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Restoration of manifold-valued images by half-quadratic minimization
1. | Department of Mathematics, University of Kaiserslautern, Paul-Ehrlich-Str. 31, 67663 Kaiserslautern, Germany, Germany, Germany |
2. | Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong, China |
3. | Faculty of Mathematics, University of Chemnitz, Reichenhainer Str. 39, 09107 Chemnitz, Germany |
References:
[1] |
P.-A. Absil, R. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton and Oxford, Princeton University Press, 2008.
doi: 10.1515/9781400830244. |
[2] |
B. L. Adams, S. I. Wright and K. Kunze, Orientation imaging: The emergence of a new microscopy, Journal Metallurgical and Materials Transactions A, 24 (1993), 819-831.
doi: 10.1007/BF02656503. |
[3] |
A. D. Aleksandrov, A theorem on triangles in a metric space and some of its applications, in Trudy Mat. Inst. Steklov., v 38, Izdat. Akad. Nauk SSSR, Moscow, 38 (1951), 5-23. |
[4] |
M. Allain, J. Idier and Y. Goussard, On global and local convergence of half-quadratic algorithms, IEEE Transactions on Image Processing, 2 (2002), 633-836.
doi: 10.1109/ICIP.2002.1040080. |
[5] |
M. Bačák, R. Bergmann, G. Steidl and A. Weinmann, A second order non-smooth variational model for restoring manifold-valued images, SIAM Journal of Scientific Computing, 38 (2016), A567-A597.
doi: 10.1137/15M101988X. |
[6] |
M. Bačák, Convex Analysis and Optimization in Hadamard Spaces, vol. 22 of De Gruyter Series in Nonlinear Analysis and Applications, De Gruyter, Berlin, 2014.
doi: 10.1515/9783110361629. |
[7] |
F. Bachmann, R. Hielscher, P. E. Jupp, W. Pantleon, H. Schaeben and E. Wegert, Inferential statistics of electron backscatter diffraction data from within individual crystalline grains, Journal of Applied Crystallography, 43 (2010), 1338-1355.
doi: 10.1107/S002188981003027X. |
[8] |
F. Bachmann, R. Hielscher and H. Schaeben, Grain detection from 2d and 3d EBSD data - specification of the MTEX algorithm, Ultramicroscopy, 111 (2011), 1720-1733.
doi: 10.1016/j.ultramic.2011.08.002. |
[9] |
R. Bergmann, F. Laus, G. Steidl and A. Weinmann, Second order differences of cyclic data and applications in variational denoising, SIAM Journal on Imaging Sciences, 7 (2014), 2916-2953.
doi: 10.1137/140969993. |
[10] |
R. Bergmann and A. Weinmann, Inpainting of cyclic data using first and second order differences, in EMMCVPR2015 (eds. X.-C. Tai, E. Bai, T. Chan, S. Y. Leung and M. Lysaker), Lecture Notes in Computer Science, Springer, Berlin, 8932 (2015), 155-168.
doi: 10.1007/978-3-319-14612-6_12. |
[11] |
R. Bergmann and A. Weinmann, A second order TV-type approach for inpainting and denoising higher dimensional combined cyclic and vector space data, Journal of Mathematical Imaging and Vision, (2016), 1-27, arXiv:1501.02684.
doi: 10.1007/s10851-015-0627-3. |
[12] |
G. E. Bredon, Topology and Geometry, vol. 139 of Graduate Texts in Mathematics, Springer, New York, 1993.
doi: 10.1007/978-1-4757-6848-0. |
[13] |
R. Bürgmann, P. A. Rosen and E. J. Fielding, Synthetic aperture radar interferometry to measure earth's surface topography and its deformation, Annual Reviews Earth and Planetary Science, 28 (2000), 169-209. |
[14] |
F. Champagnat and J. Idier, A connection between half-quadratic criteria and EM algorithms, IEEE Signal Processing Letters, 11 (2004), 709-712.
