Computer Vision Project

Project 3 / Camera Calibration and Fundamental Matrix Estimation with RANSAC

Part I: Camera Projection Matrix

We can use the following equation to move from 3D world to 2D camera coordinates:

$\begin{pmatrix}u \\ v \\ 1\end{pmatrix} \cong \begin{pmatrix}u*s \\ v*s \\ s \end{pmatrix}= \begin{pmatrix}m_{11} & m_{12} & m_{13} & m_{14} \\ m_{21} & m_{22} & m_{23} & m_{24} \\ m_{31} & m_{32} & m_{33} & m_{34} \end{pmatrix} \begin{pmatrix}X \\ Y \\ Z \\ 1 \end{pmatrix}$

And the camera center can be calculated using the following equation:

$M = (Q | m_4 )$ , and $C = -Q^{-1} m_4$

The results are shown in the following images:

Part II: Fundamental Matrix Estimation

We can solve for the Fundamental matrix using the following equation:

$\begin{pmatrix}u' & v' & 1\end{pmatrix}\begin{pmatrix}f_{11}u + f_{12}v + f_{13} \\ f_{21}u + f_{22}v + f_{23} \\ f_{31}u + f_{32}v + f_{33} \end{pmatrix} = 0$

The results (drawing epilolar line) are shown below:

Part III: Fundamental Matrix with RANSAC

With the max iteration number set to 1000, the outlier ratio set to 0.99, and the error threshold set to 0.005, using the RANSAC algorithm, we get the following results.

Hongrui Zheng

Project 3 / Camera Calibration and Fundamental Matrix Estimation with RANSAC

Part I: Camera Projection Matrix

Part II: Fundamental Matrix Estimation

Part III: Fundamental Matrix with RANSAC

Epipolar lines

Matching features using only SIFT and using RANSAC