Diffusion Geometry in Shape Analysis

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Spectral methods in deformable shape analysis

Alex Bronstein, Michael Bronstein, Umberto Castellani

March 14, 2012

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Dimensions of media

Alex Bronstein, Michael Bronstein, Umberto Castellani Spectral methods in shape analysis

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Dimensions of media

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Dimensions of media

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Dimensions of media

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3D shapes vs. images

Array of pixels

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3D shapes vs. images

Array of pixels

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3D shapes vs. images

Affine, projective

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3D shapes vs. images

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3D shapes vs. images

1 2 + ¹ ₂ =

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3D shapes vs. images

1 2 + ¹ ₂ =

1 2 + ¹ ₂ = ?

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Setting

Endow shapes with a structre

Similarity, correspondence, retrieval, etc. = similarity and correspondence between structures

Invariance under bending, scale, affine transformations, etc.

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Setting

Endow shapes with a structre

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Structure

Global structure Metric space

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Structure

Local structure Point descriptors

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Structure

Glocal structure Stable regions

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Structure

Glocal structure Stable regions

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Agenda

Diffusion processes on surfaces Spectral point of view

Global structure: diffusion geometry Local structure: diffusion kernel descriptors

Semi-local structure: maximally stable components Extensions

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Diffusion processes on surfaces

Heat equation

∆_X + ∂

∂t

u = 0

governs heat propagation on manifold X

Solution u(x,t): heat distribution at pointx at time t Initial condition u0(x): heat distribution at time t = 0 Boundary condition if manifold has a boundary

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Diffusion processes on surfaces

Heat equation

∆_X + ∂

∂t

u = 0

governs heat propagation on manifold X

Solution u(x,t): heat distribution at pointx at time t Initial condition u0(x): heat distribution at time t = 0 Boundary condition if manifold has a boundary

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Laplace-Beltrami operator ∆

_X

For two smooth functionsf,g :X →Rand standard inner product onX

hf,gi= Z

X

f(x)g(x)da the Laplacian satisfies the following properties:

Constant eigenfunction: ∆Xf = 0 if f =const

Symmetry: h∆_Xf,gi=h∆_Xg,fi

Locality: (∆Xf)(x) does not depend onf(x⁰) at anyx⁰6=x Linear precision: ifX as a plane andf =ax+by+c, then

∆Xf = 0

Positive semi-definiteness: h∆_Xf,fi ≥0

Maximum principle: functions satisfying ∆Xf = 0 (harmonic) have no minima/maxima in the interior ofX

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Laplace-Beltrami operator ∆

_X

hf,gi= Z

X

Constant eigenfunction: ∆Xf = 0 if f =const Symmetry: h∆_Xf,gi=h∆_Xg,fi

Locality: (∆Xf)(x) does not depend onf(x⁰) at anyx⁰6=x Linear precision: ifX as a plane andf =ax+by+c, then

∆Xf = 0

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Laplace-Beltrami operator ∆

_X

hf,gi= Z

X

Locality: (∆Xf)(x) does not depend onf(x⁰) at anyx⁰ 6=x

Linear precision: ifX as a plane andf =ax+by+c, then

∆Xf = 0

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Laplace-Beltrami operator ∆

_X

hf,gi= Z

X

Locality: (∆Xf)(x) does not depend onf(x⁰) at anyx⁰ 6=x Linear precision: ifX as a plane andf =ax+by+c, then

∆Xf = 0

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Laplace-Beltrami operator ∆

_X

hf,gi= Z

X

∆Xf = 0

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Laplace-Beltrami operator ∆

_X

hf,gi= Z

X

∆Xf = 0

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Discretization of the Laplacian

SurfaceX isdiscretized at n points{x1, . . . ,xn} and points are connected to form a triangular mesh

Function f :X →Rrepresented a vector f ∈Rⁿ,fi =f(xi) Discrete version of the Laplacian

(LXf)i = 1 ai

X

j

wij(fi−fj)

wij – edge weights,ai vertex normalization coefficients. In matrix notation

LXf =A⁻¹Lf where A=diag{ai} and (L)ij =diag





 X

k6=i

wik







−wij

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Discretization of the Laplacian

Function f :X →Rrepresented a vector f ∈Rⁿ,fi =f(xi)

Discrete version of the Laplacian (LXf)i = 1

ai

X

j

wij(fi−fj)

wij – edge weights,ai vertex normalization coefficients. In matrix notation





 X

k6=i

wik







−wij

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Discretization of the Laplacian

(LXf)i = 1 ai

X

j

wij(fi−fj)

wij – edge weights,ai vertex normalization coefficients.