doi: 10.1109/LSP.2004.833511. |
[15] |
T. F. Chan, S. Kang and J. Shen, Total variation denoising and enhancement of color images based on the CB and HSV color models, Journal of Visual Communication and Image Representation, 12 (2001), 422-435.
doi: 10.1006/jvci.2001.0491. |
[16] |
P. Charbonnier, L. Blanc-Féraud, G. Aubert and M. Barlaud, Deterministic edge-preserving regularization in computed imaging, IEEE Transactions on Image Processing, 6 (1997), 298-311.
doi: 10.1109/83.551699. |
[17] |
P. A. Cook, Y. Bai, S. Nedjati-Gilani, K. K. Seunarine, M. G. Hall, G. J. Parker and D. C. Alexander, Camino: Open-source diffusion-mri reconstruction and processing, in Proc. Intl. Soc. Mag. Reson. Med. 14, Seattle, WA, USA, 2006, p2759. |
[18] |
I. Daubechies, R. DeVore and C. S. Güntürk, Iteratively reweighted least squares minimization for sparse recovery, Communications in Pure and Applied Mathematics, 63 (2010), 1-38.
doi: 10.1002/cpa.20303. |
[19] |
A. H. Delaney and Y. Bresler, Globally convergent edge-preserving regularized reconstruction: An application to limited-angle tomography, IEEE Transactions on Image Processing, 7 (1998), 204-221.
doi: 10.1109/83.660997. |
[20] |
C.-A. Deledalle, L. Denis and F. Tupin, NL-InSAR: Nonlocal interferogram estimation, IEEE Transactions on Geoscience Remote Sensing, 49 (2011), 1441-1452.
doi: 10.1109/TGRS.2010.2076376. |
[21] |
D. Geman and G. Reynolds, Constrained restoration and the recovery of discontinuities, IEEE Transactions on Pattern Analysis and Machine Intelligence, 14 (1992), 367-383.
doi: 10.1109/34.120331. |
[22] |
D. Geman and C. Yang, Nonlinear image recovery with half-quadratic regularization, IEEE Transactions on Image Processing, 4 (1995), 932-946.
doi: 10.1109/83.392335. |
[23] |
M. Gräf, A unified approach to scattered data approximation on $\mathbb S^{3}$ and $SO(3)$, Advances in Computational Mathematics, 37 (2012), 379-392.
doi: 10.1007/s10444-011-9214-3. |
[24] |
P. Grohs and M. Sprecher, Total variation regularization by iteratively reweighted least squares on Hadamard spaces and the sphere,, Preprint 2014-39, (): 2014.
|
[25] |
V. K. Gupta and S. R. Agnew, A simple algorithm to eliminate ambiguities in EBSD orientation map visualization and analyses: Application to fatigue crack-tips/wakes in aluminum alloys, Microscopy and Microanalysis, 16 (2010), 831-841.
doi: 10.1017/S1431927610093992. |
[26] |
J. Jost, Nonpositive Curvature: Geometric and Analytic Aspects, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1997.
doi: 10.1007/978-3-0348-8918-6. |
[27] |
R. Kimmel and N. Sochen, Orientation diffusion or how to comb a porcupine, Journal of Visual Communication and Image Representation, 13 (2002), 238-248.
doi: 10.1006/jvci.2001.0501. |
[28] |
K. Kunze, S. I. Wright, B. L. Adams and D. J. Dingley, Advances in automatic EBSP single orientation measurements, Textures and Microstructures, 20 (1993), 41-54.
doi: 10.1155/TSM.20.41. |
[29] |
R. Lai and S. Osher, A splitting method for orthogonality constrained problems, Journal of Scientific Computing, 58 (2014), 431-449.
doi: 10.1007/s10915-013-9740-x. |
[30] |
C. L. Lawson, Contributions to the Theory of Linear Least Maximum Approximation,, Ph.D. Thesis, ().
|
[31] |
J. Lellmann, E. Strekalovskiy, S. Koetter and D. Cremers, Total variation regularization for functions with values in a manifold, in IEEE ICCV 2013, 2013, 2944-2951.
