Scientific Visualization

Data Modelling for Scientific Visualization CS 5630 / 6630 August 28, 2007 Scientific Visualization

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:3D, time-varying Scalar Fields http://background.uchicago.edu/~whu/beginners/introduction.html

Types of Data in SciVis:2D Vector Fields

Types of Data in SciVis:3D 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

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 f(x) = ax + b v1 = a.0 + b = b v2 = a.1 + b = a + b b = v1 a = v2 – b = v2 - v1 f(x) = v1+ (v2 – v1).x sometimes written as f(x) = v2.x + v1.(1-x)‏ v1 v2 u 0 1

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 f(x) = a+b.x+c.x^2+d.x^3 f'(x) = b + 2cx + 3dx^2 f(0) = v1 f(1) = v2 f'(0) = (v2 – v0)/2 f'(1) = (v3 – v1)/2 ... a = v1 b = (v2-v0) / 2 c = v0 – 5.v1/2 + 2v2 – v3/2 d = -v0/2 + 3.v1/2 – 3.v2/2 + v3/2 v2 v3 v0 v1 -1 0 1 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

Scientific Visualization