Voxel-based Solid Models:

Representation, Display and Geometric

Analysis

M. Tech Dissertation

Submitted in partial fulfillment of the requirements for the award of the degree of

Master of Technology

(Manufacturing Engineering)

by

Sandeep Sudhir Patil

(Roll No. 03310002)

under the guidance of

Prof. B. Ravi

Department of Mechanical Engineering

Indian Institute of Technology, Bombay

Mumbai – 400076

June 2005

i

Abstract

‘Voxel’ stands for ‘volumetric element’ or ‘volumetric pixel’ and is a three-dimensional

element analogous to what pixels are in two-dimensions. ‘Volumetric modeling’ in the

form of exhaustive enumeration of voxels is becoming popular in CAD/CAM/CAE

world, owing to its superiority over traditional solid models especially for free form

modeling (volumetric modeling, display and analysis).

In this work, a new generic voxel based volumetric modeling and visualization system

has been developed. The binary voxel model is stored as a stack of voxel layers

represented as ‘bit-arrays’. The process of ‘voxelization’ converts polygonal solid model

to the voxel format. The voxelization gives satisfactory results in terms of memory

consumption and processing time for voxel resolutions up to 1000 – overall volume of 1

billion voxels. Voxel resolution can be set as per application requirement. A new

algorithm has been developed for visualization of the voxel model using direct point

based rendering. This uses look-up table to generate realistic rendering. The images

generated are better than simple cube face display which gives blocky appearance. Low

level functions that operate on the voxel data are written to perform geometric

computation and reasoning. With this, functions to perform sectional-display, X-ray scan

and skin-removal simulation have been developed. The algorithms have been tested for

complex free-form models like a ‘Ganehsa’ and a Pelvic bone model along with other

engineering models. The outcome is a new basic voxel based modeling framework which

can be extended and coupled to standard solid modeling systems as a supplement

providing additional capabilities to traditional surface based systems.

Keywords- Voxel, Solid modeling, Volume modeling, Free-form, Geometric reasoning.

ii

Contents Chapter No. Title Page No.

Abstract i

List of tables v

List of figures vii

1 Introduction 1-5

1.1 Solid modeling 2

1.2 Voxel based modeling 3

1.3 Report organization 5

2 Literature review 6-31

2.1 Volume representation methods 6

2.2 Volume graphics 8

2.3 Voxel model representations 9

2.3.1 Discrete Cell Enumeration 9

2.3.2 Octree representation 10

2.3.3 Array representation 11

2.4 Voxelization 12

2.4.1 Essential characteristics of Voxelization

algorithm 13

2.4.2 Scan conversion algorithm 13

2.4.3 Implicit functions 14

2.4.4 Distance shells and fields 14

2.4.5 Parity count method 15

2.4.6 Ray stabbing 15

2.5 Volumetric display (rendering) 16

2.5.1 Classifying volume rendering methods 16

2.5.2 Indirect methods 17

2.5.3 Direct methods for the display of surfaces in binary data 19

2.5.4 Display algorithms for grey-scale data 22

2.5.5 Radiosity 23

2.6 Applications of Voxel (Volume) based modeling 24

2.6.1 Scientific data set visualization 24

2.6.2 Amorphous entity visualization 26

2.6.3 Representing solid interiors 26

2.6.4 Visualizing volume data 27

iii

2.6.5 Haptic modeling and rendering 29

2.7 Summery of Literature survey 29

3 Problem definition 32-38

3.1 Motivation 32

3.2 Study of present voxel based modeling systems 33

3.2.1 FreeForm Modeling Plus 33

3.2.2 3D Doctor 34

3.2.3 MIMICS 35

3.3 Objective and scope 37

3.4 Approach 37

4 Voxel data structure and voxelization 39-48

4.1 Voxel data set 39

4.2 Voxel data structure 40

4.2.1 Bit array 41

4.2.2 Voxel data on disc 43

4.3 Voxelization 43

4.3.1 Voxelization algorithm 43

4.3.2 Modified Voxelization algorithm 46

4.3.3 Errors in voxelization 46

5 Voxel display and analysis 49-56

5.1 Intensity look-up table 49

5.2 Geometric analysis of voxel model 54

5.2.1 Progressive sectional display 55

5.2.2 Skin removal simulation 55

5.2.3 X-ray scan 56

6 Implementation and results 57-73

6.1 Voxelization 57

6.2 Voxel display 63

6.3 Voxel section display 69

6.4 Skin removal 70

6.5 X-ray scan display 71

7 Conclusions 74-75

7.1 Summary of work done 74

7.2 Future scope 75

iv

Appendix 76-78

A. ‘STL’ file format 76

B. 3D discrete topology 77

References 79-80

Acknowledgement 81

v

List of figures

Fig. No. Figure title Page No.

2.1 Volume representation methods 7

2.3.2 Quadtree and Octree architecture 11

2.5.1 Classification of volume rendering methods 17

2.5.3 Direct volumetric rendering 22

2.6.1 Volumetric analysis and visualization 24

2.6.2 Amorphous entity visualization 26

2.6.3 Mapping human body anatomy 26

2.6.4 Solid model from CT scan data slices 27

2.6.5 Haptic modeling and rendering 29

3.2.1 Modeling in FreeForm modeling Plus 7.1 33

3.2.2 Volume Rendering in 3D Doctor 35

3.2.3 MIMICS software interface 36

4.2.1 Stack of ‘planes’ holding bit-array 41

4.3.1 Steps to initialize voxelization 44

4.3.2 Voxelization algorithm 45

4.3.3 Errors in voxelization caused by thin sections 47

4.3.4 Image showing errors in voxelization 48

5.1 Face connected neighbouring voxels 50

5.2 Voxel intensities for a sphere 52

5.2.1 Progressive sectional display 55

5.2.2 Successive skin removal simulation 56

5.2.3 X-ray scan: Intensity variations proportional to thickness 56

6.1.1 Time for Voxelization of standard shapes

with different resolution 59

6.1.2 Time for Voxelization of various solid models

with different resolution 61

6.1.3 Memory required for Voxelization of standard shapes and

various solid models with different resolution respectively 62

vi

List of figures (Cont.)

Fig. No. Figure title Page No.

6.2.1 Voxel images with point-smooth Off and On (R=20) 63

6.2.2 Voxel images with Zoom and corresponding change of

point size(S) (Resolution=200) 63

6.2.3 Voxel images with different resolutions (R) 64

6.2.4 Voxel images with different angles of rotation

about Y-axis (Resolution= 200) 64

6.2.5 Voxel model from the project and from FreeForm

(Resolution= 200) 65

6.2.6 Voxel display images of various solid models with

resolution of 400 66

6.2.7 Display Images fro Ganesha model with different

resolutions (R) 67

6.2.8 Display Images of pelvis model with different resolutions(R) 67

6.2.9 Display Images of cylinder block with different resolutions(R) 68

6.3 Sectional Display Images of cylinder block and

Ganesha model with different resolutions(R) 69

6.4 Successive Skin removal Images of cylinder block

with resolution 200 70

6.5.1 X-ray scan images of Ganesha model with minimum,

maximum and total thickness options respectively (R=200) 72

6.5.2 X-ray scan images of Cylinder block model along two

directions and with different thickness options (R=200) 73

vii

List of tables Table No. Table title Page No.

1.1 Comparison of exact and approximate 3D model representations 3

4.2.1 Voxel indices in bit-array for one Z-plane with resolution 100 42

5.1 Sample bit strings from neighbouring voxel configuration 51

5.2 Intensity look-up table 53

6.1.1 Voxelization results for standard shapes 58

6.1.2 Voxelization results for various solid models 60

1

Chapter 1

Introduction

Today, the use of computers in various fields has gone well beyond mere data storage.

Especially in engineering, computers are extensively used for virtual product modeling,

simulations, manufacturing planning and control, knowledge management, automation

and other allied business processes. The practices like CAD, CAM, CAE have developed

enormously over last couple of decades, not only in terms of penetration into industry but

the underlying technologies as well. Similar trends of increasing computer applications

are seen in other areas as well, like medical modeling and simulations, geological

modeling, environmental modeling and simulations. On the other side, non-engineering

fields like animation and computer gaming, virtual reality, art, fashion and archeological

modeling are also using computers for making new generation designs. The drivers for

this growth of computer applications are innovations in the basic computer technology

and hardware, and also the ever expanding business scope and complexities. The latest

practices like collaborative engineering, product life cycle management and tele-

operations demand more advances in the computer technology to enable faster and

efficient communication across distances.

In engineering applications, a virtual model of product and assemblies is a must for all

other computer integrated processes. Solid modeling is extensively used in engineering as

well as non-engineering areas such as medical modeling. The components and assemblies

involved are getting more complex day by day. Hence, an efficient solid modeling

technology is very fundamental for successful implementation of upstream applications.

Traditionally surface representations were used for solid modeling as they are very

compact and just sufficient for many applications. But surface models lack volumetric

information and do not cater easily for volumetric analysis and display. The need for

volumetric modeling and display was first recognized when medical scanning

technologies like CT and MRI became popular. Other areas where volume modeling is

used are scientific analysis of volumetric phenomenon like environmental and geological

research.

2

1.1 Solid modeling

Solid models come under the realm of 3D modeling. 3D modeling is a computerized

representation of physical objects in some suitable mathematic or abstract from. The 3D

models from the basis for all CAE and CAM operations.

The 3D model world constitutes two fundamental aspects: Modeling and Representation.

Modeling is how the actual computer model is generated using some modeling software

package (For instance Pro-E, CATIA) or in a raw form by some 3D scanning (CAT, MRI,

Confocal Microscopy, Radar imaging). Various methods used in 3D modeling include

CSG (Constructive Solid Geometry)

Breps (Boundary representation) and

Feature based modeling

Representation is the form of representation of the 3D model and way in which it is

stored in the computer memory. Different schemes in practice for representing 3D models

are

Surface models

Wireframe models- Meshes and polygon (faceted models)

Decomposition based (Octrees, BSP)

Voxel based: ―Voxel‖ stands for ―volume element‖ or ―volume pixel‖. Voxel

based models represent solids as a three dimensional arrangement of discrete

volume elements.

The 3D representation schemes for sold models can be broadly classified as: Exact and

Approximate representations.

Exact representation is precise model of object topology and mathematically represents

all geometry. These include Wireframe, Parametric surfaces, and Solid models

represented by CSG, BRep and implicit solid models. Here the object features are

represented exclusively.

Approximate representation involves discretization of the 3D objects and use of simple

primitives to model topology and geometry. These include Facet/ Mesh representing just

surfaces, Voxel models representing volumes.

3

The comparative positives and negatives of exact and approximate representations are

given in the following table 1.1.

Table 1.1 Comparison of exact and approximate 3D model representations

Exact representation Approximate representation

Advantages

Precision: good for simulation, modeling

Lots of modeling environments

Compact

Many applications

Easy to implement

Easy to acquire (3D scanner, CT)

Easy to render

Many algorithms possible

Limitations

Complex data structures

Expensive algorithms

Wide variety of formats

Hard to acquire data

Translation required for rendering

Loss of details

Data structure sizes can get huge, for

good fidelity

Easy to break, cracks can appear

Not suitable for certain applications

1.2 Voxel based modeling

―Voxel‖ is a 3-dimensional element analogous to ―pixel‖ in 2D. In volumetric modeling

an object is represented as a collection of voxels in 3D arrangement which may be regular

or irregular. Cubic voxels in uniform grid aligned with the coordinate axes is the simplest

and commonly used representation. The same is used in this project.

―Volume visualization‖ is a very fast growing field in computer graphics driven by need

for analysing large volumetric data sets generated in scientific and medical projects.

Volume visualization systems are used to create high-quality displays from scalar and

vector datasets defined on multidimensional grids, usually for the purpose of gaining

insight into a scientific problem. Volume visualization also uses concept of voxels. But

voxels used in CAD application are little different in the sense that generally they are

binary models where voxel only represent status of object occupancy in the given volume.

4

Voxel based modeling is not new but it was not used to extent because of huge memory

and processing capability required by computers to handle voxel models of sufficient

resolution. But now with advances in computer hardware and architectures, we have

sufficiently higher processing speed and cheaper memory that voxel models can be

implemented on desktop systems as well. It is expected that voxel based modeling will be

used extensively in future in areas like CAD/CAM/CAE, medical imaging, scientific

simulations and visualizations, animations and non-engineering modeling, virtual reality

systems and tele-operations and controls etc. Some manufacturing related problems can

be better analysed using voxel based modeling [Lu, 1996] than conventional surface

based systems.

The focus of this work is on use of ‗volumetric modeling‘ in CAD/CAE application for

engineering/non-engineering components. A ‗voxel‘ based model representation scheme

has been developed. The main goal of the project was to develop a voxel representation

scheme efficient in terms of memory and processing and which serves voxel resolution up

to 1000. That is an overall volume of 1 billion voxels. Though, in general, voxel models

of resolution up to 400 are commonly used. A new visualization algorithm using point

based direct rendering is developed for displaying voxel model. The work is different

from conventional ‗Volume visualization‘ basically in terms of type of volume data

involved and the purpose of visualization.

