Yingcai Xiao

Chapter 9

Advanced Algorithms

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 Scattered Data: sample points distributed unevenly and non-uniformly throughout the volume of interest. throughout the volume of interest.

Example Data: chemical leakage at a tank-farm.Example Data: chemical leakage at a tank-farm.

Thin-plate Spline f x y z b d d c c x c y c zi i ii

n

( , ) = ( ) + + + + =

, log2

1 2 31

4

Volume Spline f x y z b d c c x c y c zi ii

n

( , ) = + + + + =

, 31 2 3

14

Shepard method f x y z

n

i

d v

n

i

di i i( , ) =

=

=

,

1 1

1 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)

TriangulationTriangulation

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.

TriangulationTriangulation

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)

TriangulationTriangulation

Formal definition of Triangulation:

N dimensional triangulation of a point set P=(P1, P2, ……,Pn) is a collection of n-dimensional 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.

TriangulationTriangulation

Example VTK Code:

vtkPolyData *p= vtkPolyData::New();

pSetPoints(pnts);

//pnts: pointer to an array of points, vtkFloatPoints

vtkDelaunay3D *d= vtkDelaunay3D::New();

dSetInput(p);

dSetTolerance(0.0001);

vtkPolyDataMapper *m= vtkPolyDataMapper::New();

mSetInput(dGetOutput);

vtkActor *a= vtkActor::New();

aSetMapper(m);

Visualizing Unstructured Points

Splatting for Point Data

Gaussian Splatting

s: scale factor

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 LinesMarching Triangles for Contour Lines

Marching Tetrahedrons for Isosurface.

8 cases, 3 unique, doted vertexes are inside: v > contour value

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();

ibAddFunction(cylinder);

ibAddFunction(sphere);

vtkClipPolyData *c =

vtkClipPolyData::New();

cSetInput(planGetOutput())

cSetClipFunction(ib);

cGenerateClippedOutputOn();

cGenerateClipScalarsOn();

cSetValue(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 SystemsParallel Coordinate Systems

• Brushing

• Minimal Example

• Reordering

• Coloring

 

x1 x2 x3 x4

Plot of a four dimensional point.

• Compositing• Progressive Rendering• Bundling

Parallel Coordinate SystemsParallel Coordinate Systemshttp://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.

 

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.