Department of Science and Technology
Institutionen för teknik och naturvetenskap
Linköpings Universitet
Linköpings Universitet
SE-601 74 Norrköping, Sweden
601 74 Norrköping
Examensarbete
LITH-ITN-MT-EX--05/020--SE
Mesh simplification of complex
VRML models
Rebecca Lagerkvist
LITH-ITN-MT-EX--05/020--SE
Mesh simplification of complex
VRML models
Examensarbete utfört i medieteknik
vid Linköpings Tekniska Högskola, Campus
Norrköping
Rebecca Lagerkvist
Handledare Pär Klingstam
Examinator Stefan Gustavson
Rapporttyp Report category Examensarbete B-uppsats C-uppsats D-uppsats _ ________________ Språk Language Svenska/Swedish Engelska/English _ ________________ Titel Title Författare Author Sammanfattning Abstract ISBN _____________________________________________________ ISRN _________________________________________________________________
Serietitel och serienummer ISSN
Title of series, numbering ___________________________________
Nyckelord
Keyword
Datum
Date
URL för elektronisk version
Avdelning, Institution
Division, Department
Institutionen för teknik och naturvetenskap Department of Science and Technology
2005-02-24
x
x
LITH-ITN-MT-EX--05/020--SE
http://www.ep.liu.se/exjobb/itn/2005/mt/020/
Mesh simplification of complex VRML models
Rebecca Lagerkvist
In a large part of their work Volvo Cars uses digital models - in the design process, geometry simulation, safety tests, presentation material etc. The models
may be used for many purposes but they are normally only produced in one level of detail. In general, the level that suits the most extreme demands. High resolution models challenge rendering performances, transmission bandwidth and
storage capacities. At Volvo Cars there is time, money and energy to be saved by adapting the the model’s level of detail to their area of usage.
The aim of this thesis is to investigate if the Volvo Cars models can be reduced to containing less than 20% of its original triangles without compromising too much in quality. In the thesis, the mesh simplification field is researched and the simplification algorithm judged to best suit the needs of Volvo Cars is implemented in a C++ program. The program is used to test and analyze the Volvo Cars’ models.
The results show that it is possible to take away more than 80% of the the model’s polygons hardly without affecting the appearance.
Upphovsrätt
Detta dokument hålls tillgängligt på Internet – eller dess framtida ersättare –
under en längre tid från publiceringsdatum under förutsättning att inga
extra-ordinära omständigheter uppstår.
Tillgång till dokumentet innebär tillstånd för var och en att läsa, ladda ner,
skriva ut enstaka kopior för enskilt bruk och att använda det oförändrat för
ickekommersiell forskning och för undervisning. Överföring av upphovsrätten
vid en senare tidpunkt kan inte upphäva detta tillstånd. All annan användning av
dokumentet kräver upphovsmannens medgivande. För att garantera äktheten,
säkerheten och tillgängligheten finns det lösningar av teknisk och administrativ
art.
Upphovsmannens ideella rätt innefattar rätt att bli nämnd som upphovsman i
den omfattning som god sed kräver vid användning av dokumentet på ovan
beskrivna sätt samt skydd mot att dokumentet ändras eller presenteras i sådan
form eller i sådant sammanhang som är kränkande för upphovsmannens litterära
eller konstnärliga anseende eller egenart.
För ytterligare information om Linköping University Electronic Press se
förlagets hemsida
http://www.ep.liu.se/
Copyright
The publishers will keep this document online on the Internet - or its possible
replacement - for a considerable time from the date of publication barring
exceptional circumstances.
The online availability of the document implies a permanent permission for
anyone to read, to download, to print out single copies for your own use and to
use it unchanged for any non-commercial research and educational purpose.
Subsequent transfers of copyright cannot revoke this permission. All other uses
of the document are conditional on the consent of the copyright owner. The
publisher has taken technical and administrative measures to assure authenticity,
security and accessibility.
According to intellectual property law the author has the right to be
mentioned when his/her work is accessed as described above and to be protected
against infringement.
For additional information about the Linköping University Electronic Press
and its procedures for publication and for assurance of document integrity,
please refer to its WWW home page:
http://www.ep.liu.se/
Abstract
In a large part of their work Volvo Cars uses digital models - in the design pro-cess, geometry simulation, safety tests, presentation material etc. The models may be used for many purposes but they are normally only produced in one level of detail. In general, the level that suits the most extreme demands. High res-olution models challenge rendering performances, transmission bandwidth and storage capacities. At Volvo Cars there is time, money and energy to be saved by adapting the the model’s level of detail to their area of usage.
The aim of this thesis is to investigate if the Volvo Cars models can be re-duced to containing less than 20% of its original triangles without compromising too much in quality. In the thesis, the mesh simplification field is researched and the simplification algorithm judged to best suit the needs of Volvo Cars is implemented in a C++ program. The program is used to test and analyze the Volvo Cars’ models.
The results show that it is possible to take away more than 80% of the the model’s polygons hardly without affecting the appearance.
Preface
The Master Thesis was conducted at Volvo Car Cooperation, Torslanda, and was initiated by Robert Jacobsson - Teamleader Geometry Simulation. The thesis is the final examination of the Master of Media Technology program at Link¨oping University1.
I would like to express my gratitude to the following people who helped me through and made this thesis possible:
- Robert Jacobsson for initiating this project and for seeing a solution where other people saw a problem.
- Johan Segeborn for putting much effort and energy into the project. - P¨ar Klingstam for giving the work the academic touch it needed.
- Hˆakan Pettersson for being there when I could not support another Visual Studio link error.
- Sven Rud´en and Jan Hallqvist for feed-back on the test results.
- Stefan Gustavsson for his unlimited competence within computer graphics and that he never stopped believing in me.
- Bj¨orn Andersson for well thought through feed-back on the report. - Michael Hemph for his great knowledge of C++.
