Mathematical Tools for 3D Shape Analysis and Descriptionsaturno.ge.imati.cnr.it/ima/personal/biasotti/SGP13/SGP_partII.pdf · 3D Shape Analysis and Description references 28 mathematical

Eurographics Workshop on 3D Object RetrievalSGP 2013 Graduate School

Mathematical Tools for 3D Shape Analysis and Description

tools and concepts, part II

Andrea Cerri and Silvia Biasotti

mathematical concepts

– basics on algebraic topology

– simplicial complexes

– homology

– Morse theory

concepts in action

– comparing shapes

– persistent topology

– Reeb graphs (by Silvia)

01/07/2013 3D Shape Analysis and Description

content

2

topological spaces and functions to model

shapes and their properties as pairs 𝑋, 𝑓 ;

suitable for dealing with stability/ robustness;

are there alternatives to isometries to solve

other sets of problems?


isometries, but not only…

3




– homology

– Morse theory

concepts in action





content

4

Associates algebraic invariants with

topological spaces;

Classifies topological spaces up to

homeomorphisms;

Homology is an option.


algebraic topology & homology

5

General idea: to distinguish the shape of

topological spaces by their holes;

Several homology theories (singular, cellular,

Čech, etc…);

We will focus on simplicial homology.


basics on homology

6




– homology

– Morse theory

concepts in action





content

7

approach: to decompose a topological

space into simple pieces easier to study;

q-simplices are the building blocks:

– combinatorial aspect: relations among simplices;

– geometric aspect: related to their embedding in

the Euclidean space.

a simplicial complex is the combinatorial

structure, generated by q-simplices;


simplicial homology

8

a 𝒒-simplex in ℝ𝑛, with 𝑛 ≥ 𝑞, is the convex

hull of 𝑞 + 1 affinely independent points.

a face of a 𝑞-simplex is the convex hull of a

subset of the set of its points.


simplices and faces

0-simplex 1-simplex 2-simplex 3-simplex

9

a (finite) simplicial complex 𝐾 is a (finite)

collection of simplices so that

– if 𝜎 ∈ 𝐾 and 𝜏 is a face of 𝜎, then 𝜏 ∈ 𝐾;

– any two simplices can only meet along a

common face.


simplicial complex

10

the dimension of 𝐾 is the highest among the

dimensions of its simplices:

– triangle meshes are 2-complexes;

– tetrahedral meshes are 3-complexes.


dimension of a simplicial complex

11




– homology

– Morse theory

concepts in action





content

12

a 𝒒-chain is a sum of all 𝑞-simplices in 𝐾,

𝜎 = 𝑖 𝜆𝑖 𝜎𝑖, 𝜆𝑖 ∈ ℤ2= {0,1}.

– a curve on a mesh is a 1-chain

– a surface patch is a 2-chain

the 𝒒-chain space 𝐶𝑞 𝐾 is the vector space

generated by all the q -simplices in K ;


chains…

13

the boundary operator extracts the

boundary of a chain

the boundary of a boundary is null


… and boundaries

0

14

𝐾 is a 𝑘- complex of ℝ𝑛; 𝐶1(𝐾), … , 𝐶𝑘(𝐾) are

the chain spaces. The boundary operator𝜕𝑞: 𝐶𝑞(𝐾) → 𝐶𝑞−1(𝐾)

is defined as:

– for a 𝒒 −simplex 𝜎 = 𝐴0, … , 𝐴𝑞 ,

𝜕𝑞 𝜎 ≝ 𝑖=0𝑞

[𝐴0, … , 𝐴𝑖 , … , 𝐴𝑞] ;

– for chain 𝜎 = 𝑖 𝜆𝑖𝜎𝑖,

𝑖 𝜆𝑖𝜎𝑖, 𝜕𝑞(𝜎) ≝ 𝑖 𝜆𝑖𝜕𝑞(𝜎𝑖).

N.B.: sums are in ℤ2, that is, 1+1=0!


boundary operator (more formally)

15

boundary of 𝜎1 = 𝐴0, 𝐴1, 𝐴2 , 𝜕2 𝜎1 and 𝜎 = 𝜎1 + 𝜎2:

𝜕2 𝜎1 = 𝐴1, 𝐴2 + 𝐴0, 𝐴2 + [𝐴0, 𝐴1];

𝜕2 𝜎 = 𝜕2 𝜎1 + 𝜕2(𝜎2) =

= 𝐴1, 𝐴2 + 𝐴0, 𝐴2 + [𝐴0, 𝐴1] + + 𝐴2, 𝐴3 + 𝐴1, 𝐴3 + 𝐴1, 𝐴2 =

= 𝐴0, 𝐴2 + [𝐴0, 𝐴1] + 𝐴2, 𝐴3 + 𝐴1, 𝐴3 .


boundary operator (example)

𝜎1

𝜎2

𝐴0

𝐴1

𝐴2

𝐴3

𝜕1 𝜕2(𝜎1) = 𝜕1( 𝐴1, 𝐴2 ) + 𝜕1( 𝐴0, 𝐴2 ) + 𝜕1([𝐴0, 𝐴1])== 𝐴2] + [𝐴1 + 𝐴2] + [𝐴0 + 𝐴1] + [𝐴0 = 0;

16

a 𝑞-chain:

– is a 𝒒-cycle if its boundary is 0;

– is a 𝒒-boundary if is a boundary of a (𝑞 + 1)-chain;

Intuition:

Homology formalizes this idea


cycles and boundaries

17

Set 𝒁𝒒 𝑲 = {𝑞-cycles in 𝐶𝑞(𝐾)};

Let 𝑩𝒒 𝑲 = {𝑞-boundaries in 𝐶𝑞(𝐾)};

Since 𝜕 ∘ 𝜕 ≡ 0, then 𝐵𝑞 𝐾 ⊆ 𝑍𝑞 𝐾 ;

The 𝒒-𝒕𝒉 simplicial homology group of 𝐾 is

𝐻𝑞 𝐾 ≝ 𝑍𝑞(𝐾)/𝐵𝑞(𝐾)


simplicial homology

18

an element of 𝐻𝑞(𝐾) is an equivalence class

containing homologous 𝑞-cycles;

the rank of 𝐻𝑞(𝐾) is the 𝑞-th Betti number of 𝐾;

the homology 𝐻∗(𝐾) is a topological invariant.


simplicial homology (II)

For 3D data:─ 𝛽0 = # components;─ 𝛽1 = # tunnels;─ 𝛽2 = # voids;

19

example: genus of a surface

Let 𝑆 be a connected, orientable surface

without boundary. The genus 𝑔 of 𝑆 is– the maximum number of cuttings along non-

intersecting closed simple curves which can be cut along the surface without disconnecting it…

– … or half of the first Betti number 𝛽1 𝑆 .

The genus is a topological invariant.

3D Shape Analysis and Description01/07/2013 20




– homology

– Morse theory

concepts in action





content

21

algebraic topology provides invariants up

to homeomorphisms;

what if we want to reason about shapes

under more invariants?

we could use critical points of functions…


algebraic topology and differential geometry

22

𝑓: 𝑀 → ℝ smooth function on a smooth

manifold 𝑀 of dimension 𝑛;

– 𝑝 ∈ 𝑀 is critical if 𝑑𝑓𝑝 is the zero map, i.e. 𝑑𝑓

𝑑𝑥1𝑝 = ⋯ =

𝑑𝑓

𝑑𝑥𝑛𝑝 = 0;

– A critical point 𝑝 ∈ 𝑀 is non-degenerate if the Hessian 𝐻𝑓 𝑝 is non-singular, i.e.


functions and critical points (I)

det 𝐻𝑓 𝑝 = 𝑑𝑒𝑡𝜕2𝑓

𝜕𝑥𝑖𝜕𝑥𝑗(𝑝) ≠ 0

23

If 𝑝 is a non-degenerate critical point of 𝑓, the index 𝜆𝑝 of 𝑝 is defined as:

𝜆𝑝 = #{𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑒𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝐻𝑓(𝑝)};

𝑓: 𝑀 → ℝ is a Morse function if all of its critical

points are non-degenerate;


functions and critical points (II)

𝜆𝑝 = 0 𝜆𝑝 = 1 𝜆𝑝 = 2

24

𝑝 non-degenerate critical point of index 𝜆𝑝, with

𝑓(𝑝) = 𝑞. Suppose 𝑓−1[𝑞 − 𝜀, 𝑞 + 𝜀] compact, with no critical points besides 𝑝. Then 𝑀𝑞−𝜀 is homotopy

equivalent to 𝑀𝑞+𝜀 with a 𝜆𝑝-dim. disk attached.

For 𝑐 ∈ ℝ , set 𝑀𝑐 = 𝑝 ∈ 𝑀: 𝑓 𝑝 ≤ 𝑐 = 𝑓−1( −∞, 𝑐 ).

How/when does the topology of 𝑀𝑐 changes?

Let 𝑎 < 𝑏. Suppose 𝑓−1 𝑎, 𝑏 = 𝑝 ∈ 𝑀: 𝑎 ≤ 𝑓 𝑝 ≤ 𝑏is compact and has no critical points. Then 𝑀𝑎 and

𝑀𝑏 are diffeomorphic.


Morse functions & critical points: Two results

25 01/07/2013 3D Shape Analysis and Description

Morse functions & critical points: Example

𝛽0 𝛽1 𝛽226

Morse theory can be used for..

to combine the topology of 𝑀 with the

quantitative measurement provided by 𝑓

– 𝑓 is the lens to look at the properties of (𝑀, 𝑓)

– different choices of 𝑓 provide different invariants

to construct a general framework for shape

characterization which if parameterized wrt

the pair (𝑋, 𝑓)


S. Willard, General Topology, Addison-Wesley Publishing Co, 1970

R. Engelking and K. Sielucki, Topology: A geometric approach, Sigma series in pure mathematics, Heldermann, Berlin, 1992

A. Fomenko, Visual Geometry and Topology, Springer-Verlag, 1995M.

W. Hirsch, Differential Topology, Springer, 1997

H. B. Griffiths, Surfaces, Cambridge University Press, 1976

V. Guillemin and A. Pollack, Differential Topology, Englewood Cliffs, NJ:Prentice Hall, 1974

W. Massey, Algebraic topology: An Introduction, Brace&World Inc., 1967

A. Hatcher, Algebraic Topology, Cambridge University Press, 2001

J. Milnor, Morse theory, Princeton University Press, New Jersey, 1963

C. Kosniowski, A First Course in Algebraic Topology, Cambridge University Press, 1966


references

28




– homology

– Morse theory

concepts in action





content

29

Comparing objects’ shapes can done with respect

to some “relevant properties”;

To model a shape we consider pairs (𝑋, 𝑓) s.t.

– X represents the object;

– 𝑓: 𝑋 → ℝ is a function and describing the relevant

properties of the object.


comparing shapes (I)

(𝑋, 𝑓) (𝑋, ℎ)(𝑋, 𝑔) (𝑋, 𝑙)

30

How can we compare two pairs (𝑋, 𝑓), (𝑌, 𝑔) ?

Idea: Use a metric able to quantify and measure the deformation of 𝑋 into 𝑌, in terms of 𝑓 and 𝑔.

comparing shapes (II)

(𝑋, 𝑓) (𝑌, 𝑔)

01/07/2013 3D Shape Analysis and Description 31

formally: Natural pseudo-distance 𝒅

𝑑( 𝑋, 𝑓 , 𝑌, 𝑔) ≝ infℎ

max𝑥∈𝑋

𝑓 𝑥 − 𝑔 ℎ 𝑥 ∞ ,

ℎ varying among all homoeomorphisms from X to Y.

natural pseudo-distance (I)

(𝑋, 𝑓) (𝑌, 𝑔)


Problem: the natural pseudo-distance 𝑑 is difficult to

compute; we need tools to get information about it;

Persistent topology allows us get lower bounds for 𝒅by means of suitable shape descriptors;

Instead of comparing pairs (space,functions), we can compare the associated descriptors.

natural pseudo-distance (II)





– homology

– Morse theory

concepts in action





content

34

a filtration is a nested sequence 𝑋1 ⊆ 𝑋2 ⊆ ⋯𝑋𝑛 = 𝑋,

e.g. the sub-level sets of a function 𝑓: 𝑋 → ℝ.

topological exploration along a filtration of X, looking for important topological events.

persistence: Filtrations


Measure the lifespan of homology classes along the

filtration:

Encode the birth level 𝑖 and the death level 𝑗 of a homology class by a point (𝑖, 𝑗).

persistence diagrams (I)


𝑓

𝑖

𝑗

36


persistence diagrams (II)

𝑓𝑋

𝛽0 𝛽1 𝛽2

37

Changing functions produces different information.


persistence diagrams: Modularity

38

Invariance properties w.r.t. transformation groups

are inherited from the function.


persistence diagrams: Invariance properties

39

Stability with respect to the bottleneck distance*:

𝑑𝐵 Dgm 𝑓 , Dgm 𝑔 ≤ infℎ

max𝑥∈𝑋

𝑓 𝑥 − 𝑔 ℎ 𝑥∞

This implies resistance to noise;

It is a lower bound for the natural pseudo-distance.

