# Licensed under a 3-clause BSD style license - see LICENSE.rst

import numpy as np

from astropy import units as u
from astropy.utils import unbroadcast
import copy

from .wcs import WCS, WCSSUB_LONGITUDE, WCSSUB_LATITUDE

__doctest_skip__ = ['wcs_to_celestial_frame', 'celestial_frame_to_wcs']

__all__ = ['add_stokes_axis_to_wcs', 'celestial_frame_to_wcs',
           'wcs_to_celestial_frame', 'proj_plane_pixel_scales',
           'proj_plane_pixel_area', 'is_proj_plane_distorted',
           'non_celestial_pixel_scales', 'skycoord_to_pixel',
           'pixel_to_skycoord', 'custom_wcs_to_frame_mappings',
           'custom_frame_to_wcs_mappings', 'pixel_to_pixel',
           'local_partial_pixel_derivatives', 'fit_wcs_from_points']


def add_stokes_axis_to_wcs(wcs, add_before_ind):
    """
    Add a new Stokes axis that is uncorrelated with any other axes.

    Parameters
    ----------
    wcs : `~astropy.wcs.WCS`
        The WCS to add to
    add_before_ind : int
        Index of the WCS to insert the new Stokes axis in front of.
        To add at the end, do add_before_ind = wcs.wcs.naxis
        The beginning is at position 0.

    Returns
    -------
    `~astropy.wcs.WCS`
        A new `~astropy.wcs.WCS` instance with an additional axis
    """

    inds = [i + 1 for i in range(wcs.wcs.naxis)]
    inds.insert(add_before_ind, 0)
    newwcs = wcs.sub(inds)
    newwcs.wcs.ctype[add_before_ind] = 'STOKES'
    newwcs.wcs.cname[add_before_ind] = 'STOKES'
    return newwcs


def _wcs_to_celestial_frame_builtin(wcs):

    # Import astropy.coordinates here to avoid circular imports
    from astropy.coordinates import (FK4, FK4NoETerms, FK5, ICRS, ITRS,
                                     Galactic, SphericalRepresentation)

    # Import astropy.time here otherwise setup.py fails before extensions are compiled
    from astropy.time import Time

    if wcs.wcs.lng == -1 or wcs.wcs.lat == -1:
        return None

    radesys = wcs.wcs.radesys

    if np.isnan(wcs.wcs.equinox):
        equinox = None
    else:
        equinox = wcs.wcs.equinox

    xcoord = wcs.wcs.ctype[wcs.wcs.lng][:4]
    ycoord = wcs.wcs.ctype[wcs.wcs.lat][:4]

    # Apply logic from FITS standard to determine the default radesys
    if radesys == '' and xcoord == 'RA--' and ycoord == 'DEC-':
        if equinox is None:
            radesys = "ICRS"
        elif equinox < 1984.:
            radesys = "FK4"
        else:
            radesys = "FK5"

    if radesys == 'FK4':
        if equinox is not None:
            equinox = Time(equinox, format='byear')
        frame = FK4(equinox=equinox)
    elif radesys == 'FK4-NO-E':
        if equinox is not None:
            equinox = Time(equinox, format='byear')
        frame = FK4NoETerms(equinox=equinox)
    elif radesys == 'FK5':
        if equinox is not None:
            equinox = Time(equinox, format='jyear')
        frame = FK5(equinox=equinox)
    elif radesys == 'ICRS':
        frame = ICRS()
    else:
        if xcoord == 'GLON' and ycoord == 'GLAT':
            frame = Galactic()
        elif xcoord == 'TLON' and ycoord == 'TLAT':
            # The default representation for ITRS is cartesian, but for WCS
            # purposes, we need the spherical representation.
            frame = ITRS(representation_type=SphericalRepresentation,
                         obstime=wcs.wcs.dateobs or None)
        else:
            frame = None

    return frame


def _celestial_frame_to_wcs_builtin(frame, projection='TAN'):

    # Import astropy.coordinates here to avoid circular imports
    from astropy.coordinates import BaseRADecFrame, FK4, FK4NoETerms, FK5, ICRS, ITRS, Galactic

    # Create a 2-dimensional WCS
    wcs = WCS(naxis=2)

    if isinstance(frame, BaseRADecFrame):

        xcoord = 'RA--'
        ycoord = 'DEC-'
        if isinstance(frame, ICRS):
            wcs.wcs.radesys = 'ICRS'
        elif isinstance(frame, FK4NoETerms):
            wcs.wcs.radesys = 'FK4-NO-E'
            wcs.wcs.equinox = frame.equinox.byear
        elif isinstance(frame, FK4):
            wcs.wcs.radesys = 'FK4'
            wcs.wcs.equinox = frame.equinox.byear
        elif isinstance(frame, FK5):
            wcs.wcs.radesys = 'FK5'
            wcs.wcs.equinox = frame.equinox.jyear
        else:
            return None
    elif isinstance(frame, Galactic):
        xcoord = 'GLON'
        ycoord = 'GLAT'
    elif isinstance(frame, ITRS):
        xcoord = 'TLON'
        ycoord = 'TLAT'
        wcs.wcs.radesys = 'ITRS'
        wcs.wcs.dateobs = frame.obstime.utc.isot
    else:
        return None

    wcs.wcs.ctype = [xcoord + '-' + projection, ycoord + '-' + projection]

    return wcs


WCS_FRAME_MAPPINGS = [[_wcs_to_celestial_frame_builtin]]
FRAME_WCS_MAPPINGS = [[_celestial_frame_to_wcs_builtin]]


class custom_wcs_to_frame_mappings:
    def __init__(self, mappings=[]):
        if hasattr(mappings, '__call__'):
            mappings = [mappings]
        WCS_FRAME_MAPPINGS.append(mappings)

    def __enter__(self):
        pass

    def __exit__(self, type, value, tb):
        WCS_FRAME_MAPPINGS.pop()