- Olle Hellgren and our soon-to-be-born for their support. Rebecca Lagerkvist
G¨oteborg, March 6, 2005
1Civilingenj¨orsutbildningen i Medieteknik (180 po¨ang) 1
Contents
1 Overview 6 1.1 Introduction . . . 6 1.2 Research approach . . . 7 1.2.1 Aim . . . 7 1.2.2 Scope . . . 7 1.2.3 Method . . . 8 2 Terminology 10 3 Frame of reference 14 3.1 What mesh simplification is not . . . 143.2 What mesh simplification is . . . 15
3.3 Geometry-based methods of mesh simplification . . . 16
3.3.1 Vertex decimation . . . 16
3.3.2 Vertex clustering . . . 17
3.3.3 Edge contraction . . . 17
3.3.4 Simplification envelopes . . . 18
3.4 Image-based methods of mesh simplification . . . 18
3.5 Major features of the mesh simplification algorithms . . . 20
3.6 VRML - Virtual Reality Modeling Language . . . 20
3.6.1 General description . . . 21
3.6.2 VRML as a scene graph . . . 21
3.6.3 How geometry is described in VRML . . . 21
3.6.4 The Volvo Cars VRML models . . . 22
4 Results - research and development 24 4.1 Researching the mesh simplification field . . . 24
4.1.1 Requirements for the mesh simplification algorithm . . . . 24
4.1.2 Choosing an algorithm . . . 25
4.1.3 Mesh simplification algorithm of choice . . . 28
4.2 Development of the application . . . 31
4.2.1 Part one . . . 32
4.2.2 Part two . . . 33
4.2.3 Part three . . . 33 2
5 Results - testings 34
5.1 The size of the file . . . 34
5.2 Result of the testings . . . 35
5.2.1 The bracket model . . . 35
5.2.2 The robot model . . . 36
5.2.3 Late testings . . . 39
6 Conclusion 43 6.1 Evaluating the algorithm chosen . . . 43
6.2 Evaluating the application created . . . 44
6.3 Evaluating the results of the testing . . . 45
7 Future work 46 7.1 Geometric error measure . . . 46
7.2 Hidden surface removal . . . 46
7.3 Replacement of complex shapes . . . 47
A VRML syntax 49 B Open Source packages 51 B.1 The VRML parser . . . 51
B.2 The Qslim package . . . 51
List of Figures
1.1 Method . . . 9
2.1 Shaded and wire frame polygon mesh . . . 11
2.2 Manifold 3D Surface . . . 12
2.3 Non manifold 3D surface . . . 12
2.4 Moebius strip . . . 13
2.5 Convex polygon . . . 13
2.6 Concave polygon . . . 13
3.1 Same model different resolution . . . 15
3.2 Two planes with different resolution but the same accuracy. . . 15
3.3 Vertex Decimation . . . 17
3.4 Vertex Clustering . . . 17
3.5 Edge Contraction . . . 18
3.6 Image-Driven Simplification . . . 19
3.7 Comparison Image-Driven/Geometry Driven . . . 20
4.1 Disconnected surfaces . . . 25
4.2 Standard edge contraction . . . 28
4.3 Non-edge contraction . . . 29
4.4 Geometric interpretation of quadratic error metrics. . . 30
4.5 Overview of the C++ program . . . 31
5.1 Bracket - original model . . . 35
5.2 Bracket - simplification 12% . . . 36
5.3 Bracket - adjusted simplification 12% . . . 36
5.4 Robot - original . . . 37
5.5 Robot - details of original . . . 38
5.6 Robot - simplification 25% . . . 39
5.7 Robot - adjusted simplification 25% . . . 40
5.8 Volvo V70 original shaded . . . 41
5.9 Volvo V70 original wire frame . . . 41
5.10 Volvo V70 simplified shaded . . . 42
5.11 Volvo V70 simplified wire frame . . . 42
5 A.1 A piece of VRML code . . . 50
Chapter 1
Overview
1.1
Introduction
In a large part of their work Volvo Cars uses digital models - in the design process, geometry simulation, safety tests, presentation material etc.
The digital models are either produced at Volvo Cars or by one of their sup-pliers. The models may be used for many purposes but they are normally only produced in one level of detail. In general, the level that suits the most extreme demands like those of collision detection simulation. Few other applications de-mand such level of detail but since there is only one version of the model the same high resolution is used no matter what the purpose is.
High resolution models challenge rendering performances, transmission band-width and storage capacities. To work with excessive resolution wastes money. It slows the work process and demands extra resources.
At Volvo Cars there are no routines around the simplification of digital models. There is no software that automatically reduce the level of detail. When simplification is needed it is done manually by adjusting the settings in the Computer Aided Engineering (CAE) programs. This is laborious work and the result is not always satisfactory.
For documentation and presentation purposes it is often needed to export the models from the CAE-programs to a file format that can be opened in the program Cosmo Player1 on a standard PC. The standard CAE-programs at Volvo Cars are RobCad and Catia ran on UNIX. The only export option from these programs are the Virtual Reality Modeling Language (VRML). The high level of detail of the models and the properties of the VRML format makes the files extremely large. Most of the time too large to be opened in Cosmo Player on standard hardware.
Consequently, the VRML exports are not used and instead the presentation materials contain print-screen pictures of the models. Using static 2D pictures imply that the interactivity is lost, there is no possibility of making animations
1Volvo Cars standard Virtual Reality Modeling Language (VRML) browser 6
CHAPTER 1. OVERVIEW 7 or to show different views of the object. To make it possible to use VRML the level of detail of the models must be lowered. Ideally, to containing less than 20% of the original polygons without losing too much in visual quality.
Presentation material is important at Volvo Cars. It is the link between separate work groups like design, simulation and testing. The clearer the com-munication between these groups the better the work flow in the entire Volvo Cars organization.
Volvo Cars needs to analyze their digital model work. There is an obvious need to lower the level of detail of the 3D models. Primarily, to be able to use VRML in presentation material but in a broader perspective also to ease the work of everyone concerned with the models.
It is important to find out if the models carry redundant information. If that is the case they can be simplified without losing precision and the approxi-mations would be good enough even for a wider use than presentation material.
1.2
Research approach
1.2.1
Aim
The primary goal of the thesis is to examine if the Volvo Cars’ 3D digital models could be reduced to containing less than 20% of its original polygons using a mesh simplification algorithm. To reach this goal the thesis focuses the development of a C++ program performing mesh simplification based on an existing algorithm. To accomplish the goal, three main research questions have been put forth. The questions are to be answered by the thesis.
Q1 Which of the presently existing mesh simplification algorithms conforms the most to the needs of Volvo Cars?
Q2 What features do a mesh simplification program able to reduce the Volvo cars’ 3D digital models need to have?
Q3 If it is not possible to reduce the Volvo Cars’ 3D models by more than a 80% using the mesh simplification program constructed, what character-istics must be added?
1.2.2
Scope
The building of software is a never ending process and it will always be possible to make improvements. To limit the extent, the thesis work has been bounded to comprise the following:
- Literature study The literature study has been limited to include mesh simplification in the sense of how to reduce the number of polygons in a generic mesh. The study will not be directed towards specific cases such as multilayer meshes or hidden surface removal.
CHAPTER 1. OVERVIEW 8 - Application development The application will be confined to accepting a limited input. It will only be developed to allow the VRML syntax that is most common to the Volvo Cars’ digital models.
As far as possible the application will be based on Open Source material. Even if the code available is not optimized for the purpose of this thesis it will be taken advantage of. Reusing other peoples code is the only possible solution in order to develop such an extensive program as the mesh simplification application within the time span of the thesis. The mesh simplification program will not include a Graphic User Interface (GUI).
- Testing The major tests will be performed on smaller models (100 000-30000 polygons). Working with large models is very time consuming. In order to speed up the iteration process (see section ?? on page ??) it has been judge wise not to include larger models in the testing until perhaps the very last iteration cycle.
1.2.3
Method
The work of the thesis can roughly be divided into three parts: - Conceptualization
- Iterative development - Final analysis
The first part of the thesis, the conceptualization, would be to define how the aim of the thesis - to examine if the Volvo Cars’ 3D digital models could be reduced to containing less than 20% of its original polygons using a mesh simplification algorithm - could be transformed into practical act.
