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Applications of CGO solutions to coupled-physics inverse problems
Optical flow on evolving sphere-like surfaces
1. | Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Straße 69, 4040 Linz, Austria |
2. | Computational Science Center, University of Vienna, Oskar-Morgenstern-Platz 1,1090 Vienna, Austria |
3. | Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Straße 69, 4040 Linz, Austria |
In this work we consider optical flow on evolving Riemannian 2-manifolds which can be parametrised from the 2-sphere. Our main motivation is to estimate cell motion in time-lapse volumetric microscopy images depicting fluorescently labelled cells of a live zebrafish embryo. We exploit the fact that the recorded cells float on the surface of the embryo and allow for the extraction of an image sequence together with a sphere-like surface. We solve the resulting variational problem by means of a Galerkin method based on vector spherical harmonics and present numerical results computed from the aforementioned microscopy data.
References:
[1] |
F. Amat, W. Lemon, D. P. Mossing, K. McDole, Y. Wan, K. Branson, E. W. Myers and P. J. Keller,
Fast, accurate reconstruction of cell lineages from large-scale fluorescence microscopy data, Nat. Meth., 11 (2014), 951-958.
doi: 10.1038/nmeth.3036. |
[2] |
F. Amat, E. W. Myers and P. J. Keller,
Fast and robust optical flow for time-lapse microscopy using super-voxels, Bioinformatics, 29 (2013), 373-380.
doi: 10.1093/bioinformatics/bts706. |
[3] |
K. Atkinson and W. Han,
Spherical Harmonics and Approximations on the Unit Sphere: An Introduction volume 2044 of Lecture Notes in Mathematics, Springer, Heidelberg, 2012.
doi: 10.1007/978-3-642-25983-8. |
[4] |
G. Aubert, R. Deriche and P. Kornprobst,
Computing optical flow via variational techniques, SIAM J. Appl. Math., 60 (2000), 156-182.
doi: 10.1137/S0036139998340170. |
[5] |
G. Aubert and P. Kornprobst,
Mathematical Problems in Image Processing, volume 147 of Applied Mathematical Sciences, Springer, New York, 2 edition, 2006. |
[6] |
S. Baker, D. Scharstein, J. P. Lewis, S. Roth, M. J. Black and R. Szeliski,
A database and evaluation methodology for optical flow, Int. J. Comput. Vision, 92 (2011), 1-31.
doi: 10.1109/ICCV.2007.4408903. |
[7] |
M. Bauer, M. Grasmair and C. Kirisits,
Optical flow on moving manifolds, SIAM J. Imaging Sciences, 8 (2015), 484-512.
doi: 10.1137/140965235. |
[8] |
M. Botsch, L. Kobbelt, M. Pauly, P. Alliez and B. Lévy,
Polygon Mesh Processing, A K Peters, 2010.
doi: 10.1201/b10688. |
[9] |
M. P. do Carmo,
Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976. |
[10] | |
[11] |
L. C. Evans and R. F. Gariepy,
Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. |
[12] |
W. Freeden and M. Schreiner,
Spherical functions of mathematical geosciences. A scalar, vectorial, and tensorial setup, Berlin: Springer, 2009. |
[13] |
D. Gilbarg and N. Trudinger,
Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer Verlag, Berlin, 2001. |
[14] |
E. Hebey,
Sobolev Spaces on Riemannian Manifolds, volume 1635 of Lecture Notes in Mathematics, SV, Berlin, 1996.
doi: 10.1007/BFb0092907. |
[15] |
E. Hebey,
Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. |
[16] |
K. Hesse, I. H. Sloan and R. S. Womersley, Numerical integration on the sphere, In
W. Freeden, M. Z. Nashed, and T. Sonar, editors, Handbook of Geomathematics, pages 1187-1219. Springer, 2010. |
[17] |
M. W. Hirsch,
Differential Topology, volume 33 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1994. |
[18] |
B. K. P. Horn and B. G. Schunck,
Determining optical flow, Artificial Intelligence, 17 (1981), 185-203.
doi: 10.1016/0004-3702(81)90024-2. |
[19] |
A. Imiya, H. Sugaya, A. Torii and Y. Mochizuki, Variational analysis of spherical images, In
A. Gagalowicz and W. Philips, editors, Computer Analysis of Images and Patterns, volume
3691 of Lecture Notes in Computer Science, pages 104-111. Springer Berlin, Heidelberg, 2005.
