Total Variation

The total variation of a function $z$ on $\Omega$ is

$$\mathrm{TV}(z)={\int }_{\Omega }|\nabla z|\mathrm{d}x(\text{generally }\sup \{{\int }_{\Omega }z\nabla \cdot \mathbf{p}:\mathbf{p}\in C_c^1(\Omega {)}^n,|\mathbf{p}|\le 1\left\}\right).$$

The supremum form also makes sense for discontinuous $z$. For a piecewise constant function, TV equals the jump height times the length of the jump set. TV therefore favours piecewise constant images with sharp edges, which fits organs or inclusions in a background. It was introduced for image denoising by Rudin, Osher and Fatemi (ROF, 1992).

Isotropic vs. anisotropic. $|\nabla z|=\sqrt{z_x^2+z_y^2}$ is isotropic. $|z_x|+|z_y|$ is the anisotropic variant: cheaper and separable, but it prefers axis-aligned edges.

Non-differentiability. $|\nabla z|$ is not differentiable where $\nabla z=0$. There are two standard remedies:

1. Smoothing: replace $|\cdot |$ by a differentiable approximation and use gradient-based methods (see Smoothed Total Variation).
2. Primal–dual / splitting: use the dual form above and solve with the Chambolle–Pock algorithm or ADMM, which handle the non-smooth term exactly.

Proximal operator (ROF problem).

$${\mathrm{prox}}_{\beta \mathrm{TV}}(y)=\arg \underset{z}{\min }\beta \mathrm{TV}(z)+\frac{\rho }{2}\Vert z-y{\Vert }_2^2.$$

This is TV denoising itself. It is used as the regulariser step in ADMM.

In EIT, TV regularisation gives sharper inclusions than Tikhonov Regularization and is robust in clinical data (Borsic et al. 2010).

In ModularEIT.jl: TotalVariationRegularizer, total_variation.

References

1. L. I. Rudin, S. Osher, E. Fatemi (1992). Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268. doi:10.1016/0167-2789(92)90242-F
2. A. Chambolle, V. Caselles, D. Cremers, M. Novaga, T. Pock (2010). An Introduction to Total Variation for Image Analysis. In: Theoretical Foundations and Numerical Methods for Sparse Recovery, de Gruyter, 263–340. doi:10.1515/9783110226157.263
3. A. Chambolle, T. Pock (2011). A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging. J. Math. Imaging Vis. 40, 120–145. doi:10.1007/s10851-010-0251-1
4. A. Borsic, B. M. Graham, A. Adler, W. R. B. Lionheart (2010). In Vivo Impedance Imaging With Total Variation Regularization. IEEE Trans. Med. Imaging 29(1), 44–54. doi:10.1109/TMI.2009.2022540
5. Y. Wang (2022). Anisotropic TV Regularization in Electrical Impedance Tomography: An Experimental Study. Engineering 14(3), 138–146. doi:10.4236/eng.2022.143013