
Previous Article
Multiview foreground segmentation via fourth order tensor learning
 IPI Home
 This Issue

Next Article
Video stabilization of atmospheric turbulence distortion
A conformal approach for surface inpainting
1.  Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China 
2.  Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong,, China 
3.  Department of Computer Sciences, Room 2425, Computer Science Building, State University of New York at Stony Brook, Stony Brook, New York 117944400 
References:
[1] 
T. F. Chan and J. Shen, Variational restoration of nonflat image features: Models and algorithms, SIAM Journal on Applied Mathematics, 61 (2001), 13381361. doi: 10.1137/S003613999935799X. 
[2] 
T. F. Chan and J. Shen, Mathematical models for local nontexture inpaintings,, SIAM Journal on Applied Mathematics, 62 (): 1019. doi: 10.1137/S0036139900368844. 
[3] 
S. Esedoglu and J. Shen, Digital inpainting based on the MumfordShahEuler image model, Eureopean Journal on Applied Mathematics, 13 (2002), 353370. doi: 10.1017/S0956792502004904. 
[4] 
T. F. Chan and J. Shen, Variational image inpainting, Communications in Pure and Applied Mathematics, 58 (2005), 579619. doi: 10.1002/cpa.20075. 
[5] 
T. F. Chan, S. H. Kang and J. Shen, Euler's elastica and curvaturebased inpainting, SIAM Journal on Applied Mathematics, 63 (2002), 564592. doi: 10.1137/S0036139901390088. 
[6] 
S. Masnou and J. Morel, Level lines based disocclusion, $5^{th}$ IEEE International Conference on Image Processing, Chicago, IL, 3 (1998), 259263. doi: 10.1109/ICIP.1998.999016. 
[7] 
M. Bertalmio, A. L. Bertozzi and G. Sapiro, NavierStokes, fluid dynamics, and image and video inpainting, Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 1 (2001), 355362. doi: 10.1109/CVPR.2001.990497. 
[8] 
T. F. Chan, J. Shen and H.M. Zhou, Total variation wavelet inpainting, Journal of Mathematical Imaging and Visions, 25 (2006), 107125. doi: 10.1007/s1085100652573. 
[9] 
J. A. Dobrosotskaya and A. L. Bertozzi, A waveletLaplace variational technique for image deconvolution and inpainting, IEEE Transaction on Image Processing, 17 (2008), 657663. doi: 10.1109/TIP.2008.919367. 
[10] 
A. Tsai, A. Yezzi, Jr. and A. S. Willsky, Curve evolution implementation of the MumfordShah functional for image segmentation, denoising, interpolation and magnification, IEEE Transaction on Image Processing, 10 (2001), 11691186. doi: 10.1109/83.935033. 
[11] 
M. Desbrun, M. Meyer, P. Schrder and A. H. Barr, Implicit fairing of irregular meshes using diffusion and curvature flow, in "Proceedings of the $26^{th}$ Annual Conference on Computer Graphics and Interactive Techniques," ACM Press/AddisonWesley Publishing Co., New York, (1999), 317324. doi: 10.1145/311535.311576. 
[12] 
G. Taubin, Geometric signal processing on polygonal meshes, Eurographics, (2000). 
[13] 
P. Smereka, Semi implicit level set methods curvature surface diffusion motion, Journal of Scientific Computing, 19 (2003), 439456. doi: 10.1023/A:1025324613450. 
[14] 
M. Desbrun, M. Meyer, P. Schröder and A. H. Barr, Anisotropic feature preserving denoising of height fields and bivariate data, Graphics Interface, (2000). 
[15] 
M. Desbrun, M. Meyer, P. Schröder and A. H. Barr, Discrete differential geometry operators in nD, in "Proceedings on Vis. Math02," SpringerVerlag, Berlin, (2002), 3557. 
[16] 
U. Clarenz, U. Diewald and M. Rumpf, Anisotropic diffusion in surface processing, in "Proceedings of IEEE Visualization 2000," (2000), 397405. doi: 10.1109/VISUAL.2000.885721. 
[17] 
C. L. Bajaj and G. Xu, Anisotropic diffusion of subdivision surfaces and functions on surfaces, ACM Transaction on Graphics, 22 (2003), 432. 
[18] 
T. Tasdizen, R. Whitaker, P. Burchard and S. Osher, Geometric surface processing via anisotropic diffusion of normals, IEEE Visualization, (2002), 125132. 
[19] 
T. Tasdizen, R. Whitaker, P. Burchard and S. Osher, Geometric surface processing via normal maps, ACM Transactions on Graphics, 22 (2003), 10121033. doi: 10.1145/944020.944024. 
[20] 
V. Caselles, G. Haro, G. Sapiro and J. Verdera, On geometric variational models for inpainting surface holes, Computer Vision and Image Understanding, 111 (2008), 351373. doi: 10.1016/j.cviu.2008.01.002. 
[21] 
J. Verdera, V. Caselles, M. Bertalmio and G. Sapiro, Inpainting surface holes, in "Proceedings of International Conference on Image Processing, Vol. 2," (2003), 903906. doi: 10.1109/ICIP.2003.1246828. 
[22] 
J. Davis, S. Marschner, M. Garr and M. Levoy, Filling holes in complex surfaces using volumetric diffusion, in "Proceedings of First International Symposium on 3D Data Processing, Visualization, and Transmission," (2002), 428441. doi: 10.1109/TDPVT.2002.1024098. 
[23] 
U. Clarenz, U. Diewald, G. Dziuk, M. Rumpf and R. Rusu, A finite element method for surface restoration with smooth boundary conditions, Computer Aided Geometric Design, 21 (2004), 427445. doi: 10.1016/j.cagd.2004.02.004. 
[24] 
A. Sharf, M. Alexa and D. Cohen, Contextbased surface completion, ACM Transaction on Graphics, 23 (2004), 878887. 
[25] 
V. Savchenko and N. Kojekine, An approach to blend surfaces, Advances in Modeling, Animation and Rendering, (2002), 139150. doi: 10.1007/9781447101031_9. 
[26] 
M. K. Hurdal and K. Stephenson, Discrete conformal methods for cortical brain flattening, NeuroImage, 45 (2009), S86S98. doi: 10.1016/j.neuroimage.2008.10.045. 
[27] 
B. Fischl, M. Sereno, R. Tootell and A. Dale, Highresolution intersubject averaging and a coordinate system for the cortical surface, Human Brain Mapping, 8 (1999), 272284. 
[28] 
X. Gu, Y. Wang, T. F. Chan, P. M. Thompson and S.T. Yau, Genus zero surface conformal mapping and its application to brain surface mapping, IEEE Transactions on Medical Imaging, 23 (2004), 949958. 
[29] 
Y. Wang, L. M. Lui, X. Gu, K. M. Hayashi, T. F. Chan, A. W. Toga, P. M. Thompson and S.T. Yau, Brain surface conformal parameterization using Riemann surface structure, IEEE Transactions on Medical Imaging, 26 (2007), 853865. doi: 10.1109/TMI.2007.895464. 
[30] 
X. Gu and S.T. Yau, Computing conformal structures of surfaces, Communication in Information System, 2 (2002), 121145. 
[31] 
X. Gu, Y. Wang and S.T. Yau, Geometric compression using Riemann surface structure, Communication in Information System, 3 (2004), 171182. 
[32] 
L. M. Lui, T. W. Wong, W. Zeng, X. Gu, P. M. Thompson, T. F. Chan and S.T. Yau, Optimization of surface registrations using Beltrami holomorphic flow, Journal of Scientific Computing, 50 (2012), 557585. doi: 10.1007/s1091501195062. 
[33] 
L. M. Lui, T. W. Wong, X. Gu, P. M. Thompson, T. F. Chan and S.T. Yau, Hippocampal shape registration using Beltrami holomorphic flow, Medical Image Computing and Computer Assisted Intervention(MICCAI), Part II, LNCS, 6362 (2010), 323330. 
