research article log-aesthetic curves for shape completion...

Research ArticleLog-Aesthetic Curves for Shape Completion Problem

R U Gobithaasan1 Yip Siew Wei1 and Kenjiro T Miura2

1 School of Informatics amp Applied Mathematics Universiti Malaysia Terengganu 21030 Kuala Terengganu Terengganu Malaysia2 Graduate School of Engineering Shizuoka University Shizuoka 432-8561 Japan

Correspondence should be addressed to R U Gobithaasan grumtedumy

Received 20 April 2014 Revised 8 July 2014 Accepted 11 July 2014 Published 11 August 2014

Academic Editor Juan Manuel Pena

Copyright copy 2014 R U Gobithaasan 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

An object with complete boundary or silhouette is essential in various design and computer graphics feats Due to various reasonssome parts of the object can bemissing hence increasing the complexity in designing process It is therefore important to reconstructthe missing parts of an object while retaining its aesthetic appearance In this paper we propose Log-Aestheic Curves (LAC) forshape completion problem We propose an algorithm to construct LAC segment and subsequently fit into the gap of the missingparts with C-shape or S-shape For C-shape completion we define LAC segment by specifying two endpoints and their respectivetangent directions between the gaps while for S-shape the user defines an inflection point in between the endpoints The finalsection illustrates three examples to showcase the efficiency of the proposed algorithm The results are further compared withKimiarsquos method to prove that the algorithm produces equally good result Additionally the proposed algorithm provides an extradegree of freedom in which the user would be able to choose the type of spiral that they desire to solve the shape completionproblem

1 Introduction

In a design environment there are situations in which theoutlines of an object are missing or occludedThese scenariosare commonly known as shape completion gap completionor curve completion problem It is important to recompletethe shape of the missing object while retaining its aestheticappearance since a complete object can be used further forapplications inmodeling mesh analysis and computer-aideddesign as well as manufacturing purposes The completeboundary or silhouette provides geometrical information ofa model which is significant in engineering analysis andmanufacturing process In order to reconstruct the missingboundaries of an object the associated curve segment mustsatisfy the boundary conditions so that the curve passesthrough the endpoints with specified tangent direction Todate there are numbers of algorithms available for shapecompletion problem (see the works of [1ndash6])

A notable work which received much attention is byKimia et al [3] where they proposed an algorithm whichutilizes Euler spiral to solve shape completion problem

Euler spiral is also known as Cornursquos spiral or clothoidThe Euler spiral is widely applied in highwayrailway orrobotic trajectories design due to its linear curvature property(curvature varies linearly with respect to arc length) Thealgorithm proposed by Kimia et al [3] is the implementationof Euler spiral to solve the shape completion problem Intheir proposed algorithm two endpoints (119875

0 1198752) and their

tangential angles (1205790 1205792) are the inputs In order to construct

Euler spiral for the given problem curvature and arc lengthof the curve are required to be identified Since the proposedEuler spiral algorithm is an iterative process initial guess forthese two parameters (curvature and arc length) is neededKimia et al [3] recommended the use of biarc to obtainthe initial curve with its corresponding curvature and arclength for the proposed algorithm Note that the goal of theirproposed algorithm is to minimalize the distance betweenlast point and resulting curve in order for the gap to be filledusing gradient descent method

Log-Aesthetic Curves (LAC) are curves where the log-arithmic curvature graph (LCG) can be represented bya straight line The idea of LCG was formulated by

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 960302 10 pageshttpdxdoiorg1011552014960302

2 Journal of Applied Mathematics

05 10 15 20 25

05

10

15

20

X

Y

(a) Two segments of Euler spiral

00 05 10 15

00

05

10

15

20

minus05minus10 minus05

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

(b) Its corresponding logarithmic curvature graph

Figure 1 Euler spiral defined 119904 isin [02 1] (solid line) and 119904 isin [1 3] (dotted) and its corresponding LCG

Harada et al (1999) when they attempted to classify all thecurves used in automobile design It was first known aslogarithmic distribution diagram of curvature (LDDC) Itis a graph to measure the relationship between curvatureradius (denoted by 120588) and arc length of the curve segment(denoted by 119904) They found that all the curves employed inautomobile design depict linear gradient of LDDC Kanaya etal [7] refined the formulation of LDDC which is now knownas LCG The horizontal axis of LCG represents log 120588 and thevertical axis represents log 120588(119889119904

10158401198891205881015840) [8] Figure 1 illustrates

the two segments of Euler spiralIn 2005 Miura et al [9] derived a fundamental formula

of LAC based on a linear representation of LCG A yearlater Yoshida and Saito [10] investigated the characteristicsand overall shape of LAC They also proposed a method toconstruct LAC segment interactively by specifying endpointsand their tangent vectors In 2009 they further extendedLAC into space curve and proposed amethod to interactivelycontrol the curves which is similar to quadratic Beziercurves

LAC has become a promising curve in graphic andindustrial design due to its monotonic curvature propertyThere are two shape parameters of LAC denoted by 120572 andΛ The shape parameter 120572 is used to determine the type ofspiral within the family of LAC and Λ is used to satisfythe 1198661 constraint during the design process LAC comprises

Euler spiral (120572 = minus1) Nielsen spiral (120572 = 0) logarithmic orequiangular spiral (120572 = 1) circle involute (120572 = 2) and others

In 2011 Gobithaasan and Miura [11] expanded the familyof LAC into Generalized Log-Aesthetic Curve (GLAC) byadding generalized Cornu spiral (GCS) to the family of LACThis includes Euler spiral logarithmic spiral circle involuteNielsen spiral LAC and GCS as the members of GLAC

