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transform

Functions to computing and working with transformations between point clouds

apply_transform(src, transform, shift)

Apply a transformation to a set of points.

Parameters:

Name Type Description Default
src NDArray[floating]

The points to transform. Should have shape (npoints, ndim).

 required
transform NDArray[floating]

The transformation matrix. Should have shape (ndim, ndim).

 required
shift NDArray[floating]

The shift to apply after the affine tranrform. Should have shape (ndim,).

 required

Returns:

Name Type Description
transformed NDArray[floating]

The transformed points. Has the same shape as src.

Raises:

Type Description
ValueError

If src is not a 2d array. If one of src's axis is not of size ndim. If affine and shift have inconsistent shapes.
Source code in megham/transform.py 
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def apply_transform(
    src: NDArray[np.floating],
    transform: NDArray[np.floating],
    shift: NDArray[np.floating],
) -> NDArray[np.floating]:
    """
    Apply a transformation to a set of points.

    Parameters
    ----------
    src : NDArray[np.floating]
        The points to transform.
        Should have shape  (npoints, ndim).
    transform: NDArray[np.floating]
        The transformation matrix.
        Should have shape (ndim, ndim).
    shift : NDArray[np.floating]
        The shift to apply after the affine tranrform.
        Should have shape (ndim,).

    Returns
    -------
    transformed : NDArray[np.floating]
        The transformed points.
        Has the same shape as src.

    Raises
    ------
    ValueError
        If src is not a 2d array.
        If one of src's axis is not of size ndim.
        If affine and shift have inconsistent shapes.
    """
    ndim = len(shift)
    if transform.shape != (ndim, ndim):
        raise ValueError(
            f"From shift we assume ndim={ndim} but transform has shape {transform.shape}"
        )
    src_shape = np.array(src.shape)
    if len(src_shape) != 2:
        raise ValueError(f"src should be a 2d array, not {len(src.shape)}d")

    transformed = src @ transform + shift
    return transformed

compose_transform(transform_1, shift_1, transform_2, shift_2)

Combine transformations to get one that is equivalent to: dst = (src@transform_1 + shift)@transform_2 + shift_2

Parameters:

Name Type Description Default
transform_1 NDArray[floating]

The first transform (affine or rotation matrix). Should have shape (ndim, ndim).

 required
shift_1 NDArray[floating]

The first shift. Should have shape (ndim,).

 required
transform_2 NDArray[floating]

The second transform (affine or rotation matrix). Should have shape (ndim, ndim).

 required
shift_2 NDArray[floating]

The second shift. Should have shape (ndim,).

 required

Returns:

Name Type Description
transform NDArray[floating]

The composed transform. Has shape (ndim, ndim).
shift NDArray[np.floating].

The composed shift. Has shape (ndim,).
Source code in megham/transform.py 
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def compose_transform(
    transform_1: NDArray[np.floating],
    shift_1: NDArray[np.floating],
    transform_2: NDArray[np.floating],
    shift_2: NDArray[np.floating],
) -> tuple[NDArray[np.floating], NDArray[np.floating]]:
    """
    Combine transformations to get one that is equivalent to:
    dst = (src@transform_1 + shift)@transform_2 + shift_2

    Parameters
    ----------
    transform_1 : NDArray[np.floating]
        The first transform (affine or rotation matrix).
        Should have shape (ndim, ndim).
    shift_1 : NDArray[np.floating]
        The first shift.
        Should have shape (ndim,).
    transform_2 : NDArray[np.floating]
        The second transform (affine or rotation matrix).
        Should have shape (ndim, ndim).
    shift_2 : NDArray[np.floating]
        The second shift.
        Should have shape (ndim,).

    Returns
    -------
    transform : NDArray[np.floating]
        The composed transform.
        Has shape (ndim, ndim).
    shift : NDArray[np.floating].
        The composed shift.
        Has shape (ndim,).
    """
    transform = transform_1 @ transform_2
    shift = shift_1 @ transform_2 + shift_2

    return transform, shift

decompose_affine(affine)

Decompose an affine transformation into its components. This decomposetion treats the affine matrix as: rotation * shear * scale.

