Figure 1.
The example with exact solution: the function $ F $ (left), the function $ G $ (middle), and the deformation $ -\vec{D}^{ex} $ (right)
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March
2021, 14(3): 851-863.
doi: 10.3934/dcdss.2020347
A new numerical method for level set motion in normal direction used in optical flow estimation
Faculty of Civil Engineering, Slovak University of Technology, Department of Mathematics and Descriptive Geometry, Radlinského 11,810 05 Bratislava, Slovak Republic |
We present a new numerical method for the solution of level set advection equation describing a motion in normal direction for which the speed is given by the sign function of the difference of two given functions. Taking one function as the initial condition, the solution evolves towards the second given function. One of possible applications is an optical flow estimation to find a deformation between two images in a video sequence. The new numerical method is based on a bilinear interpolation of discrete values as used for the representation of images. Under natural assumptions, it ensures a monotone decrease of the absolute difference between the numerical solution and the target function, and it handles properly the discontinuity in the speed due to the dependence on the sign function. To find the deformation between two functions (or images), the backward tracking of characteristics is used. Two numerical experiments are presented, one with an exact solution to show an experimental order of convergence and one based on two images of lungs to illustrate a possible application of the method for the optical flow estimation.
Mathematics Subject Classification:
Primary: 65D18, 65M25; Secondary: 35L60.
Citation:
Peter Frolkovič, Viera Kleinová. A new numerical method for level set motion in normal direction used in optical flow estimation. Discrete and Continuous Dynamical Systems - S,
2021, 14
(3)
: 851-863.
doi: 10.3934/dcdss.2020347
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References:
| [1] |
M. Bertalmío, G. Sapiro and G. Randall,
Morphing active contours, IEEE Trans. PAMI, 22 (2000), 733-737.
doi: 10.1109/34.865191. |
| [2] |
A. Bruhn, J. Weickert and C. Schnörr,
Lucas/Kanade meets Horn/Schunck: Combining local and global optic flow methods, Int. J. Comput. Vis., 61 (2005), 211-231.
doi: 10.1023/B:VISI.0000045324.43199.43. |
| [3] |
M. Burger, H. Dirks and C.-B. Schönlieb,
A variational model for joint motion estimation and image reconstruction, SIAM J. Imaging Sci., 11 (2018), 94-128.
doi: 10.1137/16M1084183. |
| [4] |
P. Colella,
Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), 171-200.
doi: 10.1016/0021-9991(90)90233-Q. |
| [5] |
V. Duay, N. Houhou and J.-P. Thiran, Atlas-based segmentation of medical images locally constrained by level sets, IEEE International Conference on Image Processing, Genova, Italy, 2005.
doi: 10.1109/ICIP.2005.1530298. |
| [6] |
P. Frolkovič and K. Mikula,
Semi-implicit second order schemes for numerical solution of level set advection equation on Cartesian grids, Appl. Math. Comput., 329 (2018), 129-142.
doi: 10.1016/j.amc.2018.01.065. |
| [7] |
F. Gibou, R. Fedkiw and S. Osher,
A review of level-set methods and some recent applications, J. Comput. Phys., 353 (2018), 82-109.
doi: 10.1016/j.jcp.2017.10.006. |
| [8] |
B. Horn and B. Schunck,
Determining optical flow, Artificial Intelligence, 17 (1981), 185-203.
doi: 10.1016/0004-3702(81)90024-2. |
| [9] |
C.-O. Lee, K. Jeon, Y. Ha and J. Hahn,
A variational approach to blending based on warping for non-overlapped images, Comput. Vis. Image Und., 105 (2007), 112-120.
doi: 10.1016/j.cviu.2006.09.001. |
| [10] |
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied
Mathematics, Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511791253. |
| [11] |
S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, 153, Springer-Verlag, New York, 2003.
doi: 10.1007/b98879. |
| [12] |
E. Rouy and A. Tourin,
A viscosity solutions approach to shape-from-shading, SIAM J. Numer. Anal., 29 (1992), 867-884.
doi: 10.1137/0729053. |
| [13] |
J. Sethian, Level Set Methods and Fast Marching Methods, Cambridge Monographs on Applied and Computational Mathematics, 3, Cambridge University Press, Cambridge, 1999. |
| [14] |
B. C. Vemuri, J. Ye, Y. Chen and C. M. Leonard,
Image registration via level-set motion: Applications to atlas-based segmentation, Medical Image Anal., 7 (2003), 1-20.
doi: 10.1016/S1361-8415(02)00063-4. |
show all references
References:
| [1] |
M. Bertalmío, G. Sapiro and G. Randall,
Morphing active contours, IEEE Trans. PAMI, 22 (2000), 733-737.
doi: 10.1109/34.865191. |
| [2] |
A. Bruhn, J. Weickert and C. Schnörr,
Lucas/Kanade meets Horn/Schunck: Combining local and global optic flow methods, Int. J. Comput. Vis., 61 (2005), 211-231.
doi: 10.1023/B:VISI.0000045324.43199.43. |
| [3] |
M. Burger, H. Dirks and C.-B. Schönlieb,
A variational model for joint motion estimation and image reconstruction, SIAM J. Imaging Sci., 11 (2018), 94-128.
doi: 10.1137/16M1084183. |
| [4] |
P. Colella,
Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), 171-200.