In matrix notation





 X

k6=i

wik







−wij

(30)

Discretization of the Laplacian

(LXf)i = 1 ai

X

j

wij(fi−fj)

wij – edge weights,ai vertex normalization coefficients.

In matrix notation

LXf =A⁻¹Lf whereA=diag{ai} and (L)ij =diag





 X

k6=i

wik







−wij

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Discretization of the Laplacian

xi

xj

xi

xj

α_ij β_ij

Discrete Laplacian wij =

1 : xj ∈ N₁(xi) 0 : else

Discretized Laplacian wij =

cotα_ij + cotβ_ij : xj ∈ N₁(xi)

0 : else

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Desired properties of discrete Laplacians

Constant eigenfunction: ∆Xf = 0 if f =const

Symmetry: Locality:

Linear precision:

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Desired properties of discrete Laplacians

Constant eigenfunction: Satisfied by construction of LX

Symmetry: Locality:

Linear precision:

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Desired properties of discrete Laplacians

Symmetry: h∆_Xf,gi=h∆_Xg,fi

Locality:

Linear precision:

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Desired properties of discrete Laplacians

Symmetry: LX =L^T_X

Locality:

Linear precision:

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Desired properties of discrete Laplacians

Symmetry: LX =L^T_X

Reason: To have real eigenvalues and orthogonal eigenvectors

Locality:

Linear precision:

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Desired properties of discrete Laplacians

Symmetry: LX =L^T_X

Reason: To have real eigenvalues and orthogonal eigenvectors Locality: (∆Xf)(x) does not depend onf(x⁰) at anyx⁰ 6=x

Linear precision:

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Desired properties of discrete Laplacians

Symmetry: LX =L^T_X

Reason: To have real eigenvalues and orthogonal eigenvectors Locality: wij = 0 ifxj 6=N₁(xi)

Linear precision:

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Desired properties of discrete Laplacians

Symmetry: LX =L^T_X

wij stand for random walk transition probabilities along the graph edges.

Linear precision:

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Desired properties of discrete Laplacians

Symmetry: LX =L^T_X

Linear precision: ifX as a plane andf =ax+by+c, then

∆_Xf = 0

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Desired properties of discrete Laplacians

Symmetry: LX =L^T_X

Linear precision: equivalently, if X is a plane, then (LXx)i =X

wij(xi −xj) = 0

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Desired properties of discrete Laplacians

Symmetry: LX =L^T_X

Linear precision: equivalently, if X is a plane, then (LXx)i =X

j

wij(xi −xj) = 0

Reason: Flat plate must have zero bending energy.

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Desired properties of discrete Laplacians

Positive weights: wij ≥0 and each vertexi has at least one wij >0

Convergence: solution of discrete PDE withLX converges to solution of continuous PDE with ∆_X as n→ ∞

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Desired properties of discrete Laplacians

Positive semi-definiteness: f^TLXf ≥0, i.e.,LX 0

Positive weights: wij ≥0 and each vertexi has at least one wij >0

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Desired properties of discrete Laplacians

Positive semi-definiteness: f^TLXf ≥0, i.e.,LX 0 Positive weights: wij ≥0 and each vertexi has at least one wij >0

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Desired properties of discrete Laplacians

Reasons: Sufficient condition to satisfy discretemaximum principle

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Desired properties of discrete Laplacians

Positive weights + Symmetry⇒ PSD

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Desired properties of discrete Laplacians

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Desired properties of discrete Laplacians

Convergence: solution of discrete PDE withLX converges to solution of continuous PDE with ∆X as n→ ∞

Indispensable for discretization of PDE solutions

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No free lunch

Discrete Laplacians are not convergent

cot weight discretized Laplacians is convergentbut has negative weights

Many attempts have been made to construct discrete Laplacians satisfying above desired properties

“No free lunch theorem” (Wardetzky et al., 2007) There is no discrete Laplacian satisfying the above properties simultaneously!

