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Background: While researching new algorithms for computer graphics, we focused on the ones that are useful in modeling biological formations. Iterated function systems (IFS) are commonly used to visualize fractals or 3D selfsimilar objects. Their main advantage is the brevity. In the simplest cases, it is enough to define the shape of the base module and the set of transformations to create a multimodular object. In the literature, models of shells, horns, and beaks are described. To model more complex formations, the modified method was introduced in which parameters of the transformation depend on the number of iteration. Methods: The presented method combines IFS with a new approach to modeling compound objects, which uses positioners. The positioner itself and its possible applications were described in papers that do not refer to IFS models. Results: We show here how to use positioners with IFS models, including branched ones (as models of the bronchial tree). The achieved models can be simplified or more accurate depending on the variant of the algorithm. Thanks to the positioners, these models have a continuous lateral surface regardless of used shape of cross-section. The algorithm is described along with the requirements for a base module. Conclusions: It is indicated that positioners simplify the work of a graphic designer. Obtained models of bronchial trees can be used (e.g. in 3D interactive visualizations for medicine students). Keywords: computer graphics; computer science; geometric modeling; iterated function system (IFS); modeling and simulation in medicine and biology; positioner. *Corresponding author: Malgorzata Prolejko, Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Sloneczna 54 Street, Olsztyn 10-710, Poland, E-mail: m.prolejko@matman.uwm.edu.pl Cezary Stpie: Institute of Computer Science, Warsaw University of Technology, Warsaw, Poland Materials and methods IFS with condensation For the IFSC requirements, we consider modular models with modules similar in the geometrical sense. To obtain a description of the shape of a given module, it is enough to describe a module called the "base module" and the set of 238Stpie and Prolejko: Branched IFS models with positioners for biological visualizations Figure 1:Example of a compound solid in subsequent iterations for q=2: (A) the shape of the base module V with transformations P1 and P2 and (B) compound solid Zi in subsequent iterations. transformations appropriate to the given module. Thanks to IFSC, we can limit this set to transformations for first iteration, and the rest can be calculated during construction process. Now we consider a method for creating modular models with IFSC. Let us predefine numbers k and q, a certain set V in the 3D space called the base module and a set of transformations that are similarities described by matrices Pj (j=1, 2, ..., q). Number k refers to the number of iterations in the procedure necessary to obtain the final model called "compound solid". The number q determines the number of branches. The compound solid is obtained by the iterative process described as follows: Zi =V Pj Zi-1 , j =1 q Figure 2:Example of disconnected compound solid made by the same transformations as in Figure 1 but different shape of the base module. where Zo=V. The process is stopped when i=k, which means that Zk is the final set (i.e. compound solid). In this process, in every iteration, the set Zi is made as an union of module V and q copies of a compound solid from the previous iteration, each transformed by a corresponding transform matrix Pj. As the result, we obtain a compound solid organized hierarchically into a tree, which has q branches in each node. Here is an example of the description of a compound solid in every iteration for q=2: Z 0 =V , Z 1 =V P1V P2V , In the IFSC, the shape of the base module V and the set of transformations {Pj} are defined independently. The obtained compound object may have some defects due to that independence [e.g. they may be disconnected (Figure 2) or the modules may overlap each other]. We present the method of constructing the shape of the module with transformations Pj. The created compound solid has its modules that do not overlap but contact in the way that the final lateral surface is continuous. To achieve that, we use positioners. Positioners were previously used for modeling unbranched plants [3] and horns [6]. Positioners In the base module V, we can distinguish a subset B, which we call its "basis". While placing the copies of the base module in the 3D space according to transformations Pj, the subsets Aj=PjB (j=1, ..., q) are bases of these copies. In the global coordinate system, that is the coordinate system of the compound solid, they have proper shape and position: placement and orientation. The sets Aj will be called "positioners" (Figure 3). Z 2 =V P1 ( V P1V P2V ) P2 ( V P1V P2V ) =V P1V P1P1V P1P2V P2V P2 P1V P2 P2V In the example shown in Figure 1, the transformation P1 consists of a sequence of elementary transformations: scale with factor 0.75, rotation of 45° around the center of the basis, and the vertical translation by 9 units; transformation P2 consists of scale with factor 0.6, rotation of 60°, and vertical translation by 5 units and horizontal translation by 2 ones. Basic method Let us consider a set S as a sphere in 3D space and q+1 its disjoint subsets with a circular shape C0, C1, ..., Cq. s s This subsets delimit spherical caps C0 , C1s , ... Cq . Let us Stpie and Prolejko: Branched IFS models with positioners for biological visualizations239 they are not in V. The set V is open in the neighborhood of positioners. The union of V with every PjV gives a continuous surface in each point in the positioner Aj. An example of 3D model is shown on Figure 5. The transformations are the same as in Figure 1. Figure 3:Basis B of the base module and positioners A1 and A2. Extended method In the basic method, we assumed that the base module V is homeomorphic with T. Here we extend the class of base modules in the way that V is a union of sets V1, ..., VN being homeomorphisms of T. In that case, the basis of V does not have to be a single Jordan curve but a set of N such curves. The same applies to positioners. This type of modules was not described in the literature. In consequence, we can create base modules with much more sophisticated shapes, and at the same time, we can benefit from the simple and fast way of constructing compound solids. Figure 6 presents the model of a bronchial tree constructed with the extended method. The base module consists of three subsets colored differently, each of a circular cross-section. The biggest one refers to a bronchus, and the red and blue ones refer to blood vessels. The basis of the module is a set of three circles: one big and two smaller, which are the bases of the subsequent parts. The positioners are constructed in the same manner. For a model shown in Figure 6, the transformations P1 and P2 are matrix products of elementary matrices: (a) Rx, Ry, Rz are the rotations of an appropriate angle around the axis indicated by the respective index; (b) S is proportional scaling; and (c) Tx, Ty, Tz are translations. They are described by formulas: P1=Tx1Ty1Tz1SRx1Ry1Rz1 and P2=Tx2Ty2Tz2SRx2Ry2Rz2. In this notation, elementary transformations are applied from right to left. The values of parameters of elementary transformations are placed in Table 1. The method of creating entries of elementary matrices from the transformation parameters can be found in Ref. [4]. Figure 4:The topology of V. (A) A sphere with holes and (B) and (C) examples of its homeomorphisms. s s remove the caps C0 , C1s , ..., Cq and circles C1, ..., Cq from the surface S. We get T = S\ of a sphere with holes. Note that the edge C0 belongs to T. Below, we assume that the base module V is a homeomorphic transformation of T. Circles Cj from T become Jordan curves in the base module. We denote them by C ~ . j Examples of V sets are presented in Figure 4. Now we narrow down the class of sets V by applying additional properties resulting from positioners. ~ Let us assume that C ~ Aj ,( j = 1, 2, ..., q ) and C0 B. j It means that each hole excluding the basis has its shape, location, and orientation identical to the shape, location, and orientation of an appropriate positioner and the ~ hole C0 is identical to the basis. The set V is closed in the neighborhood of the edge B. Let us consider the following important feature of the set V. The shape of a module is such that, in each neighborhood of any point belonging to the positioner Aj (j=1, 2, ..., q), there exists a point belonging to V. It means that the positioners Aj are subsequent edges of surface V, but q j =1 C j j=0 C js in the form Figure 5:A compound solid with its continuous lateral surface: (A) set of neighboring modules placed with positioners P1 and P2 and (B) 3D model. 