# Backward-compatibility
custom_frame_mappings = custom_wcs_to_frame_mappings


class custom_frame_to_wcs_mappings:
    def __init__(self, mappings=[]):
        if hasattr(mappings, '__call__'):
            mappings = [mappings]
        FRAME_WCS_MAPPINGS.append(mappings)

    def __enter__(self):
        pass

    def __exit__(self, type, value, tb):
        FRAME_WCS_MAPPINGS.pop()


def wcs_to_celestial_frame(wcs):
    """
    For a given WCS, return the coordinate frame that matches the celestial
    component of the WCS.

    Parameters
    ----------
    wcs : :class:`~astropy.wcs.WCS` instance
        The WCS to find the frame for

    Returns
    -------
    frame : :class:`~astropy.coordinates.baseframe.BaseCoordinateFrame` subclass instance
        An instance of a :class:`~astropy.coordinates.baseframe.BaseCoordinateFrame`
        subclass instance that best matches the specified WCS.

    Notes
    -----

    To extend this function to frames not defined in astropy.coordinates, you
    can write your own function which should take a :class:`~astropy.wcs.WCS`
    instance and should return either an instance of a frame, or `None` if no
    matching frame was found. You can register this function temporarily with::

        >>> from astropy.wcs.utils import wcs_to_celestial_frame, custom_wcs_to_frame_mappings
        >>> with custom_wcs_to_frame_mappings(my_function):
        ...     wcs_to_celestial_frame(...)

    """
    for mapping_set in WCS_FRAME_MAPPINGS:
        for func in mapping_set:
            frame = func(wcs)
            if frame is not None:
                return frame
    raise ValueError("Could not determine celestial frame corresponding to "
                     "the specified WCS object")


def celestial_frame_to_wcs(frame, projection='TAN'):
    """
    For a given coordinate frame, return the corresponding WCS object.

    Note that the returned WCS object has only the elements corresponding to
    coordinate frames set (e.g. ctype, equinox, radesys).

    Parameters
    ----------
    frame : :class:`~astropy.coordinates.baseframe.BaseCoordinateFrame` subclass instance
        An instance of a :class:`~astropy.coordinates.baseframe.BaseCoordinateFrame`
        subclass instance for which to find the WCS
    projection : str
        Projection code to use in ctype, if applicable

    Returns
    -------
    wcs : :class:`~astropy.wcs.WCS` instance
        The corresponding WCS object

    Examples
    --------

    ::

        >>> from astropy.wcs.utils import celestial_frame_to_wcs
        >>> from astropy.coordinates import FK5
        >>> frame = FK5(equinox='J2010')
        >>> wcs = celestial_frame_to_wcs(frame)
        >>> wcs.to_header()
        WCSAXES =                    2 / Number of coordinate axes
        CRPIX1  =                  0.0 / Pixel coordinate of reference point
        CRPIX2  =                  0.0 / Pixel coordinate of reference point
        CDELT1  =                  1.0 / [deg] Coordinate increment at reference point
        CDELT2  =                  1.0 / [deg] Coordinate increment at reference point
        CUNIT1  = 'deg'                / Units of coordinate increment and value
        CUNIT2  = 'deg'                / Units of coordinate increment and value
        CTYPE1  = 'RA---TAN'           / Right ascension, gnomonic projection
        CTYPE2  = 'DEC--TAN'           / Declination, gnomonic projection
        CRVAL1  =                  0.0 / [deg] Coordinate value at reference point
        CRVAL2  =                  0.0 / [deg] Coordinate value at reference point
        LONPOLE =                180.0 / [deg] Native longitude of celestial pole
        LATPOLE =                  0.0 / [deg] Native latitude of celestial pole
        RADESYS = 'FK5'                / Equatorial coordinate system
        EQUINOX =               2010.0 / [yr] Equinox of equatorial coordinates


    Notes
    -----

    To extend this function to frames not defined in astropy.coordinates, you
    can write your own function which should take a
    :class:`~astropy.coordinates.baseframe.BaseCoordinateFrame` subclass
    instance and a projection (given as a string) and should return either a WCS
    instance, or `None` if the WCS could not be determined. You can register
    this function temporarily with::

        >>> from astropy.wcs.utils import celestial_frame_to_wcs, custom_frame_to_wcs_mappings
        >>> with custom_frame_to_wcs_mappings(my_function):
        ...     celestial_frame_to_wcs(...)

    """
    for mapping_set in FRAME_WCS_MAPPINGS:
        for func in mapping_set:
            wcs = func(frame, projection=projection)
            if wcs is not None:
                return wcs
    raise ValueError("Could not determine WCS corresponding to the specified "
                     "coordinate frame.")


def proj_plane_pixel_scales(wcs):
    """
    For a WCS returns pixel scales along each axis of the image pixel at
    the ``CRPIX`` location once it is projected onto the
    "plane of intermediate world coordinates" as defined in
    `Greisen & Calabretta 2002, A&A, 395, 1061 <https://ui.adsabs.harvard.edu/abs/2002A%26A...395.1061G>`_.