A CAD model is usually defined in B-Rep format which is essentially a surface

representation. Such model is first converted into a ‗volumetric model‘ representation by

process of ―Voxelization‖. The voxel data generated is basically binary data in uniform

Cartesian grid. As this is a simple and low level representation of only geometry of

object, there are no data values (either scalar or vector) in volumetric form. Hence the

display algorithm sought was to just suffice efficient visualization of object geometry

with sufficient accuracy and realism. The availability of volumetric representation and

visualization in a CAD/CAE system allows great many possibilities in geometrical

processing and analysis which are not easy to implement if not impossible, with

conventional surface representations. Along with basic display functions, few low level

geometric reasoning functions are also written to operate on voxel model. Using such

functions, section display, X-ray scan and skin-removal simulation module is also

developed.

5

1.3 Report organization

The detailed literature review regarding volumetric modeling and volume graphics is

presented in chapter two. The chapter begins with fundamentals of volumetric modeling

and its advantages and limitations. Various algorithms and methods proposed and

developed by researchers for voxel model representation and display are described.

Description of various voxelization algorithms is also included. The chapter ends with

applications of voxel based modeling and summary of literature review.

Chapter three includes the problem definition, objectives and scope of the work. A short

description of study of few existing voxel based modeling systems is included. The

chapter ends with the description of approach followed in the work.

Chapters four and five include detailed description of the major work done in the project.

The new voxel model representation scheme and voxelization algorithm are explained in

chapter four. While, chapter five describes the new voxel display algorithm developed.

The geometric reasoning and analysis functions implemented are included in chapter five.

Detail results regarding time and memory required for voxelization are presented in

chapter six. The results include testing on various engineering and non-engineering

models. The chapter also includes numerous images of display of various voxel models

using the developed visualization algorithm with different settings. The outcome of the

geometric analysis functions is presented at the end of the chapter.

Conclusion and scope for future work is presented at end of the report.

6

Chapter 2

Literature review

Volume graphics and volumetric modeling are separate streams of research, but have

considerable interlink. Over the last couple of decades, these fields have evolved

enormously in terms of variety of technologies and depth as well. The chapter begins with

evaluation of volumetric modeling and voxel based methods. This is followed by various

voxel based model representation techniques presently used. Then a detailed review of

various voxelization and volumetric display algorithms is presented.

2.1 Volume representation methods

Volume representation for engineering models is essentially space occupancy information

for a given object. The information is ‗Binary‘. It means that the space is either occupied

by the object or is void. Following are the various methods used for volume

representation which are described in Figure 2.1.

(a) Sweep volume

Sweeping a 2D area along a trajectory creates a new 3D object. The sweep may be

Translational, Tapered, Rotational or General 3D curve. Thus each volumetric segment is

defined by one ‗section curve‘ which is a closed planar curve swept along ‗sweep curve‘.

(b) Binary Space Partitions (BSP)

BSP involves partitioning 3D space by a sequence of planes successively dividing the

space into two halves each time. The tree representation of BSP is a binary tree with two

child nodes added as leaf nodes. A leaf node represents a homogeneous portion of space-

either ―in‖ or ―out‖. Regions decrease in size along any tree path. It is easy to determine if

a point lies inside our outside the region by traversing the BSP tree.

7

Translational

Tapered

Slanted

Rotational

Solids by Sweep CSG

Voxel enumeration BSP

(c) Constructive Solid Geometry (CSG)

Combines simple primitives using Boolean operations and represents as a binary tree. The

Boolean operations include Union, Intersection and Difference. To generate the object the

tree is processed in a depth-first pass. The CSG representation may not be unique.

(d) Boundary Representation (B-Rep)

A closed 2D surface defines a 3D object. At ach point on the boundary there is an ―in‖

and ―out‖ side. Boundary representations can be defined in two ways, ‗Primitive based': a

collection of primitives forming the boundary (polygons, for example) and ‗Freeform

based‘ (splines, parametric surfaces, implicit forms).

(e) Spatial Occupancy Enumeration

This is a voxel based method. It is an extension of the concept of pixels into 3D. Space is

considered as a regular array of cells (usually cubes). Each cell is called ―voxel‖. A 3D

object is represented as a list of object filled Voxels

Fig. 2.1 Volume representation methods

8

2.2 Volume graphics

Volume graphics is an emerging subfield of computer graphics. It is concerned with the

synthesis, modeling, manipulation, and rendering of volumetric geometric objects, stored

in a volume buffer of voxels. A volume representation that has general applicability is the

exhaustive enumeration of the occupancy of elementary cells (voxels) that lie on a

uniform 3D grid. For each cell, either a binary value indicating whether that cell is inside

or outside the object (binary voxel model) or a numerical value representing some

physical quantity such as density (gray-value voxel model) is maintained. Usually a

spatial ordering is imposed on the occupancy values by storing them in a 3D array. A 3D

array (called also volume buffer, cubic frame buffer, 3D raster) is typically used to store

these values [Kaufman, 1993].

Desired features of a modeling system

Selection of a representation scheme for solid models for any application involves

evaluating the following things [Jense, 1989]

Storage cost

Display complexities

Desired manipulations

Data availability

Representation conversions

Typical manipulations that may be performed on geometric data are

Geometric transformations (translation, rotation and scaling)

Modeling operations (Boolean set operations, re-slicing, spatial editing)

Numerical analysis (extraction and quantification of parameters such as surface

area, volume)

From the above listed parameters, requirements of a good solid model representation

could be identified as

Accurate Efficient Boolean operations

Concise Efficient display

Affine invariant Incorporate volume information

Easy acquisition Easy to implement algorithms

Guaranteed validity Good resolution variability

9

Voxel based modeling

Using voxel based methods for solid model representation provides following advantages

and limitations. [Kaufman, 1993]

Advantages

Simple, intuitive, unambiguous and unique representation

Same complexity for all objects

Natural acquisition for some applications (CT, MRI)

Trivial Boolean operations (CSG)

Heterogeneity and anisotropy of models can be easily incorporated into analysis

(which may be the case in objects made by powder metallurgy of recently by

Nanotechnology)

Display is viewpoint independent, insensitive to scene and object complexity

Limitations

Approximate representation: inherent to discretization

Large memory requirements

Larger processing times (compared to surface models in terms of display)

Aliasing

Lack of geometric representation

2.3 Voxel model representations

There are many methods proposed for voxelization of polygon models [Nooruddin, 2003;

Fang, 2000] and CSG models [Fang, 2000]. As there are many method of voxelization, so

there are many representation schemes for voxel models. Mostly the voxel file formats

are application specific. Few voxel model representations in use are as follows.

2.3.1 Discrete Cell Enumeration

There are many formats used by commercial software developers which have a specific

data field format specified for every voxel and the file essentially is enumeration of these

values in some sequence of voxels. For instance the following voxel representation

formats.

10

(a) Vox1999a file format

The proposed voxel file format consists of three logical sections. The file begins with a

file header that contains descriptions and copyright notices pertaining to the data in the

file. The header is followed by a volume description section that describes the first

volume and the voxel data in it. The volume data section follows immediately after the

end of the description section. Each datum in the data section represents a voxel

containing one or more fields (for example, an RGBA voxel). The volume description

and volume data sections may repeat to provide for multiple volumes in one file

[TeraRecon Inc., 2002].

(b) Gaussian Cube File format

The cube file describes volumetric data as well as atom positions. The file consists of a

header which includes the atom information and the size as well as orientation of the

volumetric data. This is followed by the volumetric data, one scalar per voxel element.

All aspects of the file are in text format.

Header- The header has the number of atoms included in the file followed by the position

of the origin of the volumetric data. Then it includes the number of voxels along each axis

(x, y, z) followed by the axis vector. This means the volume need not be aligned with the

coordinate axis, also means it may be sheared. The last section in the header is one line

for each atom consisting of 5 numbers, the first is the atom number, the last three are the

x, y, z coordinates of the atom center.

Volumetric data- The volumetric data is straightforward, one floating point number for

each volumetric element.

2.3.2 Octree representation

Octree is a data structure enabling efficient storage of 3D data. Octree is a 3D

generalization of a Quadtree (for 2D). Each node in an Octree has eight children. Octree

describes a recursive partitioning of a volume into cells that are completely full or empty

[Prakash, 1990]. Octree methods achieve data compression by storing voxel information

in a hierarchical tree structure, which is built in a top-down fashion by recursively

subdividing inhomogeneous regions of the volume into eight sub-regions until each

terminal node of the tree corresponds to a region of the volume in which all voxels share

the same value.

11

Actual implementation of octree uses pointers for representing connections between

parent and sibling nodes at every stage of partitioning (Fig. 2.3.2).

Fig. 2.3.2 Quadtree and Octree architecture

2.3.3 Array representation

In spite of the many advantages associated with an octree representation of solids, the

main disadvantage would be that of storage. This is because for every node in the octree,

we must store pointers to its parent and eight siblings. Instead, if the octree was

implemented as an array, we would have to allocate space for a full tree amounting to

[(8n+1

- 1) / 7] nodes, for a tree of depth n. However, we would not need to store pointers

to siblings and parent since their location is implicit in the array index of any given node.

Further, for a realistic reproduction of irregular solids, it is presumable that the octree

would grow close to its full size. Consequently, we find no storage efficiency in storing

the tree in a pointer based structure as opposed to an array based structure. In fact we

would find that in realistic modeling of sculptured surfaces and solids, a pointer based

structure would actually consume more memory space due to the pointer storage

overhead corresponding to each of the eight sons and parent.

Thus it can be seen that as the object shape becomes more complex, the depth of octree

increases and it becomes more inefficient in terms of memory. Instead a simple

exhaustive enumeration format of voxel data implemented as array will have size

independent of object shape complexity. Also it will avoid the overhead of storing

pointers. Also, the advantage can be taken of the regularity in the arrangement of voxels

in the given volume to enforce correspondence between the actual voxel in space and its

Octree

12

index in the array. There are many possible variations in array representation of voxel

models. Many commercial software packages have their own custom formats. The

simplest array implementation would be for a binary voxel model wherein a single bit can

be used to represent state of each voxel. For grayscale voxel model, depending upon the

range of variable at each voxel, an array of appropriate data type can be used.

Here is one such model employing arrays to store voxel information. A node in the voxel

model is a number called V-value as shown below.

V-value = vn vn-1..... v1 where 0 ≤ vi ≤ 8.

Each digit vi denotes the path to be taken at level i, (0=terminal, 1,2,3,4,5,6,7,8 are the

directions of the sibling nodes along the eight octants) to reach the node from root which

is assigned a level 0. The root carries a V-value 00...0. V-value gives the unique path

from root to any node in the tree and can also be obtained for a given path. Each element

of this array contains 2 bits (call Co, Cl) of information. These represent the color-type of

the node as follows.

type color-type = (white, gray, not-used, black);

C0 Cl Color

0 0 White

0 I Gray

1 0 Not-used

1 1 Black

A white or black node is a terminal node and has no children. However, with regard to the

array representation of voxel tree, the nodes allocated for their children are labeled not-

used. A gray node is a non-terminal node and has children which are black, white or gray.

2.4 Voxelization

‗Voxelization‘ is the process of converting geometric objects from their continuous

geometric representation into a set of voxels that ―best‖ approximates the continuous

object [Kaufman, 1993]. As this process mimics the scan conversion process that

pixelizes (rasterizes) 2D geometric objects, it is also referred to as ‗3D scan –conversion‘.

However, the voxelization process does not render the voxels but merely generates a

database of the discrete digitization of the continuous object.

13

2.4.1 Essential characteristics of Voxelization algorithm

A 3D scan-converter is required to obey some fidelity, connectivity, and efficiency

requirements [Kaufman, 1987]. The requirements assume viewing from a major axis, and

would have to be changed slightly for omni directional viewing.

(a) Fidelity

The discrete points, for which the region contained by them is entirely inside the

continuous object, are in the converted discrete object.

The discrete points, for which the region contained by them is entirely outside the

continuous object, are not in the converted discrete object.

Obviously, some discrete points will not belong to either of the above cases, and more

guidelines are necessary.

If the object is 3D (volume), its ―inside‖ will be converted according to requirements

1 and 2. Other points will be treated by majority decision - the discrete point is in the

object if more than half its region is in the continuous object.

(b) Connectivity

For a 1D object, 6-connectivity or 26-connectivity is required, depending on

implementation needs.

For a 2D object ―lack of tunnels‖ of certain connectivity is required. Again, 6-

connected or 26-connected tunnels can be disallowed depending on implementation

requirements.

