import math
import textwrap
from abc import ABC, abstractmethod

import numpy as np
from scipy import spatial

from .._shared.utils import safe_as_int


def _affine_matrix_from_vector(v):
    """Affine matrix from linearized (d, d + 1) matrix entries."""
    nparam = v.size
    # solve for d in: d * (d + 1) = nparam
    d = (1 + np.sqrt(1 + 4 * nparam)) / 2 - 1
    dimensionality = int(np.round(d))  # round to prevent approx errors
    if d != dimensionality:
        raise ValueError('Invalid number of elements for '
                         f'linearized matrix: {nparam}')
    matrix = np.eye(dimensionality + 1)
    matrix[:-1, :] = np.reshape(v, (dimensionality, dimensionality + 1))
    return matrix


def _center_and_normalize_points(points):
    """Center and normalize image points.

    The points are transformed in a two-step procedure that is expressed
    as a transformation matrix. The matrix of the resulting points is usually
    better conditioned than the matrix of the original points.

    Center the image points, such that the new coordinate system has its
    origin at the centroid of the image points.

    Normalize the image points, such that the mean distance from the points
    to the origin of the coordinate system is sqrt(D).

    If the points are all identical, the returned values will contain nan.

    Parameters
    ----------
    points : (N, D) array
        The coordinates of the image points.

    Returns
    -------
    matrix : (D+1, D+1) array_like
        The transformation matrix to obtain the new points.
    new_points : (N, D) array
        The transformed image points.

    References
    ----------
    .. [1] Hartley, Richard I. "In defense of the eight-point algorithm."
           Pattern Analysis and Machine Intelligence, IEEE Transactions on 19.6
           (1997): 580-593.

    """
    n, d = points.shape
    centroid = np.mean(points, axis=0)

    centered = points - centroid
    rms = np.sqrt(np.sum(centered ** 2) / n)

    # if all the points are the same, the transformation matrix cannot be
    # created. We return an equivalent matrix with np.nans as sentinel values.
    # This obviates the need for try/except blocks in functions calling this
    # one, and those are only needed when actual 0 is reached, rather than some
    # small value; ie, we don't need to worry about numerical stability here,
    # only actual 0.
    if rms == 0:
        return np.full((d + 1, d + 1), np.nan), np.full_like(points, np.nan)

    norm_factor = np.sqrt(d) / rms

    part_matrix = norm_factor * np.concatenate(
            (np.eye(d), -centroid[:, np.newaxis]), axis=1
            )
    matrix = np.concatenate(
            (part_matrix, [[0,] * d + [1]]), axis=0
            )

    points_h = np.vstack([points.T, np.ones(n)])

    new_points_h = (matrix @ points_h).T

    new_points = new_points_h[:, :d]
    new_points /= new_points_h[:, d:]

    return matrix, new_points


def _umeyama(src, dst, estimate_scale):
    """Estimate N-D similarity transformation with or without scaling.

    Parameters
    ----------
    src : (M, N) array_like
        Source coordinates.
    dst : (M, N) array_like
        Destination coordinates.
    estimate_scale : bool
        Whether to estimate scaling factor.

    Returns
    -------
    T : (N + 1, N + 1)
        The homogeneous similarity transformation matrix. The matrix contains
        NaN values only if the problem is not well-conditioned.

    References
    ----------
    .. [1] "Least-squares estimation of transformation parameters between two
            point patterns", Shinji Umeyama, PAMI 1991, :DOI:`10.1109/34.88573`

    """
    src = np.asarray(src)
    dst = np.asarray(dst)

    num = src.shape[0]
    dim = src.shape[1]

    # Compute mean of src and dst.
    src_mean = src.mean(axis=0)
    dst_mean = dst.mean(axis=0)

    # Subtract mean from src and dst.
    src_demean = src - src_mean
    dst_demean = dst - dst_mean

    # Eq. (38).
    A = dst_demean.T @ src_demean / num

    # Eq. (39).
    d = np.ones((dim,), dtype=np.float64)
    if np.linalg.det(A) < 0:
        d[dim - 1] = -1

    T = np.eye(dim + 1, dtype=np.float64)

    U, S, V = np.linalg.svd(A)

    # Eq. (40) and (43).
    rank = np.linalg.matrix_rank(A)
    if rank == 0:
        return np.nan * T
    elif rank == dim - 1:
        if np.linalg.det(U) * np.linalg.det(V) > 0:
            T[:dim, :dim] = U @ V
        else:
            s = d[dim - 1]
            d[dim - 1] = -1
            T[:dim, :dim] = U @ np.diag(d) @ V
            d[dim - 1] = s
    else:
        T[:dim, :dim] = U @ np.diag(d) @ V

    if estimate_scale:
        # Eq. (41) and (42).
        scale = 1.0 / src_demean.var(axis=0).sum() * (S @ d)
    else:
        scale = 1.0

    T[:dim, dim] = dst_mean - scale * (T[:dim, :dim] @ src_mean.T)
    T[:dim, :dim] *= scale

    return T


class _GeometricTransform(ABC):
    """Abstract base class for geometric transformations."""

    @abstractmethod
    def __call__(self, coords):
        """Apply forward transformation.

        Parameters
        ----------
        coords : (N, 2) array_like
            Source coordinates.

        Returns
        -------
        coords : (N, 2) array
            Destination coordinates.

        """

    @property
    @abstractmethod
    def inverse(self):
        """Return a transform object representing the inverse."""

    def residuals(self, src, dst):
        """Determine residuals of transformed destination coordinates.

        For each transformed source coordinate the Euclidean distance to the
        respective destination coordinate is determined.

        Parameters
        ----------
        src : (N, 2) array
            Source coordinates.
        dst : (N, 2) array
            Destination coordinates.

        Returns
        -------
        residuals : (N, ) array
            Residual for coordinate.

        """
        return np.sqrt(np.sum((self(src) - dst)**2, axis=1))


class FundamentalMatrixTransform(_GeometricTransform):
    """Fundamental matrix transformation.

    The fundamental matrix relates corresponding points between a pair of
    uncalibrated images. The matrix transforms homogeneous image points in one
    image to epipolar lines in the other image.

    The fundamental matrix is only defined for a pair of moving images. In the
    case of pure rotation or planar scenes, the homography describes the
    geometric relation between two images (`ProjectiveTransform`). If the
    intrinsic calibration of the images is known, the essential matrix describes
    the metric relation between the two images (`EssentialMatrixTransform`).

    References
    ----------
    .. [1] Hartley, Richard, and Andrew Zisserman. Multiple view geometry in
           computer vision. Cambridge university press, 2003.

