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Augmented Lagrangian method for an Euler's elastica based segmentation model that promotes convex contours
1. | Norwegian Defence Research Establishment, P.O. Box 25, NO-2027 Kjeller, Norway |
2. | Department of Mathematics, University of Bergen, P.O. Box 7803, NO-5020 Bergen, Norway |
3. | Department of Mathematics, University of Alabama, Box 870350, Tuscaloosa, AL 35487, USA |
In this paper, we propose an image segmentation model where an $L^1$ variant of the Euler's elastica energy is used as boundary regularization. An interesting feature of this model lies in its preference for convex segmentation contours. However, due to the high order and non-differentiability of Euler's elastica energy, it is nontrivial to minimize the associated functional. As in recent work on the ordinary $L^2$-Euler's elastica model in imaging, we propose using an augmented Lagrangian method to tackle the minimization problem. Specifically, we design a novel augmented Lagrangian functional that deals with the mean curvature term differently than in previous works. The new treatment reduces the number of Lagrange multipliers employed, and more importantly, it helps represent the curvature more effectively and faithfully. Numerical experiments validate the efficiency of the proposed augmented Lagrangian method and also demonstrate new features of this particular segmentation model, such as shape driven and data driven properties.
References:
[1] |
L. Ambrosio and S. Masnou,
A direct variational approach to a problem arising in image reconstruction, Interfaces Free Bound, 5 (2003), 63-81.
|
[2] |
L. Ambrosio and S. Masnou,
On a variational problem arising in image reconstruction, Free boundary problems (Trento, 2002), Internat. Ser. Numer. Math., Birkh{a"}user, Basel,, 147 (2004), 17-26.
|
[3] |
L. Ambrosio and V. M. Tortorelli,
Approximation of functionals depending on jumps by elliptic functionals via Gamma convergence., Comm. Pure Appl. Math., 43 (1990), 999-1036.
doi: 10.1002/cpa.3160430805. |
[4] |
E. Bae, J. Shi and X.-C. Tai,
Graph Cuts for Curvature Based Image Denoising, IEEE Transactions on Image Processing, 20 (2011), 1199-1210.
doi: 10.1109/TIP.2010.2090533. |
[5] |
K. Bredies, T. Pock and B. Wirth,
A convex, lower semicontinuous approximation of Euler's elastica energy, SIAM J. Math. Anal., 47 (2015), 566-613.
doi: 10.1137/130939493. |
[6] |
C. Brito-Loeza and K. Chen,
Multigrid algorithm for high order denoising, SIAM J. Imaging. Sciences, 3 (2010), 363-389.
doi: 10.1137/080737903. |
[7] |
V. Caselles, R. Kimmel and G. Sapiro,
Geodesic active contours, Int. J. Comput. Vision, 22 (1997), 61-79.
doi: 10.1109/ICCV.1995.466871. |
[8] |
T. Chan and L. A. Vese,
Active contours without edges, IEEE Trans. Image Process., 10 (2001), 266-277.
doi: 10.1109/83.902291. |
[9] |
T. Chan and S. Esedoglu,
Aspects of total variation regularized L1 function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837.
doi: 10.1137/040604297. |
[10] |
T. Chan, S. Esedoglu and M. Nikolova,
Algorithms for finding global minimizers of denoising and segmentation models, SIAM J. Appl. Math., 66 (2006), 1632-1648.
doi: 10.1137/040615286. |
[11] |
T. Chan, S. H. Kang and J. H. Shen,
Euler's elastica and curvature based inpaintings, SIAM J. Appl. Math., 63 (2002), 564-592.
|
[12] |
T. Chan and W. Zhu,
Level set based shape prior segmentation, Proc. of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, (2005), 1164-1170.
doi: 10.1109/CVPR.2005.212. |
[13] |
Y. Chen, H. Tagare, S. Thiruvenkadam, F. Huang, D. Wilson, K. Gopinath, R. Briggs and E. Geiser,
Using prior shapes in geometric active contours in a variational framework, Int. J. Comput. Vision, 50 (2002), 315-328.