    .. note::
        This function is concerned **only** about the transformation
        "image plane"->"projection plane" and **not** about the
        transformation "celestial sphere"->"projection plane"->"image plane".
        Therefore, this function ignores distortions arising due to
        non-linear nature of most projections.

    .. note::
        In order to compute the scales corresponding to celestial axes only,
        make sure that the input `~astropy.wcs.WCS` object contains
        celestial axes only, e.g., by passing in the
        `~astropy.wcs.WCS.celestial` WCS object.

    Parameters
    ----------
    wcs : `~astropy.wcs.WCS`
        A world coordinate system object.

    Returns
    -------
    scale : ndarray
        A vector (`~numpy.ndarray`) of projection plane increments
        corresponding to each pixel side (axis). The units of the returned
        results are the same as the units of `~astropy.wcs.Wcsprm.cdelt`,
        `~astropy.wcs.Wcsprm.crval`, and `~astropy.wcs.Wcsprm.cd` for
        the celestial WCS and can be obtained by inquiring the value
        of `~astropy.wcs.Wcsprm.cunit` property of the input
        `~astropy.wcs.WCS` WCS object.

    See Also
    --------
    astropy.wcs.utils.proj_plane_pixel_area

    """
    return np.sqrt((wcs.pixel_scale_matrix**2).sum(axis=0, dtype=float))


def proj_plane_pixel_area(wcs):
    """
    For a **celestial** WCS (see `astropy.wcs.WCS.celestial`) returns pixel
    area of the image pixel at the ``CRPIX`` location once it is projected
    onto the "plane of intermediate world coordinates" as defined in
    `Greisen & Calabretta 2002, A&A, 395, 1061 <https://ui.adsabs.harvard.edu/abs/2002A%26A...395.1061G>`_.

    .. note::
        This function is concerned **only** about the transformation
        "image plane"->"projection plane" and **not** about the
        transformation "celestial sphere"->"projection plane"->"image plane".
        Therefore, this function ignores distortions arising due to
        non-linear nature of most projections.

    .. note::
        In order to compute the area of pixels corresponding to celestial
        axes only, this function uses the `~astropy.wcs.WCS.celestial` WCS
        object of the input ``wcs``.  This is different from the
        `~astropy.wcs.utils.proj_plane_pixel_scales` function
        that computes the scales for the axes of the input WCS itself.

    Parameters
    ----------
    wcs : `~astropy.wcs.WCS`
        A world coordinate system object.

    Returns
    -------
    area : float
        Area (in the projection plane) of the pixel at ``CRPIX`` location.
        The units of the returned result are the same as the units of
        the `~astropy.wcs.Wcsprm.cdelt`, `~astropy.wcs.Wcsprm.crval`,
        and `~astropy.wcs.Wcsprm.cd` for the celestial WCS and can be
        obtained by inquiring the value of `~astropy.wcs.Wcsprm.cunit`
        property of the `~astropy.wcs.WCS.celestial` WCS object.

    Raises
    ------
    ValueError
        Pixel area is defined only for 2D pixels. Most likely the
        `~astropy.wcs.Wcsprm.cd` matrix of the `~astropy.wcs.WCS.celestial`
        WCS is not a square matrix of second order.

    Notes
    -----

    Depending on the application, square root of the pixel area can be used to
    represent a single pixel scale of an equivalent square pixel
    whose area is equal to the area of a generally non-square pixel.

    See Also
    --------
    astropy.wcs.utils.proj_plane_pixel_scales

    """
    psm = wcs.celestial.pixel_scale_matrix
    if psm.shape != (2, 2):
        raise ValueError("Pixel area is defined only for 2D pixels.")
    return np.abs(np.linalg.det(psm))


def is_proj_plane_distorted(wcs, maxerr=1.0e-5):
    r"""
    For a WCS returns `False` if square image (detector) pixels stay square
    when projected onto the "plane of intermediate world coordinates"
    as defined in
    `Greisen & Calabretta 2002, A&A, 395, 1061 <https://ui.adsabs.harvard.edu/abs/2002A%26A...395.1061G>`_.
    It will return `True` if transformation from image (detector) coordinates
    to the focal plane coordinates is non-orthogonal or if WCS contains
    non-linear (e.g., SIP) distortions.

    .. note::
        Since this function is concerned **only** about the transformation
        "image plane"->"focal plane" and **not** about the transformation
        "celestial sphere"->"focal plane"->"image plane",
        this function ignores distortions arising due to non-linear nature
        of most projections.

    Let's denote by *C* either the original or the reconstructed
    (from ``PC`` and ``CDELT``) CD matrix. `is_proj_plane_distorted`
    verifies that the transformation from image (detector) coordinates
    to the focal plane coordinates is orthogonal using the following
    check:

    .. math::
        \left \| \frac{C \cdot C^{\mathrm{T}}}
        {| det(C)|} - I \right \|_{\mathrm{max}} < \epsilon .

    Parameters
    ----------
    wcs : `~astropy.wcs.WCS`
        World coordinate system object

    maxerr : float, optional
        Accuracy to which the CD matrix, **normalized** such
        that :math:`|det(CD)|=1`, should be close to being an
        orthogonal matrix as described in the above equation
        (see :math:`\epsilon`).

