# 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.

1. Review the figure from last week.
2. Can you fine three orthogonal sets of parallel lines in the figure?
3. 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

1. Project the figure into 2D using an ideal projection. (Note that we do not use any affine or projective transformation at this stage.)
2. Are the parallel object lines parallel in the figure?
3. 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

1. Select three pairs of parallel lines, so that lines in different pairs are orthogonal, and identify their co-ordinates (e.g. end-points).
2. 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

1. Identify the three vanishing points, i.e. one per pair of parallel lines.
• See equation (6.51) in the textbook
2. Review the figure (2D projection).
Where should the vanishing points be by visual judgement?
3. Do your calculation seem correct?

## Step 5. Bonus Challenge.

1. Redo the exercise, but add an intrinsic camera calibration matrix $$K$$ to the projection. You should choose $$s_\theta=0$$.
2. 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 $$s_x/s_y$$