Parameters:

Name Type Description Default
affine NDArray[floating]

The (ndim, ndim) affine transformation matrix.

 required

Returns:

Name Type Description
scale NDArray[floating]

The (ndim,) array of scale parameters.
shear NDArray[floating]

The (ndim*(ndim - 1)/2,) array of shear parameters.
rot NDArray[floating]

The (ndim, ndim) rotation matrix. If ndim is 2 or 3 then decompose_rotation can be used to get euler angles.

Raises:

Type Description
ValueError

If affine is not ndim by ndim.
Source code in megham/transform.py 
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def decompose_affine(
    affine: NDArray[np.floating],
) -> tuple[NDArray[np.floating], NDArray[np.floating], NDArray[np.floating]]:
    """
    Decompose an affine transformation into its components.
    This decomposetion treats the affine matrix as: rotation * shear * scale.

    Parameters
    ----------
    affine : NDArray[np.floating]
        The (ndim, ndim) affine transformation matrix.

    Returns
    -------
    scale : NDArray[np.floating]
        The (ndim,) array of scale parameters.
    shear : NDArray[np.floating]
        The (ndim*(ndim - 1)/2,) array of shear parameters.
    rot: NDArray[np.floating]
        The (ndim, ndim) rotation matrix.
        If ndim is 2 or 3 then decompose_rotation can be used to get euler angles.

    Raises
    ------
    ValueError
        If affine is not ndim by ndim.
    """
    ndim = len(affine)
    if affine.shape != (ndim, ndim):
        raise ValueError("Affine matrix should be ndim by ndim")
    # Use the fact that rotation matrix times its transpose is the identity
    no_rot = affine.T @ affine
    # Decompose to get a matrix with just scale and shear
    no_rot = la.cholesky(no_rot).T

    scale = np.diag(no_rot)
    shear = (no_rot / scale[:, None])[np.triu_indices(len(no_rot), k=1)]
    rot = affine @ la.inv(no_rot)

    return scale, shear, rot

decompose_rotation(rotation)

Decompose a rotation matrix into its xyz rotation angles. This currently won't work on anything higher than 3 dimensions.

Parameters:

Name Type Description Default
rotation NDArray[floating]

The (ndim, ndim) rotation matrix.

 required

Returns:

Name Type Description
angles NDArray[floating]

The rotation angles in radians. If the input is 3d then this has 3 angles in xyz order, if 2d it just has one.

Raises:

Type Description
ValueError

If affine is not ndim by ndim. If ndim is not 2 or 3.
Source code in megham/transform.py 
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def decompose_rotation(rotation: NDArray[np.floating]) -> NDArray[np.floating]:
    """
    Decompose a rotation matrix into its xyz rotation angles.
    This currently won't work on anything higher than 3 dimensions.

    Parameters
    ----------
    rotation : NDArray[np.floating]
        The (ndim, ndim) rotation matrix.

    Returns
    -------
    angles : NDArray[np.floating]
        The rotation angles in radians.
        If the input is 3d then this has 3 angles in xyz order,
        if 2d it just has one.

    Raises
    ------
    ValueError
        If affine is not ndim by ndim.
        If ndim is not 2 or 3.
    """
    ndim = len(rotation)
    if ndim > 3:
        raise ValueError("No support for rotations in more than 3 dimensions")
    if ndim < 2:
        raise ValueError("Rotations with less than 2 dimensions don't make sense")
    if rotation.shape != (ndim, ndim):
        raise ValueError("Rotation matrix should be ndim by ndim")
    _rotation = np.eye(3)
    _rotation[:ndim, :ndim] = rotation
    angles = R.from_matrix(_rotation).as_euler("xyz")

    if ndim == 2:
        angles = angles[-1:]
    return angles

decompose_transform(transform, shift, transform_1, shift_1)

