Tools for Modeling and Analysis of Non-manifold Shapes

Tools for Modeling and Analysis of Non-manifold
Shapes
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 either
an 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.
2 v
1 t 2 v
1 t
df e e
1 t2 t

Introduction II
Non-manifold Shapes
non-manifold singularities, i.e., points at which the manifold
condition 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 & Contributions
General 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 structure
I the Generalized Indexed data structure with Adjacencies (IA).
Mangrove TDS framework
I rapid prototyping of data structures for arbitrary simplicial complexes
Research Area II - Decompositions and Structural Models
Manifold-Connected (MC) Decomposition - Hui and De Floriani, 2007
I the Exploded MC-Graph (hyper-graph)
I the Pairwise MC-Graph
I 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 is the linear combination of p + 1
points V = [v0; : : : ; vp] in any Euclidean space En.
Face of a Simplex
Any subset of (k + 1)-vertices in V generates a k-face 0 of , with k p.
Euclidean Simplicial complex
An Euclidean simplicial complex is a set of
simplices in En of dimension at most d, with
0 d n such that:
contains all the faces of each simplex
two simplices in can be either distinct, or
can share a face
C
e
Valid Not Valid
Geometric realizations of abstract simplicial complexes, NOT necessarily embedded in En.

Some Combinatorial Concepts
Given a p-simplex in a simplicial d-complex , with 0 p d:
Boundary
Collection B() of k-faces of , with 0 k p
Star
Collection St() of simplices with in their boundary
(incident at )
w v
f
t
St(v) = fw; f ; tg, plus their faces
incident at v
Link
Collection Lk() formed by faces of simplices in
St(), which are not incident at
v' w v
f
t
e
f
t
f
e
Lk(v) = fv0; ef ; ftg
Top Simplex
If is not on boundary of other simplices.
w is a top 1-simplex
f is a top 2-simplex
t is a top 3-simplex

Combinatorial Manifolds
Objective
Provide a combinatorial characterization of topological manifolds.
Key Idea
Discrete neighborhood of a simplex is characterized
by St() in a simplicial d-complex
Combinatorial Manifold (p + 1)-simplex
St() 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 j the collection of j-simplices
in , and Rk;m k m, then:
e
v 6
v
v
v
v
v
1
2
3
4
5
e
e
e
e
7
8
9
10
f
f
f
f
f
1
2
3
4
5
e
e
e
e
e
1
2
3
4
5
Boundary relations
Rk;m(; 0) if 0 2 B(), with k m
R2;0(f1) = fv; v1; v2g, R2;1(f1) = fe1; e6; e10g
Co-boundary relations
Rk;m(; 0) if 0 2 St(), with k m
R0;1(v) = fe6; : : : ; e10g, R1;2(e10) = ff1; f2g
Adjacency relations
Rk;k (; 0), if and 0 shares a (k 1)-simplex,
with k6= 0
R0;0(; 0), if an edge connects and 0
R0;0(v) = fv1; : : : ; v5g, R2;2(f1) = ff2; f5g
Topological Data Structures
Subset of topological entities (simplices) and topological relations

Directed Graph Representation (Mangrove) for a
Topological Data Structure
A topological data structure can be represented as a directed graph G = (N;A):
each node n in N describes a simplex
each arc (n; n0 ) describes a topological relation Rk;m(; 0)
Boundary Arc (n; n0 )
If Rk;m(; 0) is a boundary relation
Boundary Graph
Formed by nodes in N + boundary arcs
Co-boundary Arc (n; n0 )
If Rk;m(; 0) is a co-boundary relation
Co-boundary Graph
Formed by nodes in N + co-boundary arcs
Adjacency Arc (n; n0 )
If Rk;k (; 0) is an adjacency-relation
Adjacency Graph
Formed by nodes in N + adjacency arcs

Data Structures for Simplicial Complexes
There 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, restricted
to simplicial complexes, of the
Incidence-Graph (IG), Edelsbrunner,1987
Generalized Indexed data structure with
Adjacencies (IA)
Dimension-independent variant, specific for
non-manifolds, of the IA data structure,
Paoluzzi et al., 1993

