1. Numerical Meshes from Seismic Images Karl Apaza Agüero Paulo Roma Cavalcanti Antonio Oliveira Claudio Esperança COPPE – Sistemas - UFRJ

2. Goal Creation of numerical meshes from seismic images. Integrates several techniques: Image Processing. Physical Modeling. Optimization. Computational Geometry.

3. Seismic Methods 3 Based on the emission of acoustic waves onto the surface of the earth or the sea. Reflection Method: ●Acquisition. ●Processing. ●Interpretation.

4. Traditional Approach Seismic  Geometric Model  Mesh 4  Geometric Model = Set of curves and surfaces ● Horizons: separating surfaces between geological layers. ● Faults: discontinuities produced by sliding of layers. Geometric Model

5. Alternative Approach IDEA: Seismic  Mesh 5  Generates meshes directly from seismic images.  Avoids the creation of an intermediary geometric model.  Extracts horizons and faults directly from the mesh. Mesh

6. Method  Enhance the important features. ● Image processing techniques. Volumetric Visualization Seismic Image Seismic Data Enhancement of the Important Features Initial lattice Generation Minimization of the Potential Energy Atom Connection Aligned Mesh Simulation  Generate an initial lattice of atoms based on the important features. ● Interaction force between atoms. ● Pseudo regular lattice.  Minimize the total potential energy function. ● Interaction force between atoms. ● Steepest Descent Method.  Connect atoms. ● Delaunay Triangulation / Voronoi Tessellation.

7. Atoms  An atom is an image point subjected to forces exerted by its neighbors.  Influence zone depends on a threshold distance D.  An inter-atomic force must satisfy: 7 Interaction among atoms

8. Properties  Be null beyond a certain distance, limiting the influence zone of an atom.  Be a continuous function of the inter-atomic force.  Be repulsive (positive) to avoid atoms very close to each other.  Be attractive (negative) to avoid large empty spaces, when atoms are far away from each other.

9. Nominal Distance  Nominal distance, d, is the distance where attraction forces turn into repulsion forces.

10. Force Model  Interaction force among atoms is a piecewise polynomial function: 10 d: nominal distance., normalized distance.

11. Scalar Potential  To employ minimization techniques: force is defined as the negative of the gradient of an scalar potential field.

12. Atomic Potential Energy  Is the weighted sum of each atom energy in the system.  The atom energy is the sum of the forces exerted onto it by its neighbors: 12

13. Image Potential Energy  Is the sum of the potential field of the image pixels associated to atoms.  The potential field of a point, b(x i ), depends on the pixel value (grey level) associated to the image.

14. Total Potential Energy  Is the weighted sum of the atomic potential energy and the image potential energy:  The scale, ß, determines the relative contribution of A and B. ß=0  atoms create a regular lattice, not necessarily aligned to the important features. ß=1  atoms are sensitive only to the important features, producing a highly irregular lattice.  Depends on the type of atom connection. 14

15. Enhancement of the features  Sobel Detector  Identifies important features on the image. Image differentiation. Image smoothing. 15  Morphological Operators  Dilation and erosion operators enhance the important features.  Thicken or thin important features. Erosion: 3 x 3 mask

16. Initial lattice 16  The initial lattice of atoms should have the following characteristics: Minimize, locally, the atomic potential energy. Be highly regular. Be consistent with the nominal distance function.

17. Nominal Distance Function  For a constant function, it is easy to obtain a regular initial lattice holding the previous properties.  A rectangular lattice is the simplest choice.  An hexagonal lattice is the best solution for an initial lattice of points.  A non-constant function poses some difficulties.

18. 18 Seismic Image Nominal distance function d min = 6 (black pixels) d max = 12 (white pixels)  Make an array of boolean flags, w(x)=false  Create an empty list of atoms  Create an empty queue of atom positions  Add to the queue the position of the image centre  While the queue is not empty Get, and remove from the queue, the first position x i If x i is onto the image limits Make an sphere with centre x i and diameter d(x i ) If the sphere contains positions with w(x)=false Do for all positions inside the sphere w(x)=true; Add to the list an atom with coordinates x i ; Add at the end of the queue ideal positions for neighbors Algorithm: Pseudo-regular lattice of atoms

19. Minimization of The Energy Function  After the creation of the initial lattice, the atoms must be moved to a configuration that minimizes the total potential energy, P.  The Steepest Descent Algorithm (SDA) is used to minimize the total potential energy function, which may possess several local minima.  The search is repeated until the best minimum is found.

20. Lattice Optimizer 20 The threshold Є controls the iterations until the decreasing in P is negligible. Seismic Image Disturbance = 0.2 x d  Get the initial lattice x 1, x 2,..., x n  Compute the total potential energy of the initial lattice, P  Do { P 0 = P Disturb x 1, x 2,..., x n Do { P i = P One step of the SDA algorithm } While P i – P > Є |P i | } While P 0 – P > Є |P 0 |

21. Delaunay Triangulation  The optimized lattice of atoms is structured through a Delaunay triangulation or a Voronoi tessellation.  Both schemes tend to create edges (in 2D) and faces (in 3D) aligned to the important features of the image.  A Delaunay triangulation always connects atoms to its closest neighbors. 21 Delaunay Triangulation: 545 atoms

22. Voronoi Tessellation 22  Voronoi connects circumcentres of Delaunay triangles.  Atoms concentrate near the boundaries of the important features.

23. 23 Voronoi Tessellation 652 atoms Initial lattice d min = 5 d max = 10 Optimized lattice Dist. = 0.2 x d Voronoi Tessellation onto optimized lattice.

24. Results 24 Delaunay Triangulation 495 atoms dmin = 10 (black pixels) dmax = 20 (white pixels) Disturbance = 0.1 x d Voronoi Tessellation 775 atoms dmin = 8 (black pixels) dmax = 16 (white pixels) Disturbance = 0.1 x d

25. Results 25 Sobel 3 x 3 Dilation 3 x 3 Brain

26. Results 26 Final Mesh pixel color = triangle circumcentre Optimized lattice dmin = 3 (black pixels) dmax = 9 (white pixels) Disturbance = 0.2 x d Mesh 1700 atoms

27. Results 27 Delaunay Triangulation generated for the seismic volume of The Stratton Field, South of Texas.

28. Conclusions 28  The enhancement of the important features is fundamental to the presented method, in order that the point optimizer produce good results.  For an image with smooth luminance, the method is able to align the mesh to the important features.

29. Conclusions 29  The presented parameters can be applied to a great number of images.  If the input image do not allow the closing of regions, maybe because it was not filtered appropriately, the method does not close the “holes".  The method can be used to segment a large range of images.

30. Main References 30 [Hale2001] “Atomic images – A Method for Meshing Digital Images”. Proceedings of the 10 th International Meshing Roundtable, pp. 185-196. 2001. [Hale2002] “Atomic meshes: from seismic imaging to reservoir simulation”. Proceedings of the 8th European Conference on the Mathematics of Oil Recovery. 2002. [Jalba2004] “CPM: A Deformable Model for Shape Recovery and Segmentation Based on Charged Particles”. IEEE Transactions on Pattern Analysis and Machine Intelligence. 2004.