EssentialMatrix - GTSAM Docs

The two-view geometry between two calibrated cameras can be captured in a five-dimensional object known as the essential matrix. EssentialMatrix represents it as a relative rotation and unit translation direction, discarding the unobservable translation scale.

Authors: Frank Dellaert, et al. (see THANKS for the full author list)

See LICENSE for the license information

try:
    import google.colab
    %pip install --quiet gtsam-develop
except ImportError:
    pass

import gtsam
import numpy as np

Initialization¶

Construct directly from Rot3 and Unit3, or use FromPose3() to discard the translation magnitude from a relative pose.

relative_pose = gtsam.Pose3(
    gtsam.Rot3.Yaw(0.15), np.array([2.0, 0.5, 0.0])
)
essential = gtsam.EssentialMatrix.FromPose3(relative_pose)
print("rotation:")
print(essential.rotation().matrix())
print("translation direction:", essential.direction().unitVector())

Matrix and epipolar error¶

matrix() returns the rank-two matrix [t]×R. For corresponding normalized homogeneous points xa and xb, the epipolar constraint is xa.T @ E @ xb = 0. error() provides GTSAM’s two-dimensional geometric residual for a correspondence.

E = essential.matrix()
print("singular values:", np.linalg.svd(E, compute_uv=False))
assert np.linalg.matrix_rank(E, tol=1e-9) == 2

Manifold operations¶

The tangent vector has three rotation coordinates and two coordinates on the translation-direction sphere. retract() applies an increment and localCoordinates() recovers it locally.

delta = np.array([0.01, -0.02, 0.01, 0.005, -0.004])
perturbed = essential.retract(delta)
np.testing.assert_allclose(
    essential.localCoordinates(perturbed), delta, atol=1e-8
)

Source¶

EssentialMatrix.h

AI assistance caveat¶

AI was used to help draft this documentation, and inaccuracies could be present.