________________* Cohen-Steiner et al. (2005), Chazal et al. (2009), d’Amico et al. (2010)...


persistence diagrams: stability

(𝑋, 𝑓) (𝑌, 𝑔)

40

system for content-based retrieval of figurative;

Dataset containing more than 10000 trademarks;

Goal: For an image, find in the dataset the most similar ones.


Retrieval of trademark images [C., Ferri, Giorgi 2006]

41

Approach:

– for each image, battery of 25 descriptors;

– each distance between descriptors of the same type induces a metric on the database;

– combine all 25 metrics (max, sum…) to obtain a

final similarity score.


Retrieval of trademark images [C., Ferri, Giorgi 2006]

42

Point cloud data analysis [Chazal et al. ‘09]

3D segmentation [Skraba et al. ‘10]


Other possible applications…

43

data simplification [Bauer, Lange, Wardetzky ‘12]

3D (textured) shape analysis and comparison [Biasotti et al. ‘13]


Other possible applications…

44

Size functions [Frosini ‘91];

Persistent homology groups [Edelsbrunner, Letscher, Zomorodian

‘02];

Vines and vineyards [Cohen-Steiner, Edelsbrunner, Morozov ‘06];

Interval persistence [Dey, Wegner ‘07];

Zig-Zag persisetnce [Carlsson, de Silva, Morozov ‘09];

Persistent cohomology [de Silva, Morozov, Vejdemo-Johansson];

…

Multidimensional setting (𝑓: 𝑋 → ℝ𝑘):

multidimensional persistent homology group (Carlsson, Zomorodian ‘07);

Multidimensional size functions (Biasotti et al. ‘08);

Persistence spaces (C., Landi ‘13);

…


an historical perspective

45

Reeb graph

Reeb graphs store the evolution of the level

sets of the mapping function f

M

f


let M be a compact n-manifold, f: M→R a simple Morse function, and given “~” the

equivalence relation

(P, f(P)) ~ (Q, f(Q)) f(P) = f(Q) and P and Q are in the same connected component of 𝑓−1(𝑓(𝑃))

the quotient space on Mxℝ is a finite, connected simplicial complex K of dimension 1,

such that

– the counter-image of each vertex of K is a singular

connected component of the level sets of f

– the counter-image of the interior of each simplex of

dim 1 is homeomorphic to the topological product of one connected component of the level sets by ℝ


Reeb graph definition


overview of RGs when the function f varies

height bounding sphere

center

integral

geodesic

curvature

extrema

barycenter

f values

48

quiz

draw the Reeb graph with respect to the

height function f of the following shapes

f


quiz - solutions

f



RG properties

it provides a 1D structure of the shape

it describes the shape of an object under

the lens of the function f

nodes and arcs depend on the number of

critical points of f

it may be computed in operations )log( nnO

51 01/07/2013 3D Shape Analysis and Description

Reeb graph based representations

Reeb graph variations

– contour trees (simply-connected domains)

– component trees (gray-level images)

– centerline skeletons (geodesic distance from a point)

for shape matching

– Multiresolution Reeb graph (MRG), Hilaga et al. 2001

– augmented Multiresolution Reeb graph (aMRG), Tung&Schmitt 2005

– Extended Reeb graph (ERG), Biasotti et al. 2000

52

extension to volume data

f

distance

from the

barycentre


geometric embedding

nodes of the ERG correspond to regions (eithersurfacic or volumic)

arcs code the adjacency among the parts

each arc can be oriented using the growing direction of the mapping function: the RG is a direct acyclic graph

with each ERG node, store spatial attributes measuring properties of the part associated to it– node embedding (Cartesian coordinates, x, y, z)

– average value of f in the part

– volume

– area

– some radii: min, max, average


rmin

ravrmax

54

geometric embedding

each arc can be oriented using the increasing direction of f: the RG is a direct acyclic graph

nodes and arcs correspond to surface parts

node and arc attributes are stored in terms of the spherical harmonic indices of the parts

