=Paper=
{{Paper
|id=Vol-1710/paper15
|storemode=property
|title=The Bichromatic Excitable Schrodinger Metamedium
|pdfUrl=https://ceur-ws.org/Vol-1710/paper15.pdf
|volume=Vol-1710
|authors=Valeri Labunets,Ivan Artemov,Ekaterina Ostheimer
|dblpUrl=https://dblp.org/rec/conf/aist/LabunetsAO16
}}
==The Bichromatic Excitable Schrodinger Metamedium==
https://ceur-ws.org/Vol-1710/paper15.pdf
The bichromatic excitable Schrodinger
metamedium
Ekaterina Ostheimer1 , Valery Labunets2 , and Ivan Artemov2
1
Capricat LLC, Florida, USA
katya@capricat.com
2
Ural Federal University, Yekaterinburg, Russia
vlabunets05@yahoo.com
Abstract. In this work, we apply quantum cellular automata (QCA)
to study pattern formation and image processing in quantum-diusion
Schrodinger systems (QDSS) with generalized complex diusion coe -
cients. Generalized complex numbers have the real part and imaginary
2 2
part with the imaginary unit i = −1 (classical case), i = +1 (dou-
2
ble numbers) and i = 0 (dual numbers). They form three 2-D complex
algebras. Discretization of the Schrodinger equation gives lattice based
metamaterial models with various complexvalued physical parameters.
The process of excitation in these media is described by the Schrodinger
equations with the wave functions that have values in algebras of the
complex, dual, double numbers. If a traditional computer is thought of
as a programmable object, QDSS in the form of QCA is a computer
of new kind and is better visualized as a programmable material. The
purpose of this work is to introduce new metamedium in the form of
cellular automata. The cells are placed in a 2-D array and they are capa-
ble of performing basic complex operating (in dierent complex algebras)
and exchanging messages about their state. Cellular automata like archi-
tectures have been successfully used for computer vision problems and
grey-level image processing. Such media possess large opportunities in
processing of bichromatic images in comparison with the ordinary diu-
sion media with the real-valued diusion coe cients. The latter media
are used for creation of the eye-prosthesis (so called the silicon eye). The
medium suggested can serve as the prosthesis prototype for perception
of the bichromatic images.
Keywords: image processing, Schrodinger equation, complex diusion,
cellular automata, Minkowski geometry, Galilean geometry, metamedium
Introduction
We examine a graphical representation of 2D Schrodinger equation solution
found with performing several cellular automata iterations. Initial conditions
for the equation are the same every time: we use a black square image with a
white point in the middle (it is the only cell in the automata that is not equal
to zero in the beginning). The only exception is the experiment with moving
particle, that will be described in the last section. More basic information about
cellular automata, such as it's application and properties, can be found in [1]
Cellular automata is a more fast, demonstrative and easy way of modeling
a metamedium than the method of the straight equation solving. We aren't the
rst men who use a cellular automata for such purposes. For example, in [2] and
[3] it is described how to use a quantum cellular automata for image processing.
Of course, we can achieve good results in image segmentation and edge detection
using QCA, but nobody proved that QCA with a "true imaginary" diusion
coe cient D will provide the best output. That's why it is important to start
researching QCAs with more general properties, where D is an ordinary complex
number with the phase that isn't equal to 90◦ .
We use well known expression for Schrodinger equation as the basis:
d2 d2
d
φ=D· φ + φ , (1)
dt dx2 dy 2
where φ(x, y, t) is a complex value of a cell, that is located at point with coordi-
nates (x, y) at time (or iteration number) t; D is a complex diusion coe cient.
For our discrete cellular implementation it useful to represent the Laplacian
in the braces in a dierent way:
d2
φ = φ(x + 1, y, t) + φ(x − 1, y, t) − 2φ(x, y, t), (2)
dx2
d2
φ = φ(x, y + 1, t) + φ(x, y − 1, t) − 2φ(x, y, t), (3)
dy 2
d
φ = φ(x, y, t + 1) − φ(x, y, t). (4)
dt
As a result we can obtain the expression for a cell state at the next iteration:
φ(x, y, t + 1) = φ(x, y, t) + D · (φ(x + 1, y, t)+
(5)
+φ(x − 1, y, t) + φ(x, y + 1, t) + φ(x, y − 1, t) − 4φ(x, y, t)).
It can be seen, that the last expressions implicitly contains a matrix of weight co-
e cients, that is frequently used in image processing for edge detection purposes
in so-called Laplacian lter (see [4]):
0 · φ(x − 1, y + 1, t) 1 · φ(x, y + 1, t) 0 · φ(x + 1, y + 1, t) 0 1 0
1 · φ(x − 1, y, t) −4 · φ(x, y, t) 1 · φ(x + 1, y, t) → 1 −4 1 . (6)
0 · φ(x − 1, y − 1, t) 1 · φ(x, y − 1, t) 0 · φ(x + 1, y − 1, t) 0 1 0
In this matrix we use only four closes neighbors of the central cell in the au-
tomata. More accurate and complex matrix masks can be found for the approx-
imation of laplacian (for example, see 'diamond mask' in [5]).
