Abstract
We propose an adaptive implementation of a Crouzeix–Raviart-based discretization of the total variation, which has the property of approximating from below the total variation, with metrication errors only depending on the local curvature, rather than on the orientation as is usual for other approaches.
Similar content being viewed by others
Notes
That is, for any \(x\in \partial E\), there are balls \(B(y,R)\subset E\), \(B(z,R)\subset E^\complement \), with \(\{x\}=\partial B(y,R)\cap \partial B(z,R)\); in particular \(|\kappa _{\partial E}|\le 1/R\).
References
Acosta, G., Apel, T., Durán, R.G., Lombardi, A.L.: Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra. Math. Comp. 80(273), 141–163 (2011)
Alter, F., Caselles, V., Chambolle, A.: Evolution of characteristic functions of convex sets in the plane by the minimizing total variation flow. Interfaces Free Bound. 7(1), 29–53 (2005)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, New York (2000)
Anzellotti, G.: Pairings between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl. (4) 135, 293–318 (1983)
Bartels, S.: Total variation minimization with finite elements: convergence and iterative solution. SIAM J. Numer. Anal. 50(3), 1162–1180 (2012)
Bartels, S.: Error control and adaptivity for a variational model problem defined on functions of bounded variation. Math. Comput. 84(293), 1217–1240 (2015)
Bartels, S., Milicevic, M.: Stability and experimental comparison of prototypical iterative schemes for total variation regularized problems. Comput. Methods Appl. Math. 16(3), 361–388 (2016)
Bartels, S., Nochetto, R.H., Salgado, A.J.: A total variation diminishing interpolation operator and applications. Math. Comput. 84(296), 2569–2587 (2015)
Berkels, B., Effland, A., Rumpf, M.: A posteriori error control for the binary Mumford–Shah model. Math. Comput. 86(306), 1769–1791 (2017)
Boykov, Y., Kolmogorov, V.: Computing geodesics and minimal surfaces via graph cuts. In: Proceedings Ninth IEEE International Conference on Computer Vision, vol. 1, pp. 26–33 (2003)
Boykov, Y., Kolmogorov, V., Cremers, D., Delong, A.: An integral solution to surface evolution PDEs via Geo-Cuts. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) European Conference on Computer Vision (ECCV), Volume 3953 of LNCS, pp. 409–422. Springer, Graz (2006)
Braides, A.: \(\Gamma \)-Convergence for Beginners. Oxford Lecture Series in Mathematics and Its Applications, vol. 22. Oxford University Press, Oxford (2002)
Brenner, S.C.: Forty years of the Crouzeix–Raviart element. Numer. Methods Partial Differ. Equ. 31(2), 367–396 (2015)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, Volume 15 of Texts in Applied Mathematics, 3rd edn. Springer, New York (2008)
Cai, J.-F., Dong, B., Osher, S., Shen, Z.: Image restoration: total variation, wavelet frames, and beyond. J. Am. Math. Soc. 25(4), 1033–1089 (2012)
Caillaud, C., Chambolle, A.: Error estimates for finite differences approximations of the total variation (2020). (in preparation)
Carstensen, C., Liu, D.J.: Nonconforming FEMs for an optimal design problem. SIAM J. Numer. Anal. 53(2), 874–894 (2015)
Caselles, V., Chambolle, A., Novaga, M.: The discontinuity set of solutions of the TV denoising problem and some extensions. Multiscale Model. Simul. 6(3), 879–894 (2007)
Caselles, V., Chambolle, A.: Anisotropic curvature-driven flow of convex sets. Nonlinear Anal. 65(8), 1547–1577 (2006)
Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20(1–2), 89–97 (2004). Special issue on mathematics and image analysis
Chambolle, A., Levine, S.E., Lucier, B.J.: An upwind finite-difference method for total variation-based image smoothing. SIAM J. Imaging Sci. 4(1), 277–299 (2011)
Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vision 40(1), 120–145 (2011)
Chambolle, A., Pock, T.: On the ergodic convergence rates of a first-order primal-dual algorithm. Mathematical Programming, pp. 1–35 (2015). (Online first)
Chambolle, A., Pock, T.: A remark on accelerated block coordinate descent for computing the proximity operators of a sum of convex functions. SMAI J. Comput. Math. 1, 29–54 (2015)
Chambolle, A., Pock, T.: An introduction to continuous optimization for imaging. Acta Numer. 25, 161–319 (2016)
Chambolle, A., Tan, P., Vaiter, S.: Accelerated alternating descent methods for Dykstra-like problems. J. Math. Imaging Vis. 59(3), 481–497 (2017)
Condat, L.: Discrete total variation: new definition and minimization. SIAM J. Imaging Sci. 10(3), 1258–1290 (2017)
Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7(R–3), 33–75 (1973)
Dal Maso, G.: An Introduction to \(\Gamma \)-Convergence, Volume 8 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston Inc., Boston (1993)
Darbon, J., Sigelle, M.: Exact optimization of discrete constrained total variation minimization problems. In: Combinatorial Image Analysis, Volume 3322 of Lecture Notes in Computer Science, pp. 548–557. Springer, Berlin (2004)
Di Pietro, D.A., Lemaire, S.: An extension of the Crouzeix–Raviart space to general meshes with application to quasi-incompressible linear elasticity and Stokes flow. Math. Comput. 84(291), 1–31 (2015)
Elliott, C.M., Smitheman, S.A.: Numerical analysis of the TV regularization and \(H^{-1}\) fidelity model for decomposing an image into cartoon plus texture. IMA J. Numer. Anal. 29(3), 651–689 (2009)
Feng, X., Prohl, A.: Analysis of total variation flow and its finite element approximations. M2AN Math. Model. Numer. Anal. 37(3), 533–556 (2003)
Feng, X., von Oehsen, M., Prohl, A.: Rate of convergence of regularization procedures and finite element approximations for the total variation flow. Numer. Math. 100(3), 441–456 (2005)
Henao, D., Mora-Corral, C., Xianmin, X.: A numerical study of void coalescence and fracture in nonlinear elasticity. Comput. Methods Appl. Mech. Eng. 303, 163–184 (2016)
Hintermüller, M., Rautenberg, C.N., Hahn, J.: Function-analytic and numerical issues in splitting methods for total variation-based image reconstruction. Inverse Probl. 30(5), 055014 (2014)
Hong, Q., Lai, M.-J., Messi, L.M., Wang, J.: Galerkin method with splines for total variation minimization. J. Algorithms Comput. Technol. 13, 16 (2019)
Kirisits, C., Pöschl, C., Resmerita, E., Scherzer, O.: Finite-dimensional approximation of convex regularization via hexagonal pixel grids. Appl. Anal. 94(3), 612–636 (2015)
Lai, M.-J., Lucier, B., Wang, J.: The Convergence of a Central-Difference Discretization of Rudin–Osher–Fatemi Model for Image Denoising, pp. 514–526. Springer, Berlin (2009)
Lai, M.-J., Messi, L.M.: Piecewise linear approximation of the continuous Rudin–Osher–Fatemi model for image denoising. SIAM J. Numer. Anal. 50(5), 2446–2466 (2012)
Ortner, C.: Nonconforming finite-element discretization of convex variational problems. IMA J. Numer. Anal. 31(3), 847–864 (2011)
Ortner, C., Praetorius, D.: On the convergence of adaptive nonconforming finite element methods for a class of convex variational problems. SIAM J. Numer. Anal. 49(1), 346–367 (2011)
Raviart, P.-A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical aspects of finite element methods. Lecture Notes in Mathematics, vol. 606, pp. 292–315. Springer, Berlin Heidelberg, Berlin, Heidelberg (1977)
Repin, S.I.: A variation-difference method for solving problems with functionals of linear growth. Zh. Vychisl. Mat. i Mat. Fiz. 29(5), 693–708 (1989). 798
Rother, C., Kolmogorov, V., Blake, A.: ‘GrabCut’: interactive foreground extraction using iterated graph cuts. ACM Trans. Graph. 23(3), 309–314 (2004)