doi: 10.1109/ICCV.2013.366. |
[32] |
M. Moakher and P. G. Batchelor, Symmetric positive-definite matrices: From geometry to applications and visualization, in Visualization and Processing of Tensor Fields (eds. J. Weickert and H. Hagen), Springer Berlin Heidelberg, Berlin, Heidelberg, 452 (2006), 285-298.
doi: 10.1007/3-540-31272-2_17. |
[33] |
M. Nikolova and R. H. Chan, The equivalence of half-quadratic minimization and the gradient linearization iteration, IEEE Transactions on Image Processing, 16 (2007), 1623-1627.
doi: 10.1109/TIP.2007.896622. |
[34] |
M. Nikolova and M. K. Ng, Analysis of half-quadratic minimization methods for signal and image recovery, SIAM Journal on Scientific Computing, 27 (2005), 937-966.
doi: 10.1137/030600862. |
[35] |
J. F. Nye, Some geometrical relations in dislocated crystals, Acta Metallurgica, 1 (1953), 153-162.
doi: 10.1016/0001-6160(53)90054-6. |
[36] |
X. Pennec, P. Fillard and N. Ayache, A Riemannian framework for tensor computing, International Journal of Computer Vision, 66 (2006), 41-66.
doi: 10.1007/s11263-005-3222-z. |
[37] |
M. H. Quang, S. H. Kang and T. M. Le, Image and video colorization using vector-valued reproducing kernel Hilbert spaces, Journal of Mathematical Imaging and Vision, 37 (2010), 49-65.
doi: 10.1007/s10851-010-0192-8. |
[38] |
M. Raptis and S. Soatto, Tracklet descriptors for action modeling and video analysis, in ECCV 2010, Springer, 6311 (2010), 577-590.
doi: 10.1007/978-3-642-15549-9_42. |
[39] |
J. G. Rešetnjak, Non-expansive maps in a space of curvature no greater than $K$, Akademija Nauk SSSR. Sibirskoe Otdelenie. Sibirskiĭ Matematičeskiĭ Žurnal, 9 (1968), 918-927. |
[40] |
G. Rosman, M. Bronstein, A. Bronstein, A. Wolf and R. Kimmel, Group-valued regularization framework for motion segmentation of dynamic non-rigid shapes, in Scale Space and Variational Methods in Computer Vision, Springer, 6667 (2012), 725-736.
doi: 10.1007/978-3-642-24785-9_61. |
[41] |
G. Rosman, X.-C. Tai, R. Kimmel and A. M. Bruckstein, Augmented-Lagrangian regularization of manifold-valued maps, Methods and Applications of Analysis, 21 (2014), 105-121.
doi: 10.4310/MAA.2014.v21.n1.a5. |
[42] |
L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[43] | |
[44] |
S. Sra and R. Hosseini, Conic geometric optimization on the manifold of positive definite matrices, SIAM J. Optim., 25 (2015), 713-739, arXiv:1320.1039v3.
doi: 10.1137/140978168. |
[45] |
G. Steidl, S. Setzer, B. Popilka and B. Burgeth, Restoration of matrix fields by second order cone programming, Computing, 81 (2007), 161-178.
doi: 10.1007/s00607-007-0247-x. |
[46] |
E. Strekalovskiy and D. Cremers, Total variation for cyclic structures: Convex relaxation and efficient minimization, in IEEE CVPR 2011, IEEE, 2011, 1905-1911.
doi: 10.1109/CVPR.2011.5995573. |
[47] |
E. Strekalovskiy and D. Cremers, Total cyclic variation and generalizations, Journal of Mathematical Imaging and Vision, 47 (2013), 258-277.
doi: 10.1007/s10851-012-0396-1. |
[48] |
K. T. Sturm, Probability measures on metric spaces of nonpositive curvature, heat kernels and analysis on manifolds, graphs, and metric spaces, Contemporary Mathematics, 338 (2003), 357-390.
