Chapter 9
Advanced Algorithms
Yingcai Xiao
Constructing Topology
•
Data collected in the real world
① Usually scattered, not confirming to a grid.
② Can be interpolated into a grid. But problem rises when we use
global interpolation methods.
Scattered Data: sample points distributed unevenly and non-uniformly
throughout the volume of interest.
Example Data: chemical leakage at a tank-farm.
n
Thin-plate Spline
f ( x, y, z ) =
bd
i
i =1
2
i
log ( d i ) + c1 + c 2 x + c 3 y + c 4 z
Volume Spline
n
f ( x, y, z ) =
bd
i
i =1
3
i
+ c1 + c 2 x + c 3 y
+ c4 z
Shepard method
f ( x, y, z )
=
 n

1
  di vi
 i =1







 n


1 
  di 
i = 1



•
Data collected in the real world
① Can we use local interpolation with the closest points?
② Need to find and connect the closest points.
③ Demo in 2D.
Visualizing Unstructured Points
Triangulation: Connect points to form a
triangulated topological structure.
(topology generation)
Triangulation
Use Edge Swapping to make a Delaunay Triangulation.
1. Find the minimum bounding triangle. (may need to add a fake point)
2. Add one point at a time, do triangulation.
3. After each point is added, check if the joined triangles are optimal or not; if
not swap the joining edge (edge swapping). Check all other triangles and
make sure they are still optimal after adding the current point.
4. Repeat step 3 to add each point and we when all points are added and all
triangles are optimal.
5. Clean up: remove all fake points and related triangles.
Triangulation
Properties of Delaunay Triangulation:
1. Circumsphic property: no other points in the
circumspheres.
2. Optimal property: The minimum interior angle of a triangle
is greater than or equal to the minimum interior angle of
any other triangulation
3. It is the dual of the Diritchlet tessellation
① Voronoi cell, Centroid
② A collection of connected Voronoi cells tessellete the
area (partition, coverage)
Triangulation
Formal definition of Triangulation:
N dimensional triangulation of a point set P=(P1, P2, ……,Pn) is a collection of ndimensional simplexes (triangles) whose defining points lie in P.
Optimal Triangulation: a triangulation that generates maximized minimum
angles.
Delaunay Trianglation: an optimal triangulation, which satisfies the
circumshpere condition.
Circumsphere Condition: the circumshpere of any n-dementional simplexe
contains no other points of p except the n+1 points defining the simplex.
Triangulation
Example VTK Code:
vtkPolyData *p= vtkPolyData::New();
pSetPoints(pnts);
//pnts: pointer to an array of points, vtkFloatPoints
vtkDelaunay3D *d= vtkDelaunay3D::New();
dSetInput(p);
dSetTolerance(0.0001);
vtkPolyDataMapper *m= vtkPolyDataMapper::New();
mSetInput(dGetOutput);
vtkActor *a= vtkActor::New();
aSetMapper(m);
Visualizing Unstructured Points
Splatting for Point Data
Gaussian Splatting
s: scale factor
SF(r) = se
- f (r/R)2
R: radius of influence
Data value assigned to grid nodes within the radius of
influence of the data point.
Scalar Algorithms
Marching Triangles for Contour Lines
8 cases,
3 unique,
doted vertexes
are inside:
v > contour
value
Marching Tetrahedrons for Isosurface.
Dividing Cubes
• Generate contour surfaces using dense point clouds.
• For each voxel, interpolate into it if the contour going
through it, display points around the interpolated
surface.
• Subdivide the voxel to have each subvoxel cover one
pixel. Draw a point in the subvoxel if it is near the
isosurface (nodal values brace the threshold values).
• Display points are much faster than surfaces.
• The resulting surface is not continuous.
Dividing Cubes
Carpet Plots
•
Display a 2D grid (a slice of a 3D grid) as a 3D extruded
surface based on the data values
•
The extrusion is perpendicular to the 2D slice and is
proportional to the data values.
Carpet Plots
V(r) = e-r cos(10r);
Clipping Geometry with a Scalar Field
Original Geometry
Clipped Geometry