Decompose transformations to get one with the other removed. This is solving for transform_2 and shift_2 in the following equation: dst = src@transform + shift = (src@transform_1 + shift)@transform_2 + shift_2

Parameters:

Name Type Description Default
transform NDArray[floating]

The composed transform (affine or rotation matrix). Should have shape (ndim, ndim).

 required
shift NDArray[floating]

The composed shift. Should have shape (ndim,)

 required
transform_1 NDArray[floating]

The transform (affine or rotation matrix) to remove. Should have shape (ndim, ndim).

 required
shift_1 NDArray[floating]

The shift to remove. Should have shape (ndim,)

 required

Returns:

Name Type Description
transform_2 NDArray[floating]

The transform with the first transform removed. Has shape (ndim, ndim).
shift_2 NDArray[np.floating].

The shift with the first transform removed. Has shape (ndim,).
Source code in megham/transform.py 
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def decompose_transform(
    transform: NDArray[np.floating],
    shift: NDArray[np.floating],
    transform_1: NDArray[np.floating],
    shift_1: NDArray[np.floating],
) -> tuple[NDArray[np.floating], NDArray[np.floating]]:
    """
    Decompose transformations to get one with the other removed.
    This is solving for transform_2 and shift_2 in the following equation:
    dst = src@transform + shift = (src@transform_1 + shift)@transform_2 + shift_2

    Parameters
    ----------
    transform : NDArray[np.floating]
        The composed transform (affine or rotation matrix).
        Should have shape (ndim, ndim).
    shift : NDArray[np.floating]
        The composed shift.
        Should have shape (ndim,)
    transform_1 : NDArray[np.floating]
        The transform (affine or rotation matrix) to remove.
        Should have shape (ndim, ndim).
    shift_1 : NDArray[np.floating]
        The shift to remove.
        Should have shape (ndim,)

    Returns
    -------
    transform_2 : NDArray[np.floating]
        The transform with the first transform removed.
        Has shape (ndim, ndim).
    shift_2 : NDArray[np.floating].
        The shift with the first transform removed.
        Has shape (ndim,).
    """
    transform_2 = np.linalg.inv(transform_1) @ transform
    shift_2 = shift - shift_1 @ transform_2

    return transform_2, shift_2

get_affine(src, dst, weights=None, center_dst=True, force_svd=False, **kwargs)

Get affine transformation between two point clouds. It is assumed that the point clouds have the same registration, ie. src[i] corresponds to dst[i].

Transformation is dst = src@affine + shift.

Parameters:

Name Type Description Default
src NDArray[floating]

A (npoints, ndim) array of source points.

 required
dst NDArray[floating]

A (npoints, ndim) array of destination points.

 required
weights Optional[NDArray[floating]]

(npoints,) array of weights to use. If provided a weighted least squares is done instead of an SVD.

 None
center_dst bool

If True, dst will be recentered at the origin before computing transformation. This is done with get_shift, but weights will not be used if provided.

 True
force_svd bool

If True the SVD is used even if there are a small number of points or weights are present.

 False
**kwargs

Arguments to pass to get_shift.

 {}

Returns:

Name Type Description
affine NDArray[floating]

The (ndim, ndim) transformation matrix.
shift NDArray[floating]

The (ndim,) shift to apply after transformation.

Raises:

Type Description
ValueError

If the input point clouds have different shapes. If the input point clouds don't have enough points.
Source code in megham/transform.py 
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def get_affine(
    src: NDArray[np.floating],
    dst: NDArray[np.floating],
    weights: Optional[NDArray[np.floating]] = None,
    center_dst: bool = True,
    force_svd: bool = False,
    **kwargs,
) -> tuple[NDArray[np.floating], NDArray[np.floating]]:
    """
    Get affine transformation between two point clouds.
    It is assumed that the point clouds have the same registration,
    ie. src[i] corresponds to dst[i].

    Transformation is dst = src@affine + shift.