doi: 10.1090/conm/338/06080. |
[49] |
S. Sun, B. Adams and W. King, Observation of lattice curvature near the interface of a deformed aluminium bicrystal, Philosophical Magazine A, 80 (2000), 9-25.
doi: 10.1080/01418610008212038. |
[50] |
O. Tuzel, F. Porikli and P. Meer, Learning on Lie groups for invariant detection and tracking, in CVPR 2008, IEEE, 2008, 1-8.
doi: 10.1109/CVPR.2008.4587521. |
[51] |
L. Vese and S. Osher, Numerical methods for p-harmonic flows and applications to image processing, SIAM Journal on Numerical Analysis, 40 (2002), 2085-2104.
doi: 10.1137/S0036142901396715. |
[52] |
C. Villani, Topics in Optimal Transportation, AMS, Providence, 2003.
doi: 10.1007/b12016. |
[53] |
C. R. Vogel and M. E. Oman, Iterative method for total variation denoising, SIAM Journal on Scientific Computing, 17 (1996), 227-238.
doi: 10.1137/0917016. |
[54] |
C. R. Vogel and M. E. Oman, Fast, robust total variation-based reconstruction of noisy, blurred images, IEEE Transactions on Image Processing, 7 (1998), 813-824.
doi: 10.1109/83.679423. |
[55] |
J. Weickert, C. Feddern, M. Welk, B. Burgeth and T. Brox, PDEs for tensor image processing, in Visualization and Processing of Tensor Fields (eds. J. Weickert and H. Hagen), Springer, Berlin, 2006, 399-414.
doi: 10.1007/3-540-31272-2_25. |
[56] |
A. Weinmann, L. Demaret and M. Storath, Total variation regularization for manifold-valued data, SIAM Journal on Imaging Sciences, 7 (2014), 2226-2257.
doi: 10.1137/130951075. |
[57] |
M. Welk, C. Feddern, B. Burgeth and J. Weickert, Median filtering of tensor-valued images, in Pattern Recognition (eds. B. Michaelis and G. Krell), Lecture Notes in Computer Science, 2781, Springer, Berlin, 2003, 17-24.
doi: 10.1007/978-3-540-45243-0_3. |
show all references
References:
[1] |
P.-A. Absil, R. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton and Oxford, Princeton University Press, 2008.
doi: 10.1515/9781400830244. |
[2] |
B. L. Adams, S. I. Wright and K. Kunze, Orientation imaging: The emergence of a new microscopy, Journal Metallurgical and Materials Transactions A, 24 (1993), 819-831.
doi: 10.1007/BF02656503. |
[3] |
A. D. Aleksandrov, A theorem on triangles in a metric space and some of its applications, in Trudy Mat. Inst. Steklov., v 38, Izdat. Akad. Nauk SSSR, Moscow, 38 (1951), 5-23. |
[4] |
M. Allain, J. Idier and Y. Goussard, On global and local convergence of half-quadratic algorithms, IEEE Transactions on Image Processing, 2 (2002), 633-836.
doi: 10.1109/ICIP.2002.1040080. |
[5] |
M. Bačák, R. Bergmann, G. Steidl and A. Weinmann, A second order non-smooth variational model for restoring manifold-valued images, SIAM Journal of Scientific Computing, 38 (2016), A567-A597.
doi: 10.1137/15M101988X. |
[6] |
M. Bačák, Convex Analysis and Optimization in Hadamard Spaces, vol. 22 of De Gruyter Series in Nonlinear Analysis and Applications, De Gruyter, Berlin, 2014.
doi: 10.1515/9783110361629. |
[7] |
F. Bachmann, R. Hielscher, P. E. Jupp, W. Pantleon, H. Schaeben and E. Wegert, Inferential statistics of electron backscatter diffraction data from within individual crystalline grains, Journal of Applied Crystallography, 43 (2010), 1338-1355.
