Camera intrinsics, extrinsics, and the essential matrix

What's really being asked

Whether you understand camera calibration and can recover relative pose from two views using epipolar geometry.

The full answer

Intrinsics describe the camera's internal projection: focal lengths, principal point, and skew, packaged in a calibration matrix that maps 3D points in the camera frame to pixel coordinates. Extrinsics describe the camera's pose in the world, a rotation and a translation that transform world coordinates into the camera frame. To relate two views of a static scene, first find point correspondences across the images, for example with a feature detector and descriptor matching. If intrinsics are known you can normalize the matched pixels into calibrated camera coordinates and estimate the essential matrix; the five-point algorithm needs as few as five correspondences and the eight-point algorithm is simpler, both typically wrapped in RANSAC to reject outlier matches. The essential matrix encodes the relative rotation and translation between the two cameras, with translation recoverable only up to scale because monocular geometry cannot fix absolute size. It enforces the epipolar constraint, so a point in one image must lie on a corresponding epipolar line in the other, and it can be decomposed into rotation and translation, with four candidate solutions disambiguated by requiring reconstructed points to lie in front of both cameras.

The mistakes people make

Confusing the essential matrix with the fundamental matrix, which relates raw pixel coordinates without known intrinsics; claiming you can recover absolute translation scale from two views alone; forgetting RANSAC and the need to reject mismatches; saying intrinsics describe pose or extrinsics describe focal length.

What usually comes next

How does the essential matrix relate to the fundamental matrix through the intrinsics? Why exactly four pose candidates after decomposition? How do you resolve scale ambiguity, for example with stereo baseline or known object size? What degenerate configurations break estimation?

A concrete example

Two calibrated photos of a building from slightly different positions yield hundreds of matched corners. Running the five-point algorithm in RANSAC produces the essential matrix; decomposing it gives the rotation and the translation direction between the two camera positions, letting you triangulate a sparse 3D point cloud of the building, accurate in shape but unknown in absolute size until you supply a real-world measurement.