Rotation

class menpo.transform.homogeneous.rotation.Rotation(rotation_matrix, skip_checks=False)

Bases: DiscreteAffine, Similarity

Abstract n_dims rotation transform.

Parameters:

rotation_matrix : (D, D) ndarray

A valid, square rotation matrix

apply(x, **kwargs)

Applies this transform to x.

If x is Transformable, x will be handed this transform object to transform itself non-destructively (a transformed copy of the object will be returned).

If not, x is assumed to be an ndarray. The transformation will be non-destructive, returning the transformed version.

Any kwargs will be passed to the specific transform _apply() method.

Parameters:

x : Transformable or (n_points, n_dims) ndarray

The array or object to be transformed.

kwargs : dict

Passed through to _apply().

Returns:

transformed : type(x)

The transformed object or array

apply_inplace(x, **kwargs)

Applies this transform to a Transformable x destructively.

Any kwargs will be passed to the specific transform _apply() method.

Parameters:

x : Transformable

The Transformable object to be transformed.

kwargs : dict

Passed through to _apply().

Returns:

transformed : type(x)

The transformed object

as_vector(**kwargs)

Returns a flattened representation of the object as a single vector.

Returns:

vector : (N,) ndarray

The core representation of the object, flattened into a single vector. Note that this is always a view back on to the original object, but is not writable.

axis_and_angle_of_rotation()

Abstract method for computing the axis and angle of rotation.

Returns:

axis : (D,) ndarray

The unit vector representing the axis of rotation

angle_of_rotation : double

The angle in radians of the rotation about the axis. The angle is signed in a right handed sense.

compose_after(transform)

A Transform that represents this transform composed after the given transform:

c = a.compose_after(b)
c.apply(p) == a.apply(b.apply(p))

a and b are left unchanged.

This corresponds to the usual mathematical formalism for the compose operator, o.

An attempt is made to perform native composition, but will fall back to a TransformChain as a last resort. See composes_with for a description of how the mode of composition is decided.

Parameters:

transform : Transform

Transform to be applied before self

Returns:

transform : Transform or TransformChain

If the composition was native, a single new Transform will be returned. If not, a TransformChain is returned instead.

compose_after_inplace(transform)

Update self so that it represents this transform composed after the given transform:

a_orig = deepcopy(a)
a.compose_after_inplace(b)
a.apply(p) == a_orig.apply(b.apply(p))

a is permanently altered to be the result of the composition. b is left unchanged.

Parameters:

transform : composes_inplace_with

Transform to be applied before self

Raises:

ValueError :

If transform isn’t an instance of composes_inplace_with

compose_before(transform)

A Transform that represents this transform composed before the given transform:

c = a.compose_before(b)
c.apply(p) == b.apply(a.apply(p))

a and b are left unchanged.

An attempt is made to perform native composition, but will fall back to a TransformChain as a last resort. See composes_with for a description of how the mode of composition is decided.

Parameters:

transform : Transform

Transform to be applied after self

Returns:

transform : Transform or TransformChain

If the composition was native, a single new Transform will be returned. If not, a TransformChain is returned instead.

compose_before_inplace(transform)

Update self so that it represents this transform composed before the given transform:

a_orig = deepcopy(a)
a.compose_before_inplace(b)
a.apply(p) == b.apply(a_orig.apply(p))

a is permanently altered to be the result of the composition. b is left unchanged.

Parameters:

transform : composes_inplace_with

Transform to be applied after self

Raises:

ValueError :

If transform isn’t an instance of composes_inplace_with

copy()

An efficient copy of this Homogeneous family transform (i.e. one with the same homogeneous matrix).

If you need all state to be perfectly replicated, consider using deepcopy() instead.

Returns:

transform : type(self)

A copy fo the transform with the same h_matrix.

d_dx(points)

The first order derivative of this Affine transform wrt spatial changes evaluated at points.

The Jacobian for a given point (for 2D) is of the form:

Jx = [(1 + a),     -b  ]
Jy = [   b,     (1 + a)]
J =  [Jx, Jy] = [[(1 + a), -b], [b, (1 + a)]]

where a and b come from:

W(x;p) = [1 + a -b tx] [x]

[b 1 + a ty] [y]

[1]

Hence it is simply the linear component of the transform.

Parameters:

points: ndarray shape (n_points, n_dims) :

The spatial points at which the derivative should be evaluated.

Returns:

d_dx: (1, n_dims, n_dims) ndarray :

The jacobian wrt spatial changes.

d_dx[0, j, k] is the scalar differential change that the j’th dimension of the i’th point experiences due to a first order change in the k’th dimension.

Note that because the jacobian is constant across space the first axis is length 1 to allow for broadcasting.

decompose()

A DiscreteAffine is already maximally decomposed - return a copy of self in a list.

Returns:

transform : DiscreteAffine

Deep copy of self.

from_vector(vector)

Build a new instance of the object from it’s vectorized state.

self is used to fill out the missing state required to rebuild a full object from it’s standardized flattened state. This is the default implementation, which is which is a deepcopy of the object followed by a call to from_vector_inplace(). This method can be overridden for a performance benefit if desired.

Parameters:

vector : (n_parameters,) ndarray

Flattened representation of the object.

Returns:

transform : type(self)

An new instance of this class.

from_vector_inplace(p)

Returns an instance of the transform from the given parameters, expected to be in Fortran ordering.

Supports rebuilding from 2D parameter sets.

2D Rotation: 1 parameter:

[theta]

Parameters:

p : (1,) ndarray

The array of parameters.

Returns:

transform : Rotation2D

The transform initialised to the given parameters.

pseudoinverse_vector(vector)

The vectorized pseudoinverse of a provided vector instance. Syntactic sugar for:

self.from_vector(vector).pseudoinverse.as_vector()

Can be much faster than the explict call as object creation can be entirely avoided in some cases.

Parameters:

vector : (n_parameters,) ndarray

A vectorized version of self

Returns:

pseudoinverse_vector : (n_parameters,) ndarray

The pseudoinverse of the vector provided

set_h_matrix(value, copy=True, skip_checks=False)

Updates h_matrix, optionally performing sanity checks.

Note that it won’t always be possible to manually specify the h_matrix through this method, specifically if changing the h_matrix could change the nature of the transform. See h_matrix_is_mutable for how you can discover if the h_matrix is allowed to be set for a given class.

Parameters:

value : ndarray

The new homogeneous matrix to set

copy : bool, optional

If False do not copy the h_matrix. Useful for performance.

skip_checks : bool, optional

If True skip checking. Useful for performance.

Raises:

NotImplementedError :

If h_matrix_is_mutable returns False.

composes_with

Any Homogeneous can compose with any other Homogeneous.

linear_component

The linear component of this affine transform.

Type:(n_dims, n_dims) ndarray

pseudoinverse

The inverse rotation matrix.

Type:(D, D) ndarray

rotation_matrix

The rotation matrix.

Type:(D, D) ndarray

translation_component

The translation component of this affine transform.

Type:(n_dims,) ndarray