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---
title: Introductory Session to Machine Learing
categories: session 3D
---
# Reading
+ Ma 2004 Chapter 1.
# Session
1. **Briefing** [Introduction](http://www.hg.schaathun.net/talks/maskinsyn/intro.html)
2. **Exercise** [Python Setup](#practical) (below)
3. **Debrief**
+ Recap on linear algebra (introductory slides continued)
+ Lecture Notes on [Change of Basis]()
# Practical: Python Setup {#practical}
The most important task today is to install and test a number
of software packages needed.
## Install Python
We will use Python 3 in this module.
You need to install the following.
1. [Python](https://www.python.org/downloads/)
2. [pip](https://packaging.python.org/tutorials/installing-packages/).
(the package manager for python)
3. [iPython](https://ipython.org/install.html)
(a more convenient interactive interpreter)
How you install these three packages depends on your OS.
In most distroes you can use the package system to install all of this,
for instance in Debian/Ubuntu:
```sh
sudo apt-get install ipython3 python3-pip
```
(Python is installed automatically as a dependency of iPython3.
Note, you have to specify version 3 in the package names,
lest you get Python 2.)
## Install Python Packages
Python packages are installed most easily using python's own
packaging tool, pip, which is independent of the OS.
It is run from the command line.
Depending on how you installed pip, it may be a good idea to upgrade
```sh
pip3 install --upgrade pip
```
Then we install the libraries we need.
You can choose to install either in user space or as root.
User space:
```sh
pip3 install --user matplotlib numpy opencv-python
```
As root:
```sh
sudo pip3 install matplotlib numpy opencv-python
```
+ numpy is a standard library for numeric computations.
In particular it provides a data model for matrices with
the appropriate arithmetic functions.
+ matplotlib is a comprehensive library for plotting, both in 2D and 3D.
+ [OpenCV](https://opencv.org/)
is a Computer Vision library, written in C++ with bindings for
several different languages.
A third installation alternative is to use
[Virtual Environments](https://docs.python.org/3/tutorial/venv.html),
which allows you to manage python versions and dependencies separately
for each project. This may be a good idea if you have many python
projects, but if this is your first one, it is not worth the hassle.
## Run iPython
Exactly how you run iPython may depend on you OS.
In Unix-like systems we can run it straight from the command line:
```sh
ipython3
```
This should look something like this:
```
georg$ ipython3
Python 3.7.3 (default, Jul 25 2020, 13:03:44)
Type 'copyright', 'credits' or 'license' for more information
IPython 7.3.0 -- An enhanced Interactive Python. Type '?' for help.
In [1]: print("Hello World")
Hello World
In [2]: import numpy as np
In [3]: np.sin(np.pi)
Out[3]: 1.2246467991473532e-16
In [4]: np.sqrt(2)
Out[4]: 1.4142135623730951
In [5]:
```
## Some 3D Operations
In this chapter, we will define a simple 3D Object and display it in python.
The 3D object is an irregular tetrahedron, which has four corners and four
faces.
```python
import numpy as np
from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d.art3d import Poly3DCollection
```
Firtsly, we define the three corners of the tetrahedron.
```python
corners = [ [-1,0.5,0.5], [+1,0.5,0.5], [0,-0.5,0.5], [0,0.5,-0.5] ]
```
Each face is adjacent to three out of the four corners, and can
also be defined by these corners.
```python
face1 = [ corners[0], corners[1], corners[2] ]
face2 = [ corners[0], corners[1], corners[3] ]
face3 = [ corners[0], corners[2], corners[3] ]
face4 = [ corners[1], corners[2], corners[3] ]
```
To represent the 3D structure for use in 3D libraries,
we juxtapose all the faces and cast it as a matrix.
```python
vertices = np.array(face1+face2+face3+face4,dtype=float)
print(vertices)
```
Observe that the vertices (corners) are rows of the matrix.
The mathematical textbook model has the corners as columns, and
this is something we will have to deal with later.
We define the 3D object `ob` as follows.
```python
ob = Poly3DCollection(vertices, linewidths=1, alpha=0.2)
```
The `alpha` parameter makes the object opaque.
You may also want to play with colours:
```python
ob.set_facecolor( [0.5, 0.5, 1] )
ob.set_edgecolor([0,0,0])
```
To display the object, we need to create a figure with axes.
```python
plt.ion()
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
plt.show()
```
Note the `plt.ion()` line. You do not use this in scripts, but in
`ipython` it means that control returns to the prompt once the
figure is shown. It is necessary to continue modifying the plot after
it has been created.
Now, we can add our object to the plot.
```python
ax.add_collection3d(ob)
```
Quite likely, the object shows outside the range of the axes.
We can fix this as follows:
```python
s = [-2,-2,-2,2,2,2]
ax.auto_scale_xyz(s,s,s)
```
These commands make sure that the axes are scalled so that the two
points `(-2,-2,-2)` and `(2,2,2)` (defined in the list `s`)
are shown within the domain.
## Rotation and Translation of 3D objects
Continuing on the previous section, our 3D object `ob` is defined
by the `vertices` matrix, where all the rows are points in space.
