nifty_core.py

# NIFTY (Numerical Information Field Theory) has been developed at the
# Max-Planck-Institute for Astrophysics.
##
# Copyright (C) 2013 Max-Planck-Society
##
# Author: Marco Selig
# Project homepage: <http://www.mpa-garching.mpg.de/ift/nifty/>
##
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
##
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
# See the GNU General Public License for more details.
##
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.

"""
    ..                  __   ____   __
    ..                /__/ /   _/ /  /_
    ..      __ ___    __  /  /_  /   _/  __   __
    ..    /   _   | /  / /   _/ /  /   /  / /  /
    ..   /  / /  / /  / /  /   /  /_  /  /_/  /
    ..  /__/ /__/ /__/ /__/    \___/  \___   /  core
    ..                               /______/

    .. The NIFTY project homepage is http://www.mpa-garching.mpg.de/ift/nifty/

    NIFTY [#]_, "Numerical Information Field Theory", is a versatile
    library designed to enable the development of signal inference algorithms
    that operate regardless of the underlying spatial grid and its resolution.
    Its object-oriented framework is written in Python, although it accesses
    libraries written in Cython, C++, and C for efficiency.

    NIFTY offers a toolkit that abstracts discretized representations of
    continuous spaces, fields in these spaces, and operators acting on fields
    into classes. Thereby, the correct normalization of operations on fields is
    taken care of automatically without concerning the user. This allows for an
    abstract formulation and programming of inference algorithms, including
    those derived within information field theory. Thus, NIFTY permits its user
    to rapidly prototype algorithms in 1D and then apply the developed code in
    higher-dimensional settings of real world problems. The set of spaces on
    which NIFTY operates comprises point sets, n-dimensional regular grids,
    spherical spaces, their harmonic counterparts, and product spaces
    constructed as combinations of those.

    References
    ----------
    .. [#] Selig et al., "NIFTY -- Numerical Information Field Theory --
        a versatile Python library for signal inference",
        `A&A, vol. 554, id. A26 <http://dx.doi.org/10.1051/0004-6361/201321236>`_,
        2013; `arXiv:1301.4499 <http://www.arxiv.org/abs/1301.4499>`_

    Class & Feature Overview
    ------------------------
    The NIFTY library features three main classes: **spaces** that represent
    certain grids, **fields** that are defined on spaces, and **operators**
    that apply to fields.

    .. Overview of all (core) classes:
    ..
    .. - switch
    .. - notification
    .. - _about
    .. - random
    .. - space
    ..     - point_space
    ..     - rg_space
    ..     - lm_space
    ..     - gl_space
    ..     - hp_space
    ..     - nested_space
    .. - field
    .. - operator
    ..     - diagonal_operator
    ..         - power_operator
    ..     - projection_operator
    ..     - vecvec_operator
    ..     - response_operator
    .. - probing
    ..     - trace_probing
    ..     - diagonal_probing

    Overview of the main classes and functions:

    .. automodule:: nifty

    - :py:class:`space`
        - :py:class:`point_space`
        - :py:class:`rg_space`
        - :py:class:`lm_space`
        - :py:class:`gl_space`
        - :py:class:`hp_space`
        - :py:class:`nested_space`
    - :py:class:`field`
    - :py:class:`operator`
        - :py:class:`diagonal_operator`
            - :py:class:`power_operator`
        - :py:class:`projection_operator`
        - :py:class:`vecvec_operator`
        - :py:class:`response_operator`

        .. currentmodule:: nifty.nifty_tools

        - :py:class:`invertible_operator`
        - :py:class:`propagator_operator`

        .. currentmodule:: nifty.nifty_explicit

        - :py:class:`explicit_operator`

    .. automodule:: nifty

    - :py:class:`probing`
        - :py:class:`trace_probing`
        - :py:class:`diagonal_probing`

