kornia.geometry.quaternion¶

class kornia.geometry.quaternion.Quaternion(data)[source]¶

Base class to represent a Quaternion.

A quaternion is a four dimensional vector representation of a rotation transformation in 3d. See more: https://en.wikipedia.org/wiki/Quaternion

The general definition of a quaternion is given by:

\[Q = a + b \cdot \mathbf{i} + c \cdot \mathbf{j} + d \cdot \mathbf{k}\]

Thus, we represent a rotation quaternion as a contiguous torch.Tensor structure to perform rigid bodies transformations:

\[Q = \begin{bmatrix} q_w & q_x & q_y & q_z \end{bmatrix}\]

Example

>>> q = Quaternion.identity(batch_size=4)
>>> q.data
tensor([[1., 0., 0., 0.],
        [1., 0., 0., 0.],
        [1., 0., 0., 0.],
        [1., 0., 0., 0.]])
>>> q.real
tensor([1., 1., 1., 1.])
>>> q.vec
tensor([[0., 0., 0.],
        [0., 0., 0.],
        [0., 0., 0.],
        [0., 0., 0.]])

__add__(right)[source]¶

Add a given quaternion, scalar, or torch.Tensor.

Parameters:: right (Union[Quaternion, Tensor, float]) – the quaternion, scalar, or torch.Tensor to add.
Return type:: Quaternion

Example

>>> q1 = Quaternion.identity()
>>> q2 = Quaternion(torch.tensor([2., 0., 1., 1.]))
>>> q3 = q1 + q2
>>> q3.data
tensor([3., 0., 1., 1.])

__init__(data)[source]¶

Construct a quaternion from torch.Tensor or parameter data.

Parameters:: data (Union[Tensor, Parameter]) – torch.Tensor or parameter containing the quaternion data with the shape of \((B, 4)\).

Example

>>> # Create with torch.tensor(no gradients tracked by default)
>>> data = torch.tensor([1., 0., 0., 0.])
>>> q1 = Quaternion(data)
>>> # Create with parameter (gradients tracked)
>>> param_data = torch.nn.Parameter(torch.tensor([1., 0., 0., 0.]))
>>> q2 = Quaternion(param_data)

__neg__()[source]¶

Inverts the sign of the quaternion data.

Return type:: Quaternion

Example

>>> q = Quaternion.identity()
>>> -q.data
tensor([-1., -0., -0., -0.])

__pow__(t)[source]¶

Return the power of a quaternion raised to exponent t.

Parameters:: t (float) – raised exponent.
Return type:: Quaternion

Example

>>> q = Quaternion(torch.tensor([1., .5, 0., 0.]))
>>> q_pow = q**2

__radd__(left)[source]¶

Right addition (left + self) where left is a scalar or torch.Tensor.

Return type:: Quaternion

__rdiv__(left)[source]¶

Right division (left / self) where left is a scalar or torch.Tensor.

Return type:: Quaternion

__repr__()[source]¶

Return repr(self).

Return type:: str

__rmul__(left)[source]¶

Right multiplication (left * self) where left is a scalar or torch.Tensor.

Return type:: Quaternion

__rsub__(left)[source]¶

Right subtraction (left - self) where left is a scalar or torch.Tensor.

Return type:: Quaternion

__rtruediv__(left)[source]¶

Right division (left / self) where left is a scalar or torch.Tensor.

Return type:: Quaternion

__sub__(right)[source]¶

Subtract a given quaternion, scalar, or torch.Tensor.

Parameters:: right (Union[Quaternion, Tensor, float]) – the quaternion, scalar, or torch.Tensor to subtract.
Return type:: Quaternion

Example

>>> q1 = Quaternion(torch.tensor([2., 0., 1., 1.]))
>>> q2 = Quaternion.identity()
>>> q3 = q1 - q2
>>> q3.data
tensor([1., 0., 1., 1.])

property coeffs: Tuple[Tensor, Tensor, Tensor, Tensor]¶: Return a tuple with the underlying coefficients in WXYZ order.

conj()[source]¶

Compute the conjugate of the quaternion.

Return type:: Quaternion
Returns:: The conjugate quaternion, with the vector part negated.

Example

>>> q = Quaternion(torch.tensor([1., 2., 3., 4.]))
>>> q_conj = q.conj()

property data: Tensor¶: Return the underlying data with shape \((B, 4)\).

classmethod from_axis_angle(axis_angle)[source]¶

Create a quaternion from axis-angle representation.

Parameters:: axis_angle (Tensor) – rotation vector of shape \((B, 3)\).
Return type:: Quaternion

Example

>>> axis_angle = torch.tensor([[1., 0., 0.]])
>>> q = Quaternion.from_axis_angle(axis_angle)
>>> q.data
tensor([[0.8776, 0.4794, 0.0000, 0.0000]])

classmethod from_coeffs(w, x, y, z)[source]¶

Create a quaternion from the data coefficients.

Parameters:

w (float) – a float representing the \(q_w\) component.
x (float) – a float representing the \(q_x\) component.
y (float) – a float representing the \(q_y\) component.
z (float) – a float representing the \(q_z\) component.

Return type:

Quaternion

Example

>>> q = Quaternion.from_coeffs(1., 0., 0., 0.)
>>> q.data
tensor([1., 0., 0., 0.])

classmethod from_euler(roll, pitch, yaw)[source]¶

Create a quaternion from Euler angles.

Parameters:

roll (Tensor) – the roll euler angle.
pitch (Tensor) – the pitch euler angle.
yaw (Tensor) – the yaw euler angle.