[34] 
L. M. Lui, K. C. Lam, T. W. Wong and X. Gu, Texture map and video compression using Beltrami representatio, to appear in SIAM Journal on Imaging Sciences, (2013). 
[35] 
S. Haker, S. Angenent, A. Tannenbaum, R. Kikinis, G. Sapiro and M. Halle, Conformal surface parameterization for texture mapping, IEEE Transaction of Visualization and Computer Graphics, 6 (2000), 181189. 
[36] 
L. M. Lui, T. W. Wong, X. Gu, T. F. Chan and S.T. Yau, Compression of surface diffeomorphism using Beltrami coefficient, IEEE Computer Vision and Pattern Recognition(CVPR), (2010), 28392846. 
[37] 
W. Zeng, L. M. Lui, F. Luo, T. F. Chan, S.T. Yau and X. Gu, Computing quasiconformal maps using an auxiliary metric and discrete curvature flow, Numerische Mathematik, 121 (2012), 671703. doi: 10.1007/s002110120446z. 
[38] 
L. M. Lui, K. C. Lam, S.T. Yau and X. Gu, Teichmüller extremal mapping and its applications to landmark matching registration,, , (). 
[39] 
W. Zeng and X. Gu, Registration for 3D surfaces with large deformations using quasiconformal curvature flow, IEEE Conference on Computer Vision and Pattern Recognition (CVPR1), Jun 2025, (2011), 24572464. 
[40] 
H. Zhao, S. Osher, B. Merriman and and M. Kang, Implicit and nonparametric shape reconstruction from unorganized points using variational level set method, Computer Vision and Image Understanding, 80 (2000), 295319. 
[41] 
J. Liang, F. Park and H. Zhao, Robust and efficient implicit surface reconstruction of point clouds based on convexified image segmentation, Journal of Scientific Computing, 54 (2013), 577602. doi: 10.1007/s1091501296748. 
[42] 
A. Wong and H. Zhao, Computation of quasiconformal surface maps using discrete Beltrami flow,, UCLA CAM report 1285., (): 12. 
show all references
References:
[1] 
T. F. Chan and J. Shen, Variational restoration of nonflat image features: Models and algorithms, SIAM Journal on Applied Mathematics, 61 (2001), 13381361. doi: 10.1137/S003613999935799X. 
[2] 
T. F. Chan and J. Shen, Mathematical models for local nontexture inpaintings,, SIAM Journal on Applied Mathematics, 62 (): 1019. doi: 10.1137/S0036139900368844. 
[3] 
S. Esedoglu and J. Shen, Digital inpainting based on the MumfordShahEuler image model, Eureopean Journal on Applied Mathematics, 13 (2002), 353370. doi: 10.1017/S0956792502004904. 
[4] 
T. F. Chan and J. Shen, Variational image inpainting, Communications in Pure and Applied Mathematics, 58 (2005), 579619. doi: 10.1002/cpa.20075. 
[5] 
T. F. Chan, S. H. Kang and J. Shen, Euler's elastica and curvaturebased inpainting, SIAM Journal on Applied Mathematics, 63 (2002), 564592. doi: 10.1137/S0036139901390088. 
[6] 
S. Masnou and J. Morel, Level lines based disocclusion, $5^{th}$ IEEE International Conference on Image Processing, Chicago, IL, 3 (1998), 259263. doi: 10.1109/ICIP.1998.999016. 
[7] 
M. Bertalmio, A. L. Bertozzi and G. Sapiro, NavierStokes, fluid dynamics, and image and video inpainting, Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 1 (2001), 355362. doi: 10.1109/CVPR.2001.990497. 
[8] 
T. F. Chan, J. Shen and H.M. Zhou, Total variation wavelet inpainting, Journal of Mathematical Imaging and Visions, 25 (2006), 107125. doi: 10.1007/s1085100652573. 
