1. Craig Schroeder October 26, 2004
Representation and Reconstruction of Porous Objects using Deformable Models
Craig Schroeder
October 26, 2004

2. Outline Problem General Approach Implementation Further Work
Measuring Distribution
Determining Next Move
Further Work

3. Problem Generate a 3D porous object in layers
Given properties desired at each layer
Given a seed layer

4. Constraints on Output Adjacent layers should be connected
Adjacent layers should be “similar”
No disconnected material or pore

5. General Approach Same basic layerwise goals
Start with a seed layer
Modify each layer to obtain the next layer
Use spherical contact distribution

6. Internal Representation
Do not use voxels
Use a set of polygons
Polygons are disjoint
Polygons are “swept” from layer to layer
Polygons may merge or split
No plans for insertion or deletion of polygons at this time

7. Deformable Models Use deformable models
Currently used in computer graphics and computer vision
Used for picking out outlines of objects from 3D data
Find the boundary of organs in the body

8. Deformable Models
Can be represented as a closed polygon or set of polygons
Deformed by moving vertices
Move in direction of local normal
Normal at vertex is the average of the norms of the two edges beside it
May move vertex inward or outward

9. Overview of Algorithm For each layer
Start with previous layer (seed first time)
While layer is not close enough to properties
Measure the distribution
Determine how the layer should change to improve the fit
Update the layer
“Stitch” the layer together with the previous layer to create a band of triangles
Output the resulting mesh

10. Measuring the Distribution
Must determine the spherical contact distribution of a layer
Quite a bit more difficult than for voxels
Requires a tradeoff of time and difficulty

11. Idea There are not a finite number of elements Problems
Sample the area
Compute the distance for each element
Accumulate into the distributions
Problems
Computing the distance from a particular point is not easy
Only approximate
Getting a good approximation may take too long

12. Exact Computation Compute the distribution exactly
More difficult to implement
More efficient
Exact measurement is possible

13. Breaking Up the Problem
Break the layer into a Voronoi diagram
Splits the layer into tiles
Each tile contains all points closer to one vertex than any other
Cheap to compute – O(n log n)

14. Benefits of Voronoi
Can compute spherical contact distribution of a tile independently of all other tiles
Sites are convex – makes life easier
Tiles are on average of constant size

15. Complication We want this to wrap
This changes the topology of space and causes problems
Solution – make eight copies of everything around yourself, compute the diagram, and throw everything else out

16. Properties of a Tile Tile contains points inside and outside
Some points are actually closer to an edge of the polygon than it the tile’s site
Makes it a bit tricky to measure distribution

17. Split the Tile Split a Voronoi tile into regions
Is the point inside the polygon or outside?
Is the point closest to the site, the entering polygon edge, or the leaving polygon edge?
Six regions: in-site, in-enter, in-leave, out-site, out-enter, out-leave

18. About the Regions In-site or out-site must be empty
Both empty of both edges collinear
Each region is easier to measure since the nearest object is known

19. Measuring Edge Region Perform a line sweep
Sweep a line from the edge on the polygon across the region
Collect area in each bucket as we go

20. Sweep Events
Vertices
The sweep encounters a discontinuity at each vertex in the region and needs an event here

21. Sweep Events Bin boundaries
The region is being divided into areas according to its distance. Where a bin boundary occurs, the bin receiving credit for the area changes, so an event is needed

22. At Each event Compute area bounded by Sweep line at previous event
Sweep line at new event
Sides of region
Region is convex, so there is one area per event
Update the appropriate bin by that area

23. Measuring Site Region In-site or Out-site Perform a “line sweep”
Since the distance is to a point, the fronts are actually circular arcs
Has same vertex and bin events

24. Additional Event We need another type of event
Each point where the circular front is tangent to the region is an event
This event results in multiple areas per event
Not “monotone” with respect to a circle

25. Combine the Pieces
Sum up the contributions to each bin for each region and tile
Divide accumulated area by total area

26. Determining Change
We need to know how a change in vertex affects the distribution
Compute a “derivative” of the distribution with respect to a change in each vertex

27. Distribution Derivative
Compute the distribution
Offset site face slightly
Recompute
Subtract and divide by offset
Approximate, but good enough

28. Computing Change
Consider G as the goal distribution for the current layer, arranged in a vector
Let M be the measured distribution
Let D be the matrix whose columns are the derivatives as vectors
Di xi is the approximate change in current distribution where a change xi is made to vertex with derivative Di

29. Computing Change
D x is the sum of the change resulting from offsetting each vertex by the corresponding amount in vector x
M + D x is approximately the new distribution that would result
We want M + D x = G
Least squares
Minimize norm of M + D x - G

30. Making the Move Solve the least squares problem for x
Components of x tell how much to move corresponding vertices
Apply the offset

31. Potential Problems
What if two polygons intersect as a result of this move?
Unite the polygons into one (merge)
What if a polygon intersects itself?
Split the polygon up into simple pieces
What if a polygon wraps around the layer?
This is a problem; will it happen?

32. Converge Will this converge? Do not know, not able to run tests yet
Intuition
If the current configuration is not too far off, it should converge quickly
If far off, it will get caught in a local minimum
Distribution should not change much from layer to layer, so this may not be a huge problem

33. Future Work Finish !! Consider insertion and deletion of polygons
Consider ways to leave local minima; insertion and deletion might be useful for this

34. Future Work
Do this in 3D
The Voronoi decomposition is still applicable here, as is the tiling workaround
Many more regions
Closest to site, edge, or face
Sweeps will be similar, but much uglier and harder to implement
No layering necessary
Entire problem in one run