01/07/2013 3D Shape Analysis and Description 55 01/07/2013 3D Shape Analysis and Description

part correspondence

models with similar appearance

56


part correspondence

objects with dissimilar global appearance

57

shape retrieval

performance on the SHREC'07 watertight

database


ERG

ERG

ERG

58

shape approximation and compression

given the Reeb graph RG (M,f) of f Morse

and simple, a rough shape chartification is

obtained

1-strip

contains one max or min

one boundary component

3-strips

contains one saddle

three boundary components

f may assume different values

on each of these three

boundary components

RG (M,f)


v=3.8K

138K

v=35K

t =69K

2,6MB


references

P. Frosini, A distance for similarity classes of submanifolds of a Euclidean space, Bull. Australian Mathematical Society 42 , 407–416, 1990

P. Frosini, Measuring shapes by size functions, SPIE, Intelligent Robots and Computer Vision X, Vol. 1607, pp. 122-133, 1991

P. Frosini and C. Landi, Size functions and formal series, Appl. Algebra Engrg. Comm. Comput., Vol. 12, pp. 327-349, 2001

H. Edelsbrunner, J. Harer. Computational Topology - an Introduction. AMS , I-XII, 1-24 1, 2010

H. Edelsbrunner , J. Harer, Persistent homology: a survey. Surveys on discrete and computational geometry, vol. 453 of Contemp. Math AMS, 2008

R. Ghrist. Barcodes: the persistent topology of data, Bull. Amer. Math. Soc., 45(1):61-75, 2008

S Biasotti, A Cerri, P Frosini, D Giorgi, C Landi. Multidimensional size functions for shape comparison, J. of Math. Imaging and Vision 32 (2), 161-179

S. Biasotti, A. Cerri, P. Frosini, D. Giorgi, A new algorithm for computingthe 2-dimensional matching distance between size functions, Patt. Rec. Letters 32 (14), 1735-1746


references

G. Reeb, Sur les point sunguilèrs d’une form del Pfaff complètement intégrable oud’une fonction munèrique, Comptes Rendu de l’Academie des Sciences, 222:847-849, 1946

Y. Shinagawa, T. Kunii, Y. Kergosien, Surface coding based on Morse Theory, IEEE Computer Graphics and Applications, 11(5):66-78, 1991

M. Hilaga, Y. Shinagawa, T. Komura, T. Kunii, Topology matching for fullyautomatic similarity estimation of 3D shapes, Proc. SIGGRAPH 2001, pp. 203-212, 2001

S. Biasotti, L. De Floriani, B. Falcidieno, P. Frosini, D. Giorgi, C. Landi, L. Papaleo,M. Spagnuolo, Describing shapes by geometrical-topological properties of real functions, ACM Computing Surveys, 40(4):1-87, 2008

S. Biasotti, D. Giorgi, M. Spagnuolo, B. Falcidieno, Reeb graphs for shape analysis and applications, Theoretical Computer Science, Elsevier, 392(1-3):5-22, 2008

S. Biasotti, G. Patanè, M. Spagnuolo, B. Falcidieno, G. Barequet, Shape approximation by differential properties of scalar functions, Computers & Graphics, 34(3):252-262, 2010

T. Dey, Y. Wang. Reeb graphs: approximation and persistence, Proc. Symposium comput. Geom., 226-235, 2011

V. Barra, S. Biasotti, 3D shape retrieval using Kernels on Extended Reeb Graphs, Pattern Recognition, 46(11):2985-2999, 2013



Mathematical Tools for 3D Shape Analysis and Description

wrap up

Silvia Biasotti

to sum up: theory says …

we have seen:– how to represent a shape (topological spaces,

manifolds, metric spaces, simplicial complexes…)

– how to analyze, describe, compare shapes(geodesics, curvature, diffusion geometry, Morse theory, algebraic machinery…)

– examples of different applications

we have investigated:– space, invariance, property…

– theoretical assumptions vs real world assumptions

but if we go back to «real» shapes and problems, are we sure that everything works ok?– services which could use the theories discussed

