efficient 3d volume reconstruction from a point cloud using a...

Research ArticleEfficient 3D Volume Reconstruction from a Point Cloud Usinga Phase-Field Method

Darae Jeong 1 Yibao Li2 Heon Ju Lee3 Sang Min Lee3 Junxiang Yang1 Seungwoo Park1

Hyundong Kim1 Yongho Choi 1 and Junseok Kim 1

1Department of Mathematics Korea University Seoul 02841 Republic of Korea2School of Mathematics and Statistics Xirsquoan Jiaotong University Xirsquoan 710049 China3ROKIT Inc Seoul 08512 Republic of Korea

Correspondence should be addressed to Junseok Kim cfdkimkoreaackr

Received 22 November 2017 Accepted 1 January 2018 Published 6 February 2018

Academic Editor Costica Morosanu

Copyright copy 2018 Darae Jeong et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We propose an explicit hybrid numerical method for the efficient 3D volume reconstruction from unorganized point cloudsusing a phase-field method The proposed three-dimensional volume reconstruction algorithm is based on the 3D binary imagesegmentation method First we define a narrow band domain embedding the unorganized point cloud and an edge indicatingfunction Second we define a good initial phase-field function which speeds up the computation significantly Third we usea recently developed explicit hybrid numerical method for solving the three-dimensional image segmentation model to obtainefficient volume reconstruction from point cloud data In order to demonstrate the practical applicability of the proposed methodwe perform various numerical experiments

1 Introduction

In this paper we propose an efficient and robust algorithm forvolume reconstruction from a point cloud Reconstructingthe three-dimensional model from a point cloud is importantin medical applications Surface reconstruction from a pointcloud is a process of finding a surface model that approxi-mates an unknown surface for a given set of sample pointslying on or near the unknown surface [1]

Hoppe et al developed an algorithm to reconstruct asurface in the three-dimensional space from unorganizedpoints scattered on or near the unknown surface The algo-rithm is based on the idea of determining the zero levelset of a signed distance function [2] Kazhdan proposeda surface reconstruction method which takes an orientedpoint set and returns a solid model The method uses Stokesrsquotheorem to calculate the characteristic function (one insidethe model and zero outside of it) of the solid model [3] Toreconstruct implicit surfaces from scattered unorganized dataset Li et al presented a novel numerical method for surfaceembedding narrow volume reconstruction fromunorganized

points [4 5] Yang et al proposed a 3D reconstruction tech-nique from nonuniform point clouds via local hierarchicalclustering [6] Zhao et al developed a fast sweeping levelset and tagging methods [7] Yezzi Jr et al proposed a newmedical image segmentation based on feature-based metricson a given image [8]

Benes et al used the Allen-Cahn equation with a forcingterm to achieve image segmentation [9] Caselles et alproposed a model for active contours which could extractsmooth shapes and could be adapted to find several contourssimultaneously [10] Methods using geometric active contourwere introduced in [11ndash14] Zhang et al developed a weightedsparse penalty and a weighted grouping effect penalty inmodeling the subspace structure [15] Chen used an ICKFCMmethod (ICA analysis and KFCM algorithm) in medicalimage segmentation andmade a good result in extracting thecomplicated images [16] Zhang et al proposed a novel fuzzylevel set method based on finding the minimum of energyfunction to locate the true object boundaries effectively [17]Other numerical studies based on level set method were alsointroduced in [18 19]

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 7090186 9 pageshttpsdoiorg10115520187090186

2 Mathematical Problems in Engineering

050 1 15minus1 minus05minus15minus15

minus1

minus05

0

05

1

15

(a)

minus15

0

15

0 15minus15(b)

Xl

Ωl

(c) (d)

minus15

y0

1515

0x

g(x

y)

05

1

0minus15

(e)

Figure 1 Schematic of defined distance function (a) Point cloud data (b) mesh grid covering the point cloud and local mesh grid (c) localmesh gridΩ119897 embedding a point X119897 (d) narrow band domain and (e) edge indicator function 119892(x)

Mathematical Problems in Engineering 3

In this article we propose an explicit hybrid algorithmfor volume reconstruction from a point cloud Therefore itdoes not need implicit solvers such as multigrid methodsThe computation is fast and efficient because the proposedalgorithm uses a narrow band domain and a good initialcondition

