Figure 1.
(a) Occluded image; (b) T1 inpainting; (c) Anistoropic T1 inpainting
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April
2017, 11(2): 247-262.
doi: 10.3934/ipi.2017012
A two-step mixed inpainting method with curvature-based anisotropy and spatial adaptivity
| 1. | Instituto de Investigación en Señales, Sistemas e Inteligencia Computacional, sinc(ⅰ), FICH-UNL/CONICET, Argentina |
| 2. | Ciudad Universitaria, CC 217, Ruta Nac. No 168, km 472.4, (3000) Santa Fe, Argentina |
| 3. | Instituto de Matemática Aplicada del Litoral, IMAL, CONICET-UNL, Centro Científico |
| 4. | Tecnológico CONICET Santa Fe, Colectora Ruta Nac. 168, km 472, Paraje "El Pozo", 3000, Santa Fe, Argentina and Departamento de Matemática |
| 5. | Facultad de Ingeniería Química, Universidad Nacional del Litoral, Santa Fe, Argentina |
The image inpainting problem consists of restoring an image from a (possibly noisy) observation, in which data from one or more regions are missing. Several inpainting models to perform this task have been developed, and although some of them perform reasonably well in certain types of images, quite a few issues are yet to be sorted out. For instance, if the image is expected to be smooth, the inpainting can be made with very good results by means of a Bayesian approach and a maximum a posteriori computation [
In this work we present a two-step inpainting process. The first step consists of using a CDD inpainting to build a pilot image from which to infer a-priori structural information on the image gradient. The second step is inpainting the image by minimizing a mixed spatially variant anisotropic functional, whose weight and penalization directions are based upon the aforementioned pilot image. Results are presented along with comparison measures in order to illustrate the performance of this inpainting method.
Mathematics Subject Classification:
Primary: 94A08, 68U10; Secondary: 65F22.
Citation:
Francisco J. Ibarrola, Ruben D. Spies. A two-step mixed inpainting method with curvature-based anisotropy and spatial adaptivity. Inverse Problems and Imaging,
2017, 11
(2)
: 247-262.
doi: 10.3934/ipi.2017012
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References:
| [1] |
R. Acar and C. R. Vogel,
Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, 10 (1994), 1217-1229.
doi: 10.1088/0266-5611/10/6/003. |
| [2] |
D. Calvetti, F. Sgallari and E. Somersalo,
Image inpainting with structural boostrap priors, Image and Vision Computing, 24 (2006), 782-793.
doi: 10.1016/j.imavis.2006.01.015. |
| [3] |
T. Chan, S. Kang and J. Shen,
Euler's elastica and curvature based inpaintings, SIAM J. on Applied Mathematics, 63 (2002), 564-592.
doi: 10.1137/S0036139901390088. |
| [4] |
T. Chan and J. Shen,
Mathematical models for local nontexture inpaintings, SIAM J. on Applied Mathematics, 62 (2002), 1019-1043.
doi: 10.1137/S0036139900368844. |
| [5] |
F. Ibarrola and R. Spies,
Image restoration with a half-quadratic approach to mixed weighted smooth and anisotropic bounded variation regularization, SOP Transactions on Applied Mathematics, 1 (2014), 57-95.
|
| [6] |
J. Idier,
Bayesian Approach to Inverse Problems, John Wiley & Sons, 2008.
doi: 10.1002/9780470611197. |
| [7] |
F. Li, Z. Li and L. Pi,
Variable exponent functionals in image restoration, Applied Mathematics and Computation, 216 (2010), 870-882.
doi: 10.1016/j.amc.2010.01.094. |
| [8] |
G. L. Mazzieri, R. D. Spies and K. G. Temperini,
Mixed spatially varying $ L^2$-$ BV$ regularization of inverse ill-posed problems, Journal of Inverse and Ill-posed Problems, 23 (2015), 571-585.
doi: 10.1515/jiip-2014-0034. |
| [9] |
R. Rockafellar, Convex Analysis, Princeton University Press, 1970.
|
| [10] |
L. I. Rudin, S. Osher and E. Fatemi,
Nonlinear total variation based noise removal algorithms (proceedings of the 11th annual international conference of the center for nonlinear studies), Physica D, 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
show all references
References:
| [1] |
R. Acar and C. R. Vogel,
Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, 10 (1994), 1217-1229.
doi: 10.1088/0266-5611/10/6/003. |
| [2] |
D. Calvetti, F. Sgallari and E. Somersalo,
Image inpainting with structural boostrap priors, Image and Vision Computing, 24 (2006), 782-793.
doi: 10.1016/j.imavis.2006.01.015. |
| [3] |
T. Chan, S. Kang and J. Shen,
Euler's elastica and curvature based inpaintings, SIAM J. on Applied Mathematics, 63 (2002), 564-592.
doi: 10.1137/S0036139901390088. |
| [4] |
T. Chan and J. Shen,
Mathematical models for local nontexture inpaintings, SIAM J. on Applied Mathematics, 62 (2002), 1019-1043.
doi: 10.1137/S0036139900368844. |
| [5] |
F. Ibarrola and R. Spies,
Image restoration with a half-quadratic approach to mixed weighted smooth and anisotropic bounded variation regularization, SOP Transactions on Applied Mathematics, 1 (2014), 57-95.
|
| [6] |
J. Idier,
Bayesian Approach to Inverse Problems, John Wiley & Sons, 2008.
doi: 10.1002/9780470611197. |
| [7] |
F. Li, Z. Li and L. Pi,
Variable exponent functionals in image restoration, Applied Mathematics and Computation, 216 (2010), 870-882.
doi: 10.1016/j.amc.2010.01.094. |
| [8] |
G. L. Mazzieri, R. D. Spies and K. G. Temperini,
Mixed spatially varying $ L^2$-$ BV$ regularization of inverse ill-posed problems, Journal of Inverse and Ill-posed Problems, 23 (2015), 571-585.
doi: 10.1515/jiip-2014-0034. |
| [9] |
R. Rockafellar, Convex Analysis, Princeton University Press, 1970.
|
| [10] |
L. I. Rudin, S. Osher and E. Fatemi,
Nonlinear total variation based noise removal algorithms (proceedings of the 11th annual international conference of the center for nonlinear studies), Physica D, 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
Figure 2.
Occluded image (a); TV inpainting (b)
Figure 3.
(a) Occluded image; (b) TV inpainting
Figure 4.
(a) Occluded image; (b) CDD inpainting
Figure 5.
(a) Occluded image; (b) Isotropic T1 inpainting; (c) CDD inpainting; (d) Mixed anisotropic T1-TV inpainting
Figure 6.
(a) Occluded noisy image; (b) Isotropic T1 inpainting; (c) CDD inpainting; (d) Anisotropic T1-TV inpainting
Figure 7.
(a) Occluded noisy image; (b) Isotropic T1 inpainting; (c) CDD inpainting; (d) Anisotropic T1-TV inpainting
Figure 8.
Left: CKS inpaintings; right: Anisotropic T1-TV inpaintings
Figure 9.
(a) Original image; (b) Occluded (masked) image; (c) T1 inpainting; (d) CDD inpainting; (e) T1-TV inpainting
| Isotropic T1 | CDD | Anisotropic T1-TV | |
| 20.140 | 35.497 | 36.329 |
| Isotropic T1 | CDD | Anisotropic T1-TV | |
| 20.140 | 35.497 | 36.329 |
| Isotropic T1 | CDD | Anisotropic T1-TV | |
| 29.127 | 29.952 | 30.868 |
| Isotropic T1 | CDD | Anisotropic T1-TV | |
| 29.127 | 29.952 | 30.868 |
| Isotropic T1 | CDD | Anisotropic T1-TV | |
| 20.568 | 21.360 | 22.112 |
| Isotropic T1 | CDD | Anisotropic T1-TV | |
| 20.568 | 21.360 | 22.112 |
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