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A Variational Framework for Exemplar-Based Image Inpainting


Non-local methods for image denoising and inpainting have gained considerable attention in recent years. This is in part due to their superior performance in textured images, a known weakness of purely local methods. Local methods on the other hand have demonstrated to be very appropriate for the recovering of geometric structures such as image edges. The synthesis of both types of methods is a trend in current research. Variational analysis in particular is an appropriate tool for a unified treatment of local and non-local methods. In this work we propose a general variational framework for non-local image inpainting, from which important and representative previous inpainting schemes can be derived, in addition to leading to novel ones. We explicitly study some of these, relating them to previous work and showing results on synthetic and real images.

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  1. Aharon, M., Elad, M., & Bruckstein, A. M. (2006). The K-SVD: an algorithm for designing of overcomplete dictionaries for sparse representation. IEEE Transactions on Signal Processing, 54(11), 4311–4322.

  2. Almansa, A., Caselles, V., Haro, G., & Rougé, B. (2006). Restoration and zoom of irregularly sampled, blurred, and noisy images by accurate total variation minimization with local constraints. Multiscale Modeling & Simulation, 5(1), 235–272.

  3. Arias, P., Caselles, V., & Sapiro, G. (2009). A variational framework for non-local image inpainting. In Lecture notes in computer science. EMMCVPR (pp. 345–358). Berlin: Springer.

  4. Aujol, J.-F., Ladjal, S., & Masnou, S. (2010). Exemplar-based inpainting from a variational point of view. SIAM Journal on Mathematical Analysis, 42(3), 1246–1285.

  5. Awate, S. P., & Whitaker, R. T. (2005). Higher-order image statistics for unsupervised, information-theoretic, adaptive, image filtering. In Proc. of CVPR (pp. 44–51).

  6. Ballester, C., Bertalmío, M., Caselles, V., Sapiro, G., & Verdera, J. (2001). Filling-in by joint interpolation of vector fields and gray levels. IEEE Transactions on IP, 10(8), 1200–1211.

  7. Barnes, C., Shechtman, E., Finkelstein, A., & Goldman, D. B. (2009). PatchMatch: a randomized correspondence algorithm for structural image editing. In Proc. of SIGGRAPH (pp. 1–11). New York: ACM.

  8. Bertalmío, M., Sapiro, G., Caselles, V., & Ballester, C. (2000). Image inpainting. In Proc. of SIGGRAPH (pp. 417–424). New York: ACM.

  9. Bertalmío, M., Vese, L., Sapiro, G., & Osher, S. J. (2003). Simultaneous structure and texture inpainting. IEEE Transactions on IP, 12(8), 882–889.

  10. Bornard, R., Lecan, E., Laborelli, L., & Chenot, J.-H. (2002). Missing data correction in still images and image sequences. In Proc. ACM int. conf. on multimedia.

  11. Bornemann, F., & März, T. (2007). Fast image inpainting based on coherence transport. Journal of Mathematical Imaging and Vision, 28(3), 259–278.

  12. Brox, T., Kleinschmidt, O., & Cremers, D. (2008). Efficient nonlocal means for denoising of textural patterns. IEEE Transaction on IP, 17(7), 1057–1092.

  13. Buades, A., Coll, B., & Morel, J.-M. (2005). A non local algorithm for image denoising. In Proc. of the IEEE conf. on CVPR (Vol. 2, pp. 60–65).

  14. Candes, E. J., & Wakin, M. B. (2008). An introduction to compressive sampling. IEEE Signal Processing Magazine, 25(2), 21–30.

  15. Cao, F., Gousseau, Y., Masnou, S., & Pérez, P. (2009). Geometrically guided exemplar-based inpainting.

  16. Chambolle, A. (2004). An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision, 20(1–2), 89–97.

  17. Chan, T., & Shen, J. H. (2001). Mathematical models for local nontexture inpaintings. SIAM Journal on Applied Mathematics, 62(3), 1019–1043.

