Approaches

to

Uncertainty

Visualization

Alex

T.

Pang

Craig

M.

Wittenbrink

and

Suresh

K.

Lodha

Computer

Science

Department

University

of

California

Santa

Cruz

CA

USA

Hewlett-Packard

Laboratories

Page

Mill

Road

Palo

Alto

CA

USA

September

Abstract

Visualized

data

often

have

dubious

origins

and

quality

Different

forms

of

uncertainty

and

errors

are

also

introduced

as

the

data

are

derived

transformed

interpolated

and

finally

rendered

In

the

absence

of

integrated

presentation

of

data

and

uncertainty

the

analysis

of

the

visualization

is

incomplete

at

best

and

often

leads

to

inaccurate

or

incorrect

conclusions

This

paper

surveys

techniques

for

presenting

data

together

with

uncertainty

These

uncertainty

visualization

techniques

present

data

in

such

a

manner

that

users

are

made

aware

of

the

locations

and

degree

of

uncertainties

in

their

data

so

as

to

make

more

informed

analyses

and

decisions

The

techniques

include

adding

glyphs

adding

geometry

modifying

geometry

modifying

attributes

animation

sonification

and

psycho-visual

approaches

We

present

our

results

in

uncertainty

visualization

for

environmental

visualization

surface

interpolation

global

illumination

with

radiosity

flow

visualization

and

figure

animation

We

also

present

a

classification

of

the

possibilities

in

uncertainty

visualization

and

locate

our

contributions

within

this

classification

Introduction

With

few

exceptions

most

of

the

visualization

research

has

ignored

or

separated

the

presentation

of

uncertainty

from

data

Part

of

the

reason

is

the

inherent

difficulty

in

defining

characterizing

and

controlling

the

uncertainty

in

the

visualization

process

Another

reason

is

the

lack

of

methods

that

present

uncertainty

and

data

We

have

seen

this

as

an

opportunity

for

research

with

great

potential

in

a

wide

variety

of

applications

Some

examples

are

comparative

visualization

of

experimental

and

simulation

data

quantitative

and

visual

analysis

of

image

compression

algorithms

comparison

of

volume

rendering

algorithms

and

volumetric

data

sets

finite

element

analysis

data

assimilation

ensemble

forecasting

as

well

as

those

presented

in

this

paper

The

common

underlying

problem

in

these

areas

is

visually

mapping

data

and

uncertainty

together

into

a

holistic

view

As

a

possible

solution

one

might

consider

setting

free

parameters

to

uncertainty

values

using

existing

surface

volume

flow

and

multi-dimensional

visualization

methods

-LSB-

-RSB-

In

fact

we

do

start

with

existing

methods

However

even

with

the

simple

task

of

designing

glyphs

or

icons

that

incorporate

uncertainty

information

-LSB-

-RSB-

the

process

is

sometimes

counter-intuitive

For

example

while

a

glyph

may

appear

appropriate

by

itself

the

user

's

perception

of

the

glyph

may

be

different

when

a

group

of

them

is

presented

in

various

scales

and

locations

Thus

while

some

of

the

methods

we

have

examined

are

not

necessarily

new

they

must

be

able

to

render

and

convey

the

data

in

complete

accordance

with

the

facts

This

has

been

recognized

and

is

often

stated

as

a

worthy

goal

in

scientific

visualization

-LRB-

e.g.

in

the

IEEE

Visualization

discussions

on

How

to

Lie

with

Visualization

IEEE

Visualization

panels

and

reports

-LSB-

-RSB-

and

the

NCGIA

initiative

on

Visualization

of

Spatial

Data

Quality

-LSB-

-RSB-

-RRB-

but

it

has

rarely

been

pursued

or

realized

In

our

investigation

of

uncertainty

visualization

our

approach

has

been

to

look

at

the

needs

of

different

application

areas

and

to

develop

methods

to

address

them

We

found

that

in

many

instances

applications

are

orthogonal

to

methods

That

is

a

method

developed

for

an

application

may

be

applicable

in

other

areas

At

the

same

time

an

application

may

provide

ideas

for

a

visualization

method

that

may

not

have

been

apparent

without

the

application

context

A

simple

example

would

be

animation

as

a

means

to

convey

uncertainty

information

and

developing

uncertainty

visualization

methods

for

comparing

modeled

versus

motion-captured

animation

data

Because

of

this

synergy

new

applications

provide

ideas

for

more

methods

The

methods

presented

here

represent

significant

steps

toward

achieving

the

goals

of

uncertainty

visualization

This

paper

is

organized

as

follows

section

defines

uncertainty

visualization

and

identifies

the

different

sources

of

uncertainty

section

classifies

methods

for

uncertainty

visualization

section

presents

our

new

uncertainty

visualization

methods

we

then

conclude

this

paper

with

some

more

ideas

for

applications

and

methods

of

uncertainty

visualization

What

is

Uncertainty

Visualization

Uncertainty

visualization

strives

to

present

data

together

with

auxiliary

uncertainty

information

These

visualizations

present

a

more

complete

and

accurate

rendition

of

data

for

users

to

analyze

The

methods

employed

in

uncertainty

visualization

may

range

from

overloading

of

visual

parameters

such

as

those

commonly

found

in

multivariate

visualization

to

verity

visualization

-LSB-

-RSB-

where

the

display

of

both

data

and

uncertainty

is

inseparable

within

the

same

picture

Applications

which

can

benefit

from

uncertainty

visualization

are

those

where

there

is

a

chance

for

uncertainty

to

be

introduced

in

the

visualization

process

and

where

such

uncertainty

matters

Depending

on

the

intent

or

purpose

of

the

visualization

these

uncertainty

information

may

be

presented

in

a

subdued

manner

to

serve

as

a

subtle

reminder

of

the

presence

of

uncertainty

to

the

users

or

these

uncertainty

information

may

be

highlighted

and

even

exaggerated

to

help

in

data

comparison

tasks

The

ultimate

goal

of

uncertainty

visualization

is

to

provide

users

with

visualizations

that

incorporate

and

reflect

uncertainty

information

to

aid

in

data

analysis

and

decision

making

What

is

Uncertainty

We

define

uncertainty

to

include

statistical

variations

or

spread

errors

and

differences

minimummaximum

range

values

noisy

or

missing

data

This

broad

umbrella

is

intended

to

capture

most

if

not

all

the

possible

types

and

sources

of

uncertainty

in

data

NIST

has

written

a

standards

report

-LSB-

-RSB-

which

identifies

four

ways

of

expressing

uncertainty

For

the

discussion

in

this

paper

we

consider

three

types

of

uncertainty

statistical

-LCB-

either

given

by

the

estimated

mean

and

standard

deviation

which

can

be

used

to

calculate

a

confidence

interval

or

an

actual

distribution

of

the

data

error

-LCB-

a

difference

or

an

absolute

valued

error

among

estimates

of

the

data

or

between

a

known

correct

datum

and

an

estimate

and

range

-LCB-

an

interval

in

which

the

data

must

exist

but

which

can

not

be

quantified

into

either

the

statistical

or

error

definitions

Note

that

the

term

data

quality

has

an

inverse

relationship

with

data

uncertainty

-LSB-

-RSB-

and

hence

can

also

take

advantage

of

the

techniques

presented

in

this

paper

Sources

of

Uncertainty

In

order

to

understand

what

is

overlooked

in

visualization

we

quickly

review

the

sources

of

uncertainty

errors

and

ranges

within

data

Fig.

illustrates

the

three

major

blocks

in

a

visualization

pipeline

leading

to

the

analysis

of

the

visualization

output

It

is

clear

that

different

forms

of

uncertainty

are

introduced

into

the

pipeline

as

data

are

acquired

transformed

and

visualized

UUUCollectDeriveVisualizeDataVisualization

Figure

This

visualization

pipeline

shows

the

introduction

of

data

uncertainty

from

models

and

measurements

derived

uncertainty

from

transformation

processes

and

visualization

uncertainty

from

the

visualization

process

itself

Uncertainty

in

acquisition

Starting

with

the

data

acquisition

stage

one

will

note

that

nearly

all

data

sets

whether

from

instrument

measurements

numerical

models

or

data

entry

have

a

statistical

variation

-LSB-

-RSB-

With

instruments

there

is

an

experimental

variability

whether

the

measurements

are

taken

by

a

machine

or

by

a

scientist

The

more

times

the

measurement

is

taken

the

more

confident

the

measurement

But

there

will

be

a

statistical

variation

in

these

measurements

The

same

is

true

for

data

from

numerical

models

and

human

observations

or

inputs

In

numerical

modeling

the

model

and

its

parameters

have

been

decided

by

a

domain

specialist

and

are

inherently

a

simplification

-LRB-

e.g.

linearization

of

a

nonlinear

system

-RRB-

of

the

system

being

modeled

In

addition

to

model

simplification

and

sensitivity

of

these

models

to

input

parameters

numerical

calculations

performed

on

these

models

also

introduce

errors

due

to

the

choice

of

integration

algorithms

and

the

limited

precision

of

the

computing

machinery

Likewise

there

is

variability

in

human

observations

both

in

terms

of

difference

in

perception

among

individuals

and

also

to

slight

differences

when

asked

to

perform

a

task

repeatedly

Uncertainty

in

transformation

Often

times

raw

data

are

not

rendered

directly

but

are

first

subjected

to

further

transformations

with

or

without

the

knowledge

of

the

person

doing

the

visualization

task

These

data

transformation

operations

may

be

as

simple

as

conversion

from

one

unit

of

measure

to

another

or

may

involve

some

algorithm

to

fuse

one

or

more

types

of

data

together

to

derive

a

new

data

type

Data

transformation

operations

may

occur

as

early

as

the

data

acquisition

stage

or

later

in

the

visualization

stage

Likewise

data

may

be

rescaled

resampled

quantized

etc.