    Returns
    -------
    distorted : bool
        Returns `True` if focal (projection) plane is distorted and `False`
        otherwise.

    """
    cwcs = wcs.celestial
    return (not _is_cd_orthogonal(cwcs.pixel_scale_matrix, maxerr) or
            _has_distortion(cwcs))


def _is_cd_orthogonal(cd, maxerr):
    shape = cd.shape
    if not (len(shape) == 2 and shape[0] == shape[1]):
        raise ValueError("CD (or PC) matrix must be a 2D square matrix.")

    pixarea = np.abs(np.linalg.det(cd))
    if (pixarea == 0.0):
        raise ValueError("CD (or PC) matrix is singular.")

    # NOTE: Technically, below we should use np.dot(cd, np.conjugate(cd.T))
    # However, I am not aware of complex CD/PC matrices...
    I = np.dot(cd, cd.T) / pixarea
    cd_unitary_err = np.amax(np.abs(I - np.eye(shape[0])))

    return (cd_unitary_err < maxerr)


def non_celestial_pixel_scales(inwcs):
    """
    Calculate the pixel scale along each axis of a non-celestial WCS,
    for example one with mixed spectral and spatial axes.

    Parameters
    ----------
    inwcs : `~astropy.wcs.WCS`
        The world coordinate system object.

    Returns
    -------
    scale : `numpy.ndarray`
        The pixel scale along each axis.
    """

    if inwcs.is_celestial:
        raise ValueError("WCS is celestial, use celestial_pixel_scales instead")

    pccd = inwcs.pixel_scale_matrix

    if np.allclose(np.extract(1-np.eye(*pccd.shape), pccd), 0):
        return np.abs(np.diagonal(pccd))*u.deg
    else:
        raise ValueError("WCS is rotated, cannot determine consistent pixel scales")


def _has_distortion(wcs):
    """
    `True` if contains any SIP or image distortion components.
    """
    return any(getattr(wcs, dist_attr) is not None
               for dist_attr in ['cpdis1', 'cpdis2', 'det2im1', 'det2im2', 'sip'])


# TODO: in future, we should think about how the following two functions can be
# integrated better into the WCS class.

def skycoord_to_pixel(coords, wcs, origin=0, mode='all'):
    """
    Convert a set of SkyCoord coordinates into pixels.

    Parameters
    ----------
    coords : `~astropy.coordinates.SkyCoord`
        The coordinates to convert.
    wcs : `~astropy.wcs.WCS`
        The WCS transformation to use.
    origin : int
        Whether to return 0 or 1-based pixel coordinates.
    mode : 'all' or 'wcs'
        Whether to do the transformation including distortions (``'all'``) or
        only including only the core WCS transformation (``'wcs'``).

    Returns
    -------
    xp, yp : `numpy.ndarray`
        The pixel coordinates

    See Also
    --------
    astropy.coordinates.SkyCoord.from_pixel
    """

    if _has_distortion(wcs) and wcs.naxis != 2:
        raise ValueError("Can only handle WCS with distortions for 2-dimensional WCS")

    # Keep only the celestial part of the axes, also re-orders lon/lat
    wcs = wcs.sub([WCSSUB_LONGITUDE, WCSSUB_LATITUDE])

    if wcs.naxis != 2:
        raise ValueError("WCS should contain celestial component")

    # Check which frame the WCS uses
    frame = wcs_to_celestial_frame(wcs)

    # Check what unit the WCS needs
    xw_unit = u.Unit(wcs.wcs.cunit[0])
    yw_unit = u.Unit(wcs.wcs.cunit[1])

    # Convert positions to frame
    coords = coords.transform_to(frame)

    # Extract longitude and latitude. We first try and use lon/lat directly,
    # but if the representation is not spherical or unit spherical this will
    # fail. We should then force the use of the unit spherical
    # representation. We don't do that directly to make sure that we preserve
    # custom lon/lat representations if available.
    try:
        lon = coords.data.lon.to(xw_unit)
        lat = coords.data.lat.to(yw_unit)
    except AttributeError:
        lon = coords.spherical.lon.to(xw_unit)
        lat = coords.spherical.lat.to(yw_unit)

    # Convert to pixel coordinates
    if mode == 'all':
        xp, yp = wcs.all_world2pix(lon.value, lat.value, origin)
    elif mode == 'wcs':
        xp, yp = wcs.wcs_world2pix(lon.value, lat.value, origin)
    else:
        raise ValueError("mode should be either 'all' or 'wcs'")

    return xp, yp


def pixel_to_skycoord(xp, yp, wcs, origin=0, mode='all', cls=None):
    """
    Convert a set of pixel coordinates into a `~astropy.coordinates.SkyCoord`
    coordinate.

    Parameters
    ----------
    xp, yp : float or ndarray
        The coordinates to convert.
    wcs : `~astropy.wcs.WCS`
        The WCS transformation to use.
    origin : int
        Whether to return 0 or 1-based pixel coordinates.
    mode : 'all' or 'wcs'
        Whether to do the transformation including distortions (``'all'``) or
        only including only the core WCS transformation (``'wcs'``).
    cls : class or None
        The class of object to create.  Should be a
        `~astropy.coordinates.SkyCoord` subclass.  If None, defaults to
        `~astropy.coordinates.SkyCoord`.