        .. currentmodule:: nifty.nifty_explicit

        - :py:class:`explicit_probing`

    .. currentmodule:: nifty.nifty_tools

    - :py:class:`conjugate_gradient`
    - :py:class:`steepest_descent`

    .. currentmodule:: nifty.nifty_explicit

    - :py:func:`explicify`

    .. currentmodule:: nifty.nifty_power

    - :py:func:`weight_power`,
      :py:func:`smooth_power`,
      :py:func:`infer_power`,
      :py:func:`interpolate_power`

"""
from __future__ import division
import numpy as np
import pylab as pl

from d2o import distributed_data_object,\
                STRATEGIES as DISTRIBUTION_STRATEGIES

from nifty_paradict import space_paradict,\
    point_space_paradict

from nifty.config import about,\
                         nifty_configuration as gc,\
                         dependency_injector as gdi

from nifty_random import random
import nifty.nifty_utilities as utilities

POINT_DISTRIBUTION_STRATEGIES = DISTRIBUTION_STRATEGIES['global']


class space(object):
    """
        ..     _______   ______    ____ __   _______   _______
        ..   /  _____/ /   _   | /   _   / /   ____/ /   __  /
        ..  /_____  / /  /_/  / /  /_/  / /  /____  /  /____/
        .. /_______/ /   ____/  \______|  \______/  \______/  class
        ..          /__/

        NIFTY base class for spaces and their discretizations.

        The base NIFTY space class is an abstract class from which other
        specific space subclasses, including those preimplemented in NIFTY
        (e.g. the regular grid class) must be derived.

        Parameters
        ----------
        dtype : numpy.dtype, *optional*
            Data type of the field values for a field defined on this space
            (default: numpy.float64).
        datamodel :

        See Also
        --------
        point_space :  A class for unstructured lists of numbers.
        rg_space : A class for regular cartesian grids in arbitrary dimensions.
        hp_space : A class for the HEALPix discretization of the sphere
            [#]_.
        gl_space : A class for the Gauss-Legendre discretization of the sphere
            [#]_.
        lm_space : A class for spherical harmonic components.
        nested_space : A class for product spaces.

        References
        ----------
        .. [#] K.M. Gorski et al., 2005, "HEALPix: A Framework for
               High-Resolution Discretization and Fast Analysis of Data
               Distributed on the Sphere", *ApJ* 622..759G.
        .. [#] M. Reinecke and D. Sverre Seljebotn, 2013, "Libsharp - spherical
               harmonic transforms revisited";
               `arXiv:1303.4945 <http://www.arxiv.org/abs/1303.4945>`_

        Attributes
        ----------
        para : {single object, list of objects}
            This is a freeform list of parameters that derivatives of the space
            class can use.
        dtype : numpy.dtype
            Data type of the field values for a field defined on this space.
        discrete : bool
            Whether the space is inherently discrete (true) or a discretization
            of a continuous space (false).
        vol : numpy.ndarray
            An array of pixel volumes, only one component if the pixels all
            have the same volume.
    """

    def __init__(self):
        """
            Sets the attributes for a space class instance.

            Parameters
            ----------
            dtype : numpy.dtype, *optional*
                Data type of the field values for a field defined on this space
                (default: numpy.float64).
            datamodel :

            Returns
            -------
            None
        """
        self.paradict = space_paradict()

    @property
    def para(self):
        return self.paradict['default']

    @para.setter
    def para(self, x):
        self.paradict['default'] = x

    def __hash__(self):
        return hash(())

    def _identifier(self):
        """
        _identiftier returns an object which contains all information needed
        to uniquely idetnify a space. It returns a (immutable) tuple which
        therefore can be compared.
        """
        return tuple(sorted(vars(self).items()))

    def __eq__(self, x):
        if isinstance(x, type(self)):
            return self._identifier() == x._identifier()
        else:
            return False

    def __ne__(self, x):
        return not self.__eq__(x)

    def __len__(self):
        return int(self.get_dim())

    def copy(self):
        return space(para=self.para,
                     dtype=self.dtype)

    def getitem(self, data, key):
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'getitem'."))

    def setitem(self, data, key):
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'getitem'."))

    def apply_scalar_function(self, x, function, inplace=False):
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'apply_scalar_function'."))

    def unary_operation(self, x, op=None):
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'unary_operation'."))

    def binary_operation(self, x, y, op=None):
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'binary_operation'."))

    def get_shape(self):
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'shape'."))

    def get_dim(self):
        """
            Computes the dimension of the space, i.e.\  the number of pixels.