240Stpie and Prolejko: Branched IFS models with positioners for biological visualizations Figure 6:Creation of bronchial tree. (A) Base module created with the extended method: three colors indicate three subsets: a bronchus and blood vessels (enlarged view) and (B) model of a bronchial tree created with the base module from (A). Table 1:Values of coefficients of transformations used in the model in Figure 6. Elementary transformation Symbol Rz Ry Rx S Tz Ty Tx Parameter Angle Value for P1 30° 45° 0° 75% 8 0 2 Value for P2 30° 45° 0 75% 8 0 2 Rotation around the z axis Rotation around the y axis Rotation around the x axis Scaling with respect to the axes x, y, z Translation along the z axis Translation along the y axis Translation along the x axis Scale coefficient Translation (in arbitrary units) The elementary transformations are applied in order from top to bottom of the table. Algorithm Below, we show the algorithm relating to the basic method. (1) The data that should be fixed before the start: (1.1) The number of branches q and the number of iterations k; (1.2) The shape of B being a basis of the module V; B must be a Jordan curve; (1.3) Parameters of transformations Pj (j=1, 2, ..., q). (2) Start: (3) Construct the base module V fulfilling the following conditions: (3.1) The basis of V is a hole identical as B; (3.2) The set V has q additional holes (positioners) whose shapes are described in formulas: Aj=PjB (j=1, 2, ..., k); (3.3) The base module should possibly meet additional requirements, configuring him the biological original. (4) Construct the compound solid Zk using classical IFSC process: (4.1) Zo=V; q (4.2) for (i=1; i k; i + +) Zi =V j=1 Pj Zi-1 (5) End To obtain the algorithm relating to the extended method, one should replace the lines: (1.2), (3.1), and (3.2), respectively, with (1.2a), (3.1a), and (3.2a) as follows: (1.2a)The shape of B being a basis of the module V; B must be a set of N Jordan curves; (3.1a)The basis of V is a set of N holes identical as B; (3.2a)The set V has q additional sets of holes (positioners) whose shapes are described in formulas: Aj=PjB (j=1, 2, ..., k). Results and discussion In this paper, we discussed the way of use positioners for constructing geometrical models with a tree structure. We showed that, thanks to positioners, we can create compound solids with continuous lateral surface and proposed two methods of creating base modules: the basic one and the extended one. We proved the usefulness of these methods by an example of a bronchial tree (Figure 6) having a continuous lateral surface. This method simplifies generating models of tree structures. The method allows the usage of various modifications resulting from IFSC, such as making transformations Stpie and Prolejko: Branched IFS models with positioners for biological visualizations241 Figure 7:Traveling inside a bronchial tree. Pj depending on the iteration number. In that case, it is important to apply the same transformations for both the positioners and the module. The method does not introduce restrictions on the shape of the modules. It allows both modules with smooth surfaces and those that have wrinkles as yellow part from Figure 6. Thus, the operator is free to choose the graphic tools in the shape editing process. Our study focused on ensuring continuity of the lateral surface of a compound solid. This property is essential in constructing modular models. In the near future, we plan to publish the results of the research on creating modular models with smooth surfaces in the neighborhood of positioners. The proposed method is suitable for modern graphical tools. The models created with it can be used, for example, for the computer simulation of bronchoscopy procedure suitable for medicine students. An example of a simple bronchoscopy simulator is shown in Figure 7. The system uses a popular graphical tool -- Autodesk 3ds Max [7]. The operator is asked to drive the probe inside a bronchial tree to reach the selected module, marked in red. The probe position is indicated by a blue triangle. He can use the orthographic projection of the tree (the right window) and a view from a virtual camera placed on the probe (the left one). The characteristic features of the method are the possibility of dependence transformations from iteration index and lake of limitations in the shape of the base module. Therefore, one can create models more similar to the originals but at the cost of increased workload or, on the contrary, simplified models of low workload. We believe that this is an advantage of the presented approach. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission. Research funding: None declared. Employment or leadership: None declared. Honorarium: None declared. Competing interests: The funding organization(s) played no role in the study design; in the collection, analysis, and interpretation of data; in the writing of the report; or in the decision to submit the report for publication.
Bio-Algorithms and Med-Systems – de Gruyter
Published: Dec 1, 2015
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