(c) Efficiency

Use integer or fixed point arithmetic whenever possible, and floating point only when

impossible otherwise.

Use simple operations (addition) rather than complex ones (multiplication), and avoid

altogether more complex operations (square root) within inner algorithm loops.

Operate in parallel so that the temporal complexity is less than the computational

complexity, without too much hardware complexity.

Various methods used for voxelization are as follows.

2.4.2 Scan conversion algorithm

This is also called as ―volume sampling‖. The basic method for converting objects is

spatial occupancy. A regular grid of voxels is placed over the domain of the source

object, and for each voxel a binary decision is made as to whether each voxel is inside or

14

on, or outside the object [Kaufman, 1987; Huang, 1998]. The recognized problem with

this method is that the representation produces a discrete surface which is recognizably

blocky. Simple solutions have been to increase the resolution of the data set, but this in

turn increases the memory requirements and rendering times [Jones, 2001]. These

algorithms do scan-conversion with computational complexity which is linear in the

number of voxels written to the CFB (Cubic Frame Buffer) [Kaufman, 1987]. All

algorithms are incremental and use only additions, subtractions, tests and simpler

operations inside the inner algorithm loops. Since the algorithms are basically sequential,

the temporal complexity is also linear. There are algorithms proposed for voxelizing lines

and polygons [Wang, 1993].

2.4.3 Implicit functions

Sampling is a valid method for producing voxelised objects, but it suffers from resolution

problems (blocky images), and restricted orientation of normals. One type of object for

which this volume sampling is not a problem is that of implicit functions [Jones, 2001].

Here the voxel grid is calculated by evaluating the function of the implicit surface – for

example, for a sphere of radius r, f (x, y, z) = x2 +y

2 +z

2 - r

2.

Normals from such data produce naturally shaded objects. The implicit functions encoded

as above can be considered as ‗space filled‘ voxelization – there is a value at every point

in the domain of the object. The sampled voxelizations only produce values in the vicinity

of the object and can be considered to be ‗shell‘ voxelizations.

2.4.4 Distance shells and fields

If the interest is just in encoding the surface and do not need a space filled distance field,

we can voxelize the object just in the vicinity of the surface. Such a voxelization is called

‗a distance shell‘. Here the computational expense is significantly less, and the surface

representation is far superior when compared to the over sampling method which takes an

equivalent time. The distance shell adequately represents the object, and is a valid method

for voxelizing objects where only the surface needs to be encoded. As a shell voxelization

it benefits from the advantage of requiring less memory to store. Space filled distance

fields [Jones, 2001] can be calculated from distance shells and gives more realistic

modeling with many application areas.

15

2.4.5 Parity count method

For polygon mesh models, classify a voxel V by counting the number of times that a ray

with its origin at the center of V intersects polygons of the model [Nooruddin, 2003]. An

odd number of intersections mean that V is interior to the model and an even number

means it is outside. This is simply the 3D extension to the parity count method of

determining whether a point is interior to a polygon in 2D. For manifold models, the

direction of the ray is unimportant and advantage can be taken of this to speed up the

voxel classification. Many parallel rays can be cast through the polygonal model and each

one of these rays classifies all of the voxels along the ray. For an N x N x N volume, we

need to cast only N x N rays, with each ray passing through N voxel centers. Although

the parity count method works well for manifold models, many polygonal models have

various degeneracies that require modifying the voxelization process. One common

problem is for a model to have small cracks or holes in the surface. To voxelize such

models, the parity count method is extended by using k different directions of

orthographic projection and by scan-converting the model once for each direction. Each

of the k projections gives the classification of a voxel (interior or exterior), and the

majority vote is the voxel‘s final classification. For watertight models, all of the votes will

agree.

2.4.6 Ray stabbing

The ray-stabbing method also makes use of orthographic projections of a polygonal

model. It differs from the parity count method, however, in the way it interprets the depth

samples of a ray. The ray stabbing method only retains the first and last depth sample

along each ray [Nooruddin, 2003]. In effect, each ray only keeps those points of

intersection where the ray first stabs the surface of the model. Keeping both the first and

last depth samples is equivalent to stabbing the surface from two directions at once, at no

extra cost. A voxel is classified by a ray to be interior if the voxel lies between these two

extreme depth samples; otherwise, it is classified as an exterior voxel. For a single

direction of projection, this can cause some voxels to be misclassified as being interior to

the surface. To avoid this, several projections in different directions are taken. If any of

the projections classify a voxel as exterior, it is given an exterior classification. Only

those voxels that are classified as interior for all projections are given the final

classification of interior.

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2.5 Volumetric display (rendering)

Traditional surface graphics is not ideal for rendering volumetric objects which have

volume data associated; for example sampled data sets from CT, MRI and amorphous

data like gases, fire smoke, cloud. Also, when operations such as peeling, cutting,

sculpting are to be performed on solid models which expose the interiors of object,

surface graphics finds difficulty [Acker, 1993].

Advantages of Volume graphics

Suitable for sampled data and amorphous phenomena

Insensitive to scene complexity

Insensitive to surface type

Allows block (volumetric) operations

Hierarchical representation

Limitations of Volume graphics

Large memory and processing power required

Object-space aliasing

Discrete transformations

Lack of geometric representation

2.5.1 Classifying Volume rendering methods

Several methods exist for the rendering of volumes. One class of these methods converts

the voxel representation into surface and line primitives [Elvins, 1992]. However,

methods of this class suffer from various disadvantages that mainly arise from the

ambiguity of determining the exact position of the surface. As an alternative, direct

methods have been developed to render volumes. These methods can be classified as

object order or image order as shown in figure 2.5.1. Object-order methods require the

enumeration of all voxels of a volume and the determination of the affected pixels on a

screen. Image order techniques, on the other hand, determine all the voxels of a volume

which affect a given pixel on the screen. Hybrid methods exist in which the volume is

traversed in both object and image order.

17

Various rendering methods can be differentiated using the following three characteristics,

(1) whether the explicit creation of an intermediate surface representation is required (if

so, then refer to the method as "indirect")

(2) whether the method is designed to operate on binary or on grey-scale data, and

(3) for methods that operate on grey-scale data, whether a binary decision must be made

about the existence of a surface at any given location in the volume.

Voxel data

Binary Grey-scale

Direct rendering Indirect rendering

Object order Image order

Fig. 2.5.1 Classification of volume rendering methods

The following sections contain short description of various direct and indirect rendering

methods for binary voxel data as per the classification described above.

2.5.2 Indirect methods

(a) Tiling method

This one of the earliest approaches used to create images from volume data sets involved

the construction of three-dimensional polygonally-tiled surfaces from planar contour

curves defined on successive two-dimensional slices through a volume. In the simplest

cases, a single closed contour (either hand-drawn or computed using an automatic

boundary-detection algorithm) is used to outline the object of interest separately on each

slice and contour points from adjacent slices are then joined to form triangular tiles.

(b) Indirect, voxel-based methods

In addition to the tiling methods, a number of voxel-based methods have been introduced

which compute and display surfaces from volume data. Most of these methods assume,

18

for algorithmic simplicity, that all data points in the volume are equidistant. If the data

has been sampled at unequal rates in the different dimensions (for example, if the distance

between data points within a slice is less than the distance between data points in adjacent

slices), then some kind of interpolation is generally used to obtain the necessary

intermediate values. In most grey-scale data sets, the value at a voxel represents a point

sample of a continuous three-dimensional distribution, which is referred to as "intensity".

In the cuberille method, the volume occupied by each voxel is modelled as a cube of

constant intensity consisting of six polygonal faces, and object surfaces are constructed

from appropriately selected connected subsets of these cube faces. Despite the large

number of primitives, images can be displayed fairly quickly once the set of cube faces

comprising the surface has been found because the shading and hidden surface

computations can be optimized to take advantage of the regularity of the polygonal

representation. The quality of an image produced by this method will be highly dependent

on the accuracy of both the surface localization and the estimation of surface normals.

Because the cuberille method relies on an intermediate representation in which polygons

may only appear in one of three possible orientations, the images produced by this

method will exhibit sharp discontinuities in surface brightness if shading is based solely

on the polygonal approximation. Attempts to mitigate the effects of these blocky artifacts

by low-pass filtering the final image, antialiasing using sub-pixel sampling, or averaging

surface normals from adjacent polygons have met with little success.

(c) Indirect, sub-voxel based methods

One of the most popular voxel-based methods is the "marching cubes" algorithm

[Lorensen, 1987] which uses subvoxel-sized triangular elements to represent isovalue

surfaces in a volume. Unlike other methods, in which each data point, or voxel, is

represented as a small cube, the basic unit of operation in this algorithm is the cubic

volume defined between eight neighboring data samples. Each such cube in the data set is

examined and if the desired isovalue surface is found to be contained within that volume,

a trianglular tesselation of the enclosed portion of the surface is derived. As a first step,

the voxel intensity (or some precomputed object inclusion likelihood value) at each of the

eight cube vertices is compared with a threshold value to determine whether that voxel

lies inside or outside of the desired surface. The cube is then assigned one of 256 possible

19

index values based on the configuration of its vertices. These 256 cube configurations can

be mapped by rotation into 15 topologically distinct arrangements, and a triangular

tesselation is defined for each of these canonical cases. Vertex coordinates are computed

for each triangle in the tesselation by linearly interpolating between the endpoints of the

enclosing cube edge, according to the difference between the threshold value and the

values at each of the endpoints. To achieve the best possible results in shading, normals at

the triangle vertices are computed not from the surface's polygonal approximation but

rather from the intensity gradients of the underlying voxel data; gradient values found at

each of the eight cube vertices, using central differences, are linearly interpolated along

cube edges to the triangle vertex locations.

The marching cubes algorithm has a number of features that recommend it. Because

subvoxel elements of arbitrary orientation are used in the polygonal representation,

surfaces produced by this method generally fit the underlying data more closely than

surfaces obtained through either a cuberille or tiling approach, and virtual memory

requirements are kept to a minimum since the algorithm operates on only two slices of

data at a time. A marching-cubes surface will usually take longer to render than a

cuberille or tiled surface, however, due to the larger number of polygons and also to the

loss of the regularity in polygonal orientation that allowed simplified shading and hidden

surface computations to be used in the rendering of cuberille data.

2.5.3 Direct methods for the display of surfaces in binary data

In the mid-1980's a number of new methods for visualizing volume data sets were

proposed, which did not require the extraction of surfaces as a first step.

(a) Octree methods

One of the earliest direct methods uses octree encoding to model objects defined by a

binary-valued volume obtained from the grey-scale data through thresholding or some

other segmentation procedure. Images can be generated from an octree data set by

establishing a correspondence between pixels in the image and face-connected strings of

voxels in the volume, then recursively traversing the octree in a back-to-front order and

whenever a non-empty voxel is encountered overwriting the corresponding pixel value

with the color of the current voxel. Pseudo-shading can be implemented by defining top,

bottom, left and right illumination factors which are added to a voxel's color when its

20

neighbor in the corresponding direction is empty. Front-to-back octree display algorithms

have also been developed for octree data sets.

The primary disadvantages of octree encoding binary data are the time it takes to build

the tree and the complexity of the tree-traversal operations. In addition, the octree

representation of a data set is highly dependent on the initial position and orientation of

the data; two data sets that differ only by a small translation or rotation may have

substantially different octree representations.

(b) Object-space methods for the display of surfaces in binary data

Algorithms which create images by projecting volume information onto the image plane

are commonly referred to as "object-space" methods. An important feature of these

algorithms is that they do not require that the complete data set be resident in memory at

all times. One of the earliest object space approaches renders images from binary-valued

voxel data sets by traversing slices of the arbitrarily-oriented volume from back to front,

beginning within each slice at the most distant corner and progressing along either rows

or columns, projecting a value derived from the depth of each voxel perpendicularly onto

an individual pixel of the final image.

‗Splatting‘ also referred as ―forward projection‖ is an object-order approach. Only

existing objects in the volume are processed, which makes this approach time-efficient.

This method, although straightforward, is not optimal for a variety of reasons. The

resulting images also suffer from the same shading problems that were encountered in the

cuberille approach, and because a given pixel may be colored and re-colored many times

before a final value is assigned, efficiency is sacrificed. In addition, this method is limited

to performing parallel planar projections; a perspective view can be achieved only by pre-

warping the volume data before rendering is begun. Moreover, due to the poor quality of

the reconstruction algorithm, small hole artifacts can appear in the rendered image when

the volume is viewed at certain angles or magnifications. If voxel data is projected onto

the image plane in order of increasing distance from the viewer, only the first value

received by each pixel needs to be retained and all subsequent attempts to write to the

same pixel must be disallowed. A front-to-back version of the above method which

produces equivalent images in less time avoids pixel re-writes through the use of a

dynamic data structure that keeps track of unwritten scan-line segments. The algorithm

21

first ensures that face connected voxels in a single row of the volume will project onto

adjacent pixels in a single scan-line of the two-dimensional image by requiring that all

rotations be done in a specific sequence. As each row of voxels is projected onto the

image plane, the endpoints of the newly generated spans can then be compared with the

endpoints of the unwritten portions of the scan-line, which eliminates the need to test for

overwrites at each individual pixel.