    Parameters
    ----------
    matrix : (3, 3) array_like, optional
        Fundamental matrix.

    Attributes
    ----------
    params : (3, 3) array
        Fundamental matrix.

    """

    def __init__(self, matrix=None, *, dimensionality=2):
        if matrix is None:
            # default to an identity transform
            matrix = np.eye(dimensionality + 1)
        else:
            matrix = np.asarray(matrix)
            dimensionality = matrix.shape[0] - 1
            if matrix.shape != (dimensionality + 1, dimensionality + 1):
                raise ValueError("Invalid shape of transformation matrix")
        self.params = matrix
        if dimensionality != 2:
            raise NotImplementedError(
                f'{self.__class__} is only implemented for 2D coordinates '
                '(i.e. 3D transformation matrices).'
            )

    def __call__(self, coords):
        """Apply forward transformation.

        Parameters
        ----------
        coords : (N, 2) array_like
            Source coordinates.

        Returns
        -------
        coords : (N, 3) array
            Epipolar lines in the destination image.

        """
        coords = np.asarray(coords)
        coords_homogeneous = np.column_stack([coords, np.ones(coords.shape[0])])
        return coords_homogeneous @ self.params.T

    @property
    def inverse(self):
        """Return a transform object representing the inverse.

        See Hartley & Zisserman, Ch. 8: Epipolar Geometry and the Fundamental
        Matrix, for an explanation of why F.T gives the inverse.
        """
        return type(self)(matrix=self.params.T)

    def _setup_constraint_matrix(self, src, dst):
        """Setup and solve the homogeneous epipolar constraint matrix::

            dst' * F * src = 0.

        Parameters
        ----------
        src : (N, 2) array_like
            Source coordinates.
        dst : (N, 2) array_like
            Destination coordinates.

        Returns
        -------
        F_normalized : (3, 3) array
            The normalized solution to the homogeneous system. If the system
            is not well-conditioned, this matrix contains NaNs.
        src_matrix : (3, 3) array
            The transformation matrix to obtain the normalized source
            coordinates.
        dst_matrix : (3, 3) array
            The transformation matrix to obtain the normalized destination
            coordinates.

        """
        src = np.asarray(src)
        dst = np.asarray(dst)
        if src.shape != dst.shape:
            raise ValueError('src and dst shapes must be identical.')
        if src.shape[0] < 8:
            raise ValueError('src.shape[0] must be equal or larger than 8.')

        # Center and normalize image points for better numerical stability.
        try:
            src_matrix, src = _center_and_normalize_points(src)
            dst_matrix, dst = _center_and_normalize_points(dst)
        except ZeroDivisionError:
            self.params = np.full((3, 3), np.nan)
            return 3 * [np.full((3, 3), np.nan)]

        # Setup homogeneous linear equation as dst' * F * src = 0.
        A = np.ones((src.shape[0], 9))
        A[:, :2] = src
        A[:, :3] *= dst[:, 0, np.newaxis]
        A[:, 3:5] = src
        A[:, 3:6] *= dst[:, 1, np.newaxis]
        A[:, 6:8] = src

        # Solve for the nullspace of the constraint matrix.
        _, _, V = np.linalg.svd(A)
        F_normalized = V[-1, :].reshape(3, 3)

        return F_normalized, src_matrix, dst_matrix

    def estimate(self, src, dst):
        """Estimate fundamental matrix using 8-point algorithm.

        The 8-point algorithm requires at least 8 corresponding point pairs for
        a well-conditioned solution, otherwise the over-determined solution is
        estimated.

        Parameters
        ----------
        src : (N, 2) array_like
            Source coordinates.
        dst : (N, 2) array_like
            Destination coordinates.

        Returns
        -------
        success : bool
            True, if model estimation succeeds.

        """

        F_normalized, src_matrix, dst_matrix = \
            self._setup_constraint_matrix(src, dst)

        # Enforcing the internal constraint that two singular values must be
        # non-zero and one must be zero.
        U, S, V = np.linalg.svd(F_normalized)
        S[2] = 0
        F = U @ np.diag(S) @ V

        self.params = dst_matrix.T @ F @ src_matrix

        return True

    def residuals(self, src, dst):
        """Compute the Sampson distance.

        The Sampson distance is the first approximation to the geometric error.

        Parameters
        ----------
        src : (N, 2) array
            Source coordinates.
        dst : (N, 2) array
            Destination coordinates.

        Returns
        -------
        residuals : (N, ) array
            Sampson distance.

        """
        src_homogeneous = np.column_stack([src, np.ones(src.shape[0])])
        dst_homogeneous = np.column_stack([dst, np.ones(dst.shape[0])])

        F_src = self.params @ src_homogeneous.T
        Ft_dst = self.params.T @ dst_homogeneous.T

        dst_F_src = np.sum(dst_homogeneous * F_src.T, axis=1)

        return np.abs(dst_F_src) / np.sqrt(F_src[0] ** 2 + F_src[1] ** 2
                                           + Ft_dst[0] ** 2 + Ft_dst[1] ** 2)


class EssentialMatrixTransform(FundamentalMatrixTransform):
    """Essential matrix transformation.

    The essential matrix relates corresponding points between a pair of
    calibrated images. The matrix transforms normalized, homogeneous image
    points in one image to epipolar lines in the other image.

    The essential matrix is only defined for a pair of moving images capturing a
    non-planar scene. In the case of pure rotation or planar scenes, the
    homography describes the geometric relation between two images
    (`ProjectiveTransform`). If the intrinsic calibration of the images is
    unknown, the fundamental matrix describes the projective relation between
    the two images (`FundamentalMatrixTransform`).

    References
    ----------
    .. [1] Hartley, Richard, and Andrew Zisserman. Multiple view geometry in
           computer vision. Cambridge university press, 2003.

    Parameters
    ----------
    rotation : (3, 3) array_like, optional
        Rotation matrix of the relative camera motion.
    translation : (3, 1) array_like, optional
        Translation vector of the relative camera motion. The vector must
        have unit length.
    matrix : (3, 3) array_like, optional
        Essential matrix.

    Attributes
    ----------
    params : (3, 3) array
        Essential matrix.