This paper is organized as follows In Section 2 wedescribe a mathematical model and a numerical solutionalgorithm for volume reconstruction from a point cloudWe present the numerical results for several examples inSection 3 In Section 4 we conclude

2 Mathematical Model and NumericalSolution Algorithm

Now we propose an explicit hybrid numerical method forvolume reconstruction from a point cloud using a phase-fieldmethod For X119897 = (119883119897 119884119897) in the two-dimensional space orX119897 = (119883119897 119884119897 119885119897) in the three-dimensional space 119878 = X119897 |1 le 119897 le 119873 denote the point cloud in the two- or the three-dimensional space respectivelyThe geometric active contourmodel based on the mean curvature motion is given by thefollowing evolution equation [20]

120597120601 (x 119905)120597119905= 119892 (x) [minus1198651015840 (120601 (x 119905))1205982 + Δ120601 (x 119905) + 120582119865 (120601 (x 119905))]

(1)

where 119892(x) is an edge indicator function 119865(120601) = 025(1206012 minus1)2 and 120598 is a constant which is related to the phasetransition width Note that here we use a different edgeindicator function and efficient explicit numerical algo-rithm

For simplicity of exposition we first discretize (1) in thetwo-dimensional space Ω = (119886 119887) times (119888 119889) Let ℎ = (119887 minus119886)(119873119909 minus 1) = (119889 minus 119888)(119873119910 minus 1) be the uniform mesh sizewhere 119873119909 and 119873119910 are the number of grid points Let Ωℎ =x = (119909119894 119910119895) 119909119894 = 119886 + (119894 minus 1)ℎ 119910119895 = 119888 + (119895 minus 1)ℎ 1 le119894 le 119873119909 1 le 119895 le 119873119910 be the discrete domain Let 120601119899119894119895 beapproximations of 120601(119909119894 119910119895 119899Δ119905) where Δ119905 is the time stepLet 119889(x) = dist(x 119878) = min1le119897le119873|X119897 minus x| be the distance to thedata 119878 where X119897 = (119883119897 119884119897) In fact we will use the distancefunction as an edge indicator function 119892(x) In practice for119897 = 1 119873 we define a local domain Ω119897 which embedsthe point X119897 and set the minimum value at the grid pointbetween the point and the grid point For exampleΩ119897 is a 3times3gridThen the computational narrowband domain is definedas

Ωnb =119873⋃119897=1

Ω119897 (2)

Outside the narrow band domain we set a large value to theedge indicator function see Figure 1 for the procedure

In this study we apply the simplest sequential splittingprocedure We split (1) into two equations by using theoperator splitting method

120597120601 (x 119905)120597119905 = 119892 (x) [Δ120601 (x 119905) + 120582119865 (120601 (x 119905))] (3)

120597120601 (x 119905)120597119905 = minus119892 (x) 1198651015840 (120601 (x 119905))1205982

= 119892 (x) 120601 (x 119905) minus 1206013 (x 119905)1205982 (4)

For a good initial configuration we set 1206010119894119895 = 1 inside on thenarrow band domain and set 1206010119894119895 = minus1 outside the narrowband domain

Given 120601119899 we solve (3) on the narrow band domain Ωnbby using the explicit Euler method

120601lowast119894119895 minus 120601119899119894119895Δ119905 = 119892119894119895 [Δ 119889120601119899119894119895 + 120582119865 (120601119899119894119895)] (5)

where 120601lowast119894119895 is the intermediate value which is defined at point(119909119894 119910119895) The initial values at inside and outside region of thenarrowbanddomainΩnb act asDirichlet boundary conditionin computing the discrete Laplace operator Δ 119889

Then we analytically solve (4) by the method of separa-tion of variables [21 22] That is 120601119899+1119894119895 = 120595(Δ119905) by analyticallysolving

119889120595 (119905)119889119905 = 119892119894119895

120595 (119905) minus 1205953 (119905)1205982

with the initial condition 120595 (0) = 120601lowast119894119895(6)

The analytic solution is given as

120601119899+1119894119895 = 120601lowast119894119895radic119890minus2119892119894119895Δ1199051205982 + (120601lowast119894119895)2 (1 minus 119890minus2119892119894119895Δ1199051205982)