either

prior

to

or

as

part

of

the

visualization

stage

The

key

point

is

that

these

transformations

alter

the

data

from

its

original

form

and

have

the

potential

of

introducing

some

uncertainty

Uncertainty

in

visualization

What

is

more

interesting

and

perhaps

not

self

evident

is

that

uncertainty

is

also

introduced

in

the

visualization

stage

itself

For

instance

in

global

illumination

of

scenes

radiosity

algorithms

use

approximations

for

calculating

form

factors

Some

recent

work

in

this

area

addressed

the

issue

of

controlling

the

errors

-LSB-

-RSB-

As

these

researchers

also

pointed

out

the

rendering

process

introduces

uncertainty

arising

from

the

data

collection

process

algorithmic

errors

and

computational

accuracy

and

precision

Similarly

there

are

different

approaches

to

direct

volume

rendering

of

data

sets

-LSB-

-RSB-

resulting

in

slightly

different

renderings

of

the

same

data

set

The

differences

in

the

resulting

images

may

be

due

to

different

ray

traversal

methods

or

the

different

filter

functions

used

in

splatting

or

they

may

be

due

to

tetrahedralization

or

resampling

processes

or

they

may

be

simply

due

to

the

tradeoffs

between

speed

and

image

quality

Uncertainty

introduced

in

the

visualization

process

is

not

limited

to

radiosity

and

volume

rendering

but

are

also

present

in

more

routine

operations

For

example

the

use

of

interpolation

is

quite

prevalent

in

taking

slices

through

data

sets

in

contouring

as

well

as

isosurface

algorithms

-LSB-

-RSB-

to

name

a

few

Surface

approximation

and

interpolation

is

used

in

dealing

with

scattered

data

sets

-LSB-

-RSB-

Here

a

variety

of

tradeoffs

exist

in

performance

and

accuracy

and

there

is

no

ideal

surface

in

many

cases

because

of

the

many

free

parameters

available

-LSB-

-RSB-

In

many

cases

the

data

that

are

to

be

interpolated

have

numerous

errors

and

may

even

lack

topology

information

-LSB-

HDD

-RSB-

Similar

difficulties

and

range

of

choices

produce

uncertainty

in

flow

visualization

methods

For

example

different

integration

methods

step

sizes

orders

and

seeding

strategies

lead

to

slightly

different

flow

visualization

results

Effects

of

uncertainty

are

more

pronounced

in

the

vicinity

of

or

on

critical

points

in

the

flow

field

These

differences

may

at

times

result

in

drastically

different

flow

visualizations

-LSB-

-RSB-

Animation

allows

visualization

to

include

an

additional

parameter

usually

time

Again

there

are

several

opportunities

for

uncertainty

to

be

introduced

The

process

of

in-betweening

to

fill

in

frames

between

key

frames

is

analogous

to

surface

interpolation

and

though

no

method

is

correct

there

are

many

methods

available

and

all

of

them

will

result

in

slight

variations

Aside

from

differences

arising

from

interpolation

of

positional

information

potentially

more

serious

differences

may

arise

from

interpolation

of

orientation

information

depending

on

whether

quaternions

or

Euler

angles

are

used

While

we

have

tried

to

identify

the

common

and

major

culprits

of

how

uncertainty

are

introduced

in

the

visualization

pipeline

this

is

by

no

means

an

exhaustive

list

Hopefully

this

quick

enumeration

will

draw

the

attention

of

visualization

designers

and

users

to

the

potential

pitfalls

of

blindly

using

visualization

methods

without

fully

understanding

the

limitations

and

assumptions

of

each

method

We

next

turn

our

attention

to

classifying

visualization

approaches

then

uncertainty

visualization

approaches

and

finally

presenting

several

new

methods

for

uncertainty

visualization

Classification

of

Methods

To

classify

uncertainty

visualization

approaches

we

first

consider

more

general

classifications

Keller

and

Keller

-LSB-

-RSB-

classify

visualization

by

using

a

taxonomy

of

visualization

goals

Tufte

-LSB-

-RSB-

classifies

visualizations

by

developing

evaluation

and

analysis

methods

such

as

data-ink

maximization

Carswell

-LSB-

-RSB-

and

Cleveland

-LSB-

-RSB-

use

evaluation

as

a

basis

for

the

theory

of

specifiers

that

fundamental

parameters

length

area

ratios

etc.

describe

and

determine

the

effectiveness

of

visualization

Bergeron

and

Grinstein

-LSB-

-RSB-

introduce

a

classification

that

uses

lattice

arrangements

of

data

Brodlie

-LSB-

BCE

-RSB-

describes

a

classification

based

on

the

dimensionality

of

the

entity

that

is

being

visualized

Application

specific

visualization

classifications

have

been

done

by

Hesselink

et

al.

-LSB-

-RSB-

for

vector

and

tensor

field

visualization

We

create

a

classification

system

that

is

similar

to

the

systems

of

Brodlie

and

Hesselink

et

al.

in

certain

aspects

but

is

extended

to

accommodate

uncertainty

visualization

techniques

Brodlie

classifies

techniques

using

two

characteristics

First

he

uses

the

underlying

field

where

data

may

be

grouped

as

dependent

or

independent

variables

and

classified

as

having

ordinal

or

nominal

values

Second

he

uses

the

view

which

is

an

operation

on

the

underlying

field

to

produce

a

pictorial

representation

For

example

generating

a

set

of

contour

lines

for

a

given

height

field

or

generating

the

surface

representation

of

the

height

field

Hesselink

et

al.

classify

techniques

using

three

characteristics

First

they

use

the

order

of

the

data

scalar

vector

or

tensor

-LRB-

three

data

orders

-RRB-

Second

they

use

the

spatial

domain

of

the

visualization

objects

point

line

surface

or

volume

-LRB-

four

domains

-RRB-

Third

they

introduce

the

information

level

of

the

data

elementary

local

and

global

-LRB-

three

information

levels

-RRB-

However

these

classifications

do

not

account

for

different

types

of

uncertainty

information

or

techniques

In

order

to

incorporate

uncertainty

information

into

visualization

schemes

we

propose

a

classification

with

five

characteristics

value

location

data

extent

visualization

extent

and

axes

mapping

Each

of

these

concepts

are

defined

below

Value

of

datum

and

its

associated

value

uncertainty

-LRB-

scalar

vector

tensor

multivariate

-RRB-

The

data

value

y

may

be

characterized

as

a

scalar

a

vector

a

tensor

or

a

multivariate

variable

A

multivariate

variable

has

several

components

each

of

which

can

be

a

scalar

a

vector

or

a

tensor

In

our

classification

scheme

value

includes

possible

uncertainties

associated

with

each

data

value

For

example

y

-LSB-

ym

-RSB-

can

be

used

to

represent

data

values

that

are

multi-valued

scalars

vectors

or

tensors

Likewise

Uy

-LSB-

uym

-RSB-

can

be

used

to

represent

corresponding

uncertainties

in

data

value

Value

corresponds

to

the

dependent

variables

of

Brodlie

's

underlying

field

and

the

order

of

Hesselink

et

al.

Location

of

datum

and

its

associated

positional

uncertainty

-LRB-

time

etc.

-RRB-

This

characteristic

identifies

the

dimensionality

of

the

space

in

which

the

data

resides

It

specifies

how

many

independent

variables

are

used

to

describe

each

data

value

For

example

D

nD

and

time

In

our

classification

scheme

this

description

is

extended

to

include

positional

uncertainty

For

example

if

an

n-dimensional

datum

has

a

value

y

it

can

be

specified

as

y

f

-LRB-

xn

-RRB-

while

its

corresponding

positional

uncertainty

may

be

specified

by

Ux

-LSB-

uxn

-RSB-

Location

corresponds

to

the

independent

variables

of

Brodlie

's

underlying

field

Extent

of

datum

location

and

value

-LRB-

discrete

or

continuous

-RRB-

The

data

extent

corresponds

to

the

distribution

range

interval

or

period

over

which

data

is

valid

For

example

if

a

highly

fluctuating

variable

such

as

wind

velocity

was

measured

and

averaged

over

time

the

time

window

would

specify

the

time

extent

-LRB-

extent

of

location

-RRB-

the

distribution

of

wind

readings

and

the

range

of

wind

readings

over

this

time

period

would

specify

the

velocity

extent

-LRB-

extent

of

value

-RRB-

On

the

other

hand

extent

may

also

be

used

to

specify

an

interval

of

acceptable

values

-LRB-

e.g.

-RRB-

for

each

datum

That

is

value

extent

of

y

can

be

expressed

as

Ey

-LSB-

eym

-RSB-

where

each

eyi

can

be

an

interval

or

a

distribution

defined

over

the

location

extent

Ex

-LSB-

exn

-RSB-

where

the

location

extent

itself

can

be

a

range

or

distribution

or

some

more

complicated

function

of

the

data

location

Varying

data

extents

are

common

for

sampled

medical

data

in

terms

of

the

sample

spacing

and

common

in

environmental

data

in

terms

of

the

time

averaging

involved

For

our

purposes

we

distinguish

between

two

types

of

data

extents

-LCB-

discrete

and

continuous

A

discrete

extent

implies

that

data

are

valid

at

discrete

domain

values

only

while

a

continuous

extent

implies

data

are

valid

through

a

continuous

domain

An

example

of

a

discrete

data

extent

is

the

population

count

associated

with

U.S.

cities

while

an

example

of

a

continuous

data

extent

is

the

surface

of

the

earth

-LCB-

where

population

can

also

be

represented

Data

extent

corresponds

roughly

to

the

information

level

of

Hesselink

et

al.

Visualization

extent

-LRB-

discrete

or

continuous

-RRB-

The

visualization

primitive

extent

determines

whether

individual

datums

are

indicated

or

whether

a

continuous

range

of

data

are

indicated

The

visualization

extents

are

grouped

into

discrete

-LRB-

includes

points

and

glyphs

-RRB-

and

continuous

-LRB-

includes

curves

surfaces

volumes

-RRB-

Animation

is

orthogonal

to

these

two

That

is

animation

can

be

used

with

both

discrete

or

continuous

visualization

extents

The

choice

of

the

visualization

primitive

extent

is

independent

of

the

actual

underlying

extent

of

the

data

For

example

fitting

a

continuous

line

through

a

plot

of

heights

of

students

in

a

class

is

a

continuous

visualization

extent

of

a

discrete

data

set

The

data

extent

is

independent

of

the

visualization

extent

though

some

mappings

are

more

natural

and

useful

than

others

It

is

important

to

note

that

discrete

visualization

extents

such

as

glyphs

can

be

used

with

continuous

data

and

that

continuous

visualization

extents

such

as

parallel

coordinate

display

-LSB-

-RSB-

can

be

used

with

discrete

data

Visualization

extent

corresponds

with

views

of

Brodlie

and

domain

of

Hesselink

et

al.

Axes

mapping

defines

visualization

mapping

-LRB-

experiential

or

abstract

-RRB-

It

allows

different

variables

or

grouping

of

variables

to

be

mapped

together

or

to

different

axes

For

example

one

can

use

data

values

in

place

of

location

values

for

coordinate

axes

such

as

in

a

scatter

plot

in

order

to

investigate

cause

and

effect

relationships

or

investigate

the

location

using

a

spatial

plot

with

the

dependent

variable

-LRB-

s

-RRB-

being

visualized

humidity

and

temperature

values

can

be

treated

as

vector

components

etc.

Axes

mapping

allows

two

basic

approaches

to

visualization

experiential

rendering

is

to

replicate

the

viewer

's

experience

with

the

visualized

phenomenon

Abstract

rendering

is

to

plot

the

data

in

a

non-experience

based

mapping

which

may

result

in

additional

insight

and

understanding

Both

these

visualization

approaches

may

be

used

for

uncertainty

visualization

to

show

what

the

true

variations

in

the

mapped

axes

are

Essentially

in

multi-valued

data

sets

some

projection

of

the

n-dimensional

data

set

must

be

used

to

produce

a

typically

dimensional

visualization

perhaps

over

time

and

there

is

considerable

flexibility

in

the

mapping

beyond

the

experiential/abstract

demarcation

we

use

in

this

section

Value

Visualization

Extent

discrete

continuous

scalar

glyphs

-LRB-

error

bars

box

plots

pseudo-coloring

difference

images

Tufte

quartile

plots

-RRB-

side-by-side

contour

lines

blinking

multivariate

Chernoff

faces

scatter

plots

side-by-side

difference

images

vector

glyphs

modified

streamlines/ribbons/tubes

-LRB-

modified

tensor

probes

-RRB-

modified

LIC

tensor

glyphs

modified

hyperstreamlines

-LRB-

modified

tensor

probes

-RRB-

Table

Existing

and

likely

uncertainty

visualization

techniques

Table

uses

two

characteristics

-LRB-

value

and

visualization

extent

-RRB-

to

classify

existing

uncertainty

visualization

methods

The

other

characteristics

explain

and

demarcate

the

space

in

which

visualization

methods

can

be

classified

The

methods

provided

in

the

table

can

be

shared

across

the

other

characteristics

where

deemed

appropriate

A

complete

table

for

all

the

five

characteristics

is

hard

to

display

in

two

dimensions

Therefore

for

sake

of

simplicity

of

presentation

we

have

chosen

the

two

most

representative

characteristics

that

separate

methods

The

listing

of

all

of

the

characteristics

is

the

complete

classification

and

must

be

used

to

quantify

a

given

uncertainty

visualization

technique

Table

's

upper

left

cell

contains

the

most

thoroughly

researched

statistical

visualization

work

the

visualization

of

a

scalar

value

and

its

uncertainty

such

as

the

median

quartiles

and

outliers

of

a

statistically

evaluated

variable

If

visualized

with

a

discrete

visualization

extent

the

variety

of

statistical

plotting

tools

is

impressive

including

glyphs

with

various

attributes

set

to

denote

values

where

attributes

are

the

shape

color

etc.

of

a

graphical

specifier

We

discuss

attributes

in

more

details

in

Section

Error

bars

can

show

the

range

of

the

data

and

represent

a

glyph

to

show

uncertainty

that

works

well

with

the

discrete

visualization

extent

A

discrete

visualization

extent

can

be

formed

either

directly

from

discrete

data

or

densely

sampled

data

that

is

sub-sampled

or

continuous

phenomena/data

that

is

sub-sampled

The

mapping

of

data

makes

it

possible

to

visualize

a

given

data

set

with

many

techniques

converting

discrete

data

to

continuous

representation

and

vice

versa

Scalar

data

may

also

be

visualized

with

methods

that

are

continuous

visualization

extents

meaning

they

impart

the

effect

that

the

phenomena

is

not

discrete

For

example

a

discrete

representation

of

data

in

a

plot

is

to

plot

with

points

while

a

continuous

representation

of

those

same

points

is

to

plot

with

a

smoothly

varying

line

that

passes

through

the

points

For

this

reason

the

scalar

row

continuous

visualization

extent

cell

in

the

upper

right

of

the

table

shows

attributes

of

lines

contours

etc.

to

map

the

uncertainty

to

color

or

line

style

Note

that

the

dimension

of

the

data

domain

has

not

been

chosen

for

the

table

because

higher

dimensional

domains

have

nice

analogues

such

as

the

transition

from

lines

to

contours

to

isosurfaces

for

and

The

rest

of

the

table

shows

those

methods

that

deal

with

higher

order

data

including

multivariate

vector

and

tensors

A

multivariate

variable

is

simply

a

number

n

of

scalar

values

that

are

to

be

simultaneously

visualized

and

encompasses

the

traditional

multiple

valued

visualization

methods

in

addition

to

visualizing

their

uncertainty

Complex

-LRB-

real

plus

imaginary

-RRB-

data

are

typically

handled

with

this

approach

as

well

The

vector

is

distinguished

from

the

multivariate

because

it

is

less

general

and

there

are

data

sets

that

are

directly

vector

data

sets

Flow

visualization

wind

currents

etc.

are

of

this

type

and

the

assumptions

in

the

visual

mappings

and

the

techniques

used

are

different

than

for

the

multivariate

Tensor

uncertainty

visualization

takes

the

meaning

of

the

multiple

values

and

their

uncertainty

to

be

tensors

Because

the

most

general

method

is

the

multivariate

vector

and

tensor

data

sets

can

use

the

multivariate

methods

The

reverse

is

also

true

but

the

interpretation

may

not

be

correct

as

in

visualizing

financial

data

as

vectors

There

are

not

many

existing

techniques

for

higher

order

data

with

continuous

visualization

extents

We

now

describe

some

existing

uncertainty

visualization

methods

and

how

they

fit

into

our

classification

Many

researchers

are

fully

aware

of

the

uncertainty

usually

in

the

form

of

errors

in

their

data

These

are

usually

displayed

using

some

straightforward

method

such

as

side

by

side

comparison

or

differencing

For

example

Lischinski

et

al.

-LSB-

-RSB-

used

line

plots

to

render

uncertainty

Greene

et

al.

-LSB-

-RSB-

used

difference

images

and

Arvo

et

al.

-LSB-

-RSB-

used

norms

for

the

entire

image

In

surface

interpolation

Hagen

et

al.

-LSB-

HHS

-RSB-

used

pseudo-coloring

of

the

surface

curvature

and

other

properties

of

the

surface

This

is

an

example

of

scalar

value

-LRB-

first

row

-RRB-

continuous

extent

visualization

-LRB-

second

column

-RRB-

in

Table

In

geographic

and

information

systems

-LRB-

GIS

-RRB-

researchers

are

aware

of

the

statistical

variation

and

have

employed

a

range

of

techniques

to

display

this

information

For

example

aside

from

pseudo-coloring

areas

of

maps

to

represent

value

uncertainty

they

may

also

use

contour

lines

to

indicate

regions

of

similar

confidence

levels

For

cartographers

the

contours

may

be

for

areas

of

similar

spatial

distortions

from

projections

Fisher

-LSB-

-RSB-

proposed

animation

techniques

such

as

blinking

data

points

to

represent

data

uncertainty

Gershon

-LSB-

-RSB-

proposed

animation

for

the

display

of

uncertainty

in

fuzzily

classified

regions

These

are

examples

of

scalar

value

-LRB-

first

row

-RRB-

continuous

visualization

extent

-LRB-

second

column

-RRB-

We

have

not

seen

any

visualization

methods

designed

for

presenting

uncertainty

information

for

vector

or

tensor

data

However

some

existing

vector

and

tensor

visualization

methods

can

be

modified

to

include

uncertainty

information

For

example

tensor

probes

-LSB-

-RSB-

with

discrete

visualization

extents

can

be

easily

modified

to

incorporate

uncertainty

information

For

continuous

visualization

extents

line

integral

convolution

-LRB-

LIC

-RRB-

-LSB-

-RSB-

can

use

the

uncertainty

information

to

modulate

the

texture

Likewise

adding

more

variables

into

existing

flow

visualization

methods

such

as

streamlines

result

in

streamballs

-LSB-

BHR

-RSB-

and

to

hyperstreamlines

-LSB-

-RSB-

as

well

The

taxonomy

of

existing

methods

of

displaying

uncertainty

are

summarized

in

Table

Our

classification

of

uncertainty

visualization

techniques

demonstrates

that

only

the

scalar

value

discrete

visualization

extent

or

upper

left

entry

in

Table

has

been

adequately

explored

where

the

uncertainty

may

be

shown

with

economy

using

Tukey

's

box

plots

-LSB-

-RSB-

Tufte

's

quartile

plots

-LSB-

-RSB-

and/or

Cleveland

's

framed

rectangles

-LSB-

-RSB-

What

we

describe

in

the

following

section

are

new

methods

for

displaying

higher

dimensional

uncertainty

-LRB-

e.g.

