Bases: `AbstractRegularization`

An adaptive regularization scheme (regularization is described in the Regularization class above).

For the weighted regularization scheme, each pixel is given an ‘effective regularization weight’, which is applied when each set of pixel neighbors are regularized with one another. The motivation of this is that different regions of a pixelization’s mesh require different levels of regularization (e.g., high smoothing where the no signal is present and less smoothing where it is, see (Nightingale, Dye and Massey 2018)).

Unlike `Constant` regularization, neighboring pixels must now be regularized with one another in both directions (e.g. if pixel 0 regularizes pixel 1, pixel 1 must also regularize pixel 0). For example:

B = [-1, 1] [0->1]

[-1, -1] 1 now also regularizes 0

For `Constant` regularization this would NOT produce a positive-definite matrix. However, for the weighted scheme, it does!

The regularize weight_list change the B matrix as shown below - we simply multiply each pixel’s effective regularization weight by each row of B it has a -1 in, so:

regularization_weights = [1, 2, 3, 4]

B = [-1, 1, 0 ,0] # [0->1]

[0, -2, 2 ,0] # [1->2] [0, 0, -3 ,3] # [2->3] [4, 0, 0 ,-4] # [3->0]

If our -1’s werent down the diagonal this would look like:

B = [4, 0, 0 ,-4] # [3->0]

[0, -2, 2 ,0] # [1->2] [-1, 1, 0 ,0] # [0->1] [0, 0, -3 ,3] # [2->3] This is valid!

Parameters
• coefficients – The regularization coefficients which controls the degree of smoothing of the inversion reconstruction in high and low signal regions of the reconstruction.

• signal_scale (`float`) – A factor which controls how rapidly the smoothness of regularization varies from high signal regions to low signal regions.

Methods

 `regularization_matrix_from` Returns the regularization matrix of this regularization scheme. `regularization_weights_from` Returns the regularization weights of this regularization scheme.
regularization_weights_from(linear_obj)[source]#

Returns the regularization weights of this regularization scheme.

The regularization weights define the level of regularization applied to each parameter in the linear object (e.g. the `pixels` in a `Mapper`).

For standard regularization (e.g. `Constant`) are weights are equal, however for adaptive schemes (e.g. `AdaptiveBrightness`) they vary to adapt to the data being reconstructed.

Parameters

linear_obj (`LinearObj`) – The linear object (e.g. a `Mapper`) which uses these weights when performing regularization.

Return type

The regularization weights.

regularization_matrix_from(linear_obj)[source]#

Returns the regularization matrix of this regularization scheme.

Parameters

linear_obj (`LinearObj`) – The linear object (e.g. a `Mapper`) which uses this matrix to perform regularization.

Return type

The regularization matrix.