[9] 
J. A. Dobrosotskaya and A. L. Bertozzi, A waveletLaplace variational technique for image deconvolution and inpainting, IEEE Transaction on Image Processing, 17 (2008), 657663. doi: 10.1109/TIP.2008.919367. 
[10] 
A. Tsai, A. Yezzi, Jr. and A. S. Willsky, Curve evolution implementation of the MumfordShah functional for image segmentation, denoising, interpolation and magnification, IEEE Transaction on Image Processing, 10 (2001), 11691186. doi: 10.1109/83.935033. 
[11] 
M. Desbrun, M. Meyer, P. Schrder and A. H. Barr, Implicit fairing of irregular meshes using diffusion and curvature flow, in "Proceedings of the $26^{th}$ Annual Conference on Computer Graphics and Interactive Techniques," ACM Press/AddisonWesley Publishing Co., New York, (1999), 317324. doi: 10.1145/311535.311576. 
[12] 
G. Taubin, Geometric signal processing on polygonal meshes, Eurographics, (2000). 
[13] 
P. Smereka, Semi implicit level set methods curvature surface diffusion motion, Journal of Scientific Computing, 19 (2003), 439456. doi: 10.1023/A:1025324613450. 
[14] 
M. Desbrun, M. Meyer, P. Schröder and A. H. Barr, Anisotropic feature preserving denoising of height fields and bivariate data, Graphics Interface, (2000). 
[15] 
M. Desbrun, M. Meyer, P. Schröder and A. H. Barr, Discrete differential geometry operators in nD, in "Proceedings on Vis. Math02," SpringerVerlag, Berlin, (2002), 3557. 
[16] 
U. Clarenz, U. Diewald and M. Rumpf, Anisotropic diffusion in surface processing, in "Proceedings of IEEE Visualization 2000," (2000), 397405. doi: 10.1109/VISUAL.2000.885721. 
[17] 
C. L. Bajaj and G. Xu, Anisotropic diffusion of subdivision surfaces and functions on surfaces, ACM Transaction on Graphics, 22 (2003), 432. 
[18] 
T. Tasdizen, R. Whitaker, P. Burchard and S. Osher, Geometric surface processing via anisotropic diffusion of normals, IEEE Visualization, (2002), 125132. 
[19] 
T. Tasdizen, R. Whitaker, P. Burchard and S. Osher, Geometric surface processing via normal maps, ACM Transactions on Graphics, 22 (2003), 10121033. doi: 10.1145/944020.944024. 
[20] 
V. Caselles, G. Haro, G. Sapiro and J. Verdera, On geometric variational models for inpainting surface holes, Computer Vision and Image Understanding, 111 (2008), 351373. doi: 10.1016/j.cviu.2008.01.002. 
[21] 
J. Verdera, V. Caselles, M. Bertalmio and G. Sapiro, Inpainting surface holes, in "Proceedings of International Conference on Image Processing, Vol. 2," (2003), 903906. doi: 10.1109/ICIP.2003.1246828. 
[22] 
J. Davis, S. Marschner, M. Garr and M. Levoy, Filling holes in complex surfaces using volumetric diffusion, in "Proceedings of First International Symposium on 3D Data Processing, Visualization, and Transmission," (2002), 428441. doi: 10.1109/TDPVT.2002.1024098. 
[23] 
U. Clarenz, U. Diewald, G. Dziuk, M. Rumpf and R. Rusu, A finite element method for surface restoration with smooth boundary conditions, Computer Aided Geometric Design, 21 (2004), 427445. doi: 10.1016/j.cagd.2004.02.004. 
[24] 
A. Sharf, M. Alexa and D. Cohen, Contextbased surface completion, ACM Transaction on Graphics, 23 (2004), 878887. 
[25] 
V. Savchenko and N. Kojekine, An approach to blend surfaces, Advances in Modeling, Animation and Rendering, (2002), 139150. doi: 10.1007/9781447101031_9. 
