Transcript Lecture 11 - Scientific Visualization: Introduction & Scalar 1D Data

Envisioning Information
Lecture 11 – Scientific Visualization
Introduction
Scalar 1D Data
Ken Brodlie
[email protected]
ENV 2006
11.1
Data Visualization =
Scientific Vis + Information Vis
•
Information Visualization
•
Scientific Visualization
– Numeric and non-numeric
data
– Numerical data from
science, engineering and
medicine
Automobile web site
- visualizing links
Ozone layer around earth
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11.2
Scientific Visualization – What is it?
Reality
Observation
Simulation
Data
Visualization
Images, animation
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11.3
A Simple Example
This table shows the observed oxygen levels in
the flue gas, when coal undergoes combustion
in a furnace
TIME (mins)
0
2
4
OXYGEN (%) 20.8
8.8
4.2
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10
28
30
32
0.5 3.9
6.2
9.6
11.4
Visualizing the Data - but is this what we want to see?
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11.5
Estimating behaviour between the data - but is this believable?
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Now it looks believable… but something is wrong
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At least this is credible..
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11.8
What Have We Learnt?
• It is not only the data that we wish to visualize - it is also the bits
inbetween!
• The data are samples from some underlying ‘field’ which we
wish to understand
• First step is to create from the data a ‘best’ estimate of the
underlying field - we shall call this a MODEL
• This needs to be done with care and may need guidance from
the scientist
• The process of fitting a continuous curve (or surface, or volume)
through given data is known as INTERPOLATION
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Data Enrichment
• This process is sometimes called ‘data enrichment’ or
‘enhancement’
• If data is sparse, but accurate, we INTERPOLATE to get
sufficient data to create a meaningful representation of our
model
• If sparse, but in error, we may need to APPROXIMATE
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11.10
The Visualization Process
• Overall the Visualization Process can be divided into four
logical operations:
– DATA SELECTION: choose the portion of data we want
to analyse
– DATA ENRICHMENT: interpolating, or approximating
raw data - effectively creating a model
– MAPPING: conversion of data into a geometric
representation
– RENDERING: assigning visual properties to the
geometrical objects (eg colour, texture) and creating an
image
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11.11
Back to the simple example
Data
Select
Enrich
Map
Render
Extract part of data we are interested in
Interpolate to create model
Select a line graph as technique
and create line segments from
enriched data
Draw line segments on display in
suitable colour, line style and width
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11.12
Visualization Software Systems
• This model has become the basis of a family of visualization
software systems – developed from late 1980s
• Visualization seen as a pipeline of simple processes
eg contouring
– read in data
– create contour lines
– draw contour lines
• Systems provide modules implementing these steps…
• .. Plus a ‘visual programming’ environment to allow user to
connect modules together
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Visual Programming - IRIS Explorer
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11.14
Visual programming systems
• Visual programming allows easy experimentation which is what
one needs in visualization
• Examples are:
– IRIS Explorer
www.nag.co.uk
– AVS
www.avs.com
– OpenDX (grown from IBM Visualization Data Explorer)
www.opendx.org
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11.15
Classification of mapping techniques
• The mapping stage is where we decide which visualization
technique to apply to our ‘enriched’ data
• There are a bewildering range of these techniques - how do we
know which to choose?
• First step is to classify the data into sets and associate different
techniques with different sets.
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11.16
Back to the simple example
• The underlying field is a function F(x)
– F represents the oxygen level and is the DEPENDENT variable
– x represents the time and is the INDEPENDENT variable
• It is a one dimensional scalar field because
– the independent variable x is 1D
– the dependent variable F is a scalar value
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General classification scheme
• The underlying field can be regarded as a function of many
variables: say
F(x)
where F and x are both vectors:
F = (F1, F2, ... Fm)
x = (x1, x2, ... xn)
• The dimension is n
• The dependent variable can be scalar (m=1) or vector (m>1)
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A Simple Notation
• This leads to a simple classification of data as:
EnS/V
• So the simple example is of type:
E1S
• Flow within a volume can be classed as:
E3V3
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11.19
Exercise
• Can you give suitable techniques for the following classes:
• ES1
• ES2
• ES3
• EV33
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11.20
Overview of Visualization Techniques
Different techniques for different types of data
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11.21
Scalar 1D data: ES1
•
•
•
Scalar: 1 value
1D: value is measured in terms
of 1 other variable
The humble graph!
A nice example of web-based
visualization….
http://fx.sauder.ubc.ca/plot.html
•
Clear overlap here between
SciVis and InfoVis…
http://fx.sauder.ubc.ca/plot.html
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Scalar 2D Data: ES2
•
Here is yesterday’s temperature
over USA
http://weather.unisys.com/surface/
•
Can you use a 1D technique for
this sort of data?
•
Can you improve this
visualization?
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Scalar 2D Data: ES2
•
•
Here is a surface view of the
tsunami…
For movies, see:
http://www.pmel.noaa.gov/tsunami/indo_1204.
html
movie
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Scalar 3D Data: ES3
•
As dimension increases, it
becomes harder to visualize on
a 2D surface
•
Here we see a lobster within
resin.. where the resin is
represented as semitransparent
•
Technique known as volume
rendering
Image from D. Bartz and M. Meissner
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Vector 2D Data:EV22
• This is a flow field in two
dimensions
• Simple technique is to use
arrows..
• What are the strengths and
weaknesses of this
approach?
• During the module, we will
discover better techniques
for this
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Vector 3D Data:EV33
•
•
•
This is flow in a volume
Arrows become extremely
cluttered
Here we are tracing the path of
a particle through the volume
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11.27
Visualization Techniques –
Scalar 1D Data
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11.28
1D Interpolation – The Problem
f
4
3
2
1
0
1
1.75 2
3
4
x
Given (x1,f1), (x2,f2), (x3,f3), (x4,f4) - estimate the value of
f at other values of x - say, x*. Suppose x*=1.75
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11.29
Nearest Neighbour
f
4
3
2
1
0
1
1.75 2
3
4
x
Take f-value at x* as f-value of nearest data sample.
So if x* = 1.75, then f estimated as 3
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11.30
Linear Interpolation
f
4
3
2.5
2
1
0
1
1.75 2
3
4
x
Join data points with straight lines- read off
f-value corresponding to x*.. in the case that
x*=1.75, then f estimated as 2.5
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Linear Interpolation – Doing the Calculation
Suppose x* lies between x1 and x2. Then apply the
transformation:
f1
f*
f2
t = (x-x1)/(x2-x1)
so that t goes from 0 to 1.
x
x
1
0
x*
t*
f(x*) = (1-t*) f1 + t* f2
t*=(1.75-1)/(2-1)=0.75
2
1
f(1.75)=0.25*1+0.75*3
=2.5
The functions j(t)=1-t and k(t)=t are basis functions.
OR, saving a multiplication:
f(x*) = f1 + t* ( f2 - f1 )
f(1.75)=1+0.75*(3-1)
=2.5
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Nearest Neighbour and Linear Interpolation
• Nearest Neighbour
– Very fast : no arithmetic involved
– Continuity : discontinuous value
– Bounds : bounds fixed at data extremes
• Linear Interpolation
– Fast : one multiply, one divide
– Continuity : value only continuous, not slope (C0)
– Bounds : bounds fixed at data extremes
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11.33
Drawing a Smooth Curve
• Rather than straight line between points, we create a curve
piece
We estimate the
slopes g1 and g2
at the data points,
and construct curve
which has these values
and these slopes at
end-points
g1
f1
g2
f2
x1
x2
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11.34
Slope Estimation
• Slopes estimated as some average of the slopes of adjacent
chords - eg to estimate slope at x2
g2
g2 usually
arithmetic mean
(ie average) of 
f2

f1

f3
 =(f2-f1)/(x2-x1)
x1
x2
x3
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Piecewise Cubic Interpolation
Once the slopes at x1 and x2 are known, this
is sufficient to define a unique cubic polynomial
in the interval [x1,x2]
f(x) = c1(x) * f1
+ c2(x) * f2
+ h*(d1(x) * g1
- d2(x) * g2)
g1
f1
g2
f2
x1
x2
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ci(x), di(x) are
cubic Hermite
basis functions,
h = x2 – x1.
11.36
Cubic Hermite Basis Functions
Here they are:
Again set t = (x - x1)/(x2 – x1)
c1 (t) = 3(1-t)2 - 2(1-t)3
c2 (t) = 3t2 - 2t3
d1 (t) = (1-t)2 - (1-t)3
d2 (t) = t2 - t3
Check the values
at x = x1, x2 (ie t=0,1)
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Coal data - cubic interpolation
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Piecewise Cubic Interpolation
• More computation needed than with nearest neighbour or linear
interpolation.
• Continuity: slope continuity (C1) by construction - and cubic
splines will give second derivative continuity (C2)
• Bounds: bounds not controlled generally - eg if arithmetic mean
used in slope estimation...
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Shape Control
• However special choices for slope estimation do give control
over shape
• If the harmonic mean is used
1/g2 = 0.5 ( 1/1 + 1/2)
then we find that f(x) lies within the bounds of the data
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Coal data – keeping within the bounds of the data
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Rendering Line Graphs
• The final rendering step is
straightforward
• We can assume that the
underlying graphics system
will be able to draw straight
line segments
• Thus the linear interpolation
case is trivial
• For curves, we do an
approximation as sequence
of small line segments
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