Pole Height Overview

Single$-$view metrology corresponds to a set of techniques for making $3-$D measurements from a single $2-$D image. Such measurements have

several applications including, for example, measuring the height of a suspect in a crime$-$scene image. We will consider a simple scenario of

single$-$view metrology estimating $3-$D measurements of objects in a rendered $2-$D image.

Description

Consider the image shown in Figure $1.$ You are given the following information about the $3-$D scene. There are three poles colored yellow, green,

and blue, and a ground plane in this scene. The three poles are perpendicular to the ground plane. The ground plane is parallel to the $x-Z$ plane,

and the camera's optical axis is parallel to the Z$-$axis. Standing at $4$ meters, the yellow and green poles are of the same height in the $3-$D scene.

The distance between these two poles is $17$ meters. The green and blue poles are at the same, but unknown, distance from the camera. Finally,

the resolution of the image sensor of the camera is $53,333$ pixels per meter.

Recall the perspective projection equation that tells us the relationship between a point's location in the $3-$D world $(x,Y,Z),$ the camera focal

length $($f$),$ and the projection of that point in the image $(x,y)$:

$\frac{x}{x}=\frac{f}{Z}$

$\frac{y}{Y}=\frac{f}{Z}$

Write a python script to measure the heights and distances between the poles in image coordinates. You may use the pixel locations provided

below to estimate the location of the pole endpoints in the image:

yellow $=\left[\begin{array}{ccc}518& 391& 987\end{array}\right]$

$512$

green $=\left[\begin{array}{ccc}733& 462& 771\end{array}\right]$

$732$

blue $=\left[\begin{array}{ccc}1189& 311& 769\end{array}\right]$

$1189$

From these image measurements and the $3-$D quantities specified above, you will be able to determine the camera focal length and $3-$D height

of the blue pole.

Deliverables:

Using the above perspective projection equations, write code to compute and report:

Camera focal length $f($in mm $)$

$3-$D height $(Y)$ of blue pole $($in $m)$