1 In uence of diusion coe cient's phase for Euclidean
geometry
We will use a xed D's absolute value, because it only has an impact on vi-
sualization. We needed to nd such |D|, that will provide quite fast propaga-
tion processes, but will not cause an over ow because of extremely high values.
|D| = 0.11 is very convenient for our purposes.
For a standard complex plane geometry we use ordinary formulas, that we
show only for a comparison with dierent "exotic" geometries. Let Z be a com-
plex number, R - a real number, φ - an angle in radians, i−1 = i : i2 = −1
then
q
Z = R · (cos(φ) + i−1 · sin(φ)); |Z| =
2
(Re{Z}) + (Im{Z}) ;
2
(7)
Re{Z} = R · cos(φ); Im{Z} = R · sin(φ). (8)
On Fig. 1 and Fig. 2 you can see the modeling results for complex coe cient
D with it's dierent phase values. Obviously, if arg{D} = 0◦ and we assign a
◦ ◦
Fig. 1. 128th iteration of complex diusion for D 's phases 0 (left image) and 5
2
(right image) using i = −1
value Zexcited = 1+j ·0 to the cell, that is just excited, then we will not have any
nonzero values for imaginary and phase components of all cells. It corresponds
to the heat equation realization (see [6]).
The nal images were inverted to decrease the amount of black color for
better visual perception, so big values correspond darker points.
Each image on our gures consists of four quads, that represent absolute
values of complex numbers in cells (top left quad), their phases (top right one),
real parts (bottom left one) and imaginary parts (bottom right quad).
It can be seen on Fig. 2 (right part), that real and imaginary parts are
uctuating in antiphase (white rings in the bottom quads aren't at the same
position: they are located between each other). It causes a smooth decreasing
complex module picture: there are no white rings of zero absolute value on it. If
we take a 'slice' of the left top quads then we will see, that cell's absolute value
is decreasing under the law of the Gaussian curve (see Fig. 3).
◦ ◦ 2
Fig. 2. 128th iteration of complex diusion for D 's phases 25 and 60 using i = −1
Fig. 3. A Gaussian curve showing normalized cell's absolute value distribution
2 Schrodinger equation for Minkowski geometry
For Minkowski geometry we have to use hyperbolic functions instead of ordinary
trigonometric ones [7]. If i+1 = i : i2 = +1 then
q
Z = R · (cosh(φ) + i+1 · sinh(φ)); |Z| =
2
(Re{Z}) − (Im{Z}) ;
2
(9)
Re{Z} = R · cosh(φ); Im{Z} = R · sinh(φ). (10)
The unit circle for a complex plane with Minkowski geometry have an abso-
lutely dierent shape. It has the form of a hyperbola (see Fig. 4). That's why we
have to use hyperbolic functions to get a complex number's coordinate on 2-D
plane. Also it is very important, that now we have a subtraction operation in the
expression for Z 's absolute value. It leads to the possibility of complex-valued
modules. For uniform visualization purposes, we take either real or imaginary
part of |Z| respectively.
On Fig. 5 you can see the modeling results for the same initial conditions (a
single excited central cell) in Minkowski geometry for small phase values.
Note, that there are no more waves in real and phase parts of our cellular
array. The amount of waves in the imaginary parts section isn't increasing with
growth of D's phase. The ring of high phase values appears in the top right quad.
Fig. 4. Hyperbola is the unit "circle" for the Minkowski geometry
◦ ◦ 2
Fig. 5. 128th iteration of complex diusion for D 's phases 5 and 20 using i = +1
3 Complex diusion coe cient for Galilean geometry
Galilean geometry provides us with the simplest formulas for complex numbers
at new complex plane [8]. If i0 = i : i2 = 0 then
Z = R · (1 + i0 · tan(φ)); |Z| = Re{Z}; (11)
Re{Z} = R · 1; Im{Z} = R · tan(φ). (12)
In Galilean geometry the unit circle turns into a simple vertical line. It causes
the |Z|'s and Re{Z}'s independence from angle φ. Also notice that Im{Z} can
take enormously big values because of tangent function in it's expression. We
used one joint normalization for quads, that represent real and imaginary parts
of cells' values to show the balance between these two components of complex
numbers.
On Fig. 6 you can see the modeling results for Galilean geometry case.
As you can see, in this case we have the same ring of equal high phases,
but it is forming more slowly, and the values in the real parts section start to
fade. When we increase arg{D}, it can be seen that while arg{D} < 45◦ , overall
imaginary component of cells is quite small with respect to the real part, but if
arg{D} > 45◦ imaginary part starts to overbalance.