Return type:

Quaternion

Example

>>> roll, pitch, yaw = torch.tensor(0), torch.tensor(1), torch.tensor(0)
>>> q = Quaternion.from_euler(roll, pitch, yaw)
>>> q.data
tensor([0.8776, 0.0000, 0.4794, 0.0000])

classmethod from_matrix(matrix)[source]¶

Create a quaternion from a rotation matrix.

Parameters:: matrix (Tensor) – the rotation matrix to convert of shape \((B, 3, 3)\).
Return type:: Quaternion

Example

>>> m = torch.eye(3)[None]
>>> q = Quaternion.from_matrix(m)
>>> q.data
tensor([[1., 0., 0., 0.]])

classmethod identity(batch_size=None, device=None, dtype=None)[source]¶

Create a quaternion representing an identity rotation.

Parameters:

batch_size (Optional[int], optional) – the batch size of the underlying data. Default: None
device (Union[None, str, device], optional) – device to place the result on. Default: None
dtype (Optional[dtype], optional) – dtype of the result. Default: None

Return type:

Quaternion

Example

>>> q = Quaternion.identity()
>>> q.data
tensor([1., 0., 0., 0.])

inv()[source]¶

Compute the inverse of the quaternion.

Return type:: Quaternion
Returns:: The inverse quaternion.

Example

>>> q = Quaternion.identity()
>>> q_inv = q.inv()

matrix()[source]¶

Convert the quaternion to a rotation matrix of shape \((B, 3, 3)\).

Return type:: Tensor

Example

>>> q = Quaternion.identity()
>>> m = q.matrix()
>>> m
tensor([[1., 0., 0.],
        [0., 1., 0.],
        [0., 0., 1.]])

norm(keepdim=False)[source]¶

Compute the norm (magnitude) of the quaternion.

Parameters:: keepdim (bool, optional) – whether to retain the last dimension. Default: False
Return type:: Tensor
Returns:: The norm of the quaternion(s) as a torch.Tensor.

Example

>>> q = Quaternion.identity()
>>> q.norm()
tensor(1.)

normalize()[source]¶

Return a normalized (unit) quaternion.

Return type:: Quaternion
Returns:: The normalized quaternion.

Example

>>> q = Quaternion(torch.tensor([2., 1., 0., 0.]))
>>> q_norm = q.normalize()

property polar_angle: Tensor¶

Return the polar angle with shape \((B,1)\).

Example

>>> q = Quaternion.identity()
>>> q.polar_angle
tensor(0.)

property q: Tensor¶

Return the underlying data with shape \((B, 4)\).

Alias for data()

classmethod random(batch_size=None, device=None, dtype=None)[source]¶

Create a random unit quaternion of shape \((B, 4)\).

Uniformly distributed across the rotation space as per: http://planning.cs.uiuc.edu/node198.html

Parameters:

batch_size (Optional[int], optional) – the batch size of the underlying data. Default: None
device (Union[None, str, device], optional) – device to place the result on. Default: None
dtype (Optional[dtype], optional) – dtype of the result. Default: None

Return type:

Quaternion

Example

>>> q = Quaternion.random()
>>> q = Quaternion.random(batch_size=2)

property real: Tensor¶

Return the real part with shape \((B,)\).

Alias for :func: ~kornia.geometry.quaternion.Quaternion.w

property scalar: Tensor¶

Return a scalar with the real with shape \((B,)\).

Alias for :func: ~kornia.geometry.quaternion.Quaternion.w

property shape: Tuple[int, ...]¶: Return the shape of the underlying data with shape \((B, 4)\).

slerp(q1, t)[source]¶

Return a unit quaternion spherically interpolated between quaternions self.q and q1.

See more: https://en.wikipedia.org/wiki/Slerp

Parameters:

q1 (Quaternion) – second quaternion to be interpolated between.
t (float) – interpolation ratio, range [0-1]

Return type:

Quaternion

Example

>>> q0 = Quaternion.identity()
>>> q1 = Quaternion(torch.tensor([1., .5, 0., 0.]))
>>> q2 = q0.slerp(q1, .3)

squared_norm()[source]¶

Compute the squared norm (magnitude) of the quaternion.

Return type:: Tensor
Returns:: The squared norm of the quaternion(s) as a torch.Tensor.

Example

>>> q = Quaternion.identity()
>>> q.squared_norm()
tensor(1.)

to(*args, **kwargs)[source]¶

Move and/or cast the quaternion data.

Parameters:

*args (Any) – Arguments to pass to torch.Tensor.to()
**kwargs (Any) – Keyword arguments to pass to torch.Tensor.to()

Return type:

Quaternion

Returns:

A new Quaternion with converted data.

to_axis_angle()[source]¶

Convert the quaternion to an axis-angle representation.

Return type:: Tensor

Example

>>> q = Quaternion.identity()
>>> axis_angle = q.to_axis_angle()
>>> axis_angle
tensor([0., 0., 0.])

to_euler()[source]¶

Convert the quaternion to a triple of Euler angles (roll, pitch, yaw).

Return type:: Tuple[Tensor, Tensor, Tensor]

Example

>>> q = Quaternion(torch.tensor([2., 0., 1., 1.]))
>>> roll, pitch, yaw = q.to_euler()
>>> roll
tensor(2.0344)
>>> pitch
tensor(1.5708)
>>> yaw
tensor(2.2143)

property vec: Tensor¶: Return the vector with the imaginary part with shape \((B, 3)\).

property w: Tensor¶: Return the \(q_w\) with shape \((B,)\).

property x: Tensor¶: Return the \(q_x\) with shape \((B,)\).

property y: Tensor¶: Return the \(q_y\) with shape \((B,)\).

property z: Tensor¶: Return the \(q_z\) with shape \((B,)\).