METAMODEL FOR SOFTWARE SOLUTIONS IN COMPUTED TOMOGRAPHY

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1 VOL. 10, NO 22, DECEBER, 2015 ISSN ETAODEL FOR SOFTWARE SOLUTIONS IN COPUTED TOOGRAPHY Vitaliy ezhyev Faclty of Compter Systems and Software Engineering, Universiti alaysia Pahang, Gambang, alaysia E-ail: ABSTRACT etamodelling approach is now widely sed to atomate development of software for the general prposes. This paper expands a classical metamodeling approach to the design of software for mathematical modelling. Paper defines the metamodel to solve different problems in compted tomography. The proposed approach was applied for the reconstrction of the strctre of three-dimensional objects on a system of their traces on mtally perpendiclar planes. It was also sed for generation of software, intended for the search of illegal items dring cstoms control. Keywords: metamodel, domain-specific mathematical modelling, compted tomography, spline interpolation. INTRODUCTION In [1] the methodology for Domain-Specific athematical odelling (DS) is proposed. It enhances Domain-Specific odelling [2], sed for the model driven software systems development. DS allows s development of metamodels in the different mathematical semantics, which becomes possible de to introdction of an additional level of the metamodelling architectre [3]. DS tools [4] allow their sers to define mathematical metatypes and apply them for the development of domain specific types. One of the prominent examples is a geometrical meta-metamodel, which defines in the terms of sets the basic (corresponding to the dimensions of the space) geometrical metatypes (point, line, srface and 3D region). Geometrical meta-metamodel was sed for the design of cyber-physical systems [5] and mathematical modelling srfaces of celestial bodies on the base of radiolocation data [6]. Its application for the design of software tools for physical modelling and simlation was considered in [7]. Formally, we define metamodel as a triple, which inclde an alphabet of types, a grammar, sed to combine instances of the types, and applicable operations. Geometrical meta-metamodel incldes alphabet of the metatypes, being the common reslt of abstraction from the geometrical strctre of the physical objects. In this paper on the base of geometrical meta-metamodel we define etamodel for Tomography (T). Alphabet of T incldes the type of geometrical plane, created on the base of the metatype of a geometrical srface. To prodce domain specific types, instances of the plane are attribted by specific data (fnctions of distribtion of physical properties) obtained in the process of tomographic scan. Operations of the geometrical meta-metamodel are bilt on the base of the methods of interpolation, interlination and interflatation [8]. In this paper, to restore a strctre of three-dimensional objects we propose new interpolation operators, bild on the base on interflatation of fnctions. Paper considers a practical implementation of the approach - compter tool Brain Reconstrction, which bilds a three-dimensional image of brains by the system of their parallel sections. Conclsion and references list finalize the paper. ATERIAL AND ETHODS In tomographic research there is often a problem arose by the set of given sections to obtain an image of a body in its any intersection [9]. This problem, for example, occrs in medicine and biology, at the stdy of the hman or experimental animal s cerebral cortex. Typically, the data, containing properties of a body, are obtained in the form of photographic images of the sections that are carriers of the flat distribtion of intensity of the light wave. From it follows the appropriateness of the geometrical plane as a basic type to prodce instances of compter objects as carriers of distribtion of some physical property (density, light intensity, etc.). The type of the plane in the metamodel T is based on the metatype of geometric srface of the metametamodel G [5]. In terms of analytical geometry, we define the type of the plane by setting restrictions on the eqation of the srface: ( xyz,, ) 0 (1) The eqation of a plane srface is of the first order. In Cartesian coordinates Ax By Cz D 0 (2) where A, BCD,, are some constants

2 VOL. 10, NO 22, DECEBER, 2015 ISSN To define the type of the plane we pt restrictions on the possible vales of xyz,,. It is sfficient to set restrictions on two from the three variables, e.g. x1 x x2 y y y 1 2 Here the mathematical part of the type definition is done. Next stage is to develop a domain specific type. The idea is to combine the mathematical type with domain specific properties, e.g. a fnction of a distribtion of physical vales. Sch the distribtions typically come from a tomograph as images in raster format, e.g. bitmap. Let s se this format de to the mathematical simplicity of the definition of fnction of distribtion of physical data. To do it, each point of the plane with coordinates (x, y) we will consider as a parameter of the fnction, representing color in some model (RGB, CYB, HSB). Let s develop the method to stdy distribtion of a physical property (x, y,z) of an internal strctre of a three-dimensional body (e.g., density) in any section. Withot loss of generality we assme the body is flly located in the nit cbe D 0,1 3. The sorce of information abot the fnction (x, y, z), i.e. the internal strctre of the body, is its traces on the system of mtally perpendiclar planes: i x, i 0, 1 1 j y, j 0, 2 2 k z, k 0, 3 3 i.e. the set of fnctions i, y, z,i 0, ; 1 1 j x,, z, j 0, 2; 2 k x, y,, k 0, 3. 3 To restore the internal strctre of a body we develop operators of spline interpolation O 1, O2, O 3 of the fnction by the variables x, y, z, correspondingly (3) (4) (5) O1(x, O2(x, O3(x, where y, z) y, z) y, z) Sp 1 Sp 1,i i0 2 Sp2,j j0 3 Sp3,k k0 1, i x x y i 1 x,, y, z, j 2, z, k 3 z x, y,, m m 1,2,3 having the properties Sp p,i i,p, i, p 0, (6) are the basic splines of degree Similarly we define splines Sp 2, j, Sp 3, k. Interlination (interflatation) of a fnction of many variables is recovery (possibly, approximate) of this fnction by the help of its traces and traces of its derivatives p to the given order on the system of lines (or srfaces, respectively) [8]. The operator of spline-interflatation is defined as L x, y, z x, y, z (7) O O O O O O1O2 O2O3 O1O2O (8) 3 It has the following properties: i i L, y, z, y, z, i 0, 1; 1 1 j j L x,, z x,, z, j 0, 2; 2 2 k k L x, y, x, y,, k 0, In the real case, if experimental data are given with some accracy, the following theorem can be formlated. Theorem. If the experimental data i j k, y, z, i 0, ; x,, z, j 0, ; x, y,, k 0, (9) Are given with maximm error, and is a random variable niformly distribted in D, i.e

3 VOL. 10, NO 22, DECEBER, 2015 ISSN i 1, i( y, z), y, z, i 0, 1; 1 j 2, j( x, z) x,, z, j 0, 2; 2 k 3, k ( x, y) x, y,, k 0, 3 3 the operator O O O OO OO Lx, y, z x, y, z OO 2 3 OOO will have error Rx, y, z x, y, z L ~ x, y, z, satisfying the following relationship m1 m2 m3 R O O. C Proof. R x, y, z D x, y, z L ~ x, y,z x, y,z Lx, y,z x, y,z x, y,z Lx, y,z Lx, y, z. From it follows ineqality R L C D CD L CD, giving the proof. PRACTICAL IPLEENTATION Practical implementation of T was done in the software tool Brain Reconstrction. Note, that the software framework, considered in [4; 7] incldes Brain Reconstrction as a partial case. DS tools allows their sers to define arbitrary metamodel within a general metametamodel G, inclding that for solving problems of compted tomography. Figre-1 illstrates the interface of Brain Reconstrction, where we se the method of spline interflatation to reconstrct the three-dimensional image of hman brains by the system of its parallel sections. Brain Reconstrction ses one type of the metamodel a plane - to specify the distribtion of physical properties (here, intensity of colors). The distribtion a ser does by attribtisation of the plane by the fnction (x, y, z). This attribted plane is sed for sbseqent instantiation of objects that are directly sed to bild the model, i.e. the three-dimensional image of the cerebral cortex. At the level of the metamodel, the attribting is a method that assigns to each element of the set of geometrical points on the plane the vale of the fnction (x, y, z). The inpt for Brain Reconstrction is a set of raster images (files in bitmap format), depicting the sections of a cerebral cortex on a system of mtally perpendiclar planes x x i, y y j, z z k, i 1..1, j 1.. 2, k Names of files indicates their belonging to one of three parallel sections (for example, «x1.bmp», «x2.bmp», «x3.bmp» are the names of images, representing brain sections in a plane, perpendiclar to the axis OX). Brain Reconstrction reads the files that meet the following criteria: a) The file format corresponds to the BP standard. b) The name begins with Latin letters x, y or z, reflecting a section belonging to one of mtally perpendiclar planes. c) The next (after Latin letters) characters are Arabic nmbers, which are sed to determine the serial nmber of the Figre in the corresponding system of planes. All correct files are recorded in the array of figres - instances of the geometrical type plane, and their nmber - in the relevant internal variables М 1, М 2, М 3. Ths, the strctre of the model is atomatically bilt