How to determine distance of objects from camera using Epipolar Plane Image?

Question

I am working on converting 2d images to 3d environment. The images were collected from a video made in a lateral motion. Then the images were placed one behind the other, so it would be easy to find the correspondences between the two images. This is called a spatial-temporal volume.

Next I take a slice from the spatiotemporal volume. That slice is called the Epipolar Plane Image.

Using the Epipolar Plane Image, I want to calculate the depth of the objects in the scene and make a 3D enviornment. I have listed the reference but I have not been able to figure out the math described in the paper. Can someone help me figure this out? Any help is appreciated.

Reference

Epipolar-Plane Image Analysis: An Approach to Determining Structure from Motion* !

Answer 1

The math in this situation is easy and straight forward.

First let's define two the coordinate systems for two overlapping images taken by the same camera with the focal length with the following schema:

Let us say that first camera position is defined as follows:

While it's orientation by using three Euler angles is:

By using this definition the corresponding rotation matrix is the identity matrix

The second camera position can be defined as follows:

And since the orientation is the same as the first camera, all Euler angles remain zero:

Which also means that the corresponding rotation matrix is the identity matrix.

If the images overlap and the orientation is the same, the situation in the image space looks like this:

Here the image coordinates and their measurement accuracy are defined as follows:

This geometrical situation can be described by using the Intercept Theorem:

As you see it's not complicated. But be aware that this solution is certainly not the best, since it's base assumption that all orientation angles are the same can't be fulfilled in reality.

If you need to be accurate then you have to perform an bundle adjustment. However, this equations are often used to determine the approximated solution for this geometric situation, where the values are used to linearize the collinearity equations.