    Parameters
    ----------
    src : NDArray[np.floating]
        A (npoints, ndim) array of source points.
    dst : NDArray[np.floating]
        A (npoints, ndim) array of destination points.
    weights : Optional[NDArray[np.floating]], default: None
        (npoints,) array of weights to use.
        If provided a weighted least squares is done instead of an SVD.
    center_dst : bool, default: True
        If True, dst will be recentered at the origin before computing transformation.
        This is done with get_shift, but weights will not be used if provided.
    force_svd : bool, default: False
        If True the SVD is used even if there are a small number of points
        or weights are present.
    **kwargs
        Arguments to pass to get_shift.

    Returns
    -------
    affine : NDArray[np.floating]
        The (ndim, ndim) transformation matrix.
    shift : NDArray[np.floating]
        The (ndim,) shift to apply after transformation.

    Raises
    ------
    ValueError
        If the input point clouds have different shapes.
        If the input point clouds don't have enough points.
    """
    if src.shape != dst.shape:
        raise ValueError("Input point clouds should have the same shape")

    msk = np.isfinite(src).all(axis=1) * np.isfinite(dst).all(axis=1)
    if np.sum(msk) < src.shape[1] + 1:
        raise ValueError("Not enough finite points to compute transformation")

    # When we have a small number of points lstsq is better than SVD
    # Condition is a bit arbitrary for now
    if force_svd is False and weights is None and np.sum(msk) < 50 * src.shape[1]:
        weights = np.ones(len(src))

    _dst = dst[msk].copy()
    if center_dst:
        _dst += get_shift(_dst, np.zeros(1), **kwargs)
    _src = src[msk].copy()
    init_shift = get_shift(_src, _dst, weights=weights, **kwargs)

    if force_svd or weights is None:
        M = np.vstack((_src.T, (_dst - init_shift).T)).T
        *_, vh = la.svd(M)
        vh_splits = [
            quad
            for half in np.split(vh.T, 2, axis=0)
            for quad in np.split(half, 2, axis=1)
        ]
        affine = np.dot(vh_splits[2], la.pinv(vh_splits[0])).T
        shift = init_shift
    else:
        rt_weight = np.sqrt(weights[msk])[..., None]
        wsrc = rt_weight * _src
        wdst = rt_weight * (_dst - init_shift)
        x, *_ = la.lstsq(
            np.column_stack((wsrc, np.ones(len(wsrc)))), wdst, check_finite=False
        )
        affine = x[:-1]
        shift = x[-1] + init_shift

    transformed = src[msk] @ affine + shift
    shift += get_shift(transformed, dst[msk], **kwargs)

    return affine, shift

get_affine_two_stage(src, dst, weights)

Get affine transformation between two point clouds with a two stage solver. This first uses the SVD to do an intitial alignment and then uses weighted least squares to compute a correction on top of that.

Transformation is dst = affine@src + shift

Parameters:

Name Type Description Default
src NDArray[floating]

A (npoints, ndim) array of source points.

 required
dst NDArray[floating]

A (npoints, ndim) array of destination points.

 required
weights NDArray[floating]

(npoints,) array of weights to use. If provided a weighted least squares is done instead of an SVD.

 required

Returns:

Name Type Description
affine NDArray[floating]

The (ndim, ndim) transformation matrix.
shift NDArray[floating]

The (ndim,) shift to apply after transformation.
Source code in megham/transform.py 
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def get_affine_two_stage(
    src: NDArray[np.floating],
    dst: NDArray[np.floating],
    weights: NDArray[np.floating],
) -> tuple[NDArray[np.floating], NDArray[np.floating]]:
    """
    Get affine transformation between two point clouds with a two stage solver.
    This first uses the SVD to do an intitial alignment and
    then uses weighted least squares to compute a correction on top of that.

    Transformation is dst = affine@src + shift

    Parameters
    ----------
    src : NDArray[np.floating]
        A (npoints, ndim) array of source points.
    dst : NDArray[np.floating]
        A (npoints, ndim) array of destination points.
    weights : NDArray[np.floating]
        (npoints,) array of weights to use.
        If provided a weighted least squares is done instead of an SVD.