by analysing the inpt information (files of images). Under the strctre of the model, we refer to the dimension of the space and the nmber of elements in the array of images. Based on this information the model of the space 3 D 0,1 is defined and the distances between separate planes are calclated. Or method allows a ser to recover the threedimensional strctre of the body in arbitrary distances between its parallel sections. For simplification of explanations and to be close to the practice of scanning tomographic images (technically it is easier to obtain intersections with a constant step), let s assme that the distance between intersections is the same for any of the systems of planes. Operators of spline-interflatation (6) are implemented as software fnctions O1, O2, O3, O12, O13, O23, O123, parameters of which are variables x, y, z of high accracy (the long doble type of C++). Fnctions O12, O13, O23, O123 se the vales that retrn operator fnctions O1, O2, O3. Fnctions O1, O2, O3 compte the closest to the point with coordinates (x, y, z) layers i, j, k and sing them refer to specific elements of the array of images

4 VOL. 10, NO 22, DECEBER, 2015 ISSN Figre-1. Interface of the program Brain Reconstrction. For example, if the calclated nmber of the layer is i and this is OZ intersection, then the reference to the element of the array Images is as follows: Images[1][i]->Pixels[x][y] Every image is a two-dimensional array of pixels (pictre elements), storing an attribte - color. We se RGB color model, according to which any color can be represented as a sperposition of three basic colors - Red, Green and Ble. Each component of the color - Red, Green or Ble is given by one byte, that can reflect 2 8 =256 shades of the color (the intensity varies from 0 to 255, the 255 reflects the largest intensity of the color). The total nmber of colors that can be set by 3 bytes is = , which is close to the color recognition possibility of a hman eye (so called Tre Color model). For frther discssion let s se hexadecimal nmber system, which sign in C++ is a combination of characters 0x. Flly set bytes (FF) show the highest intensity of the corresponding colors component (Table- 1). Table-1. The vale of bytes in the RGB colors model. Color 1 byte 2 byte 3 byte Vale Red FF 0x0000FF Green 00 FF 00 0x00FF00 Ble FF xFF0000 Ths, 0xFFFFFF is the vale with the highest intensity of all colors components and so giving white, the vale 0x is black. The range between 0x and 0xFFFFFF having eqal vales of each color component allows s operate with shades of the grey color (e.g., 0x or 0xAAAAAA). Software implementation of operators O1, O2, O3 allocate each of the RGB colors components separately. The fnctions O1, O2, O3 perform a selection of individal colors component at this point. E.g. R = Images[1][i]->Pixels[x][y]&0x0000FF performs selection of the red color component of a pixel that belongs to the perpendiclar to the axis OZ plane i and has coordinates (x, y). To separate individal colors components we se bitwise and operation (&), which clears all other bits besides given by the second operand (in above sample it is 0x000000FF). To explain this operation let s se binary system. For example, we have the following color vale in the binary form Taking that FF 16 = (each byte is set by 8 binary digits, i.e. to set the color as a combination of three components we need 24 bits). To set a single nmber in hexadecimal format we need 4 bits. Using bitwise and (&) operation we allocate a separate component of a color In sch a way, the selection of components of a color is made (in this example the right byte, which corresponding to red color)

5 VOL. 10, NO 22, DECEBER, 2015 ISSN For the green color component we may se G= Images[1][i]->Pixels[x][y]&0x0000FF00 After calclation of all separate components, we can form the color again Color = R + G + B ETHODS OF 3D GRAPHICS The metamodel T also incldes methods of three-dimensional graphics to spport visalisation and analyses. To bild a three-dimensional perspective view the set of coordinates of the real object are transformed into the screen coordinates. This process is carried ot in two stages: a) Real (world) coordinates of each point (x, y, z) of an object are transformed into specific coordinates (x v, y v, z v). b) Specific coordinates (x v, y v, z v) of each point of an object sing a perspective transformation are converted into screen coordinates (x e, y e). To spport a ser in soltion of this problem, the metamodel T gives the software fnctions (API), defining relationship between sets of the real and specific coordinates. Geometrically, to bild an image of a