Abstract

A technique to generate a photorealistic three-dimensional (3D) image and color-textured model using a dedicated optical digitizer is presented. The proposed technique is started with the range and texture image acquisition from different viewpoints, followed by the registration and integration of multiple range images to get a complete and nonredundant point cloud that represents a real-life object. The accuracy of the range image and the precision of correspondence between the range image and texture image are guaranteed by sensor system calibration. Based on the point cloud, a geometric model is established by considering the connectivity of adjacent range image points. In order to enhance the photorealistic effect, we suggest a texture blending technique that utilizes a composite-weight strategy to blend the texture images within the overlapped region. This technique allows more efficient removal of the artifacts existing in the registered texture image, leading to a 3D image with photorealistic quality and color-texture modeling. Experimental results are also presented to testify to the validity of the proposed method.

© 2012 Optical Society of America

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2011

R. Tutsch, M. Petz, and M. Fischer, “Optical three-dimensional metrology with structured illumination,” Opt. Eng. 50, 101507–101510 (2011).
[CrossRef]

F. Uccheddu, A. Pelagotti, and F. Picchioni, “A greedy multiresolution method for quasi automatic texture mapping,” Proc. SPIE 8085, 80850M (2011).
[CrossRef]

2008

I. Stamos, L. Liu, C. Chen, G. Wolberg, G. Yu, and S. Zokai, “Integrating automated range registration with multiview geometry for the photorealistic modeling of large-scale scenes,” Int. J. Comput. Vis. 78, 237–260 (2008).
[CrossRef]

2006

2005

V. Rankov, R. J. Locke, R. J. Edens, P. R. Barber, and B. Vojnovic, “An algorithm for image stitching and blending,” Proc. SPIE 5701, 190–199 (2005).
[CrossRef]

2004

F. Blais, “Review of 20 years of range sensor development,” J. Electron. Imaging 13, 231–240 (2004).
[CrossRef]

G. Guidi, J. A. Beraldin, and C. Atzeni, “High-accuracy 3D modeling of cultural heritage: the digitizing of Donatello’s ‘Maddalena’,” IEEE Trans. Image Process. 13, 370–380 (2004).
[CrossRef]

2003

X. Peng, Z. Yang, and H. Niu, “Multi-resolution reconstruction of 3-D image with modified temporal unwrapping algorithm,” Opt. Commun. 224, 35–44 (2003).
[CrossRef]

2002

F. Bernardini, H. Rushmeier, I. M. Martin, J. Mittleman, and G. Taubin, “Building a digital model of Michelangelo’s Florentine Pieta,” IEEE Comp. Grap. Appl. 22, 59–67 (2002).
[CrossRef]

2000

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[CrossRef]

C. Reich, R. Ritter, and J. Thesing, “3-D shape measurement of complex objects by combining photogrammetry and fringe projection,” Opt. Eng. 39, 224–231 (2000).
[CrossRef]

1998

M. Petrov, A. Talapov, T. Robertson, A. Lebedev, A. Zhilyaev, and L. Polonskiy, “Optical 3D digitizers: bringing life to the virtual world,” IEEE Comp. Grap. Appl. 18, 28–37 (1998).
[CrossRef]

1996

R. Bergevin, M. Soucy, H. Gagnon, and D. Laurendeau, “Towards a general multi-view registration technique,” IEEE Trans. Pattern Anal. Machine Intell. 18, 540–547 (1996).
[CrossRef]

1992

P. J. Besl and H. D. McKay, “A method for registration of 3-D shapes,” IEEE Trans. Pattern Anal. Machine Intell. 14, 239–256 (1992).
[CrossRef]

1987

R. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE Trans. Robotic. Autom. 3, 323–344 (1987).
[CrossRef]

1984

Atzeni, C.

G. Guidi, J. A. Beraldin, and C. Atzeni, “High-accuracy 3D modeling of cultural heritage: the digitizing of Donatello’s ‘Maddalena’,” IEEE Trans. Image Process. 13, 370–380 (2004).
[CrossRef]

Barber, P. R.

