# zbMATH — the first resource for mathematics

On a variational theory of image amodal completion. (English) Zbl 1150.49023
Digital images can be represented as gray level functions $$u (x, y)$$ defined on a simple open subset of $$\mathbb R^ 2$$. Digital images are given as a discrete set of samples, but there are standard interpolation methods to get back to a smooth image, e.g. a trigonometric polynomial by Shannon interpolation. There is no substantial difference between digital images and what we know of retinal images as rough data: in both cases, images are band-limited by an optical device and then sampled on a grid. So most questions in visual perception theory are easily translated into computer vision problems. This opens the way to a mathematical formalization and numerical experiments. In this paper, the authors study a variational model for image amodal completion, i.e., the recovery of missing or damaged portions of a digital image by technics inspired by the well–known amodal completion process in human vision. Representing the image by a real-valued function, the authors find a set of interpolating level lines, the approach that is optimal with respect to an appropriate criterion. It is proven that this method is theoretically well–founded and equivalent to a more classical method based on a direct interpolation of the function.

##### MSC:
 49Q15 Geometric measure and integration theory, integral and normal currents in optimization 49J45 Methods involving semicontinuity and convergence; relaxation 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory 92C99 Physiological, cellular and medical topics
Full Text:
##### References:
 [1] L. ALVAREZ - Y. GOUSSEAU - J.-M. MOREL, The size of objects in natural and artificial images, Advances in Imaging and Electron Physics, 111 (1999) pp. 167-242. [2] L. AMBROSIO - V. CASELLES - S. MASNOU - J.-M. MOREL, Connected components of sets of finite perimeter and applications to image processing, J. Eur. Math. Soc., 3 (2001), pp. 39-92. Zbl0981.49024 MR1812124 · Zbl 0981.49024 · doi:10.1007/s100970000028 [3] L. AMBROSIO - N. FUSCO - D. PALLARA, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, 2000. Zbl0957.49001 MR1857292 · Zbl 0957.49001 [4] L. AMBROSIO - S. MASNOU, A direct variational approach to a problem arising in image reconstruction, Interfaces and Free Boundaries, 5 (2003), pp. 63-81. Zbl1029.49037 MR1959769 · Zbl 1029.49037 · doi:10.4171/IFB/72 [5] C. BALLESTER - M. BERTALMIO - V. CASELLES - G. SAPIRO - J. VERDERA, Filling-in by joint interpolation of vector fields and gray levels, IEEE Trans. On Image Processing, 10(8) (2001), pp. 1200-1211. Zbl1037.68771 MR1851781 · Zbl 1037.68771 · doi:10.1109/83.935036 [6] G. BELLETTINI, Variational approximation of functionals with curvatures and related properties, J. of Conv. Anal., 4(1) (1997), pp. 91-108. Zbl0882.49013 MR1459883 · Zbl 0882.49013 · emis:journals/JCA/vol.4_no.1/ · eudml:228786 [7] G. BELLETTINI - G. DAL MASO - M. PAOLINI, Semicontinuity and relaxation properties of a curvature depending functional in 2D, Ann. Scuola Norm. Sup. Pisa Cl. Sci, (4) 20, no 2 (1993), pp. 247-297. Zbl0797.49013 MR1233638 · Zbl 0797.49013 · numdam:ASNSP_1993_4_20_2_247_0 · eudml:84148 [8] G. BELLETTINI - L. MUGNAI, Characterization and representation of the lower semicontinuous envelope of the elastica functional, Ann. Inst. H. Poincaré, Anal. Non Lin., 21(6) (2004), pp. 839-880. Zbl1110.49014 MR2097034 · Zbl 1110.49014 · doi:10.1016/j.anihpc.2004.01.001 · numdam:AIHPC_2004__21_6_839_0 · eudml:78642 [9] G. BELLETTINI - L. MUGNAI, A varifolds representation of the relaxed elastica functional, Submitted. Zbl1127.49032 · Zbl 1127.49032 [10] M. BERTALMIO - A. BERTOZZI - G. SAPIRO, Navier-Stokes, fluid dynamics, and image and video inpainting, In Proc. IEEE Int. Conf. on Comp. Vision and Pattern Recog., HawaõÈ, 2001. [11] M. BERTALMIO - G. SAPIRO - V. CASELLES - C. BALLESTER, Image inpainting, In Proc. ACM Conf. Comp. Graphics (SIGGRAPH), New Orleans, USA (2000), pp. 417-424. [12] M. BERTALMIO - L. VESE - G. SAPIRO - S. OSHER, Simultaneous structure and texture image inpainting, IEEE Transactions on Image Processing, 12(8) (2003), pp. 882-889. [13] R. BORNARD - E. LECAN - L. LABORELLI - J.