◦ ◦ 2
Fig. 6. 128th iteration of complex diusion for D 's phases 50 and 80 using i = 0
4 Complex diusion process for smoothly changing
imaginary unit's square value
We can generalize our experiment by using not only imaginary units, with their
squared value −1, 0 or 1. We can de ne new variable ik = i : i2 = k and use
some new expressions, that are right for every geometry kind:
(13)
p
Z = a + ik · b; |Z| = a2 − k · b2 ;
sink (φ) b
tank (φ) = = , (14)
cosk (φ) a
where we de ne some generalized trigonometric functions:
a b
Z = |Z| · + ik · 2 = |Z| · (cosk (φ) + ik · sink (φ)). (15)
a2 − k · b2 a − k · b2
The similar approach can be seen in [9]. There are modeling results for
dierent k values on Fig. 7. As you can see, the circle of zero phases, that we got
Fig. 7. Phase values of cells (i.e. only top right quad is shown) for k = −0.25, k = −0.05
◦
and k = 0 for constant arg{D} = 40.5 , |D| = 0.07 on 128th iteration
for Galilean geometry (K = 0), is also the rst inner ring for usual Euclidean
complex geometry. So Fig. 7 can bring us the conclusion: by decreasing the value
of k, i2k = k we increase the distance between rings of zero phases for a spot that
we obtain from one excited point.
5 Experiments with interference
Until that moment we were testing the behavior of a single point, that was
excited in our metamedia. Now let us see what will happen if we consider the
interaction of two points, that are excited at the same moment.
Fig. 8 shows the results of such experiments for the simplest case: D ∈ R, i2 =
−1. This corresponds to diusion equation (or the heat equation) solving, where
we can see a simple spot blending without any wave processes and uctuations.
◦
Fig. 8. Simple interference for two points in the media with arg{D} = 0 , i2 = −1
More interesting results can be seen on Fig. 9, where we used a complex
metamedia with D ∈ C, arg{D} = 6 0◦ . In this case complex diusion coe cient's
phase is equal to 90 = 2 , so the results can be considered as Schrodinger
◦ π
equation solution.
Fig. 9. Interference pictures for two points, that are located relatively close (left image)
◦
to each other and far from each other (right one). Euclidean geometry, arg{D} = 90
It can be seen, that the output depends on the range between two points.
Also a dierent result can be achieved if two points are excited with a relative
delay (not at the same moment). It means that the phases of uctuations inside
of spots are displaced.
On Fig. 10 you can see the result of interference for metamedia with Galilean
complex geometry (i2 = 0). Note, that the white rings of zero phases don't
intersect between each other. They have a smooth connection instead.
◦
Fig. 10. Interference pictures for two points. Galilean geometry, arg{D} = 60
6 Particle movement modeling for dierent phases of
diusion coe cient
Very unusual and interesting results can be achieved, if we create the sequence
of white points, that are lying on a circle trajectory step by step instead of using
only one point at the center as an initial condition. We use well known equation
for the evaluation of coordinates (x(t), y(t)) for next excited point's position on
2-D plane:
x2 (t) + y 2 (t) = R2 ;
x(t) = R · cos(V · t), (16)
y(t) = R · sin(V · t),
where R is the radius of our circle and V is the parameter of rotation speed.
This algorithm can be used for particle movement simulation within our excitable
media with unusual laws.
Fig. 11 and Fig. 12 shows the algorithm functioning in action for high and
low D's complex phase values. Euclidean geometry is used for this experiment,
because it provides the most signi cant results.
The trail in phase and imaginary part quads is being formed when we increase
D's phase value. Notice, that at rst there are no wave interference processes in
the quads, that represent real parts and absolute values of cells. When arg{D}
reaches 60◦ or more, we get a new process' detail. A white line, that divides a
new part of a particle's trail from the older parts, comes into sight in both real
and module quads.
The further increase of D's phase leads to the same interference processes
between a particle's train parts, as we could see at Fig. 9. Note that the module
◦
Fig. 11. Particle movement simulation on the xed iteration for D 's phases 0 (left
◦
image) and 30 (right image)
◦
Fig. 12. Particle movement simulation taken on step 128 for D 's phases 60 (left image)
◦
and 90 (right image)
values of cells, that are located close to our circle trajectory aren't constant: they
are fading. It is because the uctuations aren't in-phase. But we can suppose
that it is possible to nd such value of rotation speed V , that will cause mutual
"maintenance" of all trail parts and will show us a steady-state.
7 Conclusion
As we can see, excitable media's reaction on the same impact is heavily depen-
dent on complex diuse coe cient's parameters, such as its phase. The choice of
geometry type is even more important: switching its kind from one to another
drastically changes the experiments' results. "Exotic" excitable medias that we
have introduced in this work can provide new possibilities for quantum image
processing approaches.
The future work will be related to digital image processing. Many edge detec-
tion, pattern recognition and denoising algorithms, that are implemented with
Schrodinger equation for ordinary complex plane type, should be tested with
Galilean and Minkowski geometry and with dierent D's phase values. It is
possible to achieve better processing results with our new metamedia.
8 Acknowledgement
This work was supported by the Ural Federal University's Center of Excellence in
"Quantum and Video Information Technologies: from Computer Vision to Video
Analytics" (according to the Act 211 Government of the Russian Federation,
contract 02.A03.21.0006)
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