Diffusion generated methods for denoising target-valued images

Braxton Osting; Dong Wang

doi:10.3934/ipi.2020010

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April 2020, 14(2): 205-232. doi: 10.3934/ipi.2020010

Diffusion generated methods for denoising target-valued images

Braxton Osting ^, and Dong Wang

Department of Mathematics, University of Utah, Salt Lake City, UT, USA

^* Corresponding author: Braxton Osting

Received November 2018 Revised October 2019 Published February 2020

Fund Project: The first author is supported by NSF DMS 16-19755 and DMS 17-52202.

We consider the inverse problem of denoising an image where each point (pixel) is an element of a target set, which we refer to as a target-valued image. The target sets considered are either (ⅰ) a closed convex set of Euclidean space or (ⅱ) a closed subset of the sphere. The energy for the denoising problem consists of an $ L^2 $-fidelity term which is regularized by the Dirichlet energy. A relaxation of this energy, based on the heat kernel, is introduced and the associated minimization problem is proven to be well-posed. We introduce a diffusion generated method which can be used to efficiently find minimizers of this energy. We prove results for the stability and convergence of the method for both types of target sets. The method is demonstrated on a variety of synthetic and test problems, with associated target sets given by the semi-positive definite matrices, the cube, spheres, the orthogonal matrices, and the real projective line.

Keywords: denoising, image analysis, Merriman-Bence-Osher (MBO) diffusion generated motion, manifold-valued data, DT-MRI, line field.

Mathematics Subject Classification: 65M12, 65K10, 35K05, 49Q99.

Citation: Braxton Osting, Dong Wang. Diffusion generated methods for denoising target-valued images. Inverse Problems & Imaging, 2020, 14 (2) : 205-232. doi: 10.3934/ipi.2020010

References:

