pylops.optimization.sparsity.omp

pylops.optimization.sparsity.omp(Op, y, niter_outer=10, niter_inner=40, sigma=0.0001, rtol=0.0, rtol1=0.0, normalizecols=False, Opbasis=None, optimal_coeff=False, engine='scipy', show=False, itershow=(10, 10, 10), callback=None, preallocate=False)

Orthogonal Matching Pursuit (OMP).

Solve an optimization problem with \(L^0\) regularization function given the operator Op and data y. The operator can be real or complex, and should ideally be either square \(N=M\) or underdetermined \(N<M\).

Parameters:

Oppylops.LinearOperator

Operator to invert

ynumpy.ndarray

Data

niter_outerint, optional

Number of iterations of outer loop

niter_innerint, optional

Number of iterations of inner loop. By choosing niter_inner=0, the Matching Pursuit (MP) algorithm is implemented.

sigmafloat, optional

Maximum \(L_2\) norm of residual. When smaller stop iterations.

rtolfloat, optional

Relative tolerance on residual norm wrt initial residual norm. Stops the solver when the ratio of the current residual norm to the initial residual norm is below this value.

rtol1float, optional

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

normalizecolslist, optional

Normalize columns (True) or not (False). Note that this can be expensive as it requires applying the forward operator \(n_{cols}\) times to unit vectors (i.e., containing 1 at position j and zero otherwise); use only when the columns of the operator are expected to have highly varying norms.

Opbasispylops.LinearOperator

Operator representing the basis functions. If not provided, the entire operator used for inversion Op is used.

optimal_coeffbool, optional

Estimate optimal coefficient that minimizes the norm of the residual \(\mathbf{r} - c * \mathbf{Op}^j) norm (``True`\)) or use the directly the value from the inner product \(\mathbf{Op}_j^H\,\mathbf{r}_k\).

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.

callbackcallable, optional

Function with signature (callback(x, cols)) to call after each iteration where x contains the non-zero model coefficient and cols are the indices where the current model vector is non-zero

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.

Returns:

xinvnumpy.ndarray

Inverted model

niter_outerint

Number of effective outer iterations

costnumpy.ndarray

History of cost function

See also

ISTA

Iterative Shrinkage-Thresholding Algorithm (ISTA).

FISTA

Fast Iterative Shrinkage-Thresholding Algorithm (FISTA).

SPGL1

Spectral Projected-Gradient for L1 norm (SPGL1).

SplitBregman

Split Bregman for mixed L2-L1 norms.

Notes

See pylops.optimization.cls_sparsity.OMP

Examples using pylops.optimization.sparsity.omp

MP, OMP, ISTA and FISTA

MP, OMP, ISTA and FISTA