Epipolar Geometry

Epipolar geometry describes the geometric relationship between two camera views of the same 3D scene.

It is the main geometric framework behind:

• stereo vision
• two-view 3D reconstruction
• structure from motion

Intuition §

Suppose a 3D point is seen by two cameras.

It projects to:

• in the first image
• in the second image

Once is known, the matching point in the second image is not free to appear anywhere. It must lie on a specific line called the epipolar line.

This is what makes correspondence search much easier in practice.

Epipolar Plane §

The first camera center, the second camera center, and the 3D point determine a plane called the epipolar plane.

This plane intersects each image plane in a line:

• one epipolar line in image 1
• one epipolar line in image 2

So every matched pair of points is associated with one epipolar plane and a pair of epipolar lines.

Epipoles §

The line joining the two camera centers is called the baseline.

Where this baseline intersects an image plane is called the epipole of that image.

Important fact:

• all epipolar lines in an image pass through that image’s epipole

Epipolar Constraint §

If is a point in image 1 and is its matching point in image 2, then they satisfy

where:

• and are image points in homogeneous coordinates
• is the fundamental matrix

This is the main algebraic expression of epipolar geometry.

Epipolar Lines §

The epipolar line in the second image corresponding to is

Similarly, the epipolar line in the first image corresponding to is

So:

• a point in one image maps to a line in the other image
• the correct matching point must lie on that line

Why It Matters for 3D Reconstruction §

Epipolar geometry is important because it constrains which image points can correspond to the same 3D point.

This helps reconstruction in three major ways:

• it reduces the search space for feature matching
• it rejects incorrect correspondences
• it provides the geometry needed before triangulation

In practice, this means epipolar geometry is one of the key steps between raw images and a reconstructed 3D scene.

Fundamental Matrix and Essential Matrix §

Two matrices are commonly used to represent epipolar geometry:

• Fundamental Matrix for uncalibrated cameras and pixel coordinates
• Essential Matrix for calibrated cameras and normalized image coordinates

They encode the same underlying two-view geometry, but at different levels of camera knowledge.

In Stereo Vision §

A common stereo pipeline is:

1. detect features in both images
2. match candidate features
3. estimate epipolar geometry
4. keep only matches consistent with the epipolar constraint
5. recover camera pose
6. triangulate points to estimate 3D structure

Summary §

Epipolar geometry is the geometry of two-view imaging.

Its core idea is:

• a point in one image corresponds to a line in the other image, not an arbitrary point

That single fact is what makes two-view matching and 3D reconstruction possible.