Scientific Visualization

Sampling:

• Converts continuous data into discrete samples

Reconstruction:

• Recovers (approximate) continuous data from sampled data

• Often done using interpolation

Interpolation:

• Achieved via a weighted sum of basis functions (interpolation functions)

• Basis functions are often orthonormal (orthogonal + normal)

• Choice of basis function affects:

  • Computational complexity

  • Quality of reconstruction

Grid or mesh

• Subdivision of the domain into cells or elements

Example:

• Piecewise constant interpolation = nearest-neighbour interpolation

Constant basis functions:

• Very low computational cost

• Only low-quality approximation

Higher-order basis functions:

• Enable more continuous and accurate reconstruction

Linear basis functions:

• Good compromise between complexity and quality

• Require knowledge of the cell type in the grid

• Example: Trilinear interpolation in a cube-like cell (8 vertices)

Different cell types exist:

• Not restricted to cube-like cells

• Common types

  • Vertex, Line, Triangle. Quad. Rectangle. Tetrahedron. Hexahedron

Vertices:

• Represent sample points for interpolation

Uniform grids

• Equal spacing in all directions

• Simple structure but not always optimal for data representation

Rectilinear grids

• Non-uniform spacing, but still aligned with coordinate axes

• Resolution can vary along axes

Structured grids

• Arbitrary spacing of points while maintaining topological order

• Can model more complex shapes than uniform/rectilinear grids

Unstructured grids

• No topological constraints

• Maximum flexibility for complex shapes and surfaces

Uniform grids

• Advantage: Fast computation

• Limitation: Inefficient for complex shapes

Rectilinear grids

• Advantage: Allows axis-aligned resolution variation

• Limitation: Still limited in shape representation

Structured grids

• Advantage: More freedom in specifying points

• Limitation: Still constrained by grid topology

Unstructured grids

• Advantage: Full flexibility in shape modeling

• Limitation: Higher computational and storage complexity

Definition: Assign a scalar value (e.g., temperature, pressure) to each point in space

Grid dimensions: Can be 1D, 2D, 3D, or nD

Visualization methods:

• Map scalar values to graphical representations

• Use intensity, color, texture, isosurfaces

• Highlight features such as contours

• Combine with vector/tensor field visualization

Considerations:

• Sampling and reconstruction

• Choice of cell and grid types

A 3D analog of a contour line (like in topographic maps)

Represents all points within a volume that share the same scalar value

Extracted from scalar fields such as density, intensity, or temperature

Commonly used to visualize boundaries or regions of interest (e.g., tissue in medical imaging)

Typically visualized using surface extraction algorithms like Marching Cubes

Marching squares: 2D version of marching cubes (used in volume rendering)

Isoline extraction:

• Marching squares → finds isolines (e.g., equal height in heightmaps)

• Marching cubes → finds isosurfaces (e.g., equal density in volumes)

Ambiguity:

• Exists in both isolines (marching squares) and isosurfaces (marching cubes)

• Resolution of ambiguity depends on application and implementation

Cases:

• Marching squares: 4 unique cases (from 16 total)

• Marching cubes: 15 unique cases (from 256 total)

Advanced alternatives:

• Marching tetrahedra

• Marching triangles

Slicing and contouring operations are commutative

Equivalence:

• Slicing a volume at a plane and then computing contour lines = Computing the isosurface and slicing it at the same plane

Key implication:

• 3D isosurfaces can be constructed from a set of 2D isolines

• Done by slicing the volume into parallel planes and extracting isolines per slice

Vector field: A tuple of n scalars (usually 2 or 3)

Use case: Many applications require visualization of vector fields

Key domain: Computational Fluid Dynamics (CFD)

CFD solution:

• Typically multiple datasets, one per time step

• Each time step includes attributes like:

  • Velocity, Pressure, Density

Vector field visualization complements scalar visualization

Visual icons to represent vector fields

Encode properties: position, orientation, direction, size, color

Common types: lines (hedgehogs), arrows, cones

Displayed at subsampled field points to reduce clutter

Larger glyphs → lower sampling rate → less detail

Too dense glyphs may intersect and reduce readability

Discrete representation requires cognitive interpolation

Regular sampling may interfere visually → use random sampling

Extend 2D glyph concepts to 3D vector fields

Occlusion is a key challenge in 3D

Transparency helps reduce occlusion

Grayscale + transparency improves interpretability

Can be applied to surfaces, volumes, or along isosurfaces

Sampling: Density and regularity of seed points.

Discretization: Step size Δt affects accuracy and clutter.

Both parameters can be locally adapted for optimal results.

Δt is an integration parameter, not real time.

Tensor attributes generalize vectors to higher dimensions.

For planar curves, curvature at a point is the 2nd derivative at that point.

For surfaces, curvature in direction s is computed via intersecting a plane (spanned by s and the normal) with the surface.

There are infinitely many directions s at each surface point to compute curvature.

The Hessian matrix H contains 2nd-order partial derivatives of a function.

It enables calculation of the rate of gradient variation and normal curvature: s^T H s.

It helps classify critical points (minima, maxima, saddles).

For implicit surfaces (e.g., f(x,y,z)=0), a global 3×3 Hessian is used.

Tensors describe more than curvature, including diffusion, stress, etc., and are ranked by dimension.

PCA finds extreme directions (e.g., max/min curvature or diffusion) in tensor fields.

For 2×2 symmetric tensors, PCA yields two eigenvectors s_1, s_2 (directions) and eigenvalues \lambda_1, \lambda_2 (magnitudes).

These represent principal directions and values of curvature or diffusion.

For symmetric matrices, the principal directions are perpendicular to each other.

Tensor glyphs generalize vector glyphs and represent eigenvectors (directions) and eigenvalues (magnitudes).

They encode:

• Direction → via glyph orientation (typically 2 directions)

• Magnitude → via glyph shape/size in those directions

Glyphs can also be color-coded (e.g., based on direction).

Shapes include ellipsoids, cuboids, cylinders, superquadrics.

They share visualization issues with vector glyphs (e.g., clutter, occlusion).

Tracks neural fibers by constructing streamlines along high anisotropic regions (major eigenvectors).

Streamlines follow dominant diffusion directions in tensor fields.

Strong distinction between major and other eigenvectors to reduce noise.

Appropriate seed point selection is crucial and requires experience.

Current streamlines do not encode eigenvalues.

Hyper-streamlines are stream tubes constructed along the major eigenvector field (𝑒₁).

At each point, an elliptical cross-section represents the medium (𝑒₂) and minor (𝑒₃) eigenvectors.

Ellipse radii are scaled by the corresponding eigenvalues (λ₂, λ₃), encoding tensor shape and orientation.