Minimal Surfaces using Watershed and Graph-Cuts Jean Stawiaski, Etienne Decencière 8 th International Symposium on Mathematical Morphology

2 Outline Minimal Surfaces Cauchy-Crofton Formula Combining Watershed and Graph-Cuts Conclusion

3 Minimal Surfaces Variational Methods (Snakes, Geodesics Active Contours, Continuous Maximal Flow, etc.) Graph Based Methods ( Graph Cuts, GeoCuts ) Combination of Graph Based Methods with a Morphological Segmentation.

4 Minimal Surfaces Markers We are looking for a curve that separates the markers and such that the gray levels sum under the curve is minimum.

5 is the Euclidean Length of the curve. is the arc length on the curve. is a striclty positive decreasing function. is the gradient operator. is an image and a curve. Minimal Surfaces The problem is formalized as the minimization of the energy:

6 Minimal Surfaces The Riemannian length of a curve is given by : Where D is the following metric tensor :

7 Link with graph cuts C How to set the arcs capacities such that the value of the graph cut equals the length of the curve ? Markers A curve separating the markers. The capacity of the graph cut equals the sum of the arcs capacities being cut in the graph.

8 Cauchy-Crofton Formulas is a line of R 2 is a set of lines The Cauchy-Crofton formulae establishes that : C L1 L3 L2 y x

9 Cauchy-Crofton Formulas

10 Cauchy-Crofton Formulas C

11 Cauchy-Crofton Formulas Extension of the formulae to 2D Riemannian Spaces: Extension of the formulae to 3D Riemannian Spaces :

12 Example Extension of formulas to 2D Riemannian spaces :

13 Markers Example Arcs weights: Original Image Watershed Graph Cuts

14 Watershed and Graph-Cuts Boundary between regions :

15 Watershed and Graph-Cuts By using the regions adjacency graph we are looking in the space of curves defined by the borders between regions. A cut in the graph defines a curve formed by borders of regions

16 Watershed and Graph-Cuts Let us define a strictly positive decreasing function : The Riemannian length of the border between two regions is then approximated by :

17 Watershed and Graph-Cuts The extension to 3D is straightforward : Using the regions adjacency graph instead of the pixels graph we can define more complex geometric functionals.

18 Watershed and Graph-Cuts Geodesic defined in the graph of pixels Markers In good situations our approximations is close to the true geodesic. Geodesic defined in the space of curve formed by borders between regions.

19 Results We have developed a full software for: - 3D image visualization (3D rendering, surface rendering, multiple volume overlays, volume slices) - Image segmentation : - Morphological tools (filtering, watershed) - Graph tools (min-max spanning forest, Shortest path forest, min cuts, multi-way cuts, etc.)

20 Results

21 Results

22 Results

23 Results Our method Graph Cut on the pixel graph. Marker- Controlled Watershed

24 Results ImageSizePixel GraphRegion Graph 3D Image*69x69x sec.1.36 sec. Heart MRI86x128x sec.4.2 sec. Liver CT256x193x sec.41.6 sec. Laptop, Intel Core Duo 2.16 Ghz, 1Gb Memory.

25 Conclusion Our method reduces the computational cost of classical methods without affecting, in practice, the quality of the segmentation. The method can be used interactively. Energy based on regional properties can take into account more complex geometric functionnals (curvature). The same approach was also used for different image segmentation problems (markov random field).

26 Acknowledgments Our work is funded by the Canceropôle of Île-de-France : Some Informations on our webpage : Do you want to try our software :