To note GCS is proposed by Ali et al [12] where it becomesEuler spiral when the shape parameter of GCS is zero GLAChas an extra shape parameter compared to LAC in which itcan be used to satisfy additional constraints that occur duringdesign process [13] The curvature function of GLAC is alsomonotonic similar to LAC In 2012 they further extendedGLAC into space curve

In 2012 Ziatdinov et al [14] defined LAC in the formof incomplete gamma functions They claimed that this newexpression of LAC is capable of reducing the computationtime up to 13 times faster In 2013 the problem of1198662 Hermiteinterpolation was dealt with by Miura and Gobithaasan [15]by introducing log-aesthetic splines which consists of tripleLAC segments Readers are referred to [16] for a detailedreview on aesthetic curves

In this paper we introduce the implementation of 1198661

LAC algorithm to fit the gap in between the outlines ofmissingocclusion objects We compare the performance ofthis algorithm with the results obtained by Kimia et alrsquos[3] method on the same examples in order to show theeffectiveness of LAC to solve shape completion problem

The rest of this paper is organized as follows First the for-mulation of LAC is reviewed followed by a detailed explana-tion on the LAC algorithm for solving the shape completionproblem In Section 3 we demonstrate the implementationof the proposed algorithm with various 120572 values with threeexamples to show the implementation of this method Thedata and calculation used in the examples are also depictedin this sectionThe output of shape completion using Kimiarsquosmethod is also depicted for comparison purposes We alsomake comparison of the results obtained using Kimia et alrsquos[3] method and our algorithm We further show some shapevariations which can be obtained by using S-shape LAC in

Journal of Applied Mathematics 3

Pc

Pa

Pb

120579f

120579d

(a) Original position of controlpoints

P0

P1P2

120579d998400120579f998400

(b) Position of control points after transformation

Figure 2 The configuration of triangle consists of three control points for case 120572 gt 1 The curve inside the triangle represents the generatedLAC segment

which the designer tweaks the segment in order to obtain thedesired shape

2 Log-Aesthetic Curves in Shape Completion

In this section we show the parametric expression of LACand the algorithm used to obtain the solution for shapecompletion problem with various 120572 values

21 The Analytical Formulas and Parameters There are twoshape parameters used to define LAC segment 120572 and Λ Theshape parameter 120572 determines the type of spiral andΛ is usedto control the curve so that the constraints are satisfied Let 120588and 120579 represent radius of curvature and tangent angle of thecurve respectively

Equation (1) shows the radius of curvature function interms of its turning angle To note (1) is used to define theparametric expression of LAC in terms of its turning angle asshown in (2) as follows

120588 (120579) =

119890Λ120579

if 120572 = 1

((120572 minus 1) Λ120579 + 1)1(120572minus1)

otherwise(1)

LAC (120579) = 119875119886+ int

120579

0

119890119894119906120588 (119906) 119889119906 (2)

where 119875119886represents starting point and 119894 and 119890 represent the

imaginary unit and exponent respectively LAC can also berepresented in the arc length parameter 119904 as follows

LAC (119904) =

119875119886+ int

119904

0

119890119894((1Λ)(1minus119890

minusΛ119906

))119889119906 if 120572 = 0

119875119886+ int

119904

0

119890119894(log[Λ119906+1]Λ)

119889119906 if 120572 = 1

119875119886+ int

119904

0

119890119894(((Λ120572119906+1)

(120572minus1)120572

minus1)Λ(120572minus1))119889119906 otherwise

(3)

Both (2) and (3) generate the same LAC The curve canbe drawn in complex plane by the 119909-axis representing realnumbers and the 119910-axis representing imaginary numbersReaders are referred to [10] for a detailed derivation of theparametric expression of LAC

22 The LAC Algorithm

C-Shape We customize the algorithm of constructing LACsegment to fill the gap by specifying two endpoints its tangentvector and120572 valueTheC-shape LAC curve is a direct processin which the user may interactively input the endpointsand the tangent direction of those points We can furtherautomate the identification of tangents at the endpoints byusing three-point circular arc approximation around theendpoints as explained in Steps 1 and 2 in the algorithmbelowUsing these tangents we may calculate the midpoint Thesethree points are then used to fit LAC segment in the gapfor shape completion problem The idea of this method isto find a LAC segment defined in a triangle composed of


Pc

Pa

Pb

120579d

120579e


P0 P1

P2

120579d998400120579e998400


Figure 3 The configuration of triangle consists of three control points for case 120572 le 1 The curve inside the triangle represents the generatedLAC segment

three control points as proposed by Yoshida and Saito [10]Therefore by using thismethod it is only possible to generatea C-curve Figures 2 and 3 show the configuration of thetriangle formed by three control points for cases 120572 gt 1 and120572 le 1 respectively Step 5 in the algorithm below elaboratesthese configurations

S-Shape Basically a S-shape LAC segment consists of two C-shape curves joined with 119866

1 continuity In order to constructthe S-curve for a given gap the inflection point is essential Asimple approach used is that users are given the freedom todefine the inflection point so that the algorithm is carried outtwice consisting of two C-shape segments of LAC Thus thisapproach needs two endpoints its tangent vectors 120572 valueand an inflection point

The process involves breaking the S-curve into two C-curves so the first piece of curve is constructed betweenstarting point (black) and inflection point (gray) while thesecond piece is between inflection point (gray) and endpoint(black) Figure 4 illustrates the inflection point that we definefor one of the numerical examples in this paper