-H. CHENOT, Missing data correction in still images and image sequences, In Proc. 10th ACM Int. Conf. on Multimedia, (2002), pp. 355-361. [14] T.F. CHAN - S.H. KANG - J. SHEN, Euler’s elastica and curvature based inpainting, SIAM Journal of Applied Math., 63 (2) (2002), pp. 564-592. Zbl1028.68185 MR1951951 · Zbl 1028.68185 · doi:10.1137/S0036139901390088 [15] T.F. CHAN - J. SHEN, Mathematical models for local deterministic inpaintings, SIAM Journal of Applied Math., 62(3) (2001), pp. 1019-1043. Zbl1050.68157 MR1897733 · Zbl 1050.68157 · doi:10.1137/S0036139900368844 [16] T.F. CHAN - J. SHEN, Non-texture inpainting by curvature-driven diffusion (CDD), Journal of Visual Comm. and Image Rep., 12(4) (2001), pp. 436-449. [17] G. CITTI - A. SARTI, A cortical based model of perceptual completion in the roto-translation space, Submitted. Zbl1088.92008 · Zbl 1088.92008 [18] A. CRIMINISI - P. PÉREZ - K. TOYAMA, Object removal by exemplar-based inpainting, In IEEE Int. Conf. Comp. Vision and Pattern Recog., vol. 2, (2003), pp. 721-728. [19] G. DAL MASO, An Introduction to G-convergence, Birkhaüser, 1993. Zbl0816.49001 MR1201152 · Zbl 0816.49001 [20] A. EFROS - T. LEUNG, Texture synthesis by non-parametric sampling, In Proc. Int. Conf. on Comp. Vision, vol. 2, Kerkyra, Greece (1999), pp. 1033-1038. [21] M. ELAD - J.-L. STARCK - D. DONOHO - P. QUERRE, Simultaneous cartoon and texture image inpainting using Morphological Component Analysis, Applied and Comp. Harmonic Analysis, 2005, to appear. Zbl1081.68732 MR2186449 · Zbl 1081.68732 · doi:10.1016/j.acha.2005.03.005 [22] S. ESEDOGLU - S. RUUTH - R. TSAI, Threshold dynamics for shape reconstruction and disocclusion, Technical report, UCLA CAM Report 05-22, April 2005, In Proc. of IEEE Int. Conf. on Image Processing 2005. [23] S. ESEDOGLU - J. SHEN, Digital image inpainting by the Mumford-ShahEuler image model, European J. Appl. Math., 13 (2002), pp. 353-370. Zbl1017.94505 MR1925256 · Zbl 1017.94505 · doi:10.1017/S0956792502004904 [24] C. FANTONI - W. GERBINO, Contour interpolation by vector field combination, Journal of Vision, 3 (2003), pp. 281-303. [25] Y. GOUSSEAU, La distribution des formes dans les images naturelles, PhD thesis, CEREMADE, Université Paris IX, 2000. [26] R. GRZIBOVSKIS - A. HEINTZ, A convolution-thresholding scheme for the Willmore flow, Preprint, 2003. Zbl1147.53054 · Zbl 1147.53054 · doi:10.4171/IFB/183 · www.ems-ph.org [27] R. HLADKY - S. PAULS, Minimal surfaces in the roto-translation group with applications to a neuro-biological image completion model, Submitted. [28] G. KANIZSA, Organization in Vision: Essays on Gestalt Perception, Präger, 1979. [29] S. MASNOU, Filtrage et Désocclusion d’Images par Méthodes d’Ensembles de Niveau, PhD thesis, Université Paris-IX Dauphine, France, 1998. [30] S. MASNOU, Disocclusion: a variational approach using level lines, IEEE Trans. Image Processing, 11 (2002), pp. 68-76. MR1888912 [31] S. MASNOU, Euler spirals for image amodal completion, In preparation. [32] S. MASNOU - J.-M. MOREL, Level lines based disocclusion, In 5th IEEE Int. Conf. on Image Processing, Chicago, Illinois, October 4-7, 1998. [33] G. MATHERON, Random Sets and Integral Geometry, John Wiley, N.Y., 1975. Zbl0321.60009 MR385969 · Zbl 0321.60009 [34] Y. MEYER, Oscillating patterns in image processing and nonlinear evolution equations, In Amer. Math. Soc., editor, University Lecture Series, volume Col. 22, 2001. Zbl0987.35003 MR1852741 · Zbl 0987.35003 [35] D. MUMFORD, Elastica and computer vision, In C.L. Bajaj, editor, Algebraic Geometry and its Applications, Springer-Verlag, New-York (1994), pp. 491-506. Zbl0798.53003 MR1272050 · Zbl 0798.53003 [36] M. NITZBERG - D. MUMFORD, The 2.1-D Sketch, In Proc. 3rd Int. Conf. on Computer Vision, Osaka, Japan (1990), pp. 138-144. [37] J. PETITOT, Neurogeometry of V1 and Kanizsa contours, Axiomathes, 13 (2003), pp. 347-363. [38] E. VILLÉGER - G. AUBERT - L. BLANC-FÉRAUD, Image Disocclusion Using a Probabilistic Gradient Orientation, In Proc. Int. Conf. on Pattern Recog. (ICPR), Cambridge, U.K., 2004. [39] L. YAROSLAVSKY - M. EDEN, Fundamentals of Digital Optics, Birkhäuser, 1996. Zbl0877.94006 MR1430120 · Zbl 0877.94006 [40] W.P. ZIEMER, Weakly differentiable functions, Springer Verlag, 1989. Zbl0692.46022 MR1014685 · Zbl 0692.46022 · doi:10.1007/978-1-4612-1015-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.