Variable Voxel Computing Method

Innovative Approaches to Reduce the Computing Load in Voxel-based and New Representation Methods

Sezgi Yalçınkaya1, Burak Delikanlı2 1,2Istanbul Technical University 1,2{yalcinkayas18|burak.delikanli}@itu.edu.tr

The concept of ``voxel'' refers to a volumetric element or volumetric and corresponds to the smallest piece that can be computed in solid and complex model analysis. The voxel-based solid modeling commonly used by geometry-based CAD (Computer-Aided Design) applications. Whilst other geometry-based modeling methods, it uses as the smallest unit instead of dots or vectors. However, the size of the data contained in the smallest unit causes problems such as computing load and representation inaccuracies. This study fundamentally aims to find a fast and effective method for voxel-based solid modeling. While doing that it presents a new visualization algorithm. During the research, the transformation of a geometric model into voxels, then the reproduction of these voxels, and finally, the representation method were practiced and compared. In this process, three complex models were developed and compared by their complexity, their voxelization time, and the amount of time that spend during the formation. As a result, the study proposes new representation methods for voxel-based solid modeling.

Keywords: Voxel-based Modeling, Solid Models, Representation Methods, Computing Load

INTRODUCTION Aided Design (CAD), Computer-Aided Manufactur- Nowadays, computers are not just machines that col- ing (CAM), and Computer-Aided Engineering (CAE) lect, distribute, and visualize data but they are in applications existing and developing in the industry. a completely different position. Meanwhile, it has Accurate visual representation of geometric models been used as a production planning, control, and dis- through computers has become a problem for math- tribution center in fields such as industry and busi- ematicians, designers, manufacturers, and engineers ness, as modeling and representation tools in the as well as developers. fields of architecture and art, and as a production Geometric modeling represents a collection of and simulation tool in the fields of engineering. Dif- methods that can be used in computer-based appli- ferent model representations are used in Computer- cations to synthesize, representation, and analysis of

D1.T4.S2. MAKING THROUGH CODE –BUILT BY DATA AND THE ARCHITECTURAL ILLUSTRATION - Volume 1 - eCAADe 38 | 663 shape and other geometric information (Agarwal and plications, findings, outcomes, and evaluations were Waggenspack 1992). Modeling systems that we used discussed. in CAD, CAM, or CAE applications are usually geo- metrical model-based. There are three different cat- SOLID MODELS AND VOXEL-BASED MOD- egories of geometric modeling methods: wireframe ELING or line, surface, and solid modeling (Al-Ahmari et al. 2016). These modeling methods have advantages Solid Models A three-dimensional model is a computational repre- and disadvantages compared with each other, and sentation of any object that is intended to be trans- these criteria determine which method will be used ferred, by referring volumetric data in x, y, and z coor- in the representation. While solid modeling is the dinates. Modeling is a way of production of a model most comprehensive modeling method in represent- by using some modeling software packages (e.g Pro- ing a shape, it has disadvantages compared to wire- E, CATIA) or three-dimensional scanning tools (CAT, frame and surface models due to excess of the com- MRI, etc.) to produce a row model in the three- puting load. However, solid modeling includes vol- dimensional space environment. Constructive Solid ume information as well as edges and surfaces to rep- Geometry, Boundary Representation, Feature-based resent a shape. Solid models can be considered the modeling can be listed as examples of modeling best way to represent a shape, but the multiplicity of types. Representation of the solid models usually operations limits their use. represented by the surfaced based modeling soft- Solid modeling is widely used in medical model- ware.“The surface-based modeling method has the ing as well as in the fields of engineering and design. advantage of being more efficient interns of capac- The number of information contained in the repre- ity and (sometimes) productivity on the computation sentation of the models in these areas increases the because an object can be defined easier by the sur- preference of solid models. The increasing complex- face than volume elements.” (Toennics and Tronnier, ity of the assembly process and the number of parts 2010). Therefore, this modeling method is fast and and components increases the need for efficient solid practical and is considered sufficient for many mod- modeling methods in production and design. Also, els. However, Surface models are lack of volumetric production-oriented applications in this area are of knowledge and cannot be easily prepared for volu- great significance. This study aims to find solutions metric analysis and imaging. For this reason, it car- to reduce the computing load by analyzing the repre- ries great importance to express the differentiation of sentation methods of solid modeling and proposing volumetric density differences, and these occupancy innovative approaches. In addition, the frame of solid vacancy rates in complex models are transferred well, modeling is examined and the voxel-based model- easily, and quickly. There are different schemes to ing method is explained. Then, analyzes are made represent three-dimensional models. These are Sur- on the computing algorithms created with the new face models, Lattice models - networks and approaches proposed. For this approach three dif- (directional models), decomposition-based (, ferent models were selected and developed in a vir- BSP), and voxel-based representation schemes. tual field then they were compared and discussed ac- cording to their responses to those algorithms. Their reactions to transformation into voxel models were shown in the charts. Their transformation speed and their reaction to execute their model conversion to another volumetric model were examined in detail. Finally, in the conclusion section, possible future ap-