a

vector

of

uncertainty

parameters

-RRB-

in

surfaces

and

in

animation

applications

The

primary

methods

we

discuss

are

adding

glyphs

-LRB-

Section

-RRB-

adding

geometry

-LRB-

Section

-RRB-

geometry

modification

-LRB-

Section

-RRB-

attribute

modification

-LRB-

Section

-RRB-

animation

-LRB-

Section

-RRB-

sonification

-LRB-

Section

-RRB-

and

psycho-visual

approaches

-LRB-

Section

-RRB-

Technique

Value

Location

Data

extent

Vis

extent

Axes

mapping

side-by-side

any

or

any

any

experiential

difference

image

scalar

or

time

any

continuous

experiential

pseudo-color

scalar

or

any

any

experiential

contour

lines

scalar

any

continuous

experiential

blinking

scalar

any

any

experiential

scatter

plot

multivariate

nD

any

discrete

abstract

Table

Taxonomy

of

some

existing

uncertainty

visualization

methods

as

used

in

different

appli

cations

Uncertainty

Visualization

Methods

We

have

developed

a

variety

of

new

uncertainty

visualization

methods

These

are

organized

into

a

table

showing

general

approach

versus

application

domain

-LRB-

Table

-RRB-

as

well

as

their

classification

according

to

the

five

characteristics

listed

in

the

previous

section

-LRB-

Table

-RRB-

Entries

in

Table

indicate

methods

that

can

be

used

for

an

application

The

presentation

below

is

organized

by

general

approach

with

detailed

description

of

how

a

particular

method

and

its

relationship

to

our

classification

scheme

in

Table

But

first

we

describe

the

four

different

applications

that

we

have

investigated

focusing

on

their

relevance

to

uncertainty

visualization

and

the

type

of

uncertainty

in

each

case

Approach

Application

Radiosity

Animation

Interpolation

Flow

Add

glyphs

spherical

ladders

uncertainty

ellipsoidal

Add

geometry

snow

angels

fat

surfaces

bumps

ribbons

Modify

geometry

affine

transform

IFS

displacement

Modify

attributes

reflectivity

textures

bump

mapping

pseudo-color

Animation

magnitude

frequency

oscillate

oscillate

batons

ranking

Sonification

pitch

instrument

duration

Psycho-Visual

left/right

subliminal

Table

Some

uncertainty

visualization

methods

for

a

range

of

applications

Radiosity

There

are

several

motivations

for

comparing

results

from

different

radiosity

algorithms

The

main

motivation

is

that

like

direct

volume

rendering

radiosity

calculations

can

be

expensive

and

hence

numerous

works

in

this

area

have

focused

on

different

approximations

to

speed

up

the

calculations

Competing

results

are

often

displayed

side

by

side

and

the

burden

is

placed

on

the

user

to

identify

the

regions

and

extent

of

difference

Since

rendering

is

an

integral

part

of

the

visualization

process

we

need

to

understand

how

these

different

forms

of

approximation

influence

the

final

picture

Another

motivation

for

this

application

is

its

similarity

to

other

applications

That

is

radiosity

is

concerned

with

calculation

of

surface

radiosities

of

a

static

scene

The

ability

to

highlight

differences

between

competing

methods

on

these

surfaces

can

benefit

other

applications

such

as

finite

element

analyses

of

structural

components

Animation

In

this

particular

application

we

are

interested

in

comparing

two

sets

of

animation

data

modeled

human

motion

versus

motion-captured

human

motion

Positional

data

are

available

for

key

points

of

the

body

such

as

head

hips

extremities

etc.

However

the

data

could

just

as

well

be

from

other

sources

such

as

comparing

animations

using

different

interpolation

methods

or

from

other

applications

such

as

different

paths

of

streamlines

in

flow

visualization

From

a

modeling

point

of

view

it

is

important

to

understand

how

the

modeled

motions

differ

from

actual

human

motions

-LRB-

or

how

one

streamline

is

different

from

another

streamline

etc.

-RRB-

in

order

to

improve

the

animation

model

Interpolation

In

dealing

with

environmental

data

sets

one

often

encounters

sparse

and

scattered

data

The

dispersion

and

sparsity

of

the

data

points

are

often

attributed

to

the

accessibility

of

the

sites

and

the

cost

of

the

instruments

that

collect

them

In

an

ideal

situation

the

field

would

be

populated

with

enough

instrumentation

to

sample

the

field

at

sufficient

resolution

to

capture

the

phenomenon

of

interest

In

reality

we

have

to

approximate

and/or

interpolate

the

field

based

on

the

limited

number

of

data

points

Because

the

choice

of

interpolation

algorithm

can

substantially

influence

the

appearance

of

the

interpolated

field

it

is

important

to

understand

how

the

distribution

of

data

location

as

well

as

data

values

influence

the

performance

of

different

algorithms

Flow

visualization

There

are

several

flow

visualization

software

packages

where

users

are

given

the

option

of

selecting

different

integration

methods

integration

time

interval

integration

directions

etc.

This

flexibility

may

actually

be

detrimental

if

the

user

does

not

fully

understand

the

impact

of

these

choices

on

the

resulting

flow

visualization

Table

shows

the

four

application

areas

discussed

as

columns

and

a

classification

of

uncertainty

visualization

methods

along

rows

We

have

split

techniques

into

seven

fundamental

areas

which

provide

a

separation

of

means

by

which

information

are

encoded

into

a

visualization

The

following

sections

describe

each

area

and

relate

them

to

the

applications

Add

Glyphs

A

glyph

is

a

geometrically

plotted

specifier

that

encodes

data

values

Glyphs

encode

information

through

their

shape

and/or

color

An

example

would

be

arrow

glyphs

which

are

used

to

visualize

magnitude

and

directional

information

in

vector

fields

Glyphs

can

also

be

used

to

visualize

uncertainty

in

a

variety

of

ways

We

have

investigated

glyphs

in

the

following

applications

radiosity

-LSB-

-RSB-

vector

fields

-LSB-

WSF

-RSB-

surface

interpolation

-LSB-

-RSB-

flow

visualization

-LSB-

-RSB-

and

key-framed

animation

-LSB-

-RSB-

The

primary

issues

in

using

glyphs

for

visualizing

uncertainty

are

the

sampling

frequency/location

and

the

placement

orientation

-LRB-

e.g.

-LSB-

-RSB-

-RRB-

For

most

of

our

applications

to

date

we

have

been

able

to

specify

the

orientation

through

the

nature

of

the

data

being

presented

For

example

in

interpolation

of

a

height

field

there

is

going

to

be

uncertainty

in

the

heights

and

the

glyphs

are

therefore

oriented

vertically

to

indicate

this

height

variation

Glyphs

in

radiosity

have

included

spheres

whose

radii

are

scaled

to

the

difference

in

different

radiosity

solutions

For

multivariate

uncertainty

information

ellipsoidal

or

more

complicated

Technique

Value

Location

Data

extent

Vis

extent

Axes

mapping

material

prop

scalar

discrete

surface

experiential

texture

mapping

scalar

discrete

surface

volume

experiential

spherical

scalar

discrete

point

experiential

affine

transform

scalar

discrete

surface

experiential

mag/freq

scalar

discrete

surface

and

anim

experiential

left/right

scalar

discrete

surface

experiential

bump

mapping

multivariate

and

time

discrete

surface

experiential

snow

angels

vector

and

time

discrete

surface

experiential

oscillation

multivariate

and

time

discrete

point

curve

anim

experiential

uncertainty

glyph

vector

or

continuous

point

experiential

fat

surfaces

scalar

or

continuous

surface

volume

experiential

IFS

multivariate

or

discrete

continuous

experiential

displacement

scalar

or

continuous

surface

experiential

instrument

scalar

discrete

point

abstract

subliminal

nominal

or

discrete

surface

experiential

ellipsoidal

multivariate

and

time

continuous

point

experiential

ribbons

multivariate

and

time

continuous

surface

experiential

batons

multivariate

and

time

continuous

point

curve

anim

experiential

ranking

multivariate

and

time

continuous

curve

and

anim

experiential

duration

scalar

or

time

discrete

curve

or

anim

experiential

Table

New

uncertainty

visualizations

methods

as

applied

to

our

applications

glyphs

can

be

used

together

with

color

Fig.

shows

an

example

where

the

differences

from

two

radiosity

solutions

are

visualized

with

spherical

glyphs

Glyphs

in

vector

fields

have

been

custom

designed

to

encode

the

magnitude

and

bearing

uncertainties

We

have

worked

with

maximum

likelihood

output

from

ocean

radars

wind

sensing

radars

and

interpolated

winds

from

scattered

stations

Fig.

shows

how

uncertainty

vector

glyphs

are

used

to

visualize

vector

fields

with

derived

magnitude

and

directional

uncertainty

information

Glyphs

in

surface

interpolation

have

been

used

to

compare

interpolants

such

as

bilinear

multi-quadric

and

Shepard

's

The

length

of

the

glyph

is

scaled

to

the

difference

between

the

interpolants

Fig.

shows

how

such

line

glyphs

do

pairwise

comparisons

of

the

interpolated

surfaces

Glyphs

in

flow

visualization

have

been

used

to

show

the

deviation

of

flow

solvers

Fig.