doi: 10.1007/11556121_14. |
[20] |
P. J. Keller, Imaging morphogenesis: Technological advances and biological insights Science, 340 (2013), 1234168.
doi: 10.1126/science.1234168. |
[21] |
C. B. Kimmel, W. W. Ballard, S. R. Kimmel, B. Ullmann and T. F. Schilling,
Stages of embryonic development of the zebrafish, Devel. Dyn., 203 (1995), 253-310.
doi: 10.1002/aja.1002030302. |
[22] |
C. Kirisits, L. F. Lang and O. Scherzer, Optical flow on evolving surfaces with an application
to the analysis of 4D microscopy data, In A. Kuijper, K. Bredies, T. Pock, and H. Bischof,
editors, SSVM'13: Proceedings of the fourth International Conference on Scale Space and
Variational Methods in Computer Vision, volume 7893 of Lecture Notes in Computer Science,
pages 246-257, Berlin, Heidelberg, 2013. Springer-Verlag.
doi: 10.1007/978-3-642-38267-3_21. |
[23] |
C. Kirisits, L. F. Lang and O. Scherzer,
Decomposition of optical flow on the sphere, GEM. Int. J. Geomath., 5 (2014), 117-141.
doi: 10.1007/s13137-013-0055-8. |
[24] |
C. Kirisits, L. F. Lang and O. Scherzer,
Optical flow on evolving surfaces with space and time regularisation, J. Math. Imaging Vision, 52 (2015), 55-70.
doi: 10.1007/s10851-014-0513-4. |
[25] |
J. M. Lee,
Riemannian Manifolds volume 176 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1997.
doi: 10.1007/b98852. |
[26] |
J. M. Lee,
Introduction to Smooth Manifolds, volume 218 of Graduate Texts in Mathematics, Springer, New York, 2 edition, 2013. |
[27] |
J. Lefévre and S. Baillet,
Optical flow and advection on 2-Riemannian manifolds: A common framework, IEEE Trans. Pattern Anal. Mach. Intell., 30 (2008), 1081-1092.
|
[28] |
S. G. Megason and S. E. Fraser,
Digitizing life at the level of the cell: High-performance laser-scanning microscopy and image analysis for in toto imaging of development, Mech. Dev., 120 (2003), 1407-1420.
doi: 10.1016/j.mod.2003.07.005. |
[29] |
C. Melani, M. Campana, B. Lombardot, B. Rizzi, F. Veronesi, C. Zanella, P. Bourgine, K. Mikula, N. Peyriéras and A. Sarti,
Cells tracking in a live zebrafish embryo, In Proceedings of the 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBS 2007), (2007), 1631-1634.
doi: 10.1109/IEMBS.2007.4352619. |
[30] |
V. Michel,
Lectures on Constructive Approximation. Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and The Ball, New York, NY: Birkhäuser, 2013.
doi: 10.1007/978-0-8176-8403-7. |
[31] |
T. Mizoguchi, H. Verkade, J. K. Heath, A. Kuroiwa and Y. Kikuchi,
Sdf1/Cxcr4 signaling controls the dorsal migration of endodermal cells during zebrafish gastrulation, Development, 135 (2008), 2521-2529.
doi: 10.1242/dev.020107. |
[32] |
M. A. Penna and K. A. Dines,
A simple method for fitting sphere-like surfaces, IEEE Trans. Pattern Anal. Mach. Intell., 29 (2007), 1673-1678.
doi: 10.1109/TPAMI.2007.1114. |
[33] |
P. Quelhas, A. M. Mendonça and A. Campilho, Optical flow based arabidopsis thaliana root
meristem cell division detection, In A. Campilho and M. Kamel, editors, Image Analysis and
Recognition, volume 6112 of Lecture Notes in Computer Science, pages 217-226. Springer
Berlin Heidelberg, 2010.
doi: 10.1007/978-3-642-13775-4_22. |
[34] |
B. Schmid, G. Shah, N. Scherf, M. Weber, K. Thierbach, C. Campos Pérez, I. Roeder, P. Aanstad and J. Huisken, High-speed panoramic light-sheet microscopy reveals global endodermal cell dynamics Nat. Commun. , 4 (2013), p2207.
doi: 10.1038/ncomms3207. |
[35] |
Ch. Schnörr,
Determining optical flow for irregular domains by minimizing quadratic functionals of a certain class, Int. J. Comput. Vision, 6 (1991), 25-38.
