Visualization of Large Scale Volumetric Datasets - Computing Science

VISUALIZATION OF LARGE SCALE VOLUMETRIC

DATASETS

by

Hamidreza Younesy Aghdam

B.Sc., Sharif University of Technology, Tehran, Iran, 2002

a thesis submitted in partial fulfillment

of the requirements for the degree of

Master of Science

in the School

of

Computing Science

c© Hamidreza Younesy Aghdam 2005

SIMON FRASER UNIVERSITY

Summer 2005

All rights reserved. This work may not be

reproduced in whole or in part, by photocopy

or other means, without the permission of the author.

APPROVAL

Name: Hamidreza Younesy Aghdam

Degree: Master of Science

Title of thesis: Visualization of Large Scale Volumetric Datasets

Examining Committee: Dr. Arthur (Ted) Kirkpatrick, Assistant Professor,

Computing Science, Simon Fraser University

Chair

Dr. Torsten Moller, Assistant Professor,


Senior Supervisor

Dr. Richard Zhang, Assistant Professor,


Supervisor

Dr. Daniel Weiskopf, Assistant Professor,


Supervisor

Dr. Hamish Carr, Lecturer, Computer Science, Uni-

versity College Dublin

External Examiner

Date Approved:

ii

Abstract

In this thesis, we address the problem of large-scale data visualization from two aspects,

dimensionality and resolution. We introduce a novel data structure called Differential Time-

Histogram Table (DTHT) for visualization of time-varying (4D) scalar data. The proposed

data structure takes advantage of the coherence in time-varying datasets and allows efficient

updates of data necessary for rendering during data exploration and visualization while

guaranteeing that the scalar field visualized is within a given error tolerance of the scalar

field sampled.

To address the high-resolution datasets, we propose a hierarchical data structure and

introduce a novel hybrid framework to improve the quality of multi-resolution visualization.

For more accurate rendering at coarser levels of detail, we reduce aliasing artifacts by ap-

proximating data distribution with a Gaussian basis at each level of detail and we reduce

blurring by using transparent isosurfaces to capture high-frequency features usually missed

in coarse resolution renderings.

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I dedicate this work to my parents

for their unconditional love and continuous supports

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“Let the beauty of what you love be what you do.”

— Jalal ad-Din Rumi (Persian Poet and Mystic, 1207-1273)

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Acknowledgments

I would like to express my deepest gratitude to my supervisor Dr. Torsten Moller whose

expertise, inspiring ideas, understanding, and patience, added considerably to my graduate

experience.

I would like to thank the other members of my committee, Dr. Hamish Carr for his

generous support and invaluable remarks on my work, and to Dr. Richard Zhang and Dr.

Daniel Weiskopf for the assistance they provided at all levels of the research project.

Many thanks to Dr. Arman Rahmim, Hamidreza Vajihollahi, and my colleagues and

friends at Simon Fraser University and GrUVi Lab for their assistance on countless occasions,

exchanges of knowledge, and venting of frustration during my graduate program, which

helped enrich the experience.

Finally, I wish to thank my parents and family in Iran for all the sacrifices and support

they provided me through my entire life.

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Contents

Approval ii

Abstract iii

Dedication iv

Quotation v

Acknowledgments vi

Contents vii

List of Tables ix

List of Figures x

1 Introduction 1

2 State of the Art 5

2.1 State of the Art in Time-Varying Data Visualization . . . . . . . . . . . . . . 5

2.2 State of the Art in Multi-resolution Techniques . . . . . . . . . . . . . . . . . 7

3 Visualization of Time-Varying Volumetric Data 9

3.1 Data Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Case Studies: Data Characteristics of Isabel & Turbulence . . . . . . . . . . . 12

3.2.1 Temporal Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2.2 Histogram Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.3 Spatial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

vii

3.3 Differential Time-Histogram Table . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.1 Computing the DTHT . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.2 Queries in the DTHT . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Multi-resolution Volume Rendering 36

4.1 Rendering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1.1 Accurate Multi-resolution Transfer Functions . . . . . . . . . . . . . . 38

4.1.2 Translucent Isosurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.1.3 Combined Isosurface and Volumetric Rendering . . . . . . . . . . . . . 43

4.1.4 Progressive Refinement Using Image Caches . . . . . . . . . . . . . . . 44

4.2 Hybrid Branch-on-Need Octree (HYBON) . . . . . . . . . . . . . . . . . . . . 45

4.2.1 Hybrid Octree Representation . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.2 Hybrid Octree Optimization . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3 Implementation and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3.1 Memory and storage overhead . . . . . . . . . . . . . . . . . . . . . . . 51

5 Conclusions and Future Work 74

5.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Bibliography 76

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List of Tables

3.1 Size (in GB and percent of original dataset) of DTHT using lists and octrees

with all or only the first active set stored. Storing all active sets is expensive

storage-wise, but it enables us to access any time step directly. As an alter-

native we can store only the first active set and use differentials to obtain

the active sets in other time steps. This improves the storage requirement

substantially but does not allow direct access to time steps. . . . . . . . . . . 23

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List of Figures

3.1 Temporal Coherence: Number of samples that change by more than an error

bound of λ = 0%, 0.3%, 1%, 3% for the temperature and pressure scalars of

the Isabel dataset as well as the jets and vortex datasets. . . . . . . . . . . . 14

3.2 Temporal coherence and histogram distribution for the temperature dataset:

In this figure and following three figures, the striped yellow plot shows the

number of samples that change by more than the error bound λ. The blue

plot represents the number of samples per bin (histogram), at each time step.

The isovalue range is divided into 100 isovalue bins. . . . . . . . . . . . . . . 16

3.3 Temporal coherence and histogram distribution for the pressure dataset. Note

that in b, c and d, the y-axis has been scaled by 10 to show the details of the

diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4 Temporal coherence and histogram distribution for the jet dataset. Note that

the y-axis have different scales to show the details of the diagrams. . . . . . . 18

3.5 Temporal coherence and histogram distribution for the vortex dataset. . . . . 19

3.6 The Differential Time-Histogram Table (DTHT). Each bin holds the active

set (a) and differentials between the bin and adjacent bins (arrows). Compo-

nents not used for our experiments are shown in a lighter shade. . . . . . . . 20

3.7 Data and file structure of a branch-on-need octree (in 2D): (a) dataset (b)

tree structure (c) depth-first order file structure: byte codes represent active

children of a node. A 0000 code represents a leaf node containing a data block. 21

3.8 Active DTHT cells for a given transfer function. . . . . . . . . . . . . . . . . 24

x

3.9 λ = 1.0% Rendering and data loading performance for DTHT using active

and differential files. The reported times are averaged costs while navigating

over multiple time steps. The yellow overlay shows the ratio of active voxels

to all voxels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.10 λ = 1.0% Performance for DTHT using active and differential files, showing

also the ratio of active voxels to all voxels (in yellow). A transfer function

spanning 10% of the data range was used with different centre points. Results

are averaged over several time steps. . . . . . . . . . . . . . . . . . . . . . . . 27

3.11 Ratio between DTHT differential updating time and DTHT direct loading

time for different error bounds, for different transfer function widths. Ideally,

the ratio should be above 1 (dotted line) all the time. . . . . . . . . . . . . . 28

3.12 Ratio between DTHT differential updating time and DTHT direct loading

time for different error bounds and different placements of a transfer function

of constant width (10%). Ideally, the ratio should be above 1 (dotted line)

all the time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.13 Loading time of incremental transfer function changes. The graphs show the

time to update the active sets when the active bins are changed in a constant

time step. Increasing the transfer function width (solid-red), only causes

additions to the active set; thus, there is only one peak in the diagram when

a large number of voxels are added to the active set. Changing the placement

of the transfer functions (dotted-blue), requires some voxels to be added and

some voxels to be removed; thus there is one peak in the diagram when a

large number of voxels are added and one peak later when those voxels are

removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.14 Rendered images with representative transfer functions for Isabel tempera-

ture dataset. In this figure and the following three figures the left column

shows the results for increasing transfer function width and the right column

shows the results for moving transfer function placement, corresponding to

our experiments in this section. The wired frame shows the octree structure

and the blocks containing active voxels. . . . . . . . . . . . . . . . . . . . . . 32

3.15 Rendered images with representative transfer functions for Isabel pressure

dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

xi

3.16 Rendered images with representative transfer functions for Turbulent Jets

Flow dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.17 Rendered mages with representative transfer functions for Turbulent Vortex

Flow dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1 Renderings of the Visible Human Male Head at different levels of details.

Top row: using mean value for approximation. Bottom row: using Gaussian

function for approximation. From left to right, each image represents an

additional level of coarsening by a factor of two in each dimension. . . . . . . 37

4.2 Renderings of a synthetic spherical distance dataset based on the Euclidean

distance to the centre of the volume, at different levels of details. Top row:

using mean value. Bottom row: using Gaussian basis approximation. From

left to right, each image represents an additional level of coarsening by a

factor of two in each dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3 Bottom: a transfer function with the opacities as the vertical axis (V ). Top:

the computed Gaussian basis transfer function; The horizontal and vertical

axes are respectively mean (µ) and standard deviation (σ) and each point

has been drawn with the color and opacity computed for the related µ and σ. 42

4.4 Adding isosurfaces to reveal sharp features in the Richtmyer-Meshkov insta-

bility dataset. Left: splatting. Right: splatting + transparent isosurface. . . . 43

4.5 The structure and indexing of HYBON tree (in 2D). An index of a node

encodes the index of the parent node (2D: index / 4) and the child index of

the node (2D: index % 4) within the parent node. . . . . . . . . . . . . . . . . 46

4.6 Creating cell structures in subvolumes by overlapping border voxels. . . . . . 47

4.7 A high-frequency transfer function (top) and a low-frequency transfer func-

tion (bottom) used for our experiments. . . . . . . . . . . . . . . . . . . . . . 50

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4.8 Effects of downsampling distributions with a box low-pass filter on the spher-

ical distance function, rendered using a high-frequency transfer function. Top

most images are highest resolution (643) and quality, bottom images are low-

est resolution (83) and quality. For the high-frequency transfer function cho-

sen, downsampling should blur the thin colored shells into a grey-like color.

