure 1)

In Figure 1 the 119887 value is the crisp fuzzy number whichhas full membership function that is it is equal to 1

Before we define the uncertain data points we mustdefine the fuzzy relation which is used as a converter fromthe definition of fuzzy number to the definition of fuzzy datapoints in real numbers Then the definition of fuzzy relationand also the definition of the relation between two fuzzypoints can be given in Definition 3 till Definition 5

Definition 3 Let119883119884 sube 119877 be universal sets then

larrrarr119877 = ((119909 119910) 120583larrrarr

119877

(119909 119910)) | (119909 119910) sube 119883 times 119884 (2)

is called a fuzzy relation on119883 times 119884 [4 21]

Definition 4 Let 119883119884 sube 119877 and larrrarr119860 = 119909 120583larrrarr119860

(119909) | 119909 isin 119883


(119910) | 119910 isin 119884 are two fuzzy sets Thenlarrrarr119877 =[(119909 119910) 120583larrrarr

119877

(119909 119910)] (119909 119910) isin 119883 times 119884 is a fuzzy relation onlarrrarr119860

andlarrrarr119861 if 120583larrrarr119877

(119909 119910) le 120583larrrarr119860

(119909) forall(119909 119910) isin 119883times119884 and 120583larrrarr119877

(119909 119910) le


(119910) forall(119909 119910) isin 119883 times 119884 [4]

Definition 5 Let 119883119884 sube 119877 with larrrarr119872 = 119909 120583larrrarr119872

(119909) | 119909 isin 119883


(119910) | 119910 isin 119884 represent two fuzzy data Then

the fuzzy relation between both fuzzy data is given bylarrrarr119875 =[(119909 119910) 120583larrrarr

119875

(119909 119910)] (119909 119910) isin 119883 times 119884The uncertain data points can be defined after fuzzy

number and fuzzy relation had been defined This uncertaindata point becomes fuzzy data point (FDP) as shown inDefinition 6


Crisp data point


0

02

04

06

08

1

0 1 2 3 4 5 60

02

04

06

08

1

0 1 2 3 4 5 60

02

04

06

08

1

x

x x120583A(x)

120583A(x)

120583A(x)







119894 120583119863(119863119894)) | 119863


120583119875(119863119894) =

0 if 119863119894notin 119877

119888 isin (0 1) if 119863119894

larrrarrisin 119877

1 if 119863119894isin 119877

(3)

with 120583119863(119863119894) = ⟨120583

119875(119863larr119894) 120583119875(119863119894) 120583119875(119863rarr119894)⟩ where 120583

119863(119863larr119894)



larrrarr119863 =

larrrarr119863119894= (119909119894 119910119894) | 119894 = 0 1 119899 (4)


119894 119863119894larrrarr119863rarr


larr

119894119863119894 andlarrrarr119863

rarr

119894being




0 1 2 3 4 5 6 7 8

02

46

0

1

y

x

120583(xy)

08

06

04

02



larrrarr119863 =

larrrarr119863119894= (119909119894 119910119894 119911119894) | 119894 = 0 1 119899 (5)




harrP0

harrP1

harrP2

harrP3

harrP4

(a)

harrP(00)

harrP(10)

harrP(20)

harrP(12)

harrP(22)

harrP(33)

harrP(32)harr

P(31)harrP(30)

(b)




larrrarr119861119904 (119905) =

119896+ℎminus1

sum119894=1

larrrarr119875119894119861119894ℎ(119905) (6)







larrrarr119861119904119878 (119904 119905) =

119898

sum119894=0

119899

sum119895=0

larrrarr119875(119894119895)119873119894119901(119904)119873119895119902(119905) (7)

where (i) 119873119894119901(119904) and 119873



(119894119895)= ⟨larrrarr119875larr

(119894119895) 119875(119894119895)larrrarr119875rarr

(119894119895)⟩ are fuzzy









(119894119895)120572119896


119896isin (0 1] with 119896 = 1 2 119897

larrrarr119875(119894119895)120572119896


(119894119895)120572119896

119875(119894119895)larrrarr119875rarr

(119894119895)120572119896

⟩


(119894119895)) 120572119896 +larrrarr119875larr

(119894119895)] 119875(119894119895)



(119894119895)]⟩

(8)



and 119875(119894119895)120572119896

canbe given as

119875120572119896

= 119875(119894119895)120572119896

where 119875(119894119895)120572119896

=1

3sum119894=0119895=0

⟨119875larr

(119894119895)120572119896

119875(119894119895) 119875rarr

(119894119895)120572119896

⟩

(9)