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No free lunch

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No free lunch

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No free lunch

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Eigendecomposition of Laplacian

Oncompact domains Laplacian admitscountable orthogonal eigendecomposition

∆_Xφ_i =λ_iφ_i

λ_i – eigenvalues;φ_i(x) – corresponding eigenfunctions

Discrete generalized eigendecompositionfor LX =A⁻¹L AΦ = ΛLΦ

Λ =diag{λ₁, . . . , λk} – diagonal matrix of firstk eigenvalues Φ – n×k matrix of corresponding eigenvectors

Spectral decomposition theorem (∆_Xf)(x) =X

i≥0

λ_iφ_i(x)· hφ_i,fi Discrete equivalent: LXf =X

i≥0

λ_iφ_iφ^T_i f

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Eigendecomposition of Laplacian

∆_Xφ_i =λ_iφ_i

λ_i – eigenvalues;φ_i(x) – corresponding eigenfunctions Discrete generalized eigendecompositionfor LX =A⁻¹L

AΦ = ΛLΦ

Λ =diag{λ₁, . . . , λ_k} – diagonal matrix of firstk eigenvalues Φ – n×k matrix of corresponding eigenvectors

i≥0

λ_iφ_iφ^T_i f

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Eigendecomposition of Laplacian

∆_Xφ_i =λ_iφ_i

λ_i – eigenvalues;φ_i(x) – corresponding eigenfunctions Discrete generalized eigendecompositionfor LX =A⁻¹L

AΦ = ΛLΦ

Λ =diag{λ₁, . . . , λ_k} – diagonal matrix of firstk eigenvalues Φ – n×k matrix of corresponding eigenvectors

i≥0

λ_iφ_iφ^T_i f

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Finite elements

Discretize {λ_i, φ_i}directly!

Weak form eigendecomposition

h∆_Xφ, αi=λhφ, αi

Fix a sufficiently regular basis {α₁, . . . , αm} spanning a subspace of L²(X)

φ(x)≈u1α1(x) +· · ·umαm(x)

Write a system of equations for k = 1, . . . ,m

m

X

i=1

uih∆_Xα_i, α_ji

| {z }

aij

=

m

X

i=1

uihα_i, α_ji

| {z }

bij

In matrix notation: Au=λBu

(58)

Finite elements

φ(x)≈u1α1(x) +· · ·umαm(x)

m

X

i=1

uih∆_Xα_i, α_ji

| {z }

aij

=

m

X

i=1

uihα_i, α_ji

| {z }

bij

(59)

Finite elements

φ(x)≈u1α1(x) +· · ·umαm(x)

m

X

i=1

uih∆_Xα_i, α_ji

| {z }

aij

=

m

X

i=1

uihα_i, α_ji

| {z }

bij

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Finite elements

φ(x)≈u1α1(x) +· · ·umαm(x)

m

X

i=1

uih∆_Xα_i, α_ji

| {z }

aij

=

m

X

i=1

uihα_i, α_ji

| {z }

bij

(61)

Finite elements

φ(x)≈u1α1(x) +· · ·umαm(x)

m

Xuih∆_Xα_i, α_ji =

m

Xuihα_i, α_ji

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Finite elements

φ(x)≈u1α1(x) +· · ·umαm(x)

m

X

i=1

uih∆_Xα_i, α_ji

| {z }

aij

=

m

X

i=1

uihα_i, α_ji

| {z }

bij

(63)

To see the sound

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Chladni plates

Solutions to stationary Helmholtz equation

∆_Xf =λf

Laplacian eigenfunctions = plate vibration modes

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Chladni plates

Solutions to stationary Helmholtz equation

∆ f =λf

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Shape DNA

(Reuteret al., 2006) use Laplacian spectrum{λ_i} as an isometry-invariant shape descriptor –shape DNA

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Shape DNA

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“Can we hear the shape of the drum?”

Isometric shapes areisospectral

Are isospectral shapes isometric?

Can one hear the shape of the drum? (Mark Kac)

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“Can we hear the shape of the drum?”

Isometric shapes areisospectral Are isospectral shapes isometric?

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“Can we hear the shape of the drum?”

Isometric shapes areisospectral Are isospectral shapes isometric?

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“Can we hear the shape of the drum?”

The following shape properties can be recovered (“heard”) from the spectrum of the Laplacian:

Area

Euler characteristic (genus) Total Gaussian curvature Can we hear the metric?

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“Can we hear the shape of the drum?”

Area

Euler characteristic (genus)

Total Gaussian curvature Can we hear the metric?

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“Can we hear the shape of the drum?”

Area

Euler characteristic (genus) Total Gaussian curvature

Can we hear the metric?

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“Can we hear the shape of the drum?”

Area

Euler characteristic (genus) Total Gaussian curvature Can we hear the metric?

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One cannot hear the shape of the drum!

