pylops.optimization.sparsity.splitbregman

pylops.optimization.sparsity.splitbregman(Op, y, RegsL1, x0=None, niter_outer=3, niter_inner=5, RegsL2=None, dataregsL2=None, mu=1.0, epsRL1s=None, epsRL2s=None, tol=1e-10, tau=1.0, restart=False, show=False, itershow=[10, 10, 10], show_inner=False, callback=None, **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

y : numpy.ndarray

Data

RegsL1 : list

\(L_1\) regularization operators

x0 : numpy.ndarray, optional

Initial guess

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)

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

itershow : list, optional

Display set log for the first N1 steps, last N2 steps, and every N3 steps in between where N1, N2, N3 are the three element of the list.

show_inner : bool, optional

Display inner iteration logs of lsqr

callback : callable, optional

Function with signature (callback(x)) to call after each iteration where x is the current model vector

**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

cost : numpy.ndarray, optional

History of cost function through iterations

Notes

See pylops.optimization.cls_sparsity.SplitBregman

Examples using pylops.optimization.sparsity.splitbregman