Vision is the inverse problem of image formation

Briefing

The actual briefing will extensively use blackboard drawings and improvisation. Hence the lecture notes below are not complete.

• Slides are available.
• Image Formation Notes are more rudimentary but also provide some additional material.

Learning Outcomes

During this session, the goal is to learn to master the following concepts and models:

• The image as a sampled function
• Projection from 3D to 2D, as it occurs in a camera
  • thin lens equation
  • vanishing point
• The thin lens model
  • aperture, focus
• The pinhole model

Exercises

Exercises are from Ma 2004 page 62ff.

I recommend to discuss the following problems in small groups. Use figures and diagrams as basis for your discussion where possible.

1. (Based on Exercise 3.1.) Show that any point on the line through \(o\) (optical centre) and \(p\) projects onto the same image co-ordinates as \(p\).

   Both geometric and algebraic arguments are possible, and it is useful to do both. The geometric argument starts with a drawing of the pinhole model. The algebraic argument starts with the ideal projection formula. You should make both arguments and reflect on the relationship between them.

2. (Exercise 3.2) Consider a thin lens imaging a plane parallel to the lens at a distance \(z\) from the focal plane. Determine the region of this plane that contributes to the image \(I\) at the point \(x\). (Hint: consider first a one-dimensional imaging model, then extend to a two-dimensional image.)

   Note You should start by drawing the model, and you may have to add more parameters. The question makes sense if you assume that the plane is out of focus, which is not possible in the pinhole model but is in a more generic thin lens model.

3. Exercise 3.8

4. Exercise 3.3 Part 1-2. Part 3-4 depends on the cameara calibration which we discuss in the next session.

Debrief

1. Questions and Answers
2. Recap as required