In this work, we use quantum color cellular automata to study pattern formation and image processing in quantum-diffusion Schrödinger systems with triplet-valued (color-valued) diffusion coefficients. Triplet numbers have the real part and two imaginary parts (with two imaginary units 1 and 2 , where 3 1 ). They form 3-D triplet algebra. Discretization of the Schrödinger equation gives quantum color cellular automata with various triplet-valued physical parameters. The process of excitation in these media is described by the color Schrödinger equations with the wave functions that have values in triplet algebras. The color Schrödinger metamedia can be used for creation of the eye-prosthesis. The color metamedium suggested can serve as the prosthesis prototype for perception of the color images.
In this work, we study properties of the color Schrödinger excitable metamedium in the form of a cellular automaton. The more detailed information about cellular automata can be found in [7]. The automaton's cells are located inside a 2D array. They can perform basic operations with triple (color) numbers (in color algebra A 3 (R | 1, 1 , 2 ) ). These cells are able to inform the neighboring cells about their states. Such media possess large opportunities in processing of color images in comparison with the ordinary diffusion media with the real-valued diffusion coefficient.
The rest of the paper is organized as follows: in Section 2, the object of the study (the color Schrödinger equation) is described. In Section 3, a brief introduction to mathematical background (color algebra A3 A3 (R | 1, 1 , 2 ) of triplet numbers C r g 1 b 2 ) is given (subsection 3.1) in order to understand the concept behind the proposed method. In subsection 3.2, the proposed method based on color Schrödinger equations is explained. Next, we defined Schrödinger transform of color image, discussed its properties. In Section 4, the basic color metamedia (the color Schrödinger-Euclidean, color SchrödingerMinkowskian, color Schrödinger-Galilean and color 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.
In this work, we apply quantum cellular automata to study pattern formation and image processing in color quantumdiffusion Schrödinger metamedia with triplet-valued diffusion coefficients: ( x , y , t ) t
2 ( x , y , t ) D x 2 2 ( x, y , t )
f x, y , t ,
y 2
where (x, y,t) r (x, y,t) g (x, y,t)1 b(x, y,t) 2 is a color wave function that describes excitement of medium,
f x , y , t f r ( x , y , t ) f g ( x , y , t ) 1 f b ( x , y , t ) 2 is an exciting color source (input color signal) and D r g 1 b 2 is
colorvalued diffusion coefficient. It describes the process of excitement in the so-called color Schrodinger metamedium with A3 ( )
valued (color) wave function (x, y, t) . Discretization of the color Schrödinger equation gives a color quantum Schrödinger
(
Image Processing, Geoinformation Technology and Information Security / V. Labunets et al. cellular automaton with various triple–valued physical parameters. Their microelectronic realizations appear to be a programmable Schrodinger metamedia [8]. The main purpose of this work is the investigation of time evolution for color Schrödinger metamedia in the form of quantum cellular automata with triplet diffusion coefficients. The automaton's cells are located inside a 2D array. They can perform basic operations with triple (color) numbers (in color algebra A 3 (R | 1, 1 , 2 ) ). These cells are able to inform the neighboring cells about their states. Such media possess large opportunities in processing of color images in comparison with the ordinary diffusion media with the real-valued diffusion coefficients. 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 color images [9-15].
C r g 1 b 2 | r, g,b R .
C (r g b2 2) are given by [6]: 2 2 2 C1 C2 r1 g1 1 b1 2 r2 g2 1 b2 2 r1 r2 g1 g2 1 b1 b2 2 ,
C1 C2 ( r1 g1 b1 2 ) ( r2 g 2 b2 2 ) r1r2 b1 g 2 g1b2 g1r2 r1 g 2 b1b2 b1r2 g1 g 2 r1b2 2 . It is useful to introduce the following triplet numbers elum : 1 2 / 3 ,
E chr : (1 3 2 32 ) / 3, where 3 exp i 2 / 3 . It is easy to check el2um elum , Ec2hr =Echr , elum Echr Echr elum 0 . Hence, elum , Echr are orthogonal idempotents (projectors) and every triplet (color) number C r g b 2 can be represented in the form of the linear combination of a "scalar" alum elum and "complex" zchr Echr components C alum elum zchr Echr alum , zchr in the idempotent basis elum , E chr , where alum elum = C elum , zchr Echr C Echr , because We will call real numbers alum R the luminance numbers and complex numbers zchr C - the chromatic numbers. Obviously, C elu (alu elu zch Ech ) elu alu el2u zch Echelu alu elu =, C Ech (alu elu zch Ech ) Ech alu eluEch Zch Ec2h zch Ech. alum elum C elum (r g 1 b 2 ) 1 1 2 (r g b) 3 1 1 2 3 , zchr Echr C Echr (r g 1 b 2 ) 1 1 1 2 2 (r g1 b 2 ) 1 1 1 2 2 .