Rudin, L., Osher, S.J., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60, 259–268 (1992). [Also in Experimental Mathematics: Computational Issues in Nonlinear Science (Proc. Los Alamos Conf. 1991)]
Wang, J., Lucier, B.J.: Error bounds for finite-difference methods for Rudin–Osher–Fatemi image smoothing. SIAM J. Numer. Anal. 49(2), 845–868 (2011)
Xianmin, X., Henao, D.: An efficient numerical method for cavitation in nonlinear elasticity. Math. Models Methods Appl. Sci. 21(8), 1733–1760 (2011)
Acknowledgements
The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the program “Variational methods, new optimisation techniques and new fast numerical algorithms” (Sept.-Oct., 2017), when this work was undertaken. It has benefitted the support of the EPSRC Grant N. EP/K032208/1. The work of A.C. was partially supported by a grant of the Simons Foundation. T.P. acknowledges support by the Austrian science fund (FWF) under the project EANOI, No. I1148, and the ERC starting Grant HOMOVIS, No. 640156. The authors were sharing Mila Nikolova’s office at the Isaac Newton Institute and were very happy to enjoy many interesting discussions with her on this project. We dedicate this paper to her memory.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was funded by EPSRC Grant N. EP/K032208/1 (Isaac Newton Institute, Cambridge), the Simons Foundation, FWF project EANOI, No. I1148, ERC Grant HOMOVIS, No. 640156.
Appendices
A Proof of the Duality Theorem 1
The goal of this section is to provide a proof of the duality theorem, Theorem 1. We prove a more general result for the CR approximation of Sobolev semi-norms.
1.1 A.1 Almost Constant Crouzeix–Raviart Functions
To simplify, we drop the scale parameter h which is useless in this section. We consider a mesh \({\mathcal {T}}\) of d-dimensional simplices in a polygonal domain \(\Omega \subset {\mathbb {R}}^d\). We denote \(V({\mathcal {T}})\) the non-conforming P1 functions (Crouzeix–Raviart), defined in (10). We then let
It is the space of the functions which are 0 on average in each simplex (in other words, the average of the middle values of the facets vanishes, equivalently the value at the center of each simplex is zero).
We define \(P0({\mathcal {T}})^d\approx ({\mathbb {R}}^d)^{\mathcal {T}}\) as usual as the space of “P0” functions which are constant on each \(T\in {\mathcal {T}}\). Endowed with the topology of \(L^2(\Omega ;{\mathbb {R}}^d)\), it is a Euclidean space with the weighted scalar product: for \(p,q\in P0({\mathcal {T}})\),
Then, we consider the gradients
We want to characterize this space and its orthogonal. In order to do this, we consider the space \(RT0({\mathcal {T}})\) of the first-order Raviart–Thomas vector fields subject to the mesh \({\mathcal {T}}\) (cf Sect. 2.3, these are defined by their fluxes through the edges of the elements \(T\in {\mathcal {T}}\)). As before we also let \(RT0_0({\mathcal {T}})\subset RT0({\mathcal {T}})\) the RT0 fields with zero flux through \(\partial \Omega \).
We know that cf Lemma 2.4:
More generally, if \(u\in V({\mathcal {T}})\) and \(\sigma \in RT0_0({\mathcal {T}})\), one has thanks to (11):
As Du and \(\hbox {div}\,\sigma \) are constant on each triangle, this can also be written:
where \(c_T\) is the center of mass of the simplex T (so that given any affine function a(x), \(\int _T a(x)\mathrm{d}x = |T|a(c_T)\)).
Hence in particular, for any \({\mathbf {p}}\in GV^0({\mathcal {T}})\) and \(\sigma \in RT0_0({\mathcal {T}})\), \(\int _\Omega \sigma \cdot {\mathbf {p}}\mathrm{d}x = 0\). A natural question is whether this is an if and only if; that is, if the orthogonal of \(\{ (\sigma (c_T))_T\in P0({\mathcal {T}}): \sigma \in RT0_0({\mathcal {T}})\}\) is \(GV^0({\mathcal {T}})\).
