#
Stereo epipolar geometry

##
Programming assignment #2(b)

CS 223b - Introduction to Computer Vision

Winter Quarter, 2002

By:

Mitul Saha & Rohit Singh

**Objective:**Given a pair of uncalibrated stereo images, we are required to
plot the corresponding epipolar lines using the knowledge of projective
geometry.

**Procedure:** Two images of the same scene are related by the epipolar geometry.
To determine the epipolar geometry and hence plot the epipolar lines we need
to estimate the 3x3 singular matrix known as the Fundamnetal matrix.
Since the camera is uncalibrated we need to find point corresspondences
between the two images to estimate the Fundamental matrix. Hence we start
with manually matching some points between the two images.

The following two images shows 20 points marked by red cross choosen for matching:
As per 8-point Algorithm we need atleast 8 points to estimate the Fundamental
matrix. But only 8 points are not enough to get an accurate estimate of the Fundamental
matrix for the entire
image. So choose to match 20 points to get a good estimate of the fundamental
matrix. Also the points chosen should be well scattered in the image so that the
Fundamental matrix estimate is uniformly accurate. Say If we choose points only
around the statue then the resulting Fundamental matrix does not map the far
away points accurately. Also the corner points are easy to match manually
with good accuracy.

After matching the images we use the 8-point algorithm to estimate the Fundamental
matrix. After estimating the Fundamnetal matrix we plot the epipolar lines fpr
a given point in the left image.

Now the corresponding epipolar line in the left image is the line joining the
clicked point and the epipole of the left image. The epipole of the left image
is the third column of the V matrix obtained from SVD of F (Page 157: Trucco). Hence now
we have the epipolar lines in both the images.

The following images shows the plot of the epipolar lines in the right
image corresponding to the clicked points on the left image.

The left image with some clicked points marked by red *: (The lines passing
through the clicked points on the left image are the
epipolar lines corresponding to the left image for the clicked points)

The right image with epipolar lines corresponding to the clicked points
in the left image: (The red
line image is the epipolar line corresponding to the clicked point marked by * on the
red line of the above left image and so on.)

### Matlab Source Codes: