# TGVDenoiser
### *class* deepinv.models.TGVDenoiser(verbose=False, n_it_max=1000, crit=1e-5, x2=None, u2=None, r2=None, ths=None)
Bases: [`Denoiser`](https://deepinv.org/api/stubs/deepinv.models.Denoiser.html.md#deepinv.models.Denoiser)
Proximal operator of (2nd order) Total Generalized Variation operator.
Adapted from Bredies *et al.*[1](#footcite-bredies2010total).
This algorithm converges to the unique image $x$ (and the auxiliary vector field $r$) minimizing
$$
\underset{x, r}{\arg\min} \; \frac{1}{2}\|x-y\|_2^2 + \lambda_1 \|r\|_{1,2} + \lambda_2 \|J(Dx-r)\|_{1,F}
$$
where $D$ maps an image to its gradient field and $J$ maps a vector field to its Jacobian.
For a large value of $\lambda_2$, the TGV behaves like the TV.
For a small value, it behaves like the $\ell_1$-Frobenius norm of the Hessian.
The problem is solved with an over-relaxed Chambolle-Pock algorithm, see Condat[2](#footcite-condat2013primal).
Code (and description) adapted from Laurent Condat’s matlab version ([https://lcondat.github.io/software.html](https://lcondat.github.io/software.html)) and
Daniil Smolyakov’s [code](https://github.com/RoundedGlint585/TGVDenoising/blob/master/TGV%20WithoutHist.ipynb).
#### NOTE
The regularization term $\|r\|_{1,2} + \|J(Dx-r)\|_{1,F}$ is implicitly normalized by its Lipschitz
constant, i.e. $\sqrt{72}$, see e.g. K. Bredies et al., “Total generalized variation,” SIAM J. Imaging
Sci., 3(3), 492-526, 2010.
* **Parameters:**
* **verbose** ([*bool*](https://docs.python.org/3.12/library/functions.html#bool)) – Whether to print computation details or not. Default: False.
* **n_it_max** ([*int*](https://docs.python.org/3.12/library/functions.html#int)) – Maximum number of iterations. Default: 1000.
* **crit** ([*float*](https://docs.python.org/3.12/library/functions.html#float)) – Convergence criterion. Default: 1e-5.
* **x2** ([*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor) *,* *None*) – Primary variable. Default: None.
* **u2** ([*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor) *,* *None*) – Dual variable. Default: None.
* **r2** ([*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor) *,* *None*) – Auxiliary variable. Default: None.
* **ths** ([*float*](https://docs.python.org/3.12/library/functions.html#float) *,* [*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)) – Regularization parameter. Can also be passed to `forward`. Default: None
* **References:**
* **[1]** Kristian Bredies, Karl Kunisch, and Thomas Pock. Total generalized variation. *SIAM Journal on Imaging Sciences*, 3(3):492–526, 2010.
* **[2]** Laurent Condat. A primal–dual splitting method for convex optimization involving lipschitzian, proximable and linear composite terms. *Journal of optimization theory and applications*, 158(2):460–479, 2013.
#### *static* epsilon(I)
Applies the jacobian of a vector field.
#### *static* epsilon_adjoint(G)
Applies the adjoint of the jacobian of a vector field.
#### forward(y, ths=None, \*\*kwargs)
Computes the proximity operator of the TGV norm.
* **Parameters:**
* **y** ([*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)) – Noisy image. Assumes a tensor of shape (B, C, H, W) (2D data) or (B, C, D, H, W) (3D data).
* **ths** ([*float*](https://docs.python.org/3.12/library/functions.html#float) *,* [*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)) – Regularization parameter. Takes priority over `ths` passed at initialization.
* **Returns:**
Denoised image.
#### *static* nabla(x)
Applies the finite differences operator associated with tensors of the same shape as x.
#### *static* nabla_adjoint(x)
Applies the adjoint of the finite difference operator.
## Examples using `TGVDenoiser`: