The Jump Set under Geometric Regularization. Part 1: Basic Technique and First-Order Denoising


Let $u \in \textup{BV}(\Omega)$ solve the total variation (TV) denoising problem with $L^2$-squared fidelity and data $f$. Caselles, Chambolle, and Novaga [Multiscale Model. Simul., 6 (2008), pp. 879--894] have shown the containment $\mathcal{H}^{m-1}(J_u \setminus J_f)=0$ of the jump set $J_u$ of $u$ in that of $f$. Their proof unfortunately depends heavily on the co-area formula, as do many results in this area, and as such is not directly extensible to higher-order, curvature-based, and other advanced geometric regularizers, such as total generalized variation and Euler's elastica. These have received increased attention in recent times due to their better practical regularization properties compared to conventional TV or wavelets. We prove analogous jump set containment properties for a general class of regularizers. We do this with novel Lipschitz transformation techniques and do not require the co-area formula. In the present Part 1 we demonstrate the general technique on first-order regularizers, while in Part 2 we will extend it to higher-order regularizers. In particular, we concentrate in this part on $\textup{TV}$ and, as a novelty, Huber-regularized $\textup{TV}$. We also demonstrate that the technique would apply to nonconvex $\textup{TV}$ models as well as the Perona--Malik anisotropic diffusion, if these approaches were well-posed to begin with.


  1. total variation
  2. jump set
  3. regularization
  4. Lipschitz
  5. Huber
  6. Perona--Malik

