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, rtol=0.0, rtol1=0.0, tau=1.0, restart=False, engine='scipy', show=False, itershow=(10, 10, 10), show_inner=False, callback=None, preallocate=False, **kwargs_lsqr)

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

ynumpy.ndarray

Data

RegsL1list

\(L_1\) regularization operators

x0numpy.ndarray, optional

Initial guess

niter_outerint

Number of iterations of outer loop

niter_innerint

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.

RegsL2list

Additional \(L_2\) regularization operators (if None, \(L_2\) regularization is not added to the problem)

dataregsL2list, 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)

mufloat, optional

Data term damping

epsRL1slist

\(L_1\) Regularization dampings (must have the same number of elements as RegsL1)

epsRL2slist

\(L_2\) Regularization dampings (must have the same number of elements as RegsL2)

tolfloat, optional

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

rtolfloat, optional

Relative tolerance on total cost function wrt initial total cost function. Stops the solver when the ratio of the current total cost function to the initial total cost function is below this value.

rtol1float, optional

Relative tolerance on total cost function wrt to data. Stops the solver when the ratio of the current total cost function to the data norm is below this value.

taufloat, optional

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

restartbool, optional

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

enginestr, optional

Solver to use (scipy or pylops)

showbool, optional

Display iterations log

itershowtuple, 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_innerbool, optional

Display inner iteration logs of lsqr

callbackcallable, optional

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

preallocatebool, optional

Added in version 2.6.0.

Pre-allocate all variables used by the solver. Note that if y is a JAX array, this option is ignored and variables are not pre-allocated since JAX does not support in-place operations.

**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:

xinvnumpy.ndarray

Inverted model

itn_outint

Iteration number of outer loop upon termination

costnumpy.ndarray, optional

History of the total cost function through iterations

Notes

See pylops.optimization.cls_sparsity.SplitBregman

Examples using pylops.optimization.sparsity.splitbregman

Total Variation (TV) Regularization

Total Variation (TV) Regularization

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05. Image deblurring

16. CT Scan Imaging

16. CT Scan Imaging