    Returns
    -------
    affine : NDArray[np.floating]
        The (ndim, ndim) transformation matrix.
    shift : NDArray[np.floating]
        The (ndim,) shift to apply after transformation.
    """
    # Do an initial alignment without weights
    affine_0, shift_0 = get_affine(src, dst, force_svd=True)
    init_align = apply_transform(src, affine_0, shift_0)
    # Now compute the actual transform
    affine, shift = get_affine(init_align, dst, weights)
    # Compose the transforms
    affine, shift = compose_transform(affine_0, shift_0, affine, shift)
    # Now one last shift correction
    transformed = apply_transform(src, affine, shift)
    shift += get_shift(transformed, dst, "mean", weights)

    return affine, shift

get_rigid(src, dst, center_dst=True, **kwargs)

Get rigid transformation between two point clouds. It is assumed that the point clouds have the same registration, ie. src[i] corresponds to dst[i].

Transformation is dst = src@rot + shift.

Parameters:

Name Type Description Default
src NDArray[floating]

A (npoints, ndim) array of source points.

 required
dst NDArray[floating]

A (npoints, ndim) array of destination points.

 required
center_dst bool

If True, dst will be recentered at the origin before computing transformation. This is done with get_shift, but weights will not be used if provided.

 True
**kwargs

Arguments to pass to get_shift.

 {}

Returns:

Name Type Description
rotation NDArray[floating]

The (ndim, ndim) rotation matrix.
shift NDArray[floating]

The (ndim,) shift to apply after transformation. If point are in col basis will be returned as a column vector.

Raises:

Type Description
ValueError

If the input point clouds have different shapes. If the input point clouds don't have enough points.
Source code in megham/transform.py 
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def get_rigid(
    src: NDArray[np.floating],
    dst: NDArray[np.floating],
    center_dst: bool = True,
    **kwargs,
) -> tuple[NDArray[np.floating], NDArray[np.floating]]:
    """
    Get rigid transformation between two point clouds.
    It is assumed that the point clouds have the same registration,
    ie. src[i] corresponds to dst[i].

    Transformation is dst = src@rot + shift.

    Parameters
    ----------
    src : NDArray[np.floating]
        A (npoints, ndim) array of source points.
    dst : NDArray[np.floating]
        A (npoints, ndim) array of destination points.
    center_dst : bool, default: True
        If True, dst will be recentered at the origin before computing transformation.
        This is done with get_shift, but weights will not be used if provided.
    **kwargs
        Arguments to pass to get_shift.

    Returns
    -------
    rotation : NDArray[np.floating]
        The (ndim, ndim) rotation matrix.
    shift : NDArray[np.floating]
        The (ndim,) shift to apply after transformation.
        If point are in col basis will be returned as a column vector.

    Raises
    ------
    ValueError
        If the input point clouds have different shapes.
        If the input point clouds don't have enough points.
    """
    if src.shape != dst.shape:
        raise ValueError("Input point clouds should have the same shape")

    msk = np.isfinite(src).all(axis=1) * np.isfinite(dst).all(axis=1)
    ndim = src.shape[1]
    if np.sum(msk) < ndim * (ndim - 1) / 2:
        raise ValueError("Not enough finite points to compute transformation")

    _dst = dst[msk].copy()
    if center_dst:
        _kwargs = kwargs.copy()
        _kwargs.update({"weights": None})
        _dst += get_shift(_dst, np.zeros(1), **_kwargs)
    _src = src[msk].copy()
    _src += get_shift(_src, _dst, **kwargs)

    M = _src.T @ (_dst)
    u, _, vh = la.svd(M)
    v = vh.T
    uT = u.T

    corr = np.eye(ndim)
    corr[-1, -1] = la.det((v) @ (uT))
    rot = v @ corr @ uT
    rot = rot.T

    transformed = src[msk] @ rot
    shift = get_shift(transformed, dst[msk], **kwargs)

    return rot, shift

get_shift(src, dst, method='median', weights=None)

Get shift between two point clouds. Shift can be applied as dst = src + shift.