To examine if the Volvo Cars’ models could be reduced, a tool to perform the testing with is needed. The primer work of this thesis was therefore to create such a tool. Hence, the conceptualization consisted in trying to give a concrete form to how the mesh simplification algorithm could be implemented. The conceptualization would be done primarily through researching the field of mesh simplification software. To look at existing software - both commercial and open source - in order to create an idea of what the difficulties and actual problems with developing one would be. A classical literature study, with the aim of finding theories that apply to the research questions of section 1.2.1, would also be a part of the initial phase.
The conceptualization phase was to be keept short. The thesis is an edu-cational project which means that there is little experience brought into it. To design software on an abstract basis demands a lot of know-how, more than was available in this project. Hence, the ambition was to early move into the development phase and there build the software by trial and error.
CHAPTER 1. OVERVIEW 9 To successfully build something by trial and error there must be several trials. The development phase was therefor made iterative. What the development phase was to consist in can be seen in Fig. 1.1 on page 9.
Figure 1.1: The thesis’ three phases. The development phase is shaded in gray. The thought was to have outer and inner loops. The inner loop would consist of programming, testing and validation and would be repeated more often than the two outer loops. In the beginning of the development phase the iterations would mostly follow the upper outer loop and go through literature study, modeling and back to the inner loop. As the project advanced the idea was that the work would move on to the path of the lower outer loop and focus more on documentation. Nevertheless, if needed it would be possible to go back and do more literature studies even rather late in the development.
The project was planned according to Fig. 1.1 to make it possible to bring in new knowledge even during the development phase.
The testings of the inner loops would follow a simple methodology. Since computer graphics is much about what looks good is good the only validation would be to look at the models to see if their appearance seemed okay. The tests merely would consist in running the Volvo Cars’ models through the program created and examine the outcome, if there would be an outcome.
Once the development phase would be over all the testing and programming would be ended and the results analyzed and put into the documentation.
Chapter 2
Terminology
The following terminology will be used through out this master thesis report without further explanation.
Rendering
Rendering is the action of drawing the 3D computer graphics model on to the 2D screen. The following terminology can be associated with rendering. Frames per second
Rendering speed is often measured in frames per second and it refers to how many pictures of the model that can be drawn on the screen per second. GPU rendering vs. CPU rendering
To render large models could be a time consuming process. Over the years the speed of rendering has been improved mainly be the introduction of hardware rendering performed on the GPU (Graphics Processing Unit) instead of the CPU (Central Processing Unit). The GPU:s have many parallel processors optimized for rendering purposes. Using the GPU’s processors correctly can speed up rendering by hundreds or even thousands of frames per seconds.
To use GPU rendering the program performing the rendering must support it. Both OpenGL and DirectX, the most widespread graphic programming packages, support GPU rendering.
Wireframe
When a model is rendered in wire frame no surfaces properties are being shown. This is illustrated in Fig. 2.1 on page 11.
CHAPTER 2. TERMINOLOGY 11
Polygon Mesh
A polygon mesh is assumed to be a collection of vertices in three dimensional space, and an associated collection of polygons composed of those vertices. The polygons could be seen as the building blocks of the model (see Fig 2.1 on page 11). A simplification algorithm is intended to simplify a polygonal mesh by reducing the number of polygons in the mesh, while retaining as much fidelity to the model as possible. The following terminology could be associated with a polygon mesh.
Figure 2.1: 3D computer graphics model. In the model to the right, which is rendered in wire frame, the triangles constituting the model can clearly be seen.
Vertices and coordinates
In this thesis report the term vertices will be used for x, y and z-values that constitutes the points which are linked together to form the polygons.
The term coordinates refer to the integer numbers telling how the vertices will be connected into polygons. This terminology is not standard in all litera-ture and it might be that the two terms are used differently elsewhere.
Polygons and triangles
In this thesis report the term polygon refer the shapes that are formed when the vertices are linked together. There is no limit to how many vertices each polygon may have. As said, the polygons constitute the mesh.
Triangles are virtually the same thing as polygons with the only difference that triangles can only have three vertices.
CHAPTER 2. TERMINOLOGY 12 Tessellated
A tessellated surface is a surface described by polygons as opposed to a surface defined mathematically through NURBS or implicit functions.
Manifold
A manifold polygon mesh consists of a collection of edges, vertices and triangles connected such that each edge is shared by at most two triangles. In a manifold without boundaries, each edge is shared by exactly two triangles.
Figure 2.2: Manifold 3D Surface
Non-manifold
A non-manifold polygon mesh consists of a collection of edges, vertices and tri-angles connected such that each edge may be shared by more than two tritri-angles. The triangles may also intersect each other arbitrarily.
Figure 2.3: Non manifold 3D surface
Orientable
A surface is said to be orientable if all the of the face normals of a mesh may be oriented consistently. It is easier to understand the concept of orientable from the concept of non-orientable (see next paragraph).
Non-orientable
The most common example of a non-orientable surface is the moebius strip (Fig. 2.4 on page 13). It has only one surface and only one edge. Sliding a surface normal along its surface the direction of the surface normal will have changed
CHAPTER 2. TERMINOLOGY 13 with an 180 degrees when it reaches the point it started from. This is what makes the surface non-orientable.
Figure 2.4: Moebius strip
Topology
The topology of a mesh deals with the dependencies between its polygons. Sim-plified it could be said that the topology of a models tells how many holes there are in the model.
Convex polygons
A planar polygon is said to be convex if it contains all the line segment con-necting any pair of its points. Circling a convex polygon all the turns around the corners will be in the same directions.
Figure 2.5: Convex polygon
Concave polygons
A concave polygon is a polygon that is not convex. It must have at least four sides and one internal angle that is greater than 180◦.
Chapter 3
Frame of reference
3.1
What mesh simplification is not
Before digging deeper into mesh simplification it is necessary to put clear some things that are often misconceived.
When speaking about mesh simplification one speaks about the reduction of the complexity of the 3D computer graphics model itself to a lower level of detail approximation of the original one.
Reducing the entire 3D model should not be confounded with reducing the complexity of the rendered image of the model. Today’s 3D model viewers are usually very efficient when transforming a 3D model to a 2D image seen on the screen. Level of detail is adjusted according to the distance from which the model is looked at, the z-buffer makes sure that objects hidden behind other objects are never drawn on the screen etc. These techniques are concerned with the mapping of the model from 3D to 2D. They are dependent on knowing in which direction the model is viewed upon and they do nothing to the actual model, only to the picture of it drawn on the screen. The computer still needs to handle equal amount of data every time a calculation is performed on the model.
It is also important to keep clear the difference between reducing the model and compressing the file containing the model. There are many efficient tech-niques to compress the 3D model files. To illustrate, if a 3D model contains many of the same objects it is enough to describe the geometry of the first ob-ject and then refer back to that one for the rest of them. However, when shown on a screen the model is still drawn with the same amount of polygons as before the compression. The model is not reduced the file is compressed.
The new standard file format for 3D computer models at Volvo Cars is Jupiter (.jt). Jupiter is a light format, the files are small, because intelligent algorithms of compression are used. However, when drawing the .jt-models on the screen they will be just as heavy as any other model containing the same number of polygons.
CHAPTER 3. FRAME OF REFERENCE 15
3.2
What mesh simplification is
Many computer graphics application require complex, highly detailed models to serve their purposes. Collision detection simulation, for example, sometimes demands the models to be made to an accuracy of millimeters and represented in very high resolution. Sometimes the resolution is as high that the human eye cease to see the difference between the original model and the approximation (see Fig. 3.1 on page 15). Since the computational cost of using a model is
Figure 3.1: Little difference can be seen between the original circles (the second rendered in wire frame) to the left and the approximations, containing only 0,5 per cent of the polygons, to the right.
directly related to complexity is it often desirable to use an approximations in place of the excessively detailed originals. It must be possible to adapt the level of detail to the fit the usage of the model.
Reducing complexity of the computer graphics models can be done through mesh simplification. Mesh simplification is the taking away of triangles from the mesh . The more triangles a mesh has the higher the resolution. High resolution is not directly proportional to high accuracy. Take the example of Fig 3.2 on page 15. The plane to the right is by no means better described than the plane
Figure 3.2: Two planes with different resolution but the same accuracy. to the left. Even though the plane to the right would be simplified and triangles taken away until it only consisted of the two triangles of the plane to the left the definition of the plane would not be less exact. There is no loss in accuracy because before the simplification there was a redundancy in information. Thus there can be mesh simplification without decreasing the accuracy. The normal case is however that the taking away of triangles creates a coarser approximation of the original model.
CHAPTER 3. FRAME OF REFERENCE 16 Over the last 10 to 15 years there have been a large number of published articles on the subject of mesh simplification. Every method consists of two parts - how to decide which triangles to remove and how to remove them. Deciding which triangles to remove is called calculating the cost of reduction or simplification and removing the triangles is simply called simplification or reduction.
Roughly the different methods presented can be divided into two categories based on the manner in which the cost is calculated - Geometry-based and Image-based. Infinitely more research has been put into the Geometry-based field than into the Image-based which appeared just recently.
3.3
Geometry-based methods of mesh
simplifi-cation
Common to so called geometry-based simplification processes is the use of a ge-ometric 3D distance measure between the original and the simplified model to guide the simplification process. Minimizing this distance is equivalent to min-imizing the error of the approximation. Consequently, the goal of the geometry based methods is geometric fidelity. In reality there are very few applications in which geometric fidelity is of much importance. Normally it is more interesting that the approximation is visually similar to the original.
Nevertheless, research has been confined to geometry based methods. It was not until July 2000 when Lindstrom and Turk released their article on Image-Driven Simplification [10] that anything different appeared (see section 3.4 on page 18).
Even though visual similarity might be more interesting than geometric fi-delity it is not said that the geometry-based methods are not useful.
Geometry-based methods have been the subject of research since the sev-enties. Down below, an overview of what is judged to be the most important categories of these methods is presented. However, due to the abundance of publications within this field the list is bound to be incomplete.
3.3.1
Vertex decimation
Initially described by Schroeder et al. [4]. The method iteratively selects a vertex for removal, remove all adjacent triangles, and re-triangulates the result-ing hole (see Fig. 3.3 on page 17). The first algorithm presented by Schroeder in 1992 carefully preserved topology, which is restricting for multi resolution rendering systems. However, in 1997 Schroeder presented a supplement to his article [9] describing a vertex decimation technique that did not maintain the topology of the models and could handle non-manifold surfaces, something that the first one could not. The cost of removing triangles or as Schroeder calls it -the error (e) - is based on -the triangle’s area (a):
ei=
√ ai
CHAPTER 3. FRAME OF REFERENCE 17 This error measure has the effect of removing small, isolated triangles first.
Schroeder’s decimation technique has O(n) time complexity which makes it suitable for large models. It also preserves high-frequency information such as sharp edges and corners.
Figure 3.3: Vertex Decimation - The triangles are taken away and the hole is re-triangulated.
3.3.2
Vertex clustering
Algorithm initially described by Rossignac and Borrel [5]. A bounding box is placed around the original model and it is divided into a grid. Within each cell, the cell’s vertices are clustered together into a single vertex, and the model’s triangles are updated accordingly (see Fig. 3.4 on page 17). Instead of an uniform grid an adaptive structure such as an octree1can be used. The process
of vertex clustering can be very fast but it can also make drastic topological changes to the mesh. The size of the grid provide an error bound. The error bound gives a measurement of the geometric difference between the original model and the approximation yielded.
Figure 3.4: Vertex Clustering - model is divided into a grid and the vertices within the same cell is clustered to one.
3.3.3
Edge contraction
An edge of a triangle is contracted into a single point. The triangles supported by that edge become degenerate and are removed during the contraction (see 1Octree - It is a hierarchical representation of 3D objects, designed to use less memory
than representing every voxel of the object explicitly. Octrees are based on subdividing the full voxel space containing the represented object into 8 octets by planes perpendicular to the three coordinate axes.
CHAPTER 3. FRAME OF REFERENCE 18 Fig. 3.5 on page 18). The process is continued iteratively. Several researchers have made use of this method in their algorithms and the biggest difference between them lies in how the edge to be contracted is chosen. Examples of edge contracting algorithms have been presented by Hoppe [6] and Ronfard and Rossignac [7]. Garland and Heckbert later developed the edge contraction technique even further [2]. They called it pair contraction and their algorithm is capable of joining disconnected surfaces.
Figure 3.5: Edge Contraction - The two triangles charing one edge in the middle are taken away
3.3.4
Simplification envelopes
Cohen et al [8] has described this simplification technique. The mesh is encapsu-lated between an inner and an outer surface. The surfaces define the boundaries that the mesh must be contained within during the simplification process. There are a number of assumptions that the simplification surface has to fulfill which restricts the use of the algorithm. There is a local as well as a global version to the algorithm. Both algorithms try to construct new triangles from existing vertices by combining them so that the new triangles fit within the bounding envelopes. The local algorithm provides a fast method for generating approxi-mations of large input meshes while as the complexity of the global algorithm is at least O(n2) which makes it unfit to apply on large models. An advantage
of the global algorithm is that the bounding envelopes supply an error bound. The simplification envelop algorithm can only handle manifold surfaces.
3.4
Image-based methods of mesh simplification
The notion of Image-Based or Image-Driven simplification was recently put forward by Peter Lindstrom and Greg Turk. With their article Image-Driven Simplification [10] they introduced a new concept of controlling the simplifica-tion process. Instead of using the geometric 3D distance measured between the original model and the approximation Lindstrom and Turk guided their algo-rithm through measuring the difference between pictures of the original model and the approximated model. Turk and Lindstrom were the first to introduce such an approach and so far they seem to be the only ones researching it.
CHAPTER 3. FRAME OF REFERENCE 19 As many geometry-based methods Lindstrom’s and Turk’s uses the edge collapse operator to make incremental changes to the model. To evaluate which edge to collapse, i.e the cost of taking it away, they use an image metric - a function that give a measure of the distance between two images. Lindstrom and Turk use the root mean square (RMS) error to evaluate the difference between the pixels in two images. Since it is virtually impossible to capture the entire appearance of an object in a single image the object is capture from 20 different directions (see Fig. 3.6 on page 19) The corresponding images of the original and the approximation are compared in order to evaluate which edges to collapse in the following iteration.
