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” SIGGRAPH ’87 Marschner and Lobb. “An evaluation of Reconstruction Filters for Volume Rendering” Vis ’94: Proceedings of the conference on Visualization ’94 Parker et al. “Interactive Ray Tracing for Volume Visualization” IEEE Transactions on Visualization and Computer Graphics Rossl et al. “Visualization of Volume Data with Quadratic Super Splines” VIS. ’03: Proceedings of the 14 th 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 Only this will depend on (a,b,c) BT Volumes - Reconstruction Why does this work? Consider volume reconstruction within an arbitrary tetrahedron, T New weight term

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 (64 3 ) Engine (128 3 ) Bucky ball (32 3 )

19 Results We are able to render BT Volumes up to 64 3 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

23 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 Bezier Tetrahedra (cont) Cubic Bezier Tetrahedra are a family of Bezier solids defined by a tetrahedron and a set of 20 weights 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 don’t 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 Proof: Lets take the inverse transform so becomes and therefore when we have a constant term which can combine with c 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