            Parameters
            ----------
            split : bool, *optional*
                Whether to return the dimension split up, i.e. the numbers of
                pixels in each direction, or not (default: False).

            Returns
            -------
            dim : {int, numpy.ndarray}
                Dimension(s) of the space.
        """
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'dim'."))

    def get_dof(self):
        """
            Computes the number of degrees of freedom of the space.

            Returns
            -------
            dof : int
                Number of degrees of freedom of the space.
        """
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'dof'."))

    def cast(self, x, verbose=False):
        """
            Computes valid field values from a given object, trying
            to translate the given data into a valid form. Thereby it is as
            benevolent as possible.

            Parameters
            ----------
            x : {float, numpy.ndarray, nifty.field}
                Object to be transformed into an array of valid field values.

            Returns
            -------
            x : numpy.ndarray, distributed_data_object
                Array containing the field values, which are compatible to the
                space.

            Other parameters
            ----------------
            verbose : bool, *optional*
                Whether the method should raise a warning if information is
                lost during casting (default: False).
        """
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'cast'."))

    # TODO: Move enforce power into power_indices class
    def enforce_power(self, spec, **kwargs):
        """
            Provides a valid power spectrum array from a given object.

            Parameters
            ----------
            spec : {scalar, list, numpy.ndarray, nifty.field, function}
                Fiducial power spectrum from which a valid power spectrum is to
                be calculated. Scalars are interpreted as constant power
                spectra.

            Returns
            -------
            spec : numpy.ndarray
                Valid power spectrum.

            Other parameters
            ----------------
            size : int, *optional*
                Number of bands the power spectrum shall have (default: None).
            kindex : numpy.ndarray, *optional*
                Scale of each band.
            codomain : nifty.space, *optional*
                A compatible codomain for power indexing (default: None).
            log : bool, *optional*
                Flag specifying if the spectral binning is performed on
                logarithmic
                scale or not; if set, the number of used bins is set
                automatically (if not given otherwise); by default no binning
                is done (default: None).
            nbin : integer, *optional*
                Number of used spectral bins; if given `log` is set to
                ``False``;
                integers below the minimum of 3 induce an automatic setting;
                by default no binning is done (default: None).
            binbounds : {list, array}, *optional*
                User specific inner boundaries of the bins, which are preferred
                over the above parameters; by default no binning is done
                (default: None).
            vmin : {scalar, list, ndarray, field}, *optional*
                Lower limit of the uniform distribution if ``random == "uni"``
                (default: 0).

        """
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'enforce_power'."))

    def check_codomain(self, codomain):
        """
            Checks whether a given codomain is compatible to the space or not.

            Parameters
            ----------
            codomain : nifty.space
                Space to be checked for compatibility.

            Returns
            -------
            check : bool
                Whether or not the given codomain is compatible to the space.
        """
        if codomain is None:
            return False
        else:
            raise NotImplementedError(about._errors.cstring(
                "ERROR: no generic instance method 'check_codomain'."))

    def get_codomain(self, **kwargs):
        """
            Generates a compatible codomain to which transformations are
            reasonable, usually either the position basis or the basis of
            harmonic eigenmodes.

            Parameters
            ----------
            coname : string, *optional*
                String specifying a desired codomain (default: None).
            cozerocenter : {bool, numpy.ndarray}, *optional*
                Whether or not the grid is zerocentered for each axis or not
                (default: None).
            conest : list, *optional*
                List of nested spaces of the codomain (default: None).
            coorder : list, *optional*
                Permutation of the list of nested spaces (default: None).

            Returns
            -------
            codomain : nifty.space
                A compatible codomain.
        """
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'get_codomain'."))

    def get_random_values(self, **kwargs):
        """
            Generates random field values according to the specifications given
            by the parameters.

            Returns
            -------
            x : numpy.ndarray
                Valid field values.