MSC codes

  1. 26B30
  2. 49Q20
  3. 65J20

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W. K. Allard, Total variation regularization for image denoising, I. Geometric theory, SIAM J. Math. Anal., 39 (2008), pp. 1150--1190.
F. Alter, V. Caselles, and A. Chambolle, A characterization of convex calibrable sets in $\R^n$, Math. Ann., 332 (2005), pp. 329--366.
L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York, 2000.
M. Benning and M. Burger, Ground states and singular vectors of convex variational regularization methods, Methods Appl. Anal., 20 (2013), pp. 295--334.
A. L. Bertozzi and J. B. Greer, Low-curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes, Commun. Pure Appl. Math., 57 (2004), pp. 764--790.
K. Bredies, K. Kunisch, and T. Pock, Total generalized variation, SIAM J. Imaging Sci., 3 (2011), pp. 492--526.
K. Bredies, K. Kunisch, and T. Valkonen, Properties of $L^1$-${TGV}^2$: The one-dimensional case, J. Math. Anal. Appl., 398 (2013), pp. 438--454.
M. Burger, M. Franek, and C.-B. Schönlieb, Regularized regression and density estimation based on optimal transport, Appl. Math. Res. eXpress, 2012 (2012), pp. 209--253.
V. Caselles, A. Chambolle, and M. Novaga, The discontinuity set of solutions of the TV denoising problem and some extensions, Multiscale Model. Simul., 6 (2008), pp. 879--894.
A. Chambolle, An algorithm for mean curvature motion, Interfaces Free Bound., 6 (2004), pp. 195--218.
A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), pp. 167--188.
T. Chan, A. Marquina, and P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), pp. 503--516.
T. F. Chan and S. Esedoglu, Aspects of total variation regularized $L^1$ function approximation, SIAM J. Appl. Math., 65 (2005), pp. 1817--1837.
T. F. Chan, S. H. Kang, and J. Shen, Euler's elastica and curvature-based inpainting, SIAM J. Appl. Math., (2002), pp. 564--592.
B. Dacorogna and J. Moser, On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), pp. 1--26.
G. Dal Maso, I. Fonseca, G. Leoni, and M. Morini, A higher order model for image restoration: The one-dimensional case, SIAM J. Math. Anal., 40 (2009), pp. 2351--2391.
S. Didas, J. Weickert, and B. Burgeth, Properties of higher order nonlinear diffusion filtering, J. Math. Imaging Vis., 35 (2009), pp. 208--226.
V. Duval, J. F. Aujol, and Y. Gousseau, The TV$L_1$ model: A geometric point of view, Multiscale Model. Simul., 8 (2009), pp. 154--189.
I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$ Spaces, Springer-Verlag, New York, 2007.
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Math., Springer, New York, 2001.
P. Guidotti, Anisotropic diffusions of image processing from Perona-Malik on, Adv. Stud. Pure Math., to appear.
M. Hintermüller and G. Stadler, An infeasible primal-dual algorithm for total bounded variation--based inf-convolution-type image restoration, SIAM J. Sci. Comput., 28 (2006), pp. 1--23.
M. Hintermüller and T. Wu, Nonconvex TV$^q$-models in image restoration: Analysis and a trust-region regularization--based superlinearly convergent solver, SIAM J. Imaging Sci., 6 (2013), pp. 1385--1415.
M. Hintermüller and T. Wu, A superlinearly convergent R-regularized Newton scheme for variational models with concave sparsity-promoting priors, Comput. Optim. Appl., 57 (2014), pp. 1--25.
M. Hintermüller, T. Valkonen, and T. Wu, Limiting aspects of non-convex $\mbox{TV}^\varphi$ models, submitted.
J. Huang and D. Mumford, Statistics of natural images and models, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Vol. 1, 1999.
J. Lellmann, D. Lorenz, C.-B. Schönlieb, and T. Valkonen, Imaging with Kantorovich-Rubinstein discrepancy, SIAM J. Imaging Sci., 7 (2014), pp. 2833--2859.
M. Lysaker, A. Lundervold, and X.-C. Tai, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12 (2003), pp. 1579--1590.
Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, AMS, Providence, RI, 2001.
M. Nikolova, Minimizers of cost functions involving nonsmooth data-fideltiy terms. Application of processing of outliers, SIAM J. Numer. Anal., 40 (2002), pp. 965--994.
P. Ochs, A. Dosovitskiy, T. Brox, and T. Pock, An iterated $l_1$ algorithm for non-smooth non-convex optimization in computer vision, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2013.
K. Papafitsoros and K. Bredies, A Study of the One Dimensional Total Generalised Variation Regularisation Problem, eprint, 2013.
K. Papafitsoros and C.-B. Schönlieb, A combined first and second order variational approach for image reconstruction, J. Math. Imaging Vis., 48 (2014), pp. 308--338.
K. Papafitsoros and T. Valkonen, Asymptotic behaviour of total generalised variation, in Proceedings of the Fifth International Conference on Scale Space and Variational Methods in Computer Vision (SSVM), 2015.
P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Machine Intelligence, 12 (1990), pp. 629--639.
F. Rindler and G. Shaw, Strictly Continuous Extensions of Functionals with Linear Growth to the Space BV, preprint, 2013.
W. Ring, Structural properties of solutions to total variation regularization problems, ESAIM Math. Model. Numer. Anal., 34 (2000), pp. 799--810.
L. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259--268.
J. Shen, S. Kang, and T. Chan, Euler's elastica and curvature-based inpainting, SIAM J. Appl. Math., 63 (2003), pp. 564--592.
T. Valkonen, Transport equation and image interpolation with SBD velocity fields, J. Math. Pures Appl., 95 (2011), pp. 459--494.
T. Valkonen, The jump set under geometric regularisation. Part 2: Higher-order approaches, submitted.
T. Valkonen, A primal-dual hybrid gradient method for non-linear operators with applications to MRI, Inverse Problems, 30 (2014), 055012.
L. A. Vese and S. J. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing, J. Sci. Comput., 19 (2003), pp. 553--572.
J. Weickert, Anisotropic Diffusion in Image Processing, ECMI series, B.G. Teubner, Stuttgart, 1998.
W. Yin, D. Goldfarb, and S. Osher, The total variation regularized $L^1$ model for multiscale decomposition, Multiscale Model. Simul., 6 (2007), pp. 190--211.

Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 2587 - 2629
ISSN (online): 1095-7154


Submitted: 7 July 2014
Accepted: 13 April 2015
Published online: 2 July 2015


  1. total variation
  2. jump set
  3. regularization
  4. Lipschitz
  5. Huber
  6. Perona--Malik

MSC codes

  1. 26B30
  2. 49Q20
  3. 65J20



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