Visualization Fundamentals

1
Visualization Fundamentals

• Interpolation Functions
• Basic 2D Scalar Data Visualization Techniques
• Color Maps
• High Fields
• Contours

2
Interpolation (1)

• Visualization deals with discrete data

Values defined only at cell vertices
3
Interpolation (2)

• We often need information at positions other than
  cell vertices

4
Interpolation (3)

• Three essential information
• Cell type
• Data values at cell vertices
• Parametric coordinates of the point p

di cell point value Wi weight (S wi 1) D
interpolated result
5
Interpolation (4)
Parametric Coordinates Used to specify
the location of a point within a cell
W0 (1-r) W1 r
r
6
Interpolation (5)
(b) Triangle
s
W0 1-r-s W1 r W2 s
p2
rs 1 (why?)
r0
Why?
r
p0
p1
s0
7
Interpolation (6)
(C) Pixel
s
s1
p2
p3
W0 (1-r)(1-s) W1 r(1-s) W2 (1-r)s W3 rs
(s,t)
r0
r 1
r
p0
p1
s0
Why?
This is also called bi-linear interpolation
8
Interpolation (7)
(D) Polygon
2
p3
2
p4
Wi (1/ri) / S(1/ri)
r3
p2
p5
Weighted distance function
p0
p1
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Interpolation (8)
(D) Tetrahedron
t
s
p3
p2
p0
r
p1
10
Interpolation (9)
(D) Cube (voxel)
t
W0 (1-r)(1-s)(1-r) W1 r(1-s)(1-t) W2
(1-r)s(1-t) W3 rs(1-r) W4 (1-r)(1-s)t W5
r(1-s)t W6 (1-r)st W7 rst
p6
p7
p4
p5
s
p2
p3
r
p1
p0
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Interpolation (10)
The interpolation function can be used to
calculate the geometric position as well. That
is, given (r,s,t), calculate the global
coordinates Local to global coordinate
transformation P S Wi Pi
n-1
i0
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Interpolation (11)

• How to get (r,s,t) ?
• Line, Pixel, Cube are all trivial
• Triangle, Tetrahedron can be solved analytically
• Qudrilateral or Hexahedra need numerical method

P S Wi Pi
n-1
Known P, Pi Unkown Wi (i.e. r,s,t)
i0
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Color Mapping
R G B
v1 v2 v3 ...
Values at vertices Color lookup
Result
table
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High Fields
Used to visualize digital elevation maps

• Triangulate
• Lift the height
• of each point
• Connect the points
• as triangles

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Contours
3
7
10
4
C 6