In this work, we apply quantum cellular automata (QCA) to study pattern formation and image processing in quantum-diffusion Schrödinger metamedia with generalized complex diffusion coefficients. 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 Schrödinger equation gives the quantum Schrödinger cellular automaton with various complex-valued physical parameters. The process of excitation in these media is described by the Schrödinger equations with the wave functions that have values in algebras of the generalized complex numbers. This medium can be 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.
The metamedia (metamaterials), in which the electro dynamical, thermal and other physical parameters have "exotic" values
(negative, imaginary, complex or quaternion ones), shows us the wonderful diversity of dynamic behavior and self-organization
types. It is becoming more and clearer that such systems are not exclusive: when researchers try to investigate the nature of
complex systems - chemical, biological or physical, - they find many of certain examples. In particular, this fact mainly refers to
biological systems, because these systems are always quite far from stable state and their parameters frequently have exotic
values. A theoretical quantum brain model was proposed in [
Linear and nonlinear Schrodinger equations [
In this work, we apply quantum cellular automata to study pattern formation and image processing in quantum-diffusion Schrodinger metamedia with generalized complex diffusion coefficients. 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 Schrödinger equation gives the quantum Schrödinger cellular automaton with various complex–valued 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 generalized complex numbers. This medium can be 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.
The rest of the paper is organized as follows: in Section 2, the object of the study (the Schrödinger equation) is described. In Section 3, a brief introduction to mathematical background (algebra A2 (R | i), i2 1, 0 of generalized complex numbers z a ib ) is given (subsection 3.1) in order to understand the concept behind the proposed method. In subsection 3.2, the proposed method based on Schrödinger equations is explained. Next, we defined Schrödinger transform of image, discussed its properties and the properties of the Schrödinger transforms are analyzed. In Section 4, the basic metamedia (the SchrödingerEuclidean, Schrödinger-Minkowskian, Schrödinger-Galilean, Schrödinger-Yaglom) are devised and analyzed in detail. The simulation result and algorithm complexity are demonstrated too. Finally, we gave our conclusion in Section 5.
2 (x, y, t) D x2 2 (x, y, t)
f x, y, t , y2 (1) where (x, y, t) is a function describing the media's excitement, f x, y,t is an exciting source (input signal) and D is a diffusion coefficient (real number).
The main purpose of this work is the investigation of derivative laws for Schrödinger metamedia with generalized complex
diffusion coefficient in the form of quantum cellular automata. The generalized complex numbers [
Excitation of waves in metamedia are described by three Schrödinger equations with a A2 (R | i) -valued wave functions
(x, y, t) . The discretization of the Schrödinger equations gives us a metamedia models in the form of three excitable cellular
automata. Their microelectronic realizations appear to be a programmable Schrodinger metamedia [
In this work, we study properties of the Schrödinger excitable metamedium in the form of a cellular automaton. The more
detailed information about cellular automata can be found in [
are different [
We consider the algebraic and geometric properties of three 2-D complex algebras A2 (R | i) : z x iy | x, y R, where i i ,i0 ,i . Additions for all three algebra are identical: z1 z2 x1 iy1 x2 iy2 x1 x2 i y1 y2 , but multiplications x1x2 y1 y2 i x1 y2 x1 y2 , i2 1, z1z2 x1 iy1 x2 iy2 x1x2 y1 y2 i x1 y2 x1 y2 , i2 1,
x1x2 i x1 y2 x1 y2 , i2 0.
The conjugation operation can be defined for A2 (R | i) . It maps each number z x iy to the number z x iy : x iy . It is possible to define a pseudo norm using conjugation.
Definition 1. The quadratic norm x2 y 2 , z A2 (R | i ), z zz x2 i2 y 2 x2 y 2 , z A2 (R | i ),
x2 , z A2 (R | i0 ).
The conjugation operation can be defined is called the pseudonorm of the number z x iy . It is easy to check that N z1z2 N z1 N z2 .
Definition 2. The value of an arithmetical square root of the product of numbers zz N z is called an absolute value of a generalized complex number z and can be denoted as norm x2 y2 , z A2 (R | i ), z zz x2 i2 y2 x2 y2 , z A2 (R | i ), (2) x , z A2 (R | i0 ).