doi: 10.1016/0021-9991(90)90233-Q. |
| [5] |
V. Duay, N. Houhou and J.-P. Thiran, Atlas-based segmentation of medical images locally constrained by level sets, IEEE International Conference on Image Processing, Genova, Italy, 2005.
doi: 10.1109/ICIP.2005.1530298. |
| [6] |
P. Frolkovič and K. Mikula,
Semi-implicit second order schemes for numerical solution of level set advection equation on Cartesian grids, Appl. Math. Comput., 329 (2018), 129-142.
doi: 10.1016/j.amc.2018.01.065. |
| [7] |
F. Gibou, R. Fedkiw and S. Osher,
A review of level-set methods and some recent applications, J. Comput. Phys., 353 (2018), 82-109.
doi: 10.1016/j.jcp.2017.10.006. |
| [8] |
B. Horn and B. Schunck,
Determining optical flow, Artificial Intelligence, 17 (1981), 185-203.
doi: 10.1016/0004-3702(81)90024-2. |
| [9] |
C.-O. Lee, K. Jeon, Y. Ha and J. Hahn,
A variational approach to blending based on warping for non-overlapped images, Comput. Vis. Image Und., 105 (2007), 112-120.
doi: 10.1016/j.cviu.2006.09.001. |
| [10] |
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied
Mathematics, Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511791253. |
| [11] |
S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, 153, Springer-Verlag, New York, 2003.
doi: 10.1007/b98879. |
| [12] |
E. Rouy and A. Tourin,
A viscosity solutions approach to shape-from-shading, SIAM J. Numer. Anal., 29 (1992), 867-884.
doi: 10.1137/0729053. |
| [13] |
J. Sethian, Level Set Methods and Fast Marching Methods, Cambridge Monographs on Applied and Computational Mathematics, 3, Cambridge University Press, Cambridge, 1999. |
| [14] |
B. C. Vemuri, J. Ye, Y. Chen and C. M. Leonard,
Image registration via level-set motion: Applications to atlas-based segmentation, Medical Image Anal., 7 (2003), 1-20.
doi: 10.1016/S1361-8415(02)00063-4. |
Figure 2.
Comparison of the exact deformation $ \vec{D}^{ex} $ (the blue arrows) with the numerical one $ \vec{D} $ (the red arrows) for the discretization steps $ h = 0.1 $ (top left), $ h = 0.05 $ (top right), $ h = 0.025 $ (bottom left), and $ h = 0.0125 $ (bottom right)
Figure 3.
The images of lungs scan: the source image $ F $ (left), the target image $ G $ (middle), the difference image $ |G-F| $ (right)
Figure 4.
The plot and the table of the normalized norm $ e^n $ in (36)
Figure 5.
The image given by the values $ F(x_{ij}-\vec{D}_{ij}) $ (left) and the difference image given by the values $ |G_{ij}-F(x_{ij}-\vec{D}_{ij})| $ (right)
Figure 6.
The plot of deformation $ \vec{D} $ for the example with the images of lungs
Figure 7.
The plot of deformation $ \vec{D} $ for the example with the images of lungs. Only arrows in the points where $ |G_{ij}-F_{ij}| > E_{crit} $ are plotted
Table 1.
The error norm (35) and the corresponding $ EOC $ for the example with exact solution
| EOC | EOC | ||||
| 10 | 1 | 0.00703 | - | 0.00312 | - |
| 20 | 2 | 0.00401 | 0.81 | 0.00171 | 0.87 |
| 40 | 4 | 0.00235 | 0.77 | 0.000893 | 0.94 |
| 80 | 8 | 0.00160 | 0.55 | 0.000458 | 0.96 |
| 160 | 16 | 0.00281 | -0.81 | 0.000232 | 0.98 |
| EOC | EOC | ||||
| 10 | 1 | 0.00703 | - | 0.00312 | - |
| 20 | 2 | 0.00401 | 0.81 | 0.00171 | 0.87 |
| 40 | 4 | 0.00235 | 0.77 | 0.000893 | 0.94 |
| 80 | 8 | 0.00160 | 0.55 | 0.000458 | 0.96 |
| 160 | 16 | 0.00281 | -0.81 | 0.000232 | 0.98 |
Table 2.
The $ \text{L}_1 $ -norms and the experimental rates of convergence for the example with exact solution
| EOC | EOC | ||||
| 10 | 1 | 0.003120 | - | 0.004433 | - |
| 20 | 2 | 0.001307 | 1.2556 | 0.002379 | 0.8985 |
| 40 | 4 | 0.000528 | 1.3075 | 0.001259 | 0.9175 |
| 80 | 8 | 0.000220 | 1.2667 | 0.000659 | 0.9339 |
| 160 | 16 | 0.000096 | 1.1947 | 0.000339 | 0.9590 |
| EOC | EOC | ||||
| 10 | 1 | 0.003120 | - | 0.004433 | - |
| 20 | 2 | 0.001307 | 1.2556 | 0.002379 | 0.8985 |
| 40 | 4 | 0.000528 | 1.3075 | 0.001259 | 0.9175 |
| 80 | 8 | 0.000220 | 1.2667 | 0.000659 | 0.9339 |
| 160 | 16 | 0.000096 | 1.1947 | 0.000339 | 0.9590 |
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