The Incidence Graph (IG) Edelsbrunner, 1987
Abstract simplicial d-complex
Dimension-independent
For each p-simplex :
I boundary relation Rp;p1()
I co-boundary relation Rp;p+1()
Global mangrove (IG-graph)
IG Boundary/Co-boundary Arcs
Correspond to Rp;p1 and Rp;p+1
IG Boundary/Co-boundary Graph
Nodes + IG Boundary/Co-boundary Arcs
v'=5 w v=0
2
1
f
t
e
3 4
0,1,2,3
0,3,4
0,4
0,1,3 0,2,3 0,1,2 1,2,3
0,5 3,4 0,3 0,2 0,1 2,3 1,3 1,2
4 5 0 3 2 1
IG
Boundary Graph
0,1,2,3
0,3,4
0,4
0,1,3 0,2,3 0,1,2 1,2,3
0,5 3,4 0,3 0,2 0,1 2,3 1,3 1,2
4 5 0 3 2 1
IG
Co-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;p1() Directly encoded O(1)
Rp;q (), p q Recursively combine Rp;p1, Rp1;p2, and so on O(1)
Rp;p+1() Directly encoded O(1)
Rp;q (), p q Recursively combine Rk;k+1 and Rk+1;k , for k p O(kRp;q ()k)
R0;0() Combine R0;1 and R1;0 O(kR0;0()k)
Rp;p(), with p6= 0 Combine Rp;p1 and Rp1;p O(kRp;p()k)
Storage Cost
2
Xd
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
For each p-simplex :
I boundary relation Rp;p1()
I partial co-boundary relation Rp
;p+1()
Global Mangrove (IS-Graph)
Partial co-boundary relation Rp
;p+1()
One arbitrary (p + 1)-simplex for each
connected component in Lk().
v'=5 w v=0
2
1
f
t
e
3 4
R 0;1(v) = fw; eg
Important
Rd
1;d Rd1;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;p1
IS Boundary Graph IG Boundary Graph
Nodes + IS Boundary Arcs
IS Co-boundary Arcs
Correspond to Rp
;p+1
IS Co-boundary Graph
Nodes + IS Co-boundary Arcs
v'=5 w v=0
2
1
f
t
e
3 4
0,1,2,3
0,3,4
0,4
0,1,3 0,2,3 0,1,2 1,2,3
0,5 3,4 0,3 0,2 0,1 2,3 1,3 1,2
4 5 0 3 2 1
IS
Boundary Graph IG Boundary Graph
0,1,2,3
0,3,4
0,4
0,1,3 0,2,3 0,1,2 1,2,3
0,5 3,4 0,3 0,2 0,1 2,3 1,3 1,2
4 5 0 3 2 1
IS
Co-boundary Graph
IS-Graph is more compact than IG-Graph

Storage Cost of the IS Data Structure
Boundary Relations
Xd
p=1
sp(p + 1)
sp : number of p-simplices
+
Partial Co-boundary Relations
Xd
p=1
X
2p
H
Xd
p=1
sp(p + 1)
H : #connected components in Lk()
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

Storage Cost of the IS Data Structure (cont’d)
For Manifolds:
;p+1 contains:
only one (p + 1)-simplex, if p d
one or two d-simplices, if p = d 1
Remark
Rd
1;d Rd1;d
v v
e
All edges in R0;1(v) versus one edge in R0
;1(v)
Boundary Relations
Xd
p=1
sp(p + 1)
+
Partial Co-boundary
Relations
dX2
p=0
sp + (d + 1)sd

