---
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

## 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