Reconstructing Porous Structures from a Statistical Representation Craig Schroeder CSGSC October 6, 2004 Outline Why store porous structures? Why store them statistically? Choosing a statistical description Brute force reconstruction Brute force layerwise reconstruction Alternative layerwise approaches 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 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 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 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 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 Our Choice Local property, can be efficiently locally updated Isotropic no information about direction 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 3D Reconstruction Brute force entire solid at once Initialize Move material around to improve fit Converge to desired properties 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 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 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 Piecewise Reconstruction Perform reconstruction on smaller pieces first Combine pieces together to create larger object More economical Reduced quality 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 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 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 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 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