3 3 Hence, alum r g b, zchr r g 1 b 2 r
g 2 b i 23 ( g b). In the new duplex representation two main arithmetic operations have the simplest form:
C B alum elum zchr Echr blum elum wchr Echr alu blu elum zchr wchr Echr ,
Consequently, a color algebra A () is the direct sum of real R and complex C fields: A 3 R e lu C E ch R C. It 3 is known that every 2-D complex number z x iy can be represented geometrically by the modulus z x iy and by the polar angle arctg x/ y . The modulus z x2 y2 is multiplicative and the polar angle is additive upon the multiplication of ordinary complex numbers. The triplet numbers introduced in this section have the form the variables r, g and b being real numbers. In a geometric representation, the triplet number is represented by the point C (r, g,b), or as a 3-D vector with coordinates (r , g ,b) in the 3-D color space R 3co l (see Fig. 1).
Let the point Obe the point of the origin of the R, G, B axes, and T A c h will be the line, which contains the points with equal coordinates r g b (it is called an achromatic diagonal). The luminance numbers alum lie on this achromatic diagonal. Also let M (alu ) be the plane r g b alum that is perpendicular to an achromatic axis T A c h ; this plane intersects it on a range alum from the point of origin O. It is called a chromatic plane. It contains chromatic numbers zchr .
a) b) Fig. 1. a) The geometrical representation of a triplet number C= r g b 2 in the form of a 3D vector C= r, g , b R c3ol or as a point C= C (r, g ,b) Rc3ol in 3-D color space R 3co l . The geometrical characteristics have the following values: alum r g b / 3, d r2 g2 b2 , Rchr d2 al2um, chr arg zchr b) the color cube, it's achromatic diagonal and chromatic plane.
Obviously, a vector C= r g b 2 (r , g , b) can be described: 1) by the projection alum of a line segment OC on a line T A c h , i.e. by the luminance component and 2) by a complex number zchr in the chromatic plane. Besides, the absolute value of this number appears to be the range z chr from C (r , g , b) to this line, i.e. it describes the saturation (which is marked by the symbol Schr zchr ) of a triplet number C r g b 2 and the azimuth angle chr arg z chr represents its color hue (which we can mark by the symbol H chr chr arg z chr ) according to Fig. 2.
Let a diffusion coefficient D r g 1 b 2 in the Schrödinger equation
( x,t y , t ) ( r g 1 b 2 ) 2 (xx,2 y , t ) 2 (xy,2 y , t ) f x, y , t , (
In case of zero initial conditions, we can write the solution of (
(x, y, t) 0t 2 D1t - 2 - - e- x-4D2 t-y- 2 f , , d d d . (
Let us introduce a 2-D regular lattice with nodes xn , ym , tk , where xn 1 xn h, ym1 ym h and tk 1 tk . Here h
and are spaces between nodes on the space Z 2Sp R 2 and time Zt R t 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 ), (
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As a result, we get the 2-D discrete color 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 Z 2Sp R 2Sp consists of a set of cells (elementary automata, or electrical circuits Aut ), which homogeneously cover a 2-D Euclidean space. Each cell is labeled by its position A ut ( xn , ym ) A ut (n, m ), n, m Z 2Sp
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 in [16] ). Each so decorated note will be called a cell A ut(n, m) and will communicate with a finite number of other cells Aut(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 Aut(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 Z2Sp , the classical neighborhoods are the von Neumann and Moore 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 ) || m ax | 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) | (m,n),(i, k) 1, respectively. To each cell A ut(n, m)
M (m, n) : (i, k) | 1 (m, n),(i,k) 1 and
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 (
We can write the Schrödinger equation (
D r g 1 b 2 Dlum elum Dchr Echr ,
the equation (
a) b) c) d) Fig. 3. The state of the color Schrodinger-Euclidean metamedium at moments a) tk 16 and b) tk 210 , when Dlum Schr , chr 0 and c) tk 13 and d) tk 120 , when Dlum Schr , chr 0.
Image Processing, Geoinformation Technology and Information Security / V. Labunets et al. d d 2 d 2 dt lum ( x, y, t) Dlum dx2 lum ( x, y, t) dy2 lum ( x, y, t) ,
d d 2 d 2 dt chr ( x, y, t) Dchr dx2 chr (x, y, t ) dy 2 chr ( x, y, t ) .
one for a luminance and other - for a chromatic components.
