1. Visualization Fundamentals
Interpolation Functions
Basic 2D Scalar Data Visualization Techniques
Color Maps
High Fields
Contours

2. Interpolation (1) Visualization deals with discrete data
(a) vertex
(b) Polyvertex
(c) line
(d) polyline
(e) triangle
(e) Quadrilateral
(e) Polygon
(f) Tetrahedron
(f) Hexahedron
Values defined only at cell vertices

3. Interpolation (2)
We often need information at positions other than cell vertices p
P = ? 10 13 9 12
Interpolation: compute data
from known points

4. Interpolation (3) Three essential information: Cell type
Data values at cell vertices
Parametric coordinates of the point p
D = S Wi * di
i=0
n-1 di
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
(a) line
r = 0
r = 1
0 <= r <=1 d0 d1
W0 = (1-r)
W1 = r
D = S Wi * di
i=0
n-1 r

6. Interpolation (5) (b) Triangle s W0 = 1-r-s W1 = r W2 = s p2
r+s = 1 (why?)
r=0
Why? r p0 p1
s=0

7. Interpolation (6) (C) Pixel s s=1 p2 p3 W0 = (1-r)(1-s) W1 = r(1-s)
(s,t)
r=0
r =1 r p0
s=0 p1
Why?
This is also called bi-linear interpolation

8. Interpolation (7) (D) Polygon p3 p4 Wi = (1/ri) / S(1/ri) r3 p2 p5
Weighted distance function p0 p1

9. Interpolation (8) (D) Tetrahedron t s p3 W0 = 1-r-s-t p2 W1 = r W2 = s
W3 = t 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 p0 p1

11. 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
i=0

12. 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)
i=0

13. Color Mapping
R G B v1 v2 v3
...
Values at vertices Color lookup Result
table

14. High Fields Used to visualize digital elevation maps Triangulate
Lift the height
of each point
Connect the points
as triangles

15. Contours 3 7 10 4
C = 6