The Bichromatic Excitable
Schrodinger Metamedium
Valeri Labunets,Ivan Artemov
and Ekaterina Ostheimer
2
The excitable metamedium
The ways of modeling
Directly search for solutions of equations
Simulate with acellular automata
- hard, - slow,+ accurate
+ easy, + fast,- approx.
Cellular automata organization
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Transition function in it’s general form:
Current cell and it’s neighboring cells
Eliminating the edge effects
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Schrodinger transition function
Schrodinger equation in 2D space:
Expanding the Laplacian expression:
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Diffusion coefficient as a real number
In that case all cells also contain real numbers.
Propagation Interference Particle motion
Schrodinger equation turns intodiffusion equation
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Our motivation
Tasks
Physical Mathematical
Imageprocessin
g
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Diffusion coefficient as a complex numberThen 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
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The experiments for a complex D – particle motion
Step #19 Step #40 Step #70
Step #95 Step #125 Step #150
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The experiments for a complex D – interference
Step #10 Step #30 Step #50
Step #75 Step #95 Step #125
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The experiments for a complex D
Interference
Particlemotion
Close points Distant points
Arg{D} = 30 deg. Arg{D} = 60 deg. Arg{D} = 90 deg.
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Unusual geometries
Euclideangeometry
Minkowskigeometry
Galileangeometry
These have different definitions for imaginary unit’s square:
i2 = -1 i2 = +1 i2 = 0
12
Minkowski geometry
Formulas for the new kind of space:
Formulas for the Euclidean geometry (for the comparison):
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Galilean geometry
Propagation Interference
The color excitableSchrodinger metamedium
Valeri Labunets,Ivan Artemov
andEkaterina Ostheimer
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
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Color representations’ connection
RGB-format. A cell contains three real numbersHSL-format. A cell contains one real and one complex number
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Elementary cells’ componentsfor triplet-valued diffusion coefficient
Complex numberReal number
Single cell
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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
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Simple operations for triplet numbersTwo 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
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Advanced operation for triplet numbers
The multiplication of two triplet numbers:
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Visualization in our program
Visualizing Hue:
RGBPicture
Lightness
HueSaturation
An example of a color blending in the
program
Processing result’s output structure
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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
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Component-wise blurringTriplet numbers and real images
High Saturation of D High Lightness
of D
Images after some steps of
cellular automata’s
work
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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:
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Particle motion for color DWe 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
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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
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What if a chromatic planeuses 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
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Color D and the Galilean geometry for ZchrExcited cell progress (spot propagation) – D’s Hue
is 70°:
Step # 5 Step # 20 Step # 35
Step # 65 Step # 115 Step # 200
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Color D and the Galilean geometry for ZchrNow we are locking the work step and changing D’s
Hue valueOnly RGB results (top parts) and Hue components are
shown.
5° 20° 40° 60° 70° 80° 89° 90°
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Color D and Minkowski geometry for Zchr
1° 5° 45° 45°
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Questions?
Thank you for attention!