pylops.optimization.sparsity.SplitBregman

pylops.optimization.sparsity.SplitBregman(Op, RegsL1, data, niter_outer=3, niter_inner=5, RegsL2=None, dataregsL2=None, mu=1.0, epsRL1s=None, epsRL2s=None, tol=1e-10, tau=1.0, x0=None, restart=False, show=False, **kwargs_lsqr)[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:
Op : pylops.LinearOperator

Operator to invert

RegsL1 : list

\(L^1\) regularization operators

data : numpy.ndarray

Data

niter_outer : int

Number of iterations of outer loop

niter_inner : int

Number of iterations of inner loop of first step of the Split Bregman algorithm. A small number of iterations is generally sufficient and for many applications optimal efficiency is obtained when only one iteration is performed.

RegsL2 : list

Additional \(L^2\) regularization operators (if None, \(L^2\) regularization is not added to the problem)

dataregsL2 : list, optional

\(L^2\) Regularization data (must have the same number of elements of RegsL2 or equal to None to use a zero data for every regularization operator in RegsL2)

mu : float, optional

Data term damping

epsRL1s : list

\(L^1\) Regularization dampings (must have the same number of elements as RegsL1)

epsRL2s : list

\(L^2\) Regularization dampings (must have the same number of elements as RegsL2)

tol : float, optional

Tolerance. Stop outer iterations if difference between inverted model at subsequent iterations is smaller than tol

tau : float, optional

Scaling factor in the Bregman update (must be close to 1)

x0 : numpy.ndarray, optional

Initial guess

restart : bool, optional

The unconstrained inverse problem in inner loop is initialized with the initial guess (True) or with the last estimate (False)

show : bool, optional

Display iterations log

**kwargs_lsqr

Arbitrary keyword arguments for scipy.sparse.linalg.lsqr solver used to solve the first subproblem in the first step of the Split Bregman algorithm.
Returns:
xinv : numpy.ndarray

Inverted model

itn_out : int

Iteration number of outer loop upon termination

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:

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

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:

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

and solved as follows:

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

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.

Examples using pylops.optimization.sparsity.SplitBregman