Figure 3.6: Image-Driven Simplification is guided by the comparison between images of the original model and the approximated one. In the test accounted for in their article [10] Turk and Lindstrom used 20 images from different angles. Image-driven simplification removes detail that has little effect on rendered images of a model. When some part of a model is invisible in all views it will be drastically simplified. This feature is highly usable to CAE-models that are to be reduced for visualization purposes since they usually contain many layers of objects. To take away hidden interior objects is otherwise a difficult task.
The biggest drawback of the image-driven algorithm of Turk and Lindstrom is that it is slow. In their article Turk and Lindstrom compare with the Garland and Heckbert algorithm (see section 4.1.3 on page 28) and the image-driven one shows many times slower Fig. 3.7 on page 20 shows a model of a donut reduced to 512 triangles. The result of the two algorithm is similar but the processing time different. The Turk and Lindstrom algorithm took almost five times as long (5.33 seconds compared to 1.04).
To the image-driven algorithm’s defense it should be said that when com-paring it with a geometry-based algorithm plus an algorithm that takes away interior objects the time difference probably would not be that large.
CHAPTER 3. FRAME OF REFERENCE 20
Figure 3.7: To the left the Qslim algorithm by Garland and Heckbert, in the middle Image-driven by Turk and Lindstrom and to the right the original model.
3.5
Major features of the mesh simplification
al-gorithms
This section will summon up the major features of the above described mesh simplification algorithms. The features are put into a table in order to make it easy to compare the different methods. Even though they were presented together in section 3.3 the Pair Contraction algorithm of Michael Garland and Paul Heckbert is put separate from the Edge Contraction algorithms. This reason is that the Pair Contraction algorithm contains one crucial feature that Edge Contraction algorithms do not have.
The heading Error measurement in the below table refers to if there is a measure of the geometric difference between the original model and the approx-imation produced by the algorithm. If the quality can be told by a geometric measure.
The heading fast refers to if the algorithm is reasonably fast. A No in this column means that one of the significant features of the algorithm is that it is slow.
If the table says Depends the answer can be both Yes and No depending on which algorithm within that category of algorithms that is referred to. For example, there are Edge Contraction algorithms that can handle arbitrary input and there are those who cannot.
3.6
VRML - Virtual Reality Modeling Language
VRML is the format of the export of the Volvo Cars’ CAE-program. In order to build an application that take VRML as an input one must understand its syntax. To easier understand this chapter it is recommended to take a look at the schematic picture of the VRML syntax in the Appendix A.1 on page 50.
CHAPTER 3. FRAME OF REFERENCE 21
Table 3.1: The table shows the major features of the mesh simplification algo-rithms examined Method Takes arbitrary input Joins dis-connected surfaces Error measure-ment Maintains topology Fast Vertex Decimation Yes No No No Yes Vertex Clustering Depends Yes Yes No Yes Edge Contraction Depends No No Depends Yes Pair Contraction Yes Yes No No Yes Simplification
en-velops
No Yes Depends No Depends Imaged-Driven Yes Yes No No No
3.6.1
General description
The Virtual Reality Modeling Language (VRML) is a textual language describ-ing 3D scenes. VRML differs from other programmdescrib-ing language like C/C++ in that instead of describing how the computer should act through a series of commands, it describes how a 3D scene should look. At the highest level of abstraction, VRML is just a way for objects to read and write themselves. VRML defines a set objects useful for doing 3D graphics. These objects are called nodes. The basic node categories are: Shape, Camera/Light, Property, Transformation, Grouping and WWW. Within each categories there are several subtypes defined. To illustrate, a Transformation node can be of the subtype Transform, Rotation or Translation. Each containing different property fields.
3.6.2
VRML as a scene graph
The nodes in the VRML world are organized in a hierarchical structure called scene graph. The scene graph decides the ordering of the nodes. It also has a notion of state which implies that nodes higher up in the hierarchy can affect nodes that are lower down. A scene graph has many advantages for 3D graphics purposes.
3.6.3
How geometry is described in VRML
The basic building blocks in VRML are shapes described by nodes and their accompanying fields and field values. The basic node type that defines the geometry of an VRML-object is the Shape node. The Shape node typically contains a geometry and a appearance field. The geometry field describes the 3D-structure while as the appearance defines the surface properties. The surface properties are based on the material that could be either a color or a texture. The fields of a node could be defined either by values directly or by another node.
CHAPTER 3. FRAME OF REFERENCE 22 Predefined primitives
VRML provides several predefined primitive geometry nodes that can be used to define the objects. Such primitives are Box, Cone, Cylinder and Sphere. These objects cannot be simplified directly since the vertices and coordinates of the objects are not explicitly defined.
Vertices and coordinates
Except for the case with the predefined primitives the geometry of an object is described through vertices and coordinates. Each vertex sets a point in space - i.e. the x-, y- and z-values. The coordinates tell how the vertices are linked together into triangles, quadratics or shapes with more corners. However, ren-dering systems can only handle triangles and if the vertices are not defined as such there must be splitting before the object can be rendered.
The IndexedFaceSet node can be used as the value of the geometry field in the Shape node (see Appendix A.1 on page 50). Then the vertices are declared in the coord field and the coordinates in the coordIndex field.
There are many variations to the VRML syntax. Nevertheless, the mesh simplification application will only accept input where the geometry is described in the IndexedFaceSet node.
Normals
Normals in VRML are unit vectors used to direct the lighting and shading of the object. The normals are specified in the normal field of the Shape node. Typically the normal field value is specified by the Normal node. If the normal field is empty the normals are automatically computed for each triangle or coor-dinate in the triangle set and the normalIndex and normalPerVertex fields are ignored. Otherwise, if the field is not empty, the normalIndex field specifies the x-, y- and z-values of the normals and the normalPerVertex field tells whether the normals should be used for each face or each coordinate.
3.6.4
The Volvo Cars VRML models
The Volvo Cars 3D VRML models that will be used in the tests later on have the following properties.
- The models constitute many smaller objects linked together by the scene graph.
- The size of the files is in general very large and could easily contain millions of polygons.
- There is nothing known about the how the models are exported from the CAE-programs and thus it is better assumed that the structure of the mesh is arbitrary, i.e. it can be manifold, non-manifold, orientable, non-orientable etc.
CHAPTER 3. FRAME OF REFERENCE 23 - The models contain no textures but the different parts are of different
colors.
- The SCALE property in VRML implies that the objects will not necessar-ily have the same size as the vertices of the triangles tell since they can be declared to one size and then in the rendering process scaled to another. - The vertices and coordinates are declared in the IndexedFaceSet node.
Chapter 4
Results - research and
development
The chapter presents the results of the research and development phase of the thesis. The results shown in this chapter give the answers to research question one - Which of the presently existing mesh simplification algorithms conforms the most to the needs of Volvo Cars? - and two - What features do a mesh simplification program able to reduce the Volvo cars’ 3D digital models need to have?.
4.1
Researching the mesh simplification field
At Volvo Cars the 3D models are considered to be the originals while as the real, physical objects are the copies. Quality of the digital material is therefore important. To pick a mesh simplification algorithm that manages to reduce the resolution without compromising too much on quality is naturally the aim of the selection process.