Topological Relations in the IS Data Structure
Rp;p1() Directly encoded O(1)
Rp;q (), p q Recursively combine Rp;p1, Rp1;p2, and so on O(1)
Rd1;d () Directly encoded O(1)
Rp;q (), p q Recursively combine Rk
;k+1 and Rk+1;k , for k p O(kSt()k)
R0;0() Combine R0;1 and R1;0 O(kSt()k)
Rp;p(), with p6= 0 Combine Rp;p1 and Rp1;p O(kSt()k)
IS star-graph of a p-simplex
Subgraph G of the IS-graph:
nodes representing simplices in St()
IS boundary arcs restricted to St()
IS co-boundary arcs restricted to St()
Co-boundary relation Rp;q()
breadth-first traversal of G
examine top simplices in St()
and their faces
linear in kSt()k
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 with
Adjacencies (IA)
Key Idea
More compact encoding for a simplicial
d-complex
The Indexed data structure with
Adjacencies (IA), Paoluzzi et al., 1993
vertices, plus d-simplices
boundary relation Rd;0 for
d-simplices
adjacency relation Rd;d for
d-simplices
only for manifolds
The IA data structure
Encodes vertices and top simplices
Adjacency-based
Probably, the most compact representation
for non-manifolds (with respect to the state of
the art)
Non-manifold variant of the Extended IA
(EIA) data structure, De Floriani, et al. 2003
For manifolds, it reduces to the EIA data
structure (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 - Definition
Represents an abstract simplicial d-complex
Boundary relation Rp
;0()
Vertices of a top p-simplex ,
for 1 p d
R 1;0(w) = f1; 2g,
R2
;0(f1) = f1; 3; 4g
Adjacency relation Rp
;p()
Top p-simplices sharing a
(p 1)-simplex with a top
p-simplex , with 2 p d
R 2;2(f1) = ff2; f3; f4g,
R23
2(f5) = ff6g,
;R3(t1) = ft2g
; For Manifolds
IA reduces to EIA
5
3
6
3 2
7 4
1
2
8
9
10
f
11 13
12
14
w
f f
f
t
t
f
f
1
1
4
5
6
2
e
v
only one d-simplex in
Rd for each vertex
0
;Rd1;d : empty
at most one d-simplex
in Rd;d
p-cluster
Maximal collection of adjacent top
p-simplices
2-clusters: ff1; f2; f3; f4g; ff5; f6g
3-cluster: ft1; t2g
Partial co-boundary relation R0
;p(v)
Arbitrary top p-simplex for each
p-cluster in St(v), with 2 p d
R 0;1(v) = fwg,
R 0;2(v) = ff2; f5g, R0
;3(v) = ft1g
1;p()
Top p-simplices incident at a
(p 1)-face of a top p-simplex,
with 2 p d (more than two)
R 1;2(e) = ff1; f2; f3; f4g

The IA data structure - Non-Manifold Adjacency
Key Idea
A (p 1)-face of a top p-simplex is non-manifold if it is shared by more than two top
p-simplices.
5
3
6
3 2
7 4
1
2
8
9
10
f
11 13
12
14
w
f f
f
t
t
f
f
1
1
4
5
6
2
e
v
Manifold Adjacency - At most two top p-simplices in St()
Encode only the other top p-simplex adjacent to along
;2(f5) = ff6g, R2
R2
;2(f6) = ff5g
Non-Manifold Adjacency (Otherwise)
Encode Rp
;p() along as Rp
1;p()
;2(fi ) = R 1;2(e) = ff1; f2; f3; f4g, with i = 1; : : : ; 4
R2
Consequences
Compact encoding of Rp
;p, Rp
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 Rp
;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 Rp
;p and Rp
1;p
1,11,12,14
1,3,7
1,3
1,12,13,14
1,8,9 1,9,10 1,3,4 1,3,5 1,3,6
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 (compression
scheme)
2D manifolds, no editing

Topological Relations in the IA* Data Structure
Given a simplicial d-complex , a simplex not directly encoded is represented by its vertices :
Rp
;0() Directly encoded O(1)
Rp;q (), p q Generate faces of O(1)
R0;k (v) (top) Expand R 0;k (v) by Rk;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 (), p q Select q-simplices in St() from top simplices in St(v) O(#top simplices in St(v))
p
Rd;d () Directly encoded O(1)
R0;0(v) Combine R0;1 and R1;0 O(#top simplices in St(v))
Rp;p() Extract Rp and combine Rp;p+1 and Rp+1;p O(#top simplices in St(v))
;: v is a vertex in Rp;0()
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 each
k-cluster in R 0;k (v)
Transitive closure of Rk
;k
Linear 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)
Framework
The 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 architecture
I 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 at
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 :
BOUNDARY - boundary B()
STAR - star St()
ADJACENCY - adjacency relation
Rp;p()
LINK - link Lk()
IS_MANIFOLD - checks if is manifold
(when possible)
In this context
Mangrove dynamic plugin in the system
Current Implementations (but extensible)
IG, IS, IA data structures
TS data structure
I adjacency-based
I simplicial 2-complexes in E3
I De Floriani et al., 2003
NMIA data structure
I adjacency-based
I simplicial 3-complexes in E3
I De Floriani and Hui, 2003
SIG data structure
I incidence-based
I dimension-independent
I 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