(7)

Therefore (5) and (7) consist of an efficient and robustalgorithm for volume reconstruction from a point cloud Weshould note that the proposed numerical solution algorithmis fully explicitTherefore we do not need an iterativemethodsuch as multigrid method to solve the governing equationAlso the implementation of the algorithm is straightforward

3 Computational Experiments

31 Two-Dimensional Experiments

311 Motion by Mean Curvature To test the proposed num-erical scheme we perform a numerical experiment The testis motion by mean curvature If we set 119892(x) = 1 and 120582 = 0then the governing equation (1) becomes the original Allen-Cahn equation [23] which is a reaction-diffusion equationdescribing the process of phase separation in a binary alloy


05 100

05

1

(a)

Analytic resultNumerical result

0373

0375

038

0385

039

0395

04

0002 0004 0006 0008 001 00120001

(b)

Figure 2 Temporal evolutions of the radius with Δ119905 = 015ℎ2 up to 119905 = 5000Δ119905 in the two-dimensional space (a) Zero level contour and (b)radius 119877(119905) of circle with respect to time

0

05

1

15

2

25

3

0 1 2 3251505(a)

0

05

1

15

2

25

3

0 1 2 3251505(b)

0

05

1

15

2

25

3

0 1 2 3251505(c)

Figure 3 Temporal evolution of the interface (a) initial condition (b) 100 iterations and (c) 1000 iterations

mixture In the two-dimensional case as 120598 approaches zerothe zero level set of 120601 evolves with the following velocity

119881 = minus120581 = minus 1119877 (8)

where 119881 is the normal velocity 120581 is the curvature and 119877 isthe radius of curvature at the point of the zero level set [24]Then (8) is rewritten by 119889119877(119905)119889119905 = minus1119877(119905) with 119877(0) = 1198770Therefore analytic solution is given as 119877(119905) = radic11987720 minus 2119905

On the computational domain Ω = (0 1) times (0 1)we investigate the motion by mean curvature of the circlein the annulus narrow band domain (119909 119910) | 035 leradic(119909 minus 05)2 + (119910 minus 05)2 le 043 as shown in Figure 2(a) Wedefine an initial condition as

120601 (119909 119910 0) = tanh 1198770 minus radic(119909 minus 05)2 + (119910 minus 05)2

radic2120598 (9)

In this numerical simulation we use the following parame-ters 1198770 = 04 120598 = 0011 ℎ = 1256 Δ119905 = 015ℎ2 and 119879 =5000Δ119905 Figures 2(a) and 2(b) show the temporal evolution ofthe initial circle and its radius with respect to time respect-ively For verification of our numerical results we include theresults of the analytic solution As shown in Figure 2 theinitial circle shrinks under the motion by mean curvature

312 The Basic Working Mechanism of the Algorithm Theedge indicator function 119892(x) is close to zero where the pointcloud exists Therefore the evolution will stop or slow downin the neighborhood of the point cloud In (1) 120597120601120597119905 =minus1198651015840(120601)1205982 + Δ120601makes the phase-field shrink until it reachesthe point cloud by themean curvature flow If the geometry ofthe point cloud is not convex then the term 120582119865(120601)makes thelevel set of the phase-field further shrink For more detailsplease refer to [20] To confirm the working mechanism ofthe algorithm the temporal evolution of the interface in thetwo-dimensional space is shown in Figure 3 Here we use


(a)

Ωl

(b)

(c) (d)

Figure 4 Construction of the three-dimensional distance function 119889(x) and the narrow band domain (a) point cloud data and mesh (b)cross section of point cloud and a local meshΩ119897 of single pointX119897 (c) local mesh gridΩ119897 embedding a pointX119897 and (d) narrow band domainwhich is determined by the distance function

