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Master the geometry that relates two views of the same scene. Learn how epipolar constraints reduce stereo matching from a 2D search to a 1D scan, how the fundamental and essential matrices encode this geometry, and how to build a complete stereo depth pipeline from rectification through disparity estimation to 3D reconstruction.
Epipolar geometry is the intrinsic projective geometry between two views of the same scene. It describes the geometric constraints that arise when two cameras observe common 3D points, regardless of the scene's content. These constraints are among the most powerful tools in computer vision: they tell us where a point visible in one image must lie in the other image, dramatically simplifying correspondence search and enabling depth estimation.
What is this chapter about? We develop the mathematical framework for two-view geometry from first principles. Starting with the epipolar constraint, we derive the fundamental matrix (uncalibrated cameras) and essential matrix (calibrated cameras), learn stereo rectification to simplify matching, explore stereo matching algorithms, and build a complete pipeline from image pairs to depth maps.
Why does this matter? Stereo vision is one of the oldest and most reliable methods for recovering 3D structure from images. It is used in autonomous driving (stereo depth cameras), robotics (obstacle avoidance), AR/VR (depth sensing), and medical imaging (endoscopic 3D reconstruction). Understanding epipolar geometry is also a prerequisite for multi-view reconstruction and SLAM.
How the topics connect: We start with the epipolar constraint to establish the geometric relationship between views. The fundamental matrix algebraically encodes this relationship for arbitrary cameras. The essential matrix specializes it for calibrated cameras and enables pose recovery. Stereo rectification transforms images to exploit epipolar geometry for efficient matching. Stereo matching algorithms compute dense correspondences, producing disparity maps that convert to depth. Finally, we assemble everything into end-to-end stereo pipelines.
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The coplanarity condition — how two views of one point constrain each other to epipolar lines.
The fundamental and essential matrices
The 3x3 rank-2 matrix F that encodes epipolar geometry for uncalibrated cameras.
F for calibrated cameras — E = K'^T F K directly gives rotation and translation.
Rectification and correspondence search
Warping images so epipolar lines become horizontal — enabling efficient 1D matching.
Block matching, SGM, and cost volumes — finding correspondences along epipolar lines.
Depth estimation and complete pipelines
Disparity d = f*B/Z — converting pixel shifts to metric depth measurements.
Classical OpenCV pipelines, deep learning stereo, and practical failure modes.
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