American Institute of Mathematical Sciences

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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 and 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.

[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.

[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.

[4]

J. M. Ball, Mathematics and liquid crystals, Molecular Crystals and Liquid Crystals, 647 (2017), 1-27.  doi: 10.1080/15421406.2017.1289425.

[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.

[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.

[7]

D. P. Bertsekas, Convex Optimization Algorithms, Athena Scientific, 2015.

[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.

[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.

[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.

[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.

[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.

[13]

I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Society for Industrial and Applied Mathematics, 1999. doi: 10.1137/1.9781611971088.

[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.

[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.

[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.

[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.

[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.

[19]

B. M. Kampes, Radar Interferometry, Springer Netherlands, 2006.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[29]

B. Merriman, J. Bence and S. Osher, Diffusion generated motion by mean curvature, AMS Selected Letters, Crystal Grower's Workshop, 73-83.

[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.

[31]

B. Osting and T. H. Reeb, Consistency of Dirichlet partitions, SIAM Journal on Mathematical Analysis, 49 (2017), 4251-4274.  doi: 10.1137/16M1098309.

[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.

[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.

[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/.

[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.

[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.

[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.

[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.

[39]

L. Sciavicco and B. Siciliano, Modelling and Control of Robot Manipulators, Springer London, 2000. doi: 10.1007/978-1-4471-0449-0.

[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.

[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.

[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.

[43]

B. A. Wandell, Clarifying human white matter, Annual Review of Neuroscience, 39 (2016), 103-128. 

[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.

[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.

[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.

[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.

[48]

D. Wang and X.-P. Wang, The iterative convolution-thresholding method (ictm) for image segmentation, arXiv preprint, arXiv 1904.10917.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[4]

J. M. Ball, Mathematics and liquid crystals, Molecular Crystals and Liquid Crystals, 647 (2017), 1-27.  doi: 10.1080/15421406.2017.1289425.

[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.

[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.

[7]

D. P. Bertsekas, Convex Optimization Algorithms, Athena Scientific, 2015.

[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.

[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.

[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.

[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.

[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.

[13]

I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Society for Industrial and Applied Mathematics, 1999. doi: 10.1137/1.9781611971088.

[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.

[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.

[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.

[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.

[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.

[19]

B. M. Kampes, Radar Interferometry, Springer Netherlands, 2006.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[29]

B. Merriman, J. Bence and S. Osher, Diffusion generated motion by mean curvature, AMS Selected Letters, Crystal Grower's Workshop, 73-83.

[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.

[31]

B. Osting and T. H. Reeb, Consistency of Dirichlet partitions, SIAM Journal on Mathematical Analysis, 49 (2017), 4251-4274.  doi: 10.1137/16M1098309.

[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.

[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.

[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/.

[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.

[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.

[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.

[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.

[39]

L. Sciavicco and B. Siciliano, Modelling and Control of Robot Manipulators, Springer London, 2000. doi: 10.1007/978-1-4471-0449-0.

[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.

[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.

[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.

[43]

B. A. Wandell, Clarifying human white matter, Annual Review of Neuroscience, 39 (2016), 103-128. 

[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.

[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.

[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.

[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.

[48]

D. Wang and X.-P. Wang, The iterative convolution-thresholding method (ictm) for image segmentation, arXiv preprint, arXiv 1904.10917.

[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.

[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.

[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.

[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.

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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