[1]	G. Alberti and G. Bellettini, A non-local anisotropic model for phase transitions: Asymptotic behaviour of rescaled energies, European Journal of Applied Mathematics, 9 (1998), 261-284. doi: 10.1017/S0956792598003453. Google Scholar
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[3]	M. Bačák, R. Bergmann, G. Steidl and A. Weinmann, A second order nonsmooth variational model for restoring manifold-valued images, SIAM Journal on Scientific Computing, 38 (2016), A567-A597. doi: 10.1137/15M101988X. Google Scholar
[4]	J. M. Ball, Mathematics and liquid crystals, Molecular Crystals and Liquid Crystals, 647 (2017), 1-27. doi: 10.1080/15421406.2017.1289425. Google Scholar
[5]	R. Bergmann and D. Tenbrinck, A graph framework for manifold-valued data, SIAM Journal on Imaging Sciences, 11 (2018), 325-360. doi: 10.1137/17M1118567. Google Scholar
[6]	A. L. Bertozzi and A. Flenner, Diffuse interface models on graphs for classification of high dimensional data, SIAM Review, 58 (2016), 293-328. doi: 10.1137/16M1070426. Google Scholar
[7]	D. P. Bertsekas, Convex Optimization Algorithms, Athena Scientific, 2015. Google Scholar
[8]	F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices, vol. 13 of Progress in nonlinear differential equations and their applications, Birkhäuser Boston, 1994. doi: 10.1007/978-1-4612-0287-5. Google Scholar
[9]	L. A. Cafferelli and F. H. Lin, An optimal partition problem for eigenvalues, Journal of Scientific Computing, 31 (2007), 5-18. doi: 10.1007/s10915-006-9114-8. Google Scholar
[10]	L. Calatroni, Y. van Gennip, C.-B. Schönlieb, H. M. Rowland and A. Flenner, Graph clustering, variational image segmentation methods and hough transform scale detection for object measurement in images, Journal of Mathematical Imaging and Vision, 57 (2016), 269-291. doi: 10.1007/s10851-016-0678-0. Google Scholar
[11]	P. Cook, Y. Bai, S. Nedjati-Gilani, K. Seunarine, M. Hall, G. Parker and D. C. Alexander, Camino: Open-source diffusion-mri reconstruction and processing, http://camino.cs.ucl.ac.uk/, 2006. Google Scholar
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[14]	S. Esedoglu and F. Otto, Threshold dynamics for networks with arbitrary surface tensions, Communications on Pure and Applied Mathematics, 68 (2015), 808-864. doi: 10.1002/cpa.21527. Google Scholar
[15]	S. Esedoglu, S. Ruuth and R. Tsai, Diffusion generated motion using signed distance functions, Journal of Computational Physics, 229 (2010), 1017-1042. doi: 10.1016/j.jcp.2009.10.002. Google Scholar
[16]	S. Esedoglu, S. J. Ruuth and R. Tsai, Threshold dynamics for high order geometric motions, Interfaces and Free Boundaries, 10 (2008), 263-282. doi: 10.4171/IFB/189. Google Scholar
[17]	S. Esedoglu and Y.-H. R. Tsai, Threshold dynamics for the piecewise constant Mumford-Shah functional, Journal of Computational Physics, 211 (2006), 367-384. doi: 10.1016/j.jcp.2005.05.027. Google Scholar
[18]	P. Grohs, M. Sprecher and T. Yu, Scattered manifold-valued data approximation, Numerische Mathematik, 135 (2016), 987-1010. doi: 10.1007/s00211-016-0823-0. Google Scholar
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[20]	F. Laus, M. Nikolova, J. Persch and G. Steidl, A nonlocal denoising algorithm for manifold-valued images using second order statistics, SIAM Journal on Imaging Sciences, 10 (2017), 416-448. doi: 10.1137/16M1087114. Google Scholar
[21]	T. Laux and F. Otto, Convergence of the thresholding scheme for multi-phase mean-curvature flow,, Calculus of Variations and Partial Differential Equations, 55 (2016), Art. 129, 74 pp. doi: 10.1007/s00526-016-1053-0. Google Scholar
[22]	T. Laux and D. Swartz, Convergence of thresholding schemes incorporating bulk effects, Interfaces and Free Boundaries, 19 (2017), 273-304. doi: 10.4171/IFB/383. Google Scholar
[23]	T. Laux and N. K. Yip, Analysis of diffusion generated motion for mean curvature flow in codimension two: A gradient-flow approach, Archive for Rational Mechanics and Analysis, 232 (2018), 1113-1163. doi: 10.1007/s00205-018-01340-x. Google Scholar
[24]	C. Lenglet, J. Campbell, M. Descoteaux, G. Haro, P. Savadjiev, D. Wassermann, A. Anwander, R. Deriche, G. Pike, G. Sapiro, K. Siddiqi and P. Thompson, Mathematical methods for diffusion MRI processing, NeuroImage, 45 (2009), S111-S122. doi: 10.1016/j.neuroimage.2008.10.054. Google Scholar
[25]	C. Li, C.-Y. Kao, J. C. Gore and Z. Ding, Minimization of region-scalable fitting energy for image segmentation, IEEE Transactions on Image Processing, 17 (2008), 1940-1949. doi: 10.1109/TIP.2008.2002304. Google Scholar
[26]	C. B. Macdonald, B. Merriman and S. J. Ruuth, Simple computation of reaction-diffusion processes on point clouds, Proceedings of the National Academy of Sciences, 110 (2013), 9209-9214. doi: 10.1073/pnas.1221408110. Google Scholar