The Mangrove TDS Framework - Ghost Simplices
Ghost p-simplex
Not directly encoded in a local mangrove
Explicit Representation
Set of vertices V = fv0; : : : ; vpg
too knownledge
no efficiency for any queries
Implicit Representation
A p-simplex can be either:
a top p-simplex , or
a p-face of a top t-simplex 0, p t
GhostSimplexPointer reference
(t; i; p; pi)
i is the identifier of 0
pi is the identifier of as p-face of 0
0 pi
t + 1
pi + 1

Advantages
less knowledge is required
fixed-length representation
does not depend on an enumerations
of faces
Disadvantage?
a not unique representation
a GhostSimplexPointer reference for
each top simplex in St()

Explicit Representation of Ghost Simplices
Key Idea
A ghost p-simplex is described by (p + 1)
positions of vertices in V0 Rt;0(0) as
= [k0; : : : ; kp]
Enumeration Rule (Simplicial Homology)
The i-th (p 1)-face i of is defined as
i = [k0; : : : ; ki1; ki+1; : : : ; kp]
Consequence
Partial order relation , such that
i , i.e., i is a face of
Two Hasse diagrams for all t 1
Storage cost of Hasse diagrams
2
Xd
t=1
Xt
p=0
t + 1
p + 1

0,1,2,3
1,2,3 0,2,3 0,1,3 0,1,2
2,3 1,3 1,2 0,3 0,2 0,1
0 1 2 3
(3; 0; 2; 2): vertices in positions [0; 1; 3] in R3;0(t0)
0,1,2,3
0,1,2 0,3,4 1,3,5 2,4,5
2,3 1,3 1,2 0,3 0,2 0,1
0 1 2 3
Same lattice in terms of immediate subfaces
(3; 0; 2; 2) formed by edges [1; 3; 5]
Explicit representation of in O(1)

Experimental Results on Our Mangroves
We have compared the efficiency of queries on our six mangroves within the Mangrove TDS
Framework
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 simplex
I STAR is 35% more efficient than IS for vertices
I 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 the
expressive 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 (shape
decomposition)
Semantic model (future
work)
Structural Model for Non-Manifolds
Components joint together at
non-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 and k d:
Manifold (k 1)-path (MC-Adjacency)
Sequence of k-simplices in , where each of
simplices is adjacent through a manifold
(k 1)-simplex, bounding at most two
k-simplices
Manifold-Connected (MC) k-Complex
Formed by all k-simplices in connected by a
manifold (k 1)-path
MC-Decomposition
Collection of MC k-Complexes in
MC k-Complexes are the equivalence classes versus MC-Adjacency, and become unique if
restricted to top k-simplices in

Manifold-Connected (MC) Decomposition (cont’d)
MC-Decomposition
Decomposition of a simplicial complex 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-Decomposition
Two-level Graph-based Data Structure
the lower level describes a non-manifold shape by any mangrove (topological model)
the upper level describes the connectivity of MC-components through a graph-based data
structure (structural model)
MC-graph G = (N;A)
each node in N one MC-component (direct references to top simplices in );
each arc a = (n1; n2; : : : ; nk ) in A intersection of MC-components described by
n1; n2; : : : ; nk (common singularities, as direct references to simplices in )
Relating MC-Components and singularities
the number of MC-Components partially characterizes a
singularity
needs IS_MANIFOLD (no dimension-independent)
efficiency depends on the properties of mangrove
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: C4
3 MC-components of dimension 2: C1, C2, C3
Pairwise MC-Graph
An arc intersection of two MC-Components, formed by
a subset of singularities
(partial)
Exploded MC-Graph (Hyper-graph)
A hyper-arc a singularity , and connects all
MC-components sharing
Compact MC-Graph (Hyper-graph)
An hyper-arc corresponds to a maximal set of singularities
common to several MC-components

