env 200612.1 envisioning information lecture 12 – scientific visualization scalar 2d data ken...

ENV 2006 12.1

Envisioning Information

Lecture 12 – Scientific Visualization

Scalar 2D Data

Ken Brodliekwb@comp.leeds.ac.uk

ENV 2006 12.2

2D Interpolation – Regular Gridded Data

• Suppose we are given data on a regular rectangular grid

f given at eachgrid point; wefill out the emptyspaces byinterpolating valueswithin each cell

ENV 2006 12.3

• Straightforward extension from 1D: take f-value from nearest data sample

• No continuity• Bounds fixed at data extremes

Nearest Neighbour Interpolation

ENV 2006 12.4

• Consider one grid rectangle:– suppose corners are at (0,0), (1,0), (1,1), (0,1) ... ie a unit square

– values at corners are f00, f10, f11, f01

f00 f10

f01 f11

How do weestimate valueat a point (x,y)inside thesquare?

Bilinear Interpolation

ENV 2006 12.5

f00 f10

f01 f11

(i) interpolate in x-direction between f00,f10; and f01,f11

(ii) interpolate in y-direction

We carry out three 1D interpolations:

Exercise: Show this is equivalent to calculatingf(x,y) = (1-x)(1-y)f00+x(1-y)f10+(1-x)yf01+ xyf11

(x,y)

Bilinear Interpolation

ENV 2006 12.6

• Apply within each grid rectangle• Fast• Continuity of value, not slope (C0)• Bounds fixed at data extremes

Piecewise Bilinear Interpolation

ENV 2006 12.7

Contour Drawing

• Contouring is very common technique for 2D scalar data

• Isolines join points of equal value

– sometimes with shading added

• How can we quickly and accurately draw these isolines?

ENV 2006 12.8

• As an example, consider this data:

10 -5

1 -2

Where does the zero level contour go?

An Example

ENV 2006 12.9

• The bilinear interpolant is linear along any edge - thus we can predict where the contour will cut the edges (just by simple proportions)

10 -5

-21

10

-5

cross-section viewalong top edge

Intersections with sides

ENV 2006 12.10

Simple Approach

• A simple approach to get the contour inside the grid rectangle is just to join up the intersection points

10 -5

-21

Question:Does this always work?

Try an example whereone pair of oppositecorners are positive,other pair negative

ENV 2006 12.11

• But this does not always work - look at this data:

10 -5

1-3

Try it - it is ambiguous!

Ambiguity

ENV 2006 12.12

• We need to worry about the behaviour of the interpolant inside the grid square

• The contour of the bilinear interpolant is NOT a straight line – it is a curve -

10 -5

-21

This is curve of: f(x,y) = (1-x)(1-y)f00+x(1-y)f10+(1-x)yf01+ xyf11

= 0

BUT how can we draw it?

Joining intersectionswith straight lineswas only an approximation…drawing the contourof bilinear interpolantwill resolve ambiguity

What is the Problem?

ENV 2006 12.13

Tracking Contours

• We can track the contour in small steps through the grid rectangle – starting from intersection with the edges– Take a step, probe at equal distance to either side, then predict

next point

Next point oncontour

-0.3

0.9

Current pointon contour

Probes

-0.3

0.9

BUT THIS IS SLOW!!

Can we find an alternative which uses straight lines, but resolves the ambiguous case?

ENV 2006 12.14

Implementing Rectangle-based Contouring

• For a rectangle, there will be 24 = 16 cases

• There are 4 configurations– All same sign (no contour)– 3 same sign (one contour

piece)– 2 adjacent with same sign (one

contour piece)– 2 opposite with same sign (two

pieces, but ambiguous… function has a saddle)

+ +

+ +

+ +

+ -

+ -

+ -

+ -

- +

Note: the ambiguity can be resolvedby looking at value at saddle point:

Saddle value = (f00f11 - f01f10)/ (f00+f11-f01+f10)

ENV 2006 12.15

Solution by Decomposing Cell

• Another possibility is to split cell into four triangles

• Within a triangle, we can fit a linear model

– F(x,y) = a + bx +cy

• How do we split?• How do we calculate a,b,c?• What is the gain?

f1

f2 f3

ENV 2006 12.16

Cell Decomposition

• Problem of drawing the curved lines has been circumvented by decomposing cell into four pieces within which the contours are well defined straight lines

• How might we estimate value at centre?