    Examples
    --------
    >>> import numpy as np
    >>> import skimage as ski
    >>>
    >>> tform_matrix = ski.transform.EssentialMatrixTransform(
    ...     rotation=np.eye(3), translation=np.array([0, 0, 1])
    ... )
    >>> tform_matrix.params
    array([[ 0., -1.,  0.],
           [ 1.,  0.,  0.],
           [ 0.,  0.,  0.]])
    >>> src = np.array([[ 1.839035, 1.924743],
    ...                 [ 0.543582, 0.375221],
    ...                 [ 0.47324 , 0.142522],
    ...                 [ 0.96491 , 0.598376],
    ...                 [ 0.102388, 0.140092],
    ...                 [15.994343, 9.622164],
    ...                 [ 0.285901, 0.430055],
    ...                 [ 0.09115 , 0.254594]])
    >>> dst = np.array([[1.002114, 1.129644],
    ...                 [1.521742, 1.846002],
    ...                 [1.084332, 0.275134],
    ...                 [0.293328, 0.588992],
    ...                 [0.839509, 0.08729 ],
    ...                 [1.779735, 1.116857],
    ...                 [0.878616, 0.602447],
    ...                 [0.642616, 1.028681]])
    >>> tform_matrix.estimate(src, dst)
    True
    >>> tform_matrix.residuals(src, dst)
    array([0.4245518687, 0.0146044753, 0.1384703409, 0.1214095141,
           0.2775934609, 0.3245311807, 0.0021077555, 0.2651228318])

    """

    def __init__(self, rotation=None, translation=None, matrix=None,
                 *, dimensionality=2):
        super().__init__(matrix=matrix, dimensionality=dimensionality)
        if rotation is not None:
            rotation = np.asarray(rotation)
            if translation is None:
                raise ValueError("Both rotation and translation required")
            translation = np.asarray(translation)
            if rotation.shape != (3, 3):
                raise ValueError("Invalid shape of rotation matrix")
            if abs(np.linalg.det(rotation) - 1) > 1e-6:
                raise ValueError("Rotation matrix must have unit determinant")
            if translation.size != 3:
                raise ValueError("Invalid shape of translation vector")
            if abs(np.linalg.norm(translation) - 1) > 1e-6:
                raise ValueError("Translation vector must have unit length")
            # Matrix representation of the cross product for t.
            t_x = np.array([0, -translation[2], translation[1],
                            translation[2], 0, -translation[0],
                            -translation[1], translation[0], 0]).reshape(3, 3)
            self.params = t_x @ rotation
        elif matrix is not None:
            matrix = np.asarray(matrix)
            if matrix.shape != (3, 3):
                raise ValueError("Invalid shape of transformation matrix")
            self.params = matrix
        else:
            # default to an identity transform
            self.params = np.eye(3)

    def estimate(self, src, dst):
        """Estimate essential matrix using 8-point algorithm.

        The 8-point algorithm requires at least 8 corresponding point pairs for
        a well-conditioned solution, otherwise the over-determined solution is
        estimated.

        Parameters
        ----------
        src : (N, 2) array_like
            Source coordinates.
        dst : (N, 2) array_like
            Destination coordinates.

        Returns
        -------
        success : bool
            True, if model estimation succeeds.

        """

        E_normalized, src_matrix, dst_matrix = \
            self._setup_constraint_matrix(src, dst)

        # Enforcing the internal constraint that two singular values must be
        # equal and one must be zero.
        U, S, V = np.linalg.svd(E_normalized)
        S[0] = (S[0] + S[1]) / 2.0
        S[1] = S[0]
        S[2] = 0
        E = U @ np.diag(S) @ V

        self.params = dst_matrix.T @ E @ src_matrix

        return True


class ProjectiveTransform(_GeometricTransform):
    r"""Projective transformation.

    Apply a projective transformation (homography) on coordinates.

    For each homogeneous coordinate :math:`\mathbf{x} = [x, y, 1]^T`, its
    target position is calculated by multiplying with the given matrix,
    :math:`H`, to give :math:`H \mathbf{x}`::

      [[a0 a1 a2]
       [b0 b1 b2]
       [c0 c1 1 ]].

    E.g., to rotate by theta degrees clockwise, the matrix should be::

      [[cos(theta) -sin(theta) 0]
       [sin(theta)  cos(theta) 0]
       [0            0         1]]

    or, to translate x by 10 and y by 20::

      [[1 0 10]
       [0 1 20]
       [0 0 1 ]].

    Parameters
    ----------
    matrix : (D+1, D+1) array_like, optional
        Homogeneous transformation matrix.
    dimensionality : int, optional
        The number of dimensions of the transform. This is ignored if
        ``matrix`` is not None.

    Attributes
    ----------
    params : (D+1, D+1) array
        Homogeneous transformation matrix.

    """

    def __init__(self, matrix=None, *, dimensionality=2):
        if matrix is None:
            # default to an identity transform
            matrix = np.eye(dimensionality + 1)
        else:
            matrix = np.asarray(matrix)
            dimensionality = matrix.shape[0] - 1
            if matrix.shape != (dimensionality + 1, dimensionality + 1):
                raise ValueError("invalid shape of transformation matrix")
        self.params = matrix
        self._coeffs = range(matrix.size - 1)

    @property
    def _inv_matrix(self):
        return np.linalg.inv(self.params)

    def _apply_mat(self, coords, matrix):
        ndim = matrix.shape[0] - 1
        coords = np.array(coords, copy=False, ndmin=2)

        src = np.concatenate([coords, np.ones((coords.shape[0], 1))], axis=1)
        dst = src @ matrix.T

        # below, we will divide by the last dimension of the homogeneous
        # coordinate matrix. In order to avoid division by zero,
        # we replace exact zeros in this column with a very small number.
        dst[dst[:, ndim] == 0, ndim] = np.finfo(float).eps
        # rescale to homogeneous coordinates
        dst[:, :ndim] /= dst[:, ndim:ndim+1]

        return dst[:, :ndim]

    def __array__(self, dtype=None):
        if dtype is None:
            return self.params
        else:
            return self.params.astype(dtype)

    def __call__(self, coords):
        """Apply forward transformation.

        Parameters
        ----------
        coords : (N, D) array_like
            Source coordinates.

        Returns
        -------
        coords_out : (N, D) array
            Destination coordinates.

        """
        return self._apply_mat(coords, self.params)

    @property
    def inverse(self):
        """Return a transform object representing the inverse."""
        return type(self)(matrix=self._inv_matrix)

    def estimate(self, src, dst, weights=None):
        """Estimate the transformation from a set of corresponding points.

        You can determine the over-, well- and under-determined parameters
        with the total least-squares method.

        Number of source and destination coordinates must match.

        The transformation is defined as::

            X = (a0*x + a1*y + a2) / (c0*x + c1*y + 1)
            Y = (b0*x + b1*y + b2) / (c0*x + c1*y + 1)

        These equations can be transformed to the following form::

            0 = a0*x + a1*y + a2 - c0*x*X - c1*y*X - X
            0 = b0*x + b1*y + b2 - c0*x*Y - c1*y*Y - Y

        which exist for each set of corresponding points, so we have a set of
        N * 2 equations. The coefficients appear linearly so we can write
        A x = 0, where::

            A   = [[x y 1 0 0 0 -x*X -y*X -X]
                   [0 0 0 x y 1 -x*Y -y*Y -Y]
                    ...
                    ...
                  ]
            x.T = [a0 a1 a2 b0 b1 b2 c0 c1 c3]

        In case of total least-squares the solution of this homogeneous system
        of equations is the right singular vector of A which corresponds to the
        smallest singular value normed by the coefficient c3.

        Weights can be applied to each pair of corresponding points to
        indicate, particularly in an overdetermined system, if point pairs have
        higher or lower confidence or uncertainties associated with them. From
        the matrix treatment of least squares problems, these weight values are
        normalised, square-rooted, then built into a diagonal matrix, by which
        A is multiplied.

        In case of the affine transformation the coefficients c0 and c1 are 0.
        Thus the system of equations is::

            A   = [[x y 1 0 0 0 -X]
                   [0 0 0 x y 1 -Y]
                    ...
                    ...
                  ]
            x.T = [a0 a1 a2 b0 b1 b2 c3]

        Parameters
        ----------
        src : (N, 2) array_like
            Source coordinates.
        dst : (N, 2) array_like
            Destination coordinates.
        weights : (N,) array_like, optional
            Relative weight values for each pair of points.

        Returns
        -------
        success : bool
            True, if model estimation succeeds.

        """
        src = np.asarray(src)
        dst = np.asarray(dst)
        n, d = src.shape

        src_matrix, src = _center_and_normalize_points(src)
        dst_matrix, dst = _center_and_normalize_points(dst)
        if not np.all(np.isfinite(src_matrix + dst_matrix)):
            self.params = np.full((d + 1, d + 1), np.nan)
            return False

        # params: a0, a1, a2, b0, b1, b2, c0, c1
        A = np.zeros((n * d, (d+1) ** 2))
        # fill the A matrix with the appropriate block matrices; see docstring
        # for 2D example — this can be generalised to more blocks in the 3D and
        # higher-dimensional cases.
        for ddim in range(d):
            A[ddim*n : (ddim+1)*n, ddim*(d+1) : ddim*(d+1) + d] = src
            A[ddim*n : (ddim+1)*n, ddim*(d+1) + d] = 1
            A[ddim*n : (ddim+1)*n, -d-1:-1] = src
            A[ddim*n : (ddim+1)*n, -1] = -1
            A[ddim*n : (ddim+1)*n, -d-1:] *= -dst[:, ddim:(ddim+1)]

        # Select relevant columns, depending on params
        A = A[:, list(self._coeffs) + [-1]]

        # Get the vectors that correspond to singular values, also applying
        # the weighting if provided
        if weights is None:
            _, _, V = np.linalg.svd(A)
        else:
            weights = np.asarray(weights)
            W = np.diag(np.tile(np.sqrt(weights / np.max(weights)), d))
            _, _, V = np.linalg.svd(W @ A)

        # if the last element of the vector corresponding to the smallest
        # singular value is close to zero, this implies a degenerate case
        # because it is a rank-defective transform, which would map points
        # to a line rather than a plane.
        if np.isclose(V[-1, -1], 0):
            self.params = np.full((d + 1, d + 1), np.nan)
            return False

        H = np.zeros((d+1, d+1))
        # solution is right singular vector that corresponds to smallest
        # singular value
        H.flat[list(self._coeffs) + [-1]] = - V[-1, :-1] / V[-1, -1]
        H[d, d] = 1

        # De-center and de-normalize
        H = np.linalg.inv(dst_matrix) @ H @ src_matrix

        # Small errors can creep in if points are not exact, causing the last
        # element of H to deviate from unity. Correct for that here.
        H /= H[-1, -1]

        self.params = H

        return True

    def __add__(self, other):
        """Combine this transformation with another."""
        if isinstance(other, ProjectiveTransform):
            # combination of the same types result in a transformation of this
            # type again, otherwise use general projective transformation
            if type(self) == type(other):
                tform = self.__class__
            else:
                tform = ProjectiveTransform
            return tform(other.params @ self.params)
        else:
            raise TypeError("Cannot combine transformations of differing "
                            "types.")

    def __nice__(self):
        """common 'paramstr' used by __str__ and __repr__"""
        npstring = np.array2string(self.params, separator=', ')
        paramstr = 'matrix=\n' + textwrap.indent(npstring, '    ')
        return paramstr

    def __repr__(self):
        """Add standard repr formatting around a __nice__ string"""
        paramstr = self.__nice__()
        classname = self.__class__.__name__
        classstr = classname
        return f'<{classstr}({paramstr}) at {hex(id(self))}>'

    def __str__(self):
        """Add standard str formatting around a __nice__ string"""
        paramstr = self.__nice__()
        classname = self.__class__.__name__
        classstr = classname
        return f'<{classstr}({paramstr})>'

    @property
    def dimensionality(self):
        """The dimensionality of the transformation."""
        return self.params.shape[0] - 1


class AffineTransform(ProjectiveTransform):
    """Affine transformation.

    Has the following form::

        X = a0 * x + a1 * y + a2
          =   sx * x * [cos(rotation) + tan(shear_y) * sin(rotation)]
            - sy * y * [tan(shear_x) * cos(rotation) + sin(rotation)]
            + translation_x

        Y = b0 * x + b1 * y + b2
          =   sx * x * [sin(rotation) - tan(shear_y) * cos(rotation)]
            - sy * y * [tan(shear_x) * sin(rotation) - cos(rotation)]
            + translation_y

    where ``sx`` and ``sy`` are scale factors in the x and y directions.

    This is equivalent to applying the operations in the following order:

    1. Scale
    2. Shear
    3. Rotate
    4. Translate

    The homogeneous transformation matrix is::

        [[a0  a1  a2]
         [b0  b1  b2]
         [0   0    1]]

    In 2D, the transformation parameters can be given as the homogeneous
    transformation matrix, above, or as the implicit parameters, scale,
    rotation, shear, and translation in x (a2) and y (b2). For 3D and higher,
    only the matrix form is allowed.

    In narrower transforms, such as the Euclidean (only rotation and
    translation) or Similarity (rotation, translation, and a global scale
    factor) transforms, it is possible to specify 3D transforms using implicit
    parameters also.