Algorithm 1 demonstrates the LAC algorithm

221 Computation of Tangent Angles (Step 1ndashStep 3) To auto-mate the process of defining the tangents at the endpointswe may use circular arc approximation (osculating circle)that best fits the endpoints and points surrounding them inorder to determine the tangent angle at both endpoints Sinceit is an osculating circle we substitute the points into theequation of circleThen we identify the center (ℎ 119896) radius 119903and turning angle 119905 of circle from three points which include

Figure 4 The input for S-shape completion problem

the endpoint Next we determine the unit tangent vector atthe endpoints and subsequently calculate the correspondingtangent angle by using specified formula as indicated in thealgorithm Alternately we may also allow the user to directlydefine tangent vectors at those points

222 Identification of Second Control Point (Step 4) Once theunit tangent vector at endpoints is identified we determinethe coordinate of the second control point 119875

119887which is


REMARK Endpoints 119875119886 119875119888 inflection point 119875inflect and 120572 are user defined The tangent angle at both endpoints 120579

0 1205792 can be

obtained either from three point circular arc approximation around the endpoints or the user defines it Let (119883 119884) represent thecoordinate of a pointINPUT 119875

119886 119875119888 119875inflect 120572

OUTPUT 120579119889 Λ 119903scale LAC segment

BeginStep 1 Set 119891 (119905) larr ℎ + 119903 cos (119905) 119896 + 119903 sin (119905) where (ℎ 119896) is the center of circle 119903 is radius and 119905 represents angleStep 2 Identify unit tangent vector UTV (119905) larr 119891

1015840(119905)

10038171003817100381710038171198911015840(119905)

1003817100381710038171003817 and set (119883new 119884new) larr (119883 119884) + UTV (119905)Step 3 Calculate directional angle 120579 larr tanminus1((119884new minus 119884)(119883new minus 119883))Step 4 Solve simultaneous equations 119884 = 119898119883 + 119888 where 119898 larr (119884new minus 119884)(119883new minus 119883) to get position of second control point 119875

119887

Step 5 If 10038171003817100381710038171198751198861198751198871003817100381710038171003817 le

10038171003817100381710038171198751198871198751198881003817100381710038171003817 then 119875

119886larr 119875119886 119875119888larr 119875119888

else 119875119886larr 119875119888 119875119888larr 119875119886

If 120572 gt 1 then 119875start larr 119875119888 119875end larr 119875

119886

else 119875start larr 119875119886 119875end larr 119875

119888

Step 6 Translate 119875start = (0 0) rotate triangle such that 119875119887= (minus119883 0) for 120572 gt 1 119875

119887= (119883 0) for 120572 le 1

If 119875end = (119883 minus119884) or 119875end = (minus119883 minus119884) then reflect the triangle through 119909-axisStep 7 Compute 120579

119889larr 120587 minus cosminus1((minus 1003817100381710038171003817119875119886119875119888

1003817100381710038171003817

2

+1003817100381710038171003817119875119887119875119888

1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119887

1003817100381710038171003817

2

)(21003817100381710038171003817119875119887119875119888

1003817100381710038171003817

10038171003817100381710038171198751198861198751198871003817100381710038171003817))

Step 8 If 120572 gt 1 then 120579119891larr cosminus1((minus 1003817100381710038171003817119875119886119875119887

1003817100381710038171003817

2

+1003817100381710038171003817119875119887119875119888

1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119888

1003817100381710038171003817

2

)(21003817100381710038171003817119875119886119875119888

1003817100381710038171003817

10038171003817100381710038171198751198871198751198881003817100381710038171003817))

else 120579119890larr cosminus1((minus 1003817100381710038171003817119875119887119875119888

1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119887

1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119888

1003817100381710038171003817

2

)(21003817100381710038171003817119875119886119875119887

1003817100381710038171003817

10038171003817100381710038171198751198861198751198881003817100381710038171003817))

Step 9 Identify bound of Λ If 120572 lt 1 then 119886 larr 1 times 10minus10 119887 larr 1(120579

119889(1 minus 120572))

else if 120572 gt 1 then 119886 larr 1 times 10minus10 119887 larr 1(120579

119889(120572 minus 1))

else 119886 larr 1 times 10minus10 119887 larr 1 times 10

5Step 10 Set tolerance Tol larr 10

minus10iteration number 119873 larr (1 In 2) In (| 119886 minus 119887 | Tol)

Step 11 Calculate Λ using Bisection methodStep 12 Determine scaling factor 119903scale larr

10038171003817100381710038171198751198861198751198881003817100381710038171003817

1003817100381710038171003817119875011987521003817100381710038171003817

Step 13 Scale 119903scale to the curve and transform inversely the triangle to its original positionStep 14 Construct LAC segment using (2)Step 15 OUTPUT (120579

119889 Λ 119903scale LAC segment)

End

Algorithm 1

the intersection point of the tangential line at both endpointsby solving the equations simultaneously

223 Affirmation of Initial and Last Point Position (Step 5)We assume the distance between 119875

119886and 119875

119887is less than 119875

119887

and 119875119888as proposed by Yoshida and Saito [10] The position

of 119875119886and 119875

119888is switched if the criterion is not satisfied

Otherwise the position remains unchanged Note that thereare two cases of constructing LAC segment based on the120572 value as illustrated in Figures 2 and 3 For case 120572 le 1the initial endpoint 119875

119886is the starting point during the curve

construction process 119875start On the other hand starting pointof curve construction process for case 120572 gt 1 is the lastendpoint 119875

119888

224 Transformation Process (Step 6) The transformationprocess includes translation rotation and reflection of theoriginal triangle that consist of119875