It is also important to look at the constitution of the polygon mesh in Volvo Cars’ models. Since nothing is known about the export functions of the CAE-programs used at Volvo Cars it is better assumed that it is arbitrary, i.e mani-fold, non-manimani-fold, orientable, non-orientable etc. Thus the algorithm must be capable of handling such input.
The following section summarizes the perquisites that a mesh simplification algorithm should fulfill in order to function on the Volvo Cars digital models. To understand some of the requirements it is recommended to keep the properties of the Volvo Cars’ models in mind. These are stated in section 3.6.4 on page 22.
4.1.1
Requirements for the mesh simplification algorithm
- The algorithm should be capable of handling arbitrary input. 24
CHAPTER 4. RESULTS - RESEARCH AND DEVELOPMENT 25 - It is preferable that the algorithm can close disconnected surfaces (see Fig.
4.1 on page 25).
Figure 4.1: The original model to the left is simplified with an algorithm that disallows the joining of disconnected surfaces (middle) and with one that allows it (right).
- The algorithm should be able to handle large models. - It is preferable that the algorithm is reasonably fast.
- The algorithm should not cause any flipping over of triangles. A flipped-over triangle will not reflect light properly and appear as black in the rendered model.
- It is preferable if the algorithm has a built-in handling of hidden objects. - The algorithm should be fairly straightforward to implement.
- It is preferable that the algorithm produces a geometric measure of how much the approximation differs from the original model.
- The manner in which triangles are taken away should be intelligent and relate to the purpose of mesh simplification.
4.1.2
Choosing an algorithm
The requirements stated in the previous section automatically exclude some simplification algorithms. Which ones to leave out can be deduced by looking at the major features of the various algorithms.
To make it easier for the reader the table from section 3.5 on page 20 is included again. However, if there is a need for instructions on how to read the table the reader must turn back to 3.5.
Note that there is a difference between the Error measurement of the table and the Error metrics that is mentioned further on. Error metrics refers to what guides the algorithm. How it is decided which triangles to take away. Er-ror measurement refers to if the algorithm produces a geometric measurement telling how much the approximated model differs from the original. An Error measurement is interesting to have since it makes it possible to set boundary values in terms of a geometric measure. It is possible to say that the model
CHAPTER 4. RESULTS - RESEARCH AND DEVELOPMENT 26
Table 4.1: The table shows the major features of the mesh simplification algo-rithms examined Method Takes arbitrary input Joins dis-connected surfaces Error measure-ment Maintains topology Fast Vertex Decimation Yes No No No Yes Vertex Clustering Depends Yes Yes No Yes Edge Contraction Depends No No Depends Yes Pair Contraction Yes Yes No No Yes Simplification
en-velops
No No Depends Yes Depends Imaged-Driven Yes Yes No No No should be reduced until the the boundary value of 5 millimeters difference be-tween the original and the approximation is reached instead of just stating that 50% of the faces should be taken away.
Every algorithm has Error Metrics but not all provide an Error measure-ment.
The table tells that the Simplification envelops algorithm does not fulfill the requirement of arbitrary input. Arbitrary input is an important claim and not accepting it strongly speaks against the Simplification envelop algorithm.
The Pair Contraction algorithm is an improvement of the the Edge Contrac-tion algorithms. As can be seen from the table the Pair ContracContrac-tion algorithm complete more requirements than the Edge Contraction algorithms since it can join disconnected surfaces. In this case the Edge Contraction algorithms do not have any other advantage over the Pair Contraction and they are thus excluded in favor of Pair Contraction.
This leaves four algorithms Vertex Decimation, Vertex Clustering, Pair Con-traction and Image-Driven.
The error metrics of Vertex Decimation does not seem convincing. The triangles are taken away in oder of area going from small to large. Even though it is plausible that taking away small triangles causes less harm than taken away large, there is no guarantee. It would be more interesting if the error metrics was based on the difference between the original and the approximation.
Vertex Clustering is known to be a fast and sloppy algorithm. Sloppy in the sense that it can make drastic changes to the mesh. It does not have a thought through methodology on how to decide which triangles to take away. Instead all the vertices within a grid cell are clustered together no matter what the cost of such an operation might be. Since it is important that the approximated model differ as little as possible from their originals the rather unscientific approach of the Vertex Clustering algorithm is unappealing.
There are three major arguments that decide between the Image-Driven and the Pair Contraction algorithms. The first argument can be found in the table.
CHAPTER 4. RESULTS - RESEARCH AND DEVELOPMENT 27 The Image Driven algorithm is slow. As stated in the requirements in section 3.6.4 the input models are usually very large making the speed of the algorithm an issue. This obviously speaks against the Image-Driven algorithm. On the other hand, there is no claim that the application should be capable of working in real-time and thus the time-argument is not strong enough to solely exclude this algorithm.
The advantage of the Image-Driven algorithm over the Pair Contraction is that it has an built-in handling of hidden objects. Looking at the structure of many of the Volvo Cars’ models (see section 5.2.2 on page 36) it becomes clear that such a feature could be very useful. However, one would want to have the choice of turning the hidden surface handler on and off since there might be cases where the interior of a model should be maintained. With the Image-Driven algorithm that would be impossible. It is also perfectly possible to add a hidden surface handler to any mesh simplification algorithm. Consequently, the built-in handling of hidden objects cannot be used as an argument for the Image-Driven algorithm.
The third argument is the most important one. As described, the Image-Driven algorithm is guided by the visual comparison between the original and the approximated model. This is a excellent technique if it can be secured that it is only the visual that matters. With the Volvo Cars application there is no such warranty. If the mesh simplification application was to be used only on presentation material a visual comparison could be enough. However, there is a possibility that Volvo Cars would want to extend the use of mesh simplification and apply it on all their models. Even the ones used for geometry simulation and hence collision detection. In such a case geometry becomes more important than appearance and guiding the algorithm by what looks good would not be sufficient.
Judging by the above described arguments the algorithm that seems to best fit the requirements is the Garland and Heckbert Pair Contraction [2]. The algorithm takes arbitrary input, joins disconnected surfaces, does not maintain topology and it is reasonably fast. The one major feature missing is the geo-metric error measure. It is however perfectly possibly to add such a quality to the algorithm. Garland himself has published a paper [1] on how this could be done.
The Garland and Heckbert algorithm is guided by an error metrics that carefully measures how much the mesh would change if a certain triangle is taken away. It is actually this error metrics that gives the algorithm its cred-ibility and it is probably the reason to why the algorithm has reached such a success within the computer graphics community. The error metrics of other algorithm such as the one of Vertex Clustering or the one of Vertex Decimation are not adapted to the actual purpose as the Pair Contraction’s is. Garland and Heckbert has looked at what is really important in the context - a mini-mized difference between original and approximation - and let their algorithm be guided accordingly.
The Garland and Heckbert algorithm is presented in detail in the next sec-tion.
CHAPTER 4. RESULTS - RESEARCH AND DEVELOPMENT 28
4.1.3
Mesh simplification algorithm of choice
Garland and Heckbert first presented their algorithm in an article[2] - Surface simplification using error metrics - published on SIGGRAPH in 1997. Since then there has been some improvements made to it [3]. Primarily for the han-dling of colors and textures. However, the part of the algorithm used in this theses work is entirely from the 1997 article.