(a) (b)





5 Lakebed Modeling







harrz



larr997888997888rarr119861119904119878119871 (119904 119905) =

119898

sum119894=07

119899

sum119895=07

119871larrrarr119863(119894119895)1198731198943(119904)1198731198953(119905) (10)




larr997888997888rarr11986111990411987811987112057205

(119904 119905) =

119898

sum119894=07

119899

sum119895=07

119871larrrarr119863(119894119895)12057205

1198731198943(119904)1198731198953(119905) (11)

where 119871larrrarr119863(119894119895)12057205


119871larrrarr119863(119894119895)12057205


(119894119895)12057205

119871

119863(119894119895)119871larrrarr119863rarr

(119894119895)12057205

⟩

= ⟨[(119871


(119894119895)) 05 +119871larrrarr119863larr

(119894119895)] 119871

119863(119894119895)


(119894119895) minus119871

119863(119894119895)) 05 +


(119894119895)]⟩

(12)







11986111990411987811987112057205

(119904 119905) =

119898

sum119894=07

119899

sum119895=07

119871

119863(119894119895)12057205

1198731198943(119904)1198731198953(119905) (13)

where 119871119863(119894119895)12057205


119871

119863(119894119895)12057205

=1

3sum

119894=07119895=07


(119894119895)12057205

119863(119894119895)119871larrrarr119863rarr

(119894119895)12057205

⟩

(14)






sum119899

119896=0119899

119871119863119896119890

sum119896=0119899

119899 ( 119871119863119896119890

)where 119871119863

119896119890

=

119871119863119896minus 119871119863119896

119871119863119896

with 119896 = 0 1 2 119899 = 64

(15)



0 10 20 30 40 50 60

004

006

008

010

012

014Er

ror (

)



0 10 20 30 40 50 60

005

010

015

020


Erro

r (

)


0 10 20 30 40 50 600005

0010

0015

0020

0025


Erro

r (

)



6 Conclusion








Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Crisp data point


0

02

04

06

08

1

0 1 2 3 4 5 60

02

04

06

08

1

0 1 2 3 4 5 60

02

04

06

08

1

x

x x120583A(x)

120583A(x)

120583A(x)







119894 120583119863(119863119894)) | 119863


120583119875(119863119894) =

0 if 119863119894notin 119877

119888 isin (0 1) if 119863119894

larrrarrisin 119877

1 if 119863119894isin 119877

(3)

with 120583119863(119863119894) = ⟨120583

119875(119863larr119894) 120583119875(119863119894) 120583119875(119863rarr119894)⟩ where 120583

119863(119863larr119894)



larrrarr119863 =

larrrarr119863119894= (119909119894 119910119894) | 119894 = 0 1 119899 (4)


119894 119863119894larrrarr119863rarr


larr

119894119863119894 andlarrrarr119863

rarr

119894being




0 1 2 3 4 5 6 7 8

02

46

0

1

y

x

120583(xy)

08

06

04

02



larrrarr119863 =

larrrarr119863119894= (119909119894 119910119894 119911119894) | 119894 = 0 1 119899 (5)




harrP0

harrP1

harrP2

harrP3

harrP4

(a)

harrP(00)

harrP(10)

harrP(20)

harrP(12)

harrP(22)

harrP(33)

harrP(32)harr

P(31)harrP(30)

(b)




larrrarr119861119904 (119905) =

119896+ℎminus1

sum119894=1

larrrarr119875119894119861119894ℎ(119905) (6)







larrrarr119861119904119878 (119904 119905) =

119898

sum119894=0

119899

sum119895=0

larrrarr119875(119894119895)119873119894119901(119904)119873119895119902(119905) (7)

where (i) 119873119894119901(119904) and 119873



(119894119895)= ⟨larrrarr119875larr

(119894119895) 119875(119894119895)larrrarr119875rarr

(119894119895)⟩ are fuzzy









(119894119895)120572119896


119896isin (0 1] with 119896 = 1 2 119897

larrrarr119875(119894119895)120572119896


(119894119895)120572119896

119875(119894119895)larrrarr119875rarr

(119894119895)120572119896

⟩


(119894119895)) 120572119896 +larrrarr119875larr

(119894119895)] 119875(119894119895)



