Scientific Data Visualization

General

Rendering vs. Visualization

Rendering displays synthetic or modeled scenes.
Visualization displays measured or simulated data.
The goal is to make complicated datasets easier to interpret.
Volume rendering is better understood as volume visualization because it visualizes sampled data.

Data Types

Scalar data has one value per point.
- Examples: density, pressure, temperature, and height.
Vector data has a tuple of values per point.
- Examples: velocity, direction, and force.
Tensor data describes directional behavior with higher-dimensional attributes.
- Examples: curvature, diffusion, stress, and strain.

Sampled Data

Sampling and Reconstruction

Continuous data must be sampled before it can be visualized.
Reconstruction approximates continuous data from samples.
Interpolation uses weighted sums of basis functions.
Basis function choice is a tradeoff:
- Simple functions are cheap but low quality.
- Higher-order functions are smoother but more expensive.

Basis Functions

Constant basis functions are nearest-neighbor reconstruction.
They are cheap but produce blocky results.
Linear basis functions improve continuity.
Linear interpolation depends on the cell type.

Cells and Grids

Cell Types

Vertices are sample points.
Common cells:
- Vertex.
- Line.
- Triangle.
- Quad.
- Tetrahedron.
- Hexahedron.
Interpolation depends on the cell.
Triangles usually use linear or barycentric interpolation.
Hexahedra usually use trilinear interpolation.

Grid Types

Uniform grid:
- Equal spacing.
- Simple but not always optimal.
Rectilinear grid:
- Axis-aligned with non-uniform spacing.
- Resolution changes only along axes.
Structured grid:
- Sample points can move freely.
- Topology stays fixed.
Unstructured grid:
- No fixed topological order.
- Flexible but harder to process.

Scalar Fields

Visualization Options

Map scalar values to:
- Intensity.
- Color.
- Texture.
- Opacity.
- Contours or isolines.
- Isosurfaces.
Scalar visualization can be combined with vector and tensor visualization.

Isosurfaces

An isosurface is the set of all 3D points with the same scalar value.
It is commonly extracted with marching cubes.
It is useful when one meaningful boundary exists.

Isolines

An isoline is the set of all 2D points with the same scalar value.
Marching squares extracts isolines.
It is the 2D version of marching cubes.

Marching Squares and Marching Cubes

Marching squares has 4 corners and $2^4 = 16$ cases.
Symmetry reduces this to four unique cases.
Marching cubes has 8 corners and $2^8 = 256$ cases.
Symmetry reduces this to 15 unique cases.
Ambiguity can happen.
Alternatives such as marching triangles or marching tetrahedra can avoid some ambiguity.

Slicing vs. Contouring

Slicing and contouring are commutative.
Slicing a volume and then computing isolines is equivalent to computing an isosurface and then slicing it.
This means a 3D isosurface can be assembled from 2D isolines on parallel slices.

Vector Fields

Vector Data

Vector data usually stores 2D or 3D tuples.
A common application is computational fluid dynamics.
Per time step, the simulation can store:
- Velocity.
- Pressure.
- Density.
- Divergence.
- Vorticity.

Vector Glyphs

A glyph is a small icon placed in the field.
Glyphs can encode position, orientation, direction, size, magnitude, and color.
Examples include lines, hedgehogs, arrows, and cones.
The vector field is usually subsampled.
High sampling density shows more data but creates clutter.
Low sampling density is cleaner but can miss detail.
Random sampling can reduce artifacts from a regular sampling grid.

3D Glyphs

3D glyphs have the same meaning as 2D glyphs.
Occlusion becomes a major limitation.
Transparency can help, especially with grayscale color maps.

Stream Lines

Idea

Start at a seed point.
Follow the vector field step by step.
Connect computed points into a line.
Stop when an exit criterion is reached.

Euler Integration

p_{n+1} = p_n + \Delta t \cdot v(p_n)

$\Delta t$ is an integration parameter, not physical time for a static vector field.
Runge-Kutta integration gives better quality than Euler integration.

Quality

Streamline quality depends on seed point density, seed distribution, and step width.
Too many streamlines create clutter.
Too few streamlines miss structure.
Parameters can be adapted locally.

Stream Tubes

A stream tube sweeps a circular cross section along a streamline.
Tube diameter can encode another scalar.
Color can encode another property.
Stream tubes can be traced downstream, upstream, or in both directions.

Tensor Fields

Tensors

A tensor generalizes scalars and vectors.
Rank examples:
- Scalar: rank 0.
- Vector: rank 1.
- Matrix or Hessian: rank 2.
Tensors describe direction-dependent quantities such as curvature, diffusion, stress, and strain.

Curvature

Planar curve curvature uses the second derivative.
Surface curvature depends on direction.
For a surface point, choose direction $s$ .
Curvature in direction $s$ can be expressed with Hessian $H$ .

c(s) = s^T H s

PCA and Eigenvectors

Tensor visualization often focuses on extreme directions.
Eigendecomposition gives principal directions and magnitudes.
Eigenvectors are the principal directions.
Eigenvalues are the strengths in those directions.
Isotropic behavior has similar eigenvalues.
Anisotropic behavior has one or more dominant eigenvalues.

Tensor Glyphs

Tensor glyphs generalize vector glyphs.
They encode eigenvectors as directions.
They encode eigenvalues as size or shape.
Common glyph shapes include ellipsoids, cuboids, cylinders, and superquadrics.
They suffer from clutter, occlusion, and sampling-density problems.

Fiber Tracking

Fiber tracking is used for tensor data such as DT-MRI.
It constructs streamlines along major eigenvectors.
It needs strong anisotropy to be stable.
Seed point choice is critical.
Plain fiber streamlines show direction and path but do not encode all eigenvalues.

Hyper-Streamlines

Hyper-streamlines are stream tubes for tensor fields.
They follow the major eigenvector field.
The tube cross section is elliptical.
The ellipse axes follow the medium and minor eigenvectors.
The ellipse radii scale with the corresponding eigenvalues.