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Stability of Multidimensional Persistent Homology Groups Claudia Landi 1 , 2 1 Di.S.M.I., University of Modena and Reggio Emilia , Italy 2 ARCES - Vision Mathematics Group, University of Bologna, Italy clandi@unimore.it GETCO 2010 Geometric and

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Stability of Multidimensional Persistent Homology Groups

Claudia Landi1,2

1Di.S.M.I., University of Modena and Reggio Emilia , Italy 2ARCES - Vision Mathematics Group, University of Bologna, Italy

clandi@unimore.it

GETCO 2010 Geometric and Topological Methods in Computer Science Aalborg University, January 11-15, 2010

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 1 / 46

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Outline

1

The Persistent Topology approach to shape comparison

2

Multidimensional Persistent Homology

3

Finiteness of Rank Invariants

4

Representation of rank invariants via persistence diagrams

5

Comparison of Rank Invariants

6

Stability with respect to noisy functions

7

Stability with respect to noisy domains

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 2 / 46

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Background

1

The Persistent Topology approach to shape comparison

2

Multidimensional Persistent Homology

3

Finiteness of Rank Invariants

4

Representation of rank invariants via persistence diagrams

5

Comparison of Rank Invariants

6

Stability with respect to noisy functions

7

Stability with respect to noisy domains

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 3 / 46

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Background

What is the shape of an object? Oxford Dictionary: the external form1 or appearance of someone

  • r something as produced by their outline

1Form: Visible shape or configuration Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 4 / 46

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Background

What is the shape of an object? Oxford Dictionary: the external form1 or appearance of someone

  • r something as produced by their outline

Mathematically: no universally accepted definition

  • utline, surface

up to rigid motions, or affinities, or perspective transformations may or may not take into account color or texture information

1Form: Visible shape or configuration Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 4 / 46

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Background

What is the shape of an object? Oxford Dictionary: the external form1 or appearance of someone

  • r something as produced by their outline

Mathematically: no universally accepted definition

  • utline, surface

up to rigid motions, or affinities, or perspective transformations may or may not take into account color or texture information

Most of the proposed techniques for shape recognition are tailored for some particular interesting cases

polyhedral rigid objects planar curves point cloud data

1Form: Visible shape or configuration Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 4 / 46

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Background

What is the shape of an object? Tentative definitions are generally based on observers’ perceptions. Dependence on observers implies large subjectivity

changes due to object orientation and distance from the object changes due to light conditions

Human judgments focus on persistent perceptions

Non-persistent properties can be considered as noise. Only stable perceptions concur to give a shape to objects.

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 5 / 46

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Background

How to model observations? A set of observations can be modeled as a topological space X. The topological space depends on what the observer is

  • bserving: boundary, interior, projection, contour

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 6 / 46

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Background

How to model perceptions? Observer’s perceptions can be modeled as a function f : X → Rk. The function depends on the shape property the observer is perceiving: curvature, roundness, elongation, connectivity etc. For each observation x ∈ X, f describes x as seen by the

  • bserver.

Thus we are led to study pairs (X, f) where – X is a topological space – f : X → Rk a (continuous) function, called a measuring (filtering) function.

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 7 / 46

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Background

Shape comparison In general shape comparison amounts to giving a measure of dissimilarity between shapes. When two pairs (X1, f1) and (X2, f2) are a comparable set of

  • bservations and perceptions, it is natural to ask how dissimilar

they are. Persistent Topology proposes an approach where comparing shapes means comparing properties expressed by real functions d    + f1, + f2    =?