Obviously, chr (x, y,t) chr (x, y,t) eichr (x,y,t) S(x, y,t)eiHchr (x,y,t), where S ( x, y, t ) chr ( x, y, t ) , H chr ( x, y, t ) chr ( x, y , t )
are the saturation and the hue of a wave function, respectively. The relation between an angle chr ( x, y, t) and a color hue
H chr ( x, y, t ) is shown on a Fig. 2. The first expression in (
For the modeling results representation we will use the cellular automaton, in which the cell's states are shown as color pixels (as triplet numbers). The sum of four Dirac's delta-functions (red, green, white and blue) as an input signal f x , y , t . On Fig. 3 these functions are represented as four points of the corresponding colors. Each figure consist of four parts: the top left quarter shows the resulting RGB picture (i.e. presents wave function (x, y, t) in the RGB format), the top right part shows the luminance component lum ( x, y, t ) of a color wave function, the bottom left one shows the saturation chr ( x, y , t ) S ( x , y , t ) and the last one represents the color tone chr ( x, y, t) .
Initially we will consider time evolution of the Schrodinger-Euclidean metamedium for the "balanced" chromatic and achromatic parameters D Dlum e lum S chr e i H chr E chr where Dlum Schr , chr 0 , i.e. D D lum e lum E chr . In this case we take the equal values of a diffusion coefficient's luminance and saturation, when the chromatic phase is equal to zero: Dlum Schr , chr 0 . The results of a simulation for this case are shown on Fig. 3a ( tk 16 ) and Fig. 3b ( tk 120 ). Fig. 3c-d shows the process of a color excitement's propagation in a color metamedium, which diffusion coefficient has the low value of saturation ( Dlum Schr , chr 0 ). The achromatic components on all illustrations in this work are inverted to reduce the amount of dark colors for a better visual perception of pictures. Therefore, the darker colors mean higher values of excitement. Note that chromatic parts of all spots are spreading slower than achromatic ones: the size of spots in the top right quad (excitement's luminance representation) is bigger than in the bottom left one (excitement's saturation representation).
Results that are more interesting can be obtained when we increase the value of a color hue chr of a diffusion coefficient. As an input signal we use a single red-colored Dirac's delta-function that is affecting the central point of a cellular automaton. For a comparison, it is important to see the excitement of cellular automaton with a zero color hue chr 0 (when Dlum Schr 0.11 ). The results are presented on Fig. 4. This picture shows only resulting RGB images (the top part) and cells' chromatic phases (the bottom part). Also, note the Fig 5. c) Fig. 4. The state of time evolution of the color Schrodinger-Euclidean metamedium at moments tk 0, 10, 70, 160
a) chr 0 b chr c) chr 0 ).
Image Processing, Geoinformation Technology and Information Security / V. Labunets et al.
The chromatic plane, in which D chr D chr e i chr S chr e i H chr lays, appears to be a classic complex algebra with i2 1 . It is interesting to study a color metamedium with a chromatic plane in the form of another two complex algebra with i2 1 and i02 0 [17] , i.e. with the following chromatic components:
We will call such media the color Schrodinger-Yaglom metamedia. The Fig. 6a contains excitements for color SchrodingerGalilean metamedium at the moment of time tk 128 (for the same input signal as in the previous case) for different values of the hue of the diffusion coefficient ( chr 5 , 40, 60 ). As we can see on a Fig. 6b, the further increase of the hue (for diffusion coefficient) chr 70 , 89, 90 leads to the fast concentration and contraction of a phase circle in the middle of the bottom right square. In addition, our red-colored initial point completely turns into a spot with a pearl halo when the hue of the coefficient D reaches chr 90 . In addition, we should mention that values chr arg{Dchr } do not produce any new phenomena because of the periodic nature of trigonometric functions. Indeed, the color excitement with arg{Dchr } chr 900 turns out to be the inverted (by a color tone) excitement of a metamedium with arg{Dchr } chr 90 (see the Fig. 7a that is quite similar to Fig. 6b).
The example of the excitement of the Schrödinger-Minkowskian metamedium with a chromatic component of a diffusion coefficient in the form of a double number Dchr Dchr eichr Schr eiHch is shown on a Fig. 7b.
Let the excitement function f x , y , t in equation (
R and the center at the point x 0 , y 0 .
The source has an angular velocity : Image Processing, Geoinformation Technology and Information Security / V. Labunets et al.