Parameters:

Name Type Description Default
src NDArray[floating]

A (ndim, npoints) array of source points.

 required
dst NDArray[floating]

Nominally a (ndim, npoints) array of destination points, but really any array broadcastable with src is accepted. Some useful options are: * np.zeros(1) to align with the origin * A (ndim,) array to align with an arbitrary point

 required
method str

Method to use to align points. Current accepted values are: 'median' and 'mean'

 'median'
weights Optional[NDArray[floating]]

(npoints,) array of weights to use. If provided and method is 'mean' then a weighted average is used. If method is median this is not currently used.

 None

Returns:

Name Type Description
shift NDArray[floating]

The (ndim,) shift to apply after transformation.

Raises:

Type Description
ValueError

If an invalid method is provided
Source code in megham/transform.py 
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def get_shift(
    src: NDArray[np.floating],
    dst: NDArray[np.floating],
    method: str = "median",
    weights: Optional[NDArray[np.floating]] = None,
) -> NDArray[np.floating]:
    """
    Get shift between two point clouds.
    Shift can be applied as dst = src + shift.

    Parameters
    -----------
    src : NDArray[np.floating]
        A (ndim, npoints) array of source points.
    dst : NDArray[np.floating]
        Nominally a (ndim, npoints) array of destination points,
        but really any array broadcastable with src is accepted.
        Some useful options are:
        * np.zeros(1) to align with the origin
        * A (ndim,) array to align with an arbitrary point
    method : str, default: 'median'
        Method to use to align points.
        Current accepted values are: 'median' and 'mean'
    weights : Optional[NDArray[np.floating]], default: None
        (npoints,) array of weights to use.
        If provided and method is 'mean' then a weighted average is used.
        If method is median this is not currently used.

    Returns
    -------
    shift : NDArray[np.floating]
        The (ndim,) shift to apply after transformation.

    Raises
    ------
    ValueError
        If an invalid method is provided
    """
    if method not in ["median", "mean"]:
        raise ValueError(f"Invalid method: {method}")

    shift = np.zeros(src.shape[1])
    if method == "median":
        shift = np.median(dst - src, axis=0)
    elif method == "mean":
        if weights is None:
            shift = np.mean(dst - src, axis=0)
        else:
            wdiff = weights[..., None] * (dst - src)
            shift = np.nansum(wdiff, axis=0) / np.nansum(weights)

    return shift

invert_transform(transform, shift)

Invert a transformation. If the inverted transformation is applied to a point cloud that has already been transformed, you will recover the original point cloud.

Parameters:

Name Type Description Default
transform NDArray[floating]

The transform (affine or rotation matrix) the invert. Should have shape (ndim, ndim).

 required
shift NDArray[floating]

The shift to invert. Should have shape (ndim,)

 required

Returns:

Name Type Description
transform_inv NDArray[floating]

The inverted transformation matrix.
shift_inv NDArray[floating]

The inverted shift vector.
Source code in megham/transform.py 
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def invert_transform(
    transform: NDArray[np.floating], shift: NDArray[np.floating]
) -> tuple[NDArray[np.floating], NDArray[np.floating]]:
    """
    Invert a transformation.
    If the inverted transformation is applied to a point cloud that has already been
    transformed, you will recover the original point cloud.

    Parameters
    ----------
    transform : NDArray[np.floating]
        The transform (affine or rotation matrix) the invert.
        Should have shape (ndim, ndim).
    shift : NDArray[np.floating]
        The shift to invert.
        Should have shape (ndim,)

    Returns
    -------
    transform_inv : NDArray[np.floating]
        The inverted transformation matrix.
    shift_inv : NDArray[np.floating]
        The inverted shift vector.
    """
    transform_inv = np.linalg.inv(transform)
    shift_inv = (-1 * shift) @ transform_inv

    return transform_inv, shift_inv