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 8 / 46

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Background

About Stability (1) In order that comparisons be reliable we need: Stability in the perceptions yielding to a request for stability w.r.t. perturbations of the function

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 9 / 46

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Background

About Stability (2) Stability in the observations Buddha has genus 104, Dragon has genus 46, David’s head has genus 340. Most of these tunnels/handles are artifacts of the acquisition process of volumetric data (Guskov-Wood, Topological noise removal) yielding to a request for stability w.r.t. perturbations of the topological space

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 10 / 46

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Background

Persistent Topology Tools size functions [Frosini ’91] size homotopy groups [Frosini-Mulazzani ’99] persistent homology groups [Edelsbrunner-Letscher-Zomorodian ’00] vines and vineyards [CohenSteiner-Edelsbrunner-Morozov ’06] interval persistence [Dey-Wenger ’07] multidimensional homology groups [Carlsson-Zomorodian ’07]

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 11 / 46

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Multidimensional Persistent Homology

1

The Persistent Topology approach to shape comparison

2

Multidimensional Persistent Homology

3

Finiteness of Rank Invariants

4

Representation of rank invariants via persistence diagrams

5

Comparison of Rank Invariants

6

Stability with respect to noisy functions

7

Stability with respect to noisy domains

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 12 / 46

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Multidimensional Persistent Homology

Main idea: Given a space X, take a vector-valued function f : X → Rk; consider the collection of nested lower level sets of f; encode the scale at which a topological feature (e.g., a connected component, a tunnel, a void) is created, and when it is annihilated along this filtration using homology groups; further encode this information using a parametrized version of Betti numbers.

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 13 / 46

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Multidimensional Persistent Homology

Formally: Given a space X and a continuous function f : X → Rk, Lower level sets For every x ∈ Rk, X f u = {x ∈ X : f(x) u}. ((u1, . . . , uk) (v1, . . . , vk) means uj ≤ vj for every index j.) Definition (Carlsson&Zomorodian 2007) The multidimensional persistent homology groups of (X, f) are the groups H

u, v q

(X, f) = ImHq

  • X

f u ֒ → X f v

  • for

u ≺ v. Homology coefficients taken in a field K

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 14 / 46

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Multidimensional Persistent Homology

Definition (Carlsson&Zomorodian 2007) The rank invariants of (X, f) are functions ρ(X,

f),q : {

u ≺ v} → N ∪ {∞}, q ∈ Z, such that ρ(X,

f),q(

u, v) equals the rank of the persistent homology group H

u, v q

(X, f). Case q = 0: Rank invariants are also called size functions [Frosini et al. 1991,....]

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 15 / 46

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Multidimensional Persistent Homology

Example of Rank Invariant

  • f : X → R2,

f = (y, z), ρ(X,

f),1 : {

u ≺ v ∈ R2 × R2} → N ρ(X,

f),1(

u, v) = 1-homology classes born before u and still alive at v X

  • f

x y y z z

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 16 / 46

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Multidimensional Persistent Homology

Example of Rank Invariant

  • f : X → R2,

f = (y, z), ρ(X,

f),1 : {

u ≺ v ∈ R2 × R2} → N ρ(X,

f),1(

u, v) = 1-homology classes born before u and still alive at v X

  • f
  • u
  • v

x y y z z ρ(X,

f),1(

u, v) = 2

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 16 / 46

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Multidimensional Persistent Homology

Example of Rank Invariant

  • f : X → R2,

f = (y, z), ρ(X,

f),1 : {

u ≺ v ∈ R2 × R2} → N ρ(X,

f),1(

u, v) = 1-homology classes born before u and still alive at v X

  • f
  • u
  • v

x y y z z ρ(X,

f),1(

u, v) = 1

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 16 / 46

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Multidimensional Persistent Homology

Example of Rank Invariant

  • f : X → R2,

f = (y, z), ρ(X,

f),1 : {

u ≺ v ∈ R2 × R2} → N ρ(X,

f),1(

u, v) = 1-homology classes born before u and still alive at v X

  • f
  • u
  • v

x y y z z ρ(X,

f),1(

u, v) = 1

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 16 / 46

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Multidimensional Persistent Homology

Issues Finiteness of rank invariants; Representation of rank invariants via persistence diagrams Comparison of rank invariants; Stability of rank invariants with respect to perturbations of f; Stability of rank invariants with respect to perturbations of X;

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 17 / 46

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Finiteness of Rank Invariants