shows

the

use

of

glyphs

to

compare

Eulerian

integration

versus

Runge-Kutta

where

the

length

of

the

glyph

is

the

difference

between

two

solvers

Glyphs

in

key-framed

animation

have

been

used

in

a

similar

fashion

to

the

flow

visualization

They

are

used

to

illustrate

path

differences

arising

from

using

different

interpolation

methods

The

variety

of

applications

that

use

glyphs

for

uncertainty

visualization

as

well

as

the

variety

of

encodings

possible

with

glyphs

make

this

method

highly

successful

and

extensible

to

many

other

applications

Add

Geometry

Uncertainty

can

be

visualized

in

certain

applications

by

adding

geometry

to

rendered

scenes

While

glyphs

do

add

geometry

they

are

placed

at

discrete

locations

Adding

geometry

is

used

to

denote

a

more

continuous

representation

of

data

Techniques

include

contour

lines

isosurfaces

streamlines

and

swept

surfaces

and

volumes

Some

extensions

of

these

techniques

for

uncertainty

visualization

are

described

below

Figure

Spherical

glyphs

scaled

to

radiosity

differences

Figure

Uncertainty

vector

glyphs

over

Monterey

Bay

Figure

Line

glyphs

show

difference

be

tween

bilinear

and

multi-quadric

interpo

lated

surfaces

Figure

Line

glyphs

tile

particle

positions

along

streamlines

computed

by

two

different

integration

methods

To

show

the

differences

between

two

methods

of

scattered

data

interpolation

a

fat

surface

can

be

created

by

sweeping

one

surface

to

the

other

This

is

illustrated

in

Fig.

where

a

cross

sectional

cut

reveals

varying

thickness

across

the

surface

Figure

Fat

surfaces

showing

the

difference

between

the

bilinear

and

multi-quadric

in

terpolations

The

cross

sectional

slice

shows

how

fat

surfaces

can

be

used

to

exaggerate

the

difference

Figure

Surface

patches

are

split

into

four

pieces

and

displaced

vertically

according

to

radiosity

differences

Fig.

illustrates

a

variation

where

geometry

is

added

on

a

per

patch

basis

instead

of

over

the

entire

region

in

order

to

show

the

radiosity

differences

on

each

patch

The

patches

are

divided

into

smaller

patches

and

are

displaced

vertically

at

their

centers

by

an

amount

proportional

to

the

difference

Rather

than

displacing

them

up

or

down

according

to

the

signed

difference

a

red/green

coloring

scheme

is

used

instead

Together

with

a

movable

light

source

-LRB-

black

sphere

-RRB-

the

user

can

examine

different

areas

in

more

detail

For

flow

visualization

applications

geometry

is

added

in

the

form

of

uncertainty

ribbons

representing

the

extent

between

two

streamlines

calculated

from

two

different

integration

methods

See

Fig.

When

comparing

two

sets

of

character

animation

line

segments

are

used

to

connect

key

points

of

the

body

The

area

bordered

by

corresponding

line

segments

of

the

simulated

data

and

the

motion-captured

data

are

then

used

to

sweep

out

an

area

on

the

same

plane

as

the

animation

data

This

is

illustrated

in

Fig.

and

is

similar

to

having

each

stick

figure

do

a

snow

angel

one

after

the

other

over

the

same

location

Regions

which

have

less

depression

indicate

more

variance

Modify

Geometry

Uncertainty

can

be

visualized

by

modifying

geometry

in

a

scene

Geometry

may

be

translated

scaled

rotated

or

generally

warped

or

distorted

They

may

also

be

displaced

subdivided

or

refined

Simple

affine

transformations

such

as

translation

and

rotation

have

been

used

to

indicate

uncertainty

in

the

data

For

the

radiosity

application

the

surface

patches

are

either

translated

in

or

out

according

to

the

radiosity

difference

calculated

by

two

different

form

factor

methods

-LRB-

Fig.

-RRB-

or

rotated

up

to

a

maximum

of

degrees

-LRB-

Fig.

-RRB-

Figure

Uncertainty

ribbons

are

used

to

show

the

deviations

and

twisting

between

two

streamlines

generated

by

different

inte

gration

methods

Figure

Snow

angels

are

marks

left

on

the

ground

by

sweeping

out

limb

segments

The

amount

of

depression

which

is

mapped

to

gray

level

above

corresponds

to

the

similar

ity

in

the

two

animation

data

sets

Figure

Surfaces

patches

are

translated

in

or

out

of

their

original

positions

to

highlight

differences

Figure

Surfaces

patches

are

rotated

in

stead

of

translated

giving

a

similar

effect

Fractal

interpolation

using

iterated

function

systems

-LRB-

IFS

-RRB-

is

an

additional

method

we

have

used

to

impart

uncertainty

into

the

visualization

The

interpolation

method

used

in

the

reconstruction

of

sampled

data

sets

during

rendering

impacts

the

interpretation

of

the

data

We

have

found

that

smooth

interpolations

sometimes

impart

the

message

that

the

data

itself

represents

a

smooth

phenomenon

or

that

there

was

a

higher

density

of

data

collected

than

indicated

We

developed

interpolation

methods

to

demonstrate

uncertainty

through

the

roughness

of

the

surface

or

the

variation

in

the

volume

-LSB-

-RSB-

Fig

shows

a

fractal

interpolation

of

scalar

samples

Similar

to

Fig.

Fig.

produces

a

bumpy

looking

surface

This

is

achieved

by

displacing

the

surface

up

or

down

according

to

the

scaled

difference

between

different

surface

reconstruction

methods

The

location

and

frequency

of

the

bumps

give

an

indication

of

the

location

and

magnitude

of

deviation

between

the

two

methods

of

interpolation

Figure

IFS

surface

interpolation

of

scat

tered

data

Figure

Bumpy

looking

surface

created

by

simple

facet

displacement

Modify

Attributes

Uncertainty

can

be

visualized

by

modifying

attributes

of

geometry

in

the

rendered

scene

Attributes

include

the

bidirectional

reflectance

distribution

function

-LRB-

BRDF

-RRB-

and

other

means

of

controlling

the

shading

such

as

pseudo-coloring

The

control

of

the

shading

and

coloring

provides

several

parameters

that

can

be

mapped

to

uncertainty

The

simplest

is

to

use

a

color

lookup

table

approach

where

a

color

palette

is

used

to

map

uncertainty

values

to

different

colors

on

the

visualization

primitives

This

approach

can

be

used

to

pseudo-color

surfaces

streamlines

glyphs

and

animation

characters

Additional

parameter

control

is

possible

through

variables

in

the

shading

process

Examples

include

mapping

different

reflectivity

coefficients

such

as

specular

and

diffuse

to

uncertainty

values

in

order

to

alter

the

appearance

of

the

visualization

primitive

Another

example

would

be

manipulation

of

surface

normals

similar

to

bump

mapping

and

in

combination

with

lighting

controls

to

provide

indications

of

uncertainty

-LSB-

-RSB-

Below

are

some

examples

that

illustrate

how

modifying

attributes

can

be

used

to

incorporate

uncertainty

For

comparing

the

difference

between

the

radiosities

on

a

surface

resulting

from

two

different

methods

of

form

factor

calculations

material

properties

of

the

polygon

can

be

altered

to

make

it

either

more

or

less

diffuse

or

specular

according

to

the

difference

on

the

polygon

For

example

differences

can

be

scaled

to

range

from

to

and

then

mapped

as

diffuse

and

specular

coefficients

such

that

the

polygon

with

the

least

difference

is

assigned

diffuse

and

specular

coefficients

of

and

the

one

with

the

largest

difference

is

assigned

coefficients

of

Fig.

illustrates

how

mapping

differences

to

diffuse

coefficients

will

make

those

surface

patches

with

higher

differences

appear

brighter

Fig.

maps

differences

to

specular

coefficients

Note

that

the

surfaces

are

rendered

in

connection

with

a

movable

light

source

If

the

light

source

is

tied

to

the

camera

or

eye

position

then

altering

specular

coefficients

allows

one

to

focus

on

planes

parallel

to

the

viewing

plane

Altering

the

specular

coefficients

is

a

better

tool

if

the

user

is

interested

in

viewing

a

particular

area

but

altering

the

diffuse

coefficients

is

better

for

viewing

the

error

associated

with

the

entire

image

Figure

Altering

diffuse

coefficients

ac

cording

to

difference

Figure

Altering

specular

coefficients

ac

cording

to

difference

Aside

from

the

use

of

shading

lighting

and

coloring

parameters

to

alter

the

attributes

and

hence

the

appearance

of

visualization

primitives

for

uncertainty

visualization

embellishments

such

as

textures

can

also

be

used

to

combine

uncertainty

information

in

visualization

products

Using

the

same

data

set

Figs.

and

illustrate

how

the

differences

can

be

mapped

to

and

textures

respectively

Each

surface

patch

in

the

scene

is

texture

mapped

with

either

a

circular

pattern

or

a

conical

shape

Where

the

differences

are

smaller

several

smaller

circles

-LRB-

or

solid

textures

-RRB-

are

mapped

and

where

the

differences

are

larger

larger

circles

or

portions

of

circles

-LRB-

or

solid

textures

-RRB-

appear

Figure

Radiosity

differences

mapped

to

circular

textures

Figure

Radiosity

differences

mapped

to

solid

textures

For

comparing

simulated

and

real

animations

a

gridded

mesh

is

placed

over

the

animation

data

A

normal

coming

out

of

the

mesh

plane

at

each

grid

point

is

then

perturbed

based

on

its

distance

from

key

points

and

their

positional

errors

Using

Euclidean

distances

the

positional

errors

are

calculated

and

used

to

perturb

the

mesh

normals

Fig.

illustrates

this

technique

For

interpolation

of

vector

data

with

magnitude

and

directional

uncertainties

Fig.