|
[36] |
T. Schuster and J. Weickert,
On the application of projection methods for computing optical flow fields, Inverse Probl. Imaging, 1 (2007), 673-690.
doi: 10.3934/ipi.2007.1.673. |
[37] |
A. Torii, A. Imiya, H. Sugaya and Y. Mochizuki, Optical Flow Computation for Compound
Eyes: Variational Analysis of Omni-Directional Views, In M. De Gregorio, V. Di Maio,
M. Frucci, and C. Musio, editors, Brain, Vision, and Artificial Intelligence, volume 3704 of
Lecture Notes in Computer Science, pages 527-536. Springer Berlin, Heidelberg, 2005.
doi: 10.1007/11565123_51. |
[38] |
H. Triebel,
Theory of Function Spaces. II, volume 84 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 1992.
doi: 10.1007/978-3-0346-0419-2. |
[39] |
R. M. Warga and C. Nüsslein-Volhard,
Origin and development of the zebrafish endoderm, Development, 126 (1999), 827-838.
|
[40] |
J. Weickert, A. Bruhn, T. Brox and N. Papenberg, A survey on variational optic flow methods
for small displacements, In O. Scherzer, editor, Mathematical Models for Registration and
Applications to Medical Imaging, volume 10 of Mathematics in Industry, pages 103-136.
Springer, Berlin Heidelberg, 2006.
doi: 10.1007/978-3-540-34767-5_5. |
[41] |
J. Weickert and Ch. Schnörr,
A theoretical framework for convex regularizers in PDE-based computation of image motion, Int. J. Comput. Vision, 45 (2001), 245-264.
|
[42] |
J. Weickert and Ch. Schnörr,
Variational optic flow computation with a spatio-temporal smoothness constraint, J. Math. Imaging Vision, 14 (2001), 245-255.
|
show all references
References:
[1] |
F. Amat, W. Lemon, D. P. Mossing, K. McDole, Y. Wan, K. Branson, E. W. Myers and P. J. Keller,
Fast, accurate reconstruction of cell lineages from large-scale fluorescence microscopy data, Nat. Meth., 11 (2014), 951-958.
doi: 10.1038/nmeth.3036. |
[2] |
F. Amat, E. W. Myers and P. J. Keller,
Fast and robust optical flow for time-lapse microscopy using super-voxels, Bioinformatics, 29 (2013), 373-380.
doi: 10.1093/bioinformatics/bts706. |
[3] |
K. Atkinson and W. Han,
Spherical Harmonics and Approximations on the Unit Sphere: An Introduction volume 2044 of Lecture Notes in Mathematics, Springer, Heidelberg, 2012.
doi: 10.1007/978-3-642-25983-8. |
[4] |
G. Aubert, R. Deriche and P. Kornprobst,
Computing optical flow via variational techniques, SIAM J. Appl. Math., 60 (2000), 156-182.
doi: 10.1137/S0036139998340170. |
[5] |
G. Aubert and P. Kornprobst,
Mathematical Problems in Image Processing, volume 147 of Applied Mathematical Sciences, Springer, New York, 2 edition, 2006. |
[6] |
S. Baker, D. Scharstein, J. P. Lewis, S. Roth, M. J. Black and R. Szeliski,
A database and evaluation methodology for optical flow, Int. J. Comput. Vision, 92 (2011), 1-31.
doi: 10.1109/ICCV.2007.4408903. |
[7] |
M. Bauer, M. Grasmair and C. Kirisits,
Optical flow on moving manifolds, SIAM J. Imaging Sciences, 8 (2015), 484-512.
doi: 10.1137/140965235. |
[8] |
M. Botsch, L. Kobbelt, M. Pauly, P. Alliez and B. Lévy,
Polygon Mesh Processing, A K Peters, 2010.
doi: 10.1201/b10688. |
[9] |
M. P. do Carmo,
Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976. |
[10] | |
[11] |
L. C. Evans and R. F. Gariepy,
Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. |
[12] |
W. Freeden and M. Schreiner,
Spherical functions of mathematical geosciences. A scalar, vectorial, and tensorial setup, Berlin: Springer, 2009. |
[13] |
D. Gilbarg and N. Trudinger,
Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer Verlag, Berlin, 2001. |
[14] |
E. Hebey,
Sobolev Spaces on Riemannian Manifolds, volume 1635 of Lecture Notes in Mathematics, SV, Berlin, 1996.