1

The Persistent Topology approach to shape comparison

2

Multidimensional Persistent Homology

3

Finiteness of Rank Invariants

4

Representation of rank invariants via persistence diagrams

5

Comparison of Rank Invariants

6

Stability with respect to noisy functions

7

Stability with respect to noisy domains

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 18 / 46

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Finiteness of Rank Invariants

Observation Case q = 0 (size functions): X compact and locally connected, f : X → Rk continuous imply ρ(X,

f),0(

u, v) = rk Im H0

  • X

f u ֒ → X f v

  • < +∞

for every u < v. Definition (Cohen-Steiner et al. 2005) X triangulable space, f : X → R is tame if it has a finite number of homological critical values and Hq (Xf ≤ u) is finite-dimensional for every q ∈ Z and u ∈ R.

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 19 / 46

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Finiteness of Rank Invariants

Tame functions are not closed under the max operator: f1(u, v) =

  • v − u2 sin 1

u

u = 0, v u = 0. f1(u, v) ≤ 0

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 20 / 46

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Finiteness of Rank Invariants

Tame functions are not closed under the max operator: f1(u, v) =

  • v − u2 sin 1

u

u = 0, v u = 0. f1(u, v) ≤ 0 f2(u, v) =

  • −v − u2 sin 1

u

u = 0, −v u = 0. f2(u, v) ≤ 0

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 20 / 46

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Finiteness of Rank Invariants

Tame functions are not closed under the max operator: f1(u, v) =

  • v − u2 sin 1

u

u = 0, v u = 0. f1(u, v) ≤ 0 f2(u, v) =

  • −v − u2 sin 1

u

u = 0, −v u = 0. f2(u, v) ≤ 0 f(u, v) = max{f1, f2} f(u, v) ≤ 0

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 20 / 46

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Finiteness of Rank Invariants

Definition A topological subspace Y of Rn is called a Euclidean neighborhood retract if there is a neighborhood in Rn of which Y is a retract. Theorem (Borsuk) If Y ⊆ Rn is locally compact and locally contractible then Y is an ENR. Theorem If X ⊆ Rn is compact and locally contractible then ρ(X,

f),q(

u, v) < +∞ for every u < v, q ∈ Z, f : X → Rk continuous.

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 21 / 46

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Representation of rank invariants via persistence diagrams

1

The Persistent Topology approach to shape comparison

2

Multidimensional Persistent Homology

3

Finiteness of Rank Invariants

4

Representation of rank invariants via persistence diagrams

5

Comparison of Rank Invariants

6

Stability with respect to noisy functions

7

Stability with respect to noisy domains

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 22 / 46

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Representation of rank invariants via persistence diagrams

Persistence Diagrams (1) Case k = 1

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 23 / 46

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Representation of rank invariants via persistence diagrams

Persistence Diagrams (1) Case k = 1

x y 4 5 3 2 1

ρ(X,f),q

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 23 / 46

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Representation of rank invariants via persistence diagrams

Persistence Diagrams (1) Case k = 1

x y 4 5 3 2 1 x y 1

ρ(X,f),q ρ(X,f),q

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 23 / 46

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Representation of rank invariants via persistence diagrams

Persistence Diagrams (1) Case k = 1

x y 4 5 3 2 1 x y 1 2

ρ(X,f),q ρ(X,f),q

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 23 / 46

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Representation of rank invariants via persistence diagrams

Persistence Diagrams (1) Case k = 1

x y 4 5 3 2 1 x y 1 2 3

ρ(X,f),q ρ(X,f),q

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 23 / 46

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Representation of rank invariants via persistence diagrams

Persistence Diagrams (1) Case k = 1

x y 4 5 3 2 1 x y 1 2 3 3 4

ρ(X,f),q ρ(X,f),q

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 23 / 46

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Representation of rank invariants via persistence diagrams

Persistence Diagrams (1) Case k = 1

x y 4 5 3 2 1 x y 1 2 3 4 5

ρ(X,f),q ρ(X,f),q

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 23 / 46

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Representation of rank invariants via persistence diagrams