[27]	A. Majumdar and A. Zarnescu, Landau-de Gennes theory of nematic liquid crystals: The Oseen-Frank limit and beyond, Archive for Rational Mechanics and Analysis, 196 (2009), 227-280. doi: 10.1007/s00205-009-0249-2. Google Scholar
[28]	B. Merriman, J. K. Bence and S. Osher, Diffusion generated motion by mean curvature, UCLA CAM Report 92-18, 1992, URL ftp://ftp.math.ucla.edu/pub/camreport/cam92-18.pdf. Google Scholar
[29]	B. Merriman, J. Bence and S. Osher, Diffusion generated motion by mean curvature, AMS Selected Letters, Crystal Grower's Workshop, 73-83. Google Scholar
[30]	M. Miranda, D. Pallara, F. Paronetto and M. Preunkert, Short-time heat flow and functions of bounded variation in $r^n$, Annales-Faculte des Sciences Toulouse Mathematiques, 16 (2007), 125-145. doi: 10.5802/afst.1142. Google Scholar
[31]	B. Osting and T. H. Reeb, Consistency of Dirichlet partitions, SIAM Journal on Mathematical Analysis, 49 (2017), 4251-4274. doi: 10.1137/16M1098309. Google Scholar
[32]	B. Osting and D. Wang, A diffusion generated method for orthogonal matrix-valued fields, Mathematics of Computation, 89 (2020), 515-550. doi: 10.1090/mcom/3473. Google Scholar
[33]	B. Osting, C. D. White and E. Oudet, Minimal Dirichlet energy partitions for graphs, SIAM Journal on Scientific Computing, 36 (2014), A1635-A1651. doi: 10.1137/130934568. Google Scholar
[34]	F. Rocca, C. Prati and A. Ferretti, An overview of SAR interferometry, in Proceedings of the 3rd ERS Symposium on Space at the Service of our Environment, 1997, URL http://earth.esa.int/workshops/ers97/program-details/speeches/rocca-et-al/. Google Scholar
[35]	J. Rubinstein, P. Sternberg and J. B. Keller, Reaction-diffusion processes and evolution to harmonic maps, SIAM Journal on Applied Mathematics, 49 (1989), 1722-1733. doi: 10.1137/0149104. Google Scholar
[36]	L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F. Google Scholar
[37]	S. J. Ruuth, B. Merriman, J. Xin and S. Osher, Diffusion-generated motion by mean curvature for filaments, Journal of Nonlinear Science, 11 (2001), 473-493. doi: 10.1007/s00332-001-0404-x. Google Scholar
[38]	S. J. Ruuth and B. Merriman, A simple embedding method for solving partial differential equations on surfaces, Journal of Computational Physics, 227 (2008), 1943-1961. doi: 10.1016/j.jcp.2007.10.009. Google Scholar
[39]	L. Sciavicco and B. Siciliano, Modelling and Control of Robot Manipulators, Springer London, 2000. doi: 10.1007/978-1-4471-0449-0. Google Scholar
[40]	P. Skraba, B. Wang, G. Chen and P. Rosen, Robustness-based simplification of 2d steady and unsteady vector fields, IEEE Transactions on Visualization and Computer Graphics, 21 (2015), 930-944. doi: 10.1109/TVCG.2015.2440250. Google Scholar
[41]	Y. van Gennip, N. Guillen, B. Osting and A. L. Bertozzi, Mean curvature, threshold dynamics, and phase field theory on finite graphs, Milan Journal of Mathematics, 82 (2014), 3-65. doi: 10.1007/s00032-014-0216-8. Google Scholar
[42]	R. Viertel and B. Osting, An approach to quad meshing based on harmonic cross valued maps and the Ginzburg-Landau theory, SIAM Journal on Scientific Computing, 41 (2019), A452-A479. doi: 10.1137/17M1142703. Google Scholar
[43]	B. A. Wandell, Clarifying human white matter, Annual Review of Neuroscience, 39 (2016), 103-128. Google Scholar
[44]	D. Wang, A. Cherkaev and B. Osting, Dynamics and stationary configurations of heterogeneous foams, PLOS ONE, 14 (2019), e0215836. doi: 10.1371/journal.pone.0215836. Google Scholar
[45]	D. Wang, H. Li, X. Wei and X.-P. Wang, An efficient iterative thresholding method for image segmentation, Journal of Computational Physics, 350 (2017), 657-667. doi: 10.1016/j.jcp.2017.08.020. Google Scholar
[46]	D. Wang and B. Osting, A diffusion generated method for computing Dirichlet partitions, Journal of Computational and Applied Mathematics, 351 (2019), 302-316. doi: 10.1016/j.cam.2018.11.015. Google Scholar
[47]	D. Wang, B. Osting and X.-P. Wang, Interface dynamics for an Allen-Cahn-type equation governing a matrix-valued field, SIAM Multiscale Modelling and Simulation, 17 (2019), 1252-1273. doi: 10.1137/19M1250595. Google Scholar
[48]	D. Wang and X.-P. Wang, The iterative convolution-thresholding method (ictm) for image segmentation, arXiv preprint, arXiv 1904.10917. Google Scholar
[49]	D. Wang, X.-P. Wang and X. Xu, An improved threshold dynamics method for wetting dynamics, Journal of Computational Physics, 392 (2019), 291-310. doi: 10.1016/j.jcp.2019.04.037. Google Scholar
[50]	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
[51]	X. Xu, D. Wang and X.-P. Wang, An efficient threshold dynamics method for wetting on rough surfaces, Journal of Computational Physics, 330 (2017), 510-528. doi: 10.1016/j.jcp.2016.11.008. Google Scholar
[52]	D. Zosso and B. Osting, A minimal surface criterion for graph partitioning, AIMS Inverse Problems and Imaging, 10 (2016), 1149-1180. doi: 10.3934/ipi.2016036. Google Scholar