(c) Image-space methods for the display of surfaces in binary data

In addition to the object-space methods described above, image-space methods - which

render the data on a pixel-by-pixel basis - have also been developed to display solid

objects defined by binary-valued voxel data sets. The simplest of these methods steps

through the volume along rays cast perpendicularly from the center of each pixel until the

first non-empty voxel is encountered, shading pixels of the final image with a value based

primarily on their distance from this surface voxel. Ray casting method is sometimes

referred as ―backward projection‖ or ―resampling‖. It integrates the intensity along a ray

through the voxels (Fig. 2.5.3) [Acker, 1993]. It is an image-order approach. A ray is shot

through the volume for every pixel in the resulting image. The volume itself is processed

object oriented, where a transfer function defines the edges of an object in the volume.

The intersection point of a ray - perpendicular to the viewing plane - with an object

border is rendered by a shading function which represents lighting effects like shades and

highlights. Shading improves volume perception.

One advantage of the ‗ray-casting‘ approach is that it does not require a complete

traversal of the volume; processing can be stopped as soon as a surface voxel is

encountered along the ray from each pixel. A disadvantage of this approach is that in

order to generate images at arbitrary orientations the entire data set must generally be

accessible in memory at all times. Techniques have been developed which reduce the

time required to render a series of images by exploiting the coherence between

consecutive frames of a rotation sequence.

22

Fig. 2.5.3 Direct volumetric rendering

2.5.4 Display algorithms for grey-scale data

This section describes few of the visualization algorithms developed for rendering grey-

scale voxel data.

(a) Direct, binary methods for the display of surfaces in grey-scale data

While early methods were designed to operate on binary-valued voxel data sets, in which

surfaces have been pre-defined using some type of segmentation procedure, more recent

methods have moved towards integrating segmentation into the rendering process. This

gives twofold. First of all, retaining the grey-scale information at each voxel allows the

use of shading models which more accurately reflect the orientation of underlying object

surfaces. Because the values found at each voxel typically represent point samples of a

smoothly-varying function in three dimensions, shading methods that base their estimate

of surface orientation on a 0th-order reconstruction in which the volume is represented by

a set of cubes of uniform intensity are clearly not optimal. A much better approximation

to the surface orientation at a boundary voxel is given by the gradient of grey-scale values

across the voxels in a 3x3 neighborhood. In addition, surface position can be estimated

Slicing Isosurface rendering

Ray tracing

23

with greater precision when the possibilities are not limited to specific locations on a

regular grid.

(b) Direct, binary methods for the display of intensity distributions in grey-scale

data

A direct binary method has been developed which uses color to represent different

intensity ranges within a volume. Before three-dimensional surface or volume rendering

methods came into widespread use, volume data was typically viewed on a slice-by-slice

basis. A display method that builds upon this slice-based approach forms images using

stacks of two-dimensional slices. Data is extracted from the volume along a user-defined

axis to produce the set of slices, and on each slice data points are assigned one of several

possible colors based on the density range into which they fall. The slices are then

projected onto the screen, one at a time from back to front, with each slice slightly offset

from its predecessor to produce an oblique view. Only those points falling within the

desired density ranges are displayed on each slice, to prevent the unwanted obscuration of

data from preceding slices.

(c) Direct, non-binary methods for the display of grey-scale data

There exist a large number of other non-binary methods to display grey-scale volume data

such as Multi-planar reconstruction. But, as the volume model considered in the project is

binary, such methods are not directly relevant and hence are not discussed here.

2.5.5 Radiosity

In the radiosity approach, instead of modeling light as particles bouncing around in a

scene (as is done in ray tracing), light is modeled as energy whose distribution tends

toward a stable equilibrium. The amount of radiation leaving a body is called the radiosity

[Cohen, 1985]. It is the sum of the incident radiation which is reflected and that which is

emitted by the body.

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2.6 Applications of Voxel (Volume) based modeling

This section discusses various applications of voxel based modeling and volume graphics.

2.6.1 Scientific data set visualization

Because of inherent discrete nature of voxel models, they lend themselves for easy

scientific analysis and visualization involving volumes. As the voxel sizes become

smaller and smaller, made possible by larger and cheaper memory, the accuracy of

analysis continuously improves. More importantly, voxel models can very effectively

incorporate the non-homogeneity of solids into analysis, as the local property variations

can be associated with voxel models (Fig. 2.6.1).

Fig. 2.6.1 Volumetric analysis and visualization

In CAD/CAM/CAE applications like for volumetric (mass) property evaluation,

interference, collision detection (tool path planning), simulations, manufacturability

analysis [Yagel, 1995; Chandru, 1995] voxel models are more easy (though not precise)

to use. Voxel models can give real life rendering involving reflection, refraction and

diffractions in volumes.

25

In view of CAD/CAE applications, aspects of geometric modeling that can benefit from

adopting Voxel based approach are [Chandru, 1995]:

(a) Estimating mass properties

The voxel model lets designers evaluate the mass properties of the modeled object.

For instance, the total material volume can be obtained as a simple sum of all the

nonzero voxels of the volume buffer.

(b) Interference detection

Detecting interference among the components of an assembly is a major problem in

computer-aided design of mechanical parts. Analytical methods for computing such

interference have been proposed but they are complicated and do not easily generalize

to arbitrary geometries. This measure can be applied irrespective of the assembly‘s

complexity and is another advantage of a voxel model.

(c) Tolerancing

Tolerancing is an important aspect of mechanical. In a voxel-based modeler, the

intersection of two objects due to variations in tolerance can be visualized as a 3D

interference volume itself.

(d) CSG modeling

Computation of the Boolean operations reduces to voxel-by-voxel logical operations.

(e) Voxel-level analysis

The powerful aspect of voxel-based modeling is that the designer can selectively

modify individual voxels so that the resulting object meets the design specifications.

(f) Designing composites

A voxel-based modeler can ultimately provide the capability to design a composite

object with materials selectively placed at individual voxels.

(g) Reverse engineering

The combination of Layered Manufacturing (LM) technologies and volume scanning

devices (like CT and MRI) is a powerful reverse-engineering platform. A voxel-based

modeling package can import the scanned volume data, perform voxel-level

modifications, and then directly fabricate the object, or its tooling, using LM

equipment. Custom prostheses, replicas of archaeological artifacts, and retooling for

components where a sample is available but the design information is lost are some

areas that greatly benefit from this combination.

26

2.6.2 Amorphous entity visualization

The entities like smoke, clouds, steam, fire, etc., which are amorphous (discrete, random

volumes) in nature, involved in simulations or artistic models, computer games etc. are

impossible to represent using surface models. Voxel based models perfectly fit in for their

representation and visualization (Fig. 2.6.2).

Fig-2.6.2 Amorphous entity visualization

2.6.3 Representing solid interiors

Voxel models are very effective in representing interior details of solid models. It is

difficult to model as well as represent solid free from objects like Human body and

internal organs, free-form solid artwork like jewelry, artifacts in traditional modeling

schemes like CSG or B-rep. Voxel models prove to be best suited for such applications as

they provide means for

Slicing: display the volume data mapped to colours, along a slice plane

Iso-surfacing: generate opaque and semi-opaque surfaces

Transparency effects: volume material governs reflected and emitted light

Fig. 2.6.3 Mapping human body anatomy

Slice

Semi-transparent

material Iso-surface

27

These advantages of voxel models are effectively used in applications like

Surgery simulation and planning

An interactive surgical planning system allows a surgeon to simulate manipulations of

organs prior to an actual operation. Using patient data, acquired by means of CT scanning

for example, and an interactive 3D display system, multiple operating strategies can be

tried out in advance. Possible difficulties thus may be anticipated, while by recording the

sequence of actions taken during the simulation, an operation protocol can be produced.

Prosthetic design

Voxel models can be used to combine and manipulate both physical data (CT scan) and

artificially created objects (CAD models). The easy way in which spatial selection can be

made in a voxel model is used for the removal of parts of organs. The Boolean set

operations union and difference allow the combination of reconstructed organs with user-

created secondary objects, such as prosthetics.

2.6.4 Visualizing volume data

Some geometric data acquisition techniques like CT (Computed Tomography), MRI

(Magnetic Resonance Imaging) and PET (Positron Emission Tomography) generate

volume information. This lends the voxel based models as the most suitable visualization

tool for such data (Fig. 2.6.4).

Fig. 2.6.4 Solid model from CT scan data slices

This proves very much useful in medical applications like for modeling prosthesis or

implants. The following section briefly describes aspects of medical modeling.

28

Medical modeling

Some acquisition methods like CT scan, MRI, 3D-scan generate solids. There are few

unique aspects of 3D medical imaging that distinguish it from other graphics processing

applications. [Stytz, 1991]

Two well-known, closely related factors are data volume and computational cost. The

typical medical image contains a large amount of data. As an average CT procedure

generates more than 2 million voxels per patient examination. MRI, PET, and ultrasound

procedures produce similar amounts of data. The second factor is that the algorithms used

for 3D medical imaging have great computational cost even at moderate 3D resolution.

Additional factors, first, there are no underlying mathematical models that can be applied

to medical image data to simplify it. Second, the display is not static. Typically,

physicians, technicians, and other users want to interact with the display to perform

digital dissection and slicing. Third, for a 3D representation to provide an increase in

clinical usefulness over a 2D representation it must be capable of portraying the scene

from all points of view.

The importance of the voxel model comes from its use in the CT, MRI, SPECT, and PET

medical imaging modalities as well as for 3D medical image rendering. The voxel model

is the assumed input data format for the three major approaches to medical image object

space modeling described in this section: the contour approach, the surface approach, and

the volume approach. These three object space modeling techniques provide the input

data to the two major classes of 3D medical image rendering techniques—surface and

volume rendering. Three major approaches to portraying an object in a 3D medical image

volume have been developed as the contour, surface, and volume object space depiction

methods. The contour, lD-primitive, approach uses sets of lD contours to form a

description of the boundary of an object of interest within the individual slices that form a

3D volume. The boundary is represented in each slice as a sequence of points connected

by directed line segments. Before forming the boundary representation, the desired object

contour must be isolated using a segmentation operator. Thresholding can be used if the

boundary occurs between high contrast materials. Otherwise, boundary detection using

automatic or manual tracking is used for segmentation.

29

2.6.5 Haptic modeling and rendering

Haptic devices are now commercially available which give real time force feed-back.

These devices are very effectively combined with voxel modeling for

Interactive surgical simulation and

planning

Free-form modeling: virtual clay sculpting

(Fig.2.6.5)

Haptic based product design

Fig. 2.6.5 Haptic modeling and rendering

2.7 Summery of Literature survey

The important concepts and methods of voxel based modeling (volume graphics) have

been presented. Although volumetric representations and visualization techniques seem

more natural for sampled or computed data sets, they are very advantageous in traditional

geometric-based applications as well. This trend implies an expanding role for volume

visualization, and it has thus the potential to revolutionize the field of computer graphics,

by providing an alternative to surface graphics, called ―volume graphics‖. Volume

graphics has advantages over surface graphics by being viewpoint independent,

insensitive to scene and object complexity, and lending itself to the realization of block

operations, CSG modeling, and hierarchical representation. It is suitable for the

30

representation of sampled or simulated datasets and their intermixing with geometric

objects, and it supports the visualization of internal structures. The problems associated

with the volume data representation are, large memory requirement, larger processing

time, aliasing, and lack of geometric representation.

These computational and some algorithmic bottlenecks need to be overcome to realize the

potential offered by voxel based design. The possible areas for research aimed at

eliminating these bottlenecks are as follows.

(A) Memory

The memory requirements of voxel models are enormous. For storing a reasonable

resolution voxel model say 400x400x400 in raw form , (i.e. just as a 3D array of voxels)

requires 64 M bytes of space [Chandru, 1995]. One can trade-off computation time for

storage space by one of the following means.

Store the voxel array in compressed form and use algorithms that will operate directly

on the compressed data.

Convert the voxel array into some other more compact representation and reconvert

into voxels when required.

Retain the original geometric representation and use voxelization algorithms when

necessary. This is especially valuable since in engineering design there is a large

library of components that can be imported into the voxel based modeler.

(B) Rendering complexity

In order to be an effective design tool, a voxel based system should be able to update the

display at interactive rates. Current graphics rendering systems cannot provide a level of

rendering performance on voxel models that is comparable to their polygon rendering

performance. However this situation is likely to change in the near future since parallel

algorithms and hardware support for volume rendering are the focus of several current

research efforts.