            Other parameters
            ----------------
            random : string, *optional*
                Specifies the probability distribution from which the random
                numbers are to be drawn.
                Supported distributions are:

                - "pm1" (uniform distribution over {+1,-1} or {+1,+i,-1,-i}
                - "gau" (normal distribution with zero-mean and a given
                    standard deviation or variance)
                - "syn" (synthesizes from a given power spectrum)
                - "uni" (uniform distribution over [vmin,vmax[)

                (default: None).
            dev : float, *optional*
                Standard deviation (default: 1).
            var : float, *optional*
                Variance, overriding `dev` if both are specified
                (default: 1).
            spec : {scalar, list, numpy.ndarray, nifty.field, function},
                    *optional*
                Power spectrum (default: 1).
            pindex : numpy.ndarray, *optional*
                Indexing array giving the power spectrum index of each band
                (default: None).
            kindex : numpy.ndarray, *optional*
                Scale of each band (default: None).
            codomain : nifty.space, *optional*
                A compatible codomain with power indices (default: None).
            log : bool, *optional*
                Flag specifying if the spectral binning is performed on
                logarithmic
                scale or not; if set, the number of used bins is set
                automatically (if not given otherwise); by default no binning
                is done (default: None).
            nbin : integer, *optional*
                Number of used spectral bins; if given `log` is set to
                ``False``;
                integers below the minimum of 3 induce an automatic setting;
                by default no binning is done (default: None).
            binbounds : {list, array}, *optional*
                User specific inner boundaries of the bins, which are preferred
                over the above parameters; by default no binning is done
                (default: None).
            vmin : {scalar, list, ndarray, field}, *optional*
                Lower limit of the uniform distribution if ``random == "uni"``
                (default: 0).
            vmin : float, *optional*
                Lower limit for a uniform distribution (default: 0).
            vmax : float, *optional*
                Upper limit for a uniform distribution (default: 1).
        """
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'get_random_values'."))

    def calc_weight(self, x, power=1):
        """
            Weights a given array of field values with the pixel volumes (not
            the meta volumes) to a given power.

            Parameters
            ----------
            x : numpy.ndarray
                Array to be weighted.
            power : float, *optional*
                Power of the pixel volumes to be used (default: 1).

            Returns
            -------
            y : numpy.ndarray
                Weighted array.
        """
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'calc_weight'."))

    def get_weight(self, power=1):
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'get_weight'."))

    def calc_norm(self, x, q):
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'norm'."))

    def calc_dot(self, x, y):
        """
            Computes the discrete inner product of two given arrays of field
            values.

            Parameters
            ----------
            x : numpy.ndarray
                First array
            y : numpy.ndarray
                Second array

            Returns
            -------
            dot : scalar
                Inner product of the two arrays.
        """
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'dot'."))

    def calc_transform(self, x, codomain=None, **kwargs):
        """
            Computes the transform of a given array of field values.

            Parameters
            ----------
            x : numpy.ndarray
                Array to be transformed.
            codomain : nifty.space, *optional*
                codomain space to which the transformation shall map
                (default: self).

            Returns
            -------
            Tx : numpy.ndarray
                Transformed array

            Other parameters
            ----------------
            iter : int, *optional*
                Number of iterations performed in specific transformations.
        """
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'calc_transform'."))

    def calc_smooth(self, x, sigma=0, **kwargs):
        """
            Smoothes an array of field values by convolution with a Gaussian
            kernel.

            Parameters
            ----------
            x : numpy.ndarray
                Array of field values to be smoothed.
            sigma : float, *optional*
                Standard deviation of the Gaussian kernel, specified in units
                of length in position space (default: 0).

            Returns
            -------
            Gx : numpy.ndarray
                Smoothed array.

            Other parameters
            ----------------
            iter : int, *optional*
                Number of iterations (default: 0).
        """
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'calc_smooth'."))

    def calc_power(self, x, **kwargs):
        """
            Computes the power of an array of field values.

            Parameters
            ----------
            x : numpy.ndarray
                Array containing the field values of which the power is to be
                calculated.