664 | eCAADe 38 - D1.T4.S2. MAKING THROUGH CODE –BUILT BY DATA AND THE ARCHITECTURAL ILLUSTRATION - Volume 1 Figure 1 of voxel models can also be applied.“It is expected A Modular Volume: that voxel-based modeling will be used extensively The Bemis Cubical in future in areas like CAD/CAM/CAE, medical imag- Modular Concept ing, scientific simulations and visualizations, anima- (Russel, A.L., 2012). tions, and non-engineering modeling, virtual reality systems, and teleoperations and controls, etc. Some manufacturing-related problems can be better ana- lyzed using voxel-based modeling than conventional surface-based systems. ”(Patil and Ravi, 2005) Re- lated problems can be analyzed better than conven- Voxel-Based Modeling tional surface-based systems by using voxel-based The voxel is the name given to the three-dimensional modeling. The focus of this study on this system, elements that are the smallest part of an object in the which also offers a new visualization algorithm, is to third dimension. When modeling volumetrically, ev- create a new volumetric modeling method for en- ery regular or irregular element in any object can be gineering / non-engineering components. A voxel- considered as an element of the voxel collection. The based model representation scheme was developed simplest and most commonly used notation is cubic together for this purpose. voxels which are aligned with the coordinate axes at a uniform grid. Volume visualization in complex ge- REPRESENTATION METHODS ometries and the analysis of large data sets is a very The preparation of the representation of a model is rapidly evolving field for .“Volume very important in terms of finding the correct expres- visualization systems are used to create high-quality sion for that model. Volume graphics and volume displays from scalar and vector datasets defined on models have developed considerably over the past multidimensional grids, usually for the purpose of decade thanks to technological diversity. Represen- gaining insight into a scientific problem.”(Patil and tation methods which were prepared by making cer- Ravi, 2005). The term voxel also used by volumet- tain extraction operations from solid models, analyz- ric modeling. However, the Voxels used in CAD im- ing the volumes by breaking them into pieces over plementation are slightly different, as they are usu- their occupancy and gaps or three-dimensional mod- ally based on the volumetric occupancy and space els created by drawing certain boundaries of two- state of the voxel only in the given object. It is not dimensional shapes help the models to represent true to say that this is new for voxel-based model- and transfer the true meaning. ing methods, and we have been using this model- Figure 2 ing method for a long time under different head- Volume ings. We can see modular representation theories in Representation some architectural representations of Albert Farwell Methods Bemis in the 1930s (Figure 1). Later on, these repre- sentation theories were not frequently used as repre- sentation methods due to the lack of large memory and processing capacity required for computers to use voxel models with sufficient resolution. But now computers, along with improvements in hardware and software, have high enough processing speed and cheaper memory that representation methods