|
[14] |
D. Cremers, F. Tischhauser, J. Weickert and C. Schnorr,
Diffusion Snakes: Introducing statistical shape knowledge into the Mumford--Shah functional, Int. J. Comput. Vision, 50 (2002), 295-313.
|
[15] |
E. De Giorgi, Some remarks on Γ-convergence and least squares methods, in Composite Media
and Homogenization Theory, G. Dal Maso and G.F. Dell' Antonio (Eds.) Birkhauser, Boston,
(1991), 135–142. |
[16] |
M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc. , 1976. |
[17] |
Y. Duan, Y. Wang, X.-C. Tai and J. Hahn,
A fast augmented Lagrangian method for Euler's elastica model, SSVM 2011, LSCS, 6667 (2012), 144-156.
|
[18] |
N. El-Zehiry and L. Grady,
Fast global optimization of curvature, Proc. of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, (2010), 3257-3264.
doi: 10.1109/CVPR.2010.5540057. |
[19] |
M. Elsey and S. Esedoglu,
Analogue of the total variation denoising model in the context of geometry processing, SIAM J. Multiscale Modeling and Simulation, 7 (2009), 1549-1573.
doi: 10.1137/080736612. |
[20] |
S. Esedoglu and R. March,
Segmentation with depth but without detecting junctions, J. Math. Imaging and Vision., 18 (2003), 7-15.
doi: 10.1023/A:1021837026373. |
[21] |
M. Hintermüller, C. N. Rautenberg and J. Hahn, Functional-analytic and numerical issues
in splitting methods for total variation-based image reconstruction, Inverse Problems, 30
(2014), 055014. |
[22] |
M. Kass, A. Witkin and D. Terzopoulos,
Snakes: Active contour models, Int. J. Comput. Vision, 1 (1988), 321-331.
doi: 10.1007/BF00133570. |
[23] |
M. Leventon, W. Grimson and O. Faugeraus,
Statistical shape influence in geodesic active contours, Proc. of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, (2000), 316-323.
|
[24] |
P. L. Lions and B. Mercier,
Splitting algorithms for the sume of two nonlinear opertors, SIAM J. Numer. Amal., 16 (1979), 964-979.
doi: 10.1137/0716071. |
[25] |
R. March and M. Dozio,
A variational method for the recovery of smooth boundaries, Image and Vision Computing, 15 (1997), 705-712.
doi: 10.1016/S0262-8856(97)00002-4. |
[26] |
S. Masnou,
Disocclusion: A variational approach using level lines, IEEE Trans. Image Process., 11 (2002), 68-76.
doi: 10.1109/83.982815. |
[27] |
S. Masnou and J. M. Morel,
Level lines based disocclusion, Proc. IEEE Int. Conf. on Image Processing, Chicago, IL, (1998), 259-263.
doi: 10.1109/ICIP.1998.999016. |
[28] |
D. Mumford and J. Shah,
Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685.
doi: 10.1002/cpa.3160420503. |
[29] |
M. Myllykoski, R. Glowinski, T. Kärkkäinen and T. Rossi,
A new augmented Lagrangian approach for L1-mean curvature image denoising, SIAM J. Imaging Sci., 8 (2015), 95-125.
doi: 10.1137/140962164. |
[30] |
M. Nitzberg, D. Mumford and T. Shiota, Filering, Segmentation, and Depth, Lecture Notes
in Computer Science, 662, Springer Verlag, Berlin, 1993. |
[31] |
S. Osher and J. A. Sethian,
Fronts propagating with curvature-dependent speed-algorithm based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49.
doi: 10.1016/0021-9991(88)90002-2. |
[32] |
S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag,
New York, 2003. |
[33] |
D. Peng, B. Merriman, S. Osher, H. K. Zhao and M. Kang,
A PDE-based fast local level set method, J. Comput. Phys., 155 (1999), 410-438.
doi: 10.1006/jcph.1999.6345. |
[34] |
R. T. Rockafellar,
Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Mathematics of Operations Research, 1 (1976), 97-116.
doi: 10.1287/moor.1.2.97. |
[35] |
L. Rudin, S. Osher and E. Fatemi,
Nonlinear total variation based noise removal algorithm, Physica D, 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[36] |
T. Schoenemann, F. Kahl and D. Cremers, Curvature Regularity for Region-Based Image
Segmentation and Inpainting: A Linear Programming Relaxation, IEEE International Conference on Computer Vision (ICCV), 2009.