V. Rankov, R. J. Locke, R. J. Edens, P. R. Barber, and B. Vojnovic, “An algorithm for image stitching and blending,” Proc. SPIE 5701, 190–199 (2005).
[CrossRef]

Beraldin, J. A.

G. Guidi, J. A. Beraldin, and C. Atzeni, “High-accuracy 3D modeling of cultural heritage: the digitizing of Donatello’s ‘Maddalena’,” IEEE Trans. Image Process. 13, 370–380 (2004).
[CrossRef]

Bergevin, R.

R. Bergevin, M. Soucy, H. Gagnon, and D. Laurendeau, “Towards a general multi-view registration technique,” IEEE Trans. Pattern Anal. Machine Intell. 18, 540–547 (1996).
[CrossRef]

Bernardini, F.

F. Bernardini, H. Rushmeier, I. M. Martin, J. Mittleman, and G. Taubin, “Building a digital model of Michelangelo’s Florentine Pieta,” IEEE Comp. Grap. Appl. 22, 59–67 (2002).
[CrossRef]

Besl, P. J.

P. J. Besl and H. D. McKay, “A method for registration of 3-D shapes,” IEEE Trans. Pattern Anal. Machine Intell. 14, 239–256 (1992).
[CrossRef]

Blais, F.

F. Blais, “Review of 20 years of range sensor development,” J. Electron. Imaging 13, 231–240 (2004).
[CrossRef]

Brown, G. M.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[CrossRef]

Callieri, M.

M. Callieri, P. Cignoni, and R. Scopigno, “Reconstructing textured meshes from multiple range rgb maps,” in 7th International Fall Workshop on Vision, Modeling, and Visualization (IOS, 2002), pp. 419–426.

Chen, C.

I. Stamos, L. Liu, C. Chen, G. Wolberg, G. Yu, and S. Zokai, “Integrating automated range registration with multiview geometry for the photorealistic modeling of large-scale scenes,” Int. J. Comput. Vis. 78, 237–260 (2008).
[CrossRef]

Chen, F.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[CrossRef]

Cignoni, P.

M. Callieri, P. Cignoni, and R. Scopigno, “Reconstructing textured meshes from multiple range rgb maps,” in 7th International Fall Workshop on Vision, Modeling, and Visualization (IOS, 2002), pp. 419–426.

Cremers, D.

B. Goldluecke and D. Cremers, “Superresolution texture maps for multiview reconstruction,” in IEEE 12th International Conference on Computer Vision (IEEE, 2009), pp. 1677–1684.

Debevec, P. E.

P. E. Debevec, C. J. Taylor, and J. Malik, “Modeling and rendering architecture from photographs: a hybrid geometry- and image-based approach,” in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (1996), pp. 11–20.

Edens, R. J.

V. Rankov, R. J. Locke, R. J. Edens, P. R. Barber, and B. Vojnovic, “An algorithm for image stitching and blending,” Proc. SPIE 5701, 190–199 (2005).
[CrossRef]

Fischer, M.

R. Tutsch, M. Petz, and M. Fischer, “Optical three-dimensional metrology with structured illumination,” Opt. Eng. 50, 101507–101510 (2011).
[CrossRef]

Gagnon, H.

R. Bergevin, M. Soucy, H. Gagnon, and D. Laurendeau, “Towards a general multi-view registration technique,” IEEE Trans. Pattern Anal. Machine Intell. 18, 540–547 (1996).
[CrossRef]

Goldluecke, B.

B. Goldluecke and D. Cremers, “Superresolution texture maps for multiview reconstruction,” in IEEE 12th International Conference on Computer Vision (IEEE, 2009), pp. 1677–1684.

Guidi, G.

G. Guidi, J. A. Beraldin, and C. Atzeni, “High-accuracy 3D modeling of cultural heritage: the digitizing of Donatello’s ‘Maddalena’,” IEEE Trans. Image Process. 13, 370–380 (2004).
[CrossRef]

Halioua, M.