D1.T4.S2. MAKING THROUGH CODE –BUILT BY DATA AND THE ARCHITECTURAL ILLUSTRATION - Volume 1 - eCAADe 38 | 665 Volume Representation Methods representation of the scene of the equivalent dis- Volume representation can give a lot of information crete voxel-based expression for algorithms gener- about the size, density, or material of a model. This ated from a geometric representation of the scene. information, make it easy to represent the models All of these algorithms are called voxelization algo- that are difficult and complex to transfer with two- rithms” (Cohen-or and Kaufmen, 1995). dimensional lines. These are usually entitled ‘Surface Models’ because the prepared models are often used Basic Voxelization Algorithms in terms of visuality. Representation methods that In this section, the process of the voxelization of geo- can be used for the “surface model” can be exempli- metric models was examined. These models are the fied as; Constructive Solid Geometry (CSG), Boundary crankshaft, which is a complex engine part, a free- Representation (B-Rep), Sweep volume, Binary Space form sculpture made by hand and produced without Partitions (BSP) (Figure 2). the use of any computer-aided system, and a med- ical model, the Pelvis bone, which is convenient for reaching and examining (Figure 4). The separation Figure 3 of a geometric model into voxels, then the reproduc- Quadtree and tion of these voxels, and finally the representation method used in the computer memory with space Representation and processing facilities were examined by compar- (Patil and Ravi, ison. The computer used during the process is the 2005) Lenovo Legion Y520 laptop with 1050 ti processor Voxel Model Representations power. Algorithms were studied on Grasshopper Since voxel models are considered to be the smallest which is an interface of Rhinoceros. All algorithms parts that cover in a given volume, volumetric occu- tested and the results are shown in the tables. pancy rates should be counted in models prepared Figure 4 with these voxels. In voxel-based models, space is Studied Models considered as a regular sequence of cells (generally (Crankshaft, cubes) 3D object is represented as a list of objects Sculpture, Pelvis filled with voxels. The representation methods, com- Bone) monly used for the representation of these models can be listed as, discrete cell enumeration, octree rep- resentation, and array representation (Figure 3).

ANALYSIS Algorithm 1. The first algorithm simply places ran- Voxelization dom points in externally introduced geometry and “Voxelization is the process of transforming geomet- then assigns random cubes to each of these points. ric objects from their continuous geometric repre- This condition, in which each cube is considered a sentations into a group of voxels approaching the voxel, proceeds by increasing the number of voxels continuous object” (Kaufman, 1993). Since the small- exponentially within the same model. The simplic- est pieces, we reach as a result of two-dimensional ity of the algorithm demonstrates the importance of scanning operations are pixels, we can call the pro- distribution and voxelization from accurate control cess we reach voxels as a result of three-dimensional points in model analysis and resolution. Results and scanning.“Modeling a geometric scene in transformation were shown via charts (Table 1). requires algorithms generated from a geometric

666 | eCAADe 38 - D1.T4.S2. MAKING THROUGH CODE –BUILT BY DATA AND THE ARCHITECTURAL ILLUSTRATION - Volume 1 Table 1 Model transformation according to change of voxel size and graphical representation of first algorithm results

D1.T4.S2. MAKING THROUGH CODE –BUILT BY DATA AND THE ARCHITECTURAL ILLUSTRATION - Volume 1 - eCAADe 38 | 667 Table 2 Model transformation according to change of voxel size and graphical representation of the second algorithm result

668 | eCAADe 38 - D1.T4.S2. MAKING THROUGH CODE –BUILT BY DATA AND THE ARCHITECTURAL ILLUSTRATION - Volume 1 Table 3 Model transformation according to change of voxel size and graphical representation of the third algorithm result

D1.T4.S2. MAKING THROUGH CODE –BUILT BY DATA AND THE ARCHITECTURAL ILLUSTRATION - Volume 1 - eCAADe 38 | 669 Algorithm 2. Unlike the first, this algorithm does the Voxel Data Analysis and Results voxelization process by calculating several basic frag- The results of the analysis of this study, the reactions mentations. For this system, which is constructed us- of different geometric models with different expec- ing the octree representation method, the geometry tations are examined and explained through tables. boundaries defined from the outside identified, then These tables include an engineering model which they put into a box, and this box is divided into cubes is a crankshaft, an abstract sculpture, and a medi- within the defined grids. Cube sizes are in the hands cal model, a pelvic bone. These models were exam- of the user who uses the algorithm. After the cubes ined in three different algorithms. All of these mod- are partitioned, the geometric shape transferred and els run with the algorithms, the values of the for- the intersection between these cubes are separated mation rate, resolution value and representation for- from each other. The remaining part is the geometric mat differences were examined in this section. The model, which is voxelized. The data that considered numerical values were obtained with the Grasshop- to be the prime voxel of this model and the forma- per plugin which runs on the Rhinoceros interface tion rates of the model renewed as voxel size changes and the differences were discussed with the follow- were given in the following chart and table (Table 2). ing charts.(Figure 5, 6) Outcomes and areas to devel- The results have been followed through the whole re- oped have been discussed in the conclusion part. search. Figure 6 Algorithm 3. Finally, algorithm three maintains the Comparison of foundations of the second algorithm, and the models vocalization rates are constructed in the form of array representation. between algorithm With this algorithm, the targeted algorithm is to get 1 and 2 results faster than the results we get in the second one. Besides, the color codes in the sequence repre- sentation flow from one color to another, making it easier for the user to understand the occupancy va- cancy rates. The transformation results were shown for each model with the following chart and table (Ta- ble 3).