The essence of the Garland and Heckbert algorithm is here presented. Pair contraction
The Michael Garland and Paul S. Heckbert surface simplification algorithm is based on the iterative contraction of vertex pairs. A pair contraction written (v1, v2) → ~v, moves the vertices v1and v2 to the new position ~v, connects all
their incident edges to v1, and deletes the vertex v2. If (v1, v2) is an edge, then
one or more faces will be removed (see Fig. 4.2 on page 28) . Otherwise, two previously separate sections of the model will be joined at ~v (see Fig. 4.3 on page 29). Using pair contraction instead of edge contraction makes it possible to merge individual components into a single object.
The algorithm is based on the assumption that, in a good approximation, points do not move far from their original positions. A pair is valid for contrac-tion if either:
1. (v1, v2) is an edge
2. ||v1− v2|| < t, where t is a threshold parameter
Using a threshold of t = 0 gives a simple edge contraction algorithm (see Fig. 4.2 on page 28). Higher thresholds allow non-connected vertices to be paired (see Fig. 4.3 on page 29). If the threshold is too high, widely separated portions of the model can be connected and it could create O(n2) pairs. The set of
Figure 4.2: Standard edge contraction (t = 0) The edge shared between the two shaded triangles is contracted into a single point. The shaded triangles become degenerate and are removed.
valid pairs is chosen at initialization time. When the contraction (v1, v2) → ~v
is performed every occurrence of v2 in a valid pair is replaced by v1, and
CHAPTER 4. RESULTS - RESEARCH AND DEVELOPMENT 29
Figure 4.3: Non-edge contraction (t > 0) When the threshold t > 0 non-edge pairs can be contracted and unconnected regions joined together.
Defining the cost of a contraction
In order to select which contraction to perform Garland and Heckbert have introduced the notion of cost. To define the cost, they attempt to characterize the error at each vertex. In the original model, each vertex is the solution of the intersection of a set of planes of the triangles that meet at the vertex. A set of plane can be associated with each vertex and the error can be defined as the sum of squared distances to its planes:
∆v = ∆([vxvyvz1]T) =
X
p∈planes(v)
(pTv)2 (4.1) where p = [a b c d]T represents the plane defined by the equation (ax + by +
cz + d) = 0 where a2+ b2+ c2= 1. The error metric described above can be
rewritten in the quadratic form:
∆v = X p∈planes(v) (vTp)(pTv) = X p∈planes(v) vT(ppT)v = vT( X p∈planes(v) KP)v (4.2)
where KP is the matrix:
KP= ppT = a2 ab ac ad ab b2 bc bd ac bc c2 cd ad bd cd d2 (4.3) KP can be used to find the squared distance of any point in space to the plane
p.
The sum of all Kp can be represented with Q which then represents the
entire set of planes for each vertex. The error at vertex v = [vxvyvz1]T can then
be expressed as the quadratic form:
CHAPTER 4. RESULTS - RESEARCH AND DEVELOPMENT 30 For every contraction (v1, v2) → ~v a new matrix Q containing the
approxima-tions of the error at the new vertex ~v must be derived. Garland and Heckbert have chosen to use the simple additive rule Q = Q1+ Q2.
Selecting ~v
Before the contraction (v1, v2) → ~v can be performed a new position for ~v must
be chosen. The easy solution would be to select either v1, v2or (v1+ v2)/2
de-pending on which of them that has the lowest value of ∆(~v). However, Garland and Heckbert go farther than that and chooses the ~v that minimizes ∆(~v). The error function ∆ is quadratic and finding its minimum is thus a linear problem. ~v is found by solving ∂∆/∂x = ∂∆/∂y = ∂∆/∂z = 0 which is the same as solving for ~v in following system of equations:
q11 q12 q13 q14 q12 q22 q23 q24 q13 q23 q33 q34 0 0 0 1 ~ v = 0 0 0 1 (4.5)
Assuming that the Q matrix is invertible gives the following solution to equation 4.5: ~v = q11 q12 q13 q14 q12 q22 q23 q24 q13 q23 q33 q34 0 0 0 1 −1 0 0 0 1 (4.6) If Q is not invertible, the algorithm seeks the optimal vertex along the edge v1v2. If that also fails ~v is chosen from the endpoints and the midpoints.
Geometric interpretation of the cost
The level surface ∆(v) = , i.e. the set of all points whose error with respect to Q is , represents a quadratic surface. Geometrically these are almost always ellipsoids with ~v as the center of the ellipsoid (see Fig. 4.4 on 30).
CHAPTER 4. RESULTS - RESEARCH AND DEVELOPMENT 31
4.2
Development of the application
The primary aim of the thesis is to examine if the Volvo Cars’ 3D digital models could be reduced to containing less than 20% of its original polygons using a mesh simplification algorithm. The method (see section 1.1 on page 9) chosen to reach this aim was to create a C++ program that would take the Volvo Cars’ VRML models as an input and give simplified approximations of these as an output. The basis of the C++ program would be the implementation of an existing mesh simplification algorithm.
Section 4.1 on page 24 gives an account of which algorithm that was selected and why. As seen, the choice fell on Michael Garland’s and Paul Heckbert’s Pair Contraction simplification with Quadratic error measure [2]. The theory of the algorithm is fully presented in section 4.1.3 on page 28.
The first program created consisted of the three parts visualized in Fig. 4.5 on page 31. Part one takes care of the user input from the DOS-prompt and the parsing of the VRML file with the original model. To parse a file means to read it and to extract information from it. The file is read and the model is stored in the memory as a scene graph structure. (Section 3.6 on page 20 tells more about this structure.) Part one then runs a loop that extracts the
Figure 4.5: The C++ program created for the mesh simplification application consists of three parts and the user interface.
vertices and coordinates from the first node in scene graph. The polygons are split into triangles and the information is sent on to Part two where the actual simplification takes place. Once the vertices and coordinates are simplified to desired percentage they are sent on to Part three where they are reinserted into the three structure. The program then moves back to Part one and the next node in the hierarchy is approached. The loop goes on until all nodes has been simplified. Once finished Part three writes the new VRML structure to a file. This is the basis of the program. To yield better results the program has been altered slightly during testing. The next chapter accounts for these changes.
The specifics about each part of the application are accounted for in the three following sections.
CHAPTER 4. RESULTS - RESEARCH AND DEVELOPMENT 32
4.2.1
Part one
For Part one of the program there were three major problems that had to be solved and the essence of those will be accounted for here.
Node by node vs. entire model
The VRML-model is simplified node by node and not the entire model at the same time. This means than an object in the VRML scene graph is extracted, simplified and reinserted into the hierarchy before the next object is approached. There are advantages and disadvantages with this method. An advantage is that there is no limit to what size the entire model may have since it is divided into parts and the algorithm only treats a little at the time.
Simplifying only the vertices and coordinates of the same node does not change their surface properties (since they are the same for all of them) and thus there is no need to send that information along. It also preserves the internal order of the objects and does not break the VRML scene graph apart. Simplifying node by node also makes it possible to improve the speed of the application by creating parallel threads in the program so that nodes are read from the VRML-file, simplified and reinserted simultaneously.