    Parameters
    ----------
    matrix : (D+1, D+1) array_like, optional
        Homogeneous transformation matrix. If this matrix is provided, it is an
        error to provide any of scale, rotation, shear, or translation.
    scale : {s as float or (sx, sy) as array, list or tuple}, optional
        Scale factor(s). If a single value, it will be assigned to both
        sx and sy. Only available for 2D.

        .. versionadded:: 0.17
           Added support for supplying a single scalar value.
    rotation : float, optional
        Rotation angle, clockwise, as radians. Only available for 2D.
    shear : float or 2-tuple of float, optional
        The x and y shear angles, clockwise, by which these axes are
        rotated around the origin [2].
        If a single value is given, take that to be the x shear angle, with
        the y angle remaining 0. Only available in 2D.
    translation : (tx, ty) as array, list or tuple, optional
        Translation parameters. Only available for 2D.
    dimensionality : int, optional
        The dimensionality of the transform. This is not used if any other
        parameters are provided.

    Attributes
    ----------
    params : (D+1, D+1) array
        Homogeneous transformation matrix.

    Raises
    ------
    ValueError
        If both ``matrix`` and any of the other parameters are provided.

    Examples
    --------
    >>> import numpy as np
    >>> import skimage as ski
    >>> img = ski.data.astronaut()

    Define source and destination points:

    >>> src = np.array([[150, 150],
    ...                 [250, 100],
    ...                 [150, 200]])
    >>> dst = np.array([[200, 200],
    ...                 [300, 150],
    ...                 [150, 400]])

    Estimate the transformation matrix:

    >>> tform = ski.transform.AffineTransform()
    >>> tform.estimate(src, dst)
    True

    Apply the transformation:

    >>> warped = ski.transform.warp(img, inverse_map=tform.inverse)

    References
    ----------
    .. [1] Wikipedia, "Affine transformation",
           https://en.wikipedia.org/wiki/Affine_transformation#Image_transformation
    .. [2] Wikipedia, "Shear mapping",
           https://en.wikipedia.org/wiki/Shear_mapping
    """

    def __init__(self, matrix=None, scale=None, rotation=None, shear=None,
                 translation=None, *, dimensionality=2):
        params = any(param is not None
                     for param in (scale, rotation, shear, translation))

        # these parameters get overwritten if a higher-D matrix is given
        self._coeffs = range(dimensionality * (dimensionality + 1))

        if params and matrix is not None:
            raise ValueError("You cannot specify the transformation matrix and"
                             " the implicit parameters at the same time.")
        if params and dimensionality > 2:
            raise ValueError('Parameter input is only supported in 2D.')
        elif matrix is not None:
            matrix = np.asarray(matrix)
            if matrix.ndim != 2 or matrix.shape[0] != matrix.shape[1]:
                raise ValueError("Invalid shape of transformation matrix.")
            else:
                dimensionality = matrix.shape[0] - 1
                nparam = dimensionality * (dimensionality + 1)
            self._coeffs = range(nparam)
            self.params = matrix
        elif params:  # note: 2D only
            if scale is None:
                scale = (1, 1)
            if rotation is None:
                rotation = 0
            if shear is None:
                shear = 0
            if translation is None:
                translation = (0, 0)

            if np.isscalar(scale):
                sx = sy = scale
            else:
                sx, sy = scale

            if np.isscalar(shear):
                shear_x, shear_y = (shear, 0)
            else:
                shear_x, shear_y = shear

            a0 = sx * (
                math.cos(rotation) + math.tan(shear_y) * math.sin(rotation)
            )
            a1 = -sy * (
                math.tan(shear_x) * math.cos(rotation) + math.sin(rotation)
            )
            a2 = translation[0]

            b0 = sx * (
                math.sin(rotation) - math.tan(shear_y) * math.cos(rotation)
            )
            b1 = -sy * (
                math.tan(shear_x) * math.sin(rotation) - math.cos(rotation)
            )
            b2 = translation[1]
            self.params = np.array([[a0, a1, a2], [b0, b1, b2], [0, 0, 1]])
        else:
            # default to an identity transform
            self.params = np.eye(dimensionality + 1)

    @property
    def scale(self):
        if self.dimensionality != 2:
            return np.sqrt(np.sum(self.params ** 2, axis=0))[:self.dimensionality]
        else:
            ss = np.sum(self.params ** 2, axis=0)
            ss[1] = ss[1] / (math.tan(self.shear)**2 + 1)
            return np.sqrt(ss)[:self.dimensionality]

    @property
    def rotation(self):
        if self.dimensionality != 2:
            raise NotImplementedError(
                'The rotation property is only implemented for 2D transforms.'
            )
        return math.atan2(self.params[1, 0], self.params[0, 0])

    @property
    def shear(self):
        if self.dimensionality != 2:
            raise NotImplementedError(
                'The shear property is only implemented for 2D transforms.'
            )
        beta = math.atan2(- self.params[0, 1], self.params[1, 1])
        return beta - self.rotation

    @property
    def translation(self):
        return self.params[0:self.dimensionality, self.dimensionality]


class PiecewiseAffineTransform(_GeometricTransform):
    """Piecewise affine transformation.

    Control points are used to define the mapping. The transform is based on
    a Delaunay triangulation of the points to form a mesh. Each triangle is
    used to find a local affine transform.

    Attributes
    ----------
    affines : list of AffineTransform objects
        Affine transformations for each triangle in the mesh.
    inverse_affines : list of AffineTransform objects
        Inverse affine transformations for each triangle in the mesh.

    """

    def __init__(self):
        self._tesselation = None
        self._inverse_tesselation = None
        self.affines = None
        self.inverse_affines = None

    def estimate(self, src, dst):
        """Estimate the transformation from a set of corresponding points.

        Number of source and destination coordinates must match.

        Parameters
        ----------
        src : (N, D) array_like
            Source coordinates.
        dst : (N, D) array_like
            Destination coordinates.

        Returns
        -------
        success : bool
            True, if all pieces of the model are successfully estimated.

        """
        src = np.asarray(src)
        dst = np.asarray(dst)

        ndim = src.shape[1]
        # forward piecewise affine
        # triangulate input positions into mesh
        self._tesselation = spatial.Delaunay(src)

        success = True

        # find affine mapping from source positions to destination
        self.affines = []
        for tri in self._tesselation.simplices:
            affine = AffineTransform(dimensionality=ndim)
            success &= affine.estimate(src[tri, :], dst[tri, :])
            self.affines.append(affine)

        # inverse piecewise affine
        # triangulate input positions into mesh
        self._inverse_tesselation = spatial.Delaunay(dst)
        # find affine mapping from source positions to destination
        self.inverse_affines = []
        for tri in self._inverse_tesselation.simplices:
            affine = AffineTransform(dimensionality=ndim)
            success &= affine.estimate(dst[tri, :], src[tri, :])
            self.inverse_affines.append(affine)

        return success

    def __call__(self, coords):
        """Apply forward transformation.