            Returns
            -------
            spec : numpy.ndarray
                Power contained in the input array.

            Other parameters
            ----------------
            pindex : numpy.ndarray, *optional*
                Indexing array assigning the input array components to
                components of the power spectrum (default: None).
            kindex : numpy.ndarray, *optional*
                Scale corresponding to each band in the power spectrum
                (default: None).
            rho : numpy.ndarray, *optional*
                Number of degrees of freedom per band (default: None).
            codomain : nifty.space, *optional*
                A compatible codomain for power indexing (default: None).
            log : bool, *optional*
                Flag specifying if the spectral binning is performed on
                logarithmic
                scale or not; if set, the number of used bins is set
                automatically (if not given otherwise); by default no binning
                is done (default: None).
            nbin : integer, *optional*
                Number of used spectral bins; if given `log` is set to
                ``False``;
                integers below the minimum of 3 induce an automatic setting;
                by default no binning is done (default: None).
            binbounds : {list, array}, *optional*
                User specific inner boundaries of the bins, which are preferred
                over the above parameters; by default no binning is done
                (default: None).
            vmin : {scalar, list, ndarray, field}, *optional*
                Lower limit of the uniform distribution if ``random == "uni"``
                (default: 0).

        """
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'calc_power'."))

    def calc_real_Q(self, x):
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'calc_real_Q'."))

    def calc_bincount(self, x, weights=None, minlength=None):
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'calc_bincount'."))

    def get_plot(self, x, **kwargs):
        """
            Creates a plot of field values according to the specifications
            given by the parameters.

            Parameters
            ----------
            x : numpy.ndarray
                Array containing the field values.

            Returns
            -------
            None

            Other parameters
            ----------------
            title : string, *optional*
                Title of the plot (default: "").
            vmin : float, *optional*
                Minimum value to be displayed (default: ``min(x)``).
            vmax : float, *optional*
                Maximum value to be displayed (default: ``max(x)``).
            power : bool, *optional*
                Whether to plot the power contained in the field or the field
                values themselves (default: False).
            unit : string, *optional*
                Unit of the field values (default: "").
            norm : string, *optional*
                Scaling of the field values before plotting (default: None).
            cmap : matplotlib.colors.LinearSegmentedColormap, *optional*
                Color map to be used for two-dimensional plots (default: None).
            cbar : bool, *optional*
                Whether to show the color bar or not (default: True).
            other : {single object, tuple of objects}, *optional*
                Object or tuple of objects to be added, where objects can be
                scalars, arrays, or fields (default: None).
            legend : bool, *optional*
                Whether to show the legend or not (default: False).
            mono : bool, *optional*
                Whether to plot the monopole or not (default: True).
            save : string, *optional*
                Valid file name where the figure is to be stored, by default
                the figure is not saved (default: False).
            error : {float, numpy.ndarray, nifty.field}, *optional*
                Object indicating some confidence interval to be plotted
                (default: None).
            kindex : numpy.ndarray, *optional*
                Scale corresponding to each band in the power spectrum
                (default: None).
            codomain : nifty.space, *optional*
                A compatible codomain for power indexing (default: None).
            log : bool, *optional*
                Flag specifying if the spectral binning is performed on
                logarithmic
                scale or not; if set, the number of used bins is set
                automatically (if not given otherwise); by default no binning
                is done (default: None).
            nbin : integer, *optional*
                Number of used spectral bins; if given `log` is set to
                ``False``;
                integers below the minimum of 3 induce an automatic setting;
                by default no binning is done (default: None).
            binbounds : {list, array}, *optional*
                User specific inner boundaries of the bins, which are preferred
                over the above parameters; by default no binning is done
                (default: None).
            vmin : {scalar, list, ndarray, field}, *optional*
                Lower limit of the uniform distribution if ``random == "uni"``
                (default: 0).
            iter : int, *optional*
                Number of iterations (default: 0).

