Example #1
0
def OMP(Op,
        data,
        niter_outer=10,
        niter_inner=40,
        sigma=1e-4,
        normalizecols=False,
        show=False):
    r"""Orthogonal Matching Pursuit (OMP).

    Solve an optimization problem with :math:`L0` regularization function given
    the operator ``Op`` and data ``y``. The operator can be real or complex,
    and should ideally be either square :math:`N=M` or underdetermined
    :math:`N<M`.

    Parameters
    ----------
    Op : :obj:`pylops.LinearOperator`
        Operator to invert
    data : :obj:`numpy.ndarray`
        Data
    niter_outer : :obj:`int`
        Number of iterations of outer loop
    niter_inner : :obj:`int`
        Number of iterations of inner loop
    sigma : :obj:`list`
        Maximum L2 norm of residual. When smaller stop iterations.
    normalizecols : :obj:`list`
        Normalize columns (``True``) or not (``False``). Note that this can be
        expensive as it requires applying the forward operator
        :math:`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.
    show : :obj:`bool`, optional
        Display iterations log

    Returns
    -------
    xinv : :obj:`numpy.ndarray`
        Inverted model
    iiter : :obj:`int`
        Number of effective outer iterations
    cost : :obj:`numpy.ndarray`, optional
        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
    -----
    Solves the following optimization problem for the operator
    :math:`\mathbf{Op}` and the data :math:`\mathbf{d}`:

    .. math::
            ||\mathbf{x}||_0 \quad  subj. to \quad
            ||\mathbf{Op}\mathbf{x}-\mathbf{b}||_2 <= \sigma,

    using Orthogonal Matching Pursuit (OMP). This is a very
    simple iterative algorithm which applies the following step:

    .. math::
        \Lambda_k = \Lambda_{k-1} \cup \{ arg max_j
        |\mathbf{Op}_j^H \mathbf{r}_k| \} \\
        \mathbf{x}_k =  \{ arg min_{\mathbf{x}}
        ||\mathbf{Op}_{\Lambda_k} \mathbf{x} - \mathbf{b}||_2

    """
    Op = LinearOperator(Op)
    if show:
        tstart = time.time()
        print(
            'OMP optimization\n'
            '-----------------------------------------------------------------\n'
            'The Operator Op has %d rows and %d cols\n'
            'sigma = %.2e\tniter_outer = %d\tniter_inner = %d\n'
            'normalization=%s' % (Op.shape[0], Op.shape[1], sigma, niter_outer,
                                  niter_inner, normalizecols))
    # find normalization factor for each column
    if normalizecols:
        ncols = Op.shape[1]
        norms = np.zeros(ncols)
        for icol in range(ncols):
            unit = np.zeros(ncols)
            unit[icol] = 1
            norms[icol] = np.linalg.norm(Op.matvec(unit))
    if show:
        print(
            '-----------------------------------------------------------------'
        )
        head1 = '    Itn           r2norm'
        print(head1)

    cols = []
    res = data.copy()
    cost = np.zeros(niter_outer + 1)
    cost[0] = np.linalg.norm(data)
    iiter = 0
    while iiter < niter_outer and cost[iiter] > sigma:
        cres = np.abs(Op.rmatvec(res))
        if normalizecols:
            cres = cres / norms
        # exclude columns already chosen by putting them negative
        if iiter > 0:
            cres[cols] = -1
        # choose column with max cres
        imax = np.argwhere(cres == np.max(cres)).ravel()
        nimax = len(imax)
        if nimax > 0:
            imax = imax[np.random.permutation(nimax)[0]]
        else:
            imax = imax[0]
        cols.append(imax)

        # estimate model for current set of columns
        Opcol = Op.apply_columns(cols)
        x = lsqr(Opcol, data, iter_lim=niter_inner)[0]
        res = data - Opcol.matvec(x)
        iiter += 1
        cost[iiter] = np.linalg.norm(res)
        if show:
            if iiter < 10 or niter_outer - iiter < 10 or iiter % 10 == 0:
                msg = '%6g        %12.5e' % (iiter + 1, cost[iiter])
                print(msg)
    xinv = np.zeros(Op.shape[1], dtype=Op.dtype)
    xinv[cols] = x
    if show:
        print('\nIterations = %d        Total time (s) = %.2f' %
              (iiter, time.time() - tstart))
        print(
            '-----------------------------------------------------------------\n'
        )
    return xinv, iiter, cost
Example #2
0
def OMP(Op, data, niter_outer=10, niter_inner=40, sigma=1e-4,
        normalizecols=False, show=False):
    r"""Orthogonal Matching Pursuit (OMP).

    Solve an optimization problem with :math:`L0` regularization function given
    the operator ``Op`` and data ``y``. The operator can be real or complex,
    and should ideally be either square :math:`N=M` or underdetermined
    :math:`N<M`.

