Tools for Modeling and Analysis of Non-manifold Shapes

Tools for Modeling and Analysis of Non-manifoldShapes

David Canino

Department of Computer Science, Universitá degli Studi di Genova, Italy

PhD. Final Exam

May 7, 2012

David Canino (DISI) May 7, 2012 1 / 1

Introduction I

Manifold shapes (Topological Manifold)

Each point has a neighborhood homeomorphic to eitheran open ball (internal point), or to a closed half-ball(boundary point).

Properties

simple structure (topology)

smooth, derivable, . . .

efficient representations

many tools based on manifold shapes.

But they are a subset of all shapes.

Non-manifold Shapes

Shapes which violate manifold conditions.2t1v

2v

1t

df e e

t21t


Introduction IINon-manifold Shapes

non-manifold singularities, i.e., points at which the manifoldcondition is not satisfied

parts of different dimensions.

Idealization Process

Applied to simpler (manifold) shapes, and produce idealized shapes

Engineering component Idealized Shape FEM simulation

Remove details and simplify shapes for FEM simulations


Objectives & ContributionsGeneral Objective

Represent simplicial complexes describing non-manifold shapes.

Research Area I - Representation by Topological Data Structures

Two data structures for abstract simplicial complexes in arbitrary dimensions:I the Incidence Simplicial (IS) data structureI the Generalized Indexed data structure with Adjacencies (IA∗).

Mangrove TDS frameworkI rapid prototyping of data structures for arbitrary simplicial complexes

Research Area II - Decompositions and Structural Models

Manifold-Connected (MC) Decomposition - Hui and De Floriani, 2007I the Exploded MC-Graph (hyper-graph)I the Pairwise MC-GraphI the Compact MC-Graph (hyper-graph)

Mayer-Vietoris (MV) Algorithm for computing Z-homology, which combines:I the MC-Decomposition (Pairwise MC-Graph)I the Constructive Homology Theory - Sergeraert and Rubio, 2006


Simplicial Complexes

Euclidean simplex

Let p a non negative integer, then an Euclidean p-simplex σ is the linear combination of p + 1points Vσ = [v0, . . . , vp] in any Euclidean space En.

Face of a Simplex

Any subset of (k + 1)-vertices in Vσ generates a k -face σ′ of σ, with k < p.

Euclidean Simplicial complex

An Euclidean simplicial complex Σ is a set ofsimplices in En of dimension at most d , with0 ≤ d ≤ n such that:

Σ contains all the faces of each simplex

two simplices in Σ can be either distinct, orcan share a face

Ce

Valid Not Valid

Geometric realizations of abstract simplicial complexes, NOT necessarily embedded in En.


Some Combinatorial Concepts

Given a p-simplex σ in a simplicial d-complex Σ, with 0 ≤ p ≤ d :

Boundary

Collection B(σ) of k -faces of σ, with 0 ≤ k < p

Star

Collection St(σ) of simplices with σ in their boundary(incident at σ)

vw

ft

St(v) = {w , f , t}, plus their facesincident at v

Link

Collection Lk(σ) formed by faces of simplices inSt(σ), which are not incident at σ

f

t

v' vw

ef

tf

v

e

Lk(v) = {v ′, ef , ft}

Top Simplex

If σ is not on boundary of other simplices.

w is a top 1-simplexf is a top 2-simplext is a top 3-simplex


Combinatorial Manifolds

Objective

Provide a combinatorial characterization of topological manifolds.

Key Idea

Discrete neighborhood of a simplex σ is characterizedby St(σ) in a simplicial d-complex Σ

Combinatorial Manifold (p + 1)-simplex σ

St(σ) is homemorphic to the triangulation of the(d − p)-sphere

Combinatorial Manifold Complex

All simplices are combinatorial manifold

Combinatorial manifold

Combinatorial non-manifold

Problems & Restrictions

NOT algorithmically decidable for d ≥ 5, Nabutovski, 1996 (not dimension-independent )


Topological Relations & Data Structures

Objective

Connectivity of simplices

Let Σj the collection of j-simplicesin Σ, and Rk,m ⊆ Σk × Σm, then:

e

v

6v

v

v

v

v

1

2

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5

e

e

e

e7

8

9

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ff

ff

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1

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ff

ff

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e

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e

e

1

2

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Boundary relations

Rk,m(σ, σ′) if σ′ ∈ B(σ), with k > m

R2,0(f1) = {v , v1, v2}, R2,1(f1) = {e1, e6, e10}

Co-boundary relations

Rk,m(σ, σ′) if σ′ ∈ St(σ), with k < m

R0,1(v) = {e6, . . . , e10}, R1,2(e10) = {f1, f2}

Adjacency relations

Rk,k (σ, σ′), if σ and σ′ shares a (k − 1)-simplex,with k 6= 0

R0,0(σ, σ′), if an edge connects σ and σ′

R0,0(v) = {v1, . . . , v5}, R2,2(f1) = {f2, f5}

Topological Data Structures

Subset of topological entities (simplices) and topological relations


Directed Graph Representation (Mangrove) for aTopological Data Structure

A topological data structure can be represented as a directed graph GΣ = (NΣ,AΣ):

each node nσ in NΣ describes a simplex σ

each arc (nσ , nσ′ ) describes a topological relation Rk,m(σ, σ′)

Boundary Arc (nσ , nσ′ )

If Rk,m(σ, σ′) is a boundary relation

Boundary Graph

Formed by nodes in NΣ + boundary arcs

Co-boundary Arc (nσ , nσ′ )

If Rk,m(σ, σ′) is a co-boundary relation

Co-boundary Graph

Formed by nodes inNΣ + co-boundary arcs

Adjacency Arc (nσ , nσ′ )

If Rk,k (σ, σ′) is an adjacency-relation

Adjacency Graph

Formed by nodes in NΣ + adjacency arcs


Data Structures for Simplicial ComplexesThere are a lot of representations in the literature, De Floriani and Hui, 2005

Taxonomy (partial)

Dimension-Independent versus Dimension-Specific

Manifold versus Non-Manifold

Incidence-based versus Adjacency-based

Incidence-based (global mangrove)

all simplices

boundary and co-boundary relations

↓

Adjacency-based (local mangrove)

vertices and top simplices

adjacency relations

↓

Incidence Simplicial (IS) data structure

Dimension-independent variant, restrictedto simplicial complexes, of theIncidence-Graph (IG), Edelsbrunner,1987

Generalized Indexed data structure withAdjacencies (IA∗)

Dimension-independent variant, specific fornon-manifolds, of the IA data structure,Paoluzzi et al., 1993


The Incidence Graph (IG) Edelsbrunner, 1987

Abstract simplicial d-complex Σ

Dimension-independent

For each p-simplex σ:I boundary relation Rp,p−1(σ)I co-boundary relation Rp,p+1(σ)

Global mangrove (IG-graph)

IG Boundary/Co-boundary Arcs

Correspond to Rp,p−1 and Rp,p+1

IG Boundary/Co-boundary Graph

Nodes + IG Boundary/Co-boundary Arcs

wv'=5 v=0

2

1

3 4

ft

e

0,1,2,3

0,3,4

0,4

4 5 0 3 2 1

0,5 3,4 0,3 0,2 0,1 2,3 1,3 1,2

0,1,3 0,2,3 0,1,2 1,2,3

IGBoundary Graph

0,1,2,3

0,3,4

0,4

4 5 0 3 2 1

0,5 3,4 0,3 0,2 0,1 2,3 1,3 1,2

0,1,3 0,2,3 0,1,2 1,2,3

IGCo-boundary Graph


Properties of the IG Data Structure

Topological Relations

Can be retrieved in optimal time, i.e., linear in the number of involved simplices

Rp,p−1(σ) Directly encoded O(1)

Rp,q(σ), p > q Recursively combineRp,p−1,Rp−1,p−2, and so on O(1)

Rp,p+1(σ) Directly encoded O(1)

Rp,q(σ), p < q Recursively combineRk,k+1 andRk+1,k , for k > p O(‖Rp,q(σ)‖)

R0,0(σ) CombineR0,1 andR1,0 O(‖R0,0(σ)‖)

Rp,p(σ), with p 6= 0 CombineRp,p−1 andRp−1,p O(‖Rp,p(σ)‖)

Storage Cost

2d∑

p=1

sp(p + 1)

sp : number of p-simplices

Disadvantages

too verbose

large overhead for manifolds


The Incidence Simplicial (IS) Data Structure

Key idea: simplify the IG

Boundary relations are constant

No need full co-boundary relations



Encodes all simplices in Σ

For each p-simplex σ:I boundary relation Rp,p−1(σ)I partial co-boundary relation R∗p,p+1(σ)

Global Mangrove (IS-Graph)

Partial co-boundary relation R∗p,p+1(σ)

One arbitrary (p + 1)-simplex for eachconnected component in Lk(σ).

wv'=5 v=0

2

1

3 4

ft

e

R∗0,1(v) = {w , e}

Important

R∗d−1,d ≡ Rd−1,d

L. De Floriani, A. Hui, D. Panozzo, D. Canino, A Dimension-Independent Data Structure for Simplicial

Complexes, In S. Shontz Ed., Proceedings of the 19th International Meshing Roundtable (IMR 2010), pages