The main disadvantage with node by node simplification appear when the polygons are unevenly distributed in the model. If the entire model is simplified at the same time, polygons will be taken away primarily from the denser part of the model because the cost of removing them there is lower. When the model is reduced in parts every node will be reduced to the same extent (50%, 70%, 90% etc) no matter if it belongs to the dense part or not. This is a major disadvantage and could lower the quality considerately.
Which information to extract
The only information that is extracted from the VRML node and sent onto the next part of the program are the vertices (the x, y and z values of each point) and the coordinates (the information about how the vertices are connected together). The VRML-model is reduced node by node and thus the information about the surface properties does not need to be extracted and connected to each triangle. All the triangles that are reduced at the same time have the same surface properties and after simplification is performed they are put back into their original node where their surface properties are described.
Not even the normals are taken out and connected to the triangles. Instead they are taken away for each node in the VRML-model. It is not necessary to set the normals explicitly in VRML. If the the normals are not set the VRML-viewer will calculate new normals for each triangle.
How to split the polygons
In VRML a polygon can easily consist of more than 20 vertices. Since the Gar-land and Heckbert algorithm (see section 4.1.3 on 28) only can handle triangles
CHAPTER 4. RESULTS - RESEARCH AND DEVELOPMENT 33 the VRML polygons must be split before sent onto simplification.
From VRML1.3 and onward there is support for non-convex polygons. Nev-ertheless, the use of them is not recommended and in this application it has been assumed that all polygons are convex. Only dealing with convex polygons make splitting easier. From the sixth vertex of a polygon and onward the splitting can be done according to the following algorithm:
setCoordIndex(6thvertex, i, i + 1, −1)
To familiarize yourself with the VRML syntax please have a look at Fig A.1 on page 50
There are two exceptional cases for polygons with four and five vertices. In larger polygons the first five vertices are treated according to the algorithm of five polygons and the rest according to the above described algorithm.
4.2.2
Part two
In Part two the actual simplification takes place. The code is an implementa-tion of the Garland’s and Heckbert’s mesh simplificaimplementa-tion algorithm described in section 4.1.3 on page 28.
The code uses functions from the Qslim packages which is an Open Source software put together by Garland himself. Appendix B tells more about the Qslim package. The flow of the code in Part two can be summarized as follows:
- Compute the Q matrices for all the initial vertices - Select all valid pairs
- Compute the optimal contraction target ~v for each valid pair (v1, v2). The
error ~vT(Q1+ Q2)~v of this target vertex becomes the cost of contracting
that pair.
- Place all the pairs in a heap ordered by cost with the minimum cost pair on the top.
- Iteratively remove the pair (v1, v2) of least cost from the heap, contract
this pair, and update the cost of the pairs involving v1.
Once the vertices and coordinates been simplified they are sent on to Part three.
4.2.3
Part three
The work of Part three is actually quite simple. No major difficulties were encountered and not big problems were needed to be solved. The work of Part three is to delete the old vertices and coordinates from the VRML node and to insert the new ones. This was achieved simply by calling functions from the Cyber97VRML library. To print the VRML scene graph back to the file the print function of Cyber97VRML was used.
Chapter 5
Results - testings
The previous chapter accounted for the choice of mesh simplification algorithm and how if was implemented. Now it is the time to use the application created in order to fulfill the aim of the thesis - to examine if the Volvo Cars’ 3D digital models could be reduced to containing less than 20% of its original polygons using a mesh simplification algorithm.
The results where produced in several sets. When Garland’s Qslim pack-aged had been linked to the VRML parser and the in between interface was functioning satisfactory the first round of testing was performed. Then followed improvements of the application and thereafter new tests.
The expectation was to be able to take away more than 80% of the polygons from the original models without creating major degeneracies in the approxi-mations.
The tests were performed on models that are used in the simulation process at Volvo Cars.
In this chapter the testing and analysis of two different models will be ac-counted for. The models have distinct characteristics but are both representative for what the Volvo Cars’ models might look like.
5.1
The size of the file
It is important to realize that the size of the file containing a 3D-model is not directly proportional to the number of triangles the model contains. (This fact has been discussed previously in section 3.1 on page 14.) A file of size 25 Mb can actually contain a lighter model than a file of 20Mb. It depends very much on how compressed the writing of the file is. When splitting the polygons of more than three vertices into triangles the VRML file becomes bigger in size because expressing the polygons as triangles is a less compressed way of writing.
At Volvo Cars the size of the models is always measured in the size of the the file which is misleading.
CHAPTER 5. RESULTS - TESTINGS 35
5.2
Result of the testings
For the first tests small models of some 100 000 polygons were used. Simplifying larger models (some millions of polygons) is very time consuming and in the beginning quick feed back is important.
5.2.1
The bracket model
The first object tested was a model of a so called bracket which is a small part of the outer front of the car.
At the first testings the results were not at all satisfactory. A simplification of merely 50% already created problems and taking away more than 80% of the original polygon showed unthinkable. The first conclusion drawn was that the model was not as high resolutioned as thought from the beginning and could therefore not be simplified as much. It seemed curious however that this small model would contain as many polygons if it was not of high resolution. An application was created that counted the polygons of each node in the VRML-models. When printing this information very interesting facts were discovered. It proved that the polygons were unevenly distributed throughout the model. Some parts were as simplified as they could possibly be while as others were of extreme resolution. In this particular case the parts modeled in a very a high resolution was rather small and thus difficult to discover merely by looking at the model rendered in wire frame. This is illustrated in Fig. 5.1 on page 35, the two small holds in the upper corners of the model actually constitute 88% of the total number of polygons in the entire model.
Figure 5.1: Original model (91 638 polygons) with the high resolution hold to the right. The hold almost appear not to be rendered in wire frame due to the excessiveness in polygons.
Simplifying the model in Fig. 5.1 applying the same amount of reduction to the entire model produced a very poor result which can be seen in Fig. 5.2 on page 36. 88% of the polygons are taken away and the appearance of the approximation created is far from the original. Especially disturbing are the large holes created when simplifying already low resolution parts of the model.
CHAPTER 5. RESULTS - TESTINGS 36
Figure 5.2: Approximation containing only 12% of the original polygons. Every component is reduced to the same extent leaving the holds with superfluous polygons and the rest of the model with too few of them.
Setting the parameters differently so that only the high resolution holds were simplified while the rest of the model was left untouched generated a completely different result. Achieving the same amount of simplification (88% of the original polygons taken away) the approximation produced this time, to be seen in Fig. 5.3 on page 36, is hard to tell from the original model from Fig. 5.1.
Figure 5.3: Approximation containing only 12% of the original polygons. The holds are reduced to only containing 0.5% of the original polygons while the rest of the model is maintained intact.
The result shown in Fig. 5.3 is indeed satisfactory. However, to reach such an outcome the model had to be analyzed and the parameters of the simplification algorithm set according to the structure. Since this is not an automatized procedure it would be time consuming to do it for every single model.
5.2.2
The robot model
The next object to be examined is a model of an industrial robot. Before robots can be used in production it must be decided how they should move and what they should do. This is done through geometry simulation using the digital models.