show all references

References:

[1]	G. Alberti and G. Bellettini, A non-local anisotropic model for phase transitions: Asymptotic behaviour of rescaled energies, European Journal of Applied Mathematics, 9 (1998), 261-284. doi: 10.1017/S0956792598003453. Google Scholar
[2]	H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, Society for Industrial and Applied Mathematics, 2006, URL http://dx.doi.org/10.1137/1.9780898718782. Google Scholar
[3]	M. Bačák, R. Bergmann, G. Steidl and A. Weinmann, A second order nonsmooth variational model for restoring manifold-valued images, SIAM Journal on Scientific Computing, 38 (2016), A567-A597. doi: 10.1137/15M101988X. Google Scholar
[4]	J. M. Ball, Mathematics and liquid crystals, Molecular Crystals and Liquid Crystals, 647 (2017), 1-27. doi: 10.1080/15421406.2017.1289425. Google Scholar
[5]	R. Bergmann and D. Tenbrinck, A graph framework for manifold-valued data, SIAM Journal on Imaging Sciences, 11 (2018), 325-360. doi: 10.1137/17M1118567. Google Scholar
[6]	A. L. Bertozzi and A. Flenner, Diffuse interface models on graphs for classification of high dimensional data, SIAM Review, 58 (2016), 293-328. doi: 10.1137/16M1070426. Google Scholar
[7]	D. P. Bertsekas, Convex Optimization Algorithms, Athena Scientific, 2015. Google Scholar
[8]	F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices, vol. 13 of Progress in nonlinear differential equations and their applications, Birkhäuser Boston, 1994. doi: 10.1007/978-1-4612-0287-5. Google Scholar
[9]	L. A. Cafferelli and F. H. Lin, An optimal partition problem for eigenvalues, Journal of Scientific Computing, 31 (2007), 5-18. doi: 10.1007/s10915-006-9114-8. Google Scholar
[10]	L. Calatroni, Y. van Gennip, C.-B. Schönlieb, H. M. Rowland and A. Flenner, Graph clustering, variational image segmentation methods and hough transform scale detection for object measurement in images, Journal of Mathematical Imaging and Vision, 57 (2016), 269-291. doi: 10.1007/s10851-016-0678-0. Google Scholar
[11]	P. Cook, Y. Bai, S. Nedjati-Gilani, K. Seunarine, M. Hall, G. Parker and D. C. Alexander, Camino: Open-source diffusion-mri reconstruction and processing, http://camino.cs.ucl.ac.uk/, 2006. Google Scholar
[12]	W. E and X.-P. Wang, Numerical methods for the Landau-Lifshitz equation, SIAM Journal on Numerical Analysis, 38 (2000), 1647-1665. doi: 10.1137/S0036142999352199. Google Scholar
[13]	I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Society for Industrial and Applied Mathematics, 1999. doi: 10.1137/1.9781611971088. Google Scholar
[14]	S. Esedoglu and F. Otto, Threshold dynamics for networks with arbitrary surface tensions, Communications on Pure and Applied Mathematics, 68 (2015), 808-864. doi: 10.1002/cpa.21527. Google Scholar
[15]	S. Esedoglu, S. Ruuth and R. Tsai, Diffusion generated motion using signed distance functions, Journal of Computational Physics, 229 (2010), 1017-1042. doi: 10.1016/j.jcp.2009.10.002. Google Scholar
[16]	S. Esedoglu, S. J. Ruuth and R. Tsai, Threshold dynamics for high order geometric motions, Interfaces and Free Boundaries, 10 (2008), 263-282. doi: 10.4171/IFB/189. Google Scholar
[17]	S. Esedoglu and Y.-H. R. Tsai, Threshold dynamics for the piecewise constant Mumford-Shah functional, Journal of Computational Physics, 211 (2006), 367-384. doi: 10.1016/j.jcp.2005.05.027. Google Scholar
