Interactive Volume Isosurface Rendering Using BT Volumes

1
Interactive Volume Isosurface Rendering Using BT
Volumes

• John Kloetzli
• Marc Olano
• Penny Rheingans
• UMBC

2
Overview

• Introduction and Previous Work
• Background
• Volume Reconstruction
• Bezier Tetrahedra
• BT Volumes
• Definition
• Function Fitting
• Reconstruction
• Rendering
• Results and Conclusions

3
Introduction

• Isosurface rendering is important in many
  applications
• Medical visualization
• Chemistry
• Marching Cubes is the most popular interactive
  technique
• Very fast, BUT
• Low quality reconstruction

Marching Cubes rendering
4
Introduction (cont)

• BT Volumes can do better than MC
• Interactive frame rates
• Isosurface level changed on the fly
• High-quality reconstruction
• Many filters possible

BT Volume rendering
5
Previous Work

• Lorensen et al. Marching cubes A
  High-Resolution 3D Surface Construction
  Algorithm 1987. SIGGRAPH 87
• Marschner and Lobb. An evaluation of
  Reconstruction Filters for Volume Rendering
  1994. Vis 94 Proceedings of the conference on
  Visualization 94
• Parker et al. Interactive Ray Tracing for Volume
  Visualization 1999. IEEE Transactions on
  Visualization and Computer Graphics
• Rossl et al. Visualization of Volume Data with
  Quadratic Super Splines 2003. VIS. 03
  Proceedings of the 14th IEEE Visualization

6
Overview

• Introduction and Previous Work
• Background
• Volume Reconstruction
• Bezier Tetrahedra
• BT Volumes
• Definition
• Function Fitting
• Reconstruction
• Rendering
• Results and Conclusions

7
Background

• Discrete Convolution is the mathematical tool
  used to reconstruct a continuous function from a
  discrete sampling
• The Filter Kernel is a function used to
  combine sample points from a scalar volume

8
Bezier Tetrahedra

• Cubic Bezier Tetrahedra are a family of Bezier
  solids defined by a tetrahedron T and a set of 20
  weights
• BT can be rendered quickly using graphics hardware

Loop and Blinn, Real-Time GPU Rendering of
Piecewise Algebraic Surfaces (Siggraph
Proceedings, Boston, 2006)
9
Overview

• Introduction and Previous Work
• Background
• Volume Reconstruction
• Bezier Tetrahedra
• BT Volumes
• Definition
• Function Fitting
• Reconstruction
• Rendering
• Results and Conclusions

10
BT Volumes - Overview

• We introduce the BT Volume
• Piecewise-defined continuous 3D function
• Tetrahedral grid of Bezier Tetrahedra
• BT Volumes allow
• Direct volume fitting
• Exact filtering with approximated filters
• Interactive rendering of isosurfaces

11
BT Volumes - Definition

• In order to define a BT Volume we need to create
  a tetrahedral partition
• We use a special mapping to define how
  each voxel divides into tetrahedra
• must be applied in exactly the same way to
  each voxel
• The result is a shift invariant partition of 3D
  space
• We can create a continuous function by
  associating BT weights with each tetrahedron

12
BT Volumes - Fitting

• We can construct BT Volumes by approximating
  existing 3D functions
• We used simple least-squares method

We used the a 6-tetrahedron function for our
work
13
BT Volumes - Reconstruction

• We could use direct fitting on a large volume,
  but it is impractical
• A better technique is to fit a reconstruction
  filter as a BT Volume
• The filter will be much smaller than the volume
• Convolution of a BT Volume with a discrete scalar
  volume will produce another BT Volume
• The resulting BT Volume will retain the
  characteristics of the reconstruction filter

14
BT Volumes - Reconstruction

• This works because the function enforces that
  all BT in the same relative position will line up
  correctly
• Convolution will be the summation of scaled BT

15
BT Volumes - Reconstruction
Why does this work? Consider volume
reconstruction within an arbitrary tetrahedron, T
New weight term
Only this will depend on (a,b,c)
16
BT Volumes - Rendering

• We can render the BT Volume interactively using
  the method devised by Loop and Blinn
• Generate the BT Volume
• Store each Bezier Tetrahedron as a single vertex
• Geometry shader creates screen-space triangles
• Pixel shader solves for particular isosurface

17
Overview

• Introduction and Previous Work
• Background
• Volume Reconstruction
• Bezier Tetrahedra
• BT Volumes
• Definition
• Function Fitting
• Reconstruction
• Rendering
• Results and Conclusions

18
Results
Molecule (643)
Bucky ball (323)
Engine (1283)
19
Results

• We are able to render BT Volumes up to 643
  interactively
• BT Volumes do require a lot of space

NVIDIA 8800 GTS
20
Results

• Video

21
Conclusions

• We have presented BT Volumes as a volume
  representation format capable of
• Direct function fitting
• Convolution filtering
• Interactive rendering
• Future work
• Examination of more expressive functions
• Analysis of filter approximations
• Compression/storage methods

22
Questions?

• Acknowledgements
• The Volume Library provided volume data sets
  (www9.informatik.uni-erlangen.de/External/vollib/
  )
• Dr. Alark Joshi and Jesus Caban for their support
• The entire VANGOGH lab at UMBC for their help
• This work was funded in part by NSF grant 0121288

23
Bezier Tetrahedra (cont)

• Cubic Bezier Tetrahedra are a family of Bezier
  solids defined by a tetrahedron and a set of 20
  weights

Let be the positions of the vertices
of the tetrahedron and be the
weights. Then we can transform a point in
Euclidian space into barycentric space by
and the Bezier Tetrahedron is defined as
Loop and Blinn, Real-Time GPU Rendering of
Piecewise Algebraic Surfaces (Siggraph
Proceedings, Boston, 2006)
24
BT Volumes - Rendering

• We can exploit properties of the volume and BT to
  get early-out tests for the pixel shader
• Store the minimum and maximum BT weight values
  for the filter with each tetrahedron
• If the current isosurface level is not within
  that range, we dont have to proceed any further
  for that tetrahedron
• Space requirements 2 floating point values per
  tetrahedra
• Speedup Variable. If the volume has empty
  space, this will speed rendering up considerably

25
BT Volumes - Rendering

• Changing isosurface level
• How can we render any isosuface level?
• We want a space where the polynomial has a
  constant term which can absorb the constant
• Euclidian space works for this purpose

Proof Lets take the inverse transform so
becomes and therefore when we have a
constant term which can combine with c