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title: Image Formation
categories: session
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> Vision is the inverse problem of image formation
# Briefing
This session is prepared for **self-study**.
However, the room is available, and you should meet up
and take advantage of collaboration.
Use the **first hour** to watch and listen to the lectures
(videos and slideshows with audio). See under [Briefing](#briefing).
Use the **rest of the time** to solve the
[Exercises](#exercises) as group work.
You should discuss possible solutions between yourselves **before**
you review the [Solutions](Solutions/Image Formation).
Remember that communicating and arguing solutions to your peers is
one of the most important learning outcomes of the module.
# Briefing {#briefing}
The actual briefing will extensively use blackboard drawings and
improvisation. Hence the lecture notes below are **not complete**.
+ [Slides](http://www.hg.schaathun.net/talks/camera.html) are available.
+ [Image Formation Notes]() are more rudimentary but also provide some additional material.
+ [Image Formation Lecture]()
+ Rudimentary notes from 2022: [Image Formation Notes]() and
[Slides](http://www.hg.schaathun.net/talks/maskinsyn/camera.html)
# Learning Outcomes
## 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
+ thin lens equation
+ vanishing point
+ The thin lens model
+ aperture, focus
+ aperture, focus
+ The pinhole model
# Exercises
# Exercises {#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.
If you prefer, you may consult the
[Solutions](Solutions/Image Formation) after each individual exercise.
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 $p$.
## Equivalence of Points (Based on Exercise 3.1.)
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.
> Show that any point on the line through $o$ (optical centre) and
> $p$ projects onto the same image co-ordinates as $p$.
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.)
1. Start by drawing the lens, image, the points $p$ and $o$,
and the image point.
2. What does the drawing tell you about the problem?
Add details to the drawing as required.
3. Recall the equations which relate the $(x,y)$ co-ordinates of the
image point to the $(X,Y,Z)$ co-ordinates of $p$.
(Write it down.)
4. Consider a different point $p'$ on the same line, and add it
to your drawing. Where is its image point?
5. How does do the co-ordinates $(X',Y',Z')$ of $p'$ relate to $(X,Y,Z)$ and $(x,y)$?
5. From the above, you should have two arguments solving the
problem, one geometric and one algebraic.
Each deserves attention.
Are these arguments convincing?
Complete any details as required.
6. Reflect on the relationship between the algebraic and the
geometric argument.
**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.
## (Exercise 3.2)
2. Exercise 3.8
> 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.)
2. Exercise 3.3 Part 1-2.
Part 3-4 depends on the cameara calibration which we discuss
in the next session.
**Note**
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.
1. Always start by making a drawing of the model.
2. Add all concepts mentioned in the problem text to the figure
(as far as possible).
3. Add any additional concepts that you find important.
4. Identify the concept in question, that is the region contributing
to the point $x$ in this case.
# Debrief
## Scale Ambiguity (Exercise 3.8).
> It is common sense that with a perspective camera, one cannot
> tell an object from another object that is exactly twice
> as big but twice as far.
> This is a classic ambiguity introduced by the perspective projection.
> Use the ideal camera model to explain why this is true.
> Is the same also true for the orthographic projection? Explain.
1. You can start with the problem you drew above for Exercise 1 (Ma:3.1).
Consider an object extending between two points $p_1$ and $p_2$ in a
plane parallel to the lens. Draw this situation.
2. Imagine that both points move on a line through the optical centre $o$,
as you did in Exercise 1. What happens to the image?
What happens to the object extending between $p_1$ and $p_2$?
3. Write up an argument based on the above reflections.
## Field of View (based on Exercise 3.3 Part 1)
> How can describe the area (in 3D) observed by a camera?
Consider a camera with focal length 24 mm, and a retinal plane
(CCD array) (16 mm x 12 mm).
1. As always, start with a drawing. Draw the pinhole module.
Consider only the $x$-direction where the sensor is 16 mm.
(You can do the $y$-direction (12mm) afterwards.)
2. Write the known lengths into the figure.
2. Where are the points which are observable to the camera?
Reflect on the question.
3. You should find that the observable points fall betweeen two
lines through the focus (pinhole). Calculate the angle
$\theta$ between these two lines.
+ Note that the optical axis, through the focus, is orthogonal
and centred on the sensor array. It may be easier to calculate
the angle $\theta/2$ between the optical axis and one of the edge
lines.
4. The angle $\theta$ is known as the field of view (FoV).
Once you have calculated FoV for the specific camera,
give an expression of FoV as a function of the focal length $f$
and the radius of the sensor $r$.
## Real World and Imavge Co-ordinates (based on Exercise 3.3 Part 2)
> Given a point $(X,Y,Z)$ in 3D, what is the co-ordinates $(x,y)$ of
the image point?
Consider the same camera system and model as you used in the previous
exercise.
Consider first a point with co-ordinates $(X,Y,Z)=(6m,4m,8m)$.
1. Draw first the pinhole model in the $x$-direction and find the
$x$-co-ordinate corresponding to $X=6m$.
1. Then draw the $xy$-direction and find the
$y$-co-ordinate corresponding to $X=4m$.
# Debrief {#debrief}
See [Solutions](Solutions/Image Formation)
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