the following parameters 120598 = 00096 ℎ = 1200 and Δ119905 =28284119890 minus 532 Three-Dimensional Experiments Next we discretize (1)in the three-dimensional space that is Ω = (119886 119887) times (119888 119889) times(119890 119891) Let ℎ = (119887 minus 119886)(119873119909 minus 1) = (119889 minus 119888)(119873119910 minus 1) = (119891 minus119890)(119873119911 minus 1) be the uniform mesh size where119873119909119873119910 and119873119911are the total number of grid points LetΩℎ = x = (119909119894 119910119895 119911119896) 119909119894 = 119886+ (119894 minus 1)ℎ 119910119895 = 119888 + (119895 minus 1)ℎ 119911119896 = 119890 + (119896 minus 1)ℎ 1 le 119894 le119873119909 1 le 119895 le 119873119910 1 le 119896 le 119873119911 be the discrete domain Wedefine 120601119899119894119895119896 as approximations of 120601(119909119894 119910119895 119911119896 119899Δ119905) where Δ119905 isthe time step size Let 119889(x) = dist(x 119878) = min1le119897le119873|X119897 minus x| bethe distance to the data 119878 where X119897 = (119883119897 119884119897 119885119897)

Figure 4 represents construction of the three-dimensionaldistance function 119889(x) and the narrow band domain InFigures 4(a) and 4(b) we can see the given point clouddata on computational grid and a local mesh Ω119897 of singlepoint X119897 Here we calculate the distance between the givenpoint X119897 and the grid points x on the local mesh Ω119897 Thedistance function 119889(x) is defined by the shortest one amongthe distanceThen we obtain the narrow band domain whichis determined by the distance function

Now we can straightforwardly extend the two-dimen-sional numerical solutions (5) and (7) to the following three-dimensional solutions

120601lowast119894119895119896 minus 120601119899119894119895119896Δ119905 = 119892119894119895119896 (Δ 119889120601119899119894119895119896 + 120582119865 (120601119899119894119895119896))

120601119899+1119894119895119896 = 120601lowast119894119895119896radic119890minus2119892119894119895119896Δ1199051205982 + (120601lowast

119894119895119896)2 (1 minus 119890minus2119892119894119895119896Δ1199051205982)

(10)

321 Reconstruction from Various Point Clouds First wereconstruct volume of Happy Buddha from the given scat-tered points (119873 = 1621848) as shown in Figure 5(a) [25] Fornumerical test we use the following parameters 120598 = 00069ℎ = 00048 Δ119905 = ℎ(200radic3)119873119909 = 250119873119910 = 500119873119911 = 250Ω = (0 12) times (0 24) times (0 12) and 120582 = 100 In the firstand second rows in Figure 5 we can see the front and backviews of Happy Buddha By the proposed scheme we obtainthe numerical solution 120601 after 200 iterations (see Figure 5(c))with the initial condition in Figure 5(b)

Next we reconstruct volume of Armadillo model fromthe given scattered points (119873 = 129732) as shown inFigure 6(a) [25] For numerical test we use the followingparameters 120598 = 00127 ℎ = 00088 Δ119905 = ℎ(200radic3) 119873119909 =220119873119910 = 250119873119911 = 200 Ω = (0 22) times (0 25) times (0 2) and120582 = 1000 In the first and second rows in Figure 6 we cansee the front and back views of Armadillo model By theproposed scheme we obtain the numerical solution 120601 after400 iterations (see Figure 6(c)) with the initial condition inFigure 6(b)

As the final example we reconstruct volume of StanfordDragon from the given scattered points (119873 = 656469) asshown in Figure 7(a) [25] For numerical test we use the


(a) (b) (c)

Figure 5 Front and back views of Happy Buddha (a) point clouds (b) initial condition of 120601 and (c) numerical solution 120601 after 200 iterations

following parameters 120598 = 000432 ℎ = 0003 Δ119905 =ℎ(200radic3) 119873119909 = 400 119873119910 = 320 119873119911 = 240 Ω = (0 12) times(0 1) times (0 08) and 120582 = 100 In the first and second rowsin Figure 7 we can see the front and back views of StanfordDragon By the proposed scheme we obtain the numericalsolution 120601 after 400 iterations (see Figure 7(c)) with the initialcondition in Figure 7(b)

322 Effect of 120582 In this section we investigate the effect of 120582parameter on the three-dimensional volume reconstructionThe parameter makes the level set of 120601 shrink to the givenpoints We use the same parameters in Figure 7 except forthe 120582 value As shown in Figure 8 if the value of 120582 is smallthen the surface is oversmoothed by the motion by meancurvatureOn the other hand if it is too large then the surfaceis rough

