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---
title: Distorted Space
categories: session
---
**Briefing** [Distorted Lecture]()
**Additional Reading**
Chapter 9 in *OpenCV 3 Computer Vision with Python Cookbook* by
Alexey Spizhevoy (author).
Search for it in [Oria](https://oria.no/).
There is an e-book available.
# Exercises
+ Exercise 6.4
## An Example of a Distorted spaceo
### Step 1. Preliminaries.
1. Consider a triangle in the object space, formed by the three
vertices, $A=(0,0,10)$, $B=(20,0,10)$, and $(20,10,10)$.
Let
+ $\alpha$ be the angle between the vectors $\widevec{AB}$
and $\widevec{AC}$.
+ $\beta$ be the angle between the vectors $\widevec{BA}$
and $\widevec{BC}$.
Draw a figure to show this information.
2. Calculate the angle $\alpha$.
3. Consider an ideal perspective camera, posed such that the camera
frame matches the world frame. Calculate the image points
corresponding to $A$, $B$, and $C$. Draw an image to display
this projection.
4. Calculate the angles in the images of $\alpha$ and $\beta$.
Do they match the original angles?
5. Calculate the lengths of the edges of the triangles in the image.
6. Make a new drawing to display the quantities calculated in 4 and 5.
Keep it for later reference.
### Step 2. Distortion.
7. Suppose the camera is not idea, but instead has calibration matrix
$$K =
\begin{bmatrix}
2 & \frac1{\sqrt{3}} & 4 \\ 0 & 1 & 4 \\ 0 & 0 & 1
\end{bmatrix}$$
8. Calculate the image points corresponding to $A$, $B$, and $C$,
using the camera calibration matrix $K$.
Draw the resulting image to scale.
9. Calculate the lengths of the edges of the image of the triangle.
10. Let $\alpha'$ and $\beta'$ be the images of the angles $\alpha$
and $\beta$. Calculate these to image angles and add the information
to you figure.
11. Compare your distorted image to the original image from Step 1.
### Step 3. The Distorted Inner Product Space
## From the Textbook
+ Exercise 6.4
# Debrief