Hartley, R.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge University, 2003).

Hecker, J.

F. Pinhin, J. Hecker, D. Lischinski, D. H. Salesin, and R. Szeliski, “Synthesizing realistic facial expressions from photographs,” in Conference on Computer Graphics and Interactive Techniques (1998), pp. 75–84.

Heikkila, J.

J. Heikkila and O. Silven, “A four-step camera calibration procedure with implicit image correction,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 1997), pp. 1106–1112.

Ivanov, D.

V. Lempitsky and D. Ivanov, “Seamless mosaicing of image-based texture maps,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2007), pp. 1–6.

Laurendeau, D.

R. Bergevin, M. Soucy, H. Gagnon, and D. Laurendeau, “Towards a general multi-view registration technique,” IEEE Trans. Pattern Anal. Machine Intell. 18, 540–547 (1996).
[CrossRef]

Lebedev, A.

M. Petrov, A. Talapov, T. Robertson, A. Lebedev, A. Zhilyaev, and L. Polonskiy, “Optical 3D digitizers: bringing life to the virtual world,” IEEE Comp. Grap. Appl. 18, 28–37 (1998).
[CrossRef]

Lempitsky, V.

V. Lempitsky and D. Ivanov, “Seamless mosaicing of image-based texture maps,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2007), pp. 1–6.

Levoy, M.

S. Rusinkiewicz and M. Levoy, “Efficient variants of the ICP algorithm,” in Third International Conference on 3-D Digital Imaging and Modeling (2001), pp. 145–152.

Lischinski, D.

F. Pinhin, J. Hecker, D. Lischinski, D. H. Salesin, and R. Szeliski, “Synthesizing realistic facial expressions from photographs,” in Conference on Computer Graphics and Interactive Techniques (1998), pp. 75–84.

Liu, H. C.

Liu, L.

I. Stamos, L. Liu, C. Chen, G. Wolberg, G. Yu, and S. Zokai, “Integrating automated range registration with multiview geometry for the photorealistic modeling of large-scale scenes,” Int. J. Comput. Vis. 78, 237–260 (2008).
[CrossRef]

Locke, R. J.

V. Rankov, R. J. Locke, R. J. Edens, P. R. Barber, and B. Vojnovic, “An algorithm for image stitching and blending,” Proc. SPIE 5701, 190–199 (2005).
[CrossRef]

Malik, J.

P. E. Debevec, C. J. Taylor, and J. Malik, “Modeling and rendering architecture from photographs: a hybrid geometry- and image-based approach,” in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (1996), pp. 11–20.

Martin, I. M.

F. Bernardini, H. Rushmeier, I. M. Martin, J. Mittleman, and G. Taubin, “Building a digital model of Michelangelo’s Florentine Pieta,” IEEE Comp. Grap. Appl. 22, 59–67 (2002).
[CrossRef]

McKay, H. D.

P. J. Besl and H. D. McKay, “A method for registration of 3-D shapes,” IEEE Trans. Pattern Anal. Machine Intell. 14, 239–256 (1992).
[CrossRef]

Mittleman, J.

F. Bernardini, H. Rushmeier, I. M. Martin, J. Mittleman, and G. Taubin, “Building a digital model of Michelangelo’s Florentine Pieta,” IEEE Comp. Grap. Appl. 22, 59–67 (2002).
[CrossRef]

Niu, H.

X. Peng, Z. Yang, and H. Niu, “Multi-resolution reconstruction of 3-D image with modified temporal unwrapping algorithm,” Opt. Commun. 224, 35–44 (2003).
[CrossRef]

Pelagotti, A.

F. Uccheddu, A. Pelagotti, and F. Picchioni, “A greedy multiresolution method for quasi automatic texture mapping,” Proc. SPIE 8085, 80850M (2011).
[CrossRef]

Peng, X.