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Relation to harmonic analysis

1D signals

− d²

dx²e^inx =n²e^inx

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Relation to harmonic analysis

1D signals

− d²

e^inx n²e^inx

3D shapes

∆_Xφ_i(x) =λ_iφ_i(x)

(78)

Relation to harmonic analysis

A continuous functionf ∈L²(X) can be represented as Synthesis: f(x) =X

i≥0

F(λ_i)φ_i(x)

Analysis: F(λ_i) = Z

X

f(x)φi(x)da=hf, φ_ii λ=frequency

{F(λ_i)} =Fourier coefficients

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Relation to harmonic analysis

i≥0

F(λ_i)φ_i(x) Analysis: F(λ_i) =

Z

X

f(x)φi(x)da=hf, φ_ii

λ=frequency

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Relation to harmonic analysis

i≥0

Z

X

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Relation to harmonic analysis

i≥0

Z

X

(82)

Heat kernel

Heat equation

∆_X + ∂

∂t

u = 0

Solution given by heat operator u(x,t) = (H^tu0)(x) =

Z

X

ht(x,y)u0(y)da(y)

Heat kernel ht(x,y) : solution at pointx at time t with point heat source at y

Impulse responseof heat equation

Intrinsic quantity – invariant to inelastic (isometric) bending Heat operator can be interpreted as a non shift-invariant version of convolution

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Heat kernel

Heat equation

∆_X + ∂

∂t

u = 0

Z

X

ht(x,y)u0(y)da(y)

Heat kernel ht(x,y) : solution at pointx at time t with point heat source at y

(84)

Heat kernel

Heat equation

∆_X + ∂

∂t

u = 0

Z

X

ht(x,y)u0(y)da(y)

Heat kernel ht(x,y) : solution at point x at time t with point heat source at y

(85)

Heat kernel

Heat equation

∆_X + ∂

∂t

u = 0

Z

X

ht(x,y)u0(y)da(y)

(86)

Heat kernel

Heat equation

∆_X + ∂

∂t

u = 0

Z

X

ht(x,y)u0(y)da(y)

Intrinsic quantity – invariant to inelastic (isometric) bending

Heat operator can be interpreted as a non shift-invariant version of convolution

(87)

Heat kernel

Heat equation

∆_X + ∂

∂t

u = 0

Z

X

ht(x,y)u0(y)da(y)

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Heat kernel

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Probabilistic interpretation

Brownian motion x(t) starts at point x

x

x(t)

C

Pr(x(t)∈C) = Z

C

ht(x,y)da(y)

ht(x,y) =transition probability density fromx toy by random walk of length t

(90)

Probabilistic interpretation

x

x(t)

C

Pr(x(t)∈C) = Z

C

ht(x,y)da(y)

ht(x,y) =transition probability density fromx toy by random walk of length t

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Probabilistic interpretation

x

x(t)

C

Pr(x(t)∈C) = Z

h (x,y)da(y)

(92)

Spectral interpretation

Let ∆Xφi =λiφi be the Laplacian eigendecomposition

By spectral decomposition theorem ht(x,y) =X

i≥0

e^−λⁱ^tφi(x)φi(y)

e^−λt =frequency response of heat operatorH^t

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Spectral interpretation

Let ∆Xφi =λiφi be the Laplacian eigendecomposition By spectral decomposition theorem

ht(x,y) =X

i≥0

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Spectral interpretation

Let ∆Xφi =λiφi be the Laplacian eigendecomposition By spectral decomposition theorem

ht(x,y) =X

i≥0

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Diffusion operators

General diffusion operator (Ku)(x) =

Z

X

k(x,y)u(y)da(y)

Diffusion kernel

k(x,y) =X

i≥0

K(λi)φi(x)φi(y)

K(λ) = frequency response (lowpass filter) K^t is also a diffusion operator with response K^t(λ) A scale space of operators {K^t}_t≥0

(96)

Diffusion operators

Z

X

k(x,y)u(y)da(y) Diffusion kernel

k(x,y) =X

i≥0

K(λi)φi(x)φi(y)

K(λ) = frequency response (lowpass filter) K^t is also a diffusion operator with response K^t(λ) A scale space of operators {K^t}_t≥0

(97)

Diffusion operators

Z

X

k(x,y) =X

i≥0

K(λi)φi(x)φi(y)

K(λ) = frequency response (lowpass filter)

K^t is also a diffusion operator with response K^t(λ) A scale space of operators {K^t}_t≥0

(98)

Diffusion operators

Z

X

k(x,y) =X

i≥0

K(λi)φi(x)φi(y)