        Coordinates outside of the mesh will be set to `- 1`.

        Parameters
        ----------
        coords : (N, D) array_like
            Source coordinates.

        Returns
        -------
        coords : (N, 2) array
            Transformed coordinates.

        """
        coords = np.asarray(coords)
        out = np.empty_like(coords, np.float64)

        # determine triangle index for each coordinate
        simplex = self._tesselation.find_simplex(coords)

        # coordinates outside of mesh
        out[simplex == -1, :] = -1

        for index in range(len(self._tesselation.simplices)):
            # affine transform for triangle
            affine = self.affines[index]
            # all coordinates within triangle
            index_mask = simplex == index

            out[index_mask, :] = affine(coords[index_mask, :])

        return out

    @property
    def inverse(self):
        """Return a transform object representing the inverse."""
        tform = type(self)()
        tform._tesselation = self._inverse_tesselation
        tform._inverse_tesselation = self._tesselation
        tform.affines = self.inverse_affines
        tform.inverse_affines = self.affines
        return tform


def _euler_rotation(axis, angle):
    """Produce a single-axis Euler rotation matrix.

    Parameters
    ----------
    axis : int in {0, 1, 2}
        The axis of rotation.
    angle : float
        The angle of rotation in radians.

    Returns
    -------
    Ri : array of float, shape (3, 3)
        The rotation matrix along axis `axis`.
    """
    i = axis
    s = (-1)**i * np.sin(angle)
    c = np.cos(angle)
    R2 = np.array([[c, -s],
                   [s,  c]])
    Ri = np.eye(3)
    # We need the axes other than the rotation axis, in the right order:
    # 0 -> (1, 2); 1 -> (0, 2); 2 -> (0, 1).
    axes = sorted({0, 1, 2} - {axis})
    # We then embed the 2-axis rotation matrix into the full matrix.
    # (1, 2) -> R[1:3:1, 1:3:1] = R2, (0, 2) -> R[0:3:2, 0:3:2] = R2, etc.
    sl = slice(axes[0], axes[1] + 1, axes[1] - axes[0])
    Ri[sl, sl] = R2
    return Ri


def _euler_rotation_matrix(angles, axes=None):
    """Produce an Euler rotation matrix from the given angles.

    The matrix will have dimension equal to the number of angles given.

    Parameters
    ----------
    angles : array of float, shape (3,)
        The transformation angles in radians.
    axes : list of int
        The axes about which to produce the rotation. Defaults to 0, 1, 2.

    Returns
    -------
    R : array of float, shape (3, 3)
        The Euler rotation matrix.
    """
    if axes is None:
        axes = range(3)
    dim = len(angles)
    R = np.eye(dim)
    for i, angle in zip(axes, angles):
        R = R @ _euler_rotation(i, angle)
    return R


class EuclideanTransform(ProjectiveTransform):
    """Euclidean transformation, also known as a rigid transform.

    Has the following form::

        X = a0 * x - b0 * y + a1 =
          = x * cos(rotation) - y * sin(rotation) + a1

        Y = b0 * x + a0 * y + b1 =
          = x * sin(rotation) + y * cos(rotation) + b1

    where the homogeneous transformation matrix is::

        [[a0  b0  a1]
         [b0  a0  b1]
         [0   0    1]]

    The Euclidean transformation is a rigid transformation with rotation and
    translation parameters. The similarity transformation extends the Euclidean
    transformation with a single scaling factor.

    In 2D and 3D, the transformation parameters may be provided either via
    `matrix`, the homogeneous transformation matrix, above, or via the
    implicit parameters `rotation` and/or `translation` (where `a1` is the
    translation along `x`, `b1` along `y`, etc.). Beyond 3D, if the
    transformation is only a translation, you may use the implicit parameter
    `translation`; otherwise, you must use `matrix`.

    Parameters
    ----------
    matrix : (D+1, D+1) array_like, optional
        Homogeneous transformation matrix.
    rotation : float or sequence of float, optional
        Rotation angle, clockwise, as radians. If given as
        a vector, it is interpreted as Euler rotation angles [1]_. Only 2D
        (single rotation) and 3D (Euler rotations) values are supported. For
        higher dimensions, you must provide or estimate the transformation
        matrix.
    translation : (x, y[, z, ...]) sequence of float, length D, optional
        Translation parameters for each axis.
    dimensionality : int, optional
        The dimensionality of the transform.

    Attributes
    ----------
    params : (D+1, D+1) array
        Homogeneous transformation matrix.

    References
    ----------
    .. [1] https://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions
    """

    def __init__(self, matrix=None, rotation=None, translation=None,
                 *, dimensionality=2):
        params_given = rotation is not None or translation is not None

        if params_given and matrix is not None:
            raise ValueError("You cannot specify the transformation matrix and"
                             " the implicit parameters at the same time.")
        elif matrix is not None:
            matrix = np.asarray(matrix)
            if matrix.shape[0] != matrix.shape[1]:
                raise ValueError("Invalid shape of transformation matrix.")
            self.params = matrix
        elif params_given:
            if rotation is None:
                dimensionality = len(translation)
                if dimensionality == 2:
                    rotation = 0
                elif dimensionality == 3:
                    rotation = np.zeros(3)
                else:
                    raise ValueError(
                        'Parameters cannot be specified for dimension '
                        f'{dimensionality} transforms'
                    )
            else:
                if not np.isscalar(rotation) and len(rotation) != 3:
                    raise ValueError(
                        'Parameters cannot be specified for dimension '
                        f'{dimensionality} transforms'
                    )
            if translation is None:
                translation = (0,) * dimensionality

            if dimensionality == 2:
                self.params = np.array([
                    [math.cos(rotation), - math.sin(rotation), 0],
                    [math.sin(rotation),   math.cos(rotation), 0],
                    [                 0,                    0, 1]
                ])
            elif dimensionality == 3:
                self.params = np.eye(dimensionality + 1)
                self.params[:dimensionality, :dimensionality] = (
                    _euler_rotation_matrix(rotation)
                )
            self.params[0:dimensionality, dimensionality] = translation
        else:
            # default to an identity transform
            self.params = np.eye(dimensionality + 1)

    def estimate(self, src, dst):
        """Estimate the transformation from a set of corresponding points.

        You can determine the over-, well- and under-determined parameters
        with the total least-squares method.

        Number of source and destination coordinates must match.

