DeepGIS

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

Distance to Object: 1000cm

Camera Extrinsics

Baseline: 50cm

Toe-in Angle: 0°

Camera Intrinsics

Focal Length: 35mm

Principal Point X: 0mm

Principal Point Y: 0mm

Computed Matrices

Fundamental Matrix (F)

\[\begin{bmatrix} 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & -0.707 \\ 0.000 & 0.707 & 0.000 \end{bmatrix}\]

Essential Matrix (E)

\[\begin{bmatrix} 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & -0.707 \\ 0.000 & 0.707 & 0.000 \end{bmatrix}\]

Calibration Matrix (K)

\[\begin{bmatrix} 35.000 & 0.000 & 18.000 \\ 0.000 & 35.000 & 12.000 \\ 0.000 & 0.000 & 1.000 \end{bmatrix}\]

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.