Multi-View Geometry Fundamentals

Understanding the geometric relationships between multiple views of a 3D scene is fundamental to computer vision. This tutorial explores key concepts through interactive 3D visualizations.

Epipolar Geometry

When a 3D point is viewed from two different camera positions, it creates a geometric relationship between the two views:

  • The epipolar plane is formed by the 3D point and the two camera centers
  • The epipolar lines are the intersections of this plane with the image planes
  • The epipoles are the projections of each camera center onto the other camera's image plane

The fundamental matrix F encodes this geometry:

\[x'^T F x = 0\] where \(x\) and \(x'\) are corresponding points in the two images.

Essential Matrix

The essential matrix E represents the geometric relationship between two calibrated cameras:

\[E = K'^T F K\] \[E = [t]_{\times} R\] where:
  • \(K\) is the camera calibration matrix
  • \(R\) is the rotation matrix between cameras
  • \([t]_{\times}\) is the skew-symmetric matrix of the translation vector

Camera Calibration

The camera calibration matrix K contains the intrinsic parameters:

\[K = \begin{bmatrix} f_x & s & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1 \end{bmatrix}\] where:
  • \(f_x, f_y\) are focal lengths in pixels
  • \(s\) is the skew parameter
  • \((c_x, c_y)\) is the principal point

Left Camera View

Right Camera View

Scene

1000cm

Camera Extrinsics

50cm
0°

Camera Intrinsics

35mm
0mm
0mm

Computed Matrices

Fundamental Matrix (F)

Essential Matrix (E)

Calibration Matrix (K)

Practice Problems (Coming Soon)

Problem 1: Camera Calibration

Given a set of 3D-2D point correspondences, estimate the camera calibration matrix K.

Problem 2: Fundamental Matrix Estimation

Using point correspondences between two views, estimate the fundamental matrix F.

Problem 3: Camera Pose Estimation

Recover the relative pose (R, t) between two cameras using the essential matrix.