doi: 10.1109/ICCV.2009.5459209. |
[37] |
T. Schoenemann, F. Kahl, S. Masnou and D. Cremers,
A linear framework for region-based image segmentation and inpainting involving curvature penalization, Int. J. Comput. Vision, 99 (2012), 53-68.
doi: 10.1007/s11263-012-0518-7. |
[38] |
E. Strekalovskiy and D. Cremers,
Generalized ordering constraints for multilabel optimization, IEEE International Conference on Computer Vision, ICCV 2011, (2011), 2619-2626.
doi: 10.1109/ICCV.2011.6126551. |
[39] |
X. C. Tai, J. Hahn and G. J. Chung,
A fast algorithm for Euler's Elastica model using augmented Lagrangian method, SIAM J. Imaging Sciences, 4 (2011), 313-344.
doi: 10.1137/100803730. |
[40] |
C. Wu and X. C. Tai,
Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, Vectorial TV, and high order models, SIAM J. Imaging Sciences, 3 (2010), 300-339.
doi: 10.1137/090767558. |
[41] |
F. Yang, K. Chen and B. Yu,
Homotopy method for a mean curvature-based denoising model, Appl. Numer. Math., 62 (2012), 185-200.
doi: 10.1016/j.apnum.2011.12.001. |
[42] |
W. Zhu and T. Chan,
A variational model for capturing illusory contours using curvature, J. Math. Imaging Vision, 27 (2007), 29-40.
doi: 10.1007/s10851-006-9695-8. |
[43] |
W. Zhu and T. Chan,
Image denoising using mean curvature of image surface, SIAM J. Imaging Sciences, 5 (2012), 1-32.
doi: 10.1137/110822268. |
[44] |
W. Zhu, X. C. Tai and T. Chan,
Image segmentation using Euler's elastica as the regularization, J. Sci. Comput., 57 (2013), 414-438.
doi: 10.1007/s10915-013-9710-3. |
[45] |
W. Zhu, X. C. Tai and T. Chan,
Augmented Lagrangian method for a mean curvature based image denoising model, Inverse Probl. Imag., 7 (2013), 1409-1432.
doi: 10.3934/ipi.2013.7.1409. |
show all references
References:
[1] |
L. Ambrosio and S. Masnou,
A direct variational approach to a problem arising in image reconstruction, Interfaces Free Bound, 5 (2003), 63-81.
|
[2] |
L. Ambrosio and S. Masnou,
On a variational problem arising in image reconstruction, Free boundary problems (Trento, 2002), Internat. Ser. Numer. Math., Birkh{a"}user, Basel,, 147 (2004), 17-26.
|
[3] |
L. Ambrosio and V. M. Tortorelli,
Approximation of functionals depending on jumps by elliptic functionals via Gamma convergence., Comm. Pure Appl. Math., 43 (1990), 999-1036.
doi: 10.1002/cpa.3160430805. |
[4] |
E. Bae, J. Shi and X.-C. Tai,
Graph Cuts for Curvature Based Image Denoising, IEEE Transactions on Image Processing, 20 (2011), 1199-1210.
doi: 10.1109/TIP.2010.2090533. |
[5] |
K. Bredies, T. Pock and B. Wirth,
A convex, lower semicontinuous approximation of Euler's elastica energy, SIAM J. Math. Anal., 47 (2015), 566-613.
doi: 10.1137/130939493. |
[6] |
C. Brito-Loeza and K. Chen,
Multigrid algorithm for high order denoising, SIAM J. Imaging. Sciences, 3 (2010), 363-389.
doi: 10.1137/080737903. |
[7] |
V. Caselles, R. Kimmel and G. Sapiro,
Geodesic active contours, Int. J. Comput. Vision, 22 (1997), 61-79.
doi: 10.1109/ICCV.1995.466871. |
[8] |
T. Chan and L. A. Vese,
Active contours without edges, IEEE Trans. Image Process., 10 (2001), 266-277.