doi: 10.1107/S002188981003027X. |
[8] |
F. Bachmann, R. Hielscher and H. Schaeben, Grain detection from 2d and 3d EBSD data - specification of the MTEX algorithm, Ultramicroscopy, 111 (2011), 1720-1733.
doi: 10.1016/j.ultramic.2011.08.002. |
[9] |
R. Bergmann, F. Laus, G. Steidl and A. Weinmann, Second order differences of cyclic data and applications in variational denoising, SIAM Journal on Imaging Sciences, 7 (2014), 2916-2953.
doi: 10.1137/140969993. |
[10] |
R. Bergmann and A. Weinmann, Inpainting of cyclic data using first and second order differences, in EMMCVPR2015 (eds. X.-C. Tai, E. Bai, T. Chan, S. Y. Leung and M. Lysaker), Lecture Notes in Computer Science, Springer, Berlin, 8932 (2015), 155-168.
doi: 10.1007/978-3-319-14612-6_12. |
[11] |
R. Bergmann and A. Weinmann, A second order TV-type approach for inpainting and denoising higher dimensional combined cyclic and vector space data, Journal of Mathematical Imaging and Vision, (2016), 1-27, arXiv:1501.02684.
doi: 10.1007/s10851-015-0627-3. |
[12] |
G. E. Bredon, Topology and Geometry, vol. 139 of Graduate Texts in Mathematics, Springer, New York, 1993.
doi: 10.1007/978-1-4757-6848-0. |
[13] |
R. Bürgmann, P. A. Rosen and E. J. Fielding, Synthetic aperture radar interferometry to measure earth's surface topography and its deformation, Annual Reviews Earth and Planetary Science, 28 (2000), 169-209. |
[14] |
F. Champagnat and J. Idier, A connection between half-quadratic criteria and EM algorithms, IEEE Signal Processing Letters, 11 (2004), 709-712.
doi: 10.1109/LSP.2004.833511. |
[15] |
T. F. Chan, S. Kang and J. Shen, Total variation denoising and enhancement of color images based on the CB and HSV color models, Journal of Visual Communication and Image Representation, 12 (2001), 422-435.
doi: 10.1006/jvci.2001.0491. |
[16] |
P. Charbonnier, L. Blanc-Féraud, G. Aubert and M. Barlaud, Deterministic edge-preserving regularization in computed imaging, IEEE Transactions on Image Processing, 6 (1997), 298-311.
doi: 10.1109/83.551699. |
[17] |
P. A. Cook, Y. Bai, S. Nedjati-Gilani, K. K. Seunarine, M. G. Hall, G. J. Parker and D. C. Alexander, Camino: Open-source diffusion-mri reconstruction and processing, in Proc. Intl. Soc. Mag. Reson. Med. 14, Seattle, WA, USA, 2006, p2759. |
[18] |
I. Daubechies, R. DeVore and C. S. Güntürk, Iteratively reweighted least squares minimization for sparse recovery, Communications in Pure and Applied Mathematics, 63 (2010), 1-38.
doi: 10.1002/cpa.20303. |
[19] |
A. H. Delaney and Y. Bresler, Globally convergent edge-preserving regularized reconstruction: An application to limited-angle tomography, IEEE Transactions on Image Processing, 7 (1998), 204-221.
doi: 10.1109/83.660997. |
[20] |
C.-A. Deledalle, L. Denis and F. Tupin, NL-InSAR: Nonlocal interferogram estimation, IEEE Transactions on Geoscience Remote Sensing, 49 (2011), 1441-1452.
doi: 10.1109/TGRS.2010.2076376. |
[21] |
D. Geman and G. Reynolds, Constrained restoration and the recovery of discontinuities, IEEE Transactions on Pattern Analysis and Machine Intelligence, 14 (1992), 367-383.