Experimental Results
We 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 - Sergeraert
and Rubio, 2006
Basic idea
Combine:
homology of its MC-components
homology of the intersection of MC-components
45
nodes, 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 Approach
Associate an algebraic object, namely a chain-complex (;D), to a simplicial complex from
which we extract the Z-homology (Betti numbers, generators, torsion coefficients)
sequence of chain-groups p
Group of p-chains linear combinations of
oriented k-simplices
(;D) : 0 0
: : :
dp1
p1
dp
: : :d
0
0
Smith Normal Form (SNF), Munkres,1999
Incidence Matrix Ip is reduced through
Gaussian eliminations to its Smith Normal
Form (SNF) Np:
Np =
0
p1
0 0 0
: 0 0 Id

B@
p
0 : : :

p
l

p1
m 0 0 0
1
CA
where:
is a diagonal matrix, with i 2 Z
Ip = Pp1NpPp (basis change)
D sequence of boundary operators dp
Describes the oriented boundary Bo(
p
i ) of a
p-simplex
p
i in terms of its immediate
p1
j by the incidence matrix Ip
subfaces
Incidence Matrix Ip of order p
Ip
j;i =
8
:
p1
j62 Bo(
0 if
p
i )
p1
j 2 Bo(
1 if +
p
i )
p1
j 2 Bo(
1 if
p
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 Algorithm
Input: a simplicial d-complex discretizing a non-manifold shape
Output: the Z-homology (Betti numbers, generators, torsion-coefficients)
First step
compute SNF reductions of
all MC-components
Generic step
Given components A and B
two components such that
A B6= ;, 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 simplices
MS: maximum size of a MC-Component
MG: maximum size of the intersection of two
MC-components
Property #1
Guarantees a small size of the intersection
between two MC-Components
Property #2
Produces subcomplexes smaller than the
input shape
#
Good properties for the MV algorithm
Suitable for computations
Consequence #1
Small MC-components reduce time
complexity of the SNF reductions
Consequence #2
Small intersections make the cone
reductions possible (while merging the
MC-components)

Experimental Results
We have exploited:
the SNF algorithm provided by Moka Modeller, G. Damiand, LIRIS, Lyon, France (no
optimizations), http://moka-modeler.sourceforge.net
our Pairwise MC-Graph + IS data structure
Shape SNFs(MB) SNFt (ms) MVs(%) MVt(%) Result
Armchair 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

Experimental Results (cont’d)
MC-Decomposition + (Z; Z2; Z2) for the Twist
shape
MC-Decomposition + (Z; Z4; Z3) for the Twist
shape

Conclusions and Future Works
What 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 structure
I the Generalized Indexed data structure with Adjacencies (IA).
Mangrove TDS framework
I rapid prototyping of data structures for arbitrary simplicial complexes
Research Area II - Decompositions and Structural Models
Manifold-Connected (MC) Decomposition - Hui and De Floriani, 2007
I the Exploded MC-Graph (hyper-graph)
I the Pairwise MC-Graph
I 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 for
manifolds

My Papers
1 D. Canino, L. De Floriani, A Decomposition-based Approach to Modeling and Understanding Arbitrary
Shapes, 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 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)
3 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
4 D. Canino, A Dimension-Independent and Extensible Framework for Huge Geometric Models, 8th
Eurographics Italian Chapter Conference, Eurographics Association, 2010, Poster
5 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, pages
403-420, 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 Papers
L. 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, In
Proceedings of the 1st Eurographics Symposium on Geometry Processing (SGP ’03), pages 72-82, ACM
Press, 2003
L. De Floriani and A. Hui, Data Structures for Simplicial Complexes: an Analysis and a Comparison, In
Proceedings 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 for
Non-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, In
Proceedings 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 in
Mathematical Physics, 181:303-330, 1996.
A. Paoluzzi, F. Bernardini, C. Cattani, and V. Ferrucci, Dimension-Independent Modeling with Simplicial
Complexes, 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 533â˘A
S¸ 550.
Springer, 2011.
F. Sergeraert and J. Rubio, Constructive Homological Algebra and Applications, 2006,
http://www-fourier.ujf-grenoble.fr/sergerar/Papers/

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