TV-regularised reconstruction and TV-based denoising #596
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📚 Documentation |
| class TotalVariationRegularizedReconstruction(DirectReconstruction): | ||
| r"""TV-regularized reconstruction. | ||
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| This algorithm solves the problem :math:`min_x \frac{1}{2}||(Ax - y)||_2^2 + ||L\nabla x||_1` |
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maybe it would be good to distinguish between \lambda * | \nabla x |_1 (for scalar reg parameter) and | \Lambda \nabla x |_1 for an entire lambda-map. Also, strictly speaking, we would need to introduce the notation to have different regularization parameters for different directions, but this could get too complicated and cumbersome for this here?
| :math:`\nabla` is the finite difference operator applied to :math:`x`. | ||
| """ | ||
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| n_iterations: int |
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do we want to have n_iterations or max_iterations with a tolerance? Our PDHG allows for the second as well.
| noise | ||
| KNoise used for prewhitening. If None, no prewhitening is performed | ||
| dcf | ||
| K-space sampling density compensation. If None, set up based on kdata. The dcf is only used to calculate a |
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Do we want to restrict ourselves to this or also allow the problem min_x 1/2*||W^(1/2)(Ax-y)||_2^2?
| kdata = prewhiten_kspace(kdata, self.noise) | ||
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| # Create the acquisition model A = F S if the CSM S is defined otherwise A = F with the Fourier operator F | ||
| acquisition_operator = self.fourier_op @ self.csm.as_operator() if self.csm is not None else self.fourier_op |
| r"""TV denoising. | ||
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| This algorithm solves the problem :math:`min_x \frac{1}{2}||(x - y)||_2^2 + ||L\nabla x||_1` | ||
| by using the PDHG-algorithm. :math:`y` is the target image, :math:`L` is the strength of the regularization and |
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maybe better:
:math:y is the given noisy image
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| # %% | ||
| data_weight = 0.5 | ||
| n_adam_iterations = 4 |
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n_admm_iterations instead of n_adam_iterations
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| # by doing | ||
| # | ||
| # $x_{k+1} = argmin_x \lambda \| \nabla x \|_1 + \frac{\rho}{2}||x - z_k + u_k||_2^2$ |
| # | ||
| # using PDHG. | ||
| # | ||
| # Because we have 2D dynamic images we can apply the TV-regularization along x,y and time. |
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suggestion:
Because we have 2D dynamic images, we can apply the TV-regularization along x,y, and time.
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| # %% [markdown] | ||
| # #### TV-Regularized Reconstruction using ADMM | ||
| # In the above example we need to apply the acquisition operator during the PDHG iterations which is computationally |
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suggestion:
In the above example, PDHG repeatedly applies the acquisition operator and its adjoint during the iterations, which is computationally demanding and hence takes a long time. Another option is to use the Alternating Direction Method of Multipliers (ADMM) [S. Boyd et al, 2011], which solves the general problem
| # $u_{k+1} = u_k + x_{k+1} - z_{k+1}$ | ||
| # | ||
| # The first step is TV-based denoising of $x$, the second step is a regularized iterative SENSE update of $z$ and the | ||
| # final step updates the helper variable $u$. |
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updates the dual variable
Requires #426