1. Reconstructing Porous Structures from a Statistical Representation Craig Schroeder CSGSC October 6, 2004

2. Outline ● Why store porous structures? ● Why store them statistically? ● Choosing a statistical description ● Brute force reconstruction ● Brute force layerwise reconstruction ● Alternative layerwise approaches

3. Porous Materials ● Many common materials are porous – Bread, sponge, bone, rock structures ● Porosity is often essential to the properties and uses of an object – Makes bones lighter and more flexible – Allows water to flow through permeable rocks Osteoporotic Bone, NASA

4. Representing Porous Materials ● Exactly – Store entire geometry as a mesh or volume – Does not lose information – Overly verbose; stores information that is not needed ● Store density – Loses potentially important information ● Connectivity, strength, pore size – Concise

5. Store Statistical Properties ● Measure statistical or engineering properties of a given object and store those properties ● May retain desired properties ● Very concise ● May permit an object “similar” to the original to be reconstructed ● Statistical properties may possibly be known or determined based on requirements

6. Choosing Statistical Properties ● One could choose to store just about anything ● Some are better than others – Properties that need to be preserved – Properties that tend to describe overall structure – Properties that make measurement and reconstruction easier or more practical ● Local is in general easier than global – Properties that would be useful for designing structures directly

7. Our Choice ● A variation on the spherical contact distribution – From each point inside the object, measure the distance to the nearest pore – From each point inside a pore, measure the distance to the object – Combine these two sets of measurements into two distributions; this is what we use

8. Our Choice ● Local property, can be efficiently locally updated ● Isotropic – no information about direction

9. Goal of Reconstruction ● Use the stored statistical properties ● Construct an object with the same properties ● May also want desirable features – Can be fabricated – no disconnected material – No isolated pores in the object ● Efficient

10. 3D Reconstruction ● Brute force – entire solid at once – Initialize – Move material around to improve fit – Converge to desired properties

11. 3D Reconstruction ● Initialize – Fill a grid of voxels randomly with material to obtain the desired density ● Move material around – Swap voxels ● Preserves density – Invert voxels ● Does not preserve density ● More atomic

12. 3D Reconstruction ● Improve fit – Update the properties of the solid at each iteration – If fit has improved, keep the changes – If the fit has become worse ● Reject changes ● Reject changes with probabilistically – How much worse is the new fit? – How far along in reconstruction are we? – Allow more exploration early – Reduce exploration later for convergence

13. 3D Reconstruction ● Converge to the desired distribution – Simulated annealing (probabilistic reject) gives much better results than a strict reject – Very, very close fit to desired properties – Slow – tens of thousands of voxels requires many hours of computation time

14. Piecewise Reconstruction ● Perform reconstruction on smaller pieces first ● Combine pieces together to create larger object ● More economical ● Reduced quality

15. Layerwise Reconstruction ● Build the final volume up layer by layer ● Each layer is reconstructed using the same approach as was used for 3D ● Layers must not be independent – Reconstruct layers given the layers already constructed – Causes some overconstaint problems

16. Layerwise Reconstruction ● Measure and use properties of a layer from the original object for the reconstruction ● Stack up layers in order ● Seems to produce a good fit to 3D properties ● Has very serious layer artifacts ● Layers are too different – Consecutive layers only ~70% same ● Lacks good connectivity characteristics

17. Other Layerwise Possibilities ● Store each layer's properties separately ● Store statistical relationships between consecutive layers in addition to layer properties to improve interface between layers ● Store statistical information that may be used to construct layers from adjacent ones – Seed layer – Change one layer into next layer

18. Other Layerwise Possibilities ● Level sets? – Not troubled by topology – Can they reconstruct from statistical properties? – What about connectivity? ● Swept volumes? – Topology is a major concern here – Has potential – Connectivity is covered

19. What Lies Beyond ● Construct porous objects from engineering properties directly – Strength, connectivity, flexibility, flow – New way to design structures ● Bone scaffolds and replacements – Scaffold for live tissue