[18]	P. Grohs, M. Sprecher and T. Yu, Scattered manifold-valued data approximation, Numerische Mathematik, 135 (2016), 987-1010. doi: 10.1007/s00211-016-0823-0. Google Scholar
[19]	B. M. Kampes, Radar Interferometry, Springer Netherlands, 2006. Google Scholar
[20]	F. Laus, M. Nikolova, J. Persch and G. Steidl, A nonlocal denoising algorithm for manifold-valued images using second order statistics, SIAM Journal on Imaging Sciences, 10 (2017), 416-448. doi: 10.1137/16M1087114. Google Scholar
[21]	T. Laux and F. Otto, Convergence of the thresholding scheme for multi-phase mean-curvature flow,, Calculus of Variations and Partial Differential Equations, 55 (2016), Art. 129, 74 pp. doi: 10.1007/s00526-016-1053-0. Google Scholar
[22]	T. Laux and D. Swartz, Convergence of thresholding schemes incorporating bulk effects, Interfaces and Free Boundaries, 19 (2017), 273-304. doi: 10.4171/IFB/383. Google Scholar
[23]	T. Laux and N. K. Yip, Analysis of diffusion generated motion for mean curvature flow in codimension two: A gradient-flow approach, Archive for Rational Mechanics and Analysis, 232 (2018), 1113-1163. doi: 10.1007/s00205-018-01340-x. Google Scholar
[24]	C. Lenglet, J. Campbell, M. Descoteaux, G. Haro, P. Savadjiev, D. Wassermann, A. Anwander, R. Deriche, G. Pike, G. Sapiro, K. Siddiqi and P. Thompson, Mathematical methods for diffusion MRI processing, NeuroImage, 45 (2009), S111-S122. doi: 10.1016/j.neuroimage.2008.10.054. Google Scholar
[25]	C. Li, C.-Y. Kao, J. C. Gore and Z. Ding, Minimization of region-scalable fitting energy for image segmentation, IEEE Transactions on Image Processing, 17 (2008), 1940-1949. doi: 10.1109/TIP.2008.2002304. Google Scholar
[26]	C. B. Macdonald, B. Merriman and S. J. Ruuth, Simple computation of reaction-diffusion processes on point clouds, Proceedings of the National Academy of Sciences, 110 (2013), 9209-9214. doi: 10.1073/pnas.1221408110. Google Scholar
[27]	A. Majumdar and A. Zarnescu, Landau-de Gennes theory of nematic liquid crystals: The Oseen-Frank limit and beyond, Archive for Rational Mechanics and Analysis, 196 (2009), 227-280. doi: 10.1007/s00205-009-0249-2. Google Scholar
[28]	B. Merriman, J. K. Bence and S. Osher, Diffusion generated motion by mean curvature, UCLA CAM Report 92-18, 1992, URL ftp://ftp.math.ucla.edu/pub/camreport/cam92-18.pdf. Google Scholar
[29]	B. Merriman, J. Bence and S. Osher, Diffusion generated motion by mean curvature, AMS Selected Letters, Crystal Grower's Workshop, 73-83. Google Scholar
[30]	M. Miranda, D. Pallara, F. Paronetto and M. Preunkert, Short-time heat flow and functions of bounded variation in $r^n$, Annales-Faculte des Sciences Toulouse Mathematiques, 16 (2007), 125-145. doi: 10.5802/afst.1142. Google Scholar
[31]	B. Osting and T. H. Reeb, Consistency of Dirichlet partitions, SIAM Journal on Mathematical Analysis, 49 (2017), 4251-4274. doi: 10.1137/16M1098309. Google Scholar
[32]	B. Osting and D. Wang, A diffusion generated method for orthogonal matrix-valued fields, Mathematics of Computation, 89 (2020), 515-550. doi: 10.1090/mcom/3473. Google Scholar
[33]	B. Osting, C. D. White and E. Oudet, Minimal Dirichlet energy partitions for graphs, SIAM Journal on Scientific Computing, 36 (2014), A1635-A1651. doi: 10.1137/130934568. Google Scholar
[34]	F. Rocca, C. Prati and A. Ferretti, An overview of SAR interferometry, in Proceedings of the 3rd ERS Symposium on Space at the Service of our Environment, 1997, URL http://earth.esa.int/workshops/ers97/program-details/speeches/rocca-et-al/. Google Scholar