The histogram distribution (left), while expensive, does this correctly. The

mean distribution (right) fares poorly, while the Gaussian distribution (mid-

dle) is reasonably effective at low cost. . . . . . . . . . . . . . . . . . . . . . . 53

4.9 Effects of downsampling distributions with a tent low-pass filter on the spher-

ical distance dataset, rendered using a high-frequency transfer function. . . . 54

4.10 Effects of downsampling distributions with a cubic low-pass filter on the

spherical distance dataset, rendered using a high-frequency transfer function. 55

4.11 Effects of downsampling distributions with a box low-pass filter on the Visible

Human Male Head dataset, rendered using a high-frequency transfer function. 56

4.12 Effects of downsampling distributions with a tent low-pass filter on the Visible

Human Male Head dataset, rendered using a high-frequency transfer function. 57

4.13 Effects of downsampling distributions with a cubic low-pass filter on the Vis-

ible Human Male Head dataset, rendered using a high-frequency transfer

function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.14 Effects of downsampling distributions with a box low-pass filter on the Richtmyer-

Meshkov Instability dataset, rendered using a high-frequency transfer function. 59

4.15 Effects of downsampling distributions with a tent low-pass filter on the Richtmyer-

Meshkov Instability dataset, rendered using a high-frequency transfer function. 60


Richtmyer-Meshkov Instability dataset, rendered using a high-frequency trans-

fer function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.17 Effects of downsampling distribution with a box low-pass filter on the spheri-

cal distance function , rendered using a low-frequency transfer function. Res-

olution and quality decrease from top to bottom. All three distributions

perform adequately. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.18 Effects of downsampling distributions with a tent low-pass filter on the spher-

ical distance dataset, rendered using a low-frequency transfer function. . . . . 63

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spherical distance dataset, rendered using a low-frequency transfer function. . 64

4.20 Effects of downsampling distributions with a box low-pass filter on the Visible

Human Male Head dataset, rendered using a low-frequency transfer function. 65

4.21 Effects of downsampling distributions with a tent low-pass filter on the Visible

Human Male Head dataset, rendered using a low-frequency transfer function. 66

4.22 Effects of downsampling distributions with a cubic low-pass filter on the Visi-

ble Human Male Head dataset, rendered using a low-frequency transfer function. 67

4.23 Effects of downsampling distributions with a box low-pass filter on the Richtmyer-

Meshkov Instability dataset, rendered using a low-frequency transfer function. 68

4.24 Effects of downsampling distributions with a tent low-pass filter on the Richtmyer-

Meshkov Instability dataset, rendered using a low-frequency transfer function. 69


Richtmyer-Meshkov Instability dataset, rendered using a low-frequency trans-

fer function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.26 Errors of the mean and Gaussian approximations of the histogram for the

three test datasets. At a certain level of detail, for each method, the reported

error is the root mean squared error of the voxels’ approximated histogram in

comparison with their actual histogram, for all voxels in that level of coarsening. 71

4.27 Rendering errors of the mean and Gaussian approximations (f) of the his-

togram for the three test datasets using a high-frequency transfer function

(V ). At a certain level of detail, for each method, the reported error is the

root mean squared error of the voxels’ color computed using their approxi-

mated histogram in comparison with their color computed using their actual

histogram, for all voxels in that level of coarsening. . . . . . . . . . . . . . . . 72

4.28 Rendering errors of the mean and Gaussian approximations (f) of the his-

togram for the three test datasets using a medium-frequency transfer function

(V ). At a certain level of detail, for each method, the reported error is the

root mean squared error of the voxels’ color computed using their approxi-

mated histogram in comparison with their color computed using their actual

histogram, for all voxels in that level of coarsening. . . . . . . . . . . . . . . . 73

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Chapter 1

Introduction

“Visualization is concerned with exploring data in such a way as to gain understanding

and insight into the data” [12]. The goal of visualization is “to promote a deeper level of

understanding of the data under investigation and to foster new insight into the underlying

processes, relying on the human’s powerful ability to visualize” [4]. The resulting visual

display of data enables the scientists “to visually perceive features which are hidden in the

data but nevertheless are needed for data exploration and analysis” [40].

One of the research areas in visualization is volume visualization which deals with meth-

ods to explore, analyze and visualize volumetric data acquired in medicine, computational

physics and various other scientific disciplines. In other words, “it is a method of extracting

meaningful information from volumetric data using interactive graphics and imaging and it

is concerned with data representation, modeling, manipulation and rendering” [40].

Volumetric data are acquired by sampling (e.g. CT and MRI), simulation (e.g. com-

putational fluid dynamics) or modeling (voxelization of geometric models) techniques and

are typically represented as a 3D regular or irregular discrete grid of volume elements called

voxels. Voxels are points in 3D, sampled at the grid points and have associated values which

represent some measurable properties or dependent variables (e.g. colour, density, material,

velocity) of the real phenomenon or object residing at the sample location of that voxel.

A Cell consists of a group of neighbor voxels and its geometrical shape depends on the

sampling grid used in sampling the dataset. In a regular Cartesian grid, each group of 8

neighbor voxels on a unit cube’s corners consist a cell.

Over the years many methods have been developed to visualize volumetric data. Among

1

CHAPTER 1. INTRODUCTION 2

those, two broad categories are isosurface extraction and direct volume rendering. Isosur-

face extraction methods approximate surfaces (representing material boundaries contained

within the volumetric data) using geometric primitives, most commonly triangles, and are

usually rendered using conventional graphics accelerator hardware. The resulting surface is

called the isosurface and is defined by a scalar isovalue as a visualization parameter. Some

isosurface extraction methods include marching cubes [32] marching tetrahedra [46], divid-

ing cubes [9] and ray tracing [37]. Representing a surface contained within a volumetric

dataset using geometric primitives can be useful in many applications, particularly when

the datasets contain actual boundaries between materials in the data. However, there are

several drawbacks to this approach. Geometric primitives approximate surfaces contained

within the original data. Adequate approximations may require an excessive number of ge-

ometric primitives and since only a surface representation is used, much of the information

contained within the data is lost during the rendering process [17] (chapter 7).

In response to this, direct volume rendering techniques were developed which create a

2D image directly from 3D volumetric data [26, 54, 23]. Direct volume rendering conveys

more information in a single image than surface rendering, and is thus a valuable tool for

the exploration and analysis of data. Instead of extracting an intermediate representation,

direct volume rendering provides a method for directly rendering the volumetric data by

compositing the original samples onto the image plane. This is typically done using a trans-

fer function which maps the samples to a color and intensity. Direct volume rendering

algorithms include object-order, image-order and domain-based approaches. Object-order

volume rendering approaches such as splatting [54] and shear-warp [23] use a forward map-

ping scheme by decomposing the volume into a set of basic elements and projecting them

individually into an image plane. Image-order algorithms such as raycasting [26], use a back-

ward mapping scheme where rays are cast from each pixel in the image plane through the

volume data to determine the final pixel value. In domain rendering, the spatial 3D data is

first transformed into another domain such as the frequency or the wavelet domain. A pro-

jection is generated using the information of the intermediate domain. Frequency domain

volume rendering [50] is an example of domain rendering algorithms. Due to the increased

computational effort required and the enormous size of volumetric datasets, the ongoing

challenge of research in direct volume rendering is to achieve interactive performance.

Traditional volume visualization methods work well on small and medium-scale datasets.

However, the size of average volumetric data continues to grow as the computational power


and scanning resolution increases. Hence, different approaches are required to address the

difficulties associated with the processing of the volumetric data. Large scale volumetric

datasets are commonly characterized as volumetric datasets that exceed the resource limits

of hardware such as memory and computational power. The challenges in visualizing large

datasets are, first, whether the data can be visualized at all and then, whether they can

be processed quickly enough to allow some degree of interactivity (i.e. performance). The

ability to achieve interactive frame rates, in the range of at least a few frames/s, for large

datasets is still an open research challenge. Although various methods have been proposed

to adapt traditional methods to be used with massive parallel hardware [39], it is still a

challenging problem to employ visualization methods on commodity computers.

Major factors that determine the size of a volumetric dataset include dimensionality,

resolution and data modality. Dimensionality is the number of independent coordinates

required to uniquely specify the sample points in the data space. We define resolution

to be the number of samples along each dimension, without considering the actual size of

the sampled phenomenon. Data modality is the number of scalar values per data point.

Considering the first two factors, this thesis addresses two major categories of large-scale

datasets: datasets with many frames of medium resolution volumes (four dimensional) and

datasets with a single high resolution volume. The first category of datasets are called time-

varying volumetric datasets, and are a sequence of varying three dimensional volumetric

data acquired at consecutive time steps. The number of these time steps determines the

resolution along the time dimension. In this thesis we have used time-varying datasets

of sizes ranging from about 128 × 128 × 128 × 150 of 4-byte float samples (1.3 GB) to

500 × 500 × 100 × 50 of 4-byte float samples (4.8 GB).

The second major category is large-scale single frame datasets which have a high res-

olution in the x, y and z dimensions. Since each of the two categories mentioned above

has unique properties and characteristics, we’ve studied these categories separately in this

thesis. The size of the datasets used in this thesis are in the range of about 500×500×2000

1-byte samples (0.5 GB) to 20003 1-byte samples (8 GB).

In this thesis we present new methods to exploit the properties of large-scale volumet-

ric datasets to visualize them efficiently. An overview of algorithms for visualization of

time-varying volumetric datasets and multi-resolution rendering is given in Chapter 2. In

Chapter 3 we study time-varying volumetric datasets and we introduce a new data structure

called Differential Time-Histogram Table [59]. We show how to exploit data characteristics


to build a binned data structure to accelerate interactive exploration of such data. Chapter 4

addresses high resolution datasets and introduces a hierarchical data structure to support

multi-resolution data visualization and progressive refinement of visualizations. Extending

the existing techniques, we address the question of multi-resolution rendering quality, using

statistical information to decrease aliasing artifacts in direct volume rendering while using

transparent isosurfaces to decrease the loss of high-frequency features. Finally, Chapter 5

presents the conclusions and discusses directions for future work.

Chapter 2

State of the Art

2.1 State of the Art in Time-Varying Data Visualization

The difficulty of rendering large datasets is well-recognized in the literature, with successive

solutions addressing progressively larger datasets by exploiting various data characteristics.

These characteristics will be thoroughly discussed in Section 3.1. However, in order to

discuss the previous work, we start by defining some of the characteristics briefly in this

chapter. We consider coherence in general as a property of the functions in which small

changes in the functions domain, imply small functional changes. In volumetric datasets,

spatial coherence is coherence of values of voxels with respect to the spatial location. In time-

varying volumetric datasets, temporal coherence is coherence of values of voxels sampled at

different time steps but same spatial location. Histogram distribution determines how well

the values of voxels have been distributed in the overall range of values.

Broadly speaking, the solutions proposed by researchers for visualization of time-varying

datasets have dealt with isosurface extraction and volume rendering as separate topics.

Modern isosurface extraction and volume rendering methods are based on the recognition

that the problem is decomposable into smaller sub-problems [60]. Lorensen & Cline [32]

recognized this and decomposed the extraction of an isosurface over an entire dataset by

extracting the surface for each cell in a cubic mesh independently.

Almost simultaneously, Wyvill, McPheeters & Wyvill [57] exploited spatial coherence

for efficient isosurface extraction in their continuation method. Spatial coherence has also

been exploited for volume rendering [51].

Other researchers [8, 13, 30, 55] exploit spatial distribution for efficient extraction of

5

CHAPTER 2. STATE OF THE ART 6

isosurface active sets. In particular, Wilhelms & van Gelder [55] introduced the Branch-On-

Need Octree (BONO), a space efficient variation of the traditional octree. Octrees, however,

do not always coincide with the spatial distribution of the data, and are not easily applicable

to irregular or unstructured grids.

Livnat et al. [30] introduced the notion of span space - plotting the isovalues spanned

by each cell and building a search structure to find only cells that span a desired isovalue.

Since this represents the range of isovalues in a cell by a single entry in a data structure, it

exploits histogram distribution. Histogram distribution has also been exploited to accelerate

rendering for isosurfaces [6] by computing active set changes with respect to isovalue.

Chiang et al. [7] clustered cells in irregular meshes into meta-cells of roughly equal

numbers of cells, then built an I/O-efficient span space search structure on these meta cells,

again exploiting histogram distribution.

For time-varying data, it is impossible to load an entire dataset plus search structures

into main memory, so researchers have reduced the impact of slow disk access by building

memory-resident search structures and loading data from disk as needed.

Sutton & Hansen [48] extended branch-on-need octrees to time-varying data with Tem-

poral Branch-on-Need Octrees (T-BON), using spatial and temporal coherence to access

only those portions of search structure and data which are necessary to update the active

set.

Similarly, Shen [42] noted that separate span space structures for each time step are inef-

ficient, and proposed a Temporal Hierarchical Index (THI) Tree to suppress search structure

redundancy between time steps, exploiting both histogram distribution and temporal co-

herence.

Reinhard et al. [39] binned samples by isovalue for each time step, loading only those

bins required for a given query: this form of binning exploits histogram distribution but not

temporal coherence.

For volume rendering, Shen & Johnson [45] introduced Differential Volume Rendering,

in which they precomputed the voxels that changed between adjacent time steps, and loaded

only those voxels that changed, principally exploiting temporal coherence.

