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Computing handles and tunnels in 3D models

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### Computing handles and tunnels in 3D models

Tamal K. Dey

Joint work: Kuiyu Li, Jian Sun, D. Cohen-Steiner

Projects in Jyamiti group

- Surface reconstrucion (Cocone)
- Delaunay meshing (DelPSC)
- Shape feature processing (Segmatch, Cskel)

Use of Handles and Tunnels

- Topological simplification / feature recognition / parameterization
- El-Sana and Varshney[1997] use α-hulls to search for tunnels.
- Guskov and Wood[2001] propose a surface growing strategy to remove small handles.
- Nooruddin and Turk[2003] use morphological operations to remove small handles.
- Wood, Hoppe, Desbrun, and Schroder[2004] use Reeb graph to remove handles.
- Shattuck and Leahy[2001] use Reeb graph to compute handles.
- From WUSTL Model/surface editing (Zhou, Ju, and Hu[2007] use medial axis to detect loops) and work by Grimm et al.

Nontrivial loops as first attempt?

- Nontrivial loops can neither be handle nor tunnel.
- Handle and tunnel loops have to be non-separating.

Related work for loops on surface

- Loops on a surface
- Compute polygon schemas [Vegter-Yap90][Dey-Schipper95]
- Linear time algorithm

- Compute optimal systems of loops [VL05][EW06]
- Verdiere and Lazarus gave an algorithm for computing a system of loops which is shortest among the homotopy class of a given one.
- Erickson and Whittlesey gave a greedy algorithm to compute the shortest system of loops among all systems of loops.

- Compute optimal cut graph [EH04]
- Computing a shortest cut graph is NP-hard.

- Compute polygon schemas [Vegter-Yap90][Dey-Schipper95]

A better idea

- Intuitively, a loop on a connected, closed, surface M in R3 is:
- handle if it spans a disk (surface) in the bounded space bordered by M.
- tunnel if it spans a disk (surface) in the unbounded space bordered by M.

Cycle, Boundary and Homology

- p-chain: c=aii, i is p-simplex.
(e2+e3+e4) is a 1-chain.

- If c'=ai'i, then c+c'=(ai+ai')i.
- c + c = 0 for Z2 coefficients.
- Boundary operator
- p = [v0,v1,…,vp] = i=0p[v0,…,~vi,…,vp]
- e.g.: t = (e2+e3+e4)
- e.g.: e0+ e1+ e4 = (v0+v2)+(v0+v1)+(v1+v2) = 0

Cycle, Boundary and Homology

- p-cycle: if pc = 0.
- p-boundary: if there’s (p+1)-chain
d, s.t. c=pd .

- p-cycle group Zp = kernel (p).
- p-boundary group Bp = image (p+1).
- Hp= Zp / Bp
- Each element of Hp represents an equivalent class.
- c1 and c2 are in the same class if c1 + c2 is in Bp.
- Trivial elements in Hpbounds (p+1)-chain

Handle and tunnel loops – definitions [Dey-Li-Sun 07]

- Let M be aconnected, closed, surface in R3. M separates R3into insideIand outsideO.Both I and O have M as boundary.
- Handle:A loop onM whose homology class is trivialinH1(I) and non-trivial inH1(O).
- Tunnel:A loop onM whose homology class is trivial inH1(O) and non-trivial inH1(I).

Nasty knots

- Consider a thickened trefoil
- handle loop (green)
- tunnel loop (red)

Handle and tunnel loops – existence

- Theorem (Dey, Li, Sun[2007])
- For any connected closed surface M of genus g in R3, there exist g handle loops {hi}i=1…g forming a basis for H1(O) and g tunnel loops {ti}i=1...g forming a basis for H1(I).
- Furthermore, {hi}i=1...g and {ti}i=1...g form a basis for H1(M).

First algorithm for GR models[Dey-Li-Sun 07]

- Graph retractability assumption : A surface is graph retractable if both I and O deformation retract to a graph, denoted I and O, respectively.

Inner and outer cores

- I and O are disjoint core graphs, each with g loops : {KjI}j=1g and {KjO}j=1g and {Kj}j=12g all together.
- How to compute I and O ?

Linking number

- J and K are two disjoint loops in R3.
- In a regular projection, there are two ways in which J crosses under K.
- The linking number, lk(K, J), is the sum of these signed crossings.

Linking number Theorem

- A loop on M is a handle iff lk(, KiI) 0 for at least one inner core and lk(, KiO)=0 for all outer cores.
- A loop on M is a tunnel iff lk(, KiO) 0 for at least one outer core and lk(, KiI)=0 for all inner cores.

Minimally linked basis

- A loop is minimally linked if it links once only with one core loop.
- Theorem : There exist 2g minimally linked loops, denoted {Jj}j=12g, such that lk(Ki, Jj)=ij.
- Half of them linked with KiI’s, denoted {JjI}j=1g, are handle loops.
- Other half linked with KiO’s, denoted {JjO}j=1g, are tunnel loops.
- {[JjI]} j=1g form a basis for H1(I) and {[JjO]}j=1g form a basis for H1(O). Hence {[Jj]}j=12g form a basis for H1(M).

Topological algorithm

- Assume I and O are given.
- Step1: Compute {Ki}i=12g using the spanning tree of I and O.
- Step2: Compute a system of 2g loops on M, denoted {j}j=12g. (say by Dey-Schipper or Vegter-Yap linear time algorithms)
- Step3: Compute lk(Ki, j) for all i and j. Let A be the 2gx2g matrix {lk(Ki, j)}.
- A is the transform matrix from a minimally linked basis {[Jj]}j=12g to basis {[j]} j=12g.
- A-1 = {aji} exists and has integer entries.
- [Jj] = i=12g aji[i].

- Step4: Obtain Jj by concatenating i’s according to the above expression.

An implementation to compute system of handle and tunnel loops with small size

- Compute I and O
- Basic idea: Collapse the inside (outside) Voronoi diagram to obtain I (O).
- The curve-skeleton [Dey-Sun06] captures the geometry better.
- Each skeleton edge (e) gets associated with an additional value called geodesic size (g(e)) indicating the local size of M.
- Establish graph structure on the curve skeleton.
- The geodesic size for a graph edge, g(E) = min{g(e): e is a skeleton edge in E}.

- Issue: not always work for any graph retractable surface, e.g., a thickening of a house with two room.

Core graphs by curve-skeletons loops with small size[DS05]

An implementation to compute system of handle and tunnel loops with small size

- Compute core loops {Kj}j=12g
- Compute the maximal spanning tree for I and O using geodesic sizes as weight.
- Add the remaining edges, Ei’s, to form Ki’s.

- Compute minimally linked {Jj}j=12g
- Basic idea: compute Ji’s at different location indicated by Ei’s.
- Let e be the skeleton edge with the smallest geodesic size in Ei. Let p be one of the vertices of the dual Delaunay triangle of e.
- Compute an optimal system of loops by geodesic exploration and apply topological algorithm.
- In our experiments, one of the loop in the system of loops itself satisfies the condition to be Ji’s.

Results loops with small size

A Persistence based algorithm loops with small size[Dey-Li-Sun-Cohen-Steiner 08]

- A fast algorithm based on persistent homology introduced by Edelsbrunner, Letscher, Zomorodian[2002].
- The algorithm does not require any extra structure such as medial axis, curve skeleton, or Reeb graphs.
- It works on a larger class of models. Both surface mesh and isosurface with volume grids can be input.

Filtration in persistent homology loops with small size

- Filtration of a simplicial complex K is a nested sequence of complexes:
- 0 = K-1 <K0 <K1 <…<Kn = K
- Assume Ki - Ki-1 = i

- Persistence homology studies how the homology groups change over the filtration.
- In our algorithm, we use the pairing concept in the persistent homology.

Positive and Negative simplices loops with small size

- Once a p-simplex is added to Ki-1, it is
- Positive if it creates a non-boundary p-cycle.
- Negative if it kills an existing (p-1)-cycle.

- A negative p-simplex is always paired with a unique positive (p-1)-simplex ' where kills a (p-1)-cycle created by '.

Positive loops with small size and negative simplices

Pairing algorithm loops with small size– pair ()

1) c=p

2)d is youngest positive (p-1)-simplex in c

3)while (d is paired and c Φ) do

- Let c' be the cycle killed by the simplex paired with d
- c=c'+c
- d is the youngest positive (p-1)-simplex in c
end while

4)if c is not empty, then

is negative p-simplex and paired with d

else

is positive p-simplex.

Computing handle and tunnel loops loops with small size

- Topological algorithm
- Two refinements for improving loop quality.
- Assume M, I, and O denote the simplicial complexes of surface, inside, and outside respectively; g is the genus of M.

Topological algorithm loops with small size

- 3 Steps:
- 1) Simplices of Mare added to the filtration, generating 2g unpaired positive edges;
- 2) Simplices of Iare added, g previously unpaired edges get paired, each pair corresponds to a handle loop;
- 3) Simplices of Oare added, the rest of g unpaired edges get paired, each pair corresponds to a tunnel loop.

Topological algorithm loops with small sizecont.

Refinement 1 loops with small size

- For a negative triangle , a series of homologous loops are obtained.
- Let L1, L2,…,Lk be those that lie on M, sorted by the time when they are found. Set the handle (or tunnel) loop L()for as L1.

Refinement 2 loops with small size

- A handle (or tunnel) loop is found when a cross section of M get filled.
- Cross sections of small size will get filled first if we add triangles in I and O to the filtration in increasing order of their geodesic sizes.

Computing geodesics loops with small size– for refinement 2

- Geodesic size of an edge E =(A1, A2):geodesic distance between points A1' and A2' on M, where Ai'=Ai if Ai is on M; otherwise, Ai' is the closet point on M for Ai
- Geodesic size size of a triangle t:max {g(e)|e is in t}

Entire algorithm loops with small size

- 1) Compute geodesic size for edges and triangles in I and O;
- 2) For each simplex on M, pair();
- 3)E = unpaired positive edges on M;
- 4) In the order of increasing geodesic size, pair() for each triangle in I;
- 5) if ( is negative and its paired positive edge is in E), thenoutput a handle: hi = L();
- 6) Do similarly as 4) and 5) to get tunnels.

Entire algorithm loops with small size

Gearbox (genus 78) loops with small size

Isosurface loops with small sizeFuel (genus 8)

Isosurface (Engine, genus 20) loops with small size

Aneurysm loops with small size

Knotty cup and noise loops with small size

Experiments loops with small size

- We implemented our algorithm in C++. All experiments were done on a Dell PC with 2.8GHz Intel Pentium D CPU and 1GB RAM.

Feature Detection loops with small size

- Compute handle (tunnel) features based on handle (tunnel) loops.
- Sweep the handle (tunnel) loop until the length become too long.

Topological simplification loops with small size

Simplification (Buddha) loops with small size

Conclusions and future work loops with small size

- A simple, topologically correct algorithm for handles and tunnels.
- Mathematical proof of geometric qualities of the loops?
- What about surfaces with boundaries?
- Handletunnel software available from
http://www.cse.ohio-state.edu/~tamaldey/handle.html

Pegasus (genus 5) loops with small size

2-torus loops with small size(genus 2)

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