(C) Interactive volume sculpting

One of the premises on which a voxel based modeling and design system is based is the

availability of a powerful interaction paradigm that provides the designer with the

31

freedom to realize arbitrary shapes. Without this capability called ―interactive sculpting‖

most of the advantages of a voxel based design for rapid prototyping will be nullified.

Interactive sculpting is currently a highly active research area.

Efficient algorithms for computational analysis of voxel based models and virtual reality

interfaces for sculpting are some of the other challenging research problems.

The simplicity of designing analysis algorithms based on mass properties in voxel based

models can be used for developing many manufacturing simulations and design solutions.

For example, metal flow simulation in casting, casting solidification analysis, prediction

of defects and alike.

The progress so far in volume graphics, in computer hardware, and memory architectures,

coupled with the desire to reveal the inner structures of volumetric objects, suggests that

volume visualization and volume graphics using voxel based technologies may develop

into major trends in computer graphics and CAD/CAM.

32

Chapter 3

Problem definition

This chapter gives the basic outline of the project in terms of motivation for selecting the

problem and the scope of the work done. Also, this includes the review of few voxel

based system available commercially.

3.1 Motivation

The literature survey clearly brings out the impact of Voxel modeling on a wide range of

applications, ranging from CAD/CAM/CAE to medical field like prosthesis design,

surgery simulations and planning. Especially for the design of complex free-form shaped

objects, voxel based methods prove very effective.

As concluded from the literature survey, there is scope for working on developing

compact, application specific voxel model representations and visualization algorithms.

Even with existing voxel based methodologies available, a large world of applications

remains unexplored, specifically in manufacturing field.

With the advantages associated with voxel (volume) modeling and the developments in

computing hardware, it is predicted that the fundamental problems like memory

limitation, resolution (accuracy with voxel size) and computation times, will be soon

resolved.

Thus, it is expected that with the need for volumetric analysis increasing with very

complex and heterogeneous systems developed by latest technologies like

Nanotechnology, advanced computer graphics in Virtual Reality (VR) simulations, will

bring more and more focus on Voxel based modeling and Volume graphics.

33

3.2 Study of present voxel based modeling systems

The following software packages are installed at Prof. B. Ravi‘s lab in Mechanical

Engineering department at IIT Bombay. We have complete license for FreeForm while

educational trial licenses for MIMICS and 3D Doctor. The installations were followed by

demonstration and training sessions by the experts from respective companies.

3.2.1 FreeForm Modeling Plus

This software is developed by SensAble Technologies Inc., USA (www.sensable.com). It

comes with a haptic interface called ―Phantom‖. The FreeForm Modeling Plus system is

the ideal tool for the design of complex, free-form shapes and delivers manufacturable

models, files for rapid prototyping, and photo-realistic renderings.

Fig. 3.2.1 Modeling in FreeForm modeling Plus 7.1

This is a free-form interactive modeling technique based on the metaphor of sculpting and

painting a voxel based solid material, such as a block of clay or wax. There are two

motivations for this approach. First, modeling topologically complex and highly-detailed

34

objects are still difficult in most CAD systems. Second, sculpting has shown to be useful

in volumetric applications. Real-time human interaction is achieved in this approach,

since the actions of sculpting (as carving, sawing) and painting are localized in the

volume buffer, a localized rendering can be employed to re-project only those pixels that

are affected. Carving is the process of taking a pre-existing volume sampled tool to chip

or chisel the object bit by bit. Since both the object and tool are represented as

independent volume buffers, the process of sculpting involves positioning the tool with

respect to the object and performing a Boolean subtraction between the two volumes.

Sawing is the process of removing a whole chunk of material at once, much like a

carpenter sawing off a portion of a wood piece. We have modeled few complex non-

engineering objects like the Ganesha and some jewelry items using this system.

The main features of the system are

Interactive sculpting of digital clay with haptic feed back

Operations like adding clay, sculpt, smudge, pull, cut, combine, remove, etc along

with standard CAD operations like extrude, cut, revolve, mirror, sweep, etc

Selective clay hardness

Voxel size- Resolution adjustment (Scaling)

File format conversions with selective accuracy

3.2.2 3D Doctor

This is a volumetric visualization and modeling package developed by Able Software

Corporation (www.ablesw.com). 3D-DOCTOR creates 3D displays from CT/MRI

images. 3D-DOCTOR's vector-based editing tools allow easy image data handling,

volume measurement, and quantitative analysis. From extracted boundary data, 3D

surface models are created using the Surface Rendering command. Multiple objects can

be easily combined to create sophisticated 3D views of complex structures for

applications such as surgical planning and diagnostic imaging. Parameters such as

material properties, colors, viewing angle, and display settings, can be adjusted

interactively. Volume rendering supports several rendering modes, including

Transparency, where voxels are treated as transparent, Direct, where only surface voxels

are displayed, and Maximum Density, where only the brightest voxel is displayed along

each ray.

35

Image thresholding Surface rendering

Direct volume rendering Transparent volume rendering

Fig. 3.2.2 Volume Rendering in 3D Doctor

3.2.3 MIMICS

Materialise's Interactive Medical Image Control System (MIMICS) is an interactive tool

for the visualization and segmentation of CT images as well as MRI images and 3D

rendering of objects. In the medical field MIMICS can be used for diagnostic, operation

planning or rehearsal purposes. A very flexible interface to rapid prototyping systems is

included for building distinctive segmentation objects.

The software enables the surgeon or the radiologist to control and correct the

segmentation of CT-scans and MRI-scans. MIMICS is a general purpose segmentation

program for gray value images. It can process any number of 2D image slices

(rectangular images are allowed). The only restriction is the physical memory of

computer.

36

Fig. 3.2.3 MIMICS software interface

MIMICS and 3D Doctor are very much similar in the sense that both take CT, MRI slice

data as input in form of Di-com images or BMP images with additional data. Then the

images are spatially referenced with each other and ―thresholding‖ is done. Thresholding

is essentially a process of selecting a set of pixels with gray values within specified limits.

Then the voxelization is as if adding thickness to the plane slices. The pixels selected are

converted to voxels by adding thickness and are grouped together to form the solid. The

distance between the slices determines the voxel thickness whereas the voxel height and

the width can be appropriately set. The model can be visualized using surface or volume

rendering with required accuracy. Some standard file conversions through import/export

are available.

Going through all these software packages, following remarks can be made.

FreeForm provides for the virtual sculpting of digital clay using Boolean operations

on voxel model representation. The model display is continuously updated surface

representation.

MIMICS and 3D Doctor use voxel model as modeling base for inherent volume

data obtaind through 3D scans like CT or MRI.

37

Voxel models are very appropriate for volumetric Boolean operations and free form

shape models like human body anatomy models.

3.3 Objective and scope

The overall objective of the project was to develop an efficient voxel based modeling and

visualization system which can be easily adopted for applications like manufacturability

analysis. The idea has been to take advantage of volumetric nature of voxel models which

make them ideal for problems involving volumes like solid models in CAD/CAM/CAE.

The focus of the work is been on binary voxel models which have only space occupancy

information. Also, cubic voxels with regular arrangement aligned with the coordinate

axes are considered. The major work involved devising fast and efficient voxelization

algorithm which can convert existing solid model in faceted format into the custom

developed voxel form. Solid model in form of ASCII STL file is taken as input into the

system. The voxel model representation scheme involves model storage in memory as

well as in hard drive in file format. Minimum memory required and faster processing

were the main goals. To serve for general application requirements, the system was to be

made capable of handling 1 billion voxel models. This gives resolution of 1000 voxels

along each coordinate axes. Still provision was sought for selectively setting the voxel

resolution as per the application requirement. A fast and effective visualization algorithm

would display the voxel model. The voxel model format would enable standard Boolean

operations and also geometric reasoning to be performed on the voxelized solid.

3.4 Approach

The work is split into modules which are implemented one by one as follows.

(a) First a voxel model representation scheme was developed which would need

minimum memory and at the same time allow faster access to individual voxel. The main

features sought in such a scheme were dynamic memory allocation, provision for

selective resolution, speed of access and ease of implementation. Various alternatives

were evaluated to select the most efficient one. Having selected the voxel model

38

representation format, a file format was determined to store the voxel model on hard drive

in as compact form as possible.

(b) After finalizing the voxel representation scheme, a voxelization algorithm was

developed to convert a solid model in STL format into the voxel format. A class base to

implement the selected representation scheme for voxel into C++ environment was

developed. The algorithm and the storage format were implemented using VC++ and

validated using various solid models. The algorithm for voxelization was optimized for

quicker voxelization and tested for few complex free-form models with various

complexities and triangle counts.

(c) After considering various methods available for voxel display, a new algorithm was

developed and implemented to render voxel models. OpenGL is used as graphics

environment. The algorithm was tested for various complex models and the system was

modified for an errors found and for better performance.

(d) Then over the developed voxel model base, many low level functions were written.

These functions include functions to access and modify individual voxels, handle voxel

bodies, voxel display functions and few geometric reasoning functions.

(e) Using the developed framework of voxel based modeling, functions were written for

sectional display, X-ray scan and skin-removal simulation. The functions were tested for

various models and the results were compiled.

39

Chapter 4

Voxel data structure and voxelization

This chapter contains the in depth description of the basic work done in the project to

come up with new voxel model representation scheme. Details of the considerations made

for designing the voxel model representation scheme along with the comparative

evaluation of different options is presented. The chapter also includes the voxelization

algorithm and the modifications made to optimize it. Discussion is presented on the errors

in the voxel model, the causes and measures to reduce the errors.

4.1 Voxel data set

As voxel model is exhaustive enumeration of volume data, usually the amount of data is

very large as compared to parametric representation of solids. The size of data depends

upon the number of voxels and also the amount of information to be stored per voxel. The

number of voxels depends upon the grid used for discretization of solids. There has to be

a trade-off between the voxel resolution and the size of data. As the size of voxels

reduces, the accuracy of representation improves and we get higher resolution in analysis.

But this will correspondingly increase the number of voxels to be stored for given solid

model which may be difficult to handle in terms of available computer memory and

computational load.

Sometimes the resolution required may be dictated by the application requirement. The

actual physical size of the object also determines the number of voxels required for

expected resolution. For example, consider a cylinder block model of an automobile with

maximum dimension of around 500 mm. Then a grid of 1billion voxels (1000 x 1000 x

1000) will give voxel dimension of 0.5 mm, which may be just sufficient for application

like casting solidification simulation. The same model with a grid of 500 x 500 x 500 will

give resolution of 1 mm which may be sufficient for approximate weight calculation or

determination of C.G. Thus, it would be very useful if a voxel model representation and

storage scheme is flexible enough so that it can be easily configured for required

40

resolution depending upon particular application. So, we have incorporated such

flexibility in the design of voxel file format and representation. But, still we will limit the

maximum grid to 1000 x 1000 x 1000 (1 billion voxels) as per the current computer

memory limitations.

The information to be stored for every voxel also depends upon the application

requirements. For a simple object, it may be sufficient to store only occupancy data,

meaning only whether a voxel is an object voxel or not. Such voxel models are called

―binary voxel models‖, as they store only one binary variable per voxel. The project

currently uses only binary voxel models. But many times, some property which varies

spatially within the region of interest (object) need to be stored for corresponding voxels.

Then we have to store a variable (discrete or continuous) with every voxel as for example

local density value, intensity, type of voxel, etc. Sometimes, there might be requirement

of storing more than one property value with every voxel, for example local velocity

vector, temperature, etc. The voxel model needs to be flexible in this aspect as well to be

able to be adopted for changing requirements.

For the voxel model to be developed, Cartesian coordinate system is chosen. Presently,

voxels are assumed to be cubic, meaning the scaling factor along all three axes is the

same. The voxels are assumed to be aligned with the coordinate axes.

4.2 Voxel data structure

The voxel data needs to be stored efficiently in memory so as to allow very quick access

to any voxel information and also be very compact as voxel data volume could be very

large. As the voxel data size depends on the resolution set, the data size is variable. For a

cubic voxel volume of resolution 1000, the maximum number of voxels may go up to

1x109. But at the same time, for any further manipulation of voxel data for geometric

reasoning or analysis, each voxel needs to be easy to access and modify. Taking this into

consideration, the voxel data is stored in memory as a ―Bit-array‖ while the file on the

disc can be made compact by writing in form of run-lengths.

41

4.2.1 Bit array

A new bit array class is defined. Functions are written to set, reset, access and edit the

voxel information in the bit array. Every voxel is represented by single bit. So, it is a

binary structure, classifying voxels as either filled by the object or empty. The bit array is

created one layer at a time. So, to start with, pointers to ‗plane‘ structure are declared.

‗Plane‘ is a structure which holds a bit array to store (resolution+2)2 voxels. Assuming

maximum possible resolution of 1000, an array of pointers to structure ‗plane‘ is created

at the beginning. Then depending upon the requirement as per the object shape, new plane

is created for every Z increment and then the voxels in that plane are stored. All bits are

initially set to ‗0‘. Then as per the voxelization algorithm, the filled voxels are set to ‗1‘.