        Parameters
        ----------
        src : (N, 2) array_like
            Source coordinates.
        dst : (N, 2) array_like
            Destination coordinates.

        Returns
        -------
        success : bool
            True, if model estimation succeeds.

        """
        self.params = _umeyama(src, dst, False)

        # _umeyama will return nan if the problem is not well-conditioned.
        return not np.any(np.isnan(self.params))

    @property
    def rotation(self):
        if self.dimensionality == 2:
            return math.atan2(self.params[1, 0], self.params[1, 1])
        elif self.dimensionality == 3:
            # Returning 3D Euler rotation matrix
            return self.params[:3, :3]
        else:
            raise NotImplementedError(
                'Rotation only implemented for 2D and 3D transforms.'
            )

    @property
    def translation(self):
        return self.params[0:self.dimensionality, self.dimensionality]


class SimilarityTransform(EuclideanTransform):
    """Similarity transformation.

    Has the following form in 2D::

        X = a0 * x - b0 * y + a1 =
          = s * x * cos(rotation) - s * y * sin(rotation) + a1

        Y = b0 * x + a0 * y + b1 =
          = s * x * sin(rotation) + s * y * cos(rotation) + b1

    where ``s`` is a scale factor and the homogeneous transformation matrix is::

        [[a0  -b0  a1]
         [b0  a0  b1]
         [0   0    1]]

    The similarity transformation extends the Euclidean transformation with a
    single scaling factor in addition to the rotation and translation
    parameters.

    Parameters
    ----------
    matrix : (dim+1, dim+1) array_like, optional
        Homogeneous transformation matrix.
    scale : float, optional
        Scale factor. Implemented only for 2D and 3D.
    rotation : float, optional
        Rotation angle, clockwise, as radians.
        Implemented only for 2D and 3D. For 3D, this is given in ZYX Euler
        angles.
    translation : (dim,) array_like, optional
        x, y[, z] translation parameters. Implemented only for 2D and 3D.

    Attributes
    ----------
    params : (dim+1, dim+1) array
        Homogeneous transformation matrix.

    """

    def __init__(self, matrix=None, scale=None, rotation=None,
                 translation=None, *, dimensionality=2):
        self.params = None
        params = any(param is not None
                     for param in (scale, rotation, translation))

        if params and matrix is not None:
            raise ValueError("You cannot specify the transformation matrix and"
                             " the implicit parameters at the same time.")
        elif matrix is not None:
            matrix = np.asarray(matrix)
            if matrix.ndim != 2 or matrix.shape[0] != matrix.shape[1]:
                raise ValueError("Invalid shape of transformation matrix.")
            else:
                self.params = matrix
                dimensionality = matrix.shape[0] - 1
        if params:
            if dimensionality not in (2, 3):
                raise ValueError('Parameters only supported for 2D and 3D.')
            matrix = np.eye(dimensionality + 1, dtype=float)
            if scale is None:
                scale = 1
            if rotation is None:
                rotation = 0 if dimensionality == 2 else (0, 0, 0)
            if translation is None:
                translation = (0,) * dimensionality
            if dimensionality == 2:
                ax = (0, 1)
                c, s = np.cos(rotation), np.sin(rotation)
                matrix[ax, ax] = c
                matrix[ax, ax[::-1]] = -s, s
            else:  # 3D rotation
                matrix[:3, :3] = _euler_rotation_matrix(rotation)

            matrix[:dimensionality, :dimensionality] *= scale
            matrix[:dimensionality, dimensionality] = translation
            self.params = matrix
        elif self.params is None:
            # default to an identity transform
            self.params = np.eye(dimensionality + 1)

    def estimate(self, src, dst):
        """Estimate the transformation from a set of corresponding points.

        You can determine the over-, well- and under-determined parameters
        with the total least-squares method.

        Number of source and destination coordinates must match.

        Parameters
        ----------
        src : (N, 2) array_like
            Source coordinates.
        dst : (N, 2) array_like
            Destination coordinates.

        Returns
        -------
        success : bool
            True, if model estimation succeeds.

        """

        self.params = _umeyama(src, dst, estimate_scale=True)

        # _umeyama will return nan if the problem is not well-conditioned.
        return not np.any(np.isnan(self.params))

    @property
    def scale(self):
        # det = scale**(# of dimensions), therefore scale = det**(1/ndim)
        if self.dimensionality == 2:
            return np.sqrt(np.linalg.det(self.params))
        elif self.dimensionality == 3:
            return np.cbrt(np.linalg.det(self.params))
        else:
            raise NotImplementedError(
                'Scale is only implemented for 2D and 3D.')


class PolynomialTransform(_GeometricTransform):
    """2D polynomial transformation.

    Has the following form::

        X = sum[j=0:order]( sum[i=0:j]( a_ji * x**(j - i) * y**i ))
        Y = sum[j=0:order]( sum[i=0:j]( b_ji * x**(j - i) * y**i ))

    Parameters
    ----------
    params : (2, N) array_like, optional
        Polynomial coefficients where `N * 2 = (order + 1) * (order + 2)`. So,
        a_ji is defined in `params[0, :]` and b_ji in `params[1, :]`.

    Attributes
    ----------
    params : (2, N) array
        Polynomial coefficients where `N * 2 = (order + 1) * (order + 2)`. So,
        a_ji is defined in `params[0, :]` and b_ji in `params[1, :]`.

    """

    def __init__(self, params=None, *, dimensionality=2):
        if dimensionality != 2:
            raise NotImplementedError(
                'Polynomial transforms are only implemented for 2D.'
            )
        if params is None:
            # default to transformation which preserves original coordinates
            params = np.array([[0, 1, 0], [0, 0, 1]])
        else:
            params = np.asarray(params)
        if params.shape[0] != 2:
            raise ValueError("invalid shape of transformation parameters")
        self.params = params

    def estimate(self, src, dst, order=2, weights=None):
        """Estimate the transformation from a set of corresponding points.

        You can determine the over-, well- and under-determined parameters
        with the total least-squares method.

        Number of source and destination coordinates must match.