– discretization issues and applicability


complex shapes, complex problems

online repositories of 3D models

– Google 3D Warehousehttp://sketchup.google.com/3dwarehouse/





– 3Dvia http:// www.3dvia.com/search/


http://sketchup.google.com/3dwarehouse/


http://www.turbosquid.com/

http://www.3dvia.com/search/




– 3Dvia http:// www.3dvia.com/search/

– Turbosquid http://www.turbosquid.com/

• “…search our stock catalog to get the 3D

model you want, or use our Custom 3D modeling service for made-to-order 3D

models. Join the world's top artists who use TurboSquid 3D models in advertising, architecture, broadcast, games, training, film,

the web, and just for fun”



3D Object Retrieval & Content-based

search

– Princeton Shape Benchmark

– Digital Shape Workbench v5.0

http://visionair.ge.imati.cnr.it/


to sum up: theory says …

3D shapes maybe very complex and serviceswe can imagine for sharing, searching, and

even design new shapes are many, so…

… how to get out of this maze?

basics

– the choice of one or another shape description

should be guided by

• type of shapes and their variability/complexity

• invariants or properties to be preserved / captured

– topological spaces and functions are promising to

model and reason about similarity in a mathematically well-formulated manner


pay attention to…

… the right space– rigid transformations (rotations, translations)

• Euclidean distances

– isometries

• Riemannian metric

• curvature (but unstable to local noise/perturbations)

• geodesics, diffusion geometry, Laplacian operators, etc

– local invariance to shape parameterizations

• conformal geometry

– similarities (i.e. scale operations)

• normalized Euclidean distances

– affinity (and homeomorphisms)

• persistent topology

• Morse theory

• size theory


pay attention to…

… a suitable shape description– coarse coding (but fast)

• histograms

• matrices

– articulated shapes

•medial axes

• Reeb graphs

– overall global appearance

• silhouettes

– if shape loops are relevant

• persistent topology

• graph-based descriptions


to sum up: benchmarking

importance of benchmarking and evaluation!– not only publishing papers but also to demonstrate

real innovation and applicability potential

Shape segmentation benchmark

SHape Retrieval Contexthttp://www.aimatshape.net/event/SHREC– an annual event to to evaluate the effectiveness of

3D shape analysis algorithms

– a multi-track event spanning

• different models: from watertight objects to triangle soups, from abstract shapes to medicaldata

• different tasks: from 3D retrieval to correspondence finding and segmentation




http://www.3dvia.com/search/


http://visionair.ge.imati.cnr.it/

http://www.aimatshape.net/event/SHREC

example: SHREC'12 - SAS

to evaluate the stability of the algorithms

under variations of abstract shapes

characterized by

– smooth/sharp features,

– partial/global symmetries

– different genus

to create a new benchmark where to

measure the capability of the retrieval

methods to be invariant to

– noise addition and resampling

– deformations


dataset

4

1. addition of Gaussian noise

2. different regular sampling

3. uneven sampling

4. & .5 two stretching

6. & .7 two non-uniform dilations

8. & .9 two non-uniform erosions

9 transformations, 3 degrees each, for a total of 27 perturbations


dataset


504 watertight models without self-intersections


recall-precision curves (whole dataset)


recall-precision curves (deformations)


conclusions

the right tool for the right problem

– mathematically sound methods

– benchmarking & evaluation

geometry, structure, similarity, context

– is it possible to understand something about

functionality or affordance?

– machine learning vs geometric-reasoning

– 3D query modalities

– what if shape is influenced/modified by the context?


acknowledgements

IQmulus: A High-volume Fusion and Analysis Platform for Geospatial Point Clouds, Coverages and Volumetric Data Sets, European project “FP7 IP”, 2012-2106

VISIONAIR: Vision Advanced Infrastructure for Research, European project “FP7 INFRASTRUCTURES”, 2011-2015

MULTISCALEHUMAN: Multi-scale BiologicalModalities for Physiological Human Articulation, European project “FP7 PEOPLE” Initial Training Network, 2011-2014

Patrizio Frosini , Massimo Ferri and the Vision Mathematics group at the Dept. of Mathematics, University of Bologna

Claudia Landi, Dept. of Science and Methods of Engineering, University of Modena and Reggio Emilia



at the end…

thank you for your attention!

these course notes are available at: http://www.ge.imati.cnr.it/training

Documents

Mathematical Tools for 3D Shape Analysis and Descriptionsaturno.ge.imati.cnr.it/ima/personal/biasotti/SGP13/SGP_partII.pdf · 3D Shape Analysis and Description references 28 mathematical