[35]	J. Rubinstein, P. Sternberg and J. B. Keller, Reaction-diffusion processes and evolution to harmonic maps, SIAM Journal on Applied Mathematics, 49 (1989), 1722-1733. doi: 10.1137/0149104. Google Scholar
[36]	L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F. Google Scholar
[37]	S. J. Ruuth, B. Merriman, J. Xin and S. Osher, Diffusion-generated motion by mean curvature for filaments, Journal of Nonlinear Science, 11 (2001), 473-493. doi: 10.1007/s00332-001-0404-x. Google Scholar
[38]	S. J. Ruuth and B. Merriman, A simple embedding method for solving partial differential equations on surfaces, Journal of Computational Physics, 227 (2008), 1943-1961. doi: 10.1016/j.jcp.2007.10.009. Google Scholar
[39]	L. Sciavicco and B. Siciliano, Modelling and Control of Robot Manipulators, Springer London, 2000. doi: 10.1007/978-1-4471-0449-0. Google Scholar
[40]	P. Skraba, B. Wang, G. Chen and P. Rosen, Robustness-based simplification of 2d steady and unsteady vector fields, IEEE Transactions on Visualization and Computer Graphics, 21 (2015), 930-944. doi: 10.1109/TVCG.2015.2440250. Google Scholar
[41]	Y. van Gennip, N. Guillen, B. Osting and A. L. Bertozzi, Mean curvature, threshold dynamics, and phase field theory on finite graphs, Milan Journal of Mathematics, 82 (2014), 3-65. doi: 10.1007/s00032-014-0216-8. Google Scholar
[42]	R. Viertel and B. Osting, An approach to quad meshing based on harmonic cross valued maps and the Ginzburg-Landau theory, SIAM Journal on Scientific Computing, 41 (2019), A452-A479. doi: 10.1137/17M1142703. Google Scholar
[43]	B. A. Wandell, Clarifying human white matter, Annual Review of Neuroscience, 39 (2016), 103-128. Google Scholar
[44]	D. Wang, A. Cherkaev and B. Osting, Dynamics and stationary configurations of heterogeneous foams, PLOS ONE, 14 (2019), e0215836. doi: 10.1371/journal.pone.0215836. Google Scholar
[45]	D. Wang, H. Li, X. Wei and X.-P. Wang, An efficient iterative thresholding method for image segmentation, Journal of Computational Physics, 350 (2017), 657-667. doi: 10.1016/j.jcp.2017.08.020. Google Scholar
[46]	D. Wang and B. Osting, A diffusion generated method for computing Dirichlet partitions, Journal of Computational and Applied Mathematics, 351 (2019), 302-316. doi: 10.1016/j.cam.2018.11.015. Google Scholar
[47]	D. Wang, B. Osting and X.-P. Wang, Interface dynamics for an Allen-Cahn-type equation governing a matrix-valued field, SIAM Multiscale Modelling and Simulation, 17 (2019), 1252-1273. doi: 10.1137/19M1250595. Google Scholar
[48]	D. Wang and X.-P. Wang, The iterative convolution-thresholding method (ictm) for image segmentation, arXiv preprint, arXiv 1904.10917. Google Scholar
[49]	D. Wang, X.-P. Wang and X. Xu, An improved threshold dynamics method for wetting dynamics, Journal of Computational Physics, 392 (2019), 291-310. doi: 10.1016/j.jcp.2019.04.037. Google Scholar
[50]	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
[51]	X. Xu, D. Wang and X.-P. Wang, An efficient threshold dynamics method for wetting on rough surfaces, Journal of Computational Physics, 330 (2017), 510-528. doi: 10.1016/j.jcp.2016.11.008. Google Scholar
[52]	D. Zosso and B. Osting, A minimal surface criterion for graph partitioning, AIMS Inverse Problems and Imaging, 10 (2016), 1149-1180. doi: 10.3934/ipi.2016036. Google Scholar

