Introduction

Diffusion and Schrödinger equations (linear and nonlinear) [1,2] and real-valued Gauss, complex-valued Fresnel and Schrödinger transforms associated with them are important members of the family of methods for image processing, computer vision, and computer graphics. Schrödinger transform of image as a new tool for image analysis was first given in [2]. Neural networks and cellular automata (in form of a media) which are compatible with the theory of quantum mechanics and demonstrate the particle-wave nature of information have been analyzed in [3][4][5]. The studying of processes in such metamedia is very important for many branches of the system theory. There is no general theory of the metamedia yet, and every particular example of similar media, usually provides us with the examples of new dynamic or self-organization types.

In this work, we apply quantum cellular automata to study pattern formation and image processing in quantum-diffusion Schrödinger metamedia with triplet-valued diffusion coefficients. Triplet (color) numbers [6] contain one real and two imaginary components with two hyper-imaginary units 1  and 2  and the following property 3 1:

  1 2 r g b      C

, where , , r g b are real numbers. The numbers

           R R A A C

If the diffusion coefficient in the Fourier diffusion or Planck's constant in the Schrödinger equations are a triplet number .

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 ). 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

1 2 3 3 ( | 1, , )    R A A of triplet numbers 1 2 r g b      C

) 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 Schrödinger-Minkowskian, 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.

The object of the study

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:

  2 2 2 2 ( , , ) ( , , ) ( , , ) , , , x y t x y t x y t f x y t t x y                   D (1)

where , , ( , , ) ( , , ) ( , , )

r g b f x y t f x y t f x y t f x y t     

is an exciting color source (input color signal) and x y t



. Discretization of the color Schrödinger equation gives a color quantum Schrödinger 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

1 2 3 ( | 1, , )   R A

). 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][10][11][12][13][14][15].

Methods

Mathematical background. Triplet algebra

Let us consider the algebraic and geometric properties of the triplet algebra

    R A A   1 2 | , , r g b r g b       R C ( ) ( ) . r g b r g b r r b g g b g r r g b b b r g g r b                           C C

It is useful to introduce the following triplet numbers

  2 : 1 /3 lum      e , 22 3 3

:

(1 ) / 3, chr        E where   3 exp 2 / 3 i      . It                        e E e e E e e E e E E e E Z E E C = C

We will call real numbers lum a  R the luminance numbers and complex numbers chr z  C -the chromatic numbers. Obviously,

1 2 1 2 1 2 1 1 2 2 1 1 2 2 1 2 1 2 1 1 ( ) ( ) , 3 3 1 1 ( ) ( ) .                                            e e E E C C Hence, 1 2 3 , ( ). 2 2 lum chr g b a r g b z r g b r i g b                    

In the new duplex representation two main arithmetic operations have the simplest form:

                , .                            e E e E A is the direct sum of real R and complex C fields:   3 . lu ch        R e C E R C A It is known that every 2-D complex number

x iy   z can be represented geometrically by the modulus x iy   z

  / 3, lum a r g b      2 2 2 2 2

, , arg

chr lum chr chr d r g b R d a z        b)

the color cube, it's achromatic diagonal and chromatic plane. ) according to Fig. 2.

The generalized Schrödinger equation and cellular automata

Let a diffusion coefficient

1 2 r g b      D in the Schrödinger equation   2 2 1 2 2 2 ( , , ) ( , , ) ( , , ) ( ) , , , x y t x y t x y t r g b f x y t t x y                       (2)

be a triplet number, where

D iD i    R D A and ( , , ) ( , , ) ( , , ) cl qu x y t x y t i x y t      then (2) is

bichromatic Schrödinger equation for bichromatic quantum Schrödinger metamedium [16]. It is a generalization of both diffusion and Schrödinger metamedia.

In case of zero initial conditions, we can write the solution of (2) in the form of the Cauchy integral:

            2 2 - - - 4 - 2 0 -- 1 ( , , ) , , . 2 - x y t t x y t e f d d d t                              D D(3)

This integral we will call the color Schrödinger transform (GST) of the initial image ( , , ).

f x y t If 2 / 2 ( | ), qu iD i m i     C R A then GST is ordinary Schrödinger transform [1-5].                       (4)

As a result, we get the 2-D discrete color Schrödinger equation , ) .

  ( , ,1) ( , , ) ( 1, , ) ( 1, , ) ( , 1, ) ( , 1, ) 4 ( ,n m k n m k n m k n m k n m k n m k n m k x y t x y t x y t x y t x y t x y t x y t                     D (5)

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 2 2 Sp Sp  Z R consists of a set of cells (elementary automata, or electrical circuits ut Au ), which homogeneously cover a 2-D Euclidean space. Each cell is labeled by its position

  2 , ( , ) ( , ), n m Sp n m ut x y ut n m   Z A u A u

Regular, discrete, infinite network consisting of a large number of simple identical elements in the form of elementary automata ( , ) ut n m Au 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

( , ,1) ( , , ) n m kn m k x y t     

of a cell as a function of its interaction Von Neumann neighborhood configuration, according to (5), i.e.,

 

( , , , ) .

n m k n m k n m k n m k n m k n m k n m k                    D (6)

This rule shows us the relation between a state ( , ,

                                 e E e E                       (7)

one for a luminance and other -for a chromatic components.

Obviously, 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 0

           ).

The Schrodinger-Yaglom color metamedia

The chromatic plane, in which

ch r c h r i i H ch r ch r ch r D D e S e        

lays, appears to be a classic complex algebra with 2

. i   

It is interesting to study a color metamedium with a chromatic plane in the form of another two complex algebra with 2 . The source has an angular velocity  :

    0 0 , , cos( ), sin( ) , f x y t x R t y R t          

where     When the chromatic angle chr  values are small (low color tone) then ch r D '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).

Different results can be obtained for color media with a chromatic component in the form of a double   leads to some interesting and even more unusual consequences. The Fig. 9 shows the pictures of an excitement at the moment 100 k t  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 D chr '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. . The metamedium has broken the image onto the areas of uniformity with respect to brightness and hue at this time.

Some applications of the color Schrödinger metamedia

Let the excitement function 2 0 ( , , 0) ( , , 0) ( , , 0) ( , , 0) ( , , 0) ( , , 0) 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 128 t  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 0, 32, 64,128,160 k t  and 0,84 k t  , respectively, after an input impact in the form of an initial image.

r g b lum lum chr chr f x y t f x y f x y f x y f x y f x y           e E in(

Conclusion

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.