119886119875119887119875119888 Initially the original

triangle is translated to the origin so that the starting point119875start is located at the origin Next we rotate the triangleso that the second control point is placed on the 119909-axisFinally we ensure the last point of the curve constructionprocess 119875end is located at the positive 119910-axis region If notthe reflection process is needed to reflect the last point with

negative 119910-coordinate through the 119909-axis The configurationafter transformation is illustrated in Figures 2 and 3

225 Computation of Tangent and Turning Angle (Step 7and Step 8) We apply the law of cosine of a triangle tocalculate corresponding angles as shown in Figures 2 and 3The tangent angle 120579

119889and turning angle 120579

119890for case 120572 le 1

or 120579119891for case 120572 gt 1 will be used in the subsequent step for

calculating shape parameter Λ

226 Upper Bound of Shape Parameter Λ (Step 9) For thecasewhen120572 = 1 there is no upper bound for shape parameterΛ So the value of Λ can be arbitrarily large However thereare upper bounds for Λ when 120572 lt 1 and 120572 gt 1 ThereforetheΛ value of these two cases is restricted within the range of(119886 119887) where 119886 and 119887 are assigned a specified value as indicatedin the algorithm above

227 Shape Parameter Λ Calculation (Step 10 and Step 11)We determine the Λ value that satisfies the constraintsof endpoints and angles by using bisection method Firsttolerance and bound of Λ are used in the calculation ofiteration number for bisection methodThen the calculatingprocess will run until the iteration number is met Once theΛ


(a) Euler spiral algorithm (b) LAC algorithm with two tangents setting for s-shaped curve and 120572 = minus1

(c) LACalgorithmwith two tangents setting for s-shaped curveand 120572 = 2

Figure 5 The results of fitting Euler spiral and LAC segment for two missing parts of a jug

value of the curve is obtained it is an interpolating curve thatsatisfies boundary constraints as shape parameter Λ is usedto control the curve so that the constraints are satisfied TheobtainedΛ value ensures the generated LAC segment starts atthe starting point and ends at the last point while the shape ofcurve follows the specified tangent vectors at two endpoints

228 Computation of Scale Factor (Step 12) We calculate thescaling factor by using the formula as shown in the algorithmwhere 119875

119886and 119875

119888are endpoints of the original triangle while

1198750and 119875

2are endpoints of the transformed triangle

229 Construction of LACSegment (Step 13ndashStep 15) In orderto obtain the curve which is defined in the original trianglewe inversely transform the triangle to its original position

The curve segment is constructed using (2) with the obtainedvalue of Λ 120579

119889 119903scale

3 Numerical Examples and Discussion

This section shows the input data and numerical and graph-ical results of implementing the algorithm presented inSection 2 for three examples For comparison purpose wepresent the results of Kimiarsquos algorithm [3] algorithm on thesame examples as well

31 The Graphical Results Figure 5 shows the first exampleof fitting Euler spiral and LAC segment to two parts of a jugThe results of the second example (artifact) and third example(vase) are depicted in Figures 6 and 7 respectively


(a) Euler spiral algorithm (b) LAC algorithm when 120572 = minus1

(c) LAC algorithm when 120572 = 0

Figure 6 The results of fitting Euler spiral and LAC segment for an artifact

32 The Numerical Results The input data and the obtainedparameter values to render LAC segment for C-shaped andS-shaped gaps are shown in Tables 1 2 and 3 respectively

33 Discussion We implemented LAC algorithm with var-ious 120572 values for each example In Figures 5 6 and 7 weshowed a subfigure of LAC with 120572 = minus1 in between the gapsfor the missing part of the objectsThese examples are indeedEuler spiral similar to the work of Kimia et al [3] Users areindeed fitting a circle involute when 120572 = 2 as shown in eachfigure Nielsen spiral is fitted when 120572 = 0 and logarithmicspiral when 120572 = 1 The shape parameter 120572 definitely dictatesthe type of spiral the user desires to fit into the gap

Based on the comparison of graphical results in Figures5 6 and 7 we found that Kimiarsquos algorithm [3] and theLAC algorithm behave similarly However the proposed LACalgorithm gives more flexibility to designers as they canmanipulate the shape parameter 120572 and the tangent directionat the inflection point in order to get various S-shaped LACsegments

4 Conclusion

A LAC segment can be constructed by specifying the end-points and their tangent directions LAC is a family of aes-thetic curves where their curvature changes monotonicallyIt comprises Euler spiral logarithmic spiral Nielsen spiraland circle involute with different 120572 values The algorithmfor constructing LAC segment to solve shape completion

problem is defined in Section 2 The results of implementingboth Kimia et alrsquos [3] algorithm and the proposed LACalgorithm on the same images are depicted in Section 3LAC has an upper hand due to its flexibility although bothalgorithms are capable of solving shape completion problemsSince LAC represents a bigger family rather than Eulerspiral alone it is clear that more shapes are available to fillthe missing gap Moreover the resulting LAC segments aresatisfying119866

1 continuity where the curve segments interpolateboth endpoints in the specified tangent directions In case theuser wants a119866

2 LAC solution one may plug in the techniqueproposed by Miura et al [15] or employ 119866

2 GLAC [17] inwhich we may use shape parameter V in GLAC to satisfy thedesired curvatures

Conflict of Interests

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

Acknowledgments

The authors acknowledge Universiti Malaysia Terengganu(GGP grant) and Ministry of Higher Education Malaysia(FRGS 59265) for providing financial aid to carry out thisresearch They further acknowledge anonymous referees fortheir constructive comments which improved the readabilityof this paper


(a) Euler spiral algorithm (b) Three types of tangents at the inflection points and 120572 = minus1