This absolute value can be interpreted as a distance (elliptic, hyperbolic or parabolic) from origin to the point z . In the first case, the absolute value is called elliptic, in the second case we are dealing with a hyperbolic value (it can also take imaginary values because of the result of subtraction operation x2 y2 ) and in the third case, it is called the parabolic absolute value. The generalized complex planes are turned into a 2-D pseudo metrical space if they are equipped the following pseudo metrics: where z1 x1 iy1, z2 x2 iy2
The algebra A2 (R | i) equipped with pseudo metrics, form three metrical spaces with corresponding geometries: A2 (R | i ) is transformed into the Euclidean geometry, A2 (R | i ) - into the Minkowskian geometry and A2 (R | i0 ) - into the Galilean geometry.
Definition 3. The set of all points in the generalized complex plane A2 (R | i) satisfying the equation | z |2 x2 i2 y2 r2 is called A2 (R | i) -circle of the radius r centered at the origin.
There are three types of circles: A2 (R | i0 ) -circle is the classical Euclidean (elliptic) circle (Fig.1a), A2 (R | i ) -circle is the Minkowskian (hyperbolic) circle (Fig.1b) and A2 (R | i0 ) -circle is the Galilean (parabolic) circle in the form of two parallel lines (Fig.1c). If z x iy then the generalized complex number z0 z / z has the unit modulus if z 0. It is easily see, that (4) (7)
Consider the following 2-D Schrödinger equation d dt
d 2 (x, y, t) D dx2
d 2 dy2 (x, y, t) f x, y, t , where (x, y, t) is a wave A2 (R | i) -valued function. It describes the state (x, y, t) (in terms of generalized complex numbers) of a metamedium point with coordinates (x, y) at the moment t . In (7) D Dcl iDqu is an A2 (R | i) -valued diffusion coefficient. If D Dcl R is a real number then (7) is an ordinary diffusion (or heat) equation in the real ordinary medium (we will call one as the Fourier-Gauss medium). If D iDqu C is an imaginary number then (7) becomes an ordinary Schrödinger equation with the Plank's constant iDqu i / 2m If D Dcl iDqu D cos i sin D ei A2 (R | i), then (7) is our where x cos , if i i , y sin , if I i , x 2 y 2 x 2 y 2 cos x = x ch , if i i , sin y = y sh , if I i , (5) x 2 i 2 y 2 x 2 y 2 x 2 i 2 y 2 x 2 y 2 x 1 cg , if i i0 , y sg , if I i0 .
x x Here cos , sin are generalized trigonometric functions. In the first case ( i i ) generalized trigonometric functions coincide with classical (elliptic) functions: cos cos , sin sin . In the second case ( i i ) they are equal to hyperbolic functions cos ch , sin sh . The third case ( i i0 ) gives us new kinds of trigonometric functions: cos cg 1, sin sg which will be called parabolic (or Galilean) functions.
According to (4)-(6), an arbitrary generalized complex number with the unit modulus has the following form cos i sin , if i i , z ei cos i sin ch i sh , if i i , (6) In this work, we study the diffusion equation (or the heat equation) with a diffusion coefficient in the form of a generalized complex number and with A2 (R | i) -valued wave function. We will call such equation the generalized Schrodinger equation. 1 i0 ,
if i i0 .
Image Processing, Geoinformation Technology and Information Security / V. Labunets et al.
generalization of both diffusion and Schrodinger equations. In case of zero initial conditions, we can write the solution (7) in the
form of the Cauchy integral:
This integral we will call the generalized Schrödinger transform (GST) of the initial image f (x, y,t). If
iDqu i / 2m C A2 (R | i ), then GST is ordinary Schrödinger transform [
Let us introduce a 2-D regular lattice with nodes xn , ym , tk , where xn1 xn h, ym1 ym h and tk 1 tk . Here h and are spaces between nodes on the space Z2Sp R2 and time Zt Rt lattices, respectively. For discrete Laplacian we use the following approximation: d 2 / dx2 (xn 1, ym , tk ) (xn 1, ym , tk ) 2 (xn , ym , tk ), d 2 / dy2 (xn , ym 1, tk ) (xn , ym 1, tk ) 2 (xn , ym , tk ), d 2 / dt (xn , ym , tk 1) (xn , ym , tk ).
As a result, we get the 2-D discrete Schrödinger equation (xn , ym , tk 1) (xn , ym , tk ) D (xn 1, ym , tk ) (xn 1, ym , tk ) (xn , ym 1, tk ) (xn , ym 1, tk ) 4 (xn , ym , tk ).