        """
        raise NotImplementedError(about._errors.cstring(
            "ERROR: no generic instance method 'get_plot'."))

    def __repr__(self):
        string = ""
        string += str(type(self)) + "\n"
        string += "paradict: " + str(self.paradict) + "\n"
        return string

    def __str__(self):
        return self.__repr__()


class point_space(space):
    """
        ..                            __             __
        ..                          /__/           /  /_
        ..      ______    ______    __   __ ___   /   _/
        ..    /   _   | /   _   | /  / /   _   | /  /
        ..   /  /_/  / /  /_/  / /  / /  / /  / /  /_
        ..  /   ____/  \______/ /__/ /__/ /__/  \___/  space class
        .. /__/

        NIFTY subclass for unstructured spaces.

        Unstructured spaces are lists of values without any geometrical
        information.

        Parameters
        ----------
        num : int
            Number of points.
        dtype : numpy.dtype, *optional*
            Data type of the field values (default: None).

        Attributes
        ----------
        para : numpy.ndarray
            Array containing the number of points.
        dtype : numpy.dtype
            Data type of the field values.
        discrete : bool
            Parameter captioning the fact that a :py:class:`point_space` is
            always discrete.
        vol : numpy.ndarray
            Pixel volume of the :py:class:`point_space`, which is always 1.
    """

    def __init__(self, num, dtype=np.dtype('float'), datamodel='fftw',
                 comm=gc['default_comm']):
        """
            Sets the attributes for a point_space class instance.

            Parameters
            ----------
            num : int
                Number of points.
            dtype : numpy.dtype, *optional*
                Data type of the field values (default: numpy.float64).

            Returns
            -------
            None.
        """
        self._cache_dict = {'check_codomain': {}}
        self.paradict = point_space_paradict(num=num)

        # parse dtype
        dtype = np.dtype(dtype)
        if dtype not in [np.dtype('bool'),
                         np.dtype('int16'),
                         np.dtype('int32'),
                         np.dtype('int64'),
                         np.dtype('float32'),
                         np.dtype('float64'),
                         np.dtype('complex64'),
                         np.dtype('complex128')]:
            raise ValueError(about._errors.cstring(
                             "WARNING: incompatible dtype: " + str(dtype)))
        self.dtype = dtype

        if datamodel not in POINT_DISTRIBUTION_STRATEGIES:
            about._errors.cstring("WARNING: datamodel set to default.")
            self.datamodel = \
                gc['default_distribution_strategy']
        else:
            self.datamodel = datamodel

        self.comm = self._parse_comm(comm)
        self.discrete = True
#        self.harmonic = False
        self.distances = (np.float(1),)

    @property
    def para(self):
        temp = np.array([self.paradict['num']], dtype=int)
        return temp

    @para.setter
    def para(self, x):
        self.paradict['num'] = x[0]

    def __hash__(self):
        # Extract the identifying parts from the vars(self) dict.
        result_hash = 0
        for (key, item) in vars(self).items():
            if key in ['_cache_dict']:
                continue
            result_hash ^= item.__hash__() * hash(key)
        return result_hash

    def _identifier(self):
        # Extract the identifying parts from the vars(self) dict.
        temp = [(ii[0],
                 ((lambda x: x[1].__hash__() if x[0] == 'comm' else x)(ii)))
                for ii in vars(self).iteritems()
                if ii[0] not in ['_cache_dict']
                ]
        # Return the sorted identifiers as a tuple.
        return tuple(sorted(temp))

    def _parse_comm(self, comm):
        # check if comm is a string -> the name of comm is given
        # -> Extract it from the mpi_module
        if isinstance(comm, str):
            if gc.validQ('default_comm', comm):
                result_comm = getattr(gdi[gc['mpi_module']], comm)
            else:
                raise ValueError(about._errors.cstring(
                    "ERROR: The given communicator-name is not supported."))
        # check if the given comm object is an instance of default Intracomm
        else:
            if isinstance(comm, gdi[gc['mpi_module']].Intracomm):
                result_comm = comm
            else:
                raise ValueError(about._errors.cstring(
                    "ERROR: The given comm object is not an instance of the " +
                    "default-MPI-module's Intracomm Class."))
        return result_comm

    def copy(self):
        return point_space(num=self.paradict['num'],
                           dtype=self.dtype,
                           datamodel=self.datamodel,
                           comm=self.comm)

    def getitem(self, data, key):
        return data[key]

    def setitem(self, data, update, key):
        data[key] = update

    def apply_scalar_function(self, x, function, inplace=False):
        return x.apply_scalar_function(function, inplace=inplace)

    def unary_operation(self, x, op='None', axis=None, **kwargs):
        """
        x must be a numpy array which is compatible with the space!
        Valid operations are