  18. Chan, T., Kang, S. H., & Shen, J. H. (2002). Euler’s elastica and curvature based inpaintings. SIAM Journal on Applied Mathematics, 63(2), 564–592.

  19. Cheng, Y. (1995). Mean shift, mode seeking and clustering. IEEE Transactions on PAMI, 17(8), 790–799.

  20. Criminisi, A., Pérez, P., & Toyama, K. (2004). Region filling and object removal by exemplar-based inpainting. IEEE Transactions on IP, 13(9), 1200–1212.

  21. Csiszár, I. (2008). Axiomatic characterizations of information measures. Entropy, 10(3), 261–273.

  22. Demanet, L., Song, B., & Chan, T. (2003). Image inpainting by correspondence maps: a deterministic approach (Technical report). UCLA.

  23. Drori, I., Cohen-Or, D., & Yeshurun, H. (2003). Fragment-based image completion. In Proc. of ACM SIGGRAPH (pp. 303–312). New York: ACM.

  24. Efros, A. A., & Leung, T. K. (1999). Texture synthesis by non-parametric sampling. In Proc. of the IEEE ICCV, September 1999 (pp. 1033–1038).

  25. Elad, M., Starck, J. L., Querre, P., & Donoho, D. L. (2005). Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA). Applied and Computational Harmonic Analysis, 19(3), 340–358.

  26. Esedoglu, S., & Shen, J. H. (2002). Digital image inpainting by the Mumford-Shah-Euler image model. European Journal of Applied Mathematics, 13, 353–370.

  27. Facciolo, G., Arias, P., Caselles, V., & Sapiro, G. (2009). Exemplar-based interpolation of sparsely sampled images. In Lecture notes in computer science. EMMCVPR (pp. 331–344). Berlin: Springer.

  28. Fang, C.-W., & Lien, J.-J. J. (2009). Rapid image completion system using multiresolution patch-based directional and nondirectional approaches. IEEE Transactions on IP, 18(12), 2769–2779.

  29. Gilboa, G., & Osher, S. J. (2007). Nonlocal linear image regularization and supervised segmentation. SIAM Multiscale Modeling and Simulation, 6(2), 595–630.

  30. Gilboa, G., & Osher, S. (2008). Nonlocal operators with applications to image processing. Multiscale Modeling & Simulation, 7(3), 1005–1028.

  31. Han, J., Zhou, K., Wei, L.-Y., Gong, M., Bao, H., Zhang, X., & Guo, B. (2006). Fast example-based surface texture synthesis via discrete optimization. The Visual Computer, 22(9), 918–925.

  32. Harrison, P. (2005). Texture tools. PhD thesis, Monash University.

  33. Holtzman-Gazit, M., & Yavneh, I. (2008). A scale-consistent approach to image completion. International Journal of Multiscale Computer Engineering, 6(6), 617–628.

  34. Igehy, H., & Pereira, L. (1997). Image replacement through texture synthesis. In Proc. of the IEEE ICIP.

  35. Jaynes, E. T. (1957). Information theory and statistical mechanics. Physical Review, 106(4), 620–630.

  36. Jia, J., & Tang, C.-K. (2004). Inference of segmented color and texture description by tensor voting. IEEE Transactions on PAMI, 26(6), 771–786.

  37. Kawai, N., Sato, T., & Yokoya, N. (2009). Image inpainting considering brightness change and spatial locality of textures and its evaluation. In Ad. in image and video tech. (pp. 271–282). Berlin: Springer.

  38. Kimball, S., Mattis, P., & the GIMP Dev. Team (2009). GIMP: GNU Image Manipulation Program. Version 2.6.8 released on December 2009.

  39. Komodakis, N., & Tziritas, G. (2007). Image completion using efficient belief propagation via priority scheduling and dynamic pruning. IEEE Transactions on IP, 16(11), 2649–2661.

  40. Kwatra, V., Essa, I., Bobick, A., & Kwatra, N. (2005). Texture optimization for example-based synthesis. ACM Transactions on Graphics, 24(3), 795–802.