Persistence Diagrams (1) Case k = 1

x y 3 4 3 2 1 x y m p q 2r m+p+q+2r

ρ(X,f),q

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 23 / 46

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Representation of rank invariants via persistence diagrams

Persistence Diagrams (2) Definition The multiplicity of a point p = (u, v) s. t. u < v is the number µ(p) := lim

ǫ→0+ ρ(X,f),q(u + ǫ, v − ǫ) − ρ(X,f),q(u + ǫ, v + ǫ)

−ρ(X,f),q(u − ǫ, v − ǫ) + ρ(X,f),q(u − ǫ, v + ǫ). If µ(p) > 0 the point p is called a cornerpoint.

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 24 / 46

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Representation of rank invariants via persistence diagrams

Cornerlines Definition A vertical line r : u = k such that the number µ(r) := lim

ǫ→0+ v→+∞

ρ(X,f),q(k + ǫ, v) − ρ(X,f),q(k − ǫ, v) is strictly positive is said to be a cornerline. µ(r) is called the multiplicity of r.

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 25 / 46

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Representation of rank invariants via persistence diagrams

Persistence Diagrams (3) If ˇ Cech homology is used to define persistent homology groups, then persistence diagrams completely characterize rank invariants: Theorem (Representation Theorem) For every (¯ u, ¯ v) with ¯ u < ¯ v ≤ ∞ ˇ ρ(X,f),q(¯ u, ¯ v) =

  • (u,v): u<v≤∞

u≤¯ u,v>¯ v

µ

  • (u, v)
  • .

Lemma Using ˇ Cech homology instead of singular homology, ˇ ρ(X,f),q(u, v) is right-continuous w.r.t. both the variables u and v.

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 26 / 46

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Representation of rank invariants via persistence diagrams

Example 1 X a closed rectangle of R2 containing a Warsaw circle f : X → R the Euclidean distance from the Warsaw circle ρ(X,f),1(0, v) = 0, ρ(X,f),1(u, v) = 1, 0 < u < v suff.small, ˇ ρ(X,f),1(0, v) = 1, ˇ ρ(X,f),1(u, v) = 1, 0 < u < v suff.small.

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 27 / 46

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Representation of rank invariants via persistence diagrams

Example 2 ¯ P N 1 2 u v ρ(S,f),0 ∆+ f(¯ P) = f( ¯ Q)

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 28 / 46

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Representation of rank invariants via persistence diagrams

Back to k ≥ 1 A suitable foliation:

  • l = (l1, . . . , lk),

b = (b1, . . . , bk), with

  • l = 1, li > 0,

i bi = 0

∆+ = {( u, v) ∈ Rk × Rk : u ≺ v} is foliated by the 2D half-planes with parametric equations: π(

  • l,

b) :

  • u = s

l + b

  • v = t

l + b s, t ∈ R, s < t For every (

  • l,

b), define F

  • f

(

  • l,

b) : X → R by

F

  • f

(

  • l,

b)(x) = max i=1,...,k

fi(x) − bi li

  • .

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 29 / 46

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Representation of rank invariants via persistence diagrams

Reduction to the case k = 1 Reduction Theorem For every ( u, v) = (s l + b, t l + b) ∈ π(

  • l,

b) it holds that

ˇ ρ(X,

f),q(

u, v) = ˇ ρ(X,F

f (

  • l,

b)),q(s, t).

Rank invariants are represented by persistence diagrams leaf-by-leaf

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 30 / 46

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Representation of rank invariants via persistence diagrams

Reduction to the case k = 1 Reduction Theorem For every ( u, v) = (s l + b, t l + b) ∈ π(

  • l,

b) it holds that

ˇ ρ(X,

f),q(

u, v) = ˇ ρ(X,F

f (

  • l,

b)),q(s, t).