The actual index of the voxel to be accessed from the array is calculated properly from its

X, Y and Z coordinate values. Chapter 3 includes the results of memory required for

storing voxel models of various objects with different resolution.

The actual bit array is one bit extra on all sides. This is to enable a simple filtering

algorithm for finding neighbour configuration to be applied to all voxels, including ones

at the bounding box extremes. One layer of voxel is added on all sides of minimum

voxel-box required for a model as shown in the figure 4.2.1.

Fig.4.2.1 Stack of ‘planes’ holding bit-array

To start with, one ‗plane‘ is created to represent Z=0 layer. This bit array is set to ‗0‘

allover as it is the lowermost layer just outside the voxel box. Then, as the voxelization

starts, for every increment of Z, a new ‗plane‘ is created which can hold the required bits.

Plane (Z+1)

Plane (Z)

Plane (Z-1)

One layer of voxels added

on all sides

42

After the voxelization is over with Z reaching maximum of object, one more extra ‗plane‘

is added on top, which again is set to ‗0‘ allover.

Whenever a new ‗plane‘, that is a new bit-array is created, it is set to ‗0‘ allover, and

whenever a filled voxel is found, the corresponding bit is set to ‗1‘.Then the voxel index

in the bit array is properly determined for voxel writing as well as while finding

neighbour configuration. For example, for a resolution of 100, every ‗plane‘ created will

be an array of (100 + 2)2 = (102)

2 = 10404 bits. The array in memory would look like

0 1 2 3 4 5 6 … … … … … … 10401 10402 10403

But when considered voxels on a Z-layer, actual voxel indices would be as shown in table

4.2.1.

Table 4.2.1 Voxel indices in bit-array for one Z-plane with resolution 100

x x x x x X x x x 10402 10403

510 511 X+102

408 409 X 508 509

306 307 X-102 406 407

204 205 304 305

102 103 104 104 106 x X x x 201 202 203

0 1 2 3 4 x X x x 99 100 101

Thus the total memory required (MTOTAL) for a voxel cube of resolution ‗R‘ is equal to

MTOTAL = (MPOINTER + MPLANE ARRAY) x Total no. of planes

= (4 bytes + [(R+2)2/8] bytes) x (R+2)

The goal of using minimum memory for voxel storage is achieved by two approaches.

One is the use of smallest memory unit for representing single voxel. As in the project, a

‗bit‘ is used to represent voxel in the binary voxel model. The other approach is to use as

minimum of such units as just required for the given object. This requires dynamic

memory allocation and which in turn needs a balance between pointer overhead and

memory saving.

43

4.2.2 Voxel data on disc

As voxel data is stored as bit-array in memory for quick, direct access, in contrast, voxel

data is stored on disc as a set of run-lengths to make it compact. The file is simply a

sequence of voxel type and corresponding run-length. So, ‗0‘ indicates empty voxel.

Similarly, different types of voxels can be identified by different characters or numbers. If

the voxel model is binary, containing voxels of only one type, then, file can be further

compacted by storing only run-lengths. Then, the starting run-length is always interpreted

as of empty voxels, which could be ‗0‘ also, and then follows the run-lengths for filled

and empty voxels alternately.

4.3 Voxelization

The solid model imported is in STL ASCII format. STL stands for ‗Stereolithography

Tessellated Language‘. STL format is a kind of boundary representation scheme in which

the surface of solid is represented by triangles or ‗facets‘. The file includes the listing of

facets in random order. For every facet, three coordinate values of three vertices and an

outward normal vector is stored.

4.3.1 Voxelization algorithm

After importing STL, a structure to represent the model is created which stores the

vertices, edges, facets and connectivity between them. Also, the bounding box of the

model is determined. The voxel size is determined by dividing the maximum dimension

of this bounding box by the resolution defined. The coefficients of line and plane

equation of edges and facet are also determined.

To start with, the STL model is placed in the positive octant as shown in figure 4.3.1.

This is achieved not by actually translating the solid model but taking proper Y and Z

values for the rays passed through. Then, the first ray passed is with Y=Ystart and

Z=Zstart. After finding intersections for this ray and writing the voxel configuration, the

Y value is incremented by cell size. Thus effectively the next ray is shifted along positive

Y axis. After sufficient number of shifts along Y-axis, when the Y-span of object is

covered, Z-value is incremented by cell size. This process is repeated till Z value covers

maximum Z-coordinate of the model.

44

The voxelization process essentially passes rays along X-axis in preset order and finds it‘s

intersections with the facets. The number of intersections for any ray would always be

even provided, the STL file is free of errors and the rays completely extend beyond the

model on both sides. Then, it is just finding the voxels lying between an odd and the next

even intersection, which are occupied by the solid model body. The algorithm is shown in

figure 4.3.2.

Fig.4.3.1 Steps to initialize voxelization

Y

Z

X

Z

Y X

Voxel

siz

e

Y

Z

X

Y s

tart

Z start

Y

Z

X

Filled voxels

Z r

ange

Y range

45

Fig.4.3.2 Voxelization algorithm

Create structure of faces, edges

and vertices. Also, generates

plane equations for facets.

Essentially, find the Y and Z

coordinates for the ray

STL Ascii file

Import STL

Create a ray parallel to X-axis

Extract faces which intersect X-axis Remove faces with normal

coefficients as ‗0, y, z‘

Find the intersections of this ray

with all the facets extracted

Substitute, Y and Z values for

the ray in the plane equation of

a facet and find X coordinate of

the intersection.

Check for inclusion of the

intersection point within the facet.

If yes, carry the point forward.

Find areas of triangles made by

intersection point with all

vertices. Check for difference in

areas of parent facet and sum of

three triangles made.

Sort the intersections in ascending

order

As, Y and Z for a given ray

remain constant, only X-

coordinate values are required.

Traverse the intersection

sequence, find the run-lengths

and populate the voxel bit-array

Ensure even number of

intersections. Voxels between

an odd and the next even

intersection are marked filled.

Voxel bit array

Res

et Y

or

Z v

alues

to g

o t

o n

ext

ray o

r pla

ne

resp

ecti

vel

y

46

4.3.2 Modified Voxelization algorithm

To reduce the time required for voxelization, the voxelization algorithm shown in figure

4.3.2 is modified further. The redundancies removed from the algorithm include

following.

(1) There may be many faces which intersect X-axis but the actual facet lies completely

outside the current ray. These are the facets whose all vertices have Y or Z or both

coordinates greater or smaller than the corresponding values for the current ray.

Intersections with such facets need not be calculated.

(2) Shift the model close to origin such that its minimum X, Y, Z coordinate values

coincide with the origin. Then, the rays can be passed only up to maximum limits of the

bounding box of the model, which are the maximum values of the X, Y, Z coordinates.

Thus, for every layer, Y coordinates of rays start from minimum Y and go up to

maximum Y for the bounding box. Thus, whenever, maximum Y is reached, Z-is

incremented and voxelization continues for next layer. When maximum Z is reached,

voxelization is complete.

When the STL is imported at first, the bounding box of the model is determined. This can

be used to reduce the voxelization time further. As the voxelization is layer by layer, with

memory space created for every new layer to be added, the memory as well as time

required for voxelization will be still lesser when minimum layers are generated. Thus,

the ideal order of range for given set up would be as X-range > Y-range > Z-range. In

other way, for any ratio of ranges, one can select the direction of rays and layers so that

the above order is achieved, which will take minimum time and memory for voxelization

with current scheme.

4.3.3 Errors in voxelization

Possible errors in the voxelization are as follows.

(1) Missing thin sections.

(2) Odd number of intersection caused by errors in STL of section thinner than voxel.

(3) Approximation of shape, which is inherent to voxel models. This can be improved by

having voxel models of higher resolution.

47

The errors in voxelization are made visible in voxel model display or also in section

display and X-ray scan display. This can be identified as stray voxel lines or planes,

missing voxels and similar inconsistencies.

The maximum error possible is half the voxel size.

As shown in the figure 2.1, the first ray originates from

Y start = Yminimum of the model + (Voxel size / 2)

Z start = Zminimum of the model + (Voxel size / 2)

Then, the rays are passed with shift of voxel size along Y and/ or Z axes for further

iterations. For determining voxel along the ray, the intersections are sorted in ascending

order. Then finding the run-lengths as

Run-length = (Intersection i – Intersection (i-1)) / Voxel size

Then the run-lengths are used to write the bit array for that plane as filled or empty voxels

as the case may be.

For correction along X axis, after populating integer number of voxels between two

intersections along a ray, the remainder can be analyzed for setting the partially occupied

last voxel as filled or empty. The other possible error is when voxel size is larger than the

thinnest section in the model. Here also, by comparing the distance between the

intersection with the voxel size, error ca be minimized. When such a section lies exactly

between two rays without having any intersection as shown in the figure 4.4 below, it

may be completely missed.

Fig. 4.3.3 Errors in voxelization caused by thin sections

When the Y or Z maximum value for the model lies between two adjacent rays, the first

ray will give valid intersections, but the next will not intersect inside the facets. As shown

in the figure 4.4, the maximum error would be of half the voxel size. To include one more

ray will not give extra intersection as the intersection point will fail the facet inclusion

Z Z

X Y

48

test. If tolerance is provided on facet inclusion test, then we may get extra intersections

with adjacent triangles also. This may lead to wrong interpretation of geometry.

Following figure 4.3.4 shows the errors in voxelization as voxels in thick black colour.

The figure on the right hand side shows the corrected voxel model. The erroneous voxel

in form of one dimensional voxel structures such as a single isolated voxel, a voxel line or

a plane can be identified. These are the voxels which have specific surface connectivity

configuration like only opposite faces connected, or opposite faces free and similar. If the

voxel model is scanned for such voxels, those can be reset and this will eliminate possible

erroneous voxels to some extent. This is applicable when it is assured that the smallest

feature of the original model is bigger than two voxel dimensions. Otherwise this process

may remove some of feature details.

Fig.4.3.4 Image showing errors in voxelization

49

Chapter 5

Voxel display and analysis

There are many techniques for voxel display such as depth sorting or ray tracing, direct

rendering, ‗marching cube‘ and slicing. Many methods try to generate approximate

surfaces from the voxel data and render those surfaces with appropriate normals. These

techniques are useful for small volume data and when there is no surface information

available previously, like from medical scans with resolution of 2563.

In this project, as voxel model is generated from a STL file, which is the surface

definition of object, again trying to extract surfaces from voxel data is unnecessary. Also,

as the target resolution is of (1000)3 voxels, extracting surfaces from such huge voxel data

would be a very tedious task. Therefore a novel but direct method of displaying voxels as

points is implemented.

When number of voxels is sufficiently large, displaying them as points will create a

contiguous surface like display. The problem is, points do not have normals and shadows

like 3D objects. Therefore for a point based display, variation in intensity can not be

automatically computed using standard graphics utilities. So, voxels at various locations

over the model need to be given proper intensity values individually to create realistic

display. This is achieved using a predefined lookup table of intensity values

corresponding to neighbour configuration of every voxels.

5.1 Intensity look-up table

As the voxels are assumed to be cubic in shape, every voxel will have maximum of 26

neighbours as, voxels connected by face = 6, voxels connected by edge = 12, voxels

connected by vertex = 8. As for sufficiently higher resolution and normal solids without

very thin sections, most voxels are face connected. Also, while displaying voxel faces,

only face connectivity will determine the visibility of a particular voxel and not the edge

or vertex connectivity. Therefore, for determining display intensity of voxels, only face

50

connected neighbours are considered. Figure 5.1 shows the 6 face connected

neighbouring voxels. The voxels numbers represent the order in which corresponding bits

are read from the array and added to form an integer representing the neighbouring

configuration as a whole.

Once the voxel data is obtained in form of bit array, the configuration of neighbouring

voxels can be obtained. This is done only for filled voxels.

Fig.5.1 Face connected neighbouring voxels

If the current voxel has index ‗X‘ in the bit array of plane Z, then its neighbours have

indices as follows.

Index of ‗1‘ = X – (resolution + 2)

Index of ‗2‘ = X – 1

Index of ‗3‘ = X + 1

Index of ‗4‘ = X + (resolution + 2)

Index of ‗5‘ = X (in the plane Z-1)

Index of ‗6‘ = X (in the plane Z+1)

The status of these neighbour voxels is represented by either ‗1‘ or ‗0‘ in the array as

when one is occupied or empty respectively. These neighbours are returned in the same

sequence as 1-2-3-4-5-6. Each time the returned bit is added and then the bit string is

shifted to left, except for the last bit. Thus the final bit string looks like 1-2-3-4-5-6.