doi: 10.1109/34.120331. |
[22] |
D. Geman and C. Yang, Nonlinear image recovery with half-quadratic regularization, IEEE Transactions on Image Processing, 4 (1995), 932-946.
doi: 10.1109/83.392335. |
[23] |
M. Gräf, A unified approach to scattered data approximation on $\mathbb S^{3}$ and $SO(3)$, Advances in Computational Mathematics, 37 (2012), 379-392.
doi: 10.1007/s10444-011-9214-3. |
[24] |
P. Grohs and M. Sprecher, Total variation regularization by iteratively reweighted least squares on Hadamard spaces and the sphere,, Preprint 2014-39, (): 2014.
|
[25] |
V. K. Gupta and S. R. Agnew, A simple algorithm to eliminate ambiguities in EBSD orientation map visualization and analyses: Application to fatigue crack-tips/wakes in aluminum alloys, Microscopy and Microanalysis, 16 (2010), 831-841.
doi: 10.1017/S1431927610093992. |
[26] |
J. Jost, Nonpositive Curvature: Geometric and Analytic Aspects, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1997.
doi: 10.1007/978-3-0348-8918-6. |
[27] |
R. Kimmel and N. Sochen, Orientation diffusion or how to comb a porcupine, Journal of Visual Communication and Image Representation, 13 (2002), 238-248.
doi: 10.1006/jvci.2001.0501. |
[28] |
K. Kunze, S. I. Wright, B. L. Adams and D. J. Dingley, Advances in automatic EBSP single orientation measurements, Textures and Microstructures, 20 (1993), 41-54.
doi: 10.1155/TSM.20.41. |
[29] |
R. Lai and S. Osher, A splitting method for orthogonality constrained problems, Journal of Scientific Computing, 58 (2014), 431-449.
doi: 10.1007/s10915-013-9740-x. |
[30] |
C. L. Lawson, Contributions to the Theory of Linear Least Maximum Approximation,, Ph.D. Thesis, ().
|
[31] |
J. Lellmann, E. Strekalovskiy, S. Koetter and D. Cremers, Total variation regularization for functions with values in a manifold, in IEEE ICCV 2013, 2013, 2944-2951.
doi: 10.1109/ICCV.2013.366. |
[32] |
M. Moakher and P. G. Batchelor, Symmetric positive-definite matrices: From geometry to applications and visualization, in Visualization and Processing of Tensor Fields (eds. J. Weickert and H. Hagen), Springer Berlin Heidelberg, Berlin, Heidelberg, 452 (2006), 285-298.
doi: 10.1007/3-540-31272-2_17. |
[33] |
M. Nikolova and R. H. Chan, The equivalence of half-quadratic minimization and the gradient linearization iteration, IEEE Transactions on Image Processing, 16 (2007), 1623-1627.
doi: 10.1109/TIP.2007.896622. |
[34] |
M. Nikolova and M. K. Ng, Analysis of half-quadratic minimization methods for signal and image recovery, SIAM Journal on Scientific Computing, 27 (2005), 937-966.
doi: 10.1137/030600862. |
[35] |
J. F. Nye, Some geometrical relations in dislocated crystals, Acta Metallurgica, 1 (1953), 153-162.
doi: 10.1016/0001-6160(53)90054-6. |
[36] |
X. Pennec, P. Fillard and N. Ayache, A Riemannian framework for tensor computing, International Journal of Computer Vision, 66 (2006), 41-66.
doi: 10.1007/s11263-005-3222-z. |
[37] |
M. H. Quang, S. H. Kang and T. M. Le, Image and video colorization using vector-valued reproducing kernel Hilbert spaces, Journal of Mathematical Imaging and Vision, 37 (2010), 49-65.
doi: 10.1007/s10851-010-0192-8. |
[38] |
M. Raptis and S. Soatto, Tracklet descriptors for action modeling and video analysis, in ECCV 2010, Springer, 6311 (2010), 577-590.