Shen [43] extended Differential Volume Rendering with Time-Space Partitioning (TSP)

Trees, which exploit both spatial distribution and temporal coherence with an octree repre-

sentation of the volume. Binary trees store spatial and temporal variations for the sub-blocks


of data at the nodes of the tree. This information is used for early termination of a ray-

tracing algorithm, based on a visual error metric. Shen also cached images for each sub-block

in case little or no change occurred between frames. Ma & Shen [34] further extended this

by quantizing the isovalues of the input data and storing them in octrees before computing

differentials.

Finally, Sohn and Bajaj [47] use wavelets to compress time-varying data to a desired

error bound, exploiting temporal coherence as well as spatial and histogram distribution.

In Chapter 3 we discuss statistical characteristics of time-varying datasets and unify

those characteristics in the form of our proposed data structure called Differential Time-

Histogram Table.

2.2 State of the Art in Multi-resolution Techniques

Since it is not always possible to interactively process data in the original sampling resolu-

tion, level of detail and multi-resolution techniques have been proposed by many researchers

to balance between architectural constraints (e.g. performance and memory) and fidelity.

Multi-resolution techniques have been employed in a variety of visual data representation

approaches including geometric rendering [55, 5, 33] and volumetric data visualization [24]

especially for out-of-core applications [43].

Multi-resolution volume rendering techniques principally use spatial hierarchies to rep-

resent different levels of detail. Often the goal is to maintain a specific frame-rate [29] at the

expense of image quality. A partial solution to this is progressive refinement [25, 36, 38], in

which the quality of the rendered image is improved when the user interaction stops.

Shen et al. [43] used image caches attached to the nodes of the octree in order to exploit

temporal coherence for time-varying data. In this thesis we extend that approach and the

idea of frameless rendering [36] in an object order fashion, by re-using previously rendered

images of sub-volumes of the volumetric data in order to avoid re-rendering of those sub-

volumes.

Direct volume rendering techniques [26] use transfer functions (often multi-dimensional [20])

in order to investigate complex scalar data. Surface extraction techniques [32] use geometric

models to represent surface structures in the data. Some approaches [27, 21, 22, 18] combine

both geometric surface models and volumetric transfer functions for hybrid visualization,

improving the insight into a data structure. Often these hybrid methods fail to properly sort


the data during rendering and therefore fail to deal with transparent iso-surfaces combined

with direct volume rendering. In this thesis we deal with this problem and introduce a

framework that allows for accurate hybrid volume rendering. We also demonstrate how to

use transparent iso-surfaces effectively to add high-frequency details at coarse resolutions of

a multi-resolution rendering technique.

Most hierarchical multi-resolution schemes use the mean values of the underlying func-

tion values in order to construct the representations in different levels of detail. Notable

exceptions are wavelet-based multi-resolution schemes [14, 19], which use better filtering

methods to improve the quality of lower levels of the multi-resolution hierarchy. Guthe et

al. [16, 15] use a block based wavelet compression and employ graphics hardware to handle

large datasets at interactive speed.

A couple of the proposed multi-resolution hierarchies for volume rendering address the

underlying transfer function being used during the rendering. Ljung et al. [31] introduce the

use of transfer functions at decompression time to guide a level-of-detail selection scheme.

Wittenbrink et al. [56] have shown that interpolation of the underlying function before

the application of a transfer function is of utmost importance in order to assure the best

quality. Rottger et al. [41] have pointed out that the application of a transfer function to the

underlying scalar data can create arbitrarily high frequencies, making it difficult to render

the data properly. They have proposed Pre-integrated volume rendering to minimize the

impact of the modulation of the underlying signal by the transfer function.

None of these multi-resolution volume rendering methods addresses the proper use of

transfer function and the underlying distribution of the function values to improve the

quality of rendering. In Chapter 4 we propose a novel method to improve the quality of

multi-resolution rendering by approximating the distribution of function values at coarse

levels of detail.

Chapter 3

Visualization of Time-Varying

Volumetric Data

As computing power and scanning precision increase, scientific applications generate time-

varying volumetric data with thousands of time steps and billions of voxels, challenging

our ability to explore and visualize interactively. Although each time step can generally be

rendered efficiently, rendering multiple time steps requires efficient data transfer from disk

storage to main memory to graphics hardware.

We use the temporal coherence of sequential time frames, the spatial distribution of

data values, and the histogram distribution of the data for efficient visualization. Temporal

coherence is used to prevent loading unchanged sample values [48], spatial distribution to

preserve rendering locality between images [55], and histogram distribution to update for

incremental updates during data classification.

Of these, temporal coherence and histogram distribution are encoded in a differential

table computed in a preprocessing step. This differential table is used during visualization to

minimize the amount of data that needs to be loaded from disk between any two successive

images. Spatial distribution is then used to accelerate rendering by retaining unchanged

geometric, topological and combinatorial information between successive images rendered,

based on the differential information loaded from disk.

The major contribution of our method is to identify and analyze statistical characteristics

of large scientific datasets that permit efficient error-bounded visualization. We unify the

statistical properties of the data with the known spatial and temporal coherence of the data

9

CHAPTER 3. VISUALIZATION OF TIME-VARYING VOLUMETRIC DATA 10

in a single differential table used for updating a memory-resident version of the data.

We define coherence and distribution more formally in Section 3.1, and support our

definition of coherence and distribution in Section 3.2 with a case study. In Section 3.3

we show how to exploit data characteristics to build a binned data structure representing

the temporal coherence and spatial and histogram distribution of the data, and how to

use this structure to accelerate interactive exploration of such data. We then present our

implementation results in Section 3.4.

3.1 Data Characteristics

In order to discuss both previous work and our own contributions, we start by defining our

assumed input and the statistical properties of the data that we will exploit for accelerated

visualization: we will support these assertions empirically in our case study in Section 3.2.

Expanding the notation of [8], we define a time-varying dataset as follows: Formally,

a time-varying scalar field is a function f : <4 → <. Such data is often sampled at

fixed locations X = {x1, . . . ,xn} and fixed times T = {t1, . . . , tτ}. Since we exploit this

sampling regularity to accelerate visualization, we assume that our data is of the form

F = {(x, t, f(x, t)) | x ∈ X, t ∈ T}, where each triple (x, t, f(x, t)) is a sample.

Previous researchers report [35] that only a small fraction of F is needed for any given

image generated during visualization. Thus, any image can be thought of as the result of

visualizing a subset of F defined by a query q .

More formally, given F and a condition q (e.g. a transfer function), find the set F|qof all samples (x, t, f(x, t)) for which q is satisfied (e.g. the opacity of f(x, t) is non-zero).

Thus, F|q is the set of samples needed to render q, called the active set of q.

For a query q to extract the isosurface at isovalue h, F|q consists of all samples belonging

to cells whose isovalues span h. In direct volume rendering, the active set has a range of

isovalues for which the opacity transfer function is non-zero.

We now restate our observation: we expect ‖F|q‖ � ‖F‖ where ‖‖ denotes the size of

the set. Moreover, F|q is rarely random, nor is it evenly distributed throughout the dataset.

Instead, F|q tends to consist of large contiguous blocks of samples, due to the continuity

of the physical system underlying the data, and the inherent organization of most physical

systems being studied. We exploit both characteristics to accelerate visualization.

We also exploit the fact that human exploration of data usually involves continuous (i.e.


gradual) variation of queries. Formally, we are interested, not in a single query q , but in a

sequence q1, . . . , qk of closely-related queries, with occasional abrupt changes to a new query

sequence q ′1, . . . , q

′m. We expect that the active sets for any two sequential queries will be

nearly identical, i.e. that F|qi+1 ≈ F|qi.

In particular, we exploit coherence and distribution. Let us denote two samples σ1 =

(x1, t1, f(x1, t1)) and σ2 = (x2, t2, f(x2, t2)), and choose small values δ, λ > 0.

We consider coherence in general as a property of the functions in which small changes

in the functions domain, imply small functional changes.

Spatial coherence is coherence with respect to the spatial dimensions x, y, z of the

domain of f , and is based on the spatial continuity of the physical system. Small spatial

changes imply small functional changes, i.e. if |x1 − x2| < δ and t1 = t2, then |f1 − f2| < λ

for small λ.

Temporal coherence is coherence with respect to the time dimension t of the domain

of f and is based on the temporal continuity of the physical system. Hence, if |t1 − t2| < δ

and x1 = x2, then |f1 − f2| < λ for small λ.

Given two sequential queries qi , qi+1 (e.g. a change in isovalue or time step) which

differ only slightly in space or time, F|qi and F|qi+1 will therefore overlap significantly. We

exploit this to accelerate the task of updating the active set for rendering purposes, focusing

principally on temporal exploration. We also exploit statistical properties of the datasets in

question, which we describe in terms of data distribution:

Histogram distribution is a property of the the range of f and depends largely on

the data being studied and shows how well the function values are distributed over the

functional range. In a well distributed function, the active set for a query is usually a small

subset of the function domain (e.g. only a small number of voxels need to be considered

in order to visualize a particular isosurface). We expect f to be fairly well distributed,

although sampling may affect this.

Spatial distribution is a large scale form of spatial coherence, and is based (again)

on the physical phenomena scientists and engineers study. Although data is commonly

generated over large domains, it is often true that nothing interesting happens in most of

the domain. As a result, the active sets for queries made by the user are often clustered in

a subset of the space.

Histogram distribution is crucial where two sequential queries qi and qi+1 differ only

slightly in the functional dimension. For well-distributed data values, we expect F|qi and


F|qi+1 to be substantially identical: i.e. we can change transfer functions more rapidly by

loading only the new data values required. Moreover, the spatial distribution of the data

allows us to have spatially sparse data structures.

3.2 Case Studies: Data Characteristics of Isabel & Turbu-

lence

As noted in the previous sections, our approach to accelerating visualization depends on

mathematical, statistical and physical properties of the data being studied. Before pro-

gressing further, it is therefore necessary to demonstrate that these properties exist. We do

so by means of case studies of three different datasets.

The first and largest dataset is the Hurricane Isabel dataset provided by the National

Center for Atmospheric Research (NCAR) for the IEEE Visualization Contest 2004. This

dataset has 48 time steps of dimension 500 × 500 × 100, and contains 13 different floating

point scalar variables of which we study the temperature and pressure because they were

different kind of properties by nature and hence were expected to be almost independent

and not to have very similar properties. We have also studied the Turbulent Jets (jets) and

Turbulent Vortex Flow (vortex) datasets provided by D. Silver at Rutgers University and

R. Wilson at University of Iowa to Kwan-Liu Ma at University of California, Davis. The

jets dataset contains 150 time steps of 104× 129× 129 floats, while the vortex dataset were

originally containing 100 time steps of 1283 floating point scalars. Since frame 24 of this

dataset was corrupted, we were only able to use frames 25 through 100.

3.2.1 Temporal Coherence

We demonstrate temporal coherence by computing the difference between adjacent time

steps, and graphing the percentage of differences that exceed an error bound. The error

bound (λ%) is expressed as a percentage of the overall data range and its scalar value is

calculated as (max−min)× (λ%). As we see in Figure 3.1, at an error bound of 0%, (where

only identical values are ignored) all of the voxels differ from one time step to the next.

This may be due to numerical inaccuracies in the floating point data. It therefore follows

that 100% of the samples will need to be replaced for the memory-resident version of the

data to be numerically accurate.


However, as we see from the λ = 0.3% plot on the graph, between 40% and 60% of the

samples differ by more than 0.3%. Thus, for a 0.3% error in the isovalues displayed, we can

save as much as 60% of the load time between queries that only change the time step. As

the error bound is increased further, the number of samples to be loaded decreases further:

at 1% error, between 80% and 90% of the samples need not be reloaded, while at 3% error,

over 95% of the samples need not be reloaded.

Similar effects can be seen for pressure in Isabel (Figure 3.1(b)) and for the jets and

vortex datasets (Figure 3.1(c) & (d)). In Figure 3.1(d), we believe that the sharp periodic

dips indicate that certain time steps were inadvertently duplicated. This aside, the overall

conclusion is that relatively small error bounds permit us to avoid loading large numbers of

samples as time changes.

3.2.2 Histogram Distribution

In the last section, we saw that the statistics of temporal coherence give us the opportunity

for efficient differential updates of an active set for a given error bound. It is natural to

ask whether similar statistics are true for small changes in isovalues defining a query. We

show in Figure 3.2 to Figure 3.5 that this is true by examining the histogram distribution

for each dataset.

For these experiments, we set our error bound to 0%, 0.3%, 1% and 3% and computed

the size of the active set for individual data ranges (bins) and for different times. We also

computed the number of samples whose scalar values changed more than the error bound

from the previous time step. For example, we see in Figure 3.2 that temperature isosurfaces

in the Isabel dataset have active sets of less than 5% of the size of a single frame, and that

typically less than half of the samples in those active sets require replacement between one

time step and the next. For pressure (Figure 3.3), although nearly 25% of the samples are

in the active set in the worst case, these samples change little between timesteps, indicating

that the physical phenomenon marked by this pressure feature does not move much in space

between time steps.

In the jets dataset (Figure 3.4), the worst-case behavior is exhibited near the 75th

isovalue bin, where nearly 80% of the samples are in the active set. Although this has

implications for rendering, this set changes little with respect to time. However, in this

dataset these data values constitute most of the “air” surrounding the flow under study and

are likely to be ignored during data exploration because they are usually not interesting for


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Figure 3.1: Temporal Coherence: Number of samples that change by more than an errorbound of λ = 0%, 0.3%, 1%, 3% for the temperature and pressure scalars of the Isabel datasetas well as the jets and vortex datasets.


the scientists.

In the vortex dataset (Figure 3.5), we see again that data bins involve relatively few

samples, and that these samples have some degree of temporal coherence.

In general, our studies show that although we can’t expect all datasets to have a fairly

well distributed histogram, the temporal coherence gives us the opportunity to reduce the

updates of active sets during the changing of time steps.

3.2.3 Spatial Distribution

Unlike histogram distribution, spatial distribution is harder to test, as it is best measured in

terms of the spatial data structure used to store the data. We therefore defer consideration

of spatial distribution to the discussion of Table 3.1 in Section 3.3.

3.3 Differential Time-Histogram Table

In the previous section, we demonstrated that temporal coherence and histogram distri-

bution offer the opportunity to reduce load times for gradual variations of visualization

queries when visualizing large time-varying datasets. To achieve the hypothesized reduction

in load-time we introduce a new data structure called the Differential Time-Histogram Table

(DTHT) along with an efficient algorithm for performing regular and differential queries in

order to visualize the dataset.

To exploit temporal coherence and histogram distribution, we apply differential visual-

ization in the temporal direction and binning in the isovalue direction. Since we expect user

exploration of the data to consist mostly of gradual variations in queries, we precompute the

differences between successive queries and store them as differentials. We modify temporal

coherence by including in the differential set only those samples for which the error exceeds

a chosen bound, further reducing the number of samples to be loaded at each time step.

3.3.1 Computing the DTHT

Our DTHT, shown in Figure 3.6, stores samples in a two-dimensional table of bins defined

by the isovalue range and the time step. In each bin, we store the active set (a), and a set

of differences (arrows) between the active sets of adjacent bins.

For a given dataset with τ time steps, we create a DTHT with τ bins in the temporal


(a) Isabel - temperature (λ = 0%) (b) Isabel - temperature (λ = 0.3%)

(c) Isabel - temperature (λ = 1%) (d) Isabel - temperature (λ = 3%)

Figure 3.2: Temporal coherence and histogram distribution for the temperature dataset: Inthis figure and following three figures, the striped yellow plot shows the number of samplesthat change by more than the error bound λ. The blue plot represents the number of samplesper bin (histogram), at each time step. The isovalue range is divided into 100 isovalue bins.


(a) Isabel - pressure (λ = 0%) (b) Isabel - pressure (λ = 0.3%)

(c) Isabel - pressure (λ = 1%) (d) Isabel - pressure (λ = 3%)

Figure 3.3: Temporal coherence and histogram distribution for the pressure dataset. Notethat in b, c and d, the y-axis has been scaled by 10 to show the details of the diagrams.


(a) Turbulent Jets (λ = 0%) (b) Turbulent Jets (λ = 0.3%)

(c) Turbulent Jets (λ = 1%) (d) Turbulent Jets (λ = 3%)

Figure 3.4: Temporal coherence and histogram distribution for the jet dataset. Note thatthe y-axis have different scales to show the details of the diagrams.


(a) Turbulent Vortex Flow (λ = 0%) (b) Turbulent Vortex Flow (λ = 0.3%)

(c) Turbulent Vortex Flow (λ = 1%) (d) Turbulent Vortex Flow (λ = 3%)

Figure 3.5: Temporal coherence and histogram distribution for the vortex dataset.


aaaa

aaaa

aaaa

aaaa

a

Bin

Active Set

Differentials

Time

Iso

va

lue

Figure 3.6: The Differential Time-Histogram Table (DTHT). Each bin holds the active set(a) and differentials between the bin and adjacent bins (arrows). Components not used forour experiments are shown in a lighter shade.

direction and b bins in the isovalue direction. Large values of b will increase the size of

the table but decrease the range of values in each bin. This makes active set queries more

efficient for one bin, but not necessarily faster for a query covering a large range of bins.

Moreover, large numbers of bins will result in more storage overhead.

Before pre-processing, we choose a number b and divide the isovalue range into b bins.

The value of b is chosen empirically such that the number of voxels in each bin is smaller

than a certain ratio of the total number of voxels. We also choose a λ, the amount of error

we will tolerate in isovalues for a single sample over time. We start computing the DTHT by

loading the first time step into memory and binning the samples. We then load the second

time step into memory alongside the first and compute the difference between the two time

steps for each sample. Any sample whose difference to the sample with the same spatial

location in the previous time step, exceeds the error bound is added to the differential set

in the time direction. Any sample whose difference does not exceed the error bound is

modified so that the value in the second time step is identical to the value in the first time

step: this ensures that, over time, errors do not accumulate beyond our chosen bound.

At the same time, differentials are computed in the isovalue direction if these are desired.


In our experiments, we have chosen to render using splatting: in this case each sample

belongs to only one bin and the active sets in adjacent bins are disjoint. It follows that

the isovalue differentials are equal to the active sets, and can be omitted accordingly. They

are shown in a lighter shade in Figure 3.6. In case we want to adapt the data structure to

support isosurface extraction, each voxel may belong to a range of isovalue bins depending

on the values of its neighbors; therefore the bins will overlap and it would be reasonable to

have differentials between neighbor bins to avoid redundancy.

The first time step is then discarded, and the next time step is loaded and compared to

the second time step. This process continues until the entire dataset has been processed.

(a)

(b)

(c)

1011 0010 0000 1110 0000 0000

00000000 1100 0000

1011

0010 1110 1100

Figure 3.7: Data and file structure of a branch-on-need octree (in 2D): (a) dataset (b) treestructure (c) depth-first order file structure: byte codes represent active children of a node.A 0000 code represents a leaf node containing a data block.

We have used an octree based data structure to store the active sets and differentials.

An octree is a tree data structure mainly used for organizing 3-dimensional data points in

which the volume space is recursively divided in each dimension into smaller partitions. In

a branch-on-need octree, each dimension is divided such that it creates the largest power-of-

two sized partition. For each sample, we store location, isovalue, and normalized gradient

vector at a leaf node of a branch-on-need octree. At all times, the current active set is

stored in a branch-on-need octree, as are the active sets and the differentials for each bin.

Samples are added or removed from the current active octree by tandem traversal with the


differential octrees of the active bins. Figure 3.7 shows the octree and its file structure.

Figure 3.7(a) demonstrates a sample dataset and the space division in 2D. Figure 3.7(b)

shows the tree data structure and Figure 3.7(c) shows the corresponding file structure to

store the tree. The tree is stored and loaded in a depth first order. Starting from the root

node, a byte-code is stored to indicate non-empty children nodes. Then for the leaf nodes

the data block is stored. Otherwise the non-empty sub-trees are processed recursively.

Instead of single samples, however, we store entire blocks of data (typically 32×32×32)

at octree leaves, either in linked lists or fixed size arrays. We use linked lists if no more than

60% of the block is active, in which we store the spatial locations of the samples as well as

their values. However, if more than 60% is active, we use a fixed size array where the spatial

locations of the samples are implied from their index in the array. The threshold of 60% is

computed to be the point at which the overhead of including spatial positions (indices) of

samples in a linked list is almost equal to the overhead of keeping empty samples in an array.

We save space by storing sample locations relative to the octree cell instead of absolutely,

requiring fewer bits of precision.

Moreover, octrees are memory coherent, so updates are more efficient than linear lists,

although at the expense of a larger memory footprint, as shown in Table 3.1. We note,

however, that the memory footprint is principally determined by the size of the octrees for

the active sets.

Further reductions in storage are possible depending on the nature of the queries to

be executed. We assume that abrupt queries are few and far between, and dispense with

storing active sets explicitly except for the first time step. If an abrupt query is made at

time t, we can construct the active set by starting with the corresponding active set at time

0 and applying all differential sets up to time t. Doing so reduces the amount of storage

required, as shown in the third column of Table 3.1. As with the isovalue differentials, we

indicate this in Figure 3.6 by displaying the unused active sets in a lighter shade. We can

also store active sets in a subset of a priorly chosen key frames so that we always start from

the nearest keyframe instead of starting from the first frame. Another way of reducing the

storage is to remove the active sets and differentials for the non-interesting isovalue ranges.

For example we can identify and remove the bins that contain the values of empty spaces

surrounding the actual phenomena of interest. This can have a huge impact depending on

the nature of the dataset, e.g. for the Turbulent Jets dataset, we can remove up to 80% of

the overall data.


Structure List Octree Octree

All Active Sets 1st Active Set

λ GB %age GB %age GB %ageIsabel - temperature

0.0% 34.6 737% 27.7 591% 18.5 394%0.3% 22.4 477% 17.9 382% 8.7 186%1.0% 15.0 320% 12.0 256% 2.8 60%3.0% 12.2 260% 9.8 209% 0.59 12%Isabel - pressure

0.0% 34.6 737% 27.7 590% 18.5 394%0.3% 17.6 371% 14.1 301% 4.9 104%1.0% 13.8 294% 11.1 237% 1.9 40%3.0% 12.4 264% 9.9 211% 0.72 15%Turbulent Jets

0.0% 7.4 673% 5.0 455% 3.4 330%0.3% 3.4 310% 2.4 216% 0.71 70%1.0% 2.8 255% 1.9 173% 0.20 20%3.0% 2.6 236% 1.8 164% 0.06 6%Turbulent Vortex Flow

0.0% 4.6 731% 3.7 588% 2.5 394%0.3% 4.0 636% 3.2 509% 2.0 320%1.0% 3.5 556% 2.8 445% 1.6 252%3.0% 2.6 413% 2.1 334% 0.93 150%

Table 3.1: Size (in GB and percent of original dataset) of DTHT using lists and octrees withall or only the first active set stored. Storing all active sets is expensive storage-wise, butit enables us to access any time step directly. As an alternative we can store only the firstactive set and use differentials to obtain the active sets in other time steps. This improvesthe storage requirement substantially but does not allow direct access to time steps.