Figure 1. Results of denoising an obstructed lemniscate of Bernoulli on the sphere, $ \mathcal{S}^2 $, with $ \lambda = 0.05, 0.1 $ and $ 0.15 $, respectively. The Hausdorff distances from the denoised curve to the original curve are 0.721, 0.501 and 0.504, respectively. In this simulation, $ \tau $ is fixed as $ 10^{-3} $. See Section 4.1

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Figure 2. Results of the reconstruction of a noisy $ \mathbb{S}^2 $-valued image with $ \lambda = 0.05 $, $ 0.1 $, $ 0.15 $, and $ 0.2 $, respectively. In all the simulations, $ \tau $ is fixed as $ 10^{-3} $. See Section 4.2

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Figure 3. Results of the reconstruction of a noisy $ \mathbb{S}^2 $-valued image with different methods. See Section 4.2

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Figure 4. The original 'peppers' image and one corrupted with noise. See Section 4.3

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Figure 5. Denoising in the RGB channel on the noisy 'peppers' image (see Figure 4(b)) with $ \tau = 10^{-4} $ and $ \lambda = 0.85 $, $ 0.9 $, and $ 0.95 $. The PSNR listed below each image indicates the quality of the result. See Section 4.3

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Figure 6. Denoising in the HSV channel on the noisy 'peppers' image image (see Figure 4(b)) with $ \tau = 10^{-4} $ and $ \lambda = 0.85 $, $ 0.9 $, and $ 0.95 $. The PSNR are listed below each image to indicate the quality of the result.See Section 4.3

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Figure 7. The original and noisy SPD-valued image. See Section 4.4

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Figure 8. The result of reconstruction in Figure 7(b) with $ \tau = 10^{-3} $ and $ \lambda = 0.05 $, $ 0.1 $, and $ 0.15 $. See Section 4.4

