The Bichromatic Excitable

Schrodinger Metamedium

Valeri Labunets,Ivan Artemov

and Ekaterina Ostheimer

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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

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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!