doi: 10.1007/BFb0092907. |
[15] |
E. Hebey,
Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. |
[16] |
K. Hesse, I. H. Sloan and R. S. Womersley, Numerical integration on the sphere, In
W. Freeden, M. Z. Nashed, and T. Sonar, editors, Handbook of Geomathematics, pages 1187-1219. Springer, 2010. |
[17] |
M. W. Hirsch,
Differential Topology, volume 33 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1994. |
[18] |
B. K. P. Horn and B. G. Schunck,
Determining optical flow, Artificial Intelligence, 17 (1981), 185-203.
doi: 10.1016/0004-3702(81)90024-2. |
[19] |
A. Imiya, H. Sugaya, A. Torii and Y. Mochizuki, Variational analysis of spherical images, In
A. Gagalowicz and W. Philips, editors, Computer Analysis of Images and Patterns, volume
3691 of Lecture Notes in Computer Science, pages 104-111. Springer Berlin, Heidelberg, 2005.
doi: 10.1007/11556121_14. |
[20] |
P. J. Keller, Imaging morphogenesis: Technological advances and biological insights Science, 340 (2013), 1234168.
doi: 10.1126/science.1234168. |
[21] |
C. B. Kimmel, W. W. Ballard, S. R. Kimmel, B. Ullmann and T. F. Schilling,
Stages of embryonic development of the zebrafish, Devel. Dyn., 203 (1995), 253-310.
doi: 10.1002/aja.1002030302. |
[22] |
C. Kirisits, L. F. Lang and O. Scherzer, Optical flow on evolving surfaces with an application
to the analysis of 4D microscopy data, In A. Kuijper, K. Bredies, T. Pock, and H. Bischof,
editors, SSVM'13: Proceedings of the fourth International Conference on Scale Space and
Variational Methods in Computer Vision, volume 7893 of Lecture Notes in Computer Science,
pages 246-257, Berlin, Heidelberg, 2013. Springer-Verlag.
doi: 10.1007/978-3-642-38267-3_21. |
[23] |
C. Kirisits, L. F. Lang and O. Scherzer,
Decomposition of optical flow on the sphere, GEM. Int. J. Geomath., 5 (2014), 117-141.
doi: 10.1007/s13137-013-0055-8. |
[24] |
C. Kirisits, L. F. Lang and O. Scherzer,
Optical flow on evolving surfaces with space and time regularisation, J. Math. Imaging Vision, 52 (2015), 55-70.
doi: 10.1007/s10851-014-0513-4. |
[25] |
J. M. Lee,
Riemannian Manifolds volume 176 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1997.
doi: 10.1007/b98852. |
[26] |
J. M. Lee,
Introduction to Smooth Manifolds, volume 218 of Graduate Texts in Mathematics, Springer, New York, 2 edition, 2013. |
[27] |
J. Lefévre and S. Baillet,
Optical flow and advection on 2-Riemannian manifolds: A common framework, IEEE Trans. Pattern Anal. Mach. Intell., 30 (2008), 1081-1092.
|
[28] |
S. G. Megason and S. E. Fraser,
Digitizing life at the level of the cell: High-performance laser-scanning microscopy and image analysis for in toto imaging of development, Mech. Dev., 120 (2003), 1407-1420.
doi: 10.1016/j.mod.2003.07.005. |
[29] |
C. Melani, M. Campana, B. Lombardot, B. Rizzi, F. Veronesi, C. Zanella, P. Bourgine, K. Mikula, N. Peyriéras and A. Sarti,
Cells tracking in a live zebrafish embryo, In Proceedings of the 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBS 2007), (2007), 1631-1634.
doi: 10.1109/IEMBS.2007.4352619. |
[30] |
V. Michel,
Lectures on Constructive Approximation. Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and The Ball, New York, NY: Birkhäuser, 2013.
doi: 10.1007/978-0-8176-8403-7. |
[31] |
T. Mizoguchi, H. Verkade, J. K. Heath, A. Kuroiwa and Y. Kikuchi,
Sdf1/Cxcr4 signaling controls the dorsal migration of endodermal cells during zebrafish gastrulation, Development, 135 (2008), 2521-2529.
doi: 10.1242/dev.020107. |
[32] |
M. A. Penna and K. A. Dines,
A simple method for fitting sphere-like surfaces, IEEE Trans. Pattern Anal. Mach. Intell., 29 (2007), 1673-1678.