403-420, Springer, 2010 - Chattanooga, Tennessee, USA


The IS-Graph

IS Boundary Arcs ≡ IG Boundary Arcs

Correspond to Rp,p−1

IS Boundary Graph ≡ IG Boundary Graph

Nodes + IS Boundary Arcs

IS Co-boundary Arcs

Correspond to R∗p,p+1

IS Co-boundary Graph

Nodes + IS Co-boundary Arcs

wv'=5 v=0

2

1

3 4

ft

e

0,1,2,3

0,3,4

0,4

4 5 0 3 2 1

0,5 3,4 0,3 0,2 0,1 2,3 1,3 1,2

0,1,3 0,2,3 0,1,2 1,2,3

ISBoundary Graph ≡ IG Boundary Graph

0,1,2,3

0,3,4

0,4

4 5 0 3 2 1

0,5 3,4 0,3 0,2 0,1 2,3 1,3 1,2

0,1,3 0,2,3 0,1,2 1,2,3

ISCo-boundary Graph

IS-Graph is more compact than IG-Graph


Storage Cost of the IS Data StructureBoundary Relations

d∑p=1

sp(p + 1)

sp : number of p-simplices

+

Partial Co-boundary Relations

d∑p=1

∑σ∈Σp

Hσ ≤d∑

p=1

sp(p + 1)

Hσ : #connected components in Lk(σ)

2D Shapes

Shape IG IS ∆(%)

Armchair 127k 101k 20.5

Cone 14k 11k 21.4

Frame 15k 12k 20

Tower 221k 175k 20.8

21% more compact than IG

3D Shapes

Shape IG IS ∆(%)

Basket 113k 80k 29.2

Flasks 104k 75k 27.9

Sierpinski 917k 688k 24.9

Teapot 219k 163k 25.6

27% more compact than IG

Archive of 62 shapes publicly available at http://ggg.disi.unige.it/nmcollection/

Note

More compact with manifolds =⇒ scalability to manifolds


http://ggg.disi.unige.it/nmcollection/

Storage Cost of the IS Data Structure (cont’d)

For Manifolds:

Partial co-boundary relation R∗p,p+1 contains:

only one (p + 1)-simplex, if p < d

one or two d-simplices, if p = d − 1

Remark

R∗d−1,d ≡ Rd−1,d

v v

e

All edges in R0,1(v) versus one edge in R∗0,1(v)

Boundary Relations

d∑p=1

sp(p + 1)

+

Partial Co-boundaryRelations

d−2∑p=0

sp + (d + 1)sd


Topological Relations in the IS Data Structure

Rp,p−1(σ) Directly encoded O(1)

Rp,q(σ), p > q Recursively combineRp,p−1,Rp−1,p−2, and so on O(1)

Rd−1,d (σ) Directly encoded O(1)

Rp,q(σ), p < q Recursively combineR∗k,k+1 andRk+1,k , for k > p O(‖St(σ)‖)

R0,0(σ) CombineR0,1 andR1,0 O(‖St(σ)‖)

Rp,p(σ), with p 6= 0 CombineRp,p−1 andRp−1,p O(‖St(σ)‖)

IS star-graph of a p-simplex σ

Subgraph Gσ of the IS-graph:

nodes representing simplices in St(σ)

IS boundary arcs restricted to St(σ)

IS co-boundary arcs restricted to St(σ)

Co-boundary relation Rp,q(σ)

breadth-first traversal of Gσexamine top simplices in St(σ)and their faces

linear in ‖St(σ)‖

Co-boundary relations are optimal only for simplicial 2- and 3-complexes in E3

Experiments show that they are about less than 10% slower than in the IG


The Generalized Indexed Data Structure withAdjacencies (IA∗)

Key Idea

More compact encoding for a simpliciald-complex

The Indexed data structure withAdjacencies (IA), Paoluzzi et al., 1993

vertices, plus d-simplices

boundary relation Rd,0 ford-simplices

adjacency relation Rd,d ford-simplices

only for manifolds

The IA∗ data structure



Encodes vertices and top simplices

Adjacency-based

Probably, the most compact representationfor non-manifolds (with respect to the state ofthe art)Non-manifold variant of the Extended IA(EIA) data structure, De Floriani, et al. 2003

For manifolds, it reduces to the EIA datastructure (scalable)

Local Mangrove (IA∗-Graph)

D. Canino, L. De Floriani, K. Weiss, IA*: An Adjacency-Based Representation for Non-Manifold Simplicial

Shapes in Arbitrary Dimensions, Computer & Graphics, 35(3):747-753, Elsevier Press, Shape Modeling

International 2011 (SMI 2011), Poster


The IA∗ data structure - DefinitionRepresents an abstract simplicial d-complex

Boundary relationR∗p,0(σ)

Vertices of a top p-simplex σ,for 1 ≤ p ≤ d

R∗1,0(w) = {1, 2},R∗2,0(f1) = {1, 3, 4}

Adjacency relationR∗p,p(σ)

Top p-simplices sharing a(p − 1)-simplex with a topp-simplex σ, with 2 ≤ p ≤ d

R∗2,2(f1) = {f2, f3, f4},R∗2,2(f5) = {f6},R∗3,3(t1) = {t2}

For Manifolds

IA∗ reduces to EIA

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only one d-simplex inR∗0,d for each vertex

Rd−1,d : empty

at most one d-simplexin Rd,d

p-cluster

Maximal collection of adjacent topp-simplices

2-clusters: {f1, f2, f3, f4}, {f5, f6}3-cluster: {t1, t2}

Partial co-boundary relationR∗0,p(v)

Arbitrary top p-simplex for eachp-cluster in St(v), with 2 ≤ p ≤ d

R∗0,1(v) = {w},R∗0,2(v) = {f2, f5}, R∗0,3(v) = {t1}

Partial co-boundary relationR∗p−1,p(τ)

Top p-simplices incident at a(p − 1)-face τ of a top p-simplex,with 2 ≤ p ≤ d (more than two)

R∗1,2(e) = {f1, f2, f3, f4}


The IA∗ data structure - Non-Manifold AdjacencyKey Idea

A (p − 1)-face τ of a top p-simplex σ is non-manifold if it is shared by more than two topp-simplices.

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Manifold Adjacency - At most two top p-simplices in St(τ)

Encode only the other top p-simplex adjacent to σ along τ

R∗2,2(f5) = {f6}, R∗2,2(f6) = {f5}

Non-Manifold Adjacency (Otherwise)

Encode R∗p,p(σ) along τ as R∗p−1,p(τ)

R∗2,2(fi ) = R∗1,2(e) = {f1, f2, f3, f4}, with i = 1, . . . , 4

Consequences

Compact encoding of R∗p,p , R∗p−1,p stored only once

Partial characterization of non-manifold(p − 1)-simplices


The IA∗-Graph

Formed by nodes representing vertices, top simplices, and (some) non-manifold simplices, plus:

IA∗ Boundary Arcs (IA∗ Boundary Graph)

Correspond to R∗p,0 (vertices and top simplices)

IA∗ Co-boundary Arcs (IA∗ Co-boundary Graph)

Correspond to R∗0,p (vertices and top simplices)

IA∗ Adjacency Arcs (IA∗ Adjacency Graph)

Correspond to R∗p,p and R∗p−1,p

1,11,12,14

1,3,7

1,3

1,12,13,14

1,3,61,3,51,3,41,9,101,8,9

IA∗Adjacency Graph

IA∗Boundary Graph

IA∗Co-boundary Graph


Storage Cost of the IA∗ Data Structure

TS data structure, De Floriani et al., 2003

Variant of the IA data structure

Simplicial 2-complexes in R3

NMIA data structure, De Floriani and Hui, 2003

Variant of the IA data structure

Simplicial 3-complexes in R3

2D Shapes

Shape IS TS IA∗

Armchair 101k 69.3k 69.1k

Cone 11k 7.8k 7.8k

Frame 12k 8.1k 8.1k

Tower 175k 124k 122k

IS is 1.28 times more expensive than IA∗About 5% more compact than TS

3D Shapes

Shape IS NMIA IA∗

Basket 80k 33k 33k

Flasks 75k 29.6k 29.4k

Sierpinski 688k 197k 197k

Teapot 163k 85k 84.6k

IS is 2.4 times more expensive than IA∗Abot 5% more compact than NMIA

Results

the most compact for non-manifolds

small overhead for manifolds (EIA)

Exception: Laced Ring, Gurung et al., 2011

3 times more compact (compressionscheme)

2D manifolds, no editing


Topological Relations in the IA* Data StructureGiven a simplicial d-complex Σ, a simplex not directly encoded is represented by its vertices :

R∗p,0(σ) Directly encoded O(1)

Rp,q(σ), p > q Generate faces of σ O(1)

R0,k (v) (top) ExpandR∗0,k (v) byR∗k,k O(#top k -simplices in St(v))R0,p(v) (any) Select p-simplices in St(v) from top simplices in St(v) O(#top simplices in St(v))Rp,q(σ), p < q ∗ Select q-simplices in St(σ) from top simplices in St(v) O(#top simplices in St(v))

Rd,d (σ) Directly encoded O(1)

R0,0(v) CombineR0,1 andR1,0 O(#top simplices in St(v))Rp,p(σ) ExtractR∗p,p and combineRp,p+1 andRp+1,p O(#top simplices in St(v))

∗: v is a vertex inRp,0(σ)

Co-boundary relations are optimal only for simplicial 2- and 3-complexes in E3

Basic Operation (optimal)

Retrieving top k -simplices in St(v):

Breadth-first visit of eachk -cluster in R∗0,k (v)

Transitive closure of R∗k,kLinear in #top k -simplices in St(v)