uses

pseudo-coloring

to

map

the

magnitude

uncertainties

to

arrow

glyph

colors

and

angular

uncertainty

to

the

background

field

Figure

Vertex

normals

of

mesh

points

are

bump

mapped

in

the

direction

of

local

sampled

error

Figure

Ocean

currents

in

are

shown

with

arrow

glyphs

whose

colors

are

mapped

to

the

magnitude

uncertainty

Background

field

in

dicates

angular

uncertainty

Animation

Animation

as

a

general

approach

for

visualizing

uncertainty

is

applicable

to

most

applications

including

comparison

of

animation

data

and

techniques

Uncertainty

information

can

be

visualized

by

mapping

them

to

animation

parameters

such

as

speed

or

duration

motion

blur

range

or

extent

of

motion

We

describe

how

animation

is

used

for

uncertainty

visualization

in

the

context

of

our

application

domains

For

comparing

surface

radiosities

animation

is

used

together

with

geometry

modification

-LRB-

translation

and

rotation

-RRB-

There

are

two

possible

mappings

The

first

-LRB-

amplitude

mapping

-RRB-

is

to

translate

or

rotate

the

surface

patches

proportional

to

the

differences

on

each

patch

All

the

patches

are

then

animated

in

synchrony

from

their

original

position

-LRB-

and

orientation

-RRB-

to

their

maximum

translated

position

-LRB-

and

rotated

orientation

-RRB-

This

gives

an

undulating

motion

where

the

extent

of

motion

or

rotation

is

proportional

to

uncertainty

The

second

mapping

-LRB-

frequency

mapping

-RRB-

is

to

assign

a

fixed

translation

or

rotation

amount

to

all

surface

patches

These

are

then

animated

in

varying

frequencies

proportional

to

the

differences

on

each

patch

This

gives

a

more

confusing

animation

but

the

viewer

's

attention

is

more

easily

drawn

to

fast

moving

patches

Depth

perception

using

both

methods

of

animating

surface

patches

in

are

enhanced

with

the

aid

of

stereo

glasses

-LRB-

e.g.

Crystal

Eyes

-RRB-

Several

animation

techniques

can

also

be

used

for

comparing

animation

data

Examples

that

we

have

used

to

indicate

uncertainty

include

random

motion

perturbation

motion

blurring

and

particle

systems

Fig.

shows

particles

being

generated

traced

and

aged

from

key

points

in

the

human

motion

data

Initially

a

set

of

particles

are

randomly

distributed

around

the

key

points

The

subsequent

positions

velocities

and

accelerations

were

calculated

by

gravitationally

attracting

them

towards

their

matching

key

points

Fig.

shows

how

motion

blurring

can

be

used

to

indicate

the

range

of

motion

paths

between

two

path

interpolation

methods

-LRB-

linear

and

cubic

-RRB-

over

a

M-shaped

set

of

key

points

Figure

Particles

are

used

to

trace

out

the

path

of

each

joint

Figure

Animation

with

motion

blurring

to

indicate

uncertainty

For

surface

interpolation

animation

is

combined

with

other

methods

such

as

fat

surfaces

and

glyphs

Specifically

fat

surfaces

are

made

to

oscillate

between

their

extents

The

animation

gives

an

impression

of

a

thin

surface

growing

thicker

and

then

thinner

Regions

where

the

surface

appears

to

grow

thick

very

rapidly

catch

the

user

's

attention

and

are

indicative

of

places

of

greater

differences

Alternatively

glyphs

can

be

added

and

removed

randomly

to

give

a

lively

presentation

of

surface

differences

Areas

where

the

differences

are

larger

will

have

proportionately

more

activity

than

areas

with

smaller

differences

For

flow

visualization

we

introduce

two

ways

of

using

animation

to

present

differences

in

streamlines

Glyphs

along

the

streamlines

can

be

animated

as

the

particle

is

advected

along

its

path

An

even

more

interesting

animation

is

to

rank

or

order

the

animation

according

to

some

user

specified

criteria

such

as

highest

to

lowest

uncertainty

By

presenting

the

difference

information

in

this

manner

the

user

is

immediately

alerted

to

the

areas

of

high

inaccuracy

This

technique

also

identifies

areas

of

equal

uncertainty

that

are

spatially

apart

quite

well

by

presenting

them

immediately

one

after

one

another

Fig

is

a

snapshot

using

such

an

animation

method

Animation

can

also

be

used

to

present

the

uncertainty

information

from

the

viewpoint

of

a

particle

traveling

along

a

streamline

Actually

the

viewpoint

is

taken

from

the

midpoint

between

two

streamlines

To

understand

this

imagine

yourself

being

a

particle

traveling

down

the

midpoint

between

two

streamlines

As

you

go

down

the

path

the

extremities

-LRB-

from

the

two

calculated

streamlines

-RRB-

will

change

in

length

and

orientation

There

are

at

least

two

alternative

coordinate

frames

to

use

One

is

to

move

and

orient

the

local

particle

coordinate

frame

along

the

path

trajectory

The

other

alternative

is

to

simply

translate

the

local

particle

coordinate

frame

within

the

world

coordinate

frame

That

is

the

frame

mapping

is

achieved

by

translating

the

existing

coordinate

frame

to

the

midpoint

between

the

two

streamlines

at

any

given

instant

This

simplifies

calculations

and

also

helps

the

viewer

orient

themselves

with

respect

to

the

world

Fig.

illustrates

this

method

This

view

is

reminiscent

of

a

twirling

baton

hence

the

name

The

longer

the

baton

the

higher

the

uncertainty

in

streamline

position

More

twirling

to

the

baton

the

higher

the

uncertainty

in

the

orientation

Figure

Snapshot

showing

sections

of

streamlines

with

the

same

or

higher

differ

ences

Figure

Twirling

baton

display

of

differ

ences

between

two

streamlines

Point

of

view

of

midpoint

particle

Sonification

Uncertainty

can

be

examined

by

mapping

uncertainty

to

sound

We

have

used

sound

in

conjunction

with

visualization

to

identify

uncertainty

Sonification

can

often

provide

information

about

data

that

can

not

be

seen

using

visualization

Sound

can

enhance

a

graphical

presentation

by

providing

information

about

features

of

the

data

that

may

be

hidden

or

occluded

Sound

can

also

help

the

user

to

distinguish

the

size

relationships

between

objects

that

may

be

difficult

to

determine

visually

because

of

projection

distortion

In

multivariate

mappings

where

visual

clutter

can

destroy

the

usefulness

of

the

display

the

addition

of

sound

can

provide

relief

for

the

overloaded

visual

channel

by

allowing

some

variables

to

be

presented

aurally

Also

sound

can

provide

redundancy

of

representation

and

allow

data

validation

An

excellent

discussion

of

additional

benefits

of

auditory

display

in

conjunction

with

other

displays

can

be

found

in

-LSB-

-RSB-

We

have

explored

the

use

of

sonification

in

visualizing

uncertainty

of

flow

visualization

and

surface

interpolation

using

LISTEN

-LSB-

-RSB-

LISTEN

is

a

data

sonification

system

that

allows

interactive

mapping

of

data

to

sound

parameters

such

as

pitch

duration

volume

and

timbre

Here

we

describe

two

examples

of

sonifying

uncertainty

together

with

animation

In

the

first

example

a

glyph

was

chosen

to

move

along

a

desired

path

or

curve

in

the

surface

interpolation

application

and

along

a

streamline

in

the

flow

visualization

application

The

size

-LRB-

height

or

the

radius

-RRB-

of

the

glyph

was

mapped

to

uncertainty

However

in

regions

of

low

variations

of

uncertainty

it

was

difficult

to

visually

distinguish

the

size

of

glyphs

Therefore

uncertainty

was

mapped

to

duration

thereby

slowing

the

movement

of

the

glyphs

and

prolonging

the

sound

in

regions

of

high

uncertainty

In

another

example

two

different

uncertainty

parameters

such

as

the

mean

and

Gaussian

curvatures

of

two

surface

interpolants

were

mapped

to

the

pitches

of

violin

and

drum

respectively

This

sonification

was

played

in

conjunction

with

the

presentation

of

cross-hair

glyphs

where

the

lengths

of

the

cross-hairs

were

mapped

to

uncertainty

in

the

mean

and

Gaussian

curvatures

This

overloading

of

multivariate

uncertainty

information

to

visual

and

aural

cues

helped

in

the

understanding

of

the

correlation

between

different

uncertainty

parameters

Psycho-Visual

Approaches

This

is

probably

the

least

understood

and

exploited

group

of

methods

that

can

be

used

for

uncertainty

visualization

and

visualization

in

general

Perhaps

the

main

reason

is

the

difficulty

of

eliciting

a

consistent

impression

or

response

from

the

subjects

We

have

experimented

with

two

different

methods

that

fall

under

this

category

stereopairs

and

subliminal

messages

With

the

aid

of

stereo

glasses

a

stereo

effect

can

be

achieved

by

quickly

alternating

the

presentation

of

left

and

right

views

to

the

left

and

right

eyes

Using

this

technology

we

may

compare

two

slightly

different

images

The

motivation

behind

this

is

to

make

parts

of

the

picture

blurry

while

other

parts

of

the

picture

clear

If

the

two

images

were

identical

we

would

lose

the

stereo

effect

but

would

still

see

a

clear

picture

When

this

method

was

applied

to

comparing

the

radiosities

on

surface

patches

we

achieved

the

desired

effect

of

varying

degree

of

clarity

and

blurriness

However

it

was

difficult

to

use

this

method

for

an

extended

period

of

time

as

the

visual

system

kept

trying

to

resolve

the

conflicting

information

arriving

at

both

eyes

thereby

resulting

in

eye

strain

The

idea

behind

the

use

of

subliminal

messages

was

to

see

if

we

can

open

an

additional

channel

to

send

information

to

the

user

To

this

end

we

experimented

with

flashing

textual

messages

in

between

screen

refresh

as

well

as

putting

the

messages

in

texture

maps

and

wrapping

them

onto

objects

in

the

rendered

scenes

The

textual

messages

-LRB-

e.g.