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Figure 9. Slice 28 of the original Camino DT-MRI data and the subset $ \Omega_{1} $. See Section 4.5

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Figure 10. Results of the reconstruction in Figure 9(a) with $ \tau = 10^{-4} $ and $ \lambda = 0.1 $, $ 0.2 $, and $ 0.3 $. See Section 4.5

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Figure 11. The original and noisy synthetic line-field image. See Section 4.6

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Figure 12. Denoising results for the line-field in Figure 11(b) with $ \tau = 10^{-2} $ and $ \lambda = 0.05 $, $ 0.1 $, and $ 0.15 $. See Section 4.6

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Figure 13. Denoising results for two fingerprint images with $ \tau = 10^{-2} $ and $ \lambda = 0.15 $. See Section 4.7

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Table 1. Examples of target sets, $ T $, and penalization functions, $ L $. We've grouped the examples by convex sets (top), subsets of the Euclidean sphere (middle), and other (bottom)

$ k $	$ T $	L(x)	comment	section
$ k $	$ \mathbb R^k $	$ 0 $	harmonic function
$ k $	$ T \subset \mathbb R^k $ convex	$ \frac{1}{2} \rm{dist}^2(x, T) $	convex set-valued field	§4.3
$ n^2 $	$ \rm{SPD}(n) $	$ \frac{1}{2} \rm{dist}^2\left(x, \rm{SPD}(n)\right) $	$ \rm{SPD} $ matrix-valued field	§4.4, §4.5
1	$ \{\pm 1 \} $	$ \frac{1}{4} (x^2-1)^2 $	Allen-Cahn
2	$ \mathbb S^1 $	$ \frac{1}{4} ( \|x\|^2-1)^2 $	Ginzburg-Landau	§4.3
k	$ \mathbb S^{k-1} $	$ \frac{1}{4} ( \|x\|^2-1)^2 $	sphere-valued field	§4.1, §4.2
$ n^2 $	$ O(n) $	$ \frac{1}{4} \\| x^t x - I_n \\|^2_F $	orthogonal matrix-valued field
$ k $	coordinate axes, $ \Sigma_k $	$ \frac{1}{4} \sum_{i \neq j } x_i^2 x_j^2 $	Dirichlet partitions
	$ \mathbb{RP}^1 $		line field	§4.6, §4.7

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$ k $	$ T $	L(x)	comment	section
$ k $	$ \mathbb R^k $	$ 0 $	harmonic function
$ k $	$ T \subset \mathbb R^k $ convex	$ \frac{1}{2} \rm{dist}^2(x, T) $	convex set-valued field	§4.3
$ n^2 $	$ \rm{SPD}(n) $	$ \frac{1}{2} \rm{dist}^2\left(x, \rm{SPD}(n)\right) $	$ \rm{SPD} $ matrix-valued field	§4.4, §4.5
1	$ \{\pm 1 \} $	$ \frac{1}{4} (x^2-1)^2 $	Allen-Cahn
2	$ \mathbb S^1 $	$ \frac{1}{4} ( \|x\|^2-1)^2 $	Ginzburg-Landau	§4.3
k	$ \mathbb S^{k-1} $	$ \frac{1}{4} ( \|x\|^2-1)^2 $	sphere-valued field	§4.1, §4.2
$ n^2 $	$ O(n) $	$ \frac{1}{4} \\| x^t x - I_n \\|^2_F $	orthogonal matrix-valued field
$ k $	coordinate axes, $ \Sigma_k $	$ \frac{1}{4} \sum_{i \neq j } x_i^2 x_j^2 $	Dirichlet partitions
	$ \mathbb{RP}^1 $		line field	§4.6, §4.7

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2019 Impact Factor: 1.373

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2019 Impact Factor: 1.373

American Institute of Mathematical Sciences

Diffusion generated methods for denoising target-valued images

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Diffusion generated methods for denoising target-valued images

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