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# Distorted Space

**Briefing** Distorted Lecture

**Additional Reading** Chapter 9 in *OpenCV 3 Computer Vision with Python Cookbook* by Alexey Spizhevoy (author). Search for it in Oria. There is an e-book available.

# Exercises

## An Example of a Distorted space

### Step 1. Preliminaries.

- 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 \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\).
- \(\beta\) be the angle between the vectors \(\overrightarrow{BA}\) and \(\overrightarrow{BC}\). Draw a figure to show this information.

- Calculate the angles \(\alpha\) and \(\beta\).
- 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.
- Calculate the angles in the images of \(\alpha\) and \(\beta\). Recall the formula for the angle \(\theta\) between two vectors \(u\) and \(v\) \[\theta = \tan^{-1}\frac{u\cdot v}{||u||\cdot||v||}.\] (This is found in any elementary calculus book.) Do they match the original angles?
- Calculate the lengths of the edges of the triangles in the image.
- Make a new drawing of the image plane, displaying the quantities calculated in 4 and 5. Keep it for later reference.

### Step 2. Distortion.

- Suppose the camera is not idea, but instead has calibration matrix \[K = \begin{bmatrix} 2 & \frac1{\sqrt{3}} & 4 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{bmatrix}\]
- Calculate the image points corresponding to \(A\), \(B\), and \(C\), using the camera calibration matrix \(K\). Draw the resulting image to scale.
- Calculate the lengths of the edges of the image of the triangle. For now, we use the familiar formulæ for vector norms.
- 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.
- Compare your distorted image to the original image from Step 1.

### Step 3. The Distorted Inner Product Space

Recall the textbook’s definition of an inner product in distorted space \[\langle u,v\rangle_S = u^TSv \quad\text{where } S = K^{-T}K^{-1},\] for some `camera calibration’ matrix \(K\). We are going to redo Step 2, using this distorted inner product.

- [9] Calculate the edges (norms) of the triangle, using the distorted inner product to define the norm.
- [10] Calculate the angles \(\alpha'\) and \(\beta'\), using the distorted inner product and the formula \[\theta = \tan^{-1}\frac{u\cdot v}{||u||\cdot||v||}.\]
- Make a figure illustrating the quantities calculated.
- [11] Compare the edges and angles thus calculated with the calculations from Step 2 and from Step 1.

Reflect on your observations.

### Step 4. Reflection

Think through the following:

- What does this exercise tell you about camera distortion and inner product spaces?
- Is there anything you do not quite understand? Is there something missing in the exercise, something that should have been tested?

## From the Textbook

- Exercise 6.4