Rank invariants are represented by persistence diagrams leaf-by-leaf X f

  • u = s

l + b

  • v = t

l + b x y y z z

  • b = (−a, a)
  • l

ˇ ρ(X,

f),q(

u, v) = ˇ ρ(X,F

f (

  • l,

b)),q(s, t) Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 30 / 46

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Comparison of Rank Invariants

1

The Persistent Topology approach to shape comparison

2

Multidimensional Persistent Homology

3

Finiteness of Rank Invariants

4

Representation of rank invariants via persistence diagrams

5

Comparison of Rank Invariants

6

Stability with respect to noisy functions

7

Stability with respect to noisy domains

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 31 / 46

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Comparison of Rank Invariants

Matching distance Case k = 1

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 32 / 46

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Comparison of Rank Invariants

Matching distance Case k = 1

x y x y q p m x y p’ m’ q p m p’ m’

ρ1 ρ2

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 32 / 46

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Comparison of Rank Invariants

Matching distance Case k = 1

x y x y q p m x y p’ m’ q p m p’ m’

ρ1 ρ2 dmatch (ρ1, ρ2) = min

γ:D→D′ max q∈D q − γ(q)∞

D = {m, p, q, . . .}, D′ = {m′, p′, . . .}, γ bijective.

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 32 / 46

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Comparison of Rank Invariants

Multidimensional Matching Distance Dmatch

  • ˇ

ρ(X,

f),q, ˇ

ρ(X,

g),q

  • = sup

(

  • l,

b)

min

i=1,...,k li · dmatch(ˇ

ρ(X,F

f (

  • l,

b)),q, ˇ

ρ(X,G

g (

  • l,

b)),q)

where F

  • f

(

  • l,

b)(x) = max i=1,...,k

fi(x) − bi li

  • ,

G

g (

  • l,

b)(x) = max i=1,...,k

gi(x) − bi li

  • .

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 33 / 46

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Stability with respect to noisy functions

1

The Persistent Topology approach to shape comparison

2

Multidimensional Persistent Homology

3

Finiteness of Rank Invariants

4

Representation of rank invariants via persistence diagrams

5

Comparison of Rank Invariants

6

Stability with respect to noisy functions

7

Stability with respect to noisy domains

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 34 / 46

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Stability with respect to noisy functions

X compact and locally contractible space One-Dimensional Stability Theorem: f, g : X → R continuous functions. Then dmatch(ˇ ρ(X,f),q, ˇ ρ(X,g),q) ≤ max

x∈X |f(x) − g(x)|.

Multidimensional Stability Theorem:

  • f,

g : X → Rk continuous functions. Then Dmatch

  • ˇ

ρ(X,

f),q, ˇ

ρ(X,

g),q

  • ≤ max

x∈X

f(x) − g(x)∞.

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 35 / 46

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Stability with respect to noisy functions

Example of mono-dimensional stability A curvature driven curve evolution and its 0-homology rank invariant w.r.t. the distance from the center of mass (Thanks to Frédéric Cao for curvature evolution code)

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 36 / 46

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Stability with respect to noisy functions

Example of multi-dimensional stability Experimental results on a set of 8 human models represented by triangular meshes. For each model, we take the 2-dimensional measuring function

  • f = (f1, f2) with

f1(Pi) = 1 − ||Pi − (B + w)|| maxj ||Pj − (B + w)||. where B is the center of mass of the model and

  • w =

n

i=1(Pi − B)||Pi − B||2

n

i=1 ||Pi − B||2

; similarly, f2(Pi) = 1 −

||Pi−(B− w)|| maxj ||Pj−(B− w)||.

The admissible pairs are of the form (

  • l,

b) with l = (cos θi, sin θi), θi = π

36i, i = 1, . . . , 17, and

b = (0, 0).

