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 coecients. Generalized complex numbers have the real part and imaginary part with the imaginary unit i2 = 1 (classical case), i2 = +1 (double numbers) and i2 = 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 capable of performing basic complex operating (in dierent complex algebras) and exchanging messages about their state. Cellular automata like architectures 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 diusion media with the real-valued diusion coecients. 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.
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 [
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 [
We use well known expression for Schrodinger equation as the basis: d dt = D d2 dx2 + d2 dy2 ; (1) (2) (3) (4) (5) where (x; y; t) is a complex value of a cell, that is located at point with coordinates (x; y) at time (or iteration number) t; D is a complex diusion coecient.
For our discrete cellular implementation it useful to represent the Laplacian
in the braces in a dierent way:
It can be seen, that the last expressions implicitly contains a matrix of weight
coecients, that is frequently used in image processing for edge detection purposes
in so-called Laplacian lter (see [
Inuence of diusion coecient’s phase for Euclidean geometry We will use a xed D’s absolute value, because it only has an impact on visualization. We needed to nd such jDj, that will provide quite fast propagation processes, but will not cause an overow because of extremely high values. jDj = 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 complex number, R - a real number, - an angle in radians, i 1 = i : i2 = 1 then
Z = R (cos( ) + i 1 sin( )); jZj = q (RefZg)2 + (ImfZg)2; (7)
RefZg = R cos( ); ImfZg = R sin( ): (8)
On Fig. 1 and Fig. 2 you can see the modeling results for complex coecient
D with it’s dierent phase values. Obviously, if argfDg = 0 and we assign a
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 [
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).
For Minkowski geometry we have to use hyperbolic functions instead of ordinary
trigonometric ones [
Z = R (cosh( ) + i+1 sinh( )); q jZj = (RefZg)2 (ImfZg)2; RefZg = R cosh( );
ImfZg = R sinh( ):
The unit circle for a complex plane with Minkowski geometry have an absolutely 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 jZj 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.
(9)
(10)
Galilean geometry provides us with the simplest formulas for complex numbers
at new complex plane [
Z = R (1 + i0 tan( ));
jZj = RefZg; RefZg = R 1;
ImfZg = R tan( ): (11) (12)
In Galilean geometry the unit circle turns into a simple vertical line. It causes the jZj’s and RefZg’s independence from angle . Also notice that ImfZg 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 argfDg, it can be seen that while argfDg < 45 , overall imaginary component of cells is quite small with respect to the real part, but if argfDg > 45 imaginary part starts to overbalance. D’s phases 50 and 80 using i2 = 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 dene new variable ik = i : i2 = k and use some new expressions, that are right for every geometry kind:
Z = a + ik b; jZj = pa2
k b2; sink( ) b tank( ) = = ;
cosk( ) a where we dene some generalized trigonometric functions: (13) (14) Z = jZj a2 a k b2 + ik a2 b k b2
= jZj (cosk( ) + ik sink( )): (15)
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 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.
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 2 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.
More interesting results can be seen on Fig. 9, where we used a complex metamedia with D 2 C; argfDg 6= 0 . In this case complex diusion coecient’s phase is equal to 90 = 2 , so the results can be considered as Schrodinger equation solution.
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.
Particle movement modeling for dierent phases of diusion coecient 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) + y2(t) = R2; x(t) = R cos(V t); 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 signicant 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 argfDg 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 (16) 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
As we can see, excitable media’s reaction on the same impact is heavily dependent on complex diuse coecient’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 detection, 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.
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)