0. Frank Heckel1, Olaf Konrad2, Heinz-Otto Peitgen1
Fast and Smooth Interactive Segmentation of Medical Images Using Variational Interpolation
Frank Heckel1, Olaf Konrad2, Heinz-Otto Peitgen1
1 Fraunhofer MEVIS, Germany
2 MeVis Medical Solutions, Germany

1. Overview Introduction Variational Interpolation
Interactive Segmentation using Variational Interpolation
Conclusion
Outlook

2. Introduction Automatic segmentation Good reproducibility
Not always possible
Manual segmentation
Time-consuming
Bad reproducibility
Semi-automatic segmentation
Automatic algorithm which requires user interaction
Interactive segmentation: Algorithm reacts on user in almost real-time (calculation times <1s)

3. Introduction Manual segmentation by drawing contours
Goal: Interpolation of a 3D mask from a few (user drawn) contours
Conditions:
Parallel or arbitrary oriented contours
Smooth (at least C1-continuous)
Fast (interactive calculation times)
Surface shall contain all contour points
“Extrapolation” character
Solution: Variational Interpolation

4. Variational Interpolation
Radial basis function
3D extension of thin-plate spline interpolation
Object is defined by an implicit function:
Constraints for f(x):
Smooth (C1- or C2-continuous)
Contains all contour points
“Surface constraints”
The normal on the surface corresponds to the normal in each contour point
“Normal constraints”
Surface constraint
Normal constraint

5. Variational Interpolation
Solution
Solve the linear system that describes f(x) to get the weights wj
Matrix is symmetric  Bunch-Kaufmann algorithm (LAPACK)
Global interpolation scheme

6. Interactive Segmentation using Variational Interpolation
Voxelization
Iteratively evaluate f(x) only on the surface (surface tracking similar to marching cubes)
Start points: first point of each contour (unless it is already part of a reconstructed surface)
Fill the surface slice-wise to get a segmentation of the whole object(s)

7. Interactive Segmentation using Variational Interpolation
Segmentation process
Draw / Delete / Edit Contours
Calculate Segmentation
Check Result

8. Interactive Segmentation using Variational Interpolation
Performance
Speedup
Solving the linear system is O(n³)
Fast LAPACK implementation is essential
Support 64-bit, SSE, Multi-Threading
Intel Math Kernel Library (MKL)
AMD Core Math Library (ACML)
Voxelization is O(nm)
Multi-threading can be used
Current implementation does not scale very well
Threads
CLAPACK
MKL SP 1
52.75s
10.01s
5.27 2 -
5.74s
9.19 4
3.57s
14.78 8
2.89s
18.25
Threads
Voxelization SP 1
4.15s
1.00 2
2.49s
1.67 4
1.92s
2.16 8
2.13s
1.95
Metastasis example* 17 contours, n=6246 constraints, m=23541 surface voxels
Test system: 2x Intel Xeon X5550 (2.66GHz, Turbo Boost and Hyper-Threading disabled), 12 GB RAM, Windows 7 64-bit, MKL

9. Interactive Segmentation using Variational Interpolation
Performance
Quality preserving constraint reduction to become interactive
Points that don’t contribute to the overall geometry are removed
Measure: Angle and length of adjacent edges
q=1.0
q=0.2
Quality n
LAPACK
Voxelization
Total SP
Overlap
1.0
6246
2.89s
1.92s
5.13s
1.00
100%
0.5
3608
0.75s
1.18s
2.08s
2.47
99.94%
0.2
1410
0.08s
0.55s
0.67s
7.66
99.32%
0.1
710
0.02s
0.36s
0.41s
12.51
97.05%
Test system: 2x Intel Xeon X5550 (2.66GHz, Turbo Boost and Hyper-Threading disabled), 12 GB RAM, Windows 7 64-bit, MKL (8 threads), Voxelization with 4 threads

10. Interactive Segmentation using Variational Interpolation
Robustness
Self-intersecting contours
Sign of normal constraint needs to be modified
Additional start point for surface tracking at intersection
Reduced contours
Additional constraints might have to be inserted for long line segments

11. Interactive Segmentation using Variational Interpolation
Limitations
Memory consumption is O(n²)
Performance for large objects not interactive by now
Voxelization becomes bottleneck
Liver example (14 contours, surface voxels):
Quality n
LAPACK
Voxelization
Total
Overlap
1.0
13796
23.42s
28.95s
53.28s
100%
0.5
7916
4.94s
16.79s
21.94s
99.5%
0.2
3170
0.5s
7.3s
7.82s
98.74%
0.1
1594
0.1s
8.3s
8.41s
97.68%
Test system: 2x Intel Xeon X5550 (2.66GHz, Turbo Boost and Hyper-Threading disabled), 12 GB RAM, Windows 7 64-bit, MKL (8 threads), Voxelization with 4 threads

12. Interactive Segmentation using Variational Interpolation
Limitations
Normals are calculated in 2D (based on one contour)
Contradictory contours
More complex self-intersections
Low number of constraints

13. Conclusion Accurate and smooth surface reconstruction
Contours can be arbitrarily oriented
Plausible extrapolation
Interactive for small objects if:
Fast LAPACK implementation is used (MKL or ACML)
Voxelization is parallelized
The number of constraints is reduced
Purely geometrical
Independent of modality or scanning parameters
Wide range of segmentation tasks
Can be easily extended to 4D (needs different radial basis functions)

14. Outlook Reduction of voxelization time More advanced parallelization
GPU?
Further improve robustness
Investigate local interpolation schemes
E.g., compactly supported radial basis functions by Morse et al.*
Algorithm is available in MeVisLab 2.1 (“CSOConvertTo3DMask”)
Uses only CLAPACK
* Bryan S. Morse et al., “Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions”, ACM SIGGRAPH Courses, 2005

15. Thanks to:
Thank you!

16. Variational Interpolation
Effect of the normal constraints:
G. Turk and J. F. O’Brien, “Modelling with implicit surfaces that interpolate”, ACM Transactions on Graphics, Vol. 21(4), 2002

17. Variational Interpolation
Difference to shape-based interpolation (distance transform):
Distance transform
Variational interpolation
G. Turk and J. F. O’Brien, “Shape transformation using variational implicit functions”, Proceedings of SIGGRAPH’99, 1999