doi: 10.1007/978-3-642-15549-9_42. |
[39] |
J. G. Rešetnjak, Non-expansive maps in a space of curvature no greater than $K$, Akademija Nauk SSSR. Sibirskoe Otdelenie. Sibirskiĭ Matematičeskiĭ Žurnal, 9 (1968), 918-927. |
[40] |
G. Rosman, M. Bronstein, A. Bronstein, A. Wolf and R. Kimmel, Group-valued regularization framework for motion segmentation of dynamic non-rigid shapes, in Scale Space and Variational Methods in Computer Vision, Springer, 6667 (2012), 725-736.
doi: 10.1007/978-3-642-24785-9_61. |
[41] |
G. Rosman, X.-C. Tai, R. Kimmel and A. M. Bruckstein, Augmented-Lagrangian regularization of manifold-valued maps, Methods and Applications of Analysis, 21 (2014), 105-121.
doi: 10.4310/MAA.2014.v21.n1.a5. |
[42] |
L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[43] | |
[44] |
S. Sra and R. Hosseini, Conic geometric optimization on the manifold of positive definite matrices, SIAM J. Optim., 25 (2015), 713-739, arXiv:1320.1039v3.
doi: 10.1137/140978168. |
[45] |
G. Steidl, S. Setzer, B. Popilka and B. Burgeth, Restoration of matrix fields by second order cone programming, Computing, 81 (2007), 161-178.
doi: 10.1007/s00607-007-0247-x. |
[46] |
E. Strekalovskiy and D. Cremers, Total variation for cyclic structures: Convex relaxation and efficient minimization, in IEEE CVPR 2011, IEEE, 2011, 1905-1911.
doi: 10.1109/CVPR.2011.5995573. |
[47] |
E. Strekalovskiy and D. Cremers, Total cyclic variation and generalizations, Journal of Mathematical Imaging and Vision, 47 (2013), 258-277.
doi: 10.1007/s10851-012-0396-1. |
[48] |
K. T. Sturm, Probability measures on metric spaces of nonpositive curvature, heat kernels and analysis on manifolds, graphs, and metric spaces, Contemporary Mathematics, 338 (2003), 357-390.
doi: 10.1090/conm/338/06080. |
[49] |
S. Sun, B. Adams and W. King, Observation of lattice curvature near the interface of a deformed aluminium bicrystal, Philosophical Magazine A, 80 (2000), 9-25.
doi: 10.1080/01418610008212038. |
[50] |
O. Tuzel, F. Porikli and P. Meer, Learning on Lie groups for invariant detection and tracking, in CVPR 2008, IEEE, 2008, 1-8.
doi: 10.1109/CVPR.2008.4587521. |
[51] |
L. Vese and S. Osher, Numerical methods for p-harmonic flows and applications to image processing, SIAM Journal on Numerical Analysis, 40 (2002), 2085-2104.
doi: 10.1137/S0036142901396715. |
[52] |
C. Villani, Topics in Optimal Transportation, AMS, Providence, 2003.
doi: 10.1007/b12016. |
[53] |
C. R. Vogel and M. E. Oman, Iterative method for total variation denoising, SIAM Journal on Scientific Computing, 17 (1996), 227-238.
doi: 10.1137/0917016. |
[54] |
C. R. Vogel and M. E. Oman, Fast, robust total variation-based reconstruction of noisy, blurred images, IEEE Transactions on Image Processing, 7 (1998), 813-824.
doi: 10.1109/83.679423. |
[55] |
J. Weickert, C. Feddern, M. Welk, B. Burgeth and T. Brox, PDEs for tensor image processing, in Visualization and Processing of Tensor Fields (eds. J. Weickert and H. Hagen), Springer, Berlin, 2006, 399-414.
doi: 10.1007/3-540-31272-2_25. |
[56] |
A. Weinmann, L. Demaret and M. Storath, Total variation regularization for manifold-valued data, SIAM Journal on Imaging Sciences, 7 (2014), 2226-2257.
doi: 10.1137/130951075. |
[57] |
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