[26] 
M. K. Hurdal and K. Stephenson, Discrete conformal methods for cortical brain flattening, NeuroImage, 45 (2009), S86S98. doi: 10.1016/j.neuroimage.2008.10.045. 
[27] 
B. Fischl, M. Sereno, R. Tootell and A. Dale, Highresolution intersubject averaging and a coordinate system for the cortical surface, Human Brain Mapping, 8 (1999), 272284. 
[28] 
X. Gu, Y. Wang, T. F. Chan, P. M. Thompson and S.T. Yau, Genus zero surface conformal mapping and its application to brain surface mapping, IEEE Transactions on Medical Imaging, 23 (2004), 949958. 
[29] 
Y. Wang, L. M. Lui, X. Gu, K. M. Hayashi, T. F. Chan, A. W. Toga, P. M. Thompson and S.T. Yau, Brain surface conformal parameterization using Riemann surface structure, IEEE Transactions on Medical Imaging, 26 (2007), 853865. doi: 10.1109/TMI.2007.895464. 
[30] 
X. Gu and S.T. Yau, Computing conformal structures of surfaces, Communication in Information System, 2 (2002), 121145. 
[31] 
X. Gu, Y. Wang and S.T. Yau, Geometric compression using Riemann surface structure, Communication in Information System, 3 (2004), 171182. 
[32] 
L. M. Lui, T. W. Wong, W. Zeng, X. Gu, P. M. Thompson, T. F. Chan and S.T. Yau, Optimization of surface registrations using Beltrami holomorphic flow, Journal of Scientific Computing, 50 (2012), 557585. doi: 10.1007/s1091501195062. 
[33] 
L. M. Lui, T. W. Wong, X. Gu, P. M. Thompson, T. F. Chan and S.T. Yau, Hippocampal shape registration using Beltrami holomorphic flow, Medical Image Computing and Computer Assisted Intervention(MICCAI), Part II, LNCS, 6362 (2010), 323330. 
[34] 
L. M. Lui, K. C. Lam, T. W. Wong and X. Gu, Texture map and video compression using Beltrami representatio, to appear in SIAM Journal on Imaging Sciences, (2013). 
[35] 
S. Haker, S. Angenent, A. Tannenbaum, R. Kikinis, G. Sapiro and M. Halle, Conformal surface parameterization for texture mapping, IEEE Transaction of Visualization and Computer Graphics, 6 (2000), 181189. 
[36] 
L. M. Lui, T. W. Wong, X. Gu, T. F. Chan and S.T. Yau, Compression of surface diffeomorphism using Beltrami coefficient, IEEE Computer Vision and Pattern Recognition(CVPR), (2010), 28392846. 
[37] 
W. Zeng, L. M. Lui, F. Luo, T. F. Chan, S.T. Yau and X. Gu, Computing quasiconformal maps using an auxiliary metric and discrete curvature flow, Numerische Mathematik, 121 (2012), 671703. doi: 10.1007/s002110120446z. 
[38] 
L. M. Lui, K. C. Lam, S.T. Yau and X. Gu, Teichmüller extremal mapping and its applications to landmark matching registration,, , (). 
[39] 
W. Zeng and X. Gu, Registration for 3D surfaces with large deformations using quasiconformal curvature flow, IEEE Conference on Computer Vision and Pattern Recognition (CVPR1), Jun 2025, (2011), 24572464. 
[40] 
H. Zhao, S. Osher, B. Merriman and and M. Kang, Implicit and nonparametric shape reconstruction from unorganized points using variational level set method, Computer Vision and Image Understanding, 80 (2000), 295319. 
[41] 
J. Liang, F. Park and H. Zhao, Robust and efficient implicit surface reconstruction of point clouds based on convexified image segmentation, Journal of Scientific Computing, 54 (2013), 577602. doi: 10.1007/s1091501296748. 
[42] 
A. Wong and H. Zhao, Computation of quasiconformal surface maps using discrete Beltrami flow,, UCLA CAM report 1285., (): 12. 