X. Peng, Z. Yang, and H. Niu, “Multi-resolution reconstruction of 3-D image with modified temporal unwrapping algorithm,” Opt. Commun. 224, 35–44 (2003).
[CrossRef]

Z. Zhang, X. Peng, and D. Zhang, “Transformation image into graphics,” in Integrated Image and Graphics Technologies, D. Zhang, M. Kamel, and G. Baciu, eds. (Kluwer Academic, 2004), pp. 111–129.

Petrov, M.

M. Petrov, A. Talapov, T. Robertson, A. Lebedev, A. Zhilyaev, and L. Polonskiy, “Optical 3D digitizers: bringing life to the virtual world,” IEEE Comp. Grap. Appl. 18, 28–37 (1998).
[CrossRef]

Petz, M.

R. Tutsch, M. Petz, and M. Fischer, “Optical three-dimensional metrology with structured illumination,” Opt. Eng. 50, 101507–101510 (2011).
[CrossRef]

Picchioni, F.

F. Uccheddu, A. Pelagotti, and F. Picchioni, “A greedy multiresolution method for quasi automatic texture mapping,” Proc. SPIE 8085, 80850M (2011).
[CrossRef]

Pinhin, F.

F. Pinhin, J. Hecker, D. Lischinski, D. H. Salesin, and R. Szeliski, “Synthesizing realistic facial expressions from photographs,” in Conference on Computer Graphics and Interactive Techniques (1998), pp. 75–84.

Polonskiy, L.

M. Petrov, A. Talapov, T. Robertson, A. Lebedev, A. Zhilyaev, and L. Polonskiy, “Optical 3D digitizers: bringing life to the virtual world,” IEEE Comp. Grap. Appl. 18, 28–37 (1998).
[CrossRef]

Rankov, V.

V. Rankov, R. J. Locke, R. J. Edens, P. R. Barber, and B. Vojnovic, “An algorithm for image stitching and blending,” Proc. SPIE 5701, 190–199 (2005).
[CrossRef]

Reich, C.

C. Reich, R. Ritter, and J. Thesing, “3-D shape measurement of complex objects by combining photogrammetry and fringe projection,” Opt. Eng. 39, 224–231 (2000).
[CrossRef]

Ritter, R.

C. Reich, R. Ritter, and J. Thesing, “3-D shape measurement of complex objects by combining photogrammetry and fringe projection,” Opt. Eng. 39, 224–231 (2000).
[CrossRef]

Robertson, T.

M. Petrov, A. Talapov, T. Robertson, A. Lebedev, A. Zhilyaev, and L. Polonskiy, “Optical 3D digitizers: bringing life to the virtual world,” IEEE Comp. Grap. Appl. 18, 28–37 (1998).
[CrossRef]

Rushmeier, H.

F. Bernardini, H. Rushmeier, I. M. Martin, J. Mittleman, and G. Taubin, “Building a digital model of Michelangelo’s Florentine Pieta,” IEEE Comp. Grap. Appl. 22, 59–67 (2002).
[CrossRef]

Rusinkiewicz, S.

S. Rusinkiewicz and M. Levoy, “Efficient variants of the ICP algorithm,” in Third International Conference on 3-D Digital Imaging and Modeling (2001), pp. 145–152.

Salesin, D. H.

F. Pinhin, J. Hecker, D. Lischinski, D. H. Salesin, and R. Szeliski, “Synthesizing realistic facial expressions from photographs,” in Conference on Computer Graphics and Interactive Techniques (1998), pp. 75–84.

Scopigno, R.

M. Callieri, P. Cignoni, and R. Scopigno, “Reconstructing textured meshes from multiple range rgb maps,” in 7th International Fall Workshop on Vision, Modeling, and Visualization (IOS, 2002), pp. 419–426.

Silven, O.

J. Heikkila and O. Silven, “A four-step camera calibration procedure with implicit image correction,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 1997), pp. 1106–1112.

Song, M.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[CrossRef]

Soucy, M.

R. Bergevin, M. Soucy, H. Gagnon, and D. Laurendeau, “Towards a general multi-view registration technique,” IEEE Trans. Pattern Anal. Machine Intell. 18, 540–547 (1996).
[CrossRef]

Srinivasan, V.