    Returns
    -------
    coords : `~astropy.coordinates.SkyCoord` subclass
        The celestial coordinates. Whatever ``cls`` type is.

    See Also
    --------
    astropy.coordinates.SkyCoord.from_pixel
    """

    # Import astropy.coordinates here to avoid circular imports
    from astropy.coordinates import SkyCoord, UnitSphericalRepresentation

    # we have to do this instead of actually setting the default to SkyCoord
    # because importing SkyCoord at the module-level leads to circular
    # dependencies.
    if cls is None:
        cls = SkyCoord

    if _has_distortion(wcs) and wcs.naxis != 2:
        raise ValueError("Can only handle WCS with distortions for 2-dimensional WCS")

    # Keep only the celestial part of the axes, also re-orders lon/lat
    wcs = wcs.sub([WCSSUB_LONGITUDE, WCSSUB_LATITUDE])

    if wcs.naxis != 2:
        raise ValueError("WCS should contain celestial component")

    # Check which frame the WCS uses
    frame = wcs_to_celestial_frame(wcs)

    # Check what unit the WCS gives
    lon_unit = u.Unit(wcs.wcs.cunit[0])
    lat_unit = u.Unit(wcs.wcs.cunit[1])

    # Convert pixel coordinates to celestial coordinates
    if mode == 'all':
        lon, lat = wcs.all_pix2world(xp, yp, origin)
    elif mode == 'wcs':
        lon, lat = wcs.wcs_pix2world(xp, yp, origin)
    else:
        raise ValueError("mode should be either 'all' or 'wcs'")

    # Add units to longitude/latitude
    lon = lon * lon_unit
    lat = lat * lat_unit

    # Create a SkyCoord-like object
    data = UnitSphericalRepresentation(lon=lon, lat=lat)
    coords = cls(frame.realize_frame(data))

    return coords


def _unique_with_order_preserved(items):
    """
    Return a list of unique items in the list provided, preserving the order
    in which they are found.
    """
    new_items = []
    for item in items:
        if item not in new_items:
            new_items.append(item)
    return new_items


def _pixel_to_world_correlation_matrix(wcs):
    """
    Return a correlation matrix between the pixel coordinates and the
    high level world coordinates, along with the list of high level world
    coordinate classes.

    The shape of the matrix is ``(n_world, n_pix)``, where ``n_world`` is the
    number of high level world coordinates.
    """

    # We basically want to collapse the world dimensions together that are
    # combined into the same high-level objects.

    # Get the following in advance as getting these properties can be expensive
    all_components = wcs.low_level_wcs.world_axis_object_components
    all_classes = wcs.low_level_wcs.world_axis_object_classes
    axis_correlation_matrix = wcs.low_level_wcs.axis_correlation_matrix

    components = _unique_with_order_preserved([c[0] for c in all_components])

    matrix = np.zeros((len(components), wcs.pixel_n_dim), dtype=bool)

    for iworld in range(wcs.world_n_dim):
        iworld_unique = components.index(all_components[iworld][0])
        matrix[iworld_unique] |= axis_correlation_matrix[iworld]

    classes = [all_classes[component][0] for component in components]

    return matrix, classes


def _pixel_to_pixel_correlation_matrix(wcs_in, wcs_out):
    """
    Correlation matrix between the input and output pixel coordinates for a
    pixel -> world -> pixel transformation specified by two WCS instances.

    The first WCS specified is the one used for the pixel -> world
    transformation and the second WCS specified is the one used for the world ->
    pixel transformation. The shape of the matrix is
    ``(n_pixel_out, n_pixel_in)``.
    """

    matrix1, classes1 = _pixel_to_world_correlation_matrix(wcs_in)
    matrix2, classes2 = _pixel_to_world_correlation_matrix(wcs_out)

    if len(classes1) != len(classes2):
        raise ValueError("The two WCS return a different number of world coordinates")

    # Check if classes match uniquely
    unique_match = True
    mapping = []
    for class1 in classes1:
        matches = classes2.count(class1)
        if matches == 0:
            raise ValueError("The world coordinate types of the two WCS do not match")
        elif matches > 1:
            unique_match = False
            break
        else:
            mapping.append(classes2.index(class1))

    if unique_match:

        # Classes are unique, so we need to re-order matrix2 along the world
        # axis using the mapping we found above.
        matrix2 = matrix2[mapping]

    elif classes1 != classes2:

        raise ValueError("World coordinate order doesn't match and automatic matching is ambiguous")

    matrix = np.matmul(matrix2.T, matrix1)

    return matrix


def _split_matrix(matrix):
    """
    Given an axis correlation matrix from a WCS object, return information about
    the individual WCS that can be split out.