(c) Tangents setting as in 7(b) and 120572 = minus1 (d) Tangents setting as in 7(b) and 120572 = 1

Figure 7 The results of fitting Euler spiral and LAC segment to a vase

Table 1 The input data for LAC algorithm

Figure Partgap shape Starting point Inflection point Terminating point119875119886

119875inflect 119875119888

Figure 5Outer C (47533 29067) mdash (408 512)Inner C (444 3193) mdash (4067 4907)

S (6867 198) (115139 317874) (132 44533)

Figure 6 Left C (1504 3208) mdash (1232 5896)Right C (853 307) mdash (824 583)

Figure 7C (504 126) mdash (176 120)

Right S (57667 37933) (501722 483092) (44533 598)Left S (21533 59933) (166565 4812) (94 376)


Table 2 The value of parameters obtained in LAC algorithm for C-shaped gap

Figure Part Shape parameter Angles (radian) Scale factorΛ 120579

119889120579119890or 120579119891

119903scale

Figure 5(b) Outer C 0181102 197352 109383 106443Inner C 0187822 250305 153612 577559

Figure 5(c) Outer C 0236827 197352 0879684 179903Inner C 0307163 250305 0966927 144024

Figure 6(b) Left C 0319685 155147 100107 105562Right C 0204861 235562 147579 896424

Figure 6(c) Left C 053364 155147 100107 902434Right C 0610563 235562 147579 654159

Figure 7(c) mdash 0140054 11637 0601359 271762Figure 7(d) mdash 0169061 11637 0601359 270036

Table 3 The value of parameters obtained in LAC algorithm for S-shaped gap

Figure Part Line style Shape parameter Angles (radian) Scale factorΛ 120579

119889120579119890or 120579119891

119903scale

Figure 5(b) mdashPiece 1 Thick 0646746 0739623 0451367 107501

Dashed 113587 043963 0288258 152739

Piece 2 Thick 0140372 132647 0689653 935698Dashed 0473033 102648 063682 766964

Figure 5(c) mdashPiece 1 Thick 219523 043963 0151372 568396

Dashed 107216 0739623 0288256 293721


Figure 7(c) Right S

Piece 1Thick 00000429688 0571501 0285752 227044Dashed 0698712 0871501 0585746 39271Dotted 245758 0371528 0285746 23657


Figure 7(d) Right S



Figure 7(c) Left S



Figure 7(d) Left S




References

[1] M Kolomenkin I Shimshoni and A Tal ldquoDemarcating curvesfor shape illustrationrdquo ACM Transactions on Graphics vol 27no 5 article 157 2008

[2] D Decarlo A Finkelstein S Rusinkiewicz and A SantellaldquoSuggestive contours for conveying shaperdquo ACM Transactionson Graphics vol 22 no 3 pp 848ndash855 2003

[3] B B Kimia I Frankel and A Popescu ldquoEuler spiral for shapecompletionrdquo International Journal of Computer Vision vol 54no 1ndash3 pp 159ndash182 2003

[4] T Judd F Durand and E Adelson ldquoApparent ridges for linedrawingrdquo ACM Transactions on Graphics vol 26 no 3 article19 7 pages 2007

[5] S Yoshizawa A Belyaev and H P Seidel ldquoFast and robustdetection of crest lines on meshesrdquo in Proceedings of the ACMSymposium on Solid and Physical Modeling (SPM rsquo05) pp 227ndash232 June 2005

[6] G Harary and A Tal ldquo3D Euler spirals for 3D curve comple-tionrdquo Computational Geometry Theory and Applications vol45 no 3 pp 115ndash126 2012

[7] I Kanaya Y Nakano and K Sato ldquoSimulated designerrsquoseyesmdashclassification of aesthetic surfacesrdquo in Proceedings of theInternational Conference onVirtual Systems andMultimedia pp289ndash296 2003

[8] R U Gobithaasan and K T Miura ldquoLogarithmic curvaturegraph as a shape interrogation toolrdquo Applied MathematicalSciences vol 8 no 16 pp 755ndash765 2014

[9] K T Miura J Sone A Yamashita and T Kaneko ldquoDerivationof a general formula of aesthetic curvesrdquo in Proceedings of 8thInternational Conference on Humans and Computers pp 166ndash171 Tokyo Japan 2005

[10] N Yoshida and T Saito ldquoInteractive aesthetic curve segmentsrdquoThe Visual Computer vol 22 no 9-11 pp 896ndash905 2006

[11] R U Gobithaasan and K T Miura ldquoAesthetic spiral for designrdquoSains Malaysiana vol 40 no 11 pp 1301ndash1305 2011

[12] JMAli RM Tookey J V Ball andAA Ball ldquoThe generalisedCornu spiral and its application to span generationrdquo Journal ofComputational and Applied Mathematics vol 102 no 1 pp 37ndash47 1999

[13] R U Gobithaasan R Karpagavalli and K T Miura ldquoShapeanalysis of generalized log-aesthetic curvesrdquo International Jour-nal of Mathematical Analysis vol 7 no 33 pp 1751ndash1759 2013

[14] R Ziatdinov N Yoshida and T Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012

[15] K T Miura D Shibuya R U Gobithaasan and S UsukildquoDesigning log-aesthetic splines with119866

2 continuityrdquoComputer-Aided Design and Applications vol 10 no 6 pp 1021ndash1032 2013

[16] K T Miura and R U Gobithaasan ldquoAesthetic curves andsurfaces in computer aided geometric designrdquo InternationalJournal of Automation Technology vol 8 no 3 pp 304ndash3162014

