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---
title: Image Formation
categories: session
---
> Vision is the inverse problem of image formation
- perspective
# Briefing
## The Eye Model
+ An image of the real world is projected on the Retina.
+ Modern Cameras (more or less) replicate the Eye Model.
## Image Representation
The Retina, or image sensor, is able to sense the projected rays.
A fine grid of sensors or perceptive cells is able to measure, or sample,
the light intensity falling upon it.
Let's have a look at the resulting data, using the popular
mandrill image shown to the right.
```python
import numpy as np
import cv2 as cv
from matplotlib import pyplot as plt
im = cv.imread("mandrill-grey.png")
```
Now, we have loaded the image as the object `im`.
Firstly, we observe that this is a matrix.
```
Out[55]:
array([[[ 76, 76, 76],
[ 60, 60, 60],
[ 68, 68, 68],
...,
[ 90, 90, 90],
[ 97, 97, 97],
[ 96, 96, 96]],
[[ 83, 83, 83],
[ 72, 72, 72],
[ 77, 77, 77],
...,
[ 99, 99, 99],
[ 87, 87, 87],
[106, 106, 106]],
[[ 51, 51, 51],
[ 75, 75, 75],
[117, 117, 117],
...,
[ 99, 99, 99],
[ 81, 81, 81],
[ 88, 88, 88]],
...,
[[139, 139, 139],
[140, 140, 140],
[136, 136, 136],
...,
[ 92, 92, 92],
[ 96, 96, 96],
[ 78, 78, 78]],
[[131, 131, 131],
[144, 144, 144],
[138, 138, 138],
...,
[ 85, 85, 85],
[ 98, 98, 98],
[ 90, 90, 90]],
[[109, 109, 109],
[102, 102, 102],
[109, 109, 109],
...,
[ 57, 57, 57],
[ 67, 67, 67],
[ 69, 69, 69]]], dtype=uint8)
In [56]: im.shape
Out[56]: (128, 128, 3)
```
A little confusingly, this matrix has three dimensions, as if it
were a colour (RGB) image. Since it is grey scale, we only need
one $128\times128$ matrix. As we see above, the three values in each
tripple are equal so we can take one arbitrary plane from the matrix.
```
Out[58]:
array([[ 76, 60, 68, ..., 90, 97, 96],
[ 83, 72, 77, ..., 99, 87, 106],
[ 51, 75, 117, ..., 99, 81, 88],
...,
[139, 140, 136, ..., 92, 96, 78],
[131, 144, 138, ..., 85, 98, 90],
[109, 102, 109, ..., 57, 67, 69]], dtype=uint8)
In [59]:
```
This is the **first representation** of a grey scale image, as
an $n\times m$ matrix.
OpenCV provides the functions to show the image as an image.
This is the **second representation** of the image.
```
In [60]: cv.imshow("mandrill grey", im)
In [61]: cv.waitKey(1)
Out[61]: -1
```
The matrix can also be read as a signal, sampling
values $I(x,y)$ for different values of $x$ and $y$.
This gives the **third representation**, as a 3D
surface plot.
```
In [7]: plt.ion()
Out[7]: <matplotlib.pyplot._IonContext at 0x7fb5737006d0>
In [8]: fig, ax = plt.subplots(subplot_kw={"projection": "3d"})
In [9]: xn,yn = im.shape
In [10]: xn,yn
Out[10]: (128, 128)
In [11]: X,Y=np.meshgrid(range(xn),range(yn))
In [12]: X
Out[12]:
array([[ 0, 1, 2, ..., 125, 126, 127],
[ 0, 1, 2, ..., 125, 126, 127],
[ 0, 1, 2, ..., 125, 126, 127],
...,
[ 0, 1, 2, ..., 125, 126, 127],
[ 0, 1, 2, ..., 125, 126, 127],
[ 0, 1, 2, ..., 125, 126, 127]])
In [13]: Y
Out[13]:
array([[ 0, 0, 0, ..., 0, 0, 0],
[ 1, 1, 1, ..., 1, 1, 1],
[ 2, 2, 2, ..., 2, 2, 2],
...,
[125, 125, 125, ..., 125, 125, 125],
[126, 126, 126, ..., 126, 126, 126],
[127, 127, 127, ..., 127, 127, 127]])
In [14]: ax.plot_surface(X,Y,im)
Out[14]: <mpl_toolkits.mplot3d.art3d.Poly3DCollection at 0x7fb52cd8ad00>
```
Observant readers may notice that the plot is upside down compared
to the image. This is because, conventionally, $(0,0)$ is the
top left hand pixel, while a plot would usually place $(0,0)$ at the lower
left hand side.
## Thin Lens Model
### The Focus Point
+ A convex lens collect, or focus, parallel rays into a single focus point.
+ This works as a burning glass.
- The sun is so far away that the sun rays are parallel for all
practical purposes.
+ Definitions
- **Optical Axis** is the line perpendicular on the lens,
through its centre.
- The **Focus** is a point on the Optical Axis. Rays which enter the
lense parallel to the optical axis are deflected so that they intersect
at the Focus.
- **Focal Length** is the distance between the lens and the Focus.
(We ignore the thickness of the lens.)
- The **Focal Plane** is a plane through the Focus, perpendicular
on the Optical Axis.
### The Image Plane
+ The image plane
- non-parallel rays
+ The thin lens equation
+ Points further away
+ The aperture
## The pinhole model
+ Reference frame
+ Co-ordinates
## Geometry of Image Formation
# Exercises
# Debrief
# Credits
[Introduction to Psychology](https://open.lib.umn.edu/intropsyc)
by University of Minnesota is licensed under a
[Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-nc-sa/4.0/), except where otherwise noted.
[*College Physics*](http://cnx.org/contents/031da8d3-b525-429c-80cf-6c8ed997733a/College_Physics).
Authored by: OpenStax College.
License:
[CC BY: Attribution](https://creativecommons.org/licenses/by/4.0/).
License Terms: Located at License