    The output is a list of tuples, where each tuple contains a list of
    pixel dimensions and a list of world dimensions that can be extracted to
    form a new WCS. For example, in the case of a spectral cube with the first
    two world coordinates being the celestial coordinates and the third
    coordinate being an uncorrelated spectral axis, the matrix would look like::

        array([[ True,  True, False],
               [ True,  True, False],
               [False, False,  True]])

    and this function will return ``[([0, 1], [0, 1]), ([2], [2])]``.
    """

    pixel_used = []

    split_info = []

    for ipix in range(matrix.shape[1]):
        if ipix in pixel_used:
            continue
        pixel_include = np.zeros(matrix.shape[1], dtype=bool)
        pixel_include[ipix] = True
        n_pix_prev, n_pix = 0, 1
        while n_pix > n_pix_prev:
            world_include = matrix[:, pixel_include].any(axis=1)
            pixel_include = matrix[world_include, :].any(axis=0)
            n_pix_prev, n_pix = n_pix, np.sum(pixel_include)
        pixel_indices = list(np.nonzero(pixel_include)[0])
        world_indices = list(np.nonzero(world_include)[0])
        pixel_used.extend(pixel_indices)
        split_info.append((pixel_indices, world_indices))

    return split_info


def pixel_to_pixel(wcs_in, wcs_out, *inputs):
    """
    Transform pixel coordinates in a dataset with a WCS to pixel coordinates
    in another dataset with a different WCS.

    This function is designed to efficiently deal with input pixel arrays that
    are broadcasted views of smaller arrays, and is compatible with any
    APE14-compliant WCS.

    Parameters
    ----------
    wcs_in : `~astropy.wcs.wcsapi.BaseHighLevelWCS`
        A WCS object for the original dataset which complies with the
        high-level shared APE 14 WCS API.
    wcs_out : `~astropy.wcs.wcsapi.BaseHighLevelWCS`
        A WCS object for the target dataset which complies with the
        high-level shared APE 14 WCS API.
    *inputs :
        Scalars or arrays giving the pixel coordinates to transform.
    """

    # Shortcut for scalars
    if np.isscalar(inputs[0]):
        world_outputs = wcs_in.pixel_to_world(*inputs)
        if not isinstance(world_outputs, (tuple, list)):
            world_outputs = (world_outputs,)
        return wcs_out.world_to_pixel(*world_outputs)

    # Remember original shape
    original_shape = inputs[0].shape

    matrix = _pixel_to_pixel_correlation_matrix(wcs_in, wcs_out)
    split_info = _split_matrix(matrix)

    outputs = [None] * wcs_out.pixel_n_dim

    for (pixel_in_indices, pixel_out_indices) in split_info:

        pixel_inputs = []
        for ipix in range(wcs_in.pixel_n_dim):
            if ipix in pixel_in_indices:
                pixel_inputs.append(unbroadcast(inputs[ipix]))
            else:
                pixel_inputs.append(inputs[ipix].flat[0])

        pixel_inputs = np.broadcast_arrays(*pixel_inputs)

        world_outputs = wcs_in.pixel_to_world(*pixel_inputs)

        if not isinstance(world_outputs, (tuple, list)):
            world_outputs = (world_outputs,)

        pixel_outputs = wcs_out.world_to_pixel(*world_outputs)

        if wcs_out.pixel_n_dim == 1:
            pixel_outputs = (pixel_outputs,)

        for ipix in range(wcs_out.pixel_n_dim):
            if ipix in pixel_out_indices:
                outputs[ipix] = np.broadcast_to(pixel_outputs[ipix], original_shape)

    return outputs[0] if wcs_out.pixel_n_dim == 1 else outputs


def local_partial_pixel_derivatives(wcs, *pixel, normalize_by_world=False):
    """
    Return a matrix of shape ``(world_n_dim, pixel_n_dim)`` where each entry
    ``[i, j]`` is the partial derivative d(world_i)/d(pixel_j) at the requested
    pixel position.

    Parameters
    ----------
    wcs : `~astropy.wcs.WCS`
        The WCS transformation to evaluate the derivatives for.
    *pixel : float
        The scalar pixel coordinates at which to evaluate the derivatives.
    normalize_by_world : bool
        If `True`, the matrix is normalized so that for each world entry
        the derivatives add up to 1.
    """

    # Find the world coordinates at the requested pixel
    pixel_ref = np.array(pixel)
    world_ref = np.array(wcs.pixel_to_world_values(*pixel_ref))

    # Set up the derivative matrix
    derivatives = np.zeros((wcs.world_n_dim, wcs.pixel_n_dim))

    for i in range(wcs.pixel_n_dim):
        pixel_off = pixel_ref.copy()
        pixel_off[i] += 1
        world_off = np.array(wcs.pixel_to_world_values(*pixel_off))
        derivatives[:, i] = world_off - world_ref

    if normalize_by_world:
        derivatives /= derivatives.sum(axis=0)[:, np.newaxis]

    return derivatives


def _linear_wcs_fit(params, lon, lat, x, y, w_obj):
    """
    Objective function for fitting linear terms.

    Parameters
    ----------
    params : array
        6 element array. First 4 elements are PC matrix, last 2 are CRPIX.
    lon, lat: array
        Sky coordinates.
    x, y: array
        Pixel coordinates
    w_obj: `~astropy.wcs.WCS`
        WCS object
        """
    cd = params[0:4]
    crpix = params[4:6]

    w_obj.wcs.cd = ((cd[0], cd[1]), (cd[2], cd[3]))
    w_obj.wcs.crpix = crpix
    lon2, lat2 = w_obj.wcs_pix2world(x, y, 0)

    lat_resids = lat - lat2
    lon_resids = lon - lon2
    # In case the longitude has wrapped around
    lon_resids = np.mod(lon_resids - 180.0, 360.0) - 180.0

    resids = np.concatenate((lon_resids * np.cos(np.radians(lat)), lat_resids))

    return resids


def _sip_fit(params, lon, lat, u, v, w_obj, order, coeff_names):

    """ Objective function for fitting SIP.