K^t is also a diffusion operator with response K^t(λ)

A scale space of operators {K^t}_t≥0

(99)

Diffusion operators

Z

X

k(x,y) =X

i≥0

K(λi)φi(x)φi(y)

t t

(100)

Properties of diffusion operators

Non-negativity: k(x,y)≥0

Symmetry: k(x,y) =k(y,x)

Positive semidefiniteness: for everyf(x),hKf,fi ≥0 Square integrability:

Z Z

k²(x,y)da(x)da(y)<∞ Conservation:

Z

k(x,y)da(x) = 1

(101)

Properties of diffusion operators

Non-negativity: k(x,y)≥0 Symmetry: k(x,y) =k(y,x)

Positive semidefiniteness: for everyf(x),hKf,fi ≥0 Square integrability:

Z Z

Z

k(x,y)da(x) = 1

(102)

Properties of diffusion operators

Positive semidefiniteness: for every f(x),hKf,fi ≥0

Square integrability: Z Z

Z

k(x,y)da(x) = 1

(103)

Properties of diffusion operators

Positive semidefiniteness: for every f(x),hKf,fi ≥0 Z Z

k(x,y)f(x)f(y)da(x)da(y)≥0

Square integrability: Z Z

Z

k(x,y)da(x) = 1

(104)

Properties of diffusion operators

k(x,y)f(x)f(y)da(x)da(y)≥0 Square integrability:

Z Z

k²(x,y)da(x)da(y)<∞

Conservation: Z

k(x,y)da(x) = 1

(105)

Properties of diffusion operators

k(x,y)f(x)f(y)da(x)da(y)≥0 Square integrability:

Z Z

(106)

Properties of diffusion operators

Positive semidefiniteness: for every f(x),hKf,fi ≥0

Square integrability: Conservation:

(107)

Properties of diffusion operators

Positive semidefiniteness: K(λi)≥0

Square integrability: Conservation:

(108)

Properties of diffusion operators

Positive semidefiniteness: K(λi)≥0 Square integrability:

Z Z

k²(x,y)da(x)da(y)<∞

Conservation:

(109)

Properties of diffusion operators

Positive semidefiniteness: K(λ_i)≥0 Square integrability: by Parseval’s theorem

X

i≥0

K²(λ_i)<∞

Conservation:

(110)

Properties of diffusion operators

X

i≥0

K²(λi)<∞ Conservation:

Z

k(x,y)da(x) = 1

(111)

Properties of diffusion operators

X

i≥0

K²(λ_i)<∞

Conservation: by Perron-Frobenius theorem λ_i ≤1

(112)

Diffusion geometry

Family ofdiffusion metrics

d²(x,y) = kk(x,·)−k(y,·)k²_L2(X)

= Z

X

(k(x,z)−k(y,z))²da(z)

Alternatively,

d²(x,y) = kK(λ_i)φ_i(x)−K(λ_i)φ_i(y)k_`2

= X

i≥0

K²(λi)(φi(x)−φi(y))² Equivalent due to Parseval’s theorem

{K^t}_t≥0 define a scale space of metrics {dt}_t≥0

(113)

Diffusion geometry

d²(x,y) = kk(x,·)−k(y,·)k²_L2(X)

= Z

X

(k(x,z)−k(y,z))²da(z) Alternatively,

= X

i≥0

K²(λi)(φi(x)−φi(y))²

{K^t}_t≥0 define a scale space of metrics {dt}_t≥0

(114)

Diffusion geometry

d²(x,y) = kk(x,·)−k(y,·)k²_L2(X)

= Z

X

(k(x,z)−k(y,z))²da(z) Alternatively,

= X

i≥0

K²(λi)(φi(x)−φi(y))² Equivalent due to Parseval’s theorem

{K^t}t≥0 define a scale space of metrics {dt}t≥0

(115)

Diffusion geometry

Heat diffusion metric K(λ) =e^−λt

d_t²(x,y) = X

i≥0

e^−2λⁱ^t(φi(x)−φi(y))²

Commute time metric K(λ) = ^√¹

λ

d²(x,y) =

“Connectability” of x andy by random walks of length t

“Connectability” by random walks of any length

Scale invariant!

(116)

Diffusion geometry

d_t²(x,y) = X

i≥0

λ

d²(x,y) =

Scale invariant!

(117)

Diffusion geometry

d_t²(x,y) = X

i≥0

λ

d²(x,y) =

Scale invariant!