        The transformation is defined as::

            X = sum[j=0:order]( sum[i=0:j]( a_ji * x**(j - i) * y**i ))
            Y = sum[j=0:order]( sum[i=0:j]( b_ji * x**(j - i) * y**i ))

        These equations can be transformed to the following form::

            0 = sum[j=0:order]( sum[i=0:j]( a_ji * x**(j - i) * y**i )) - X
            0 = sum[j=0:order]( sum[i=0:j]( b_ji * x**(j - i) * y**i )) - Y

        which exist for each set of corresponding points, so we have a set of
        N * 2 equations. The coefficients appear linearly so we can write
        A x = 0, where::

            A   = [[1 x y x**2 x*y y**2 ... 0 ...             0 -X]
                   [0 ...                 0 1 x y x**2 x*y y**2 -Y]
                    ...
                    ...
                  ]
            x.T = [a00 a10 a11 a20 a21 a22 ... ann
                   b00 b10 b11 b20 b21 b22 ... bnn c3]

        In case of total least-squares the solution of this homogeneous system
        of equations is the right singular vector of A which corresponds to the
        smallest singular value normed by the coefficient c3.

        Weights can be applied to each pair of corresponding points to
        indicate, particularly in an overdetermined system, if point pairs have
        higher or lower confidence or uncertainties associated with them. From
        the matrix treatment of least squares problems, these weight values are
        normalised, square-rooted, then built into a diagonal matrix, by which
        A is multiplied.

        Parameters
        ----------
        src : (N, 2) array_like
            Source coordinates.
        dst : (N, 2) array_like
            Destination coordinates.
        order : int, optional
            Polynomial order (number of coefficients is order + 1).
        weights : (N,) array_like, optional
            Relative weight values for each pair of points.

        Returns
        -------
        success : bool
            True, if model estimation succeeds.

        """
        src = np.asarray(src)
        dst = np.asarray(dst)
        xs = src[:, 0]
        ys = src[:, 1]
        xd = dst[:, 0]
        yd = dst[:, 1]
        rows = src.shape[0]

        # number of unknown polynomial coefficients
        order = safe_as_int(order)
        u = (order + 1) * (order + 2)

        A = np.zeros((rows * 2, u + 1))
        pidx = 0
        for j in range(order + 1):
            for i in range(j + 1):
                A[:rows, pidx] = xs ** (j - i) * ys ** i
                A[rows:, pidx + u // 2] = xs ** (j - i) * ys ** i
                pidx += 1

        A[:rows, -1] = xd
        A[rows:, -1] = yd

        # Get the vectors that correspond to singular values, also applying
        # the weighting if provided
        if weights is None:
            _, _, V = np.linalg.svd(A)
        else:
            weights = np.asarray(weights)
            W = np.diag(np.tile(np.sqrt(weights / np.max(weights)), 2))
            _, _, V = np.linalg.svd(W @ A)

        # solution is right singular vector that corresponds to smallest
        # singular value
        params = - V[-1, :-1] / V[-1, -1]

        self.params = params.reshape((2, u // 2))

        return True

    def __call__(self, coords):
        """Apply forward transformation.

        Parameters
        ----------
        coords : (N, 2) array_like
            source coordinates

        Returns
        -------
        coords : (N, 2) array
            Transformed coordinates.

        """
        coords = np.asarray(coords)
        x = coords[:, 0]
        y = coords[:, 1]
        u = len(self.params.ravel())
        # number of coefficients -> u = (order + 1) * (order + 2)
        order = int((- 3 + math.sqrt(9 - 4 * (2 - u))) / 2)
        dst = np.zeros(coords.shape)

        pidx = 0
        for j in range(order + 1):
            for i in range(j + 1):
                dst[:, 0] += self.params[0, pidx] * x ** (j - i) * y ** i
                dst[:, 1] += self.params[1, pidx] * x ** (j - i) * y ** i
                pidx += 1

        return dst

    @property
    def inverse(self):
        raise NotImplementedError(
            'There is no explicit way to do the inverse polynomial '
            'transformation. Instead, estimate the inverse transformation '
            'parameters by exchanging source and destination coordinates,'
            'then apply the forward transformation.')


TRANSFORMS = {
    'euclidean': EuclideanTransform,
    'similarity': SimilarityTransform,
    'affine': AffineTransform,
    'piecewise-affine': PiecewiseAffineTransform,
    'projective': ProjectiveTransform,
    'fundamental': FundamentalMatrixTransform,
    'essential': EssentialMatrixTransform,
    'polynomial': PolynomialTransform,
}


def estimate_transform(ttype, src, dst, *args, **kwargs):
    """Estimate 2D geometric transformation parameters.

    You can determine the over-, well- and under-determined parameters
    with the total least-squares method.

    Number of source and destination coordinates must match.

    Parameters
    ----------
    ttype : {'euclidean', similarity', 'affine', 'piecewise-affine', \
             'projective', 'polynomial'}
        Type of transform.
    kwargs : array_like or int
        Function parameters (src, dst, n, angle)::

            NAME / TTYPE        FUNCTION PARAMETERS
            'euclidean'         `src, `dst`
            'similarity'        `src, `dst`
            'affine'            `src, `dst`
            'piecewise-affine'  `src, `dst`
            'projective'        `src, `dst`
            'polynomial'        `src, `dst`, `order` (polynomial order,
                                                      default order is 2)

        Also see examples below.

    Returns
    -------
    tform : :class:`_GeometricTransform`
        Transform object containing the transformation parameters and providing
        access to forward and inverse transformation functions.

    Examples
    --------
    >>> import numpy as np
    >>> import skimage as ski

    >>> # estimate transformation parameters
    >>> src = np.array([0, 0, 10, 10]).reshape((2, 2))
    >>> dst = np.array([12, 14, 1, -20]).reshape((2, 2))

    >>> tform = ski.transform.estimate_transform('similarity', src, dst)

    >>> np.allclose(tform.inverse(tform(src)), src)
    True

    >>> # warp image using the estimated transformation
    >>> image = ski.data.camera()

    >>> ski.transform.warp(image, inverse_map=tform.inverse) # doctest: +SKIP

    >>> # create transformation with explicit parameters
    >>> tform2 = ski.transform.SimilarityTransform(scale=1.1, rotation=1,
    ...     translation=(10, 20))

    >>> # unite transformations, applied in order from left to right
    >>> tform3 = tform + tform2
    >>> np.allclose(tform3(src), tform2(tform(src)))
    True

    """
    ttype = ttype.lower()
    if ttype not in TRANSFORMS:
        raise ValueError(f'the transformation type \'{ttype}\' is not implemented')

    tform = TRANSFORMS[ttype](dimensionality=src.shape[1])
    tform.estimate(src, dst, *args, **kwargs)

    return tform


def matrix_transform(coords, matrix):
    """Apply 2D matrix transform.

    Parameters
    ----------
    coords : (N, 2) array_like
        x, y coordinates to transform
    matrix : (3, 3) array_like
        Homogeneous transformation matrix.

    Returns
    -------
    coords : (N, 2) array
        Transformed coordinates.

    """
    return ProjectiveTransform(matrix)(coords)
