Mathematics behind distance estimation using camera

Question

Can some one explain the logic and calculations in deriving the distance/depth(z) value in the attached research paper. Kindly make your explanation elaborate for me to understand clearly as i was not good in mathematics from school days.

Answer 1

Place your finger straight up, away from your face, directly in front of your nose. Close one eye. Open it. Close the other eye and open it. Your finger moved quite a bit. Now look at something a few meters away and try the same exercise. Not as much movement. This "movement" between one eye and the other (or one camera and another) is called disparity. By trigonometry (as in the figure you posted) disparity is inversely proportional to distance. Using the equations you posted, and drawing triangles, you can derive the relationship. If $x_L$ is the "position in a coordinate system of the left camera" and $x_R$ is the "position of a coordinate system of the right camera", then disparity $\delta = x_L - x_R$. (Different authors use different conventions, sometimes this equation appears with a minus sign).

By similar triangles, we can write the equations for $x_L$ and $x_R$:

$\displaystyle \large \frac{x_L}{f} = \frac{X+\frac{b}{2}}{Z}, \quad \frac{x_R}{f} = \frac{X-\frac{b}{2}}{Z}$

so that the disparity is:

$\large \delta = x_L -x_R = \frac{b f}{Z}$.

Here, $b$ is the baseline distance between two pinhole cameras and $f$ is the shared focal length of each camera. We see that disparity is inversely proportional to $Z$, the vertical distance from the point we are observing, and the horizontal line through the origin, $O$.