3.3.2 Queries in the DTHT

For any query, we first classify the query as gradual (a small change) or abrupt (a large

change): as noted in Section 3.1, we expect the nature of user interaction with the data will

cause most sequences of queries to be gradual in nature rather than abrupt.

We chose to implement volume rendering using point-based splatting, in which case the

active set consists only of those samples for which the opacity is non-zero. This active set

may span multiple DTHT cells, as shown in Figure 3.8, so we use the union of the active sets

for each DTHT cell. It is also possible to use other volume rendering methods that allow

runtime update of the data. For instance, hardware texture slicing would be suitable only

if the hardware allows texture updates without loading the whole texture for each update.

Abrupt queries are handled by discarding the existing active set and reloading the active

set from the corresponding bin or bins on disk. Because the data was binned, the active

set may be over-estimated for a given transfer function. This over-estimate is conservative:

it always includes the desired active set but may also include inactive samples. Moreover,

the over-estimate is sensitive to the size of the bin. Too few bins leads to large numbers

of discards for a particular query, while too many bins leads to overhead in data structures

and overhead in merging octrees.

aaaa

aaaa

Time

Iso

val

ue

TF Opacity

aaaa

aaaaActiveDTHT Cells

Figure 3.8: Active DTHT cells for a given transfer function.


Gradual queries in the temporal direction edit the active set using differentials for each

bin spanned by the transfer function. Since forward and backward differentials are reciprocal

in nature, for each arrow we only store the set of samples that needs to be added in that

direction. The set of samples that needs to be removed is then the set of samples to be

added on the reciprocal arrow between the same bins.

Gradual queries in the isovalue direction discard all samples in a bin when the transfer

function no longer spans that bin, and add all samples in a bin when the transfer function

first spans that bin. It should be noted that random access to voxels in our octree based

data structure will not be efficient since it requires top down traversing of all octrees to

access a voxel. Hence, visualizing multi-modal data would be generally difficult in the cases

where different measures are stored in separate structures. For example it would be hard

to visualize an isosurface of temperature from one structure and color it with the value of

pressure from another structure.

Similar to TSP tree [43], for faster rendering, partial images for each base level block in

the current active octree are stored as image caches, and are used when the blocks don’t

change. Since we update base level blocks in the octree only when changes exceed the

λ-bound, these partial images can survive a few time steps.

3.4 Results and Discussion

We implemented our Differential Time-Histogram Table algorithm and visualized different

scalar time-varying datasets introduced in Section 3.2, using a 3.06 GHz Intel Xeon proces-

sor, 2 GB of main memory and an nVidia AGP 8x GeForce 6800 graphics card, with data

stored on a fileserver on a 100 Mb/s Local Area Network.

In our experiments we used b = 100 isovalue bins, set the base-level brick size in the

octrees to 32 × 32 × 32, and tested four different values for error bounds (λ = 0%, 0.3%,

1.0%, 3.0%).

In our first experiment we controlled the number of active voxels by changing the width

of a left-aligned transfer function to gradually increase the range of isovalues that are set

to non-transparent opacity values. Starting with a 0% width transfer function covering no

bins, we increased the width until all data values were classified as non-transparent and all

bins were loaded. In Figure 3.9, we compared the performance of DTHT active sets alone,

with the performance of DTHT using active and differential sets, rendering each dataset


with an error bound of λ = 1.0% and transfer functions of varying widths. The performance

of the differential method, is principally driven by the width of the transfer function, and

it is clear that a speed gain in loading the data can be achieved with differential sets. The

left column of Figures 3.14 through 3.17 show representative images under this scenario.

Not surprisingly, as we had already found out in the case study, the Turbulent Vortex Flow

dataset did not have a high temporal coherence, and differential updating was slower than

direct loading (Figure 3.9(d)).

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al R

ende

ring

Cos

t (se

cond


(c) Turbulent Jets

20 40 60 80 100

0

20

40

60

80

100

Cum

ulat

ive

sum

of h

isto

gram

(%

)

0 20 40 60 80 1000

0.5

1

1.5

2


Tot

al R

ende

ring

Cos

t (se

cond



Figure 3.9: λ = 1.0% Rendering and data loading performance for DTHT using active anddifferential files. The reported times are averaged costs while navigating over multiple timesteps. The yellow overlay shows the ratio of active voxels to all voxels.

We next investigated the performance of our algorithm for different isovalue ranges,

collecting the same measurements as before for constant width (10%) transfer functions

with different placements. The results for λ = 1.0% and different datasets are shown in


Figure 3.10. The right column of Figures 3.14 through 3.17 show representative images under

this scenario. Similar to the previous experiment, because of the low temporal coherence

of the Turbulent Vortex Flow dataset, differential updating was slower than direct loading

(Figure 3.10(d)).

20 40 60 80 100

0

20

40

60

80

100

Con

volv

ed h

isto

gram

(%

)

0 20 40 60 80 1000

2

4

6

8

10

Placement of a 10% width transfer function

Tot

al R

ende

ring

Cos

t (se

cond



20 40 60 80 100

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Con

volv

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(%

)

0 20 40 60 80 1000

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4

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10

Placement of a 10% width transfer functionT

otal

Ren

derin

g C

ost (

seco

nds) DTHT direct loading

DTHT differential updatinghistogram


20 40 60 80 100

0

20

40

60

80

100

Con

volv

ed h

isto

gram

(%

)

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1


Tot

al R

ende

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Cos

t (se

cond


(c) Turbulent Jet

20 40 60 80 100

0

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volv

ed h

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(%

)

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1


Tot

al R

ende

ring

Cos

t (se

cond



Figure 3.10: λ = 1.0% Performance for DTHT using active and differential files, showingalso the ratio of active voxels to all voxels (in yellow). A transfer function spanning 10% ofthe data range was used with different centre points. Results are averaged over several timesteps.

To simplify comparison, we calculated the ratio between differential DTHT updates and

direct DTHT updates for both sets of experiments. Since our goal was to improve the loading

times, we excluded rendering time from our figures. The goal was to investigate the cases in

which updating the octree using differential files were faster than straightforward loading of

the active files. Figure 3.11 and Figure 3.12 show the ratio between the performance of the

two methods, respectively for different width transfer function and constant width transfer


function.

20 40 60 80 100

0

20

40

60

80

100

Cum

ulat

ive

sum

of h

isto

gram

(%

)

0 20 40 60 80 1000

5

10

15

20

25

Transfer function width (%)ratio

of d

irect

load

ing

to d

iffer

entia

l upd

atin

g

λ=0%λ=0.3%λ=1.0%λ=3.0%ratio = 1histogram


20 40 60 80 100

0

20

40

60

80

100

Cum

ulat

ive

sum

of h

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(%

)

0 20 40 60 80 1000

5

10

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25


of d

irect

load

ing

to d

iffer

entia

l upd

atin

g



20 40 60 80 100

0

20

40

60

80

100C

umul

ativ

e su

m o

f his

togr

am (

%)

0 20 40 60 80 1000

2

4

6

8

10


of d

irect

load

ing

to d

iffer

entia

l upd

atin

g


(c) Turbulent Jet

20 40 60 80 100

0

20

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60

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100

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of h

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(%

)

0 20 40 60 80 1000

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of d

irect

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ing

to d

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entia

l upd

atin

g



Figure 3.11: Ratio between DTHT differential updating time and DTHT direct loading timefor different error bounds, for different transfer function widths. Ideally, the ratio should beabove 1 (dotted line) all the time.

In a third experiment we investigated performance for a fixed time but varying transfer

function. Compared to the brute-force method, where all of the data is loaded, our method

takes advantage of the rendering parameters to load only the voxels participating in the

final rendering. Changing the transfer function caused new bins to be added to or removed

from the dataset. The results are shown in Figure 3.13.

Loading time of DTHT methods depends greatly on the amount of active voxels. In

the first two experiments, when a small number of voxels are active, loading the octree

directly from the active files is comparatively faster than using differential files. It is mainly

because of the fact that loading the octree directly from the active voxels files, requires


20 40 60 80 100

0

20

40

60

80

100

Con

volv

ed h

isto

gram

(%

)

0 20 40 60 80 1000

5

10

15

20

25

Placement of the 10% width transfer functionratio

of d

irect

load

ing

to d

iffer

entia

l upd

atin

gλ=0%λ=0.3%λ=1.0%λ=3.0%ratio = 1histogram


20 40 60 80 100

0

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of d

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to d

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(c) Turbulent Jet

20 40 60 80 100

0

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60

80

100

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volv

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)

0 20 40 60 80 1000

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of d

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Figure 3.12: Ratio between DTHT differential updating time and DTHT direct loading timefor different error bounds and different placements of a transfer function of constant width(10%). Ideally, the ratio should be above 1 (dotted line) all the time.

going through the octree only once per each bin, while in the differential method, it will

be twice per each bin; once to remove the invalid voxels, using voxels to remove files and

then to add the new voxels using voxels to add files. Hence, memory access time and octree

processing time overcomes the files transaction time when small amounts of data are read

from the hard drive.

As the number of active voxels increases, more data need to be loaded/updated from

the external storage. Hence, the memory-storage transactions become the bottleneck. Since

there are relatively few data in differential files, the performance of updating the octree using

differential files becomes better than direct loading. However, the ratio of this performance

highly depends on the temporal coherence of the dataset. For example since the Turbulent


Vortex Flow dataset is not as temporally coherent as the other three datasets, even with a

1.0% error bound, differential updating is slower than the direct loading because the number

of voxels to add + voxels to remove is larger than the number of active voxels.

Noise in the diagrams is mainly due to loading the data through a local area network

connection which is fast, but does not always provide a constant data rate. There are also

large variations in some parts of Figure 3.11 and Figure 3.12. It can be seen that most of

the noise is in the parts in which not many voxels are active. This is because of the fact

that as the expected loading times become smaller, the slightest activities in the hardware

or the operating system can result in more relative noise in the measured times.


0 20 40 60 80 1000

10

20

30

40

50

60

70

80

Bins

Tim

e (m

s)

resizing transfer functionmoving transfer function


0 20 40 60 80 1000

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100

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200

250

300

350

BinsT

ime

(ms)



0 20 40 60 80 1000

10

20

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Bins

Tim

e (m

s)


(c) Turbulent Jets

0 20 40 60 80 1000

5

10

15

20

Bins

Tim

e (m

s)



Figure 3.13: Loading time of incremental transfer function changes. The graphs show thetime to update the active sets when the active bins are changed in a constant time step.Increasing the transfer function width (solid-red), only causes additions to the active set;thus, there is only one peak in the diagram when a large number of voxels are added to theactive set. Changing the placement of the transfer functions (dotted-blue), requires somevoxels to be added and some voxels to be removed; thus there is one peak in the diagramwhen a large number of voxels are added and one peak later when those voxels are removed.


Figure 3.14: Rendered images with representative transfer functions for Isabel temperaturedataset. In this figure and the following three figures the left column shows the resultsfor increasing transfer function width and the right column shows the results for movingtransfer function placement, corresponding to our experiments in this section. The wiredframe shows the octree structure and the blocks containing active voxels.


Figure 3.15: Rendered images with representative transfer functions for Isabel pressuredataset.


Figure 3.16: Rendered images with representative transfer functions for Turbulent Jets Flowdataset.


Figure 3.17: Rendered mages with representative transfer functions for Turbulent VortexFlow dataset.

Chapter 4

Multi-resolution Volume Rendering

In the previous chapter, we focused on the problem of large scale dataset visualization from

the aspect of dimensionality and studied time-varying datasets which consisted of multiple

time slices of 3D volumetric data. We focused on disk access performance by reducing

the data transfer from storage to main memory during user navigation. However, more

problems arise when the size of a single time step is larger than the size we can render

in an interactive rate. In such cases, a coarser resolution version of the data is chosen

for interactive frame rates. In this chapter we will be focusing on the aspect of resolution

and we will consider another common type of large scale datasets which are single frame

high-resolution three-dimensional datasets.