Now, we give the definition of a 2-D “cellular space” (2-D regular lattice) in which the cellular automaton is defined. A regular lattice Z2Sp R2Sp consists of a set of cells (elementary automata, or electrical circuits A ut ), which homogeneously cover a 2-D 2 Euclidean space. Each cell is labeled by its position A ut(xn , ym ) A ut(n, m), n, m ZSp
Regular, discrete, infinite network consisting of a large number of simple identical elements in the form of elementary automata A ut(n, m) a copy of which will take place at each node (n, m) of the net is called the cellular automaton (see Fig.2 and Fig.3a). Each so decorated note will be called a cell A ut(n, m) and will communicate with a finite number of other cells A ut(i, k ) , which determine its neighborhood (i, k) M(m, n) , geometrically uniform M(m, n) M, M(m, n) Z2Sp . The neighborhood of the cell A ut(n, m) (including the cell itself or not, in accordance with convention) is the set of all the cells A ut(i, k ) , (i, k) M(m, n) of the network which will locally determine the evolution of A ut(n, m) . This local communication, which is deterministic, uniform and synchronous determines a global evolution of the cellular automaton, along discrete time steps tk 1 tk .
In the case of ZS2p , the classical neighborhoods are the von Neumann’s and Moore’s ones. They are known as the nearest neighbors neighborhoods, and defined according to the usual norms and the associated distances. More precisely, for (i, j) Z2Sp , || (i, j) ||1 | i | | j | and || (i, j) || max | i |,| j | will denote 1 - and -norm respectively. Let 1 and be the associated distances. Then Von Neumann and Moore neighborhoods (Fig.2) are M (m, n) : (i, k ) | 1 (m, n), (i, k ) 1 and M (m, n) : (i, k ) | (m, n), (i, k ) 1, respectively. To each cell A ut(n, m) we assign an A2 (R | i) -valued state (n, m, k ) (xn , ym , tk ) (i.e., the media's excitement). The dynamics of the cellular automaton are determined by a local transition rule, which specifies the new state (n, m, k 1) (xn , ym , tk 1 ) of a cell as a function of its interaction Von Neumann neighborhood configuration, according to (10), i.e., (8) (9) (10) (11) (n, m, k 1) (n, m, k ) D (n 1, m, k) (n 1, m, k) (n, m 1, k) (n, m 1, k) 4 (n, m, k).
This rule shows us the relation between a state (n, n, k 1) of the cell A ut(n, m) at the current moment time k 1 and the state (n, m, k ) the same cell A ut(n, m) and the states of the four neighboring cells (n 1, m, k ), (n 1, m, k ), (n, m 1, k ), (n, m 1, k ) at the previous moment time k .
a) b) Fig. 2. Examples of interaction neighborhoods (gray and black cells) for the black cell in a 2-D square lattice Z2Sp .Von Neumann neighborhoods M (m,n) and b) Moore neighborhoods M (m, n) .
The global time evolution of the cellular automaton depends on an algebraic nature of the number D . If it is a real number D Dcl R then the automaton simulates the heat propagation on a 2-D plane. According to the results of analysis, in this case an elementary medium's cell is an ordinary RC-circuit (see Fig. 3b). It is interesting to investigate the global time evolution of the Schrödinger cellular automaton with diffusion coefficient in the form of a generalized complex number
Image Processing, Geoinformation Technology and Information Security / V. Labunets et al.
D Dcl iDqu A2 (R | i) , where i2 1, 0. The analysis shows that in this case the elementary cells of a 2-D Schrödinger cellular automata are not RC-circuits, but a 2-channel filters (see Fig. 3c).
a) b) c) Fig. 3. a) The 2-D cellular automaton (the Schrödinger metamedium) and b) its equivalent electrical circuit in the form of spatially distributed RC-circuit that simulates a simple diffusion media, c) a single cell A ut (n, m) of the 2-D cellular automaton in form of a 2-channel (complex) filter.
For studying the global time evolution of the Schrödinger cellular automaton we will use the fixed absolute value of D , namely D 0.11, which provides quite fast process of diffusion propagation in the classical case with a real-valued diffusion coefficient D D0 0.11 , but will not lead to the memory overflow because of extremely high values
For a classical complex case ( i2 1 ), the diffusion coefficient can be represented in the polar form: D Dcl i Dqu Dc2l Dq2u ei D0 ei , where D0
Dc2l Dq2u , arctg Dqu / Dcl . An ordinary diffusion occurs when 0 (realvalued diffusion coefficient). The quantum diffusion (for free quantum particle) occurs when / 2 (a purely imaginary diffusion coefficient like the one in the Schrodinger equation). It is interesting to study how the global time evolution is changing when the angle runs along interval 0 / 2 . For 0 we have the Fourier-Gaussian medium (classical Newton world), and for / 2 we have the Schrödinger-Euclidean medium (quantum world). On the 's increase a classical diffusion Fourier-Gaussian medium turns into the quantum Schrödinger-Euclidean medium.