2 2 2 f x, y, t x0 R cos( t ), y0 R sin( t ) , where x0 x(t) y0 y(t) R . It means that we have a moving quantum particle in a color metamedium. Firstly, we will research the color Schrodinger-Euclidean metamedium with the chromatic component in the form of a classical complex number Dchr Dchr eichr Schr eiHchr that has a relatively low chromatic phase value chr of D chr . The Fig.8a-b demonstrates the results of modeling for this case. chr 100 , 130, 175 ; b) The excitement of the color Schrödinger-Minkowskian metamedium ( i2 1 ) at the moment tk 128 for the diffusion coefficients with hues chr 100 , 130, 175 a) chr 5 b) chr 60 c) chr 74 d) chr 85 Fig. 8. The excitement of the color Schrödinger-Euclidean metamedium by a particle moving on a circular trajectory ( tk 128 ). a) chr 70
b) chr 90
Fig. 9. The excitement of the Schrödinger-Galilean metamedium by a particle moving on a circle ( tk 100, i02 0 ).
When the chromatic angle chr values are small (low color tone) then D chr 's excitement fluctuation components, that are perpendicular to the movement trajectory, are almost absent. We only can see the parts of fluctuations that exist along the trajectory. When values of the chromatic angle chr (hue) are being increased, we can observe the excitement's fluctuations that are perpendicular to the trajectory of a movement. Also the interference of a "tail" and "head" parts becomes visible (see Fig. 8cd).
Image Processing, Geoinformation Technology and Information Security / V. Labunets et al.
Different results can be obtained for color media with a chromatic component in the form of a double Dchr Schr ei0 chr and a dual Dchr Schr ei0 chr number. For example, when we use a dual number Dchr Schr ei0 chr the alteration of chr arg{Dchr } leads to some interesting and even more unusual consequences. The Fig. 9 shows the pictures of an excitement at the moment tk 100 for the quite high values of a phase chr (the picture of an excitement changes weakly for the wide range of chr 's values). It can be seen that in the case of a Schrödinger-Galilean metamedium, the growth of a Dchr 's phase causes the increase of a violet and pearl color amounts (on condition that the moving particle has a red color). Particle trail's halo on the right part of Fig. 9 is quite bright, but there are no cells with high lightness and saturation parameter values in the investigated area. It is caused by irregular laws of the behavior of the chromatic component for this metamedia type.
Minkowskian metamedium ( i2 1) metamedia (chr 50 in both cases).
The process of excitement's interference in Schrödinger-Euclidean metamedia has a classic character. The results of a simulation for Schrodinger-Galilean ( i 2 0 ) and Schrödinger-Minkowskian ( i 2 1 ) metamedia are shown on Fig. 10a and Fig. 10b, respectively. It can be seen on Fig. 10a that in the Schrödinger-Galilean metamedium the collision of different-colored excitements produces unusual rays in the areas where an occlusion happened. There are no such phenomena in the SchrodingerEuclidean and in the Schrödinger-Minkowskian metamedia. a) b)
Fig. 11. a) The excitement function (input image) f (x, y,t0 0) of the color Schrödinger-Euclid metamedium at the initial moment t0 0 ; b) The excitement of this metamedium at the moment tk 128 . The metamedium has broken the image onto the areas of uniformity with respect to brightness and hue at this time.
Let the excitement function f ( x, y , t0 0 ) f r ( x , y , 0) f g ( x, y , 0 ) f b ( x, y , 0) 2 flum ( x , y , 0) e lum f chr ( x , y , 0 ) E chr in
(
One of the most important tasks in the digital processing of color images [18] is the distinguishing of image's parts, where some of its components have uniform values. It is the uniformity areas detection, for example, we can detect the areas with a similar brightness, saturation or color tone, etc. Usually one has to perform such operation before starting the image segmentation by some parameter. It turns out that color Schrödinger metamedia are able to implement such operations. Fig. 11b shows the excitement of a Schrödinger-Euclidean metamedia at the moment t 128 after an impact that is represented as an image, which was described previously. It is easy to see that by this time the metamedia has broken the initial image onto areas of uniformity by luminance and by color tone. Fig. 12 and Fig. 13 shows the excitements of the Schrödinger-Galilean and Schrödinger-Minkowskian metamedia at the moments tk 0, 32, 64,128,160 and tk 0,84 , respectively, after an input impact in the form of an initial image.
The metamedia with triplet (color) diffusion coefficients were first studied. Their laws of functioning are described by color Schrodinger equations. Simulation of these equations in the form of quantum cellular automata was considered. The results of modeling that were shown in this work demonstrate the complex character of the time evolution of such metamedia. Our future work will be focused on using commutative and Clifford algebras for hyperspectral image processing and pattern recognition.
This work was supported by grants the RFBR № 17-07-00886, № 17-29-03369 and by Ural State Forest University Engineering’s Center of Excellence in “Quantum and Classical Information Technologies for Remote Sensing Systems”.
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