Experimental Comparisons for Co-boundary

vertex-based: 30% faster than IS

edge-based: 10% slower than IS

face-based: 15% slower than IS


The Mangrove Topological Data Structure (TDS)FrameworkThe Mangrove TDS Framewok

Rapid prototyping of topological data structures for simplicial complexes

Satisfies completely design choices of Sieger and Botsch, 2011 for generic frameworks(probably the first in the literature, independently designed and implemented):

I flexibility - representation of topological data structures (mangroves)I efficiency - plugins-oriented architectureI easy-to-use - common interface programming)

Any data structure is supported, without restrictions, including for non-manifolds

Implicit representations of simplices not encoded in a local mangrove (ghost simplices)

The Mangrove TDS Library

Written in C++ (meta-programming techniques)

Common programming interface of the Mangrove TDS framework

We have submitted an article to an international conference, currently under review

Mangrove TDS Library will be released as GPL software athttp://sourceforge.net/projects/mangrovetds/


http://sourceforge.net/projects/mangrovetds/

The Mangrove TDS Framework - Basic Concepts

Key Idea

A topological data structure is a mangrove

Primitives customized for a p-simplex σ:

BOUNDARY - boundary B(σ)

STAR - star St(σ)

ADJACENCY - adjacency relationRp,p(σ)

LINK - link Lk(σ)

IS_MANIFOLD - checks if σ is manifold(when possible)

In this context

Mangrove ≡ dynamic plugin in the system

Current Implementations (but extensible)

IG, IS, IA∗ data structures

TS data structureI adjacency-basedI simplicial 2-complexes in E3

I De Floriani et al., 2003

NMIA data structureI adjacency-basedI simplicial 3-complexes in E3

I De Floriani and Hui, 2003

SIG data structureI incidence-basedI dimension-independentI De Floriani et al., 2004

up to now, only for simplicial complexes

extensible also for cell complexes

Current frameworks partially support non-manifolds through a predefined representation


mangrove

The Mangrove TDS Framework - Ghost Simplices

Ghost p-simplex σ

Not directly encoded in a local mangrove

Explicit Representation

Set of vertices Vσ = {v0, . . . , vp}too knownledge

no efficiency for any queries

Implicit Representation

A p-simplex σ can be either:

a top p-simplex σ, or

a p-face of a top t-simplex σ′, p ≤ t

GhostSimplexPointer reference

(t , i, p, pi)

i is the identifier of σ′

pi is the identifier of σ as p-face of σ′

0 ≤ pi <( t + 1

pi + 1

)

Advantages

less knowledge is required

fixed-length representation

does not depend on an enumerationsof faces

Disadvantage?

a not unique representation

a GhostSimplexPointer reference foreach top simplex in St(σ)


Explicit Representation of Ghost SimplicesKey Idea

A ghost p-simplex σ is described by (p + 1)

positions of vertices in Vσ′ ≡ Rt,0(σ′) asσ = [k0, . . . , kp]

Enumeration Rule (Simplicial Homology)

The i-th (p − 1)-face σi of σ is defined asσi = [k0, . . . , ki−1, ki+1, . . . , kp]

Consequence

Partial order relation <, such thatσi < σ, i.e., σi is a face of σ

Two Hasse diagrams for all t > 1

Storage cost of Hasse diagrams

2d∑

t=1

t∑p=0

( t + 1p + 1

)

0,1,2,3

0,2,31,2,3 0,1,3 0,1,2

0 1 2 3

2,3 1,3 1,2 0,3 0,2 0,1

(3, 0, 2, 2): vertices in positions [0, 1, 3] in R3,0(t0)

0,1,2,3

0,3,40,1,2 1,3,5 2,4,5

0 1 2 3

2,3 1,3 1,2 0,3 0,2 0,1

Same lattice in terms of immediate subfaces(3, 0, 2, 2) formed by edges [1, 3, 5]

Explicit representation of σ in O(1)


Experimental Results on Our Mangroves

We have compared the efficiency of queries on our six mangroves within the Mangrove TDSFramework

Our results

there is not any data structure optimal for all tasks (advantages vs disadvantages)

in any case, most of queries tend to be more efficient on the IA∗ data structure:I BOUNDARY is 30% more efficient than IS for a top simplexI STAR is 35% more efficient than IS for verticesI LINK is 3X more efficient than IS

Conversely, STAR is within 10% slower than IS for ghost simplices

These improvements are due to the GhostSimplexPointer references, which improve theexpressive power of a local mangrove


Decomposition Approach

Key Idea

Complex topology of a non-manifold shape offers valuable information for:

decomposing a shape into relevant components with a simpler topology

expose the structure of a shape (connections among components)

Topological data structure Structural model (shapedecomposition)

Semantic model (futurework)

Structural Model for Non-Manifolds

Components joint together atnon-manifold singularities

shape annotation and retrieval

identification of form features

computation of Z-homology


Manifold-Connected (MC) Decomposition Hui and De Floriani, 2007

Given a simplicial d-complex Σ and k ≤ d :

Manifold (k − 1)-path (MC-Adjacency)

Sequence of k -simplices in Σ, where each ofsimplices is adjacent through a manifold(k − 1)-simplex, bounding at most twok -simplices

Manifold-Connected (MC) k -Complex

Formed by all k -simplices in Σ connected by amanifold (k − 1)-path

MC-Decomposition

Collection of MC k -Complexes in Σ

MC k -Complexes are the equivalence classes versus MC-Adjacency, and become unique ifrestricted to top k -simplices in Σ


Manifold-Connected (MC) Decomposition (cont’d)

MC-Decomposition

Decomposition of a simplicial complex Σ into its MC-Complexes (MC-components)

Unique, decidable, and dimension-independent (also for high dimensions)

Discrete counterpart of Whitney stratification (1965);

MC-Components

decidable superclass of manifolds

contains some singularities

connected through singularities

It can be represented by a two-level graph-based data structure


Representing the MC-DecompositionTwo-level Graph-based Data Structure

the lower level describes a non-manifold shape by any mangrove Ψ (topological model)the upper level describes the connectivity of MC-components through a graph-based datastructure (structural model)

MC-graph G = (N ,A)

each node in N ≡ one MC-component (direct references to top simplices in Ψ);

each arc a = (n1, n2, . . . , nk ) in A ≡ intersection of MC-components described byn1, n2, . . . , nk (common singularities, as direct references to simplices in Ψ)

Relating MC-Components and singularities

the number of MC-Components partially characterizes asingularity

needs IS_MANIFOLD (no dimension-independent)

efficiency depends on the properties of mangrove Ψ

D. Canino, L. De Floriani, A Decomposition-based Approach to Modeling and Understanding Arbitrary Shapes,

9th Eurographics Italian Chapter Conference, Eurographics Association, 2011


Graph-based Data Structures

1 MC-component of dimension 1: C43 MC-components of dimension 2: C1, C2, C3

Pairwise MC-Graph

An arc ≡ intersection of two MC-Components, formed bya subset of singularities

(partial)

Exploded MC-Graph (Hyper-graph)

A hyper-arc ≡ a singularity σ, and connects allMC-components sharing σ

Compact MC-Graph (Hyper-graph)

An hyper-arc corresponds to a maximal set of singularitiescommon to several MC-components


Experimental ResultsWe have combined our MC-Graphs with six mangroves in our library (18 different versions)

2D shapes (Storage cost)

Shape C P E

Armchair 10.7k 10.8k 11.2k

Cone 1.2k 1.2k 1.2k

Frame 2.2k 2.7k 2.3k

Tower 20.9k 86.8k 28.6k


Shape C P E

Basket 4k 4k 4k

Flasks 4k 4.1k 4.4k

Sierpinski 180k 180k 180k

Teapot 25.6k 103.5k 26.2k

The Compact MC-Graph provides the most compact representation

2D shapes (Running Times in ms)

Shape IA∗ IS IG

Armchair 4k 10.8k 11.2k

Cone 4k 7k 15k

Frame 212 283 5.3k

Tower 8.1k 8.5k 440k

3D shapes (Running Times in ms)

Shape IA∗ IS IG

Basket 4k 8k 17k

Flasks 2.4k 6.7k 383k

Sierpinski 2.9k 7.6k 537k

Teapot 6.4k 22.3k 1M

The IA∗ data structure is the most suitable for retrieving MC-Components


Experimental Results (cont’d)


Shape C C+IA∗ IG

Armchair 10.7k 69.1k 127k

Cone 1.2k 9k 14k

Frame 2.2k 10.3k 15k

Tower 20.9k 142.9k 221k


Shape C C+IA∗ IG

Basket 4k 69.1k 127k

Cone 4k 33.4k 104k

Frame 180k 377k 917k

Tower 25.6k 110.2k 219k

The structural model Compact MC-Graph + IA∗ data structure is:

about 63% of IG for 2D shapes (37% more compact than IG)

about 39% of IG for 3D shapes (61% more compact than IG)


Iterative Computation of Z-homology

Objective

Computing Z-homology of a non-manifold shape

Mayer-Vietoris (MV) Algoritm

modular, iterative, and dimension-independent

the MC-Decomposition - Pairwise MC-Graph

the Constructive Homology Theory - Sergeraertand Rubio, 2006

Basic idea

Combine:

homology of its MC-components

homology of the intersection of MC-components

45nodes, 79 arcs→ (Z,Z27,Z5)

Joint Project with INRIA Rhone Alpes,Grenoble, France

D. Boltcheva, D. Canino, S. Merino, J.-C. Léon, L. De Floriani, F. Hétroy, An Iterative Algorithm for Homology

Computation on Simplicial Shapes, Computer-Aided Design, 43(11):1457-1467, Elsevier Press, SIAM

Conference on Geometric and Physical Modeling (GD/SPM 2011)


Classical ApproachAssociate an algebraic object, namely a chain-complex (Σ∗,D∗), to a simplicial complex Σ fromwhich we extract the Z-homology (Betti numbers, generators, torsion coefficients)

Σ∗ ≡ sequence of chain-groups Σp

Group of p-chains≡ linear combinations oforiented k -simplices

(Σ∗,D∗) : 0 0← . . .dp−1←− Σp−1

dp← . . .Σd0← 0

Smith Normal Form (SNF), Munkres,1999

Incidence Matrix Ip is reduced throughGaussian eliminations to its Smith NormalForm (SNF) Np :

Np =

σ

p0 . . . σ

pl

σp−10 0 λ 0

: 0 0 Idσ

p−1m 0 0 0

where:

λ is a diagonal matrix, with λi ∈ Z

Ip = Pp−1NpPp (basis change)

D∗ ≡ sequence of boundary operators dp

Describes the oriented boundary Bo(σpi ) of a

p-simplex σpi in terms of its immediate

subfaces σp−1j by the incidence matrix Ip

Incidence Matrix Ip of order p

Ipj,i =

0 if σp−1

j 6∈ Bo(σpi )

1 if +σp−1j ∈ Bo(σp

i )

−1 if −σp−1j ∈ Bo(σp

i ).

Problems of this approach

The Z-homology is retrieved from Np

not constructive

not feasible for large shapes

the SNF is cubic


The General Idea of the MV AlgorithmInput: a simplicial d-complex Σ discretizing a non-manifold shapeOutput: the Z-homology (Betti numbers, generators, torsion-coefficients)

First step

compute SNF reductions ofall MC-components

Generic step

Given components A and Btwo components such thatA ∩ B 6= ∅, we compute(A ∩ B)∗ from A∗, B∗, and(A ∩ B)∗

Sergeraert and Rubio, 2006

In the Pairwise MC-Graph:

store N∗ in the node describing N

collapse the arc connecting A and B

Last step

retrieve the Z-homology from the last node


Critical Properties of the MC-Decomposition

Shape s MS(%) MG(%)

Armchair 32k 38.4 0.07

Balance 24k 31.4 0.004

Bi-Twist 9k 45.5 0.6

Carter 24k 45 0.12

Chandelier 55k 11.8 0.05

Frame 4k 8 0.8

Twist 7k 65.5 0.9

s: total number of simplicesMS: maximum size of a MC-ComponentMG: maximum size of the intersection of twoMC-components

Property #1

Guarantees a small size of the intersectionbetween two MC-Components

Property #2

Produces subcomplexes smaller than theinput shape

↓

Good properties for the MV algorithmSuitable for computations

Consequence #1

Small MC-components reduce timecomplexity of the SNF reductions

Consequence #2

Small intersections make the conereductions possible (while merging theMC-components)


Experimental Results

We have exploited:

the SNF algorithm provided by Moka Modeller, G. Damiand, LIRIS, Lyon, France (nooptimizations), http://moka-modeler.sourceforge.net

our Pairwise MC-Graph + IS data structure

Shape SNFs(MB) SNFt (ms) MVs(%) MVt (%) ResultArmchair 0.6 60 88 320 (Z, 0,Z5)Bi-Twist 80 1.2× 107 73 380 (Z,Z4,Z3)

Carter 567 7.7× 107 79 450 (Z,Z27,Z5)

Twist 50 2.2× 106 55 160 (Z,Z2,Z2)

SNFs : storage cost of the SNF algorithm (MB)SNFt : running time of the SNF algorithm (ms)MVs : reduction in storage cost of the MV algorithm (% wrt SNFs)MVt : reduction in running time of the MV algorithm (% wrt SNFt )

The MV algorithm is an effective tool for computing the Z-homology↓

Reductions in storage cost and running times wrt the SNF algorithm


http://moka-modeler.sourceforge.net

Experimental Results (cont’d)

MC-Decomposition + (Z,Z2,Z2) for the Twistshape

MC-Decomposition + (Z,Z4,Z3) for the Twistshape


Conclusions and Future WorksWhat we have done

Several tools for simplicial complexes describing non-manifold shapes.

Research Area I - Representation by Topological Data Structures

Two data structures for abstract simplicial complexes in arbitrary dimensions:I the Incidence Simplicial (IS) data structureI the Generalized Indexed data structure with Adjacencies (IA∗).

Mangrove TDS frameworkI rapid prototyping of data structures for arbitrary simplicial complexes

Research Area II - Decompositions and Structural Models

Manifold-Connected (MC) Decomposition - Hui and De Floriani, 2007I the Exploded MC-Graph (hyper-graph)I the Pairwise MC-GraphI the Compact MC-Graph (hyper-graph)

Mayer-Vietoris (MV) Algorithm for computing Z-homology, which combines:I the MC-Decomposition (Pairwise MC-Graph)I the Constructive Homology Theory - Sergeraert and Rubio, 2006


Conclusions and Future Works (Cont’d)Several improvements about the different topics

Topological data structures

Extend the IS and IA∗:

towards cell complexes, like quad and hexahedral shapes (IS)

reconstructions of shapes from point data in high dimension, Rips complexes (IA∗)

editing operations (multi-resolution models for non-manifolds)

Mangrove TDS Library

release as GPL

new mangroves and new implementations of topological data structures

extension towards cell complexes

MC-Decomposition

semantic models over the MC-Decomposition

identification of 2-cycles (components bounding a void) in the shape


Conclusions and Future Works (Cont’d)

MV Algorithm

Improve the efficiency of the MV Algorithm:

shape of generators

use optimized versions of the SNF algorithm

transform MC-components into almost manifolds, and exploit more efficient methods formanifolds


My Papers

1 D. Canino, L. De Floriani, A Decomposition-based Approach to Modeling and Understanding ArbitraryShapes, 9th Eurographics Italian Chapter Conference, Eurographics Association, 2011

2 D. Boltcheva, D. Canino, S. Merino, J.-C. Léon, L. De Floriani, F. Hétroy, An Iterative Algorithm forHomology Computation on Simplicial Shapes, Computer-Aided Design, 43(11):1457-1467, ElsevierPress, SIAM Conference on Geometric and Physical Modeling (GD/SPM 2011)

3 D. Canino, L. De Floriani, K. Weiss, IA*: An Adjacency-Based Representation for Non-Manifold SimplicialShapes in Arbitrary Dimensions, Computer & Graphics, 35(3):747-753, Elsevier Press, Shape ModelingInternational 2011 (SMI 2011), Poster

4 D. Canino, A Dimension-Independent and Extensible Framework for Huge Geometric Models, 8thEurographics Italian Chapter Conference, Eurographics Association, 2010, Poster

5 L. De Floriani, A. Hui, D. Panozzo, D. Canino, A Dimension-Independent Data Structure for SimplicialComplexes, In S. Shontz Ed., Proceedings of the 19th International Meshing Roundtable, pages403-420, Springer, 2010

6 D. Canino, An Extensible Framework for Huge Geometric Models, Technical Report DISI-TR-09-08, 2009


Thank for your attention and patience. Any questions?


Interesting PapersL. De Floriani, D. Greenfieldboyce, and A. Hui, A Data Structure for Non-manifold Simplicial d-complexes,In Proceedings of the 2nd Eurographics Symposium on Geometry Processing (SGP ’04), pages 83-92,ACM Press, 2004

L. De Floriani and A. Hui, A Scalable Data Structure for Three-dimensional Non-manifold Objects, InProceedings of the 1st Eurographics Symposium on Geometry Processing (SGP ’03), pages 72-82, ACMPress, 2003

L. De Floriani and A. Hui, Data Structures for Simplicial Complexes: an Analysis and a Comparison, InProceedings of the 3rd Eurographics Symposium on Geometry Processing (SGP ’05), pages 119-128,ACM Press, 2005

L. De Floriani, P. Magillo, E. Puppo, and D. Sobrero, A Multi-resolution Topological Representation forNon-manifold Meshes, Computer-Aided Design, 36(2):141-159, 2003

H. Edelsbrunner, Algorithms in Combinatorial Geometry, Springer, 1987

A. Hui and L. De Floriani, A Two-level Topological Decomposition for Non-Manifold Simplicial Shapes, InProceedings of the ACM Symposium on Solid and Physical Modeling, pages 355-360, ACM Press, 2007

J. Munkres, Algebraic Topology, Prentice Hall, 1999

A. Nabutovsky, Geometry of the Space of Triangulations of a Compact Manifold, Communications inMathematical Physics, 181:303-330, 1996.

A. Paoluzzi, F. Bernardini, C. Cattani, and V. Ferrucci, Dimension-Independent Modeling with SimplicialComplexes, ACM Transactions on Graphics, 12(1):56-102, 1993

D. Sieger and M. Botsch, Design, Implementation, and Evaluation of the Surface_Mesh Data Structure.In S. Shontz, editor, Proceedings of the 20th International Meshing Roundtable, pages 533âAS550.Springer, 2011.

F. Sergeraert and J. Rubio, Constructive Homological Algebra and Applications, 2006,http://www-fourier.ujf-grenoble.fr/∼sergerar/Papers/


http://www-fourier.ujf-grenoble.fr/

sergerar/Papers/