The Meaning of the Terms Isotropic and Anisotropic in the Total Variation Framework

Question

The isotropic TV is defined as the estimation of 2-norm of gradients $\sqrt{(y_{i+1,j}-y_{i,j})^2+(y_{i,j+1}-y_{i,j})^2}$, while the anisotropic TV is defined as the estimation of 1-norm of gradients $|y_{i+1,j}-y_{i,j}|+|y_{i,j+1}-y_{i,j}|$.

Now I am wondering why the second one will be called anisotropic. Since from my perspective, the isotropic TV bounded the 2-norm of the gradient, and thus it can bound the norm of gradients for every directions. But for the second one, since the absolute values are bounded, it means that the 2-norm of every directions can also be bounded. Isn't it also an isotropic way to bound the gradient?

Answer 1

In the Total Variation framework we define 2 flavors:

$$ \text{Isotropic TV} \; {TV}_{ {L}_{2} } \left( X \right) = \sum_{ij} \sqrt{ { \left( {D}_{h} X \right) }_{ij}^{2} + { \left( {D}_{v} X \right) }_{ij}^{2} } $$

$$ \text{Anisotropic TV} \; {TV}_{ {L}_{1} } \left( X \right) = \sum_{ij} \sqrt{ { \left( {D}_{h} X \right) }_{ij}^{2} } + \sqrt{{ \left( {D}_{v} X \right) }_{ij}^{2} } $$

Where $ {D}_{h} $ and $ {D}_{v} $ are the horizontal / vertical finite differences operators.

Quoting Isotropic and Anisotropic Total Variation Regularization in Electrical Impedance Tomography:

It is worth noticing that referring to functional as ”anisotropic” is slightly misleading, because the weights corresponding to all coordinate directions are equal.

The term Anisotropic is used because this flavor separate Vertical and Horizontal components as done in Anisotropic Diffusion. Hence it doesn't have rotational invariant term as the Isotropic Flavor. As you know, sum of quadratic terms is connected to Ellipse and with equal weights it is a Sphere / Circle which is Isotropic.

Great reading about the subject is Laurent Condat - Discrete Total Variation - New Definition and Minimization (DOI Address). It analyze the properties of different TV definitions regarding their spatial smoothing.