# pylops.optimization.cls_sparsity.SplitBregman#

class pylops.optimization.cls_sparsity.SplitBregman(Op, callbacks=None)[source]#

Split Bregman for mixed L2-L1 norms.

Solve an unconstrained system of equations with mixed $$L_2$$ and $$L_1$$ regularization terms given the operator Op, a list of $$L_1$$ regularization terms RegsL1, and an optional list of $$L_2$$ regularization terms RegsL2.

Parameters
Oppylops.LinearOperator

Operator to invert

Notes

Solve the following system of unconstrained, regularized equations given the operator $$\mathbf{Op}$$ and a set of mixed norm ($$L^2$$ and $$L_1$$) regularization terms $$\mathbf{R}_{2,i}$$ and $$\mathbf{R}_{1,i}$$, respectively:

$J = \frac{\mu}{2} \|\textbf{y} - \textbf{Op}\,\textbf{x} \|_2^2 + \frac{1}{2}\sum_i \epsilon_{\mathbf{R}_{2,i}} \|\mathbf{y}_{\mathbf{R}_{2,i}} - \mathbf{R}_{2,i} \textbf{x} \|_2^2 + \sum_i \epsilon_{\mathbf{R}_{1,i}} \| \mathbf{R}_{1,i} \textbf{x} \|_1$

where $$\mu$$ is the reconstruction damping, $$\epsilon_{\mathbf{R}_{2,i}}$$ are the damping factors used to weight the different $$L^2$$ regularization terms of the cost function and $$\epsilon_{\mathbf{R}_{1,i}}$$ are the damping factors used to weight the different $$L_1$$ regularization terms of the cost function.

The generalized Split-Bergman algorithm [1] is used to solve such cost function: the algorithm is composed of a sequence of unconstrained inverse problems and Bregman updates.

The original system of equations is initially converted into a constrained problem:

$J = \frac{\mu}{2} \|\textbf{y} - \textbf{Op}\,\textbf{x}\|_2^2 + \frac{1}{2}\sum_i \epsilon_{\mathbf{R}_{2,i}} \|\mathbf{y}_{\mathbf{R}_{2,i}} - \mathbf{R}_{2,i} \textbf{x}\|_2^2 + \sum_i \| \textbf{y}_i \|_1 \quad \text{subject to} \quad \textbf{y}_i = \mathbf{R}_{1,i} \textbf{x} \quad \forall i$

and solved as follows:

\begin{split}\DeclareMathOperator*{\argmin}{arg\,min} \begin{align} (\textbf{x}^{k+1}, \textbf{y}_i^{k+1}) = \argmin_{\mathbf{x}, \mathbf{y}_i} \|\textbf{y} - \textbf{Op}\,\textbf{x}\|_2^2 &+ \frac{1}{2}\sum_i \epsilon_{\mathbf{R}_{2,i}} \|\mathbf{y}_{\mathbf{R}_{2,i}} - \mathbf{R}_{2,i} \textbf{x}\|_2^2 \\ &+ \frac{1}{2}\sum_i \epsilon_{\mathbf{R}_{1,i}} \|\textbf{y}_i - \mathbf{R}_{1,i} \textbf{x} - \textbf{b}_i^k\|_2^2 \\ &+ \sum_i \| \textbf{y}_i \|_1 \end{align}\end{split}
$\textbf{b}_i^{k+1}=\textbf{b}_i^k + (\mathbf{R}_{1,i} \textbf{x}^{k+1} - \textbf{y}^{k+1})$

The scipy.sparse.linalg.lsqr solver and a fast shrinkage algorithm are used within a inner loop to solve the first step. The entire procedure is repeated niter_outer times until convergence.

1

Goldstein T. and Osher S., “The Split Bregman Method for L1-Regularized Problems”, SIAM J. on Scientific Computing, vol. 2(2), pp. 323-343. 2008.

Methods

 __init__(Op[, callbacks]) callback(x, *args, **kwargs) Callback routine finalize([show]) Finalize solver run(x[, show, itershow, show_inner]) Run solver setup(y, RegsL1[, x0, niter_outer, ...]) Setup solver solve(y, RegsL1[, x0, niter_outer, ...]) Run entire solver step(x[, show, show_inner]) Run one step of solver