When this bit string is interpreted as integer, it gives a unique number for a particular

neighbouring configuration. As there are total 26 = 64 combinations possible, the result

can be stored in a variable of type ‗char‘. Now every voxel can be identified as surface or

interior. Also, amongst the surface voxels, a voxel can be categorized based on the

4

6

X

5

3 2

1 X

Y

Z

51

number of free faces where neighboring voxel is absent. This is shown in the following

table 5.1.

Table 5.1 Sample bit strings from neighbouring voxel configuration

Number

of free

faces

Sample configuration

Bit string Integer

returned

No. of similar

configurations 1 2 3 4 5 6

1

1 1 1 1 1 0 62 6

2

1 1 0 1 1 0 54 15

3

1 1 0 0 1 0 50 20

4

0 1 0 0 1 0 18 15

5

0 0 0 0 1 0 2 6

For an error free STL and sufficiently large resolution, when the voxel size is smaller than

the thinnest section in the model, the voxels of type 4 and 5 from the above table should

not be present. These are the voxels with only one or two face connected neighbours and

4

X

X 5

3 2

1

5

4

X

X 2

1

X

X

5

2

1

5

X

X 2

X

X

5

52

represent a one or two dimensional structures at voxel level. If such voxels are found,

they can be treated as errors and can be reset to ‗0‘.

If a sphere is voxelized, then voxels at various locations have configurations as shown

below in figure 5.2. The numbers outside the bracket are the integers returned from bit

string.

Fig.5.2 Voxel intensities for a sphere

Deciding voxel for a particular neighbouring configuration requires visualization and

judgment. The voxel intensity values are rated on a scale of 20, as 0 is very dark to 20

which is bright color. Table 5.2 is the intensity look-up table which holds the intensity

values for various integers returned from bit string.

For every filled voxel, the bit string value is determined. Then the returned integer is

searched for in the intensity look-up table. When a match is found, corresponding

intensity is returned and process moves to next voxel. In OpenGL , the point color is

defined as (0, currIntensity, 0) where currIntensity = (Intensity value / 20). It is taken into

consideration that while display, screen is Z = 0 plane. Therefore, considering the light

from top right hand corner, the intensities are appropriately determined.

58(18)

51(18)

54(19)

30(10)

14(10)

15(12)

46(13)

42(15)

43(16)

62(16)

X

Y 22(13)

50(20)

53

Table 5.2 Intensity look-up table

Index Integer Intensity Index Integer Intensity

1 2 20 32 26 16

2 1 12 33 19 14

3 16 20 34 49 15

4 8 14 35 41 15

5 32 20 36 13 13

6 4 10 37 21 15

7 36 18 38 50 20

8 24 16 39 42 15

9 3 14 40 14 10

10 48 20 41 22 13

11 20 14 42 27 12

12 12 20 43 39 16

13 40 16 44 60 20

14 17 14 45 15 12

15 5 16 46 43 16

16 9 12 47 51 18

17 33 14 48 23 14

18 18 20 49 46 13

19 6 18 50 58 18

20 10 16 51 54 19

21 34 20 52 30 10

22 52 14 53 45 13

23 2 20 54 57 18

24 44 20 55 53 19

25 56 16 56 29 10

26 35 14 57 61 16

27 7 16 58 62 17

28 37 16 59 47 13

29 38 18 60 55 16

30 25 12 61 31 12

31 11 12 62 59 18

Voxel with bit string corresponding to 63 is completely inside the solid body and hence

not visible.

54

5.2 Geometric analysis of voxel model

The regular arrangement of the voxels along the coordinate axes makes the addressing

individual voxels very easy as the index of a voxel is directly related to it‘s actual location

in the 3D coordinate space. Though, this imposes some limitations on the manipulations

that could be done on the voxel model such as rotation of model or handling multiple

bodies. This limitation however, can be overcome by appropriately modifying the original

STL model itself before voxelization. As the fundamental base with voxelization

algorithm and voxel model storage scheme is ready, to implement any end application

over this base, few low level functions are required which can efficiently handle the voxel

model. Such small functions include the functions to read, write individual voxel,

handling voxel bodies, display functions and others as listed below. The following set of

functions is developed which provide modularity to the system and makes it easy to

extend it‘s scope by addition of new applications over the same voxel framework.

Read voxel with location (x,y,z)

Set/reset given voxel

Get voxel array index given it‘s (x,y,z) location

Get voxel intensity code given (x,y,z)

Get voxel intensity value given intensity code

Copy entire voxel body

Delete entire voxel body

Display entire voxel body

Display a single voxel

Display a section in the voxel volume at specified location

Find neighbour to given voxel along given direction at specified distance

Find distance between two given voxels in terms of cell-distance

Remove skin- reset surface voxels

Using these low level functions, some geometric applications are developed which are as

follows.

55

5.2.1 Progressive sectional display

Planar sections can be taken normal to any of the coordinate axes through the voxel

model (fig. 5.2.1). These sections can be displayed to reveal interior of the voxel model at

specified location. The layered structure used for storing voxel model, makes the section

display very trivial along those planes. For the other two directions, little more recursions

are required for accessing respective voxels from successive layers. The section is

essentially a layer one voxel dimension thick. The advantage with voxel model is that it

can show the solid interiors in contrast to surface models.

Fig.5.2.1Progressive sectional display

5.2.2 Skin removal simulation

In the voxel model, surface voxels can be exclusively identified. Surface voxels will be

non-zero with at least one neighbour missing. This condition will give certain number of

voxel intensity codes. So, by traversing through the entire voxel model, surface voxels

can be identified and reset to simulate one skin layer removal. The problem being that

each voxel identified as surface voxel can not be reset immediately and then move to next

voxel for examination. this is because the rest voxel will change the original neighbour

configuration for the next voxel to be examined. This problem is solved by traversing the

voxel model layer by layer. The structural layers are selected for ease of implementation.

Then, first layer is examined and the indices of surface voxels are stored in temporary

array without resetting any voxel. Then the next successive layer is examined for surface

voxels and similarly the indices of surface voxels identified are stored in a temporary

array. Now before moving to the third layer, the surface voxels from the first layer can be

reset without affecting the original voxel configuration as the configuration of second

layer is already stored separately. Thus, every time a voxel layer which is one before the

previous is reset. The skin removal can be simulated one layer at a time

Successive section planes

56

Fig.5.2.2 Successive skin removal simulation

5.2.3 X-ray scan

The voxel model can be scanned along any of the coordinate axes to generate X-scan

image of the model. This X-ray scan is useful for thickness analysis of the model. This

reveals the interior thickness distribution in the model which is normally difficult to

visualize. The X-ray scan generates image wherein the intensity varies proportional to the

thickness along the scan direction. Three options are provided for determining the

intensity as the Minimum, Maximum and Total thickness along the scan direction at

every point. To generate X-ray scan image of a voxel model of resolution ‗R‘, model is

scanned along the scan direction for R2 rays. The voxel configuration along each of the

rays is determined and the intensity at the point of projection of the ray on the image

plane is determined according to the selected option.

Fig.5.2.3 X-ray scan: Intensity variations proportional to thickness

Intensity proportional

to no. of voxels

2

8

14

X-ray Voxel configuration along scan-line

MINIMUM

MAXIMUM

TOTAL

Surface voxels reset

57

Chapter 6

Implementation and results

This chapter presents the results of implementation of the developed voxelization

algorithm and the voxel model representation scheme. The memory and time required for

voxelization of various models of varying complexities is included. Chapter also includes

the display images generated by the system for various models with different resolution

settings. The images of display output from the functions like section display, skin

removal and X-ray scan are also presented.

6.1 Voxelization

The voxelization and display algorithm developed is tested for various solid models.

Following parameters are included in the results.

1) Voxelization time for different resolutions

2) Memory required for voxelization

3) Display images of different resolution, point sizes and orientations

The implementation is done on a system with following configuration.

Intel(R) Pentium, 2.40 GHz processor, 2GB RAM.

There are two sets of solid models considered for testing. One is a set of simple regular

shapes. The other includes some engineering and non-engineering objects. Tables 6.1.1

and 6.1.2 give the details of voxelization results for these solids respectively. Figures

6.1.1, 6.1.2 and 6.1.3 show the graph plots for these results.

58

Table.6.1.1 Voxelization results for standard shapes

M = Memory required for voxelization (MB), T = Time required for voxelization (sec)

Sr.

No. STL file name

Size

KB

No. of

Facets

Resolution

100 200 400 500 800 1000

M T M T M T M T M T M T

1 Pyramid 2 6 0.27 0.078 1.16 0.407 8.18 2.453 15.70 4.468 63.60 16.36 123.6 30.64

2 Prism 3 8 0.27 0.141 1.24 0.704 8.25 3.985 15.60 7.171 63.60 25.43 123.8 46.98

3 Double pyramid 3 8 0.15 0.031 1.16 0.156 8.18 0.875 15.60 1.532 63.60 5.25 123.6 9.58

4 Cube 4 12 0.24 0.172 1.23 0.984 8.27 6.297 15.60 11.73 63.60 44.08 123.6 83.42

5 Block 4 12 0.36 0.093 1.33 0.500 8.35 3.188 15.70 5.937 63.70 22.17 123.7 42.47

6 Block

(rotated) 4 12 0.30 0.141 1.30 0.719 8.50 4.015 16.10 7.125 64.10 25.56 123.6 44.17

7 Block with hole 33 112 0.60 0.125 1.70 0.656 7.40 3.844 13.06 6.984 47.4 25.31 89.5 47.22

8 Cylinder 39 136 0.70 0.171 2.10 0.906 10.00 5.500 17.96 9.922 67.00 36.50 127.9 68.11

9 Sphere 652 2352 1.80 0.219 4.00 1.016 13.50 4.922 22.30 8.610 73.80 29.39 136.2 53.31

10 Cup 788 2814 1.90 0.172 3.54 0.640 9.70 2.531 14.75 4.047 42.56 11.02 75.2 17.84

59

0

10

20

30

40

50

60

70

80

90

Pyram

id

Prism

Dou

ble

pyra

mid

Cub

e

Block

Block

(rot

ated

)

Block

with

hol

e

Cylin

der

Spher

eCup

Solid model

Vo

xelizati

on

tim

e (

sec)

100

200

400

500

800

1000

Fig.6.1.1 Time for Voxelization of standard shapes with different resolution

60

Table 6.1.2 Voxelization results for various solid models

M = Memory required for voxelization (MB), T = Time required for voxelization (sec)

Sr.

No. STL file name

Size

KB

No. of

Facets

Resolution

100 200 400 500 800 1000

M T M T M T M T M T M T

1 Valve 354 2352 0.60 0.140 1.60 0.594 6.60 2.782 11.50 4.656 41.50 14.26 78.3 24.50

2 End shield 951 3714 2.30 0.281 4.80 1.00 15.30 3.766 24.50 5.89 77.30 15.45 140.6 24.67

3 Crank 1651 6184 1.80 0.125 3.50 0.375 12.20 1.215 20.68 1.734 76.00 4.40 133 7.15

4 Body cap 1948 7430 2.43 0.187 4.57 0.532 13.50 2.250 21.50 3.531 67.80 9.219 123.6 14.81

5 Cover 2599 15776 7.10 0.656 13.10 2.578 29.60 9.281 41.40 14.47 99.10 37.00 163 58.50

6 Ganesha 4140 19298 5.30 0.328 8.50 1.062 17.35 3.610 23.80 5.531 55.50 14.11 90.55 22.58

7 Pelvis 8978 41100 10.30 0.859 15.60 2.203 30.70 6.313 41.80 9.047 97.20 20.92 158.9 32.15

8 Valve top 12355 73912 17.40 1.61 25.40 4.89 43.28 17.06 53.90 25.80 96.70 64.33 137.1 99.95

9 Test piece 12648 45892 8.90 0.453 11.90 1.078 19.85 4.047 24.95 6.078 48.40 15.84 72.75 25.91

10 Naval crest 20787 75260 11.5 0.484 13.56 1.156 19.26 3.844 22.90 5.937 35.50 14.30 47.45 22.39

61

0

20

40

60

80

100

120

Valve End shield Crank Body cap Cover Ganesha Pelvis Valve top Test piece Naval

crest

Solid model

Vo

xeli

zati

on

tim

e (

sec)

.

100

200

400

500

800

1000

Fig.6.1.2 Time for Voxelization of various solid models with different resolution

62

0

20

40

60

80

100

120

140

160

Pyr

amid

Pris

m

Tetra

hedr

on

Cub

e

Block

Cub

e(ro

tate

d)

Block

with

hole

Cylin

der

Sph

ere

Cup

Solid model

Mem

ory

req

uir

ed

(M

B)

.

100

200

400

500

800

1000

0

20

40

60

80

100

120

140

160

180

Valve

End

shield

Cra

nk

Bod

y ca

p

Cov

er

Gan

esha

Pelvis

Valve

top

Test p

iece

Nav

al cre

st

Solid modelM

em

ory

req

uir

ed

(M

B)

.

100

200

400

500

800

1000

(a) (b)

Fig.6.1.3 Memory required for Voxelization of standard shapes and various solid models with different resolution respectively

63

6.2 Voxel display

Displaying voxel model as collection of points requires point size to be controlled

properly in relation to ‗resolution‘ as well as the object display size on the screen. As a

vertex is drawn as a pixel or a group of pixels, the size in terms of pixels is properly set.

To reduce the aliasing effect and also have smooth variation in color shades, standard

antialiasing and smoothening is used from graphics utility. Figure 4.2.1 shows the images

of sphere and pyramid at resolution of 20 with Point smooth Off and On.

(a) without smooth and blend (b) with smooth and blend

Fig.6.2.1 Voxel images with point-smooth Off and On (R=20)

Figure 6.2.2 shows the images of sphere drawn for resolution of 200 with different

display sizes. Display size is determined by the ‗zoom‘ level set by the Z-value of origin

in relation to object. Here, S= size of the point in terms of pixels, which represents

individual voxel. It is observe that as the object display size increases, the image

smoothness reduces. This is because the point size also increases, and then the effect of

intensity smoothening is reduced in the interior of a point, which is a considerable area

for larger point sizes.

(a) S=2 (b) S=3.2 (c) S=4 (d) S=4.5

Fig.6.2.2 Voxel images with Zoom and corresponding change of point size(S)

(Resolution=200)

64

Figure 6.2.3 shows the images of sphere with same size of display but with different

resolutions and corresponding point size. It is quite clear that, as the resolution increases,

the intensity variation becomes very gradual and image looks smooth. This is because of

the fact that as resolution increases, the point size reduces, and moreover, the effect of

smoothening is more evident. But, this comes at a cost of more time and memory for

display.

(a) R=100 (b) R=200 (c) R=400

(d) R=800 (e) R=1000

Fig.6.2.3 Voxel images with different resolutions (R)

Figure 6.2.4 shows the sphere displayed as voxel model with resolution of 200 from

various angles of views. The variation in this case is effect of the intensity set in the

intensity look-up table for various voxel configurations.

(a) 30 (b) 60 (c) 90

(d) 120 (e) 150

Fig. 6.2.4 Voxel images with different angles of rotation about Y-axis (Resolution= 200)

65

As can be seen in figure 6.2.5, when the Ganesha model is imported in FreeForm, the

voxel model in the form of *.cly format, which is native format for FreeForm had errors.

As can be seen the trunk got connected to the body though it is not so originally. This

happened at resolutions of 200 and 400 also. But, the voxelization algorithm in this

project, that error is not generated. Also when attempted to get voxel resolution of 800 or

1000 in FreeForm, the memory requirement goes up to 500-800 MB, and the size of

simple cube model in *.cly format becomes of the order of 40-50 MB. Also, the

functioning and display becomes very slow for resolutions above 500. So, the results from

the voxelization algorithm proposed in the project are very much comparable to FreeForm

if not better. But, it is remembered that the application intended from developed voxel

model may be different and not haptic modeling as FreeForm. Still, the display of the

voxel model in the current project needs to be improved for quality and moreover time.

Currently, the time taken for display is of the order of 2minutes for cube with resolution

of 1000.

Voxel display FreeForm display

Fig.6.2.5 Voxel model from the project and from FreeForm (Resolution= 200)

Figures 6.2.6 shows the voxel display images of various solid models considered. All

images are at resolution 400, with point display size of 3 pixels and view rotated. Figures

6.2.7, 6.2.8, 6.2.9 show voxel display images of Ganesha model, pelvis model and

cylinder block model at various resolutions respectively.

66

Valve End shield Crank Body cap

Cover

Ganesha Pelvis Valve top Test piece Naval crest

Fig.6.2.6 Voxel display images of various solid models with resolution of 400

67

(a) R=100 (b) R=200 (c) R=400 (d) R=800 (e) R=1000

Fig.6.2.7 Display Images fro Ganesha model with different resolutions (R)

(a) R=100 (b) R=200 (c) R=400 (d) R=800 (e) R=1000

Fig.6.2.8 Display Images of pelvis model with different resolutions(R)

68

(a) R=100 (b) R=200

(c) R=400 (d) R=800

Fig.6.2.9 Display Images of cylinder block with different resolutions(R)

The results of the display algorithm developed are quite satisfactory as far as the

rendering quality is considered. The display is smoother and more realistic than the

simple display rendering individual voxel faces. But still the algorithm takes longer time

to render models of high resolutions. Currently it takes less than 2 seconds to generate an

image of resolution up to 200. The time required to render higher resolution images

depends upon the object shape and varies in the range 2-150 seconds. The rendering time

can be further reduced by implementing the concept of ‗display-list‘. This is simplified

listing of visible voxels which can be directly rendered. But, this is valid for voxel models

which do not change. If any change is made in the voxel configuration of the model, a

new display-list has to be generated after every such change.

69

6.3 Voxel section display

Sections of voxel model can be displayed along the coordinate axes. Sections normal to

Z-axis is very trivial as the voxel model is stored as a stack of X-Y planes. For display of

sections normal to X or Y plane, it requires more computations to find the appropriate

voxels in different Z-planes. The sections can be displayed at a specified point or also as

successive incremental sections.

Fig.6.3 Sectional Display Images of cylinder block and Ganesha model with different

resolutions(R)

70

6.4 Skin removal

This is a very novel function which allows voxel model to be pealed off layer by layer.

This operation ultimately points to the thickest portion of the model on continuous skin

removal till last filled voxel. In principal, at every skin removal, surface voxels are

identified and reset to zero. Surface voxels are those which are filled but are not

completely inside the object. Surface voxels have at least one face free.

Fig.6.4 Successive Skin removal Images of cylinder block with resolution 200

71

6.5 X-ray scan display

X-ray scan can be performed along any of the coordinate axes. In X-ray scan, there are

three options as follows.

a) Minimum thickness- the intensity at any point is proportional to the minimum section

thickness found along the corresponding scan ray.

b) Maximum thickness- the intensity at any point is proportional to the maximum section

thickness found along the corresponding scan ray.

c) Total thickness- the intensity at any point is proportional to the total of all section

thicknesses found along the corresponding scan ray.

X-ray scan is particularly useful in analyzing internal thickness variations in a model

which is difficult to visualize otherwise. The three options provided are very much useful

and eliminate the limitation of normal X-ray scan which does not identify individual

sections as minimum or maximum. Figures 6.5.1 and 6.5.2 show the X-ray scan images

generated for Ganesha model and cylinder block model. the white spots or stray lines

indicate the errors in voxelization as missing voxel segment or extra voxel segment

respectively.

72

Fig. 6.5.1 X-ray scan images of Ganesha model with minimum, maximum and total thickness options respectively (R=200)

73

Fig. 6.5.2 X-ray scan images of Cylinder block model along two directions and with different thickness options (R=200)

74

Chapter 7

Conclusions

Solid modeling is essential for all CAE implementations and other computer integrated

processes in engineering businesses. Voxel based solid modeling techniques provide

capabilities additional to conventional surface modeling which enable processing of very

complex, free-form parts and assemblies. The advances in computer technology have

made possible to implement voxel based systems on desktop PCs with sufficient accuracy

and efficiency. Though voxel based techniques may not replace surface modeling, they

offer much better solutions for certain problems, especially involving volumetric

information and analysis. The focus of the work has been on resolving the problems of

memory and time for high resolution voxel models and to develop completely new

flexible voxel based modeling environment adaptable to various applications. The aim is

to use capabilities of volumetric modeling into engineering fields as a supplement to

conventional surface modeling.

7.1 Summary of work done

The main goal of the project was to develop a voxel based model representation and

display module which can efficiently handle up to 1 billion voxel models. This gives

voxel resolution of 1000 along all coordinate axes which is sufficient for most of the

cases. A completely new and custom designed voxel based modeling environment is

developed. The basic framework to generate, store and handle voxel models of

resolutions up to 1billion voxels is implemented. The results of voxelization are

satisfactory. The developed voxel representation considers cubic voxels aligned with

coordinate axes. Spatial enumeration is used to store these binary voxel models. A stack

of bit-arrays is used to store voxel model. A new point based voxel display method is

implemented which gives refresh rate of less than 2seconds for resolutions up to 200. The

intensity variation in voxel image is achieved through intensity look-up table which gives

intensity values corresponding to various voxel configurations. The display is much more

realistic than simple cube surface display. Using this basic voxel representation and

75

display scheme, few low level geometric routines are written. These can be used to

develop high end geometric analysis functions and application. Sectional display, X-ray

scan and skin removal are the modules implemented over the developed voxel base. The

voxel modeling system is tested for numerous engineering as well as non-engineering

models.

7.2 Future scope

The main goal of the project has been successfully achieved. The system works very well

for resolutions up to 400. Though, further improvements are necessary for efficient

handling of voxel models with resolution of 1000. This may require custom designed

hardware modifications. Special memory architectures may solve the problem. This will

also allow for interactive viewing of high resolution voxel models which is currently not

possible. The voxel file format is proposed which uses concept of ‗run-lengths‘. Standard

conversion routines can be written to read and write such voxel files. On the similar basis,

a gray scale voxel based system can also be devised. Also, system could be modified to

handle multiple voxel models simultaneously. This will allow standard Boolean

operations over different voxel models.

More specifically, over the present system, immediately possible improvements may

include the following. The display could be further enhanced by tuning the intensity

values. The voxelization errors can be minimized by considering voxelization from

multiple directions and then combining the resulting voxel models. The display could be

improved to include colors and transparency. Currently, the rotation of view is

incorporated as view direction change with light positioned stationary. But, with simple

global rotation variables, a changing intensity can be implemented so as to depict object

rotation. Also, the display could be made faster by using ‗display lists‘ which will store

voxel information in more ready to display form for a fixed voxel model.

Using the basic framework developed, a voxel based application module can be

developed for engineering applications such as casting analysis. The system can be

coupled with a good user interface and controls. The voxel based module can also be

added to existing applications as a supplement.

76

Appendix A

‘STL’ file format

There are various file formats for storing solid model data like STL, IGES, STEP, etc.

The ‗.stl‘ or stereolithography format is an ASCII or binary file used in manufacturing.

STL stands for Stereolithographic Tessellated Language. It is a list of the triangular facets

that describe a computer generated solid model. This is the standard input for most rapid

prototyping machines. The ASCII .stl file starts with the lower case keyword solid and

end with endsolid. Within these keywords are listings of individual triangles that define

the faces of the solid model. Each individual triangle description defines a single normal

vector directed away from the solid's surface followed by the xyz components for all three

of the vertices.

The STL files have typical advantages of use as they constitute a simple data structure

involving facets, which is repeated throughout the file. Also, the file is easy to read and

the connectivity relations between the facets, edges and vertices are easy to store.

Following is a sample ASCII description of a single triangle within an STL file.

solid

...

facet normal 0.00 0.00 1.00

outer loop

vertex 2.00 2.00 0.00

vertex -1.00 1.00 0.00

vertex 0.00 -1.00 0.00

endloop

endfacet

...

endsolid

77

Appendix B

3D discrete topology The 3D discrete space is a set of integral grid points in 3D Euclidean space defined by

their Cartesian coordinates (x, y, z). A voxel is the unit cubic volume centered at the

integral grid point. The voxel value is mapped onto {0,1}: the voxels assigned ‗‗1‘‘ are

called the ‗‗black‘‘ voxels representing opaque objects and those assigned ‗‗0‘‘ are the

‗‗white‘‘ voxels representing the transparent background [Kaufman, 1993]. Sometimes

the voxel value is mapped onto the interval [0,1] representing either partial coverage,

variable densities, or graded opacities. Due to its larger dynamic range of values, this

approach supports 3D antialiasing and thus supports higher quality rendering.

Every voxel has 26 neighboring voxels. Two voxels are 26-adjacent if they share either a

vertex, an edge, or a face (Fig. A2). Accordingly, face-sharing voxels are defined as 6-

adjacent, and edge-sharing and face-sharing voxels are defined as 18-adjacent. A

sequence of voxels having the same value (e.g. ‗‗black‘‘) is called an N-path if all

consecutive pairs are N-adjacent. A set of voxels W is N-connected if there is an N-path

between every pair of voxels in W. An N-connected component is a maximal N-connected

set.

Given a 2D discrete 8-connected black curve, there are sequences of 8-connected white

pixels (8-component) that pass from one side of the black component to its other side

without intersecting it. This phenomenon is a discrete disagreement with the continuous

case where there is no way of penetrating a closed curve without intersecting it. To avoid

such a scenario, it has been the convention to define ‗‗opposite‘‘ types of connectivity for

the white and black sets. ‗‗Opposite‘‘ types in 2D space are 4 and 8, while in 3D space 6

is ‗‗opposite‘‘ to 26 or to 18.

78

Fig. A2 Voxel connectivity

79

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