Scientific and medical volumetric data continues to grow in scale, and visualization in-

creasingly requires processing large amounts of data. Many methods have been proposed for

optimizing performance, especially for datasets too large to fit in main memory. One pop-

ular approach uses multi-resolution representation of the data combined with hierarchical

rendering algorithms to balance the tradeoff between quality and performance.

Multi-resolution methods typically optimize the rendering speed and memory footprints.

However, few works have addressed the question of multi-resolution visual quality, i.e., the

perceptual image quality when rendering in resolutions coarser than the original resolution

of data. In this chapter, we address the question of multi-resolution visual quality using

statistical information to decrease aliasing artifacts in direct volume rendering while using

transparent isosurfaces to decrease the loss of high-frequency features. To support this

approach, we extend previous work on branch-on-need octrees [55] to hybrid branch-on-

need octrees (HYBON), which simultaneously represent the octree structure of the samples

36

CHAPTER 4. MULTI-RESOLUTION VOLUME RENDERING 37

Figure 4.1: Renderings of the Visible Human Male Head at different levels of details. Toprow: using mean value for approximation. Bottom row: using Gaussian function for ap-proximation. From left to right, each image represents an additional level of coarsening bya factor of two in each dimension.

for volume rendering and the octree structure of cubical cells for isosurface extraction.

To reduce aliasing artifacts in low-resolution volume renderings, we summarize octree

octants with Gaussian distributions rather than the standard constant mean value. To re-

duce loss of sharp features, we combine direct volume rendering with transparent isosurfaces

which add approximate high-frequency information to low-resolution images.

In addition to these fundamental improvements to low-resolution image quality, we

demonstrate performance optimizations effective during user interaction. We use a his-

togram code that improves octree performance by rapidly identifying octants not relevant

to a particular image. We also use hardware texture memory to cache output images at

different levels of detail for progressive refinement during subsequent user navigation.

The rest of this chapter is organized as follows: Section 4.1 presents the details of our

novel approach to improve the quality in multi-resolution rendering. The HYBON data

structure and the construction process are discussed in Section 4.2. Experimental results

are discussed in Section 4.3.


Figure 4.2: Renderings of a synthetic spherical distance dataset based on the Euclideandistance to the centre of the volume, at different levels of details. Top row: using meanvalue. Bottom row: using Gaussian basis approximation. From left to right, each imagerepresents an additional level of coarsening by a factor of two in each dimension.

4.1 Rendering

Given a desired level of performance, multi-resolution structures use criteria such as viewing

distance [52], projection area [29] and gaze distance [28] to identify the level of detail at

which the data must be rendered. Ideally, however, at a given level of detail, we wish to use

the highest quality image achievable at a given performance cost.

We focus on two aspects of multi-resolution rendering where image quality can be opti-

mized with relatively little impact on rendering speed: aliasing in direct volume rendering

and loss of high-frequency sharp features. We handle the first of these by improving the sta-

tistical representation of the data at coarse resolutions, and the second by using transparent

isosurfaces to add approximate high-frequency information to coarse-resolution images.

4.1.1 Accurate Multi-resolution Transfer Functions

For faster rendering, multi-resolution techniques substitute a small data set for a larger

dataset. As such, aliasing problems arise, such as the “jaggies” and discontinuities visible in

Figure 4.1 and Figure 4.2. These aliasing artifacts result from loss of detailed information


during construction of the multi-resolution representation. In particular, information is lost

here because the data in a subvolume is summarized with the mean of the samples in the

subvolume. This mean value is a good summary of the subvolume if the data values are

close to the mean, but is highly inaccurate when the standard deviation is relatively large.

This causes drastic problems in the resulting images because the transfer function often

does not have a smooth transition between colors and opacities for different values. More

sophisticated summaries of the subvolume can, however, be substituted for the mean, and

this idea forms the basis of our approach.

As an example, consider a subvolume with values {1, 1, 1, 1, 5, 5, 5, 5} and mean µ = 3.

Unless V (3), the color specified in the transfer function for 3, is near to the color produced

by blending V (1) and V (5), the summary information will misrepresent the data in the final

visual image.

For simplicity, we develop notation in one dimension: higher dimensions are essentially

identical, but with denser notation (i.e. integration along three dimensions). Assume that

the scalar field to be visualized is f(x), and that the sample points xi are evenly spaced:

i.e. that f(xi) = f(iT ) for some sampling distance T . For convenience, we use fi to refer

to f(xi).

Typically, the transfer function V is a vector-valued function mapping input isovalues

to output colors and opacity. We write this transfer function as V : IR → (R, G, B, α),

and state the rendering task in terms of visualizing V (f(x)), of which we only know the

values V (fi). A proper coarse visualization of this discrete dataset would involve smoothing

V (f) with a low-pass filter in order to avoid aliasing, then sub-sampling. For level n in the

multi-resolution hierarchy, this can be written as:

Vn+1(f(y)) =

∫ ∞

−∞Vn(f(y − x))w(x)dx (4.1)

where

f(x) =∞∑

k=−∞

f(k)h(x − k) (4.2)

where w is the (continuous) low-pass filter used for computing the (n + 1)st level of the

multi-resolution pyramid and h is a (continuous) interpolation filter used to reconstruct the

underlying function f from its discrete representation. The importance of Equation 4.2 in

volume rendering has been well documented by Wittenbrink et al. [56].

The formulation in Equation 4.1 is not particularly useful, as it requires recomputing


the multi-resolution pyramid after each change of the transfer function V . A more useful

formulation of Equation 4.1 derives from the sifting property of the Dirac delta function

δ [3]:

Vn+1(f(y)) =

∫ ∞

−∞

∫ ∞

−∞Vn(s)δ(s − f(y − x))w(x)dsdx (4.3)

=

∫ ∞

−∞Vn(s)

(∫ ∞

−∞δ(s − f(x))w(y − x)dx

)

ds

=

∫ ∞

−∞Vn(s)

(

∫

f−1(s)w(y − x)dx

)

ds (4.4)

If w is a box filter with support of size N , then∫

f−1(s) w(y − x)dx is equivalent to1

2N

∫

f−1(s)|[y−N,y+N ] dx, i.e. the restriction of the integral to the support of the filter.

In the discrete case with a box filter, this distribution is merely the histogram of a local

neighborhood of f(y) weighted by 1/2N : the histogram therefore also has support of size

2N .

Regardless of the low-pass filter chosen,∫

f−1(s) w(y−x)dx is the weighted distribution of

f in the support of the filter w, and the discrete version is the weighted histogram of f in the

support of w. Thus, for any low-pass filter w, we can guarantee accuracy of low-resolution

images within the limits of reconstruction by storing Hw(s, y) =∫

f−1(s) w(y−x)dx for each

location y, where Hw is the weighted histogram centered at y. Note that all our histograms

are local, i.e. they collect statistics about the underlying function f in the neighborhood of

y. If we were to store these weighted histograms at each multi-resolution node, we would

then compute the value at this node Vn+1(f(y)) in the following way:

Vn+1(f(y)) =

∫ ∞

−∞Vn(s)Hw(s, y)ds (4.5)

This allows us to represent our function in a multi-resolution pyramid with proper anti-

aliasing and without a priori knowledge of the transfer function. For multi-dimensional

transfer functions we would use a multi-dimensional histogram.

While the above equations were derived on the assumption of continuity, we normally

work with discrete representations and discrete histograms. We therefore need to approx-

imate f(s) at a resolution dependent on the transfer function to assure that V (f(s)) is

sampled above the Nyquist frequency. This is still an open problem: no algorithm is known

to determine such a sampling frequency. We leave this problem open, and work with the


same sampling as for the underlying scalar field fi. Given this, and assuming that we have

sampled 8-bit values ranging from 0 to 255, we can rewrite Equation 4.5 as follows:

Vn+1(fi) =255∑

j=0

Vn(j)Hw(j, i) (4.6)

with

Hw(j, i) =N∑

k=−N

δ[j − fi−k]wk (4.7)

where N is the filter support of w and δ[ ] is the discrete form of the delta impulse function.

As noted above, the second term is simply the histogram of f weighted by the filter w.

It is important to point out that we need to low-pass filter the signal fi for supersampling

the function V (f) [56]. However, for sub-sampling V (f) such as is common for multi-

resolution analysis, we need to low-pass filter V .

In practice, since histograms require a lot of storage and computation, using them slows

down rendering, negating much of the benefit from more accurate representation. To balance

these competing demands, we summarize the subvolume, not with a weighted histogram,

but with some compact approximation of the weighted histogram function. In our particular

implementation we use a Gaussian distribution as a first approximation.

We therefore store the mean (µ) and standard deviation (σ) for each voxel in the coarse

levels of detail. To find the color and opacity of a voxel with values (µ, σ), we then perform

the following integration:

Vn+1(fi) =

∫ µ+σ

µ−σ

(

1

σ√

2πe−(s−µ)2/(2σ2)

)

Vn(s)ds (4.8)

Computing this integral explicitly for each voxel is expensive. But since V generally

remains constant during rendering, we can treat the integral as a function of µ and σ and

store it in a two dimensional transfer function lookup table whose indices are the mean

isovalue and the standard deviation. Each time the user changes the transfer function, we

update the lookup table, which takes a few milliseconds. Figure 4.3 shows a sample transfer

function (bottom) and the computed transfer function based on the Gaussian basis function

(top). For the top transfer function, points have been drawn with the color and opacity

computed for the related µ and σ.

It should also be noted that the Gaussian method may fail to approximate the histogram

distributions properly if they have more than one major maximum (peak). This often hap-

pens at the borders of the sampled phenomena inside the volume with the empty space,


Figure 4.3: Bottom: a transfer function with the opacities as the vertical axis (V ). Top: thecomputed Gaussian basis transfer function; The horizontal and vertical axes are respectivelymean (µ) and standard deviation (σ) and each point has been drawn with the color andopacity computed for the related µ and σ.

where a large number of voxels with priori values are included in the filtering. One solu-

tion for this case might be to use more than one Gaussian function for approximating the

distributions when the error of a single Gaussian approximation is not negligible. Although

this thesis does not addresses this approach, we use a straight forward method to solve the

problem of the empty regions, by not considering the voxels with priori values in calculating

µ and σ.

4.1.2 Translucent Isosurfaces

In addition to generating aliasing artifacts, multi-resolution rendering methods lose high-

frequency information, generating progressively blurrier views of the data, which can be

visually disconcerting. In theory, reconstruction methods can recapture the sharp features

if the data is properly sampled. Unfortunately, the very process of preparing the data

for multi-resolution rendering requires the elimination of high-frequency information in the

coarse resolutions.

To preserve at least the appearance of sharp features, we incorporate transparent iso-

surfaces in our multi-resolution renderings. By constructing sharp geometric features, this

approach has the effect of providing approximate high- resolution information without costly

reconstruction filters, improving the visual quality of the image at relatively low cost. Al-

though extracting isosurfaces itself requires interpolation and adds processing time, this can

be minimized by using octree-based methods which render a reduced number of cells de-

pending on the current resolution level. We used the Richtmyer-Meshkov instability dataset


(2048×2048×1920) from Lawrence Livermore National Laboratory which measures entropy

in a shock-driven instability. Figure 4.4 shows images rendered in the 5th level of coarsening

(643) and 6th level of coarsening (323) with and without isosurfaces. As it can be seen, in

the hybrid rendering of the lower level of detail Figure 4.4(d), we can perceive sharp features

that don’t appear in the direct volume rendering of the higher level of detail Figure 4.4(a).