    Parameters
    ----------
    Op : :obj:`pylops.LinearOperator`
        Operator to invert
    data : :obj:`numpy.ndarray`
        Data
    niter_outer : :obj:`int`, optional
        Number of iterations of outer loop
    niter_inner : :obj:`int`, optional
        Number of iterations of inner loop. By choosing ``niter_inner=0``, the
        Matching Pursuit (MP) algorithm is implemented.
    sigma : :obj:`list`
        Maximum L2 norm of residual. When smaller stop iterations.
    normalizecols : :obj:`list`, optional
        Normalize columns (``True``) or not (``False``). Note that this can be
        expensive as it requires applying the forward operator
        :math:`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.
    show : :obj:`bool`, optional
        Display iterations log

    Returns
    -------
    xinv : :obj:`numpy.ndarray`
        Inverted model
    iiter : :obj:`int`
        Number of effective outer iterations
    cost : :obj:`numpy.ndarray`, optional
        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
    -----
    Solves the following optimization problem for the operator
    :math:`\mathbf{Op}` and the data :math:`\mathbf{d}`:

    .. math::
            ||\mathbf{x}||_0 \quad  subj. to \quad
            ||\mathbf{Op}\mathbf{x}-\mathbf{b}||_2^2 <= \sigma,

    using Orthogonal Matching Pursuit (OMP). This is a very
    simple iterative algorithm which applies the following step:

    .. math::
        \Lambda_k = \Lambda_{k-1} \cup \{ arg max_j
        |\mathbf{Op}_j^H \mathbf{r}_k| \} \\
        \mathbf{x}_k =  \{ arg min_{\mathbf{x}}
        ||\mathbf{Op}_{\Lambda_k} \mathbf{x} - \mathbf{b}||_2^2

    Note that by choosing ``niter_inner=0`` the basic Matching Pursuit (MP)
    algorithm is implemented instead. In other words, instead of solving an
    optimization at each iteration to find the best :math:`\mathbf{x}` for the
    currently selected basis functions, the vector :math:`\mathbf{x}` is just
    updated at the new basis function by taking directly the value from
    the inner product :math:`\mathbf{Op}_j^H \mathbf{r}_k`.

    In this case it is highly reccomended to provide a normalized basis
    function. If different basis have different norms, the solver is likely
    to diverge. Similar observations apply to OMP, even though mild unbalancing
    between the basis is generally properly handled.

    """
    ncp = get_array_module(data)

    Op = LinearOperator(Op)
    if show:
        tstart = time.time()
        algname = 'OMP optimization\n' if niter_inner > 0 else 'MP optimization\n'
        print(algname +
              '-----------------------------------------------------------------\n'
              'The Operator Op has %d rows and %d cols\n'
              'sigma = %.2e\tniter_outer = %d\tniter_inner = %d\n'
              'normalization=%s' %
              (Op.shape[0], Op.shape[1], sigma, niter_outer,
               niter_inner, normalizecols))
    # find normalization factor for each column
    if normalizecols:
        ncols = Op.shape[1]
        norms = ncp.zeros(ncols)
        for icol in range(ncols):
            unit = ncp.zeros(ncols, dtype=Op.dtype)
            unit[icol] = 1
            norms[icol] = np.linalg.norm(Op.matvec(unit))
    if show:
        print('-----------------------------------------------------------------')
        head1 = '    Itn           r2norm'
        print(head1)

    if niter_inner == 0:
        x = []
    cols = []
    res = data.copy()
    cost = ncp.zeros(niter_outer + 1)
    cost[0] = np.linalg.norm(data)
    iiter = 0
    while iiter < niter_outer and cost[iiter] > sigma:
        # compute inner products
        cres = Op.rmatvec(res)
        cres_abs = np.abs(cres)
        if normalizecols:
            cres_abs = cres_abs / norms
        # choose column with max cres
        cres_max = np.max(cres_abs)
        imax = np.argwhere(cres_abs == cres_max).ravel()
        nimax = len(imax)
        if nimax > 0:
            imax = imax[np.random.permutation(nimax)[0]]
        else:
            imax = imax[0]
        # update active set
        if imax not in cols:
            addnew = True
            cols.append(int(imax))
        else:
            addnew = False
            imax_in_cols = cols.index(imax)

        # estimate model for current set of columns
        if niter_inner == 0:
            # MP update
            Opcol = Op.apply_columns([int(imax), ])
            res -= Opcol.matvec(cres[imax] * ncp.ones(1))
            if addnew:
                x.append(cres[imax])
            else:
                x[imax_in_cols] += cres[imax]
        else:
            # OMP update
            Opcol = Op.apply_columns(cols)
            if ncp == np:
                x = lsqr(Opcol, data, iter_lim=niter_inner)[0]
            else:
                x = cgls(Opcol, data, ncp.zeros(int(Opcol.shape[1]),
                                                dtype=Opcol.dtype),
                         niter=niter_inner)[0]
            res = data - Opcol.matvec(x)
        iiter += 1
        cost[iiter] = np.linalg.norm(res)
        if show:
            if iiter < 10 or niter_outer - iiter < 10 or iiter % 10 == 0:
                msg = '%6g        %12.5e' % (iiter + 1, cost[iiter])
                print(msg)
    xinv = ncp.zeros(int(Op.shape[1]), dtype=Op.dtype)
    xinv[cols] = ncp.array(x)
    if show:
        print('\nIterations = %d        Total time (s) = %.2f'
              % (iiter, time.time() - tstart))
        print(
            '-----------------------------------------------------------------\n')
    return xinv, iiter, cost