1. Data Visualization Lecture 8 3D Scalar Visualization
Volume Rendering : Further Ray Casting plus Other Approaches

2. Classical Approach - Volume Rendering Integral
C(s)=light
reflected
at point s
Cast rays through image plane into volume, and measure light received
Kajiya and von Hertzen (1984)
Max (1995)
volume
m(s) =
density at
point s
image
plane s L
I = L0C(s)m(s) exp[ -s0 m(t)dt ] ds
light
density
attenuation

3. Simplifying the Integral
I = L0C(s)m(s) exp[ -s0 m(t)dt ] ds
Approximate using Riemann sums (n = number of steps)
Approximate exponential by Taylor series and introduce opacity, a, and unit spacing
Calculate recursively front-to-back as...
I = Sni=0 C(iDs)m(iDs)Ds
Pi-1j=0 exp [-m (jDs)Ds]
I = Sni=0 C(i)a(i) Pi-1j=0 (1 - a(j))
Cout = Cin + (1-ain)aiCi
aout = ain + ai(1 - ain)
{stop when
a = 1}
Compositing associative but not commutative
ie can group but cannot re-order

4. Interpolation
Sample points occur within cells, not at grid points, so we need to interpolate
Do we:
classify at grid points, then interpolate colour / opacity
interpolate data values, then classify ?
Classify - interpolate
classification done as pre-processing
smoothing effect can obscure detail
Interpolate - classify
classification now within the inner loop of the ray cast (sample points are view dependent)
in return, fine detail can be picked out

5. Classify - Interpolate
There is a danger in interpolation after classification
Naïve colour interpolation would assign 3 parts yellow, 1 part blue to centre point…
… but if opacity of bottom
left is zero?
Correct approach is to
weight according to opacity,
so colour at centre is yellow!

6. A Note on Shading
The appearance of volume rendered images depends critically on the reflectance calculation used to shade the samples...

7. A Note on Shading
Recall from isosurface rendering - that surface normal is equal to the gradient vector of f
At (xi,yj,zk), df/dx = 0.5*(f i+1,j,k - f i-1,j,k)
df/dy = 0.5*(f i,j+1,k - f i,j-1,k)
df/dz = 0.5*(f i,j,k+1 - f i,j,k-1)
This gives us normals at all grid points

8. A Note on Shading
If the classification is done at the vertices, there is a choice:
Gouraud-type: shade vertices and then interpolate to get reflected colour at sample point
Phong-type: interpolate normals at vertices to get sample normal and then calculate shading
Phong-type gives better quality - at the cost of more computational effort

9. Ray Casting Advantages Disadvantages Non binary classification
Shows structure between surfaces
Readily parallelisable
Disadvantages
Computationally expensive - cost proportional to number of voxels (compositing is expensive)
Does not take advantage of triangle rendering hardware

10. Ray Casting Examples
The following sequence of slides were produced using the ray casting technique available in the vtk software
The slides show for the bonsai tree data set how different aspects can be highlighted by control of the opacity transfer
The slides also show, by animation, a comparison of different approaches to interpolation and shading
Thanks to Chris Goodyer for creating the images

11. Legend NN = classify vertex, shade, nearest neighbour interpolate
Trilin Vertex = classify vertex, trilinear interpolate, shade
Trilin Sample = trilinear interpolate, classify, shade

12. Bonsai Tree NN – TrilinVertex – TrilinSample

13. Bonsai Tree Branches NN – TrilinVertex - TrilinSample

14. Bonsai Tree Leaves NN – TrilinVertex - TrilinSample

15. Skull NN – TrilinVertex - TrilinSample

16. Texture-based Volume Rendering
Volume rendering by ray casting is time-consuming
one ray per pixel
each ray involves tracking through volume calculating samples, and then compositing
different for each viewpoint
Alternative approach - using texture maps - can exploit graphics hardware

17. Texture Mapping
Modern graphics hardware includes facility to draw a textured polygon
The texture is an image with red, green, blue and alpha components…
… this is used in computer graphics to avoid constructing complex geometric models
… and we can exploit this in volume rendering

18. Texture-based Volume Rendering
Draw from back-to-front a set of rectangles
first rectangle drawn as an area of coloured pixels, with associated opacity, as determined by transfer function and interpolation - and merged with background in a compositing operation (supported by hardware)
successive rectangles drawn on top

19. 3D Texture-based Volume Rendering
For a given viewing direction, we would need to select slices perpendicular to this direction
This requires interpolation to get the values on the slices
Until recently this has only been possible with expensive graphics boards
volume
image
plane
3D texture mapping

20. 2D Texture-based Volume Rendering
Simpler solution - 2D texture mapping:
view volume as set of slices parallel to co-ordinate planes
-Precompute the textured planes for each of the 3 directions
-Choose the orientation best suited to viewing direction

21. Texture-based Volume Rendering

22. Comparison of Ray Casting and Texture Approaches
Texture-based
Texture-based
Ray casting

23. Close Up
Ogle: texture-based
Vlib: ray casting

24. Shear Warp Rendering
To get fast traversal, shear volume by translating each slice… then can resample as shown
Project front-to-back to get intermediate image
Then warp image

25. Example of Shear Warp Rendering

26. New Hardware Advances Holy grail: real-time volume rendering
Main searcher has been Kaufman through Cube architectures
VolumePro System first commercially available from Mitsubishi’s RealTime Visualization, now from TeraRecon
Uses shear-warp rendering

27. Splatting Another commonly used method is splatting
Fuzzy balls around each voxel projected on to image plane
Composited in the image plane
VolumeToGeom in IRIS Explorer

28. Summary of Volume Rendering Techniques
Ray casting
Splatting
Both high quality, computationally expensive
Texture-based
Shear-warp
Both lower quality, but faster

29. Reading Overviews of Volume Visualization Ray Casting - Classic Paper
Todd Elvins, `A Survey of Algorithms for Volume Visualization’, Computer Graphics, 1992
Ken Brodlie and Jason Wood, Computer Graphics Forum, 2001
Michael Meissner et al, ‘A Practical Evaluation of Popular Volume Rendering Algorithms’,
Ray Casting - Classic Paper
Marc Levoy, `Display of Surfaces from Volume Data’, IEEE CG&A, 1988
Book
Introduction to Volume Rendering, Lichtenbelt, Crane, Naqvi, Hewlett-Packard Professional Books