Figure 5 Comparison of the Transformation Time for Each Model

670 | eCAADe 38 - D1.T4.S2. MAKING THROUGH CODE –BUILT BY DATA AND THE ARCHITECTURAL ILLUSTRATION - Volume 1 CONCLUSION how voxel-based modeling methods could be used To sum up, this study proposes three different algo- more efficiently in the future. With this inspiration, rithms that translate the models which are prepared new representation methods can be obtained, which by using the geometry-based modeling method increase resolution and occupancy-vacancy rates to into voxel-based models. These models are the optimal levels for models obtained using the afore- crankshaft, which is a complex engine part, a free- mentioned voxel-based modeling methods. form sculpture made by hand and produced without the use of any computer-aided system, and a med- REFERENCES ical model, the Pelvis bone, which is convenient for Agarwal, SC and Waggenspack Jr, WN 1992, ’Decompo- reaching and examining. The algorithms were pre- sition method for extracting face topologies from pared with the Grasshopper plug-in that running on wireframe models’, Computer-Aided Design, 24(3), the Rhinoceros interface. These algorithms work on pp. 123-140 the basis of different representation methods used Akbaş, R. 2003, Geometry-based modeling and simulation of construction processes., Ph.D. Thesis, Diss. Disser- in volumetric modeling techniques, and their dif- tation, Department of Civil and Environmental En- ferences for each model are examined. The results gineering, Stanford University, Stanford, California, were shown in the tables. Depending on the num- 2003. ber of surfaces between the models, if the assigned Al-Ahmari, A, Nasr, EA and Abdulhameed, O 2016, voxel numbers and voxel sizes are changed for each Computer-Aided Inspection Planning: Theory and model, how long this process takes was taken from Practice, Crc Press Dong, Z. 2004 ’A Smart Voxelization Algorithm’, Proceed- Grasshopper data. The procedure provides informa- ings of Pacific Graphics 2004 tion about how the models are developed and repre- Landi, G., Niezgoda, S.R. and Kalidindi, S.R. 2010, sented by computer-based modeling. The best ap- ’Multi-scale modeling of elastic response of three- proach to perceive a model’s volume density and dimensional voxel-based microstructure datasets most appropriate solution procedures are discussed using novel DFT-based knowledge systems.’, Acta in the results obtained. Even though it is presumed Materialia, 58.7, pp. 2716-2725 Patil, S. and Ravi, B. 2005 ’Voxel-based representation, that every computer has a different CPU capacity display and thickness analysis of intricate shapes.’, and/or lack of storage capacity; it is clear to under- In Ninth International Conference on Computer Aided stand from this study that, the more model com- Design and Computer Graphics (CAD-CG’05) plexity increases, the longer it will take to transform Russell, A.L. 2012, ’Modularity: An interdisciplinary his- a surface-based model to a volume-based model. tory of an ordering concept.’, Information & Culture, However, we can easily see those different represen- 47.3, pp. 257-287 Subburaj, K., Patil, S. and Ravi, B. 2006, ’Voxel-based tation methods can be used to abridge this time by thickness analysis of intricate objects.’, nternational examining the differences between Algorithm 2 and Journal of CAD/CAM, 6.1, pp. 105-115 Algorithm 3. In a study based on the same algo- rithm, the conversion period in a model obtained by using the octree representation format is almost twice the rate of formation of a model created by performing array representation. This study showed us that solid models that were used in almost every field can be achieved much more quickly and effec- tively when using voxel-based modeling methods, where volumetric occupancy and vacancy rates are better understood. The study provides inspiration for

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