[17] R U Gobithaasan R Karpagavalli and K T Miura ldquoG2continuous generalized log aesthetic curvesrdquo Computer AidedDesign amp Application Accepted for publication

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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


05 10 15 20 25

05

10

15

20

X

Y

(a) Two segments of Euler spiral

00 05 10 15

00

05

10

15

20

minus05minus10 minus05

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

(b) Its corresponding logarithmic curvature graph

Figure 1 Euler spiral defined 119904 isin [02 1] (solid line) and 119904 isin [1 3] (dotted) and its corresponding LCG

Harada et al (1999) when they attempted to classify all thecurves used in automobile design It was first known aslogarithmic distribution diagram of curvature (LDDC) Itis a graph to measure the relationship between curvatureradius (denoted by 120588) and arc length of the curve segment(denoted by 119904) They found that all the curves employed inautomobile design depict linear gradient of LDDC Kanaya etal [7] refined the formulation of LDDC which is now knownas LCG The horizontal axis of LCG represents log 120588 and thevertical axis represents log 120588(119889119904

10158401198891205881015840) [8] Figure 1 illustrates

the two segments of Euler spiralIn 2005 Miura et al [9] derived a fundamental formula

of LAC based on a linear representation of LCG A yearlater Yoshida and Saito [10] investigated the characteristicsand overall shape of LAC They also proposed a method toconstruct LAC segment interactively by specifying endpointsand their tangent vectors In 2009 they further extendedLAC into space curve and proposed amethod to interactivelycontrol the curves which is similar to quadratic Beziercurves

LAC has become a promising curve in graphic andindustrial design due to its monotonic curvature propertyThere are two shape parameters of LAC denoted by 120572 andΛ The shape parameter 120572 is used to determine the type ofspiral within the family of LAC and Λ is used to satisfythe 1198661 constraint during the design process LAC comprises

Euler spiral (120572 = minus1) Nielsen spiral (120572 = 0) logarithmic orequiangular spiral (120572 = 1) circle involute (120572 = 2) and others

In 2011 Gobithaasan and Miura [11] expanded the familyof LAC into Generalized Log-Aesthetic Curve (GLAC) byadding generalized Cornu spiral (GCS) to the family of LACThis includes Euler spiral logarithmic spiral circle involuteNielsen spiral LAC and GCS as the members of GLAC

To note GCS is proposed by Ali et al [12] where it becomesEuler spiral when the shape parameter of GCS is zero GLAChas an extra shape parameter compared to LAC in which itcan be used to satisfy additional constraints that occur duringdesign process [13] The curvature function of GLAC is alsomonotonic similar to LAC In 2012 they further extendedGLAC into space curve

In 2012 Ziatdinov et al [14] defined LAC in the formof incomplete gamma functions They claimed that this newexpression of LAC is capable of reducing the computationtime up to 13 times faster In 2013 the problem of1198662 Hermiteinterpolation was dealt with by Miura and Gobithaasan [15]by introducing log-aesthetic splines which consists of tripleLAC segments Readers are referred to [16] for a detailedreview on aesthetic curves

In this paper we introduce the implementation of 1198661

LAC algorithm to fit the gap in between the outlines ofmissingocclusion objects We compare the performance ofthis algorithm with the results obtained by Kimia et alrsquos[3] method on the same examples in order to show theeffectiveness of LAC to solve shape completion problem

The rest of this paper is organized as follows First the for-mulation of LAC is reviewed followed by a detailed explana-tion on the LAC algorithm for solving the shape completionproblem In Section 3 we demonstrate the implementationof the proposed algorithm with various 120572 values with threeexamples to show the implementation of this method Thedata and calculation used in the examples are also depictedin this sectionThe output of shape completion using Kimiarsquosmethod is also depicted for comparison purposes We alsomake comparison of the results obtained using Kimia et alrsquos[3] method and our algorithm We further show some shapevariations which can be obtained by using S-shape LAC in


Pc

Pa

Pb

120579f

120579d


P0

P1P2

120579d998400120579f998400








120588 (120579) =

119890Λ120579

if 120572 = 1

((120572 minus 1) Λ120579 + 1)1(120572minus1)

otherwise(1)

LAC (120579) = 119875119886+ int

120579

0

119890119894119906120588 (119906) 119889119906 (2)



LAC (119904) =

119875119886+ int

119904

0

119890119894((1Λ)(1minus119890

minusΛ119906

))119889119906 if 120572 = 0

119875119886+ int

119904

0

119890119894(log[Λ119906+1]Λ)

119889119906 if 120572 = 1

119875119886+ int

119904

0

119890119894(((Λ120572119906+1)

(120572minus1)120572


(3)





Pc

Pa

Pb

120579d

120579e


P0 P1

P2

120579d998400120579e998400












119887which is



0 1205792 can be


119886 119875119888 119875inflect 120572



1015840(119905)

10038171003817100381710038171198911015840(119905)


119887

Step 5 If 10038171003817100381710038171198751198861198751198871003817100381710038171003817 le

10038171003817100381710038171198751198871198751198881003817100381710038171003817 then 119875

119886larr 119875119886 119875119888larr 119875119888

else 119875119886larr 119875119888 119875119888larr 119875119886


119886


119888


119887= (119883 0) for 120572 le 1



1003817100381710038171003817

2

+1003817100381710038171003817119875119887119875119888

1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119887

1003817100381710038171003817

2

)(21003817100381710038171003817119875119887119875119888

1003817100381710038171003817

10038171003817100381710038171198751198861198751198871003817100381710038171003817))