Motion is described by matrix operations on `vertices`.
### Translation
Let us define another vector in $\mathbb{R}^3$ and add it
to each point.
```python
translation = np.array( [ 1, 0, 0 ], dtype=float )
v2 = vertices + translation
print(v2)
```
Not that this operation does not make sense in conventional
mathematics. We have just added a $1\times3$ matrix to an $N\times3$ matrix.
How does python interpret this in terms of matrices?
To see what this means visually in 3D space, we can generate
a new 3D object from `v2`. We use a different face colour
for clarity.
```python
ob2 = Poly3DCollection(v2, linewidths=1, alpha=0.2)
ob2.set_facecolor( [0.5, 1, 0.5] )
ob2.set_edgecolor([0,0,0])
ax.add_collection3d(ob2)
```
How does the new object relate to the first one in 3D space?
**TODO** Check
### Rotation
In the previous test, we added a vector to the nodes in the
3D polyhedron. Let's try instead to multiply by a matrix, like this:
```python
theta = np.pi/6
R = np.array( [ [ np.cos(theta), -np.sin(theta), 0 ],
[ np.sin(theta), np.cos(theta), 0 ],
[ 0, 0, 1 ] ], dtype=float )
v3 = np.matmul(vertices,R)
print(v3)
```
This gives us a new polyhedron, like this:
```python
ob3 = Poly3DCollection(v3, linewidths=1, alpha=0.2)
ob3.set_facecolor( [1, 0.5, 0.5] )
ob3.set_edgecolor([0,0,0])
ax.add_collection3d(ob3)
```
**TODO** Check
### Removing a 3D Object
Unfortunately, `matplotlib.pyplot` is not designed for interactive
construction or animation of 3D graphics, so some things are little
bit tricky. However, it is possible to remove an existing object
from the plot. Assume we still have the objects from the last few
sections.
First of all, let's look at the objects we have plotted:
```
In [12]: ax.collections
Out[12]:
[<mpl_toolkits.mplot3d.art3d.Poly3DCollection at 0x7fdde8a074f0>,
<mpl_toolkits.mplot3d.art3d.Poly3DCollection at 0x7fdde89fe400>,
<mpl_toolkits.mplot3d.art3d.Poly3DCollection at 0x7fddf39b28b0>]
In [13]:
```
Here, `ax` is the axes system of the figure, and `ax.collections`
is a collection of all of the objects that we have plotted.
We can quite simply delete one and refresh the figure.
```python
del ax.collections[1]
plt.show()
```
What happens?
## Some Camera Operations
```python
import numpy as np
import cv2 as cv
cap = cv.VideoCapture(0)
ret, frame = cap.read()
```
Now, `ret` should be `True`, indicating that a frame has
successfully been read. If it is `False`, the following
will not work.
```python
gray = cv.cvtColor(frame, cv.COLOR_BGR2GRAY)
cv.imshow('frame', gray)
cv.waitKey(1)
```
You should see a greyscale image from your camera.
To close the camera and the window, we run the following.
```python
cap.release()
cv.destroyAllWindows()
```
This example is digested from the tutorial on
[Getting Started with Videos](https://docs.opencv.org/master/dd/d43/tutorial_py_video_display.html).
You may want to do the rest of the tutorial.
## Showing a frame (optional)
The following example may visualise rotations somewhat better
than the tetrahedron given above. Try it, and observe to learn
how it workse.
**Step 1** Load libraries and set up a figure with axis system.
```python
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d.art3d import Poly3DCollection
ax = plt.figure().add_subplot(projection='3d')
```
We can visualise just the unit vectors, using a
quiver plot. We make the `qplot` function so
that we can plot a rotated version afterwards.
```python
e1 = np.array([1,0,0])
e2 = np.array([0,1,0])
e3 = np.array([0,0,1])
def qplot(e1,e2,e3,**kw):
ax.quiver(0, 0, 0, *e1, colors="r",**kw )
ax.quiver(0, 0, 0, *e2, colors="g",**kw )
ax.quiver(0, 0, 0, *e3, colors="b",**kw )
qplot(e1,e2,e3)
```
If you want to, you can show the plot at this stage.
Now we rotate the unit vectors, and make a new quiver
plot.
```python
R = np.array([
[ 0.1729, -0.1468, 0.9739],
[ 0.9739, 0.1729, -0.1468],
[ -0.1468, 0.9739, 0.1729] ])
qplot(np.matmul(R,e1),np.matmul(R,e2),np.matmul(R,e3))
```
Finally, we need to scale and show the plot.
The `set_xlim3d` and similar functions make
arrow heads.
```python
s = [-2,-2,-2,2,2,2]
ax.auto_scale_xyz(s,s,s)
ax.set_xlim3d([-2.0, 2.0])
ax.set_xlabel('X')
ax.set_ylim3d([-2.0, 2.0])
ax.set_ylabel('Y')
ax.set_zlim3d([-2, 2])
ax.set_zlabel('Z')
plt.show()
```
## STL Files (even more optional)
If you have time, you can install the numpy-stl library as well. We shall use it [next week](3D Objects in Python#stl-files-and-the-stl-library).
