1. Domain-Modeling Techniques
Chap. 8
October 29, 2009
CDS 301
Fall, 2008
Jie Zhang
Copyright ©

2. Domain-Modeling Discrete dataset Continuous dataset
Visualization is to operate on the domain representation, e.g., the sampling points and grids, but not on the sampled attribute values.
Discrete dataset
Continuous dataset

3. Outline 8.1. Cutting 8.2. Selection
8.3. Grid Construction from Scattered Points
8.4. Grid-Processing Technique

4. Cutting
Target domain is a subset of the source domain 4

5. Brick Extracting
Target domain has the same dimension of the source domain
Extracting a volume of interest (VOI)
For structured grid with structured coordinates represented by integer number, bricking is to change the extent of each dimension 5

6. Slicing Target domain has a smaller dimension: d-1
Fix the coordinate in the slicing dimension
Along X-axis
Sagittal slice
Along Y-axis
axial slice
Along Z-axis
coronal slice 6

7. Implicit Function Cutting
Generalize the axis-aligned slicing 7

8. Selection
Cutting explicitly specifies the topology of the target domain
Selection explicitly specify the attribute value of the target dataset.
The output of selection is an unstructured grid
E.g., contouring operation, iso-surface
E.g., thresholding or segmentation 8

9. Scattered Points Gridless point cloud 12,772
3-D points represent a human face 9

10. Grid Construction Build an unstructured grid from scattered points
Most visualization software package requires data to be in a grid-based representation.
Triangulation methods are the most-used class of methods for constructing grids from scattered points 10

11. Delaunay Triangulation
Constructed triangles covers the convex hull of the point set
No point lies inside the circumscribed circle of any constructed triangles 11

12. Delaunay Triangulation
Voronoi diagram:
It is associated with Delaunay triangulation.
The vertices of the Voronoi cells are the centers of the circumscribed circles of the triangles. 12

13. Delaunay Triangulation
600 random 2-D points 13

14. Delaunay Triangulation
Angle-constrained Delaunay Triangulation.
20° < α < 140°
Added 361 extra points 14

15. Delaunay Triangulation
Area-constrained Delaunay Triangulation.
a < amax
Added 1272 extra points 15

16. Surface Reconstruction
Render a 3-D surface from a point cloud 16

17. Radial Basis Function method
Computing a 3D volumetric dataset, using the 3D RBF (radial basic function) for construction.
Find the isosurface (f=1) using marching cube algorithm
This method is problematic, e.g., partition of unity
Φ: reference RBF
R: support radius
K: decay speed coefficient
T-1: word-to-reference coordinate transform 17

18. Signed Distance method
Computing a tangent plane Ti that approximates the local surface in the neighborhood of pi
Calculate the distance function f between a given point (x) and the tangent plane at the sample point closest to (x)
The surface is simply the isosurface as f=0
Using marching cube method to obtain isosurface in 3D: a set of unstructured polygons or triangles
Using marching square method to obtain contour lines in 2-D, a set of unstructured line segment 18

19. Find Local Tangent Plane19

20. Find Local Tangent Plane20

21. Principal Component Analysis (PCA)
A matrix H: 2 X 2, or 3 X 3
Hs= λs
s: eigenvector
λ: eigenvalue
E.g., Hessian matrix
Eigenvector is of extreme
curvature

22. PCA

23. PCA If we order the eigenvalue in decreasing order:
In case of curvature:
e1 the direction of maximal curvature
e2 the direction of minimum curvature
e3: the direction of surface normal

24. Local Triangulation Method
Constructing the unstructured triangle mesh from local 2D Delaunay triangulation
Computing a tangent plane Ti that approximates the local surface in the neighborhood of pi
Project its neighbor set Ni on Ti, and compute 2D Delaunay triangulation
Add to the mesh those triangles that have pi as a vertex 24

25. Local Triangulation Method25

26. Mesh Construction
All previous methods are to create a triangular (or polygon) mesh to represent the 3-D surface: radial basis; distance; local triangulation)
Mesh detail 26

27. Surface Splatting Method
Simple, fast, “dirty” method without using mesh
Draw discs of radius Ri, centered at every sample point pi and oriented in the local tangent plane.
The disc is as a 2D transparency texture Φ: opaque (= 1), total transparent (= 0)
Transparency is calculated from the radial basis function
The texture that encodes a 2D RBF is called “splat”
Splat is a rendering element for a 3D surface that is analogous to the pixel in a 2D image. 27

28. Surface Splatting Method
Mesh reconstruction
Point-based splatting 28

29. Grid-Processing Techniques
Grid-processing techniques change
The grid geometry: location of grid sample points
The grid topology: the grid cells 29

30. Geometric Transformation
change the position of the sample points; not modify the cells, attributes, and basis functions
Affline operation: carry straight lines into straight lines and parallel lines into parallel lines.
Translation; rotation; scaling
Nonaffine operation:
Bending, twisting, and tapering
Based on attributes
Warping, height plot, and displacement plot
Based on grid
Grid-smoothing technique 30

31. Grid Simplification
Reducing the number of sampling points (but need to maintain the reconstruction quality)
Uniform sampling yields too many grid points
Adaptive sampling
fewer points in area of low surface curvature
More points in area of high surface curvature 31

32. Triangle Mesh Decimation
Recursively reduce the vertex and its triangle fan if the resulting surface lies within a user-specified distance from the un-simplified surface
36000 grid points
3600 grid points 32

33. Vertex Clustering By clustering or collapsing vertices
Assign each vertex an important value, based on the curvature (high curvature  more important)
Overlaid a grid onto the mesh
All vertices within the grid cell are collapsed to the most important vertex within the cell 33

34. Grid Refinement An opposite operation of grid simplification
Generate more sample points
A refined grid gives better results when applying grid manipulation operation
Inserting more points when the surface varies more rapidly 34

35. Grid Smoothing Question: how to reduce geometric noise?
Modify the positions of the grid points such that the reconstructed surface becomes smoother
Smoothing removes the high frequency, small scale variation, e.g, small spikes 35

36. Laplacian Smoothing
After smoothing
Before smoothing 36

37. Laplacian Smoothing
Laplacian process shifts grid points toward the barycenter of its neighboring point set 37

38. Laplacian Smoothing 38

39. End of Chap. 8
Note: