---
title: Stratified Reconstruction
categories: session
---

**Briefing** [Stratified Lecture]()


# Exercises

Again we propose an experimental exercise, trying to visualise
the projective, affine and Euclidean transformations in stratified
reconstruction.

## Step 1.  Visualisation Code.

1.  Make a simple 3D figure which you can use for testing.
    A tetrahedron is too simple.  You need some straight angles.
    A simple house with rectangular walls and sloped roof will do.
    Remember to define the corners in 3D, rather than in a 2D 
    perspective drawing.
2.  Retrieve your code to display the 3D figure from the first weeks
    of the module.  Display your figure.  Does it look right?
3.  Make an ideal projection of your house (divide each point by
    its $z$ co-ordinate), and display this in 2D.  Does it look right?

## Step 2. Euclidean transformation.

1.  Define a Euclidean transformation, e.g. rotation by $45^\circ$ around
    the $y$-axis and translation by $(0,-5,0)$.
1.  Apply the transformation to the 3D figure.
2.  Using the code from Step 1, plot both the resulting 3D figure,
    and its projection in 2D.  Do the plots look right?
4.  Try a few other transformations.  Try to find one which gives 
    a comprehensible impression in 2D.

## Step 3. Affine transformation.

1.  Define an affine transformation of the form
    $$\begin{bmatrix} K&0\\0&1 \end{bmatrix}$$
    where $K$ is upper triangular and has determinant 1.
1.  Apply the affine transformation after the Euclidean transformation
    to the 3D figure.
    Use a Euclidean transformation which looks good in 2D.
2.  Using the code from Step 1, plot both the resulting 3D figure,
    and its projection in 2D.  Do the plots look right?
4.  Try a few other transformations and compare the results.

## Step 4. Projective transformation.

1.  Define a projective transformation of the form
    $$\begin{bmatrix} I&v^T\\0&v_4 \end{bmatrix}$$
    for some vector $v^T$ and scalar $v_4$.
1.  Apply the projective transformation after the affine and
    Euclidean transformations to the 3D figure.
2.  Using the code from Step 1, plot both the resulting 3D figure,
    and its projection in 2D.  Do the plots look right?
4.  Try a few other transformations and compare the results.

## Step 5.  Reflection

1.  Compare your plots to Figure 6.10 in the textbook. 
    Do you get something similar?
    Can you get something similar by varying the transformations?
2.  Compare the exercise to Table 6.1 in the textbook.  What do you find?

# Debrief