itchy

and

sneeze

-RRB-

were

meant

to

evoke

immediate

response

from

the

viewers

Based

on

our

very

limited

experimentation

we

were

not

able

to

get

a

repeatable

and

consistent

set

of

responses

While

the

findings

in

these

two

methods

indicate

some

difficulties

that

still

need

to

be

resolved

we

think

that

psycho-visual

approaches

have

great

potential

and

need

to

be

explored

further

Summary

We

have

introduced

a

number

of

new

methods

for

uncertainty

visualization

as

presented

in

Table

These

methods

are

grouped

using

the

characteristics

from

Table

and

presented

in

Table

From

here

we

can

see

that

there

is

demand

for

research

in

techniques

for

vector

and

tensor

visualization

and

that

we

have

added

techniques

for

discrete

and

continuous

visualization

extents

From

our

exploration

of

uncertainty

visualization

techniques

we

have

found

that

continuous

visualization

extents

are

more

challenging

than

discrete

visualization

techniques

We

believe

that

our

grouping

of

methods

into

add

glyph

add

geometry

modify

geometry

modify

attributes

etc.

is

a

powerful

means

to

understanding

the

different

methods

There

are

also

many

new

techniques

that

are

yet

to

be

invented

but

will

likely

fall

into

these

categories

In

development

of

new

techniques

it

is

important

to

evaluate

them

to

determine

if

they

are

more

effective

than

existing

techniques

We

have

worked

on

evaluation

methodologies

and

have

done

a

complete

study

of

uncertainty

glyphs

The

new

techniques

that

should

survive

are

those

that

are

shown

to

be

effective

in

encoding

and

easily

decoded

by

users

The

basic

methodology

is

to

use

visual

tests

where

users

examine

visualizations

and

decode

the

information

within

the

graphic

The

amount

of

error

between

the

user

interpretation

and

the

encoding

is

statistically

evaluated

to

determine

if

the

visualizations

are

effective

There

is

a

developed

theory

of

specifiers

by

Carswell

-LSB-

-RSB-

Cleveland

-LSB-

-RSB-

and

others

which

has

shown

that

decoding

length

is

the

most

accurate

decoding

area

is

less

accurate

etc.

Decoding

of

pseudo-colors

bump

mapping

surface

textures

etc.

has

not

been

shown

to

be

straightforward

yet

The

evaluation

of

the

effectiveness

of

new

techniques

is

time

consuming

but

important

and

we

feel

it

is

a

wide

open

area

for

fundamental

research

in

visualization

In

this

section

we

have

presented

a

variety

of

tables

that

give

different

views

of

existing

visualization

methods

and

that

contrast

these

with

our

new

methods

The

emphasis

has

been

to

enumerate

the

state

of

the

art

as

it

exists

to

clearly

show

what

work

needs

to

be

done

and

to

demarcate

our

contribution

The

variety

of

uncertainty

visualization

techniques

we

have

illustrated

in

the

images

in

this

section

show

that

there

are

more

sophisticated

methods

than

side

by

side

comparison

techniques

or

ignoring

the

uncertainty

altogether

A

very

sophisticated

user

may

not

need

a

visualization

at

all

to

understand

a

system

under

analysis

but

visualization

is

for

exploration

of

systems

that

we

do

not

fully

understand

and

characterizing

the

uncertainty

visually

helps

in

more

complete

understanding

Value

Visualization

Extent

discrete

continuous

scalar

glyphs

fat

surfaces

displacement

-LRB-

spheres

lines

-RRB-

affine

transformations

magnitude/frequency

left/right

subliminal

and

multivariate

methods

below

multivariate

glyphs

-LRB-

ellipsoids

-RRB-

material

property

ribbons

IFS

batons

texture

and

bump

mapping

oscillate

ranking

batons

vector

uncertainty

glyphs

snow

angel

tensor

Table

New

methods

for

uncertainty

visualization

and

areas

for

further

research

Conclusions

In

this

paper

a

wide

variety

of

new

uncertainty

visualization

methods

were

introduced

-LRB-

adding

glyphs

adding

geometry

modifying

attributes

modifying

geometry

animation

sonification

and

psycho-visual

-RRB-

and

applied

to

many

applications

The

results

of

our

research

show

there

is

a

tremendous

variety

of

possible

means

to

map

uncertainty

into

a

scene

The

complexity

and

hard

work

in

deriving

and

understanding

the

uncertainty

in

the

first

place

remains

but

hopefully

we

have

demonstrated

tools

that

may

be

helpful

to

more

easily

investigate

uncertainty

The

approach

of

verity

visualization

where

the

technique

for

encoding

the

uncertainty

is

unambiguous

such

as

height

glyphs

for

surface

errors

is

not

possible

in

all

cases

The

resultant

data

overloading

provides

more

of

a

burden

on

the

end

user

but

the

critical

nature

of

using

data

uncertainty

while

doing

data

analysis

can

be

aided

using

our

techniques

We

have

done

psycho-visual

experiments

for

some

of

our

approaches

and

are

working

on

developing

evaluations

for

our

other

techniques

as

well

The

resulting

visualizations

of

data

and

uncertainty

are

integrated

and

present

an

accurate

depiction

to

the

user

We

also

found

that

more

than

one

uncertainty

method

can

be

used

together

and

that

application

specific

methods

are

relatively

easy

to

generalize

and

be

applied

to

different

applications

Working

with

different

applications

has

allowed

us

to

design

uncertainty

visualization

methods

that

would

not

have

been

otherwise

apparent

As

such

our

current

research

efforts

include

Comparative

visualization

of

experimental

and

simulated

data

for

validation

purposes

data

assimilation

and

ensemble

forecasting

to

resolve

measurement

and

forecast

differences

Extension

of

the

uncertainty

computation

and

visualization

from

structured

grids

to

scattered

data

sets

where

the

uncertainties

are

expected

to

be

even

more

pronounced

and

important

in

data

interpretation

Investigation

of

volume

rendering

methods

for

the

purpose

of

designing

comparative

volume

visualization

methods

We

plan

to

extend

our

work

on

surface

comparison

methods

to

volume

comparison

methods

and

apply

these

to

volume

rendered

scenes

and

images

Development

of

techniques

for

comparing

images

The

application

domain

of

images

is

large

-LRB-

e.g.

compression

algorithms

GIS

cartography

etc.

-RRB-

and

work

in

this

area

can

be

of

potential

significance

We

believe

that

these

uncertainty

visualization

methods

will

prove

valuable

to

people

who

need

to

make

informed

decisions

based

on

imperfect

data

Acknowledgements

We

wish

to

thank

the

students

involved

with

various

aspects

of

uncertainty

visualization

including

Hiroya

Chiba

Michael

Clifton

Adam

Freeman

Steve

Hodges

Chris

Oates

Elijah

Saxon

Bob

Sheehan

Alan

Tifford

and

Catherine

Wilson

We

also

wish

to

thank

colleagues

who

have

provided

data

and

feedback

including

Wendell

Nuss

Dan

Fernandez

Jessica

Hodgins

David

Lane

and

David

Kenwright

The

Santa

Cruz

Laboratory

for

Visualization

and

Graphics

-LRB-

SLVG

-RRB-

provided

a

wonderful

research

environment

and

we

would

like

to

acknowledge

that

Finally

the

authors

would

also

like

to

acknowledge

support

from

NSF

grant

ONR

grant

and

and

NASA

Cooperative

Agreement

References

-LSB-

-RSB-

James

Arvo

Kenneth

Torrance

and

Brian

Smits

A

framework

for

the

analysis

of

error

in

global

illumination

algorithms

In

Proceedings

SIGGRAPH

pages

-LCB-

Orlando

FL

July

ACM

SIGGRAPH

-LSB-

-RSB-

M.

Kate

Beard

Barbara

P.

Buttenfield

and

Sarah

B.

Clapham

NCGIA

research

initiative

Visualization

of

spatial

data

quality

Technical

Paper

National

Center

for

Geographic

Information

and

Analysis

October

-LSB-

BCE

-RSB-

Ken

W.

Brodlie

Lesley

Carpenter

Rae

A.

Earnshaw

Julian

R.

Gallop

Roger

J.

Hubbold

Anne

M.

Mumford

Chris

D.

Osland

and

Peter

Quarendon

editors

Scientific

Visualization

Techniques

and

Applications

chapter

on

Visualization

Techniques

pages

-LCB-

Springer-Verlag

-LSB-

-RSB-

R.

G.

Bergeron

and

G.

G.

Grinstein

A

reference

model

for

the

visualization

of

multi-dimensional

data

In

Proceedings

of

Eurographics

pages

-LCB-

Elsevier

Science

Publsihers

-LSB-

BHR

-RSB-

Manfred

Brill

Hans

Hagen

Hans-Christian

Rodrian

Wladimir

Djatschin

and

Stanislav

V.

Klimenko

Streamball

techniques

for

flow

visualization

In

Proceedings

Visualization

pages

-LCB-

IEEE

Computer

Society

-LSB-

-RSB-

D.

R.

Brillinger

editor

The

Collected

Works

of

John

W.

Tukey

Wadsworth

and

Brooks

-LSB-

-RSB-

C.

Melody

Carswell

Choosing

specifiers

An

evaluation

of

the

basic

tasks

model

of

graphical

perception

Human

Factors

-LRB-

-RRB-

-LCB-

October

-LSB-

-RSB-

Elizabeth

Cluff

Robert

Burton

and

William

Barrett

A

survey

and

characterization

of

multidimensional

presentation

techniques

Journal

of

Imaging

Technology

-LRB-

-RRB-

-LSB-

-RSB-

Christopher

Chatfield

Statistics

for

Technology

A

Course

In

Applied

Statistics

Chapman

and

Hall

third

edition

-LSB-

-RSB-

William

S.

Cleveland

The

Elements

of

Graphing

Data

Wadsworth

-LSB-

-RSB-

William

S.

