Restoration of manifold-valued images by half-quadratic minimization

Ronny  Bergmann; Raymond H. Chan; Ralf  Hielscher; Johannes  Persch; Gabriele  Steidl

doi:10.3934/ipi.2016001

Previous Article
Forward and backward filtering based on backward stochastic differential equations
IPI Home
This Issue
Next Article

May 2016, 10(2): 281-304. doi: 10.3934/ipi.2016001

Restoration of manifold-valued images by half-quadratic minimization

Ronny Bergmann ^1, , Raymond H. Chan ^2, , Ralf Hielscher ^3, , Johannes Persch ^1, and Gabriele Steidl ^1,

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

Received May 2015 Revised November 2015 Published May 2016

Abstract
Related Papers

The paper addresses the generalization of the half-quadratic minimization method for the restoration of images having values in a complete, connected Riemannian manifold. We recall the half-quadratic minimization method using the notation of the $c$-transform and adapt the algorithm to our special variational setting. We prove the convergence of the method for Hadamard spaces. Extensive numerical examples for images with values on spheres, in the rotation group $SO(3)$, and in the manifold of positive definite matrices demonstrate the excellent performance of the algorithm. In particular, the method with $SO(3)$-valued data shows promising results for the restoration of images obtained from Electron Backscattered Diffraction which are of interest in material science.

Keywords: half-uadratic minimization, Manifold-valued data, DT-MRI., quasi-Newton method, EBSD, variational restoration methods.

Mathematics Subject Classification: 65K10, 49Q99, 49M20, 49M3.

Citation: Ronny Bergmann, Raymond H. Chan, Ralf Hielscher, Johannes Persch, Gabriele Steidl. Restoration of manifold-valued images by half-quadratic minimization. Inverse Problems & Imaging, 2016, 10 (2) : 281-304. doi: 10.3934/ipi.2016001

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	P. Grohs and M. Sprecher, Total variation regularization by iteratively reweighted least squares on Hadamard spaces and the sphere,, Preprint 2014-39, (): 2014. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[30]	C. L. Lawson, Contributions to the Theory of Linear Least Maximum Approximation,, Ph.D. Thesis, (). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[35]	J. F. Nye, Some geometrical relations in dislocated crystals, Acta Metallurgica, 1 (1953), 153-162. doi: 10.1016/0001-6160(53)90054-6. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[43]	Z.-Z. Shi and J.-S., Lecomte, private communication, 2014. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[52]	C. Villani, Topics in Optimal Transportation, AMS, Providence, 2003. doi: 10.1007/b12016. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	P. Grohs and M. Sprecher, Total variation regularization by iteratively reweighted least squares on Hadamard spaces and the sphere,, Preprint 2014-39, (): 2014. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[30]	C. L. Lawson, Contributions to the Theory of Linear Least Maximum Approximation,, Ph.D. Thesis, (). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[35]	J. F. Nye, Some geometrical relations in dislocated crystals, Acta Metallurgica, 1 (1953), 153-162. doi: 10.1016/0001-6160(53)90054-6. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[43]	Z.-Z. Shi and J.-S., Lecomte, private communication, 2014. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[52]	C. Villani, Topics in Optimal Transportation, AMS, Providence, 2003. doi: 10.1007/b12016. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

[1]	Honglan Zhu, Qin Ni, Meilan Zeng. A quasi-Newton trust region method based on a new fractional model. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 237-249. doi: 10.3934/naco.2015.5.237
[2]	Shummin Nakayama, Yasushi Narushima, Hiroshi Yabe. Memoryless quasi-Newton methods based on spectral-scaling Broyden family for unconstrained optimization. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1773-1793. doi: 10.3934/jimo.2018122
[3]	Alice Le Brigant. Computing distances and geodesics between manifold-valued curves in the SRV framework. Journal of Geometric Mechanics, 2017, 9 (2) : 131-156. doi: 10.3934/jgm.2017005
[4]	Basim A. Hassan. A new type of quasi-newton updating formulas based on the new quasi-newton equation. Numerical Algebra, Control & Optimization, 2020, 10 (2) : 227-235. doi: 10.3934/naco.2019049
[5]	Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149
[6]	Zhi-Feng Pang, Yu-Fei Yang. Semismooth Newton method for minimization of the LLT model. Inverse Problems & Imaging, 2009, 3 (4) : 677-691. doi: 10.3934/ipi.2009.3.677
[7]	Yuhong Dai, Nobuo Yamashita. Convergence analysis of sparse quasi-Newton updates with positive definite matrix completion for two-dimensional functions. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 61-69. doi: 10.3934/naco.2011.1.61
[8]	Robert D. Sidman, Marie Erie, Henry Chu. A method, with applications, for analyzing co-registered EEG and MRI data. Conference Publications, 2001, 2001 (Special) : 349-356. doi: 10.3934/proc.2001.2001.349
[9]	Zohre Aminifard, Saman Babaie-Kafaki. Diagonally scaled memoryless quasi–Newton methods with application to compressed sensing. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021191
[10]	Lori Badea. Multigrid methods for some quasi-variational inequalities. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1457-1471. doi: 10.3934/dcdss.2013.6.1457
[11]	Tim McGraw, Baba Vemuri, Evren Özarslan, Yunmei Chen, Thomas Mareci. Variational denoising of diffusion weighted MRI. Inverse Problems & Imaging, 2009, 3 (4) : 625-648. doi: 10.3934/ipi.2009.3.625
[12]	Shuang Chen, Li-Ping Pang, Dan Li. An inexact semismooth Newton method for variational inequality with symmetric cone constraints. Journal of Industrial & Management Optimization, 2015, 11 (3) : 733-746. doi: 10.3934/jimo.2015.11.733
[13]	Cheng-Dar Liou. Note on "Cost analysis of the M/M/R machine repair problem with second optional repair: Newton-Quasi method". Journal of Industrial & Management Optimization, 2012, 8 (3) : 727-732. doi: 10.3934/jimo.2012.8.727
[14]	Kuo-Hsiung Wang, Chuen-Wen Liao, Tseng-Chang Yen. Cost analysis of the M/M/R machine repair problem with second optional repair: Newton-Quasi method. Journal of Industrial & Management Optimization, 2010, 6 (1) : 197-207. doi: 10.3934/jimo.2010.6.197
[15]	Pei Yean Lee, John B Moore. Gauss-Newton-on-manifold for pose estimation. Journal of Industrial & Management Optimization, 2005, 1 (4) : 565-587. doi: 10.3934/jimo.2005.1.565
[16]	Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013
[17]	B. S. Goh, W. J. Leong, Z. Siri. Partial Newton methods for a system of equations. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 463-469. doi: 10.3934/naco.2013.3.463
[18]	Bartomeu Coll, Joan Duran, Catalina Sbert. Half-linear regularization for nonconvex image restoration models. Inverse Problems & Imaging, 2015, 9 (2) : 337-370. doi: 10.3934/ipi.2015.9.337
[19]	Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems & Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042
[20]	Shixiu Zheng, Zhilei Xu, Huan Yang, Jintao Song, Zhenkuan Pan. Comparisons of different methods for balanced data classification under the discrete non-local total variational framework. Mathematical Foundations of Computing, 2019, 2 (1) : 11-28. doi: 10.3934/mfc.2019002

2020 Impact Factor: 1.639

American Institute of Mathematical Sciences

Restoration of manifold-valued images by half-quadratic minimization

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Restoration of manifold-valued images by half-quadratic minimization

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors