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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.

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.

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.

A Persistence based algorithm [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

- 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

- 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 '.

Pairing algorithm – 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

- 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

- 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.

Refinement 1

- 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

- 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 – 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

- 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.

Experiments

- 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

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

Conclusions and future work

- 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

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