Stamos, I.

I. Stamos, L. Liu, C. Chen, G. Wolberg, G. Yu, and S. Zokai, “Integrating automated range registration with multiview geometry for the photorealistic modeling of large-scale scenes,” Int. J. Comput. Vis. 78, 237–260 (2008).
[CrossRef]

Szeliski, R.

F. Pinhin, J. Hecker, D. Lischinski, D. H. Salesin, and R. Szeliski, “Synthesizing realistic facial expressions from photographs,” in Conference on Computer Graphics and Interactive Techniques (1998), pp. 75–84.

Talapov, A.

M. Petrov, A. Talapov, T. Robertson, A. Lebedev, A. Zhilyaev, and L. Polonskiy, “Optical 3D digitizers: bringing life to the virtual world,” IEEE Comp. Grap. Appl. 18, 28–37 (1998).
[CrossRef]

Taubin, G.

F. Bernardini, H. Rushmeier, I. M. Martin, J. Mittleman, and G. Taubin, “Building a digital model of Michelangelo’s Florentine Pieta,” IEEE Comp. Grap. Appl. 22, 59–67 (2002).
[CrossRef]

Taylor, C. J.

P. E. Debevec, C. J. Taylor, and J. Malik, “Modeling and rendering architecture from photographs: a hybrid geometry- and image-based approach,” in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (1996), pp. 11–20.

Thesing, J.

C. Reich, R. Ritter, and J. Thesing, “3-D shape measurement of complex objects by combining photogrammetry and fringe projection,” Opt. Eng. 39, 224–231 (2000).
[CrossRef]

Towers, C. E.

Towers, D. P.

Tsai, R.

R. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE Trans. Robotic. Autom. 3, 323–344 (1987).
[CrossRef]

Tutsch, R.

R. Tutsch, M. Petz, and M. Fischer, “Optical three-dimensional metrology with structured illumination,” Opt. Eng. 50, 101507–101510 (2011).
[CrossRef]

Uccheddu, F.

F. Uccheddu, A. Pelagotti, and F. Picchioni, “A greedy multiresolution method for quasi automatic texture mapping,” Proc. SPIE 8085, 80850M (2011).
[CrossRef]

Vojnovic, B.

V. Rankov, R. J. Locke, R. J. Edens, P. R. Barber, and B. Vojnovic, “An algorithm for image stitching and blending,” Proc. SPIE 5701, 190–199 (2005).
[CrossRef]

Wand, M.

M. Wand, Point-Based Multi-Resolution Rendering (Eberhard-Karls-Universität, 2004).

Wolberg, G.

I. Stamos, L. Liu, C. Chen, G. Wolberg, G. Yu, and S. Zokai, “Integrating automated range registration with multiview geometry for the photorealistic modeling of large-scale scenes,” Int. J. Comput. Vis. 78, 237–260 (2008).
[CrossRef]

Yang, Z.

X. Peng, Z. Yang, and H. Niu, “Multi-resolution reconstruction of 3-D image with modified temporal unwrapping algorithm,” Opt. Commun. 224, 35–44 (2003).
[CrossRef]

Yau, S.-T.

Yu, G.

I. Stamos, L. Liu, C. Chen, G. Wolberg, G. Yu, and S. Zokai, “Integrating automated range registration with multiview geometry for the photorealistic modeling of large-scale scenes,” Int. J. Comput. Vis. 78, 237–260 (2008).
[CrossRef]

Zhang, D.

Z. Zhang, X. Peng, and D. Zhang, “Transformation image into graphics,” in Integrated Image and Graphics Technologies, D. Zhang, M. Kamel, and G. Baciu, eds. (Kluwer Academic, 2004), pp. 111–129.

Zhang, S.

Zhang, Z.

Z. Zhang, C. E. Towers, and D. P. Towers, “Time efficient color fringe projection system for 3D shape and color using optimum 3-frequency selection,” Opt. Express 14, 6444–6455 (2006).
[CrossRef]

Z. Zhang, X. Peng, and D. Zhang, “Transformation image into graphics,” in Integrated Image and Graphics Technologies, D. Zhang, M. Kamel, and G. Baciu, eds. (Kluwer Academic, 2004), pp. 111–129.