        """
        translation = {'pos': lambda y: getattr(y, '__pos__')(),
                       'neg': lambda y: getattr(y, '__neg__')(),
                       'abs': lambda y: getattr(y, '__abs__')(),
                       'real': lambda y: getattr(y, 'real'),
                       'imag': lambda y: getattr(y, 'imag'),
                       'nanmin': lambda y: getattr(y, 'nanmin')(axis=axis),
                       'amin': lambda y: getattr(y, 'amin')(axis=axis),
                       'nanmax': lambda y: getattr(y, 'nanmax')(axis=axis),
                       'amax': lambda y: getattr(y, 'amax')(axis=axis),
                       'median': lambda y: getattr(y, 'median')(axis=axis),
                       'mean': lambda y: getattr(y, 'mean')(axis=axis),
                       'std': lambda y: getattr(y, 'std')(axis=axis),
                       'var': lambda y: getattr(y, 'var')(axis=axis),
                       'argmin_nonflat': lambda y: getattr(y, 'argmin_nonflat')(axis=axis),
                       'argmin': lambda y: getattr(y, 'argmin')(axis=axis),
                       'argmax_nonflat': lambda y: getattr(y, 'argmax_nonflat')(axis=axis),
                       'argmax': lambda y: getattr(y, 'argmax')(axis=axis),
                       'conjugate': lambda y: getattr(y, 'conjugate')(),
                       'sum': lambda y: getattr(y, 'sum')(axis=axis),
                       'prod': lambda y: getattr(y, 'prod')(axis=axis),
                       'unique': lambda y: getattr(y, 'unique')(),
                       'copy': lambda y: getattr(y, 'copy')(),
                       'copy_empty': lambda y: getattr(y, 'copy_empty')(),
                       'isnan': lambda y: getattr(y, 'isnan')(),
                       'isinf': lambda y: getattr(y, 'isinf')(),
                       'isfinite': lambda y: getattr(y, 'isfinite')(),
                       'nan_to_num': lambda y: getattr(y, 'nan_to_num')(),
                       'all': lambda y: getattr(y, 'all')(axis=axis),
                       'any': lambda y: getattr(y, 'any')(axis=axis),
                       'None': lambda y: y}

        return translation[op](x, **kwargs)

    def binary_operation(self, x, y, op='None', cast=0):

        translation = {'add': lambda z: getattr(z, '__add__'),
                       'radd': lambda z: getattr(z, '__radd__'),
                       'iadd': lambda z: getattr(z, '__iadd__'),
                       'sub': lambda z: getattr(z, '__sub__'),
                       'rsub': lambda z: getattr(z, '__rsub__'),
                       'isub': lambda z: getattr(z, '__isub__'),
                       'mul': lambda z: getattr(z, '__mul__'),
                       'rmul': lambda z: getattr(z, '__rmul__'),
                       'imul': lambda z: getattr(z, '__imul__'),
                       'div': lambda z: getattr(z, '__div__'),
                       'rdiv': lambda z: getattr(z, '__rdiv__'),
                       'idiv': lambda z: getattr(z, '__idiv__'),
                       'pow': lambda z: getattr(z, '__pow__'),
                       'rpow': lambda z: getattr(z, '__rpow__'),
                       'ipow': lambda z: getattr(z, '__ipow__'),
                       'ne': lambda z: getattr(z, '__ne__'),
                       'lt': lambda z: getattr(z, '__lt__'),
                       'le': lambda z: getattr(z, '__le__'),
                       'eq': lambda z: getattr(z, '__eq__'),
                       'ge': lambda z: getattr(z, '__ge__'),
                       'gt': lambda z: getattr(z, '__gt__'),
                       'None': lambda z: lambda u: u}

        if (cast & 1) != 0:
            x = self.cast(x)
        if (cast & 2) != 0:
            y = self.cast(y)

        return translation[op](x)(y)

    def get_shape(self):
        return (self.paradict['num'],)

    def get_dim(self):
        """
            Computes the dimension of the space, i.e.\  the number of points.