[1] 
Sebastian Acosta. A control approach to recover the wave speed (conformal factor) from one measurement. Inverse Problems and Imaging, 2015, 9 (2) : 301315. doi: 10.3934/ipi.2015.9.301 
[2] 
YoonTae Jung, SooYoung Lee, EunHee Choi. Ricci curvature of conformal deformation on compact 2manifolds. Communications on Pure and Applied Analysis, 2020, 19 (6) : 32233231. doi: 10.3934/cpaa.2020140 
[3] 
Georgi I. Kamberov. Recovering the shape of a surface from the mean curvature. Conference Publications, 1998, 1998 (Special) : 353359. doi: 10.3934/proc.1998.1998.353 
[4] 
Ali Hyder, Luca Martinazzi. Conformal metrics on $\mathbb{R}^{2m}$ with constant Qcurvature, prescribed volume and asymptotic behavior. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 283299. doi: 10.3934/dcds.2015.35.283 
[5] 
Zuxing Xuan. On conformal measures of parabolic meromorphic functions. Discrete and Continuous Dynamical Systems  B, 2015, 20 (1) : 249257. doi: 10.3934/dcdsb.2015.20.249 
[6] 
Peter Haïssinsky, Kevin M. Pilgrim. Examples of coarse expanding conformal maps. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 24032416. doi: 10.3934/dcds.2012.32.2403 
[7] 
Nicholas Hoell, Guillaume Bal. Ray transforms on a conformal class of curves. Inverse Problems and Imaging, 2014, 8 (1) : 103125. doi: 10.3934/ipi.2014.8.103 
[8] 
Tomasz Szarek, Mariusz Urbański, Anna Zdunik. Continuity of Hausdorff measure for conformal dynamical systems. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 46474692. doi: 10.3934/dcds.2013.33.4647 
[9] 
Hans Henrik Rugh. On dimensions of conformal repellers. Randomness and parameter dependency. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 25532564. doi: 10.3934/dcds.2012.32.2553 
[10] 
Mario Roy, Mariusz Urbański. Multifractal analysis for conformal graph directed Markov systems. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 627650. doi: 10.3934/dcds.2009.25.627 
[11] 
Marcelo M. Disconzi. On the existence of solutions and causality for relativistic viscous conformal fluids. Communications on Pure and Applied Analysis, 2019, 18 (4) : 15671599. doi: 10.3934/cpaa.2019075 
[12] 
Domenico Mucci. Maps into projective spaces: Liquid crystal and conformal energies. Discrete and Continuous Dynamical Systems  B, 2012, 17 (2) : 597635. doi: 10.3934/dcdsb.2012.17.597 
[13] 
Rossen I. Ivanov. Conformal and Geometric Properties of the CamassaHolm Hierarchy. Discrete and Continuous Dynamical Systems, 2007, 19 (3) : 545554. doi: 10.3934/dcds.2007.19.545 
[14] 
Juan Wang, Yongluo Cao, Yun Zhao. Dimension estimates in nonconformal setting. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 38473873. doi: 10.3934/dcds.2014.34.3847 
[15] 
Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 70317056. doi: 10.3934/dcds.2019243 
[16] 
Robert Eymard, Angela Handlovičová, Karol Mikula. Approximation of nonlinear parabolic equations using a family of conformal and nonconformal schemes. Communications on Pure and Applied Analysis, 2012, 11 (1) : 147172. doi: 10.3934/cpaa.2012.11.147 
[17] 
Jérôme Bertrand. Prescription of Gauss curvature on compact hyperbolic orbifolds. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 12691284. doi: 10.3934/dcds.2014.34.1269 
[18] 
Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks and Heterogeneous Media, 2013, 8 (1) : 922. doi: 10.3934/nhm.2013.8.9 
[19] 
Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159169. doi: 10.3934/proc.2013.2013.159 
[20] 
Chiara Corsato, Colette De Coster, Pierpaolo Omari. Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape. Conference Publications, 2015, 2015 (special) : 297303. doi: 10.3934/proc.2015.0297 
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]