10 -5

-3 1

0.75

ENV 2006 12.17

Contouring in IRIS Explorer

• Contour is a simple contouring module

• NAGContour interpolates the data and then tracks the contours

NAGContour

ENV 2006 12.18

• Often the data will be given, not on a regular grid, but at scattered locations:

f given at eachmarked point

Approach:(i) triangulate(ii) build interpolantin each triangle(iii) draw contours

2D Interpolation – Scattered Data

ENV 2006 12.19

• Triangulation is the process of forming a grid of triangles from the data points

How can we construct the triangulation?

Triangulation

ENV 2006 12.20

• We solve the DUAL problem:– Suppose a wolf is stationed at each data point. Each wolf is equally

powerful and dominates the territory closest to its own base

– What are the territories dominated by each wolf?

Tesselation

ENV 2006 12.21

Tesselation – Two or three wolves

ENV 2006 12.22

Tesselation - Two or Three Wolves

ENV 2006 12.23

• The resulting tesselation is known as the Dirichlet or Voronoi tesselation

• Given the Dirichlet tesselation for N points

P1, P2, ... PN

there is an algorithm for constructing the tesselation when an extra point is added

Dirichlet Tesselation

ENV 2006 12.24

P1

P2

P3

Tesselation forP1, P2, P3

QPoint Qadded

Dirichlet Tesselation

ENV 2006 12.25

P1

P2

P3

Q

Dirichlet Tesselation

ENV 2006 12.26

P1

P2

P3

Q

Dirichlet Tesselation

ENV 2006 12.27

– Determine polygon containing Q - here D3, surrounding P3

– Construct perpendicular bisector of P3Q and find intersection with D3 - this becomes point of modified tesselation

– Determine adjacent polygon - here D2

– Repeat the above two steps until D3 is reached again, or there is no intersection

– Remove all vertices and edges interior to the new polygon

Dirichlet Tesselation

ENV 2006 12.28

P1

P2

P3

Q

Delaunay Triangulation

ENV 2006 12.29

• Triangulation formed by joining points whose ‘territories’ share a common boundary in the tesselation

• This has the nice property that it avoids long skinny triangles• See the nice applets at:

www.cs.cornell.edu/Info/People/chew/ Delaunay.html

• Note the ‘empty circle’ property of the Delaunay triangulation

Delaunay Triangulation

ENV 2006 12.30

Contouring from Triangulated Data

• The final step is to contour from the triangulated data

• Easy – because contours of linear interpolant are straight lines – see earlier

http://www.tecplot.com

ENV 2006 12.31

Implementing Triangle-based Contouring

• Each vertex can be positive or negative (ignore zero for now)

• This gives 23 = 8 possible cases…

• … but there are only 2 distinct configurations– No contour (all same sign)– Contour (2 of one sign, 1

of the other) • Implementation:

– Determine which of 8 cases

– Select code for the appropriate configuration

f1

f2 f3

f1

f2 f3

All same sign

Two same sign

In IRIS Explorer, NAGContour will contour scattered data – using triangulation followed by a tracking method based on a nonlinear interpolant – for greater accuracy

ENV 2006 12.32

• A different mapping technique for 2D scalar data is the surface view.

• Here a surface is created in 3D space, the height representing the scalar value

• Construction is quite easy - suppose we have a rectangular grid

Surface Views

ENV 2006 12.33

Constructing a Surface View - 1

ENV 2006 12.34

Surface created as pair of triangles per grid rectangle.Rendering step is then display of triangles.

Constructing a Surface View - 2

ENV 2006 12.35

Examples - with added contours

www.tecplot.com

ENV 2006 12.36

• A further mapping technique for 2D data is the image plot• There are three variants:

– dot array : draw a dot at each data point, coloured according to the value (very fast, but low quality)

Image Plots

ENV 2006 12.37

Image Plots

Grid lines:

ENV 2006 12.38

Image Plots

Areas:

ENV 2006 12.39

IRIS Explorer – Image Plots and Surface Views

• Image views are created using:– LatToGeom: takes a lattice

(array of values) and produces a colour mosaic

– GenerateColorMap: associates colours with values

• Surface views are created using:

– DisplaceLat: takes two lattices, one to create the surface (displacement), the other to colour it (function)… and builds a 3D lattice

– LatToGeom: converts this 3D lattice to geometry

ENV 2006 12.40

Module Assessment

• Research topic (50)

• Portfolio to include:– November draft submission – Review comments– Response to comments– Final paper (35)– Presentation (15)

• Practical exam (50)– One exercise using xmdvtool

(25)– One exercise using IRIS

Explorer (25)– Task will be to explore a

dataset in each case– 2 hour supervised exam in

laboratory