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 termsRegsL1
, and an optional list of \(L^2\) regularization termsRegsL2
.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 toNone
to use a zero data for every regularization operator inRegsL2
)- 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 repeatedniter_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. - Op :