Assume \({\mathbf {p}}\in P0({\mathcal {T}})^d\) is such that \(\int _\Omega \sigma \cdot {\mathbf {p}}\mathrm{d}x = 0\) for all \(\sigma \in RT0_0({\mathcal {T}})\). Then in particular it is orthogonal to fields with zero divergence and there exists \(u\in V({\mathcal {T}})\) with \(Du={\mathbf {p}}\), thanks to Lemma 2.4. In particular, because of (41) we have
for all Raviart–Thomas field \(\sigma \) (vanishing on \(\partial \Omega \)). Now, given any inner facet \(S\subset \partial T\cap \partial T'\), \(T,T'\in {\mathcal {T}}\), \(T\ne T'\), we can introduce the field \(\sigma \) with flux 1 through S from T to \(T'\) and zero through all other facets. Then the formula becomes
This shows that u must take the same value in all the centers of the vertices. As u is defined up to a constant (in each connected component of \(\Omega \)), we can assume this value is zero, and we have shown that in \(P0({\mathcal {T}})^d\),
In particular, as a consequence, also
1.2 A.2 Duality for Discrete Sobolev Semi-norms
Let us introduce, for \(u\in V({\mathcal {T}})\), \(p\in [1,\infty )\),
and, for \({\bar{u}}\in P0({\mathcal {T}})\)
By a slight abuse of notation, we also let \(J_p^0(u) =J_p^0((u(c_T)_T)\) for \(u\in V({\mathcal {T}})\). Then:
Theorem 2
For all \(p\in ( 1,\infty )\) and any \(u\in V({\mathcal {T}})\),
where \(p'=p/(p-1)\), and for \(p=1\),
In particular, Theorem 1 corresponds to the particular case \(p=1\).
Proof
We consider the case \(p>1\). The case \(p=1\) is then recovered as a limit problem. The “\(\ge \)” inequality is obvious. To show the reverse, we assume that u is a minimizer in (45). Then, for any \(v\in V^0({\mathcal {T}})\) (cf Appendix A.1), \(u+v\) is admissible in the problem and one has \(J_p(u+v)\ge J_p(u)\). Hence, taking the derivative \(\lim _{t\downarrow 0} (J_p(u+tv)-J_p(v))/t\), it follows that
for all \(v\in V^0({\mathcal {T}})\). That is, the field \((|Du(T)|^{p-2}Du(T))_{T\in {\mathcal {T}}}\in P0({\mathcal {T}})\) is orthogonal to \(GV^0({\mathcal {T}})\), and hence thanks to (43), there exists \(\sigma \in RT0_0({\mathcal {T}})\) such that \(\sigma (c_T) = |Du(T)|^{p-2}Du(T)\) for all \(T\in {\mathcal {T}}\). Clearly, \(|\sigma (c_T)|^{p'} = |Du(T)|^p\) and the conclusion easily follows. The case \(p=1\) can be recovered as follows: one builds, for \(p>1\), a field \(\sigma _p\in RT0_0({\mathcal {T}})\) optimal in (46), and letting then \(p\rightarrow 1\), one checks that it converges to a field which is optimal in (47). \(\square \)
Remark A.1
It is quite easy to derive, as a more general result, that given \(f:{\mathbb {R}}^d\rightarrow {\mathbb {R}}\) a convex, lower semi-continuous function and any \({\bar{u}}\in P0({\mathcal {T}})\) one has
B A Variant with One Node Per Pixel
For imaging application, one drawback of our approach could be the need to introduce more nodes in the representation than the number of pixels.
Given a (gray-level) \(n\times m\) image \((u_{i,j})_{i=1,\ldots ,n}^{j=1,\ldots ,m}\) (to simplify, we assume that the scale \(h=1\) in this section), even if one rotates the grid by \(45^\circ \) and considers the pixels (i, j) as centers of edges of larger squares (for instance, (1, 1), (1, 2), (2, 1), (2, 2) would be the centers of the edges of the square \([(3/2,1/2),(5/2,3/2),(3/2,5/2),(1/2,3/2)]\)), one still needs to introduce an additional node in the center of each square (in the above example, at (3/2, 3/2)) and introduce fictitious values \(u_{i+1/2,j+1/2}\) (i, j both even or both odd) at these nodes. On average, this increases the dimension of the problems by roughly 50%.