4 Conclusions

In this article we developed an explicit hybrid numericalalgorithm for the efficient 3D volume reconstruction from

unorganized point clouds using a modified Allen-Cahnequation The 3D volume reconstruction algorithm is basedon the 3D binary image segmentation methodThe proposedalgorithm has potential to be used in various practicalindustry such as 3D model printing from scattered scanneddataThe computational results confirmed that the algorithmis very efficient and robust in reconstructing 3D volume frompoint clouds

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The first author (D Jeong) was supported by the NationalResearch Foundation of Korea (NRF) grant funded by theKorean Government (MSIP) (NRF-2017R1E1A1A03070953)The corresponding author (Junseok Kim) was supportedby Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by theMinistry


(a) (b) (c)

Figure 6 Front and back views of Armadillo model (a) initial point clouds (b) initial condition of 120601 and (c) numerical solution 120601 after 400iterations

(a) (b) (c)

Figure 7 Front and back views of Stanford Dragon model (a) initial point clouds (b) initial condition of 120601 and (c) numerical solution 120601after 400 iterations


= 0 = 10e + 2 = 10e + 5

Figure 8 Front and back views of Stanford Dragon model with various 120582 Numerical solution 120601 after 1000 iterations

of Education (NRF-2016R1D1A1B03933243) This work wassupported by the BK21 PLUS program

References

[1] F Remondino ldquoFrom point cloud to surface the modelingand visualization problemrdquo in Proceedings of the InternationalArchives of the Photogrammetry Remote Sensing and SpatialInformation Sciences vol XXXIV-5W10 2003

[2] H Hoppe T DeRose T Duchamp J McDonald andW Stuet-zle ldquoSurface reconstruction from unorganized pointsrdquo Com-puter Graphics vol 26 no 2 pp 71ndash78 1992

[3] M Kazhdan ldquoReconstruction of solid models from orientedpoint setsrdquo in Proceedings of the third Eurographics Symposiumon Geometry Processing pp 73ndash82 2005

[4] Y Li D Lee C Lee et al ldquoSurface embedding narrow volumereconstruction from unorganized pointsrdquo Computer Vision andImage Understanding vol 121 pp 100ndash107 2014

[5] Y Li and J Kim ldquoFast and efficient narrow volume reconstruc-tion from scattered datardquo Pattern Recognition vol 48 no 12article no 5459 pp 4057ndash4069 2015

[6] J Yang R Li Y Xiao and Z Cao ldquo3D reconstruction fromnon-uniform point clouds via local hierarchical clusteringrdquo inProceedings of the 9th International Conference on Digital ImageProcessing ICDIP 2017 China May 2017

[7] H-K Zhao S Osher and R Fedkiw ldquoFast surface reconstruc-tion using the level set methodrdquo in Proceedings of the IEEEWorkshop on Variational and Level Set Methods in ComputerVision VLSM 2001 pp 194ndash199 can

[8] A Yezzi Jr S Kichenassamy A Kumar P Olver and A Tan-nenbaum ldquoA geometric snakemodel for segmentation of medi-cal imageryrdquo IEEE Transactions on Medical Imaging vol 16 no2 pp 199ndash209 1997

[9] M Benes V r Chalupecky and K Mikula ldquoGeometrical imagesegmentation by the Allen-Cahn equationrdquo Applied NumericalMathematics vol 51 no 2-3 pp 187ndash205 2004

[10] V Caselles F Catte T Coll and F Dibos ldquoA geometric modelfor active contours in image processingrdquo Numerische Mathe-matik vol 66 no 1 pp 1ndash31 1993

[11] V Caselles R Kimmel and G Sapiro ldquoGeodesic active con-toursrdquo International Journal of Computer Vision vol 22 no 1pp 61ndash79 1997

[12] T F Chan and L A Vese ldquoActive contours without edgesrdquo IEEETransactions on Image Processing vol 10 no 2 pp 266ndash2772001

[13] J Hahn and C-O Lee ldquoGeometric attraction-driven flow forimage segmentation and boundary detectionrdquo Journal of VisualCommunication and Image Representation vol 21 no 1 pp 56ndash66 2010

[14] S Kichenassamy A Kumar P Olver A Tannenbaum and JYezzi ldquoConformal curvature flows from phase transitions toactive visionrdquo Archive for Rational Mechanics and Analysis vol134 no 3 pp 275ndash301 1996