On Fig. 4 and Fig. 5 the results of modeling for a complex diffusion coefficient D with different values of phase are presented. Each picture is divided onto four parts: the bottom row represents a real (x, y,t) and an imaginary (x, y,t) components of a wave excitement in the form of A2 (R | i ) -valued function (x, y, t) , the absolute value (x, y,t) is presented in the top left quarter, the phase Arg(x, y,t) is shown in the top right quad.
a) 00 b) 50 Fig. 4. The excitement of the Schrödinger-Euclidean metamedium at the time tk 128 for two values of diffusion coefficient D D0 ei , where 00 (the Fourier-Gaussian medium) and 50 (the Schrödinger-Euclidean metamedium).
At the initial moment of time a single cell A ut(x0 , y0 ) was being excited by the bichromatic Dirac delta-function (x0 , y0 , t 0) (x0 , y0 , 0) i (x0 , y0 , 0) (x0 , y0 , 0)1 i . In process of time, the excitement covers more and more cells of automaton. Fig. 4 shows the excitement of a metamedium with two diffusion coefficients: 00 (a real-valued diffusion coefficient) and 50 at the moment of time tk 128 . It can be seen that when 00 the excitement takes the form of the 2D Gaussian surface (see Fig. 4a and Fig. 5a) and describes an ordinary diffusion process. Dark intensities correspond to higher
Image Processing, Geoinformation Technology and Information Security / V. Labunets et al. number values on the mentioned figures. When 50 the excitement has not very strongly marked form of a blurred wave packet (see Fig. 4b and Fig. 5b). The wave nature denotes on the appearance of quantum properties of the Schrodinger-Euclid metamedium even with small values of an angle .
a) 00 b) 50 Fig. 5. The typical excitements a) in the form of 2-D Gaussian surface in the Fourier-Gaussian metamedium and b) in the form of a wave packet in the
Schrödinger-Euclidean metamedium ( tk 128 , 50 ).
Remark 1. Note that with small angles the absolute value of an imaginary component is significantly smaller than an absolute value of a real part: (x, y, t) (x, y, t) . For this reason in (x, y, t) (x, y, t)2 (x, y, t) 2 ( x, y, t) the real value prevails. Thereby in this case (low values of ) the real part and the absolute value of an excitement function have the form of a smooth Gaussian surface. For the visualization of low imaginary part, the normalization has been applied on a Fig 4b and Fig 5b.
When the value is being increased, firstly, a real part also begin to demonstrate a wave nature, and secondly the values of an imaginary and a real parts are flattening (x, y, t) (x, y, t) . It is shown on a Fig. 6 that when the phase is being increased, the excitement becomes more and more similar to a wave packet in form of a 2-D Gaussian surface. Inside of that surface the real and imaginary components are fluctuating in an antiphase (see interchanging black and white rings on a figure). The absolute value of an excitement has the form of a 2-D Gaussian surface.
In this case the diffusion coefficient is a double number and the wave function (x, y, t) takes its values in the algebra of double numbers A2 (R | i ) : z a ib | a,bR where i2 1. Every double number can be represented in the following polar form D Dcl i1Dqu D cosh i1 sinh D ei A2 (R | i1 ), where D D0
Dc2l Dq2u , arcth b0 / a0 . Here we consider the case when Dcl Dqu . It was done so to get an absolute value D
Dc2l Dq2u that doesn't appear to be a complex number. In addition, the values Dc2l , Dq2u were chosen so that D D0
Dc2l Dq2u 0.11. Fig. 7 contains the picture of the excitement of a cellular automaton after 128 iterations (initial excitement is the bichromatic Dirac function).
50
200
Fig. 7. The excitement of a Schrödinger-Minkowskian metamedium at the moment tk 128 for two values of a diffusion coefficient's phase D D0 ei
Unlike the previous case (when both real and imaginary values had the wave nature) in this case a real component has a smooth Gaussian form and real part has the form of a wave packet. It turned out that the frequency of fluctuations of real values' waves does not increase when the phase of a diffusion coefficient D grows. In the center of a phase image (top right quarter of pictures) the increase of an angle leads to the sharper look of a zero phases ring.
In this case the diffusion coefficient D is a dual number and wave function (x, y, t) takes its values in the algebra of dual numbers A2 (R | i0 ) : z a i0b | a,bR, where i02 0. Every dual number can be represented in the following polar form D Dcl i0 Dqu D 1 i0 Dcl ei0 A2 (R | i0 ), where D Dcl , Dqu / Dcl . On Fig. 8 we can see the form of excitement process after 128 iterations from the impact of the Dirac delta-function at the initial moment of time.