Unfortunately, it seems there is no simple strategy to eliminate this additional node. To illustrate this issue, let us first concentrate on one square. We consider the four vertices \(\{0,1\}^2\) as the middle points of the edges of the square (of area 2)
and a fifth vertex in (1/2, 1/2) in the middle, which is the middle of both the vertical and horizontal edges cutting the square into two halves. Then, given the values \(u_{\alpha ,\beta }:=u(\alpha ,\beta )\), \((\alpha ,\beta )\in \{0,1\}^2\), and c the value at the center, the Crouzeix–Raviart total variation inside the square is
(Each gradient norm is multiplied by the area 1 of the corresponding triangle, and we have used that the distance between a vertex of the cube and the center is \(1/\sqrt{2}\).) A possibility to eliminate c is to minimize this quantity with respect to c. In the “inpainting” problems of Fig. 8, this would give the same results (since this is precisely what is done automatically by the minimization in this case). Unfortunately, we have, at this point, no idea of how to solve this problem explicitly. It means that to compute the “proximity” operator of the corresponding energy on a whole image, we need to solve subproblems which keep the additional central variable.
A simpler possibility is to first optimize with respect to the value c and then, afterward, pick the best orientation. In that case, one needs to solve
A careful analysis shows that this value is given by the function
One can use (48) to define, given u defined by its pixel values \((u_{i,j})_{i=1,\ldots ,n}^{j=1,\ldots ,m}\), a discrete total variation as
We remark this is a variant of the energy defined in [24] (see also [26] for a theoretical study), which can be optimized by an efficient alternating descent method as soon as one knows how to solve explicitly the subproblems given by the proximity operator of \(J^4\), on each square.
Unfortunately, our implementation shows that it does not perform as well as the ACR technique introduced in this paper. Figure 13 compares this to the ACR result in Fig. 8: we obtain a very diffusive solution, with practically no improvement over a non-optimized Crouzeix Raviart implementation.
On the other hand, as is expected, this approximation (which in any case is still based on a hidden, underlying Crouzeix–Raviart discretization), yields to a quite precise approximation of the energy and is a reasonable regularizer for standard inverse problems, cf. Fig. 14.
C The Proximity Operator of (22)
We describe in this section how to implement the proximity operator of the function f in (22). The problem we need to solve is as follows, given \({\bar{\xi }}=({\bar{\xi }}_{mn})_{m=1,\ldots ,4}^{n=1,2}\in {\mathbb {R}}^{4\times 2}\) and \(\tau >0\):
We call \(\hbox {prox}\,_{\tau f}({\bar{\xi }})\) the solution of (50). We recall that the prox of the convex conjugate \(f^*\) is also easily recovered, once (50) is solved, using Moreau’s identity:
To solve (50), we first make the following obvious observation: denoting
(and the same for \({\bar{\xi }}\)), it is equivalent to solve:
We obtain at the minimum that \(\eta _i={\bar{\eta }}_i\), \(i=1,\ldots ,4\) and the problem boils down to
where \(|x-{\bar{x}}|^2=\sum _{i=1}^4 |x_i-{\bar{x}}_i|^2\). Remark that here, \({\bar{x}}_i\ge 0\) and it is equivalent to look for \(x\in {\mathbb {R}}_+^4\) or in \({\mathbb {R}}^4\).
We now explain how to solve this 4-dimensional convex problem. We can rewrite it as
and then we exchange \(\min \) and \(\max \). We obtain 4 problems of the form
This is well known to be solved by \(x_1=({\bar{x}}_1-\tau \mu _{12})^+\) and with value
When \({\bar{x}}_1\le \tau \mu _{12}\), this is \(|{\bar{x}}_1|^2/(2\tau )\), otherwise
We end up with the dual problem
whose optimality reads, if \(0<\mu _{12}<1\),
with \(\mu _{34}=1-\mu _{12}\).
Without loss of generality, assume that \({\bar{x}}_2\ge {\bar{x}}_1\) and \({\bar{x}}_4\ge {\bar{x}}_3\). We recast the problem as
by letting \(\mu :=\mu _{12}\) and \(\mu _{34}=1-\mu \).