Z. Zhang, “Flexible camera calibration by viewing a plane from unknown orientations,” in IEEE International Conference on Computer Vision (IEEE, 1999), pp. 666–673.

Zhilyaev, A.

M. Petrov, A. Talapov, T. Robertson, A. Lebedev, A. Zhilyaev, and L. Polonskiy, “Optical 3D digitizers: bringing life to the virtual world,” IEEE Comp. Grap. Appl. 18, 28–37 (1998).
[CrossRef]

Zisserman, A.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge University, 2003).

Zokai, S.

I. Stamos, L. Liu, C. Chen, G. Wolberg, G. Yu, and S. Zokai, “Integrating automated range registration with multiview geometry for the photorealistic modeling of large-scale scenes,” Int. J. Comput. Vis. 78, 237–260 (2008).
[CrossRef]

Appl. Opt.

IEEE Comp. Grap. Appl.

M. Petrov, A. Talapov, T. Robertson, A. Lebedev, A. Zhilyaev, and L. Polonskiy, “Optical 3D digitizers: bringing life to the virtual world,” IEEE Comp. Grap. Appl. 18, 28–37 (1998).
[CrossRef]

F. Bernardini, H. Rushmeier, I. M. Martin, J. Mittleman, and G. Taubin, “Building a digital model of Michelangelo’s Florentine Pieta,” IEEE Comp. Grap. Appl. 22, 59–67 (2002).
[CrossRef]

IEEE Trans. Image Process.

G. Guidi, J. A. Beraldin, and C. Atzeni, “High-accuracy 3D modeling of cultural heritage: the digitizing of Donatello’s ‘Maddalena’,” IEEE Trans. Image Process. 13, 370–380 (2004).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell.

P. J. Besl and H. D. McKay, “A method for registration of 3-D shapes,” IEEE Trans. Pattern Anal. Machine Intell. 14, 239–256 (1992).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Layout of dedicated 3D digitizer for range and texture image acquisition.

Fig. 2.
Fig. 2.

Comparison of different approaches for texture blending: (a) approach using average weight, (b) approach using pure normal weight, (c) approach using composite weight.

Fig. 3.
Fig. 3.

Experiment setup of a soft-lighting environment for range and texture acquisition.

Fig. 4.
Fig. 4.

Texture images taken from different viewpoints.

Fig. 5.
Fig. 5.

Experiment results: (a) partial range images, (b) complete and nonredundant geometric model, and (c) simplified model.

Fig. 6.
Fig. 6.

Textured model generated with proposed approach.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

{sLx˜L=KL[I|0]X˜,sRx˜R=KR[RLR|tLR]X˜,
{x^L=xL+δ(θL),x^R=xR+δ(θR),
{XT=RLTX+tLT,sTx˜T=KTXT,x^T=xT+δ(θT),
ijxx^(K,R,t,θ;X)2,
{RLR=RRRLT,tLR=tRRRRLTtL.
{XW1=R1XT1+t1,XWk=RkXTk+tk,XWM=RMXTM+tM,k=1,,M,
{XWk}M{Vi},i=1,,N,
M{Δj{V1j,V2j,V3j}|nj},j=1,,M.
sv˜=PkV˜,
Pk=KT[Rk|tk].
wΔj,Ik(sj,k,nj)={(sj,k·nj),Δjis valid,0,otherwise,
wΔj,Ik(lk,oΔj)={dis(lk,oΔj),Δjis valid,0,otherwise,
wΔj,Ik(Ck,OM)={1(dis(Ck,OM)D),Δjisvalid,0,otherwise,D=max{dis(Ck,OM)|k=1,2,3,,K},
wtotal=wΔj,I1+wΔj,I2+wΔj,I3++wΔj,Ik
{w˜Δj,Ik=wΔj,Ikwtotal|k=1,2,n}

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