[15] B Zhang W Wang and X Feng ldquoSubspace Clustering withSparsity and Grouping Effectrdquo Mathematical Problems in Engi-neering vol 2017 Article ID 4787039 9 pages 2017

[16] Y-T Chen ldquoMedical Image Segmentation Using IndependentComponentAnalysis-BasedKernelized Fuzzy c-MeansCluster-ingrdquoMathematical Problems in Engineering vol 2017 Article ID5892039 21 pages 2017

[17] Y Zhang J Xu and H D Cheng ldquoA Novel Fuzzy LevelSet Approach for Image Contour Detectionrdquo MathematicalProblems in Engineering vol 2016 Article ID 2602647 12 pages2016

[18] C Li C Xu C Gui andM D Fox ldquoLevel set evolution withoutre-initialization a new variational formulationrdquo in Proceedingsof the IEEE Computer Society Conference on Computer Visionand Pattern Recognition (CVPR rsquo05) pp 430ndash436 June 2005

[19] L A Vese and T F Chan ldquoA multiphase level set frameworkfor image segmentation using the Mumford and Shah modelrdquoInternational Journal of Computer Vision vol 50 no 3 pp 271ndash293 2002


[20] Y Li and J Kim ldquoA fast and accurate numerical method formedical image segmentationrdquo Journal of the Korean Society forIndustrial and Applied Mathematics vol 14 no 4 pp 201ndash2102010

[21] A M Stuart and A R Humphries Dynamical Systems andNumerical Analysis vol 2 Cambridge University Press NewYork NY USA 1998

[22] Y Li H G Lee D Jeong and J Kim ldquoAn unconditionally stablehybrid numericalmethod for solving theAllen-Cahn equationrdquoComputers amp Mathematics with Applications An InternationalJournal vol 60 no 6 pp 1591ndash1606 2010

[23] S M Allen and JW Cahn ldquoAmicroscopic theory for antiphaseboundary motion and its application to antiphase domaincoarseningrdquo Acta Metallurgica et Materialia vol 27 no 6 pp1085ndash1095 1979

[24] B Appleton and H Talbot ldquoGlobally optimal geodesic activecontoursrdquo Journal of Mathematical Imaging and Vision vol 23no 1 pp 67ndash86 2005

[25] Stanford university computer graphics laboratory httplight-fieldstanfordeduacqhtml

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


050 1 15minus1 minus05minus15minus15

minus1

minus05

0

05

1

15

(a)

minus15

0

15

0 15minus15(b)

Xl

Ωl

(c) (d)

minus15

y0

1515

0x

g(x

y)

05

1

0minus15

(e)

Figure 1 Schematic of defined distance function (a) Point cloud data (b) mesh grid covering the point cloud and local mesh grid (c) localmesh gridΩ119897 embedding a point X119897 (d) narrow band domain and (e) edge indicator function 119892(x)






120597120601 (x 119905)120597119905= 119892 (x) [minus1198651015840 (120601 (x 119905))1205982 + Δ120601 (x 119905) + 120582119865 (120601 (x 119905))]

(1)



Ωnb =119873⋃119897=1

Ω119897 (2)



120597120601 (x 119905)120597119905 = 119892 (x) [Δ120601 (x 119905) + 120582119865 (120601 (x 119905))] (3)

120597120601 (x 119905)120597119905 = minus119892 (x) 1198651015840 (120601 (x 119905))1205982

= 119892 (x) 120601 (x 119905) minus 1206013 (x 119905)1205982 (4)



120601lowast119894119895 minus 120601119899119894119895Δ119905 = 119892119894119895 [Δ 119889120601119899119894119895 + 120582119865 (120601119899119894119895)] (5)



119889120595 (119905)119889119905 = 119892119894119895

120595 (119905) minus 1205953 (119905)1205982




(7)






05 100

05

1

(a)


0373

0375

038

0385

039

0395

04

0002 0004 0006 0008 001 00120001

(b)


0

05

1

15

2

25

3

0 1 2 3251505(a)

0

05

1

15

2

25

3

0 1 2 3251505(b)

0

05

1

15

2

25

3

0 1 2 3251505(c)



119881 = minus120581 = minus 1119877 (8)




radic2120598 (9)




(a)

Ωl

(b)

