Essential Matrix

The essential matrix is used in stereo vision and epipolar geometry for calibrated cameras.

It relates a point in one image to its corresponding epipolar line in the other image.

Definition §

For corresponding points written in normalized camera coordinates, the essential matrix satisfies

where and are matching points in the two views.

Form §

If the relative pose between the two cameras is given by a rotation and translation , then

where is the skew-symmetric matrix for the cross product with .

Meaning §

The essential matrix encodes the relative motion between two calibrated cameras:

• describes how the second camera is rotated
• describes the direction of translation
• combines both into one matrix constraint on corresponding image points

Key Facts §

• it is a matrix
• it is defined up to scale
• it is used with normalized image coordinates
• it has rank 2

Relation to the Fundamental Matrix §

The essential matrix is the calibrated version of the fundamental matrix.

If and are the intrinsic calibration matrices of the two cameras, then

and equivalently

So:

• use the essential matrix when camera intrinsics are known
• use the fundamental matrix when working directly with pixel coordinates