    Parameters
    ----------
    params : array
        Fittable parameters. First 4 elements are PC matrix, last 2 are CRPIX.
    lon, lat: array
        Sky coordinates.
    u, v: array
        Pixel coordinates
    w_obj: `~astropy.wcs.WCS`
        WCS object
    """

    from ..modeling.models import SIP   # here to avoid circular import

    # unpack params
    crpix = params[0:2]
    cdx = params[2:6].reshape((2, 2))
    a_params = params[6:6+len(coeff_names)]
    b_params = params[6+len(coeff_names):]

    # assign to wcs, used for transfomations in this function
    w_obj.wcs.cd = cdx
    w_obj.wcs.crpix = crpix

    a_coeff, b_coeff = {}, {}
    for i in range(len(coeff_names)):
        a_coeff['A_' + coeff_names[i]] = a_params[i]
        b_coeff['B_' + coeff_names[i]] = b_params[i]

    sip = SIP(crpix=crpix, a_order=order, b_order=order,
              a_coeff=a_coeff, b_coeff=b_coeff)
    fuv, guv = sip(u, v)

    xo, yo = np.dot(cdx, np.array([u+fuv-crpix[0], v+guv-crpix[1]]))

    # use all pix2world in case `projection` contains distortion table
    x, y = w_obj.all_world2pix(lon, lat, 0)
    x, y = np.dot(w_obj.wcs.cd, (x-w_obj.wcs.crpix[0], y-w_obj.wcs.crpix[1]))

    resids = np.concatenate((x-xo, y-yo))

    return resids


def fit_wcs_from_points(xy, world_coords, proj_point='center',
                        projection='TAN', sip_degree=None):
    """
    Given two matching sets of coordinates on detector and sky,
    compute the WCS.

    Fits a WCS object to matched set of input detector and sky coordinates.
    Optionally, a SIP can be fit to account for geometric
    distortion. Returns an `~astropy.wcs.WCS` object with the best fit
    parameters for mapping between input pixel and sky coordinates.

    The projection type (default 'TAN') can passed in as a string, one of
    the valid three-letter projection codes - or as a WCS object with
    projection keywords already set. Note that if an input WCS has any
    non-polynomial distortion, this will be applied and reflected in the
    fit terms and coefficients. Passing in a WCS object in this way essentially
    allows it to be refit based on the matched input coordinates and projection
    point, but take care when using this option as non-projection related
    keywords in the input might cause unexpected behavior.

    Notes
    -----
    - The fiducial point for the spherical projection can be set to 'center'
      to use the mean position of input sky coordinates, or as an
      `~astropy.coordinates.SkyCoord` object.
    - Units in all output WCS objects will always be in degrees.
    - If the coordinate frame differs between `~astropy.coordinates.SkyCoord`
      objects passed in for ``world_coords`` and ``proj_point``, the frame for
      ``world_coords``  will override as the frame for the output WCS.
    - If a WCS object is passed in to ``projection`` the CD/PC matrix will
      be used as an initial guess for the fit. If this is known to be
      significantly off and may throw off the fit, set to the identity matrix
      (for example, by doing wcs.wcs.pc = [(1., 0.,), (0., 1.)])

    Parameters
    ----------
    xy : (`numpy.ndarray`, `numpy.ndarray`) tuple
        x & y pixel coordinates.
    world_coords : `~astropy.coordinates.SkyCoord`
        Skycoord object with world coordinates.
    proj_point : 'center' or ~astropy.coordinates.SkyCoord`
        Defaults to 'center', in which the geometric center of input world
        coordinates will be used as the projection point. To specify an exact
        point for the projection, a Skycoord object with a coordinate pair can
        be passed in. For consistency, the units and frame of these coordinates
        will be transformed to match ``world_coords`` if they don't.
    projection : str or `~astropy.wcs.WCS`
        Three letter projection code, of any of standard projections defined
        in the FITS WCS standard. Optionally, a WCS object with projection
        keywords set may be passed in.
    sip_degree : None or int
        If set to a non-zero integer value, will fit SIP of degree
        ``sip_degree`` to model geometric distortion. Defaults to None, meaning
        no distortion corrections will be fit.

    Returns
    -------
    wcs : `~astropy.wcs.WCS`
        The best-fit WCS to the points given.
    """

    from astropy.coordinates import SkyCoord # here to avoid circular import
    import astropy.units as u
    from .wcs import Sip
    from scipy.optimize import least_squares

    xp, yp = xy
    try:
        lon, lat = world_coords.data.lon.deg, world_coords.data.lat.deg
    except AttributeError:
        unit_sph =  world_coords.unit_spherical
        lon, lat = unit_sph.lon.deg, unit_sph.lat.deg

    # verify input
    if (type(proj_point) != type(world_coords)) and (proj_point != 'center'):
        raise ValueError("proj_point must be set to 'center', or an" +
                         "`~astropy.coordinates.SkyCoord` object with " +
                         "a pair of points.")