doi: 10.1109/83.902291. |
[9] |
T. Chan and S. Esedoglu,
Aspects of total variation regularized L1 function approximation, SIAM J. Appl. Math., 65 (2005), 1817-1837.
doi: 10.1137/040604297. |
[10] |
T. Chan, S. Esedoglu and M. Nikolova,
Algorithms for finding global minimizers of denoising and segmentation models, SIAM J. Appl. Math., 66 (2006), 1632-1648.
doi: 10.1137/040615286. |
[11] |
T. Chan, S. H. Kang and J. H. Shen,
Euler's elastica and curvature based inpaintings, SIAM J. Appl. Math., 63 (2002), 564-592.
|
[12] |
T. Chan and W. Zhu,
Level set based shape prior segmentation, Proc. of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, (2005), 1164-1170.
doi: 10.1109/CVPR.2005.212. |
[13] |
Y. Chen, H. Tagare, S. Thiruvenkadam, F. Huang, D. Wilson, K. Gopinath, R. Briggs and E. Geiser,
Using prior shapes in geometric active contours in a variational framework, Int. J. Comput. Vision, 50 (2002), 315-328.
|
[14] |
D. Cremers, F. Tischhauser, J. Weickert and C. Schnorr,
Diffusion Snakes: Introducing statistical shape knowledge into the Mumford--Shah functional, Int. J. Comput. Vision, 50 (2002), 295-313.
|
[15] |
E. De Giorgi, Some remarks on Γ-convergence and least squares methods, in Composite Media
and Homogenization Theory, G. Dal Maso and G.F. Dell' Antonio (Eds.) Birkhauser, Boston,
(1991), 135–142. |
[16] |
M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc. , 1976. |
[17] |
Y. Duan, Y. Wang, X.-C. Tai and J. Hahn,
A fast augmented Lagrangian method for Euler's elastica model, SSVM 2011, LSCS, 6667 (2012), 144-156.
|
[18] |
N. El-Zehiry and L. Grady,
Fast global optimization of curvature, Proc. of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, (2010), 3257-3264.
doi: 10.1109/CVPR.2010.5540057. |
[19] |
M. Elsey and S. Esedoglu,
Analogue of the total variation denoising model in the context of geometry processing, SIAM J. Multiscale Modeling and Simulation, 7 (2009), 1549-1573.
doi: 10.1137/080736612. |
[20] |
S. Esedoglu and R. March,
Segmentation with depth but without detecting junctions, J. Math. Imaging and Vision., 18 (2003), 7-15.
doi: 10.1023/A:1021837026373. |
[21] |
M. Hintermüller, C. N. Rautenberg and J. Hahn, Functional-analytic and numerical issues
in splitting methods for total variation-based image reconstruction, Inverse Problems, 30
(2014), 055014. |
[22] |
M. Kass, A. Witkin and D. Terzopoulos,
Snakes: Active contour models, Int. J. Comput. Vision, 1 (1988), 321-331.
doi: 10.1007/BF00133570. |
[23] |
M. Leventon, W. Grimson and O. Faugeraus,
Statistical shape influence in geodesic active contours, Proc. of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, (2000), 316-323.
|
[24] |
P. L. Lions and B. Mercier,
Splitting algorithms for the sume of two nonlinear opertors, SIAM J. Numer. Amal., 16 (1979), 964-979.
doi: 10.1137/0716071. |
[25] |
R. March and M. Dozio,
A variational method for the recovery of smooth boundaries, Image and Vision Computing, 15 (1997), 705-712.
doi: 10.1016/S0262-8856(97)00002-4. |
[26] |
S. Masnou,
Disocclusion: A variational approach using level lines, IEEE Trans. Image Process., 11 (2002), 68-76.
doi: 10.1109/83.982815. |
[27] |
S. Masnou and J. M. Morel,
Level lines based disocclusion, Proc. IEEE Int. Conf. on Image Processing, Chicago, IL, (1998), 259-263.
doi: 10.1109/ICIP.1998.999016. |
[28] |
D. Mumford and J. Shah,
Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685.
doi: 10.1002/cpa.3160420503. |
[29] |
M. Myllykoski, R. Glowinski, T. Kärkkäinen and T. Rossi,
A new augmented Lagrangian approach for L1-mean curvature image denoising, SIAM J. Imaging Sci., 8 (2015), 95-125.
doi: 10.1137/140962164. |
[30] |
M. Nitzberg, D. Mumford and T. Shiota, Filering, Segmentation, and Depth, Lecture Notes
in Computer Science, 662, Springer Verlag, Berlin, 1993. |
[31] |
S. Osher and J. A. Sethian,
Fronts propagating with curvature-dependent speed-algorithm based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49.
doi: 10.1016/0021-9991(88)90002-2. |
[32] |
S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag,
New York, 2003. |
[33] |
D. Peng, B. Merriman, S. Osher, H. K. Zhao and M. Kang,
A PDE-based fast local level set method, J. Comput. Phys., 155 (1999), 410-438.
doi: 10.1006/jcph.1999.6345. |
[34] |
R. T. Rockafellar,
Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Mathematics of Operations Research, 1 (1976), 97-116.
doi: 10.1287/moor.1.2.97. |
[35] |
L. Rudin, S. Osher and E. Fatemi,
Nonlinear total variation based noise removal algorithm, Physica D, 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[36] |
T. Schoenemann, F. Kahl and D. Cremers, Curvature Regularity for Region-Based Image
Segmentation and Inpainting: A Linear Programming Relaxation, IEEE International Conference on Computer Vision (ICCV), 2009.
doi: 10.1109/ICCV.2009.5459209. |
[37] |
T. Schoenemann, F. Kahl, S. Masnou and D. Cremers,
A linear framework for region-based image segmentation and inpainting involving curvature penalization, Int. J. Comput. Vision, 99 (2012), 53-68.
doi: 10.1007/s11263-012-0518-7. |
[38] |
E. Strekalovskiy and D. Cremers,
Generalized ordering constraints for multilabel optimization, IEEE International Conference on Computer Vision, ICCV 2011, (2011), 2619-2626.
doi: 10.1109/ICCV.2011.6126551. |
[39] |
X. C. Tai, J. Hahn and G. J. Chung,
A fast algorithm for Euler's Elastica model using augmented Lagrangian method, SIAM J. Imaging Sciences, 4 (2011), 313-344.
doi: 10.1137/100803730. |
[40] |
C. Wu and X. C. Tai,
Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, Vectorial TV, and high order models, SIAM J. Imaging Sciences, 3 (2010), 300-339.
doi: 10.1137/090767558. |
[41] |
F. Yang, K. Chen and B. Yu,
Homotopy method for a mean curvature-based denoising model, Appl. Numer. Math., 62 (2012), 185-200.
doi: 10.1016/j.apnum.2011.12.001. |
[42] |
W. Zhu and T. Chan,
A variational model for capturing illusory contours using curvature, J. Math. Imaging Vision, 27 (2007), 29-40.
doi: 10.1007/s10851-006-9695-8. |
[43] |
W. Zhu and T. Chan,
Image denoising using mean curvature of image surface, SIAM J. Imaging Sciences, 5 (2012), 1-32.
doi: 10.1137/110822268. |
[44] |
W. Zhu, X. C. Tai and T. Chan,
Image segmentation using Euler's elastica as the regularization, J. Sci. Comput., 57 (2013), 414-438.
doi: 10.1007/s10915-013-9710-3. |
[45] |
W. Zhu, X. C. Tai and T. Chan,
Augmented Lagrangian method for a mean curvature based image denoising model, Inverse Probl. Imag., 7 (2013), 1409-1432.
doi: 10.3934/ipi.2013.7.1409. |





1. Initialization: |
For |
2. Compute an approximate minimizer |
3. Update the Lagrangian multipliers |
|
|
|
4. Measure the relative residuals and stop the iteration if they are smaller than a threshold |
1. Initialization: |
For |
2. Compute an approximate minimizer |
3. Update the Lagrangian multipliers |
|
|
|
4. Measure the relative residuals and stop the iteration if they are smaller than a threshold |
1. Initialization: |
2. For fixed Lagrangian multiplier |
3. |
1. Initialization: |
2. For fixed Lagrangian multiplier |
3. |
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