doi: 10.1109/TPAMI.2007.1114. |
[33] |
P. Quelhas, A. M. Mendonça and A. Campilho, Optical flow based arabidopsis thaliana root
meristem cell division detection, In A. Campilho and M. Kamel, editors, Image Analysis and
Recognition, volume 6112 of Lecture Notes in Computer Science, pages 217-226. Springer
Berlin Heidelberg, 2010.
doi: 10.1007/978-3-642-13775-4_22. |
[34] |
B. Schmid, G. Shah, N. Scherf, M. Weber, K. Thierbach, C. Campos Pérez, I. Roeder, P. Aanstad and J. Huisken, High-speed panoramic light-sheet microscopy reveals global endodermal cell dynamics Nat. Commun. , 4 (2013), p2207.
doi: 10.1038/ncomms3207. |
[35] |
Ch. Schnörr,
Determining optical flow for irregular domains by minimizing quadratic functionals of a certain class, Int. J. Comput. Vision, 6 (1991), 25-38.
|
[36] |
T. Schuster and J. Weickert,
On the application of projection methods for computing optical flow fields, Inverse Probl. Imaging, 1 (2007), 673-690.
doi: 10.3934/ipi.2007.1.673. |
[37] |
A. Torii, A. Imiya, H. Sugaya and Y. Mochizuki, Optical Flow Computation for Compound
Eyes: Variational Analysis of Omni-Directional Views, In M. De Gregorio, V. Di Maio,
M. Frucci, and C. Musio, editors, Brain, Vision, and Artificial Intelligence, volume 3704 of
Lecture Notes in Computer Science, pages 527-536. Springer Berlin, Heidelberg, 2005.
doi: 10.1007/11565123_51. |
[38] |
H. Triebel,
Theory of Function Spaces. II, volume 84 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 1992.
doi: 10.1007/978-3-0346-0419-2. |
[39] |
R. M. Warga and C. Nüsslein-Volhard,
Origin and development of the zebrafish endoderm, Development, 126 (1999), 827-838.
|
[40] |
J. Weickert, A. Bruhn, T. Brox and N. Papenberg, A survey on variational optic flow methods
for small displacements, In O. Scherzer, editor, Mathematical Models for Registration and
Applications to Medical Imaging, volume 10 of Mathematics in Industry, pages 103-136.
Springer, Berlin Heidelberg, 2006.
doi: 10.1007/978-3-540-34767-5_5. |
[41] |
J. Weickert and Ch. Schnörr,
A theoretical framework for convex regularizers in PDE-based computation of image motion, Int. J. Comput. Vision, 45 (2001), 245-264.
|
[42] |
J. Weickert and Ch. Schnörr,
Variational optic flow computation with a spatio-temporal smoothness constraint, J. Math. Imaging Vision, 14 (2001), 245-255.
|












coordinate domain | |
time interval | |
2-sphere | |
family of sphere-like surfaces | |
tangent plane at | |
outward unit normals to | |
parametrisations of | |
gradient matrix of | |
basis for | |
basis for | |
orthonormal basis for | |
surface velocity of | |
smooth map from | |
scalar function on | |
surface gradient on | |
tangent vector fields on | |
covariant derivative of | |
radially constant extensions of | |
scalar spherical harmonic of degree | |
vector spherical harmonic of degree | |
pushforward of |
coordinate domain | |
time interval | |
2-sphere | |
family of sphere-like surfaces | |
tangent plane at | |
outward unit normals to | |
parametrisations of | |
gradient matrix of | |
basis for | |
basis for | |
orthonormal basis for | |
surface velocity of | |
smooth map from | |
scalar function on | |
surface gradient on | |
tangent vector fields on | |
covariant derivative of | |
radially constant extensions of | |
scalar spherical harmonic of degree | |
vector spherical harmonic of degree | |
pushforward of |
Figures | 9(a) | 9(b) | 9(c) | 9(d) | |||||||
10(a) | 10(b) | 10(c) | 10(d) | 11(a) | 11(c) | 11(d) | 12(a) | 12(b) | 12(c) | 12(d) | |
| | | | | | | | | | |
Figures | 9(a) | 9(b) | 9(c) | 9(d) | |||||||
10(a) | 10(b) | 10(c) | 10(d) | 11(a) | 11(c) | 11(d) | 12(a) | 12(b) | 12(c) | 12(d) | |
| | | | | | | | | | |
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G. Mastroeni, L. Pellegrini. On the image space analysis for vector variational inequalities. Journal of Industrial and Management Optimization, 2005, 1 (1) : 123-132. doi: 10.3934/jimo.2005.1.123 |
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