# Partial Scene Information

**Briefing** Partial Scene Lecture

# Exercises

This exercise builds upon the exercise on Stratified Reconstruction last week. You do not necessarily have to complete that exercise.

## Step 1. A figure with parallel lines.

- Review the figure from last week.
- Can you fine three orthogonal sets of parallel lines in the figure?
- If not, adapt it so that you can.

An example house could be designed as follows:

- Front wall: \((0,0,10),(0,2,10),(2,0,10),(2,2,10)\)
- Rear wall: \((0,0,15),(0,2,15),(2,0,15),(2,2,15)\)
- Side wall I: \((0,0,10),(0,2,10),(0,0,15),(0,2,15)\)
- Side wall II: \((2,0,10),(2,2,10),(2,0,15),(2,2,15)\)
- Roof I: \((0,2,10),(0,2,15),(1,2,15),(1,2,10)\)
- Roof II: \((2,2,10),(2,2,15),(1,2,15),(1,2,10)\)
- It should be obvious how to add the floor and the top of the front and rear walls.

This figure gives three sets of parallel lines, as follows:

- The vertical lines at the end of each wall
- The horisontal lines at the top and bottom of the side walls
- The horisontal lines at the top and bottom of the front/rear walls

## Step 2. Parallel Lines in different Projections

- Project the figure into 2D using an ideal projection. (Note that we do not use any affine or projective transformation at this stage.)
- Are the parallel object lines parallel in the figure?
- If they are, try to rotate the figure so that at least some of the parallel lines project to lines obviously not parallel.

## Step 3. Identifying the lines

- Select three pairs of parallel lines, so that lines in different pairs are orthogonal, and identify their co-ordinates (e.g. end-points).
- Identify the co-images for each of the six lines.
- Say a line connects nodes \(x_1\) and \(x_2\)
- The co-image \(\ell\) solves the two equations \(\ell x_1=\ell x_2=0\).
- Note that everything is written as 3D co-ordinates, which may be read as homogeneous co-ordinates in 2D.

## Step 4. Vanishing point

- Identify the three vanishing points, i.e. one per pair of parallel lines.
- See equation (6.51) in the textbook

- Review the figure (2D projection).

Where should the vanishing points be by visual judgement? - Do your calculation seem correct?

## Step 5. Bonus Challenge.

- Redo the exercise, but add an intrinsic camera calibration matrix \(K\) to the projection. You should choose \(s_\theta=0\).
- Recover \(K\) using the techniques explained on page 200 of the textbook. - Assume that you know that \(s_\theta=0\) as well as the aspect ration \(s_x/s_y\)