As in the previous case, a real component does not demonstrate a wave nature when an imaginary component does. What is more, when we increase the value of to 450 then the average value of a real part becomes lower than the average value of an imaginary part, and when 450 an imaginary component begins to prevail over the real one. In addition, in this case the wave nature of the absolute value of a wave function is absent because of the fact that it does not include a non-zero imaginary part. The reason of this is that for dual numbers we have an equation z x i0 y x . In this way the imaginary part of a wave function is "living on its own", it does not have an impact on an absolute value. So, it is like an invisible "ghost" that accompanies it.
Fig. 8. The excitement of a Schrödinger-Minkowskian metamedium at the moment tk 128 for two values of a diffusion coefficient's phase D D0 ei
The generalization of three algebra A2 (R | i) : z a ib | a,b R , where i i , i0 ,i , is the Yaglom algebra [
z1 z2 x1 ik y1 x2 ik y2 x1 x2 ik y1 y2 , z1z2 x1 ik y1 x2 ik y2 x1x2 k y1 y2 ik x1 y2 x1 y2 .
Image Processing, Geoinformation Technology and Information Security / V. Labunets et al.
The conjugation operation can be defined in the algebra A2 (R | ik ) . Such operation maps each number z x ik y in a new number z x ik y : x ik y . It is obvious that z zz x2 ky 2 . It can be easily seen that where In the considered case the diffusion coefficient D is the A2 (R | ik ) -valued complex number and the wave function ( x, y, t ) takes its values in the algebra A2 (R | ik ) , where ik2 k. We will call the corresponding medium the Schrodinger-Yaglom metamedium. According to (12) every A2 (R | ik ) -valued diffusion coefficient can be represented in a polar form:
Now D depends on two parameters and k . The results of modeling the Schrödinger-Yaglom metamedia for different values of k are shown on Fig. 10. It should be noted that the ring of zero phases (the bright one), which was inherent for the case with dual numbers (k = 0) also is the first inner ring of phase fluctuations for the negative values of a parameter k (on a Fig. 10 k 0, 25 and k 0, 05 ). We can see the second bright ring that is located after the first one and also after the first black ring. The second bright ring moves away from the point of origin when the absolute value of k is being decreased. When k 0 (see Fig. 10c) the mentioned ring along with the first dark one tends to infinity. (13) Fig. 9. In every plane, that crosses the vertical of the k -parameter axis, there is an algebra A2 (R | ik ) : z a ikb | a,b R . Three planes that cross this axis at three points k 1, 0 represent three algebras of complex numbers that were considered before.
a) b) c) Fig. 10. The excitement of three Schrödinger-Yaglom metamedia at the moment tk 128 for three values of the parameter k : a) k 0, 25 , b) k 0, 05 , c) k 0 for identical values arg{D} 40.5 and D 0.07 .
Because the excitement function ( x, y, t ) frequently has a wave nature, it is very interesting to study the interference picture of two excitements that appears simultaneously in the different points of a metamedium.
Image Processing, Geoinformation Technology and Information Security / V. Labunets et al.
Fig.11a shows us a superposition of two excitements when a diffusion coefficient is a real number. In this case both excitement processes appear to be 2-D Gaussian surfaces that add up with each other in process of time.
More interesting results can be seen on Fig. 11b-c for the Schrödinger-Euclidean metamedium with arg{D} / 2 i2 1. In that case the interference of excitements occurs, like it happens in a classical quantum mechanics. The results of an interference for the Schrodinger-Galilean metamedium with a dual diffusion coefficient (i2 0) are presented on Fig. 11d. Let us note that white rings of the zero phases don't add up with each other like it happens in the case of a classical interference. They are smoothly connecting instead.
The metamedia with a generalized complex diffusion coefficients were first studied. Their time evolutions are described with generalized Schrodinger equations. The implementation of such metamedia with a cellular automaton was considered. In addition, this work contains the results of modeling, which shown the complex character of such media's behavior. Our future work will be focused on using commutative and Clifford algebras for hyperspectral image processing and pattern recognition. the Schrödinger-Euclidean diffusion media (complex diffusion coefficient ): b) two closely located points were excited by the Dirac delta-functions at the initial moment of time, c) one points were located relatively far from each other, d) the interference picture of two excitements in the Schrodinger-Galilean metamedium (it has a dual diffusion coefficient).