    use_center_as_proj_point = (str(proj_point) == 'center')

    if not use_center_as_proj_point:
        assert proj_point.size == 1

    proj_codes = [
        'AZP', 'SZP', 'TAN', 'STG', 'SIN', 'ARC', 'ZEA', 'AIR', 'CYP',
        'CEA', 'CAR', 'MER', 'SFL', 'PAR', 'MOL', 'AIT', 'COP', 'COE',
        'COD', 'COO', 'BON', 'PCO', 'TSC', 'CSC', 'QSC', 'HPX', 'XPH'
    ]
    if type(projection) == str:
        if projection not in proj_codes:
            raise ValueError("Must specify valid projection code from list of "
                             + "supported types: ", ', '.join(proj_codes))
        # empty wcs to fill in with fit values
        wcs = celestial_frame_to_wcs(frame=world_coords.frame,
                                     projection=projection)
    else: #if projection is not string, should be wcs object. use as template.
        wcs = copy.deepcopy(projection)
        wcs.cdelt = (1., 1.) # make sure cdelt is 1
        wcs.sip = None

    # Change PC to CD, since cdelt will be set to 1
    if wcs.wcs.has_pc():
        wcs.wcs.cd = wcs.wcs.pc
        wcs.wcs.__delattr__('pc')

    if (type(sip_degree) != type(None)) and (type(sip_degree) != int):
        raise ValueError("sip_degree must be None, or integer.")

    # compute bounding box for sources in image coordinates:
    xpmin, xpmax, ypmin, ypmax = xp.min(), xp.max(), yp.min(), yp.max()

    # set pixel_shape to span of input points
    wcs.pixel_shape = (1 if xpmax <= 0.0 else int(np.ceil(xpmax)),
                       1 if ypmax <= 0.0 else int(np.ceil(ypmax)))

    # determine CRVAL from input
    close = lambda l, p: p[np.argmin(np.abs(l))]
    if use_center_as_proj_point:  # use center of input points
        sc1 = SkyCoord(lon.min()*u.deg, lat.max()*u.deg)
        sc2 = SkyCoord(lon.max()*u.deg, lat.min()*u.deg)
        pa = sc1.position_angle(sc2)
        sep = sc1.separation(sc2)
        midpoint_sc = sc1.directional_offset_by(pa, sep/2)
        wcs.wcs.crval = ((midpoint_sc.data.lon.deg, midpoint_sc.data.lat.deg))
        wcs.wcs.crpix = ((xpmax + xpmin) / 2., (ypmax + ypmin) / 2.)
    else:  # convert units, initial guess for crpix
        proj_point.transform_to(world_coords)
        wcs.wcs.crval = (proj_point.data.lon.deg, proj_point.data.lat.deg)
        wcs.wcs.crpix = (close(lon - wcs.wcs.crval[0], xp + 1),
                         close(lon - wcs.wcs.crval[1], yp + 1))

    # fit linear terms, assign to wcs
    # use (1, 0, 0, 1) as initial guess, in case input wcs was passed in
    # and cd terms are way off.
    # Use bounds to require that the fit center pixel is on the input image
    if xpmin == xpmax:
        xpmin, xpmax = xpmin - 0.5, xpmax + 0.5
    if ypmin == ypmax:
        ypmin, ypmax = ypmin - 0.5, ypmax + 0.5

    p0 = np.concatenate([wcs.wcs.cd.flatten(), wcs.wcs.crpix.flatten()])
    fit = least_squares(
        _linear_wcs_fit, p0,
        args=(lon, lat, xp, yp, wcs),
        bounds=[[-np.inf, -np.inf, -np.inf, -np.inf, xpmin + 1, ypmin + 1],
                [ np.inf, np.inf, np.inf, np.inf, xpmax + 1, ypmax + 1]]
    )
    wcs.wcs.crpix = np.array(fit.x[4:6])
    wcs.wcs.cd = np.array(fit.x[0:4].reshape((2, 2)))

    # fit SIP, if specified. Only fit forward coefficients
    if sip_degree:
        degree = sip_degree
        if '-SIP' not in wcs.wcs.ctype[0]:
            wcs.wcs.ctype = [x + '-SIP' for x in wcs.wcs.ctype]

        coef_names = [f'{i}_{j}' for i in range(degree+1)
                      for j in range(degree+1) if (i+j) < (degree+1) and
                      (i+j) > 1]
        p0 = np.concatenate((np.array(wcs.wcs.crpix), wcs.wcs.cd.flatten(),
                             np.zeros(2*len(coef_names))))

        fit = least_squares(
            _sip_fit, p0,
            args=(lon, lat, xp, yp, wcs, degree, coef_names),
            bounds=[[xpmin + 1, ypmin + 1] + [-np.inf]*(4 + 2*len(coef_names)),
                    [xpmax + 1, ypmax + 1] + [np.inf]*(4 + 2*len(coef_names))]
        )
        coef_fit = (list(fit.x[6:6+len(coef_names)]),
                    list(fit.x[6+len(coef_names):]))

        # put fit values in wcs
        wcs.wcs.cd = fit.x[2:6].reshape((2, 2))
        wcs.wcs.crpix = fit.x[0:2]

        a_vals = np.zeros((degree+1, degree+1))
        b_vals = np.zeros((degree+1, degree+1))

        for coef_name in coef_names:
            a_vals[int(coef_name[0])][int(coef_name[2])] = coef_fit[0].pop(0)
            b_vals[int(coef_name[0])][int(coef_name[2])] = coef_fit[1].pop(0)

        wcs.sip = Sip(a_vals, b_vals, np.zeros((degree+1, degree+1)),
                      np.zeros((degree+1, degree+1)), wcs.wcs.crpix)

    return wcs
