Scientific Visualization

Data Modelling for Scientific VisualizationCS 5630 / 6630August 28, 2007

Recap: The Vis Pipeline

Recap: The Vis Pipeline

Types of Data in SciVis: Functions

http://lambda.gsfc.nasa.gov/product/cobe/firas_image.cfm

Types of Data in SciVis: Functions on Circles

E. Anderson et al.: Towards Development of a Circuit Based Treatment for Impaired Memory

Types of Data in SciVis:2D Scalar Fields

Types of Data in SciVis:Scalar Fields on Spheres

http://lambda.gsfc.nasa.gov/product/cobe/firas_image.cfm

Types of Data in SciVis:Types of Data in SciVis:3D, time-varying Scalar Fields3D, time-varying Scalar Fields

http://background.uchicago.edu/~whu/beginners/introduction.htmlhttp://background.uchicago.edu/~whu/beginners/introduction.html

Types of Data in SciVis:2D Vector Fields

Types of Data in SciVis:Types of Data in SciVis:3D Vector Fields3D Vector Fields

Tensors

Tensors are “multilinear functions” rank 0 tensors are scalars rank 1 tensors are vectors rank 2 tensors are matrices, which transform

vectors rank 3..n tensors have no nice name, but

they transform matrices, rank-3 tensors, etc. We are not going to see these

DTI Tensors

DTI Tensors are symmetric, positive definite SPD: scale along orthogonal directions More specifically, they approximate the rate

of directional water diffusion in tissue

Types of Data in SciVis:2D, 3D Tensor Fields

Kindlmann et al. Super-Quadric Tensor Glyphs and Glyph-packing for DTI vis.

Computers like discrete data, but world is continuous

Sampling

Continuous to discrete Store properties at a finite set of points

Sampling

Continuous to discrete Store properties at a finite set of points

Sampling

Continuous to discrete Store properties at a finite set of points

Interpolation

Discrete to continuous Reconstruct the illusion of continuous data,

using finite computation

Nearest Neighbor Interpolation

Pick the closest value to you

Linear Interpolation

Assume function is linear between two samples

Linear Interpolation

Assume function is linear between two samples

v1

v2

0 1u

f(x) = ax + b

v1 = a.0 + b = bv2 = a.1 + b = a + b

b = v1a = v2 – b = v2 - v1

f(x) = v1+ (v2 – v1).x

sometimes written as

f(x) = v2.x + v1.(1-x)

Cubic Interpolation

Linear reconstruction is better than NN, but it is not smooth across sample points

Let's use a cubic Two more parameters: we need constraints Constrain derivatives

Cubic Interpolation

Same as with linear

0 1-1 2

v0v1

v3

v2

f(x) = a+b.x+c.x^2+d.x^3f'(x) = b + 2cx + 3dx^2

f(0) = v1f(1) = v2f'(0) = (v2 – v0)/2f'(1) = (v3 – v1)/2

...

a = v1b = (v2-v0) / 2c = v0 – 5.v1/2 + 2v2 – v3/2d = -v0/2 + 3.v1/2 – 3.v2/2 + v3/2

(VisTrails Demo)

Linear vs Higher-order interpolation in plotting

Might make a big difference!

Kindlmann et al. Geodesic-loxodromes... MICCAI 2007

1D vs n-D

Most common technique: separability Interpolate dimensions one at a time

(VisTrails Demo)

2D Interpolation in VTK images

Implicit vs Explicit Representations

Explicit is parametric Domain and range are “explicit”

Implicit stores domain... implicitly Zero set of a explicit domain Pro: it's easy to change topology of domain:

just change the function Con: it's harder to analyze and compute with

Implicit vs. explicit representations

Explicit:y(t) = sin(t)x(t) = cos(t)

s = (x(t), y(t)), 0 < t <= 2

Implicit:f(x,y) = x^2 + y^2 - 1

s = (x,y): f(x,y) = 0

Regular vs Irregular Data

Regular data: sampling on every point of an integer lattice

Irregular data: more general sampling

Curvilinear grid

Like a regular grid, but on curvilinear coordinates Here, radius and

angle

Triangular and Tetrahedral Meshes

Completely arbitrary samples Need to store topology: How do samples

connect with one another?

Quadrilateral and Hexahedral Meshes

Basic element is a quad or a hex Element shape is better

for computation Much, much harder to

generate

Tabular Data

Most common in information visualization Relational DBs

... etc.

Node vs cell data: do we store values on nodes (vertices) or on cells (tets and tris)?

Pure-quad vs quad-dominant: mixing types of elements

Linear vs high-order: different interpolation modes on elements