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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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?

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

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

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