By convexity of the objective, \(\mu \in [0,1]\) is optimal if and only if:
Hence, one sees that if one knows which terms are positive in the above sums, \(\mu \) is found by solving the above equations with “\(=0\)” instead of “\(\ge /\le 0\)” and then projecting the value onto the interval [0, 1]. For instance, if all values are positive,
Whenever \(\mu \in (0,1)\), of course, (51) reads
Hence, the problem is solved by exhaustion of the following cases:
-
1.
if \({\bar{x}}_2+{\bar{x}}_4\le \tau \), then clearly one can find \(\mu \in [0,1]\) such that all terms of the sums in (51) are zero, hence the solution is \(x_1=x_2=x_3=x_4=0\).
-
2.
if \({\bar{x}}_2+{\bar{x}}_4>\tau \) then:
-
(a)
either both\({\bar{x}}_2-\tau \mu >0\) and \({\bar{x}}_4-\tau +\tau \mu >0\),
-
(b)
or one side of (53) is zero so that one must be in a case of strict inequality in (51), and \(\mu \in \{0,1\}\).
The second case 2b can be first easily eliminated by checking whether \(\mu =0\) or \(\mu =1\) is a solution of the optimality condition: one has
$$\begin{aligned} {\bar{x}}_1 + {\bar{x}}_2 \le ({\bar{x}}_3-\tau )^+ + ({\bar{x}}_4-\tau )^+&\ \Leftrightarrow \ \mu =0, \\ ({\bar{x}}_1-\tau )^+ + ({\bar{x}}_2-\tau )^+ \ge {\bar{x}}_3 + {\bar{x}}_4&\ \Leftrightarrow \ \mu =1. \end{aligned}$$ -
(a)
-
3.
Otherwise, we must be in the first case 2a, where \({\bar{x}}_2-\tau \mu >0\) and \({\bar{x}}_4-\tau +\tau \mu >0\), equality (53) holds, and which is then split into four possible cases:
-
(a)
\(\mu \) given by (52), and \({\bar{x}}_1\ge \tau \mu \), \({\bar{x}}_3 \ge \tau (1-\mu )\), then \(x_1={\bar{x}}_1-\tau \mu \), \(x_2={\bar{x}}_2-\tau \mu \), \(x_3={\bar{x}}_3-\tau (1-\mu )\), \(x_4={\bar{x}}_4-\tau (1-\mu )\);
-
(b)
\(\mu =\frac{{\bar{x}}_1+{\bar{x}}_2-{\bar{x}}_4+\tau }{3\tau }\) and \({\bar{x}}_1\ge \tau \mu \), \({\bar{x}}_3\le \tau (1-\mu )\), \({\bar{x}}_3 \ge \tau (1-\mu )\), then \(x_1={\bar{x}}_1-\tau \mu \), \(x_2={\bar{x}}_2-\tau \mu \), \(x_3=0\), \(x_4={\bar{x}}_4-\tau (1-\mu )\);
-
(c)
\(\mu =\frac{{\bar{x}}_2-{\bar{x}}_3-{\bar{x}}_4+2\tau }{3\tau }\) and \({\bar{x}}_1\le \tau \mu \), \({\bar{x}}_2\ge \tau \mu \), \({\bar{x}}_3 \ge \tau (1-\mu )\), then \(x_1=0\), \(x_2={\bar{x}}_2-\tau \mu \), \(x_3={\bar{x}}_3-\tau (1-\mu )\), \(x_4={\bar{x}}_4-\tau (1-\mu )\);
-
(d)
\(\mu =\frac{{\bar{x}}_2-{\bar{x}}_4+\tau }{2\tau }\) if all the previous cases fail to hold, and then \(x_1=x_3=0\), \(x_2={\bar{x}}_2-\tau \mu \), \(x_4={\bar{x}}_4-\tau (1-\mu )\).
-
(a)
Rights and permissions
About this article
Cite this article
Chambolle, A., Pock, T. Crouzeix–Raviart Approximation of the Total Variation on Simplicial Meshes. J Math Imaging Vis 62, 872–899 (2020). https://doi.org/10.1007/s10851-019-00939-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10851-019-00939-3