(119894119895)]⟩

(8)



and 119875(119894119895)120572119896

canbe given as

119875120572119896

= 119875(119894119895)120572119896

where 119875(119894119895)120572119896

=1

3sum119894=0119895=0

⟨119875larr

(119894119895)120572119896

119875(119894119895) 119875rarr

(119894119895)120572119896

⟩

(9)


(a) (b)





5 Lakebed Modeling







harrz



larr997888997888rarr119861119904119878119871 (119904 119905) =

119898

sum119894=07

119899

sum119895=07

119871larrrarr119863(119894119895)1198731198943(119904)1198731198953(119905) (10)




larr997888997888rarr11986111990411987811987112057205

(119904 119905) =

119898

sum119894=07

119899

sum119895=07

119871larrrarr119863(119894119895)12057205

1198731198943(119904)1198731198953(119905) (11)

where 119871larrrarr119863(119894119895)12057205


119871larrrarr119863(119894119895)12057205


(119894119895)12057205

119871

119863(119894119895)119871larrrarr119863rarr

(119894119895)12057205

⟩

= ⟨[(119871


(119894119895)) 05 +119871larrrarr119863larr

(119894119895)] 119871

119863(119894119895)


(119894119895) minus119871

119863(119894119895)) 05 +


(119894119895)]⟩

(12)







11986111990411987811987112057205

(119904 119905) =

119898

sum119894=07

119899

sum119895=07

119871

119863(119894119895)12057205

1198731198943(119904)1198731198953(119905) (13)

where 119871119863(119894119895)12057205


119871

119863(119894119895)12057205

=1

3sum

119894=07119895=07


(119894119895)12057205

119863(119894119895)119871larrrarr119863rarr

(119894119895)12057205

⟩

(14)






sum119899

119896=0119899

119871119863119896119890

sum119896=0119899

119899 ( 119871119863119896119890

)where 119871119863

119896119890

=

119871119863119896minus 119871119863119896

119871119863119896

with 119896 = 0 1 2 119899 = 64

(15)



0 10 20 30 40 50 60

004

006

008

010

012

014Er

ror (

)



0 10 20 30 40 50 60

005

010

015

020


Erro

r (

)


0 10 20 30 40 50 600005

0010

0015

0020

0025


Erro

r (

)



6 Conclusion








Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










harrP0

harrP1

harrP2

harrP3

harrP4

(a)

harrP(00)

harrP(10)

harrP(20)

harrP(12)

harrP(22)

harrP(33)

harrP(32)harr

P(31)harrP(30)

(b)




larrrarr119861119904 (119905) =

119896+ℎminus1

sum119894=1

larrrarr119875119894119861119894ℎ(119905) (6)







larrrarr119861119904119878 (119904 119905) =

119898

sum119894=0

119899

sum119895=0

larrrarr119875(119894119895)119873119894119901(119904)119873119895119902(119905) (7)

where (i) 119873119894119901(119904) and 119873



(119894119895)= ⟨larrrarr119875larr

(119894119895) 119875(119894119895)larrrarr119875rarr

(119894119895)⟩ are fuzzy









(119894119895)120572119896


119896isin (0 1] with 119896 = 1 2 119897

larrrarr119875(119894119895)120572119896


(119894119895)120572119896

119875(119894119895)larrrarr119875rarr

(119894119895)120572119896

⟩


(119894119895)) 120572119896 +larrrarr119875larr

(119894119895)] 119875(119894119895)



(119894119895)]⟩

(8)



and 119875(119894119895)120572119896

canbe given as

119875120572119896

= 119875(119894119895)120572119896

where 119875(119894119895)120572119896

=1

3sum119894=0119895=0

⟨119875larr

(119894119895)120572119896

119875(119894119895) 119875rarr

(119894119895)120572119896

⟩

(9)


(a) (b)





5 Lakebed Modeling







harrz



larr997888997888rarr119861119904119878119871 (119904 119905) =

119898

sum119894=07

119899

sum119895=07

119871larrrarr119863(119894119895)1198731198943(119904)1198731198953(119905) (10)




larr997888997888rarr11986111990411987811987112057205

(119904 119905) =

119898

sum119894=07

119899

sum119895=07

119871larrrarr119863(119894119895)12057205

1198731198943(119904)1198731198953(119905) (11)

where 119871larrrarr119863(119894119895)12057205


119871larrrarr119863(119894119895)12057205


(119894119895)12057205

119871

119863(119894119895)119871larrrarr119863rarr

(119894119895)12057205

⟩

= ⟨[(119871


(119894119895)) 05 +119871larrrarr119863larr

(119894119895)] 119871

119863(119894119895)


