The bichromatic excitable Schrodinger metamedium

1. The Bichromatic
Excitable
Schrodinger Metamedium
Valeri Labunets,
Ivan Artemov
and
Ekaterina Ostheimer

2. 2
The excitable metamedium
The ways of modeling
Directly search for
solutions of equations
Simulate with a
cellular automata
- hard,
- slow,
+ accurate
+ easy,
+ fast,
- approx.

3. Cellular automata organization
3
Transition function in it’s general form:
Current cell and it’s
neighboring cells
Eliminating the edge effects

4. 4
Schrodinger transition function
Schrodinger equation in 2D space:
Expanding the Laplacian expression:

5. 5
Diffusion coefficient as a real number
In that case all cells also contain real numbers.
Propagation Interference Particle motion
Schrodinger equation turns into
diffusion equation

6. 6
Our motivation
Tasks
Physical Mathematical
Image
processing

7. 7
Diffusion coefficient as a complex number
Then all cells contain complex numbers, that have
module, phase, real part and imaginary part.
Propagation,
Arg{D} = 5 deg.
Propagation,
Arg{D} = 25 deg.
Propagation,
Arg{D} = 60 deg.
Module Phase
Real Imaginary

8. 8
The experiments for a complex D – particle motion
Step #19 Step #40 Step #70
Step #95 Step #125 Step #150

9. 9
The experiments for a complex D – interference
Step #10 Step #30 Step #50
Step #75 Step #95 Step #125

10. 10
The experiments for a complex D
Interference
Particle
motion
Close points Distant points
Arg{D} = 30 deg. Arg{D} = 60 deg. Arg{D} = 90 deg.

11. 11
Unusual geometries
Euclidean
geometry
Minkowski
geometry
Galilean
geometry
These have different definitions for imaginary unit’s square:
i2 = -1 i2 = +1 i2 = 0

12. 12
Minkowski geometry
Formulas for the new kind of space:
Formulas for the Euclidean geometry (for the comparison):

13. 13
Galilean geometry
Propagation Interference

14. The color excitable
Schrodinger metamedium
Valeri Labunets,
Ivan Artemov
and
Ekaterina Ostheimer

15. 15
Diffusion coeff. as a triplet (color) number
“D” coefficient
Red ∈ [0; 255]
Green ∈ [0; 255]
Blue ∈ [0; 255]
Hue ∈ [0°; 360°]
Saturation ∈ [0; 255]
Lightness ∈ [0; 765]
All cells will now have these components

16. 16
Color representations’ connection
RGB-format. A cell contains
three real numbers
HSL-format. A cell contains one real
and one complex number

17. 17
Elementary cells’ components
for triplet-valued diffusion coefficient
Complex number
Real number
Single cell

18. 18
Triplet (color) numbers
We have to define operations, that can be applied
to the introduced triplet numbers
Triplet num. 1 Triplet num. 2 Triplet num. 3+
x
=
Triplet num. 1 Triplet num. 2 Triplet num. 4=
xTriplet num. 1 Triplet num. 5=Real number

19. 19
Simple operations for triplet numbers
Two of the previously mentioned operations with
triplet numbers are component-wise:
The rule for sum of two triplet numbers
The multiplication of a real and a triplet number

20. 20
Advanced operation for triplet numbers
The multiplication of two triplet numbers:

21. 21
Visualization in our program
Visualizing
Hue:
RGB
Picture
Lightness
HueSaturation
An example of a color
blending in the program
Processing result’s
output structure

22. 22
The impact of luminance and saturation
The states for low Lightness of D
The states for low Saturation of D
The pairs of
metamedium states
are presented.
The neighboring
images are taken on
different steps of
cellular automata’s
functioning
Step #5
Step #5
Step #60
Step #60

23. 23
Component-wise blurring
Triplet numbers and real images
High Saturation of D High Lightness of D
Images after
some steps of
cellular
automata’s work

24. 24
The impact of a chromatic number’s phase
Some steps of cellular automata’s work
for D’s Hue = 5°
Some steps of cellular automata’s work
for D’s Hue = 30°
Top parts of all pictures represent final RGB images,
and the bottom parts represent Hue values of cells
We can change Hue of a diffusion coefficient:

25. 25
Particle motion for color D
We excite a new cell at every step of functioning (D’s Hue is 50°)
Step # 1 Step # 16 Step # 70
Step # 100 Step # 136 Step # 176
Lightness
Saturation
RGB
Hue

26. 26
Particle motion for color D
With D’s Hue = 5° With D’s Hue = 60° With D’s Hue = 75°
The represented results are taken on quite late steps
of cellular automata’s functioning
Let’s see how D’s Hue changes the particle motion process

27. 27
What if a chromatic plane
uses some unusual geometry
The the rules of composing a chromatic number
of R, G and B components will change.
Hue and Saturation (i.e. the module and the
phase) will be represented through Xchr and
Ychr in a different way.
All the previously shown experiments were
performed for an Euclidean geometry in Zchr

28. 28
Color D and the Galilean geometry for Zchr
Excited cell progress (spot propagation) – D’s Hue is 70°:
Step # 5 Step # 20 Step # 35
Step # 65 Step # 115 Step # 200

29. 29
Color D and the Galilean geometry for Zchr
Now we are locking the work step and changing D’s Hue value
Only RGB results (top parts) and Hue components are shown.
5° 20° 40° 60° 70° 80° 89° 90°

30. 30
Color D and Minkowski geometry for Zchr
1° 5° 45° 45°

31. 31
Questions?

32. Thank you for attention!