  41. Levin, A., Zomet, A., & Weiss, Y. (2003). Learning how to inpaint from global image statistics. In Proc. of IEEE ICCV.

  42. Levina, E., & Bickel, P. (2006). Texture synthesis and non-parametric resampling of random fields. Annals of Statistics, 34(4).

  43. Lezoray, O., Elmoataz, A., & Bougleux, S. (2007). Graph regularization for color image processing. Computer Vision Image Understanding, 107(1–2), 38–55.

  44. Mairal, J., Sapiro, G., & Elad, M. (2008). Learning multiscale sparse representations for image and video restoration. SIAM Multiscale Modeling and Simulation, 7(1), 214–241.

  45. Masnou, S. (2002). Disocclusion: a variational approach using level lines. IEEE Transactions on IP, 11(2), 68–76.

  46. Masnou, S., & Morel, J.-M. (1998). Level lines based disocclusion. In Proc. of IEEE ICIP.

  47. Morel, J.-M., & Yu, G. (2008). On the consistency of the SIFT method. Preprint, CMLA, 26.

  48. Pérez, P., Gangnet, M., & Blake, A. (2003). Poisson image editing. In Proc. of SIGGRAPH (pp. 313–318). New York: ACM.

  49. Pérez, P., Gangnet, M., & Blake, A. (2004). PatchWorks: example-based region tiling for image editing (Technical report). Microsoft Research.

  50. Peyré, G. (2009). Manifold models for signals and images. Computer Vision and Image Understanding, 113(2), 249–260.

  51. Peyré, G., Bougleux, S., & Cohen, L. (2008). Non-local regularization of inverse problems. In ECCV ’08 (pp. 57–68). Berlin: Springer.

  52. Peyré, G., Bougleux, S., & Cohen, L. D. (2009). Non-local regularization of inverse problems. Preprint Hal-00419791.

  53. Pizarro, L., Mrázek, P., Didas, S., Grewenig, S., & Weickert, J. (2010). Generalised nonlocal image smoothing. International Journal of Computer Vision, 90, 62–87.

  54. Protter, M., Elad, M., Takeda, H., & Milanfar, P. (2009). Generalizing the non-local-means to super-resolution reconstruction. IEEE Transactions on IP, 18(1), 36–51.

  55. Shen, J., Jin, X., & Zhou, C. (2005). Gradient based image completion by solving Poisson equation. In Ad. in multimedia information processing (pp. 257–68).

  56. Sun, J., Yuan, L., Jia, J., & Shum, H. Y. (2005). Image completion with structure propagation. In Proc. of SIGGRAPH (pp. 861–868). New York: ACM.

  57. Tong, X., Zhang, J., Liu, L., Wang, X., Guo, B., & Shum, H.-Y. (2002). Synthesis of bidirectional texture functions on arbitrary surfaces. ACM Transactions on Graphics, 21(3), 665–672.

  58. Tschumperlé, D., & Deriche, R. (2005). Vector-valued image regularization with PDE’s: a common framework for different applications. IEEE Transactions on PAMI, 27(4).

  59. Wei, L.-Y., & Levoy, M. (2000). Fast texture synthesis using tree-structured vector quantization. In Proc. of the SIGGRAPH (pp. 479–488). New York: ACM.

  60. Wexler, Y., Shechtman, E., & Irani, M. (2007). Space-time completion of video. IEEE Transactions on PAMI, 29(3), 463–476.

  61. Zhou, D., & Schölkopf, B. (2005). Regularization on discrete spaces. In Proceedings of the 27th DAGM symposium (pp. 361–368). Berlin: Springer.

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Correspondence to Pablo Arias.

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Arias, P., Facciolo, G., Caselles, V. et al. A Variational Framework for Exemplar-Based Image Inpainting. Int J Comput Vis 93, 319–347 (2011).

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  • Inpainting
  • Variational methods
  • Self-similarity
  • Non-local methods
  • Exemplar-based methods