(119894119895) minus119871

119863(119894119895)) 05 +


(119894119895)]⟩

(12)







11986111990411987811987112057205

(119904 119905) =

119898

sum119894=07

119899

sum119895=07

119871

119863(119894119895)12057205

1198731198943(119904)1198731198953(119905) (13)

where 119871119863(119894119895)12057205


119871

119863(119894119895)12057205

=1

3sum

119894=07119895=07


(119894119895)12057205

119863(119894119895)119871larrrarr119863rarr

(119894119895)12057205

⟩

(14)






sum119899

119896=0119899

119871119863119896119890

sum119896=0119899

119899 ( 119871119863119896119890

)where 119871119863

119896119890

=

119871119863119896minus 119871119863119896

119871119863119896

with 119896 = 0 1 2 119899 = 64

(15)



0 10 20 30 40 50 60

004

006

008

010

012

014Er

ror (

)



0 10 20 30 40 50 60

005

010

015

020


Erro

r (

)


0 10 20 30 40 50 600005

0010

0015

0020

0025


Erro

r (

)



6 Conclusion








Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)





5 Lakebed Modeling







harrz



larr997888997888rarr119861119904119878119871 (119904 119905) =

119898

sum119894=07

119899

sum119895=07

119871larrrarr119863(119894119895)1198731198943(119904)1198731198953(119905) (10)




larr997888997888rarr11986111990411987811987112057205

(119904 119905) =

119898

sum119894=07

119899

sum119895=07

119871larrrarr119863(119894119895)12057205

1198731198943(119904)1198731198953(119905) (11)

where 119871larrrarr119863(119894119895)12057205


119871larrrarr119863(119894119895)12057205


(119894119895)12057205

119871

119863(119894119895)119871larrrarr119863rarr

(119894119895)12057205

⟩

= ⟨[(119871


(119894119895)) 05 +119871larrrarr119863larr

(119894119895)] 119871

119863(119894119895)


(119894119895) minus119871

119863(119894119895)) 05 +


(119894119895)]⟩

(12)







11986111990411987811987112057205

(119904 119905) =

119898

sum119894=07

119899

sum119895=07

119871

119863(119894119895)12057205

1198731198943(119904)1198731198953(119905) (13)

where 119871119863(119894119895)12057205


119871

119863(119894119895)12057205

=1

3sum

119894=07119895=07


(119894119895)12057205

119863(119894119895)119871larrrarr119863rarr

(119894119895)12057205

⟩

(14)






sum119899

119896=0119899

119871119863119896119890

sum119896=0119899

119899 ( 119871119863119896119890

)where 119871119863

119896119890

=

119871119863119896minus 119871119863119896

119871119863119896

with 119896 = 0 1 2 119899 = 64

(15)



0 10 20 30 40 50 60

004

006

008

010

012

014Er

ror (

)



0 10 20 30 40 50 60

005

010

015

020


Erro

r (

)


0 10 20 30 40 50 600005

0010

0015

0020

0025


Erro

r (

)



6 Conclusion








Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















11986111990411987811987112057205

(119904 119905) =

119898

sum119894=07

119899

sum119895=07

119871

119863(119894119895)12057205

1198731198943(119904)1198731198953(119905) (13)

where 119871119863(119894119895)12057205


119871

119863(119894119895)12057205

=1

3sum

119894=07119895=07


(119894119895)12057205

119863(119894119895)119871larrrarr119863rarr

(119894119895)12057205

⟩

(14)






sum119899

119896=0119899

119871119863119896119890

sum119896=0119899

119899 ( 119871119863119896119890

)where 119871119863

119896119890

=

119871119863119896minus 119871119863119896

119871119863119896

with 119896 = 0 1 2 119899 = 64

(15)



0 10 20 30 40 50 60

004

006

008

010

012

014Er

ror (

)



0 10 20 30 40 50 60

005

010

015

020


Erro

r (

)


0 10 20 30 40 50 600005

0010

0015

0020

0025


Erro

r (

)



6 Conclusion








Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60

004

006

008

010

012

014Er

ror (

)



0 10 20 30 40 50 60

005

010

015

020


Erro

r (

)


0 10 20 30 40 50 600005

0010

0015

0020

0025


Erro

r (

)



6 Conclusion








Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









fuzzy b-spline surface modeling

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