Clipping Geometry with a Scalar Field
• Data controlled geometry change.
• To reveal the inside of a volume.
Ex.
Data values: v(x,y,z) = x + y + z – c
Isosurface: v=0 defines a plane, use it
to cut geometry.
Marching Cube Cases:
Vertex inside/outside
256=>15 Unique cases
Clipping Geometry with a Scalar Field
Marching Triangles: 8 cases, 4 unique
(doted vertexes are inside: v < 0)
Clip a plane by a cylinder and a sphere.
vtkImplicitBoolean *ib =
vtkImplicitBoolean::New();
ibAddFunction(cylinder);
ibAddFunction(sphere);
vtkClipPolyData *c =
vtkClipPolyData::New();
cSetInput(planGetOutput())
cSetClipFunction(ib);
cGenerateClippedOutputOn();
cGenerateClipScalarsOn();
cSetValue(0) // v = 0;
Vector Algorithms
Stream Tubes
Stream Lines depict flow direction of a vector fields
What about temperature? Color coding.
How about add speed, acceleration, vortisity (twist)?
Tensor Algorithm
Hyperstreamlines
High Dimensional Visualization
F(x), F(x,y), F(x,y,z), F(x1, x2, x3, ……)
•
Glyph, use each part of the glyph for a variable.
•
Use Parallel Coordinate Systems
x1
x2
x3
x4
Plot of a four dimensional point.
Parallel Coordinate Systems
•
Brushing
•
Minimal Example
•
Reordering
•
Coloring
x1
x2
• Compositing
• Progressive Rendering
• Bundling
x3
x4
Plot of a four dimensional point.
Parallel Coordinate Systems
http://www.xdat.org
http://www.cs.tau.ac.il/~aiisreal/
https://github.com/syntagmatic/parallel-coordinates
https://syntagmatic.github.io/parallel-coordinates/
http://www.ggobi.org/docs/parallel-coordinates/
http://en.wikipedia.org/wiki/Parallel_coordinates
http://stn.spotfire.com/spotfire_client_help/para/para_what_is_a_parallel_
coordinate_plot.htm
https://eagereyes.org/techniques/parallel-coordinates
http://exposedata.com/parallel/
http://www.b-eye-network.com/view/3355
http://www.parallelcoordinates.de/#toggleControls
Modeling Algorithms
Modeling Algorithms
• Visualizing Geometry:
• bundary cells
• outlines,
• wireframe (w/s in a VKT window)
•
Data Extraction: Dataset => Subset
Extract portions of data from a dataset.
• Geometry Extraction
 portion of a grid, set of cells
 subsampling: every nth data point is selected.
• Threshholding: extracting based on data values.
May change topology, e.g., uniform => unstructured.
Probing (Resampling): Dataset => Another Dataset
•
Obtains dataset attributes by sampling the original dataset with a
set of points (the probe).
•
Data values can be viewed in a particular fashion.
Swept Volume and Surfaces: Visualize Motion
A swept volume is the volume of space occupied by an object as it
moves through space along a given trajectory. A polygonal is
created from the original.
– Path: ST
– Steps: L
– Boolean Operation
Triangle Strip Generation
Triangle strips are compact representations of triangle polygons.
Greedy method: The longer the average strip length is the better.
http://en.wikipedia.org/wiki/File:Triangle_Strip_Small.png
Polygon Normal Generation
Reflection:
•
I~f(N,C) i.e., C.N.I
•
Normal may not be given by the progammer.
Facet Normal: Compute normal of a polygon from its vertexes.
Ex.
V1
V1 ´V2
N=
| V1 ´V2 |
Polygon Normal Generation:
facet / polygonal normals
Polygon Normal Generation
Vertex Normals:
Use the average of the facet normals in the use set.
Polygon Normal Generation
Vertex Normals:
Shaded using vertex normal
(average of facet normals),
too smooth.
Polygon Normal Generation
Feature Angle ~800
<900
Take the facet out of the
average computation if
the connected facets
have angle greater than
the feature angle
Decimation
Polygon Reduction (Vertex Reduction)
Threshold Distance: 
For lines:  is the distance to the line without the vertex.
Decimation
For surface:  is the distance to the surface without the
vertex.
Mesh Smoothing
No reduction, no topology change.
But the geometry changes to make the mesh smooth.
Mesh Smoothing
No reduction, no topology change.
But the geometry changes to make the mesh smooth.