1003817100381710038171003817

2

+1003817100381710038171003817119875119887119875119888

1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119888

1003817100381710038171003817

2

)(21003817100381710038171003817119875119886119875119888

1003817100381710038171003817

10038171003817100381710038171198751198871198751198881003817100381710038171003817))


1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119887

1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119888

1003817100381710038171003817

2

)(21003817100381710038171003817119875119886119875119887

1003817100381710038171003817

10038171003817100381710038171198751198861198751198881003817100381710038171003817))


119889(1 minus 120572))


119889(120572 minus 1))





10038171003817100381710038171198751198861198751198881003817100381710038171003817

1003817100381710038171003817119875011987521003817100381710038171003817



End

Algorithm 1



119886and 119875


119887


of 119875119886and 119875





119888







119890for case 120572 le 1











119886and 119875


1198750and 119875




119889 119903scale











4 Conclusion








Acknowledgments








119875inflect 119875119888


S (6867 198) (115139 317874) (132 44533)


Figure 7C (504 126) mdash (176 120)

Right S (57667 37933) (501722 483092) (44533 598)Left S (21533 59933) (166565 4812) (94 376)




119889120579119890or 120579119891

119903scale








119889120579119890or 120579119891

119903scale


Dashed 113587 043963 0288258 152739



Dashed 107216 0739623 0288256 293721


Figure 7(c) Right S



Figure 7(d) Right S



Figure 7(c) Left S



Figure 7(d) Left S




References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Pc

Pa

Pb

120579f

120579d


P0

P1P2

120579d998400120579f998400








120588 (120579) =

119890Λ120579

if 120572 = 1

((120572 minus 1) Λ120579 + 1)1(120572minus1)

otherwise(1)

LAC (120579) = 119875119886+ int

120579

0

119890119894119906120588 (119906) 119889119906 (2)



LAC (119904) =

119875119886+ int

119904

0

119890119894((1Λ)(1minus119890

minusΛ119906

))119889119906 if 120572 = 0

119875119886+ int

119904

0

119890119894(log[Λ119906+1]Λ)

119889119906 if 120572 = 1

119875119886+ int

119904

0

119890119894(((Λ120572119906+1)

(120572minus1)120572


(3)





Pc

Pa

Pb

120579d

120579e


P0 P1

P2

120579d998400120579e998400












119887which is



0 1205792 can be


119886 119875119888 119875inflect 120572



1015840(119905)

10038171003817100381710038171198911015840(119905)


119887

Step 5 If 10038171003817100381710038171198751198861198751198871003817100381710038171003817 le

10038171003817100381710038171198751198871198751198881003817100381710038171003817 then 119875

119886larr 119875119886 119875119888larr 119875119888

else 119875119886larr 119875119888 119875119888larr 119875119886


119886


119888


119887= (119883 0) for 120572 le 1



1003817100381710038171003817

2

+1003817100381710038171003817119875119887119875119888

1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119887

1003817100381710038171003817

2

)(21003817100381710038171003817119875119887119875119888

1003817100381710038171003817

10038171003817100381710038171198751198861198751198871003817100381710038171003817))


1003817100381710038171003817

2

+1003817100381710038171003817119875119887119875119888

1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119888

1003817100381710038171003817

2

)(21003817100381710038171003817119875119886119875119888

1003817100381710038171003817

10038171003817100381710038171198751198871198751198881003817100381710038171003817))


1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119887

1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119888

1003817100381710038171003817

2

)(21003817100381710038171003817119875119886119875119887

1003817100381710038171003817

10038171003817100381710038171198751198861198751198881003817100381710038171003817))


119889(1 minus 120572))


119889(120572 minus 1))





10038171003817100381710038171198751198861198751198881003817100381710038171003817

1003817100381710038171003817119875011987521003817100381710038171003817



End

Algorithm 1



119886and 119875


119887


of 119875119886and 119875





119888







119890for case 120572 le 1











119886and 119875


1198750and 119875




119889 119903scale











4 Conclusion








Acknowledgments








119875inflect 119875119888


S (6867 198) (115139 317874) (132 44533)


Figure 7C (504 126) mdash (176 120)

Right S (57667 37933) (501722 483092) (44533 598)Left S (21533 59933) (166565 4812) (94 376)




119889120579119890or 120579119891

119903scale








119889120579119890or 120579119891

119903scale


Dashed 113587 043963 0288258 152739



Dashed 107216 0739623 0288256 293721


Figure 7(c) Right S



Figure 7(d) Right S



Figure 7(c) Left S



Figure 7(d) Left S




References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Pc

Pa

Pb

120579d

120579e


P0 P1

P2

120579d998400120579e998400












119887which is



0 1205792 can be


119886 119875119888 119875inflect 120572



1015840(119905)

10038171003817100381710038171198911015840(119905)


119887

Step 5 If 10038171003817100381710038171198751198861198751198871003817100381710038171003817 le

10038171003817100381710038171198751198871198751198881003817100381710038171003817 then 119875

119886larr 119875119886 119875119888larr 119875119888

else 119875119886larr 119875119888 119875119888larr 119875119886


119886


119888


119887= (119883 0) for 120572 le 1



1003817100381710038171003817

2

+1003817100381710038171003817119875119887119875119888

1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119887

1003817100381710038171003817

2

)(21003817100381710038171003817119875119887119875119888

1003817100381710038171003817

10038171003817100381710038171198751198861198751198871003817100381710038171003817))


1003817100381710038171003817

2

+1003817100381710038171003817119875119887119875119888

1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119888

1003817100381710038171003817

2

)(21003817100381710038171003817119875119886119875119888

1003817100381710038171003817

10038171003817100381710038171198751198871198751198881003817100381710038171003817))


1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119887

1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119888

1003817100381710038171003817

2

)(21003817100381710038171003817119875119886119875119887

1003817100381710038171003817

10038171003817100381710038171198751198861198751198881003817100381710038171003817))


119889(1 minus 120572))


119889(120572 minus 1))





10038171003817100381710038171198751198861198751198881003817100381710038171003817

1003817100381710038171003817119875011987521003817100381710038171003817



End

Algorithm 1



119886and 119875


119887


of 119875119886and 119875





119888







119890for case 120572 le 1











119886and 119875


1198750and 119875




119889 119903scale











4 Conclusion








Acknowledgments








119875inflect 119875119888


S (6867 198) (115139 317874) (132 44533)


Figure 7C (504 126) mdash (176 120)

Right S (57667 37933) (501722 483092) (44533 598)Left S (21533 59933) (166565 4812) (94 376)




119889120579119890or 120579119891

119903scale








119889120579119890or 120579119891

119903scale


Dashed 113587 043963 0288258 152739



Dashed 107216 0739623 0288256 293721


Figure 7(c) Right S



Figure 7(d) Right S



Figure 7(c) Left S



Figure 7(d) Left S




References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 1205792 can be


119886 119875119888 119875inflect 120572



1015840(119905)

10038171003817100381710038171198911015840(119905)


119887

Step 5 If 10038171003817100381710038171198751198861198751198871003817100381710038171003817 le

10038171003817100381710038171198751198871198751198881003817100381710038171003817 then 119875

119886larr 119875119886 119875119888larr 119875119888

else 119875119886larr 119875119888 119875119888larr 119875119886


119886


119888


119887= (119883 0) for 120572 le 1



1003817100381710038171003817

2

+1003817100381710038171003817119875119887119875119888

1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119887

1003817100381710038171003817

2

)(21003817100381710038171003817119875119887119875119888

1003817100381710038171003817

10038171003817100381710038171198751198861198751198871003817100381710038171003817))


1003817100381710038171003817

2

+1003817100381710038171003817119875119887119875119888

1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119888

1003817100381710038171003817

2

)(21003817100381710038171003817119875119886119875119888

1003817100381710038171003817

10038171003817100381710038171198751198871198751198881003817100381710038171003817))


1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119887

1003817100381710038171003817

2

+1003817100381710038171003817119875119886119875119888

1003817100381710038171003817

2

)(21003817100381710038171003817119875119886119875119887

1003817100381710038171003817

10038171003817100381710038171198751198861198751198881003817100381710038171003817))


119889(1 minus 120572))


119889(120572 minus 1))





10038171003817100381710038171198751198861198751198881003817100381710038171003817

1003817100381710038171003817119875011987521003817100381710038171003817



End

Algorithm 1



119886and 119875


119887


of 119875119886and 119875





119888







119890for case 120572 le 1











119886and 119875


1198750and 119875




119889 119903scale











4 Conclusion








Acknowledgments








119875inflect 119875119888


S (6867 198) (115139 317874) (132 44533)


Figure 7C (504 126) mdash (176 120)

Right S (57667 37933) (501722 483092) (44533 598)Left S (21533 59933) (166565 4812) (94 376)




119889120579119890or 120579119891

119903scale








119889120579119890or 120579119891

119903scale


Dashed 113587 043963 0288258 152739



Dashed 107216 0739623 0288256 293721


Figure 7(c) Right S



Figure 7(d) Right S



Figure 7(c) Left S



Figure 7(d) Left S




References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119886and 119875


1198750and 119875




119889 119903scale











4 Conclusion








Acknowledgments








119875inflect 119875119888


S (6867 198) (115139 317874) (132 44533)


Figure 7C (504 126) mdash (176 120)

Right S (57667 37933) (501722 483092) (44533 598)Left S (21533 59933) (166565 4812) (94 376)




119889120579119890or 120579119891

119903scale








119889120579119890or 120579119891

119903scale


Dashed 113587 043963 0288258 152739



Dashed 107216 0739623 0288256 293721


Figure 7(c) Right S



Figure 7(d) Right S



Figure 7(c) Left S



Figure 7(d) Left S




References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















4 Conclusion








Acknowledgments








119875inflect 119875119888


S (6867 198) (115139 317874) (132 44533)


Figure 7C (504 126) mdash (176 120)

Right S (57667 37933) (501722 483092) (44533 598)Left S (21533 59933) (166565 4812) (94 376)




119889120579119890or 120579119891

119903scale








119889120579119890or 120579119891

119903scale


Dashed 113587 043963 0288258 152739



Dashed 107216 0739623 0288256 293721


Figure 7(c) Right S



Figure 7(d) Right S



Figure 7(c) Left S



Figure 7(d) Left S




References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119875inflect 119875119888


S (6867 198) (115139 317874) (132 44533)


Figure 7C (504 126) mdash (176 120)

Right S (57667 37933) (501722 483092) (44533 598)Left S (21533 59933) (166565 4812) (94 376)




119889120579119890or 120579119891

119903scale








119889120579119890or 120579119891

119903scale


Dashed 113587 043963 0288258 152739



Dashed 107216 0739623 0288256 293721


Figure 7(c) Right S



Figure 7(d) Right S



Figure 7(c) Left S



Figure 7(d) Left S




References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119889120579119890or 120579119891

119903scale








119889120579119890or 120579119891

119903scale


Dashed 113587 043963 0288258 152739



Dashed 107216 0739623 0288256 293721


Figure 7(c) Right S



Figure 7(d) Right S



Figure 7(c) Left S



Figure 7(d) Left S




References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article log-aesthetic curves for shape completion...

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