            Parameters
            ----------
            split : bool, *optional*
                Whether to return the dimension as an array with one component
                or as a scalar (default: False).

            Returns
            -------
            dim : {int, numpy.ndarray}
                Dimension(s) of the space.
        """
        return np.prod(self.get_shape())

    def get_dof(self, split=False):
        """
            Computes the number of degrees of freedom of the space, i.e./  the
            number of points for real-valued fields and twice that number for
            complex-valued fields.

            Returns
            -------
            dof : int
                Number of degrees of freedom of the space.
        """
        if split:
            dof = self.get_shape()
            if issubclass(self.dtype.type, np.complexfloating):
                dof = tuple(np.array(dof)*2)
        else:
            dof = self.get_dim()
            if issubclass(self.dtype.type, np.complexfloating):
                dof = dof * 2
        return dof

    def get_vol(self, split=False):
        if split:
            return self.distances
        else:
            return np.prod(self.distances)

    def get_meta_volume(self, split=False):
        """
            Calculates the meta volumes.

            The meta volumes are the volumes associated with each component of
            a field, taking into account field components that are not
            explicitly included in the array of field values but are determined
            by symmetry conditions. In the case of an :py:class:`rg_space`, the
            meta volumes are simply the pixel volumes.

            Parameters
            ----------
            total : bool, *optional*
                Whether to return the total meta volume of the space or the
                individual ones of each pixel (default: False).

            Returns
            -------
            mol : {numpy.ndarray, float}
                Meta volume of the pixels or the complete space.
        """
        if not split:
            return self.get_dim() * self.get_vol()
        else:
            mol = self.cast(1, dtype=np.dtype('float'))
            return self.calc_weight(mol, power=1)

    def cast(self, x=None, dtype=None, **kwargs):
        if dtype is not None:
            dtype = np.dtype(dtype)

        # If x is a field, extract the data and do a recursive call
        if isinstance(x, field):
            # Check if the domain matches
            if self != x.domain:
                about.warnings.cflush(
                    "WARNING: Getting data from foreign domain!")
            # Extract the data, whatever it is, and cast it again
            return self.cast(x.val,
                             dtype=dtype,
                             **kwargs)

        else:
            return self._cast_to_d2o(x=x,
                                     dtype=dtype,
                                     **kwargs)


    def _cast_to_d2o(self, x, dtype=None, **kwargs):
        """
            Computes valid field values from a given object, trying
            to translate the given data into a valid form. Thereby it is as
            benevolent as possible.

            Parameters
            ----------
            x : {float, numpy.ndarray, nifty.field}
                Object to be transformed into an array of valid field values.

            Returns
            -------
            x : numpy.ndarray, distributed_data_object
                Array containing the field values, which are compatible to the
                space.

            Other parameters
            ----------------
            verbose : bool, *optional*
                Whether the method should raise a warning if information is
                lost during casting (default: False).
        """
        if dtype is None:
            dtype = self.dtype

        # Case 1: x is a distributed_data_object
        if isinstance(x, distributed_data_object):
            to_copy = False

            # Check the shape
            if np.any(np.array(x.shape) != np.array(self.get_shape())):
                # Check if at least the number of degrees of freedom is equal
                if x.get_dim() == self.get_dim():
                    try:
                        temp = x.copy_empty(global_shape=self.get_shape())
                        temp.set_local_data(x.get_local_data(), copy=False)
                    except:
                        # If the number of dof is equal or 1, use np.reshape...
                        about.warnings.cflush(
                            "WARNING: Trying to reshape the data. This " +
                            "operation is expensive as it consolidates the " +
                            "full data!\n")
                        temp = x.get_full_data()
                        temp = np.reshape(temp