(a) 643 (15 FPS) (b) 643 (8 FPS)

(c) 323 (78 FPS) (d) 323 (29 FPS)

Figure 4.4: Adding isosurfaces to reveal sharp features in the Richtmyer-Meshkov instabilitydataset. Left: splatting. Right: splatting + transparent isosurface.

4.1.3 Combined Isosurface and Volumetric Rendering

For correct composition of direct volume rendering and multiple shaded transparent isosur-

faces, we project the volume in sorted order [18]. We do so by processing the subvolumes in

back-to-front order and traverse the cells in each subvolume in back-to-front order. For each


cell we project the isosurface(s) and the splat positioned in the center of the cell and use the

OpenGL blend function to blend the resulted colors and opacities for the isosurface(s) and

the splat. Based on the projection size of the cell, isosurfaces are extracted and rendered

either as polygons using Marching Cubes [32] or directly as iso-splats [10].

We also store an integer histogram code for each subvolume. The bits of this code

roughly represent the data ranges that are active in the subvolume. During rendering,

we construct a similar integer code from the user specified transfer function and isovalues

representing the data ranges that have a non-zero opacity and will be visible. Before loading

a subvolume into main memory for rendering, we perform a bitwise AND operation on the

subvolume’s histogram code and the visualization code. A zero result means there is no

intersection between the range of data in the subvolume and the visible range of data.

In this way we identify and ignore the subvolumes that do not participate in the visu-

alization, thus we can reduce the data transaction to improve the volume rendering speed.

We may further improve this by implementing the histogram code as a vector representing

the percentage of the data in each range and a similar vector instead of the visualization

code to represent the opacity for each range. By performing a dot product between these

two vectors, we can get an approximation of the opacity of the subvolume and ignore them

if the opacity is less than a threshold value. This does however require more run-time

computations and increases the storage and size of the lookup table.

4.1.4 Progressive Refinement Using Image Caches

The idea of progressive refinement has been broadly used in image order rendering tech-

niques such as raytracing and was introduced by Bishop et. al. [1] as Frameless Rendering.

During Frameless Rendering the frame buffer is only gradually updated with pixels that

are predicted to have changed. However, the complete frame buffer is never updated all at

once. For object order methods such as splatting and 3D texture rendering, a large percent

of subvolumes need to be re-rendered when a higher LOD is reached. Since rendering at

the higher resolutions will not be fast enough, it will not give a true progressive refinement

while maintaining the interactivity.

To solve this problem, we use image caches as we did in Chapter 3. Inspired by the Time-

Space Partitioning Tree [43], for the sub-octrees with a low temporal error, the cached image

was used rather than rendering the whole subtree. We extend this idea to multi-resolution

by storing partial renderings at each node of the tree, progressively replacing them with


higher resolution renderings when there is no user interaction. To priorize the subvolumes

to be refined, we use the same procedure as Li and Shen [29] based on the time budget and

user specified criteria. The final rendering is constructed by blending the cached images in

a back-to-front order.

Capturing partial images from the framebuffer and transferring them from main memory

to the framebuffer to composite the final rendering increases the traffic between the main

memory and graphics hardware. This decreases the rendering speed, therefore, we use the

hardware texture buffer to cache the partial renderings and to composite them later, we draw

them as textures on rectangular polygons thus minimizing the traffic between graphics card

and main memory.

Since the hardware texture is limited, we use a FIFO strategy to keep only the most

recently used images in the graphics buffer and discard or unload the least recently used

ones to main memory. We can also choose to store the partial images of only the higher

levels of the volume, thus minimizing the number of required textures.

4.2 Hybrid Branch-on-Need Octree (HYBON)

As shown in the previous section, we combine Gaussian distribution volume rendering with

transparent isosurfaces to improve image quality at coarse resolutions of a multi-resolution

dataset. In this section we describe a data structure called HYBON that provides multi-

resolution access to isosurfaces and volume rendering samples simultaneously.

4.2.1 Hybrid Octree Representation

Octrees are often used to represent the spatial relationships of volumetric data and to

exploit locality in data, accelerating operations such as accessing neighbors and traversing

the volume in a sorted order for correct composition of translucent voxels and surfaces.

However, octrees are usually used either for individual samples [25] used in volume

rendering or for entire cells used for isosurface extraction [55] in which case adjacent octants

require overlapping sets of samples.

As shown in Figure 4.5, the skeleton of our HYBON octree is a complete Branch-On-

Need octree [55] which recursively subdivides the volume spatially until all subvolumes reach

a predefined minimum size. Each node has an index that encodes the index of the parent

node (in 3D we compute the integer component of the index divided by 8) and the child


000001

000100 000101

000110 000111

011000 011001

011010 011011

011100 011101

011110 011111

......

Figure 4.5: The structure and indexing of HYBON tree (in 2D). An index of a node encodesthe index of the parent node (2D: index / 4) and the child index of the node (2D: index %4) within the parent node.

index of the node (in 3D we compute the remainder of the division of the index by 8) within

the parent node. This index is later used to create a lookup table for the pointers to the data

blocks. The hierarchical structure is constructed similar to [53, 29]. Starting from the leaf

nodes, we apply a low pass filter w as in Section 4.1 (box, tent or cubic) to each level and

downsample the data content of its child nodes, computing mean and standard deviation for

each sub-block. These two values define a Gaussian function as an approximation for the

distribution of the sample values represented by each voxel. For example, for a box filter,

the downsampled values are calculated recursively, according to the following equations:


cell

subvolume

voxel

null

Figure 4.6: Creating cell structures in subvolumes by overlapping border voxels.

µ =8∑

i=1

µi

8(4.9)

σ2 =N∑

i=1

(xi − µ)2

N(4.10)

=8∑

j=i

(µi + σi − µ)2 + (µi − σi − µ)2

2 × 8(4.11)

=1

8

8∑

i=1

(µi − µ)2 + σ2i (4.12)

In addition to the mean and standard deviation, we also precompute the normals in each

octant and quantize them as normalized vectors in polar coordinates.

The octree as described so far captures a multi-resolution version of the samples for vol-

ume rendering. For isosurface extraction, we add a cell-based structure in each subvolume.

As shown in Figure 4.6, to avoid cracks between the surfaces extracted from neighboring

subvolumes at the same level of detail, we copy the data points shared on the boundaries of

adjacent subvolumes.


4.2.2 Hybrid Octree Optimization

For rapid identification of octree cells that need to be rendered at a particular level of detail,

we apply a modified version of the span space scheme [44]. Instead of storing a minimum

and maximum isovalue for the cell, we store a quantized histogram code based on dividing

the isovalue range of the data into 32 equal bins. The isovalue span of a given cell will either

intersect a given bin or not, resulting in 32 single bits which we pack into a 32 bit variable

to compute the histogram code. At render time, we simply compute the bin into which an

isovalue of interest falls and compare the appropriate histogram code with the code for each

cell. Cells for which the corresponding bit is not set are guaranteed not to be needed for

rendering, and are immediately discarded.

Compared with a straightforward span space implementation, this histogram code has

two advantages: first, it replaces two comparisons with a single bitwise comparison, and

second, it permits simultaneous testing for a range of isovalues rather than a single isovalue.

Since we support the extraction of multiple transparent isosurfaces and generate a volume

rendering simultaneously, this significantly reduces the computation needed to determine

whether any subvolume in the octree needs processing.

As an additional optimization, we compress uniform regions of the data by setting a

threshold ε ≥ 0, and replacing any subvolume in the octree whose standard deviation is

less than ε with a single leaf node. Performed recursively through the octree, this has the

effect of compressing any large regions of little isovalue change to single entries in the octree,

again increasing rendering speed. We store the compressed data blocks containing the mean,

standard deviation and discretized normal vector, along with an lookup table that contains

pointers to the data blocks and the histogram code for each block.

Since the same partitioning policy is used in both pre-processing and run time, the

octree structure can be computed in run time for a volume with a certain resolution. In run

time, knowing a voxel’s spatial location and the level-of-detail, we can identify the index of

the block (Figure 4.5) containing that voxel. The lookup function converts the index to a

sequence number (to avoid redundant lookup entries in the table) and the sequence number

is used to get the histogram code and the pointer to the data block (or the offset to the

data block in the data file) from the lookup table. A null pointer shows the block has been

discarded in the preprocessing step. The histogram code is compared with the visualization

code (Section 4.1.3) and the data block is loaded, uncompressed and rendered if required.


4.3 Implementation and Results

We have implemented our HYBON data structure and our algorithms for high-quality multi-

resolution rendering on a 3.06 GHz Intel Xeon PC with 2 GB of main memory and an nVidia

AGP 8x GeForce 6800 graphics card. For evaluation purposes, we used three representative

datasets: a synthetic dataset, a medical dataset, and a fluid dynamics simulation dataset.

The synthetic dataset is a spherical distance function sampled at a resolution of 2563,

while the medical dataset is the visible human male CT dataset with a resolution of 512 ×512× 1878, and the simulation dataset is one frame (frame 250) of the Richtmyer-Meshkov

instability dataset from Lawrence Livermore National Laboratory which has a resolution of

2048 × 2048 × 1920.

Since the goal of our work was to provide higher-quality renderings at coarse resolutions

of a multi-resolution dataset, our first concern was the relative quality of the downsampling

and of the resultant images. We tested this by comparing V ◦ f , the function f with

transfer function V applied to it at coarse levels of resolution using three low-pass filters

for downsampling and under three different downsampling distributions. We used box, tent

(linear) and cubic (spline) low-pass filters for downsampling. The three distributions we

tested were mean (i.e. traditional BON downsampling), Gaussian distribution (mean +

standard deviation), and full filtered histogram as a benchmark to compare the former two

distributions with. Including the full histogram adds a huge overhead of 256 floating point

(or 4 byte integer) numbers per voxel (1KB) and storing the full datasets required much

more memory than our machine had available. Since it was not possible to perform the

tests at full resolution, we instead performed it at maximum resolutions of 643 and lower

for each of the datasets.

We performed two tests with the spherical distance function to illustrate the relationship

between the transfer function and the quality of the downsampling distribution. In the first

test, we applied a high-frequency transfer function which rapidly alternated between red,

green and blue, and in the second, we applied a lower-frequency transfer function which

also alternated between red, green and blue. Figure 4.7 shows these transfer functions for

reference. Each set of tests was repeated for box, tent and cubic filters for downsampling.

We show our rendering results in Figure 4.8 through Figure 4.25. In each of these figures,

the left column is the most accurate distribution: the histogram distribution, while the mid-

dle column is the Gaussian distribution, and the right column the least accurate: the mean


Figure 4.7: A high-frequency transfer function (top) and a low-frequency transfer function(bottom) used for our experiments.

distribution. We show decreasing resolution of representation from top to bottom: the top-

most images are at full resolution (643), and therefore show the function V (f(x)) accurately.

The bottom-most images are at coarsest resolution (83), and show poor approximations of

the function.

In Figure 4.8 through Figure 4.10, the transfer function is high-frequency, and the ob-

tained images show a set of thin concentric spheres of rapidly varying color. At coarse res-

olutions, we therefore expect that the high-frequency transfer function should be blurred,

resulting in a much more uniform color - i.e. grey. As we can see, the filtered histogram,

which retains the maximum amount of information about the function, has the least error

at all resolutions, while the mean-valued distribution has the worst error at all resolutions.

The Gaussian distribution performed relatively well, with much less error than the mean

distribution, but slightly more than the histogram distribution.

In comparison, in Figure 4.17 through Figure 4.19, the transfer function has low-frequency,

and the obtained images show a set of thick concentric spheres of slowly varying color. Even

at coarse resolutions, we expect that these spheres will be distinguishable, and we see that

this is happening for all three distributions from the figure.