Cleveland

and

Robert

McGill

An

experiment

in

graphical

perception

International

Journal

of

Man-Machine

Studies

-LRB-

-RRB-

-LCB-

November

-LSB-

-RSB-

T.

Delmarcelle

and

L.

Hesselink

Visualizing

second-order

tensor

fields

with

hyperstreamlines

IEEE

Computer

Graphics

and

Applications

-LRB-

-RRB-

-LCB-

July

-LSB-

-RSB-

D.L.

Darmofal

and

R.

Haimes

An

analysis

of

particle

path

integration

algorithms

Journal

of

Computational

Physics

-LRB-

-RRB-

-LCB-

January

-LSB-

-RSB-

Willem

C.

de

Leeuw

and

Jarke

J.

van

Wijk

A

probe

for

local

flow

field

visualization

In

Proceedings

of

Visualization

pages

-LCB-

IEEE

Computer

Society

Press

October

-LSB-

-RSB-

Gerald

Farin

Curves

and

Surfaces

for

Computer

Aided

Geometric

Design

A

Practical

Guide

Academic

Press

-LSB-

-RSB-

L.

K.

Forssell

and

S.

D.

Cohen

Using

line

integral

convolution

for

flow

visualization

curvilinear

grids

variable-speed

animation

and

unsteady

flows

IEEE

Transactions

on

Visualization

and

Computer

Graphics

-LRB-

-RRB-

-LCB-

June

-LSB-

-RSB-

P.

Fisher

Visualization

in

Geographical

Information

Systems

chapter

on

Animation

and

Sound

for

the

Visualization

of

Uncertain

Spatial

Information

pages

-LCB-

-LSB-

-RSB-

Nahum

D.

Gershon

Visualization

of

fuzzy

data

using

generalized

animation

In

Proceedings

of

Visualization

pages

-LCB-

IEEE

Computer

Society

Press

October

-LSB-

-RSB-

Ned

Greene

and

Michael

Kass

Error-bounded

antialiased

rendering

of

complex

environments

In

Proceedings

SIGGRAPH

pages

-LCB-

Orlando

FL

July

ACM

SIGGRAPH

-LSB-

-RSB-

A.

Globus

and

S.

Uselton

Evaluation

of

visualization

software

Computer

Graphics

pages

-LCB-

May

-LSB-

HDD

-RSB-

Hugues

Hoppe

Tony

DeRose

Tom

Duchamp

Mark

Halstead

Hugert

Jin

John

McDonald

Jean

Schweitzer

and

Werner

Stuetzle

Piecewise

smooth

surface

reconstruction

In

Proceedings

SIGGRAPH

pages

-LCB-

Orlando

FL

July

ACM

SIGGRAPH

-LSB-

HHS

-RSB-

Hans

Hagen

Stefanie

Hahmann

Thomas

Schreiber

Yasuo

Nakajima

Burkard

Wordenweber

and

Petra

Hollemann-Grundstedt

Surface

interrogation

algorithms

IEEE

Computer

Graphics

and

Applications

-LRB-

-RRB-

-LCB-

September

-LSB-

-RSB-

L.

Hesselink

F.

Post

and

J.J.

van

Wijk

Research

issues

in

vector

and

tensor

field

visualization

IEEE

Computer

Graphics

and

Applications

-LRB-

-RRB-

-LCB-

March

-LSB-

-RSB-

A.

Inselberg

and

B.

Dimsdale

Parallel

coordinates

a

tool

for

visualizing

multidimensional

geometry

In

Proceedings

of

Visualization

pages

-LCB-

IEEE

October

-LSB-

-RSB-

A.

Inselberg

The

plane

with

parallel

coordinates

The

Visual

Computer

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

Peter

Keller

and

Mary

Keller

Visual

Cues

Practical

Data

Visualization

IEEE

Computer

Society

Press

-LSB-

-RSB-

G.

Kramer

An

introduction

to

auditory

display

In

Gregory

Kramer

editor

Auditory

Display

Sonification

Audification

and

Auditory

Interfaces

pages

-LCB-

Addison-Wesley

-LSB-

-RSB-

W.

E.

Lorensen

and

H.

E.

Cline

Marching

cubes

A

high

resolution

surface

construction

algorithm

Computer

Graphics

-LRB-

-RRB-

-LCB-

-LSB-

-RSB-

S.

K.

Lodha

Scattered

data

techniques

for

surfaces

In

I.

Chakravarty

and

D.

Silver

editors

Geometry

Detection

estimation

and

Synthesis

for

Scientific

Visualization

page

to

appear

Academic

Press

-LSB-

-RSB-

Suresh

K.

Lodha

Alex

Pang

Robert

E.

Sheehan

and

Craig

M.

Wittenbrink

UFLOW

Visualizing

uncertainty

in

fluid

flow

In

Proceedings

of

Visualization

page

to

appear

October

-LSB-

-RSB-

Dani

Lischinski

Brian

Smits

and

Donald

P.

Greenberg

Bounds

and

error

estimates

for

radiosity

In

Proceedings

SIGGRAPH

pages

-LCB-

Orlando

FL

July

ACM

SIGGRAPH

-LSB-

-RSB-

Suresh

K.

Lodha

Bob

Sheehan

Alex

T.

Pang

and

Craig

M.

Wittenbrink

Visualizing

geometric

uncertainty

of

surface

interpolants

Proceedings

of

Graphics

Interface

pages

-LCB-

May

-LSB-

-RSB-

S.

K.

Lodha

C.

M.

Wilson

and

R.

E.

Sheehan

LISTEN

sounding

uncertainty

visualization

In

Proceedings

of

Visualization

IEEE

-LSB-

-RSB-

R.

Minghim

and

A.R.

Forrest

An

illustrated

analysis

of

sonification

for

scientific

visualization

In

Proceedings

of

Visualization

pages

-LCB-

IEEE

-LSB-

-RSB-

J.F.

Moreno

and

J.

Melia

An

optimum

interpolation

method

applied

to

the

resampling

of

NOAA

AVHRR

data

IEEE

Transactions

on

Geoscience

and

Remote

Sensing

-LRB-

-RRB-

-LCB-

January

-LSB-

-RSB-

Torsten

Moller

Raghu

Machiraju

Klaus

Mueller

and

Roni

Yagel

Classification

and

local

error

estimation

of

interpolation

and

derivative

filters

for

volume

rendering

In

Proceedings

Volume

Visualization

page

to

appear

San

Francisco

CA

October

IEEE

-LSB-

-RSB-

Alex

Pang

and

Naim

Alper

Bump

mapped

vector

fields

In

SPIE

IS&T

Conference

Proceedings

on

Electronic

Imaging

Vol

Visual

Data

Exploration

and

Analysis

II

pages

-LCB-

color

plate

page

SPIE

February

-LSB-

-RSB-

Alex

Pang

and

Adam

Freeman

Methods

for

comparing

surface

attributes

In

SPIE

Vol

Visual

Data

Exploration

and

Analysis

III

pages

-LCB-

SPIE

February

-LSB-

-RSB-

Alex

Pang

Jeff

Furman

and

Wendell

Nuss

Data

quality

issues

in

visualization

In

SPIE

Vol

Visual

Data

Exploration

and

Analysis

pages

-LCB-

SPIE

February

-LSB-

-RSB-

Greg

Turk

and

David

Banks

Image-guided

streamline

placement

In

Proceedings

SIGGRAPH

pages

-LCB-

New

Orleans

LA

August

ACM

SIGGRAPH

-LSB-

-RSB-

Barry

N.

Taylor

and

Chris

E.

Kuyatt

Guidelines

for

evaluating

and

expressing

the

uncertainty

of

NIST

measurement

results

Technical

report

National

Institute

of

Standards

and

Technology

Technical

Note

Gaithersburg

MD

January

-LSB-

-RSB-

Edward

R.

Tufte

The

Visual

Display

of

Quantitative

Information

Graphics

Press

-LSB-

-RSB-

Edward

R.

Tufte

Envisioning

Information

Graphics

Press

-LSB-

-RSB-

John

W.

Tukey

Exploratory

Data

Analysis

Addison

Wesley

-LSB-

-RSB-

J.K.

Udupa

and

H.-M

Hung

Surface

versus

volume

rendering

a

comparative

assessment

In

Proceedings

of

the

First

Conference

on

Visualization

in

Biomedical

Computing

pages

-LCB-

Atlanta

GA

May

IEEE

Comput

Soc

-LSB-

-RSB-

Allen

Van

Gelder

and

Jane

Wilhelms

Topological

considerations

in

isosurface

generation

ACM

Transactions

on

Graphics

-LRB-

-RRB-

October

Also

UCSC

technical

report

-LSB-

-RSB-

Craig

M.

Wittenbrink

IFS

fractal

interpolation

for

and

visualization

In

IEEE

Visualization

pages

-LCB-

Atlanta

GA

November

IEEE

-LSB-

-RSB-

Craig

M.

Wittenbrink

Alex

T.

Pang

and

Suresh

Lodha

Verity

visualization

Visual

mappings

Technical

Report

Univ.

of

Cal

Santa

Cruz

-LSB-

-RSB-

Craig

M.

Wittenbrink

Alex

T.

Pang

and

Suresh

K.

Lodha

Glyphs

for

visualizing

uncertainty

in

vector

fields

IEEE

Transactions

on

Visualization

and

Computer

Graphics

September

to

appear

-LSB-

WSF

-RSB-

Craig

M.

Wittenbrink

Elijah

Saxon

Jeff

J.

Furman

Alex

T.

Pang

and

Suresh

Lodha

Glyphs

for

visualizing

uncertainty

in

environmental

vector

fields

In

Vol

SPIE

IS&T

Conference

Proceedings

on

Electronic

Imaging

Visual

Data

Exploration

and

Analysis

pages

-LCB-

SPIE

February

classification
comparative visualization
data quality
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verity