(118)

Diffusion geometry

d_t²(x,y) = X

i≥0

λ

d²(x,y) =

Scale invariant!

(119)

Diffusion geometry

d_t²(x,y) = X

i≥0

λ

d²(x,y) = X

i≥1 1

λi(φ_i(x)−φ_i(y))²

“Connectability” of x andy

Scale invariant!

(120)

Diffusion geometry

d_t²(x,y) = X

i≥0

e^−2λⁱ^t(φ_i(x)−φ_i(y))²

λ

d²(x,y) = 1 2

Z ∞ 0

d_t²(x,y)dt

Scale invariant!

(121)

Diffusion geometry

d_t²(x,y) = X

i≥0

λ

d²(x,y) = 1 2

Z ∞ 0

d_t²(x,y)dt

Scale invariant!

(122)

Diffusion geometry

d_t²(x,y) = X

i≥0

λ

d²(x,y) = 1 2

Z ∞ 0

d_t²(x,y)dt

Scale invariant!

(123)

Diffusion maps

Map each pointx onX to asequence Φ(x) ={K(λi)φi(x)}i≥0

Φ(x) areembedding coordinatesof x in `² (“R^∞”) By Parseval’s theorem

kΦ(x)−Φ(y)k²_`2 = X

i>0

K²(λ_i)(φ_i(x)−φ_i(y))²

= d²(x,y)

Diffusion distance is represented by Euclidean distancein embedding space

(124)

Diffusion maps

Φ(x) are embedding coordinatesof x in `² (“R^∞”)

By Parseval’s theorem

kΦ(x)−Φ(y)k²_`2 = X

i>0

K²(λ_i)(φ_i(x)−φ_i(y))²

= d²(x,y)

(125)

Diffusion maps

Φ(x) are embedding coordinatesof x in `² (“R^∞”) By Parseval’s theorem

kΦ(x)−Φ(y)k²_`2 = X

i>0

= d²(x,y)

(126)

Diffusion maps

Φ(x) are embedding coordinatesof x in `² (“R^∞”) By Parseval’s theorem

kΦ(x)−Φ(y)k²_`2 = X

i>0

= d²(x,y)

Diffusion Geometry in Shape Analysis

Spectral methods in deformable shape analysis

Dimensions of media

Dimensions of media

Dimensions of media

Dimensions of media

3D shapes vs. images

3D shapes vs. images

3D shapes vs. images

3D shapes vs. images

3D shapes vs. images

1

2 + 1 2 =

3D shapes vs. images

1

2 + 1 2 =

1

2 + 1 2 = ?

Setting

Setting

Structure

Structure

Structure

Structure

Agenda

Diffusion processes on surfaces

Diffusion processes on surfaces

Laplace-Beltrami operator ∆

Laplace-Beltrami operator ∆

Laplace-Beltrami operator ∆

Laplace-Beltrami operator ∆

Laplace-Beltrami operator ∆

Laplace-Beltrami operator ∆

Discretization of the Laplacian

Discretization of the Laplacian

Discretization of the Laplacian

Discretization of the Laplacian

Discretization of the Laplacian

Desired properties of discrete Laplacians

Desired properties of discrete Laplacians

Desired properties of discrete Laplacians

Desired properties of discrete Laplacians

Desired properties of discrete Laplacians

Desired properties of discrete Laplacians

Desired properties of discrete Laplacians

Desired properties of discrete Laplacians

Desired properties of discrete Laplacians

Desired properties of discrete Laplacians

Desired properties of discrete Laplacians

Desired properties of discrete Laplacians

Desired properties of discrete Laplacians

Desired properties of discrete Laplacians

Desired properties of discrete Laplacians

Desired properties of discrete Laplacians

Desired properties of discrete Laplacians

Desired properties of discrete Laplacians

No free lunch

No free lunch

No free lunch

No free lunch

Eigendecomposition of Laplacian

Eigendecomposition of Laplacian

Eigendecomposition of Laplacian

Finite elements

Finite elements

Finite elements

Finite elements

Finite elements

Finite elements

To see the sound

Chladni plates

Chladni plates

Shape DNA

Shape DNA

“Can we hear the shape of the drum?”

“Can we hear the shape of the drum?”

“Can we hear the shape of the drum?”

“Can we hear the shape of the drum?”

“Can we hear the shape of the drum?”

“Can we hear the shape of the drum?”

2 + ¹ ₂ =

2 + ¹ ₂ =

2 + ¹ ₂ = ?