threedimensional object, we need make its projection on the plane of a screen. For this prpose, we introdce a specific coordinate system in which an object is projected on a plane X v0 vy v, and axis Z e0 e determines the direction of observation (see Figre-2). yv xv To perform the specific transformation the observation point also to be set. etamodel defines a point of observation in the spherical coordinate system by following parameters: θ = angle of the axis OX with projection of the observation line on the plane XOY; φ = polar angle of the observation line with OZ axis; ρ = distance to the observation point. Setting the triple of parameters (θ, φ, ρ) allows s to make a 3D transformation of a geometrical object. Let the position of a point in three-dimensional space set by coordinates (x, y, z). Then the specific transformation we can write in the form: x v y v z v 1 = x y z 1 V (10) where sin cos cos sin cos 0 cos cos sin sin sin 0 V (11). 0 sin cos Coordinates x v and y v of a point of an object we can already se to bild a screen image. In this case we will obtain the orthogonal projection, where each point P of an object is projected into the point P' by drawing the line, perpendiclar to the plane XOY (i.e. parallel to the observation line). Sch type of projection has place at moving the observation point into infinity, where parallel lines of a three-dimensional object remain parallel at its projection as well. x z φ θ zv Figre-2. 3D to 2D transformation. y OTHER APPLICATIONS OF THE ETAODEL The metamodel T was also sed for the development of software tools for solving problems in other domains of compted tomography. Figre-3 illstrates this approach for search of nathorized items dring cstoms control. The geometrical plane is here the basic type of the metamodel, sed to bild the data models in a software. The plane is a carrier of physical properties - obtained from a tomograph roentgen images of distribtion of density of a car, which serves as an inpt for applications of a mathematical method (fixed in the patent [10])

6 VOL. 10, NO 22, DECEBER, 2015 ISSN Figre-3. Use of metamodel to generate software for search of illegal items dring cstoms control. CONCLUSIONS Based on the geometrical meta-metamodel the metamodel T was defined, allowing design of software for solving problems of compted tomography. Alphabet of T incldes the type of geometrical plane, which inherits the metatype of the srface. Instances of the plane are attribted by fnctions of distribtion of physical properties obtained in the process of tomographic scanning. The approach was proven by development of a software tool that reconstrcts a three-dimensional image of a brain by the system of its mtally perpendiclar sections. The method restores the strctre of threedimensional objects sing interpolation operators, based on interflatation of fnctions. The proposed metamodel T was also sed to generate software for the search of illegal items dring cstoms control. REFERENCES [1] Vitaliy ezhyev ethodology of Domain Specific athematical odelling. Proceedings of the 3 rd International Congress on Natral Sciences and Engineering ICNSE 2014 (Kyoto, Japan, ay 7-9). pp [2] Steven Kelly and Jha-Pekka Tolvanen Domain-Specific odeling: Enabling Fll Code Generation. Wiley-IEEE Compter Society Pr. p [3] Vitaliy ezhyev etamodelling Architectre for odelling Domains with Different athematical Strctre. Advanced Compter and Commnication Engineering Technology. Lectre Notes in Electrical Engineering. 315: [4] Vitaliy ezhyev Architectre of software tools for Domain-Specific athematical odelling. Proceedings of 2014 International Conference on Compter, Commnication, and Control Technology (Langkawi, alaysia, September 2-4). pp [5] Vitaliy ezhyev, Refik Samet Geometrical eta-metamodel for Cyber-Physical odelling. Proceedings of International Conference Cyberworlds 2013 (Yokohama, Japan, October 21-23). pp [6] Vitaliy ezhyev, Oleg. Lytvyn, Jasni ohamad Zain etamodel for athematical odelling Srfaces of Celestial Bodies on the Base of Radiolocation Data. Indian Jornal of Science and Technology. 8(13): pp [7] Vitaliy ezhyev, Felipe Pérez-Rodrígez etamodelling approach and software tools for physical modelling and simlation. International Jornal of Software Engineering and Compter Sciences (IJSECS). 1: pp [8] Oleg. Lytvyn Interlination of fnctions and some its applications. Kharkiv: Osnova. p [9] Frank Natterer The athematics of Compterized Tomography. Classics in Applied athematics. Book 32. SIA: p [10] Patent on invention No , Ukraine. The method to restore the internal strctre of a 3D object. Ivan Sergienko, Oleg Lytvyn, Vitaliy ezhyev Decl Pbl Isse No