Numerical Meshes from Seismic Images

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

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

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

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

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

6. Seismic Data Volumetric Visualization Seismic Image Enhancement of the Important Features Initial lattice Generation Minimization of the Potential Energy Atom Connection Aligned Mesh Simulation Method • Enhance the important features. • Image processing techniques. • 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. 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: 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. 10 Force Model • Interaction force among atoms is a piecewise polynomial function: d: nominal distance. , normalizeddistance.

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

12. 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:

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(xi), depends on the pixel value (grey level) associated to the image.

14. 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.

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

16. 16 Initial lattice • 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 Algorithm: Pseudo-regular lattice of atoms • 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 xi • If xi is onto the image limits • Make an sphere with centre xi and diameter d(xi) • 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 xi; • Add at the end of the queue ideal positions for neighbors Seismic Image Nominal distance function dmin = 6 (black pixels) dmax = 12 (white pixels)

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. 20 Lattice Optimizer • Get the initial lattice x1, x2, ..., xn • Compute the total potential energy of the initial lattice, P • Do { • P0 = P • Disturb x1, x2, ..., xn • Do { • Pi = P • One step of the SDA algorithm } While Pi – P > Є |Pi| } While P0 – P > Є |P0| Seismic Image Disturbance = 0.2 x d The threshold Є controls the iterations until the decreasing in P is negligible.

21. 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. Delaunay Triangulation: 545 atoms

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

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

24. 24 Results 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. 25 Results Sobel 3 x 3 Dilation 3 x 3 Brain

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

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

28. 28 Conclusions • 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. 29 Conclusions • 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. 30 Main References [Hale2001] “Atomic images – A Method for Meshing Digital Images”. Proceedings of the 10th 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.