Volume Rendering

General

Volumetric Data

• Volume rendering visualizes 3D sampled data.
• Data can be measured, simulated, or computed procedurally.
• A volume can be understood as a stack of correlated 2D slices.
• Rendering the stack slice by slice is limited because it only works naturally from the slicing direction.

Voxels and Cells

• A voxel is one discrete sample in the volume.
• A voxel is homogeneous and stores one measured or computed value.
• A cell is the cube between 8 neighboring voxels.
• Cells are inhomogeneous because values inside them are interpolated.
• Trilinear interpolation gives smoother intermediate values than nearest-neighbor sampling.

CT Example

Scanner Idea

• An X-ray source emits rays through the object.
• A detector measures the remaining intensity.
• The measured intensity gives absorption along the ray.
• The source and detector rotate around the object.
• Many projections are reconstructed into one 2D slice.
• Moving the object through the scanner creates a stack of slices.

Reconstruction

• Absorption along a ray is a line integral.
• CT reconstruction inverts these projections.
• Important terms:
  • Radon transform maps object densities to projections.
  • Filtered backprojection reconstructs slices from projections.

Transfer Functions

Purpose

• Raw volume values are scalar data such as density, absorption, pressure, or simulation values.
• A transfer function maps scalar values to RGBA.
• It assigns color and opacity.
• Example mapping:
  • Air is transparent.
  • Soft tissue is semi-transparent.
  • Bone is opaque and bright.

Importance

• The transfer function decides what becomes visible.
• A poor transfer function hides important structures.
• A good transfer function separates materials and features.
• It can be a lookup table or a more complex function.

Texture-Based Rendering

3D Texture Slicing

• Store the volume as a 3D texture.
• Render many parallel polygons through the volume.
• Texture-map them with 3D texture coordinates.
• Render back-to-front with alpha blending.
• Slices can be axis aligned or view aligned.

Tradeoff

• The method is hardware accelerated and simple on GPUs.
• Quality depends on slice count and orientation.
• View-aligned slices need updates when the camera changes.

Alpha Blending

• Each sample has color $C$ and opacity $\alpha$.
• $\alpha=0$ means fully transparent.
• $\alpha=1$ means fully opaque.
• Colors and opacities are accumulated along the viewing direction.
• Once enough opacity is accumulated, deeper samples no longer matter.

Image-Order Rendering

Ray Casting

• Cast one ray through the volume for every pixel.
• Traverse the volume along the ray.
• Sample density, color, and opacity at positions on the ray.
• Accumulate samples into the final pixel.
• This is also called image-order rendering, pixel-space traversal, or back projection.

Ray Casting vs. Ray Tracing

• Volume ray casting usually follows one ray direction through the volume.
• It does not recursively spawn secondary rays by default.
• Ray Tracing traces secondary rays for reflection, refraction, and shadows.

Ray Transform vs. Volume Transform

Ray Transform

• Keep the volume fixed.
• Inversely transform every ray into volume space.
• For every sample:
  • Find the cell.
  • Interpolate values trilinearly.
  • Accumulate color and opacity.
• Advantage: the whole volume does not need to be resampled.
• Disadvantage: interpolation happens many times during traversal.

Volume Transform

• Transform or resample the volume first.
• Cast regular untransformed rays afterward.
• The transformation can be decomposed into shear operations.
• After each shear, voxel values need interpolation.
• Advantage: ray traversal becomes more regular.
• Disadvantage: resampling can blur data and costs memory and time.

Step Size

• Too large:
  • Small details are missed.
  • Aliasing increases.
• Too small:
  • Rendering becomes slow.
  • Too many samples are evaluated.
• Adaptive or scan-conversion techniques balance speed and quality better than a naive constant step width.

Composition and Acceleration

Composition Functions

• Average: average all sampled values.
• Maximum intensity projection: take the strongest value along a ray.
• Distance: encode distance to the first or most important hit.
• Alpha compositing: accumulate transparency and color.

Acceleration

• Early stopping:
  • Stop a ray when accumulated opacity is high enough.
• Empty-space skipping:
  • Skip regions below an opacity threshold.
  • Octrees or other space partitions can help.
• Bricking:
  • Split huge volumes into smaller chunks.
  • Load or render bricks sequentially if the full volume does not fit in memory.

Object-Order Rendering

Voxel Projection

• Traverse voxels or voxel planes.
• Project them onto the image plane.
• Accumulate into the framebuffer.
• This is also called object-order rendering, voxel-space traversal, or forward projection.

Traversal Order

• Front-to-back traversal can stop early when opacity is sufficient.
• Back-to-front traversal gives direct alpha blending and shows contributions sequentially.

Splatting

• Projecting only voxel centers can create holes.
• Splatting spreads each voxel over nearby pixels.
• A filter footprint, often Gaussian, weights the contribution.
• With orthographic projection, the same footprint can be precomputed.

Surfaces in Volumes

Gradient Shading

• If a voxel contains a surface fragment, recover a normal from the local gradient.
• The gradient direction approximates the surface normal $N$.
• If the gradient length is zero, the region is homogeneous and contains no surface.
• Use $N / |N|$ for Phong shading.
• Use $|N|$ to weight the strength of the surface contribution.

Limitation

• Gradient shading does not recover real occlusion from the light source to the surface.
• It gives a local shading look but does not extract an actual mesh.

Isosurface Extraction

Marching Cubes

• Choose an isovalue threshold.
• Process one cube or cell after another.
• Classify all 8 cube corners as above or below the threshold.
• Place polygons that separate both corner sets.
• There are $2^8 = 256$ cases.
• Symmetry reduces them to 15 unique cases.
• Interpolate along cube edges to get accurate polygon positions.
• The final mesh can be rendered normally.

When to Use

• Direct volume rendering is useful for semi-transparent and internal structures.
• Marching cubes is useful when one clear surface should become a triangle mesh.