These results are consistently reproduced for the other two datasets in Figure 4.11 to


Figure 4.16 (high-frequency transfer function) and Figure 4.20 to Figure 4.25 (low-frequency

transfer function), in which the histogram distribution consistently gives the best results,

the mean distribution the worst, and the Gaussian distribution good but not quite perfect

results.

For an analytic evaluation of the error of our approximation we plotted the root mean

squared error of mean and Gaussian approximations of the histogram distribution as the

benchmark. We computed the difference between the approximated distribution of f for each

voxel and the histogram distribution for the same voxel. As Figure 4.26 shows, the error of

the Gaussian approximation was consistently less than the error of the mean approximation

at each level of resolution, with the error usually diminishing under the Gaussian distribution

as the resolution was coarsened further. This usually occurs because the downsampled voxels

represent progressively more of the original data, with a Gaussian distribution becoming a

better fit to the data as more samples accumulate. Although using different low-pass filters,

effects the visual quality in coarser levels of detail (cubic better than linear, and linear better

that box), however it does not have a major effect on the approximations errors. This is

because the filters are not supposed to necessarily change the histograms to become more

close to Gaussian distributions.

To evaluate the effect of the Gaussian approximation on rendering, we calculated the

root mean squared error in each voxel of the function f after application of the transfer

function V . We computed the RGBA color of each voxel using the Gaussian approxima-

tion and mean approximation and compared it to the color when using the full histogram

distribution for that voxel, averaging the root mean squared errors of red, green, blue and

alpha components. A perceptual color space such as CIE Lab might be a better basis for

an insightful comparison. The results are presented in Figure 4.27 for the high-frequency

transfer function and Figure 4.28 for low-frequency transfer function. Again, as we had

already expected from the rendered figures, the average error of rendering is consistently

less for the Gaussian distribution than the mean distribution.

4.3.1 Memory and storage overhead

It is expected that encoding the standard deviation values for each voxel would add around

25% (1-byte σ added to 1-byte µ + 2-bytes gradient) overhead in terms of memory and

processing power. Surprisingly there was no significant memory or processing overhead,

presumably due to compiler optimization of data structures. Since voxel value (1 byte


mean) and normal (2 bytes) is padded to 4 bytes by the compiler, adding the standard

deviation (1 byte) adds no memory overhead in practice. And, during rendering, voxel color

was determined using a simple lookup table which was computed once for the entire dataset

at negligible cost, even with the Gaussian distribution term. The result of this was that

we did not notice any slowdown due to our algorithm. We did consider packing our data

structures to achieve a storage reduction of 20%, but found that the additional overhead for

runtime unpacking offsetted this, and reverted to an unpacked format.


Box Low-Pass Filter - High-Frequency Transfer Function

Histogram Gaussian Mean

643

323

163

83

Figure 4.8: Effects of downsampling distributions with a box low-pass filter on the sphericaldistance function, rendered using a high-frequency transfer function. Top most images arehighest resolution (643) and quality, bottom images are lowest resolution (83) and quality.For the high-frequency transfer function chosen, downsampling should blur the thin col-ored shells into a grey-like color. The histogram distribution (left), while expensive, doesthis correctly. The mean distribution (right) fares poorly, while the Gaussian distribution(middle) is reasonably effective at low cost.


Tent Low-Pass Filter - High-Frequency Transfer Function


643

323

163

83

Figure 4.9: Effects of downsampling distributions with a tent low-pass filter on the sphericaldistance dataset, rendered using a high-frequency transfer function.


Cubic Low-Pass Filter - High-Frequency Transfer Function


643

323

163

83

Figure 4.10: Effects of downsampling distributions with a cubic low-pass filter on the spher-ical distance dataset, rendered using a high-frequency transfer function.




643

323

163

83

Figure 4.11: Effects of downsampling distributions with a box low-pass filter on the VisibleHuman Male Head dataset, rendered using a high-frequency transfer function.




643

323

163

83

Figure 4.12: Effects of downsampling distributions with a tent low-pass filter on the VisibleHuman Male Head dataset, rendered using a high-frequency transfer function.




643

323

163

83

Figure 4.13: Effects of downsampling distributions with a cubic low-pass filter on the VisibleHuman Male Head dataset, rendered using a high-frequency transfer function.




643

323

163

83

Figure 4.14: Effects of downsampling distributions with a box low-pass filter on theRichtmyer-Meshkov Instability dataset, rendered using a high-frequency transfer function.




643

323

163

83

Figure 4.15: Effects of downsampling distributions with a tent low-pass filter on theRichtmyer-Meshkov Instability dataset, rendered using a high-frequency transfer function.




643

323

163

83

Figure 4.16: Effects of downsampling distributions with a cubic low-pass filter on theRichtmyer-Meshkov Instability dataset, rendered using a high-frequency transfer function.


Box Low-Pass Filter - Low-Frequency Transfer Function


643

323

163

83

Figure 4.17: Effects of downsampling distribution with a box low-pass filter on the sphericaldistance function , rendered using a low-frequency transfer function. Resolution and qualitydecrease from top to bottom. All three distributions perform adequately.


Tent Low-Pass Filter - Low-Frequency Transfer Function


643

323

163

83

Figure 4.18: Effects of downsampling distributions with a tent low-pass filter on the sphericaldistance dataset, rendered using a low-frequency transfer function.


Cubic Low-Pass Filter - Low-Frequency Transfer Function


643

323

163

83

Figure 4.19: Effects of downsampling distributions with a cubic low-pass filter on the spher-ical distance dataset, rendered using a low-frequency transfer function.




643

323

163

83

Figure 4.20: Effects of downsampling distributions with a box low-pass filter on the VisibleHuman Male Head dataset, rendered using a low-frequency transfer function.




643

323

163

83

Figure 4.21: Effects of downsampling distributions with a tent low-pass filter on the VisibleHuman Male Head dataset, rendered using a low-frequency transfer function.




643

323

163

83

Figure 4.22: Effects of downsampling distributions with a cubic low-pass filter on the VisibleHuman Male Head dataset, rendered using a low-frequency transfer function.




643

323

163

83

Figure 4.23: Effects of downsampling distributions with a box low-pass filter on theRichtmyer-Meshkov Instability dataset, rendered using a low-frequency transfer function.




643

323

163

83

Figure 4.24: Effects of downsampling distributions with a tent low-pass filter on theRichtmyer-Meshkov Instability dataset, rendered using a low-frequency transfer function.




643

323

163

83

Figure 4.25: Effects of downsampling distributions with a cubic low-pass filter on theRichtmyer-Meshkov Instability dataset, rendered using a low-frequency transfer function.


0 1 2 3 40

2

4

6

8

10

12

14

16

18

20

Level of Coarsening

App

roxi

mat

ion

Err

or (

%)

Mean − BoxMean − TentMean − CubicGaussian − BoxGaussian − TentGaussian − Cubic

(a) Spherical Distance

0 1 2 3 40

2

4

6

8

10

12

14

16

18

20

Level of Coarsening

App

roxi

mat

ion

Err

or (

%)


(b) Visible Human

0 1 2 3 40

2

4

6

8

10

12

14

16

18

20

Level of Coarsening

App

roxi

mat

ion

Err

or (

%)


(c) Richtmyer-Meshkov Instability

Figure 4.26: Errors of the mean and Gaussian approximations of the histogram for the threetest datasets. At a certain level of detail, for each method, the reported error is the rootmean squared error of the voxels’ approximated histogram in comparison with their actualhistogram, for all voxels in that level of coarsening.


0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

Level of Coarsening

Ren

derin

g E

rror

(%

)



0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

Level of Coarsening

Ren

derin

g E

rror

(%

)


(b) Visible Human

0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

Level of Coarsening

Ren

derin

g E

rror

(%

)



Figure 4.27: Rendering errors of the mean and Gaussian approximations (f) of the histogramfor the three test datasets using a high-frequency transfer function (V ). At a certain levelof detail, for each method, the reported error is the root mean squared error of the voxels’color computed using their approximated histogram in comparison with their color computedusing their actual histogram, for all voxels in that level of coarsening.


0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

Level of Coarsening

Ren

derin

g E

rror

(%

)



0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

Level of Coarsening

Ren

derin

g E

rror

(%

)


(b) Visible Human

0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

Level of Coarsening

Ren

derin

g E

rror

(%

)



Figure 4.28: Rendering errors of the mean and Gaussian approximations (f) of the histogramfor the three test datasets using a medium-frequency transfer function (V ). At a certainlevel of detail, for each method, the reported error is the root mean squared error of thevoxels’ color computed using their approximated histogram in comparison with their colorcomputed using their actual histogram, for all voxels in that level of coarsening.

Chapter 5

Conclusions and Future Work

In this thesis we studied the problems of large scale volumetric data visualization from the

aspect of data dimensionality and data resolution.

From the aspect of data dimensionality we studied the temporal coherence of sequential

time frames and histogram and spatial distribution of data in time-varying datasets and

proposed a new data structure called Differential Time-Histogram Table (DTHT) to take

advantage of these characteristics to efficiently load and render time-varying datasets.

In a preprocessing step, we encode the temporal coherence and histogram distribution

of the data in the DTHT data structure. Later, during the visualization step, we use this

differential information to accelerate updating data by minimizing the amount of data that

needs to be transfered from disk into main memory. To store the data in external storage and

to keep the active data in main memory, we use an octree data structure which accelerates

loading and processing data by exploiting spatial distribution.

To address high resolution data, we proposed a hybrid framework for volumetric multi-

resolution datasets to improve the quality of visualization. Using a Gaussian basis function

to approximate the data distribution at each level of detail, rather than the commonly used

mean approximation, we are able to composite color more accurately and reduce aliasing

artifacts. Our methods allow any low-pass filter to be applied to build the multi-resolution

pyramid without a-priori knowledge of the transfer function. We also use transparent iso-

surfaces to capture high-frequency sharp features and to reduce blurring in coarse resolution

renderings. In addition to the quality improvements we also proposed some techniques to

improve the interactivity of multi-resolution rendering by using a histogram code to bypass

the non-important data and by caching the partial renderings in the texture memory to do

74

CHAPTER 5. CONCLUSIONS AND FUTURE WORK 75

progressive refinement.

5.1 Future Work

The work presented for time-varying datasets was mainly focused on analyzing the statistical

characteristics of large datasets to reduce the data transfer between memory and external

storage during the visualization process. Further performance improvement would be pos-

sible by applying compression schemes for the data in leaf nodes and adding multi-level

information to the sub-levels of the data structure to enable multi-resolution renderings.

Currently the rendering time is dominating the overall time to produce an image. Image

caching, occlusion culling and parallel rendering will further drastically improve the render-

ing performance. Other future work includes applying proper modifications to DTHT in

order to enable the extraction of isosurfaces and to handle multi-modal datasets, investigat-

ing methods to change the error bounds (λ) during run time based on the user control and

the transfer funtion and methods to deal with multi-dimensional transfer functions.

For multi-resolution rendering, future work includes combining the framework with the

DTHT data structure to visualize multi-modal and time-varying datasets. We are also in-

vestigating the use of an adaptive approximation of the histogram distribution with different

function bases at different levels of detail for different datasets. To do so, we suggest to use

the ideas of Drew et al. [11] and use Singular Value Decompositon of a wavelet-compressed

histogram. Finally, we are planning on developing a parallel rendering pipeline to achieve

better performance with large datasets, and a compression scheme to benefit from locality

and spatial coherence in the data blocks to reduce the I/O and storage requirements.

Finally, a major extension to our work will be combining the idea of differential updat-

ing in DTHT with the multi-resolution framework in HYBON to create a multi-resolution

framework for visualization of large scale time-varying volumetric datasets.

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