(c) (d)





120601lowast119894119895119896 minus 120601119899119894119895119896Δ119905 = 119892119894119895119896 (Δ 119889120601119899119894119895119896 + 120582119865 (120601119899119894119895119896))


119894119895119896)2 (1 minus 119890minus2119892119894119895119896Δ1199051205982)

(10)





(a) (b) (c)




4 Conclusions





Acknowledgments



(a) (b) (c)


(a) (b) (c)



= 0 = 10e + 2 = 10e + 5



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of









120597120601 (x 119905)120597119905= 119892 (x) [minus1198651015840 (120601 (x 119905))1205982 + Δ120601 (x 119905) + 120582119865 (120601 (x 119905))]

(1)



Ωnb =119873⋃119897=1

Ω119897 (2)



120597120601 (x 119905)120597119905 = 119892 (x) [Δ120601 (x 119905) + 120582119865 (120601 (x 119905))] (3)

120597120601 (x 119905)120597119905 = minus119892 (x) 1198651015840 (120601 (x 119905))1205982

= 119892 (x) 120601 (x 119905) minus 1206013 (x 119905)1205982 (4)



120601lowast119894119895 minus 120601119899119894119895Δ119905 = 119892119894119895 [Δ 119889120601119899119894119895 + 120582119865 (120601119899119894119895)] (5)



119889120595 (119905)119889119905 = 119892119894119895

120595 (119905) minus 1205953 (119905)1205982




(7)






05 100

05

1

(a)


0373

0375

038

0385

039

0395

04

0002 0004 0006 0008 001 00120001

(b)


0

05

1

15

2

25

3

0 1 2 3251505(a)

0

05

1

15

2

25

3

0 1 2 3251505(b)

0

05

1

15

2

25

3

0 1 2 3251505(c)



119881 = minus120581 = minus 1119877 (8)




radic2120598 (9)




(a)

Ωl

(b)

(c) (d)





120601lowast119894119895119896 minus 120601119899119894119895119896Δ119905 = 119892119894119895119896 (Δ 119889120601119899119894119895119896 + 120582119865 (120601119899119894119895119896))


119894119895119896)2 (1 minus 119890minus2119892119894119895119896Δ1199051205982)

(10)





(a) (b) (c)




4 Conclusions





Acknowledgments



(a) (b) (c)


(a) (b) (c)



= 0 = 10e + 2 = 10e + 5



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





05 100

05

1

(a)


0373

0375

038

0385

039

0395

04

0002 0004 0006 0008 001 00120001

(b)


0

05

1

15

2

25

3

0 1 2 3251505(a)

0

05

1

15

2

25

3

0 1 2 3251505(b)

0

05

1

15

2

25

3

0 1 2 3251505(c)



119881 = minus120581 = minus 1119877 (8)




radic2120598 (9)




(a)

Ωl

(b)

(c) (d)





120601lowast119894119895119896 minus 120601119899119894119895119896Δ119905 = 119892119894119895119896 (Δ 119889120601119899119894119895119896 + 120582119865 (120601119899119894119895119896))


119894119895119896)2 (1 minus 119890minus2119892119894119895119896Δ1199051205982)

(10)





(a) (b) (c)




4 Conclusions





Acknowledgments



(a) (b) (c)


(a) (b) (c)



= 0 = 10e + 2 = 10e + 5



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





(a)

Ωl

(b)

(c) (d)





120601lowast119894119895119896 minus 120601119899119894119895119896Δ119905 = 119892119894119895119896 (Δ 119889120601119899119894119895119896 + 120582119865 (120601119899119894119895119896))


119894119895119896)2 (1 minus 119890minus2119892119894119895119896Δ1199051205982)

(10)





(a) (b) (c)




4 Conclusions





Acknowledgments



(a) (b) (c)


(a) (b) (c)



= 0 = 10e + 2 = 10e + 5



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





(a) (b) (c)




4 Conclusions





Acknowledgments



(a) (b) (c)


(a) (b) (c)



= 0 = 10e + 2 = 10e + 5



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





(a) (b) (c)


(a) (b) (c)



= 0 = 10e + 2 = 10e + 5



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





= 0 = 10e + 2 = 10e + 5



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




efficient 3d volume reconstruction from a point cloud using a...

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