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 37 / 46

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Stability with respect to noisy functions

Results Discretization of the 0-homology rank invariant of each model:

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 38 / 46

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Stability with respect to noisy functions

Results The 2D matching distance w.r.t. f = (f1, f2) and the max of the 1D matching distances w.r.t. f1 and f2:

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 39 / 46

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Stability with respect to noisy domains

1

The Persistent Topology approach to shape comparison

2

Multidimensional Persistent Homology

3

Finiteness of Rank Invariants

4

Representation of rank invariants via persistence diagrams

5

Comparison of Rank Invariants

6

Stability with respect to noisy functions

7

Stability with respect to noisy domains

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 40 / 46

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Stability with respect to noisy domains

K1,K2 = black pixels of the left and right image, resp. D = 72 × 69 pixels rectangle. Topology: 4-neighbors adjacency f : D → R, f(P) = −d(P, B), where B center of mass of K1

−40 −35 −30 −25 −20 −15 −10 −40 −35 −30 −25 −20 −15 −10 −35 −30 −25 −20 −15 −10 −5 −35 −30 −25 −20 −15 −10 −5

ρ(K1,f|K1),0 ρ(K2,f|K2),0

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 41 / 46

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SLIDE 59

Stability with respect to noisy domains

Stability w.r.t. noisy domains ❀ stability w.r.t. noisy functions: K ⊆ D ⊆ Rn ❀ dK : D → R, dk(x) = inf

y∈K y − x

  • f : D → Rk

❀

  • Φ : D → Rk+1,

Φ = (dk, f) Theorem (Stability w.r.t. noisy domains) Let K1, K2 be non-empty closed subsets of a compact and locally contractible subspace D of Rn. Take Φ1, Φ2 : D → Rk+1, Φ1 = (dK1, f1) and Φ2 = (dK2, f2). Then Dmatch

  • ˇ

ρ(D,

Φ1),q, ˇ

ρ(D,

Φ2),q

  • ≤ max
  • δH(K1, K2),

f1 − f2∞

  • .

Hausdorff distance: δH(K1, K2) = max{maxx∈K2 dK1(x), maxy∈K1 dK2(y)}

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 42 / 46

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SLIDE 60

Stability with respect to noisy domains

Retrieving information about (K, f|K) from (D, Φ = (dk, f)) K a non-empty closed and locally contractible subset of a compact and locally contractible subspace D of Rn

  • f : D → Rk a continuous function

Take Φ : D → Rk+1, Φ = (dK, f) Theorem ˇ ρ(K,

f|K ),q(

u, v) = lim

ǫ→0+ ˇ

ρ(D,

Φ),q

  • (0,

u), (ǫ, v)

  • ,

for every u, v ∈ Rk with u ≺ v.

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 43 / 46

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SLIDE 61

Stability with respect to noisy domains

−6 −4 −2 2 4 6 8 10 12 14 −6 −4 −2 2 4 6 8 10 12 14 −20 −10 10 20 30 40 50 −20 −10 10 20 30 40 50 −5 5 10 15 20 −5 5 10 15 20 −20 −10 10 20 30 40 50 60 −20 −10 10 20 30 40 50 60

ρ(D,

Φ1),0 and ρ(D, Φ2),0 restricted to the half-planes π(

  • li,

bi) of the foliation

with li = (cos i

12π, sin i 12π), i = 2, 5 and

b = (5, −5)

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 44 / 46

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SLIDE 62

Stability with respect to noisy domains

−3 −2 −1 1 2 3 4 5 6 7 −3 −2 −1 1 2 3 4 5 6 7 −4 −2 2 4 6 8 10 12 −4 −2 2 4 6 8 10 12 −2 2 4 6 8 10 −2 2 4 6 8 10 −4 −2 2 4 6 8 10 12 14 16 −4 −2 2 4 6 8 10 12 14 16

ρ(D,

Φ1),0 and ρ(D, Φ2),0 restricted to the same half-planes π(

  • li,

bi) of the

foliation and rescaled by µi = minj=1,2(li)j.

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 45 / 46

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SLIDE 63

Stability with respect to noisy domains

Conclusions Development of a paradigm for shape comparison using

Multidimensional Persistence Homology

Assessment of resistance of multidimensional homology groups under

noisy functions noisy domains

Thank you for your attention!

Claudia Landi (UniMoRE) Multidimensional Persistent Homology GETCO 2010 46 / 46