From Playstation to Hospitals
Studium Generale - 8th November 2019

Henrik Bachmann

Nagoya University
Graduate school of mathematics

www.henrikbachmann.com

About me•

Born in Hamburg (Germany)•

Studied mathematics at HamburgUniversity•

Since last month Associate Professor at
Nagoya University in the G30 Program•

Interested in Number theory

Mathematics….

Do you like mathematics?

YesNoI always liked mathematics inschool / university. It is a reallycool subject!

I accept that mathematics might
be important, but I was never a
big fan of it.

Ein Beispiel / An Example

Heute möchte ich über Mathematik sprechen
Today want

I about„Heute möchteich überMathematiksprechen“

mathtalk“Today I
want totalk aboutmath”

Google Translate

The magic machine
In general we often have something like this….

Input

OutputData we can notunderstande.g.
• Text in different
language• Something wemeasure bysensors• ….

Data we canunderstand(and manipulate)
Magic machine

Often the „inverse“ machine also exists

Today: Discuss two explicit examples

Karaoke / Playstation SingStar• Playstation SingStar is a competitive karaoke game.
• You score by singing a song in the correct pitch.

SingStar: How does it work?

SingStar machineSoundwaves
Sound is a vibration that typically propagates as an audible wave ofpressure, through a transmission medium such as a gas, liquid or solid.

Pressure

Time
10ms

A : 440Hz

C : 261Hz

SingStar: Piano version
10ms
A : 440Hz

C : 261Hz

30ms

SingStar machine
30ms

Can you be a SingStar machine?
Which three keys
are played here?

10ms

Yes:

C+G+B

No:

D+F+B

Can you be a SingStar machine?

Correct: No
D+F+B

C+G+B

Some linear algebra…
N

Movement 1 (M1): Go 1 step east
Movement 2 (M2): Go 1 step northEWS+ +- -Some linear algebra…
N

Movement 1 (M1): Go 1 step east
Movement 2 (M2): Go 1 step north+ +- -E

W+SSome linear algebra…
N

Movement 1 (M1): Go 1 step east
Movement 2 (M2): Go 1 step north+ +- -E

W+ +S

Some linear algebra…
N

Movement 1 (M1): Go 1 step east
Movement 2 (M2): Go 1 step north

EW+ +- -+ + +S

Some linear algebra…
N

Movement 1 (M1): Go 1 step eastMovement 2 (M2): Go 1 step northE

W+ +- -+ + + +S

Some linear algebra…
N

Movement 1 (M1): Go 1 step east
Movement 2 (M2): Go 1 step northE

W+ +- -+ + + + +S

Some linear algebra…
N

Movement 1 (M1): Go 1 step east
Movement 2 (M2): Go 1 step northE

W+ +- -+ + + + +S

2 (M1) + 3 (M2)

Some linear algebra…
N

Movement 1 (M1): Go 1 step east
Movement 2 (M2): Go 1 step north+ + + + +2 (M1) + 3 (M2)

Movement 3 (M3): Go 1 stepE east & 1 step south
Movement 4 (M4): Go 2 steps
west & 3 steps north

W+ +- SSome linear algebra…
N

Movement 1 (M1): Go 1 step east
Movement 2 (M2): Go 1 step north+ + + + +2 (M1) + 3 (M2)

Movement 3 (M3): Go 1 stepE east & 1 step south
Movement 4 (M4): Go 2 steps
west & 3 steps north

W+ +- S+

Some linear algebra…
N

Movement 1 (M1): Go 1 step east
Movement 2 (M2): Go 1 step north+ + + + +2 (M1) + 3 (M2)

Movement 3 (M3): Go 1 stepE east & 1 step south
Movement 4 (M4): Go 2 steps
west & 3 steps north

W+ +- S+ +

Some linear algebra…
N

Movement 1 (M1): Go 1 step east
Movement 2 (M2): Go 1 step north+ + + + +2 (M1) + 3 (M2)

Movement 3 (M3): Go 1 stepE east & 1 step south
Movement 4 (M4): Go 2 steps
west & 3 steps north

