4.1 Vis_04 Data Visualization Lecture 4 Two Dimensional Scalar Visualization Part 2: Further Contouring and Other 2D Techniques

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Data VisualizationData Visualization

Lecture 4Two Dimensional Scalar

VisualizationPart 2: Further Contouring and

Other 2D Techniques

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Return to ExampleReturn to Example

Consider this data:

10 -5

1 -2

Where does the zero level contour go?

Can we draw a straight line which willapproximate the contour line?

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Joining Intersections Directly

Joining Intersections Directly

It is tempting to approximate by joining intersections with straight lines:

10 -5

-21

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AmbiguityAmbiguity

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

10 -5

1-3

Try it - it is ambiguous!

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What is the Problem?What is the Problem?

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

This is curve of: f(x,y) = (1-x)(1-y)f00+x(1-y)f10+(1-x)yf01+ xyf11 = 0

10 -5

-21

BUT how can we draw it?

Joining intersectionswith straight lineswas only an approximation

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Next point oncontour

Tracking ContoursTracking Contours

We can track the contour in small steps through the grid rectangle - starting from the intersection with the edges– take a step, probe at an equal distance to

either side, then predict next point; and so on

-0.3

0.9

Current pointon contour

Probes

-0.3

0.9BUT THIS IS SLOW!!

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Solution by Decomposing Cell

Solution by Decomposing Cell

One technique – more efficient than tracking - 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

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Cell DecompositionCell 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

10 -5

-3 1

0.75

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Avoiding Cell DecompositionAvoiding Cell

Decomposition

As an alternative to decomposing the cell, we can try to understand how the curved contours of a bilinear interpolant behave

The difficult case is:

+

+

-

-

Opposite vertices: two +and two -

Where do the contour lines go?

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Bilinear InterpolantBilinear Interpolant

The bilinear interpolant has contours which are hyperbolas, and can be one of two forms:

In each case there is a saddle point - if saddle point is-ve, then we have case A; if +ve, case B.

+ -

- +

A+ -

- +

B

Saddle point: fx=fy=0 .. Max in one direction, min in other

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A Possible MethodA Possible Method

This suggests following method:– in ambiguous case, calculate saddle

point and join intersection points according to how bilinear behaves:

+ -

- +

A+ -

- +

B

saddle negative saddle positive

Try it on the example earlier

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Saddle PointSaddle Point

It is possible to calculate the saddle point in terms of vertex values (from fx = fy =0 ):

And the corresponding value:

x = (f00 - f01 ) / D; y = (f00 - f10 ) / D

Saddle value = (f00f11 - f01f10)/D

D = f00 + f11 - f01 -f10

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ProblemProblem

In the ambiguous case, will the earlier four triangle method always give the same result as the saddle point method?

5 -150

100-1

Try thiswith bothapproaches

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Greater AccuracyGreater Accuracy The greater understanding of the

bilinear leads to a more accurate method

A single straight line approximation can be made more accurate by using two straight line pieces– see Lopes and Brodlie paper on Web site+ -

- +A B

+ -

- +

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2D Interpolation - Scattered Data

2D Interpolation - Scattered Data

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

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TriangulationTriangulation

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

How can we construct the triangulation?

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TesselationTesselation

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?

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Tesselation - Two or Three Wolves


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Dirichlet TesselationDirichlet Tesselation

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

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P1

P2

P3

Tesselation forP1, P2, P3

QPoint Qadded

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P1

P2

P3

Q

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P1

P2

P3

Q

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– 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

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Delaunay TriangulationDelaunay Triangulation

P1

P2

P3

Q

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Delaunay TriangulationDelaunay Triangulation

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

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Contouring from Triangulated DataContouring 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

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Implementing Triangle-based Contouring

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

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Implementing Rectangle-based Contouring

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)

+ +

+ +

+ +

+ -

+ -

+ -

+ -

- +

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Surface ViewsSurface Views

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

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Constructing a Surface View - 1


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

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Examples - with added contours

Examples - with added contours

www.tecplot.com

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Image PlotsImage Plots

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)

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Grid lines:

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Areas:

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Cross SectionsCross Sections

Another option is to look at a cross-section through the data

For example, if x and y are the independent variables, we could fix y and look at f in terms of just x– then repeat for different y

– this reduces the ES2 problem to a

sequence of ES1 problems

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4.1 Vis_04 Data Visualization Lecture 4 Two Dimensional Scalar Visualization Part 2: Further Contouring and Other 2D Techniques