W+ +- S+ + +Some linear algebra…
N

Movement 1 (M1): Go 1 step east
Movement 2 (M2): Go 1 step north+ + + + +2 (M1) + 3 (M2)

Movement 3 (M3): Go 1 stepE east & 1 step south
Movement 4 (M4): Go 2 steps
west & 3 steps north

W+ +- S+ + + +Some linear algebra…
N

Movement 1 (M1): Go 1 step east
Movement 2 (M2): Go 1 step north+ + + + +2 (M1) + 3 (M2)

Movement 3 (M3): Go 1 stepE east & 1 step south
Movement 4 (M4): Go 2 steps
west & 3 steps northW

How canreach?

+ +- -? (M3) + ? (M4) = 3 (M2) + 2 (M1)
S+ + + +2 (M4) + 2 (M3)

Some linear algebra…a bit more serious
Vector Notation:

Matrix * Vector = Vector+ + + + + + + + + + + ++ + + + +Basis change
These two sets of „movements“ are examples of bases for the 2-dimensional space.
• Every point can be reached• There is a unique way to reach a point

Basis 1 (Standard basis)

Basis 2
Change of basis

Back to SingStar

D+F+B

How to we get thisresult using math?

Back to SingStar: Recording with a microphone

10ms

Sampling rate 1KHz(measures 10 values in 10ms)

SoundwaveVector

Back to SingStar: Recording with a microphone
But we can do that for each Key seperately first!

10ms

Sampling rate 1KHz
(measures 10 values in 10ms)

Soundwave

VectorSingStar: Just Linear Algebra
We obtain the same question as before=C+ D+ …

Find numbers C,D,… such that this equations holds

SingStar: Just Linear Algebra
Solution=Interpretation of the solution:

The keys D,F and B wereplayed with the same
volume(all values are the same)SingStar: How does it work?

SingStar machine

Soundwave

Vector(of pressure data)

Vector(of amplitudes, i.e.
Intensity in each frequency)

Matrix multiplication(basis change)

Discrete Fouriertransform

The highest values
gives the currentpitchFourier transform
The (discrete) Fourier transform has various applications• Digital filter
SoundFrequencies

Discrete Fouriertransform
Change / Remove
frequencies
Inverse Discrete
Fourier transform• Image processing• Data compressions (JPEG)• Appears in various areas of mathematics and physics

A little bit history
Today: Friday 8th November, 2019
124 years ago: Friday 8th November, 1895

Today 124 years ago W. C. Roentgen discovered
X-Rays (Röntgenstrahlen, レントゲン線)
For this discovery he obtained the firstNobel Prize in Physics (1901)

Wilhelm Conrad Röntgen
1845 - 1923

X-raysDifferent point of view?

Computed tomography scan (CT Scan)

CT Scanner(without cover)
Basic principle of tomography(tomos = slice/section)

X-Ray from different angles: SinogramX-Ray from different angles: Sinogram

Sinogram(taking X-rayfrom differentangels)angle (0 – 180)

Original object

Sinogram: Can you invert it?
Is this the Sinogramof S or G?

YesNo

Sinogram: Can you invert it?

Correct: Yes

CT Scan – How does it work?

CT ScanRadon transform

Radon transform (sorry a little bit math)

This transform can be inverted byusing the Fourier transform and the„Projection slice theorem“.

Backprojection: From Sinogram to the original object• One easy way: Simple backprojection

Results are blurry!

In reality filtered backprojection is used
(uses discrete Fourier transform)

Summary: Math gives magic machines

Discrete Fouriertransform

Filteredbackprojection

Some problems are too hard..
In reality, we also have problems where it is reallyhard/impossible to create a magic machine.
Example: Ramen classification
Input
Picture of Ramen?????

OutputWhat type ofRamen is it?
Taiwan Ramen?
Tonkotsu Ramen?
…

Machine learning
We can simulate “brains” and teach them!
Tonkotsu
TaiwanTaiwanNeural network

Taiwan Ramen!
… with 95% certainty!

Ramen classifier

Machine learning

In Fall 2020 I am planning to offer a „Math for machinelearning“ course in the G30 Program.

Thank you very much for your attention!