Varifolds and Surface Approximation Blanche Buet joint work with - - PowerPoint PPT Presentation

Varifolds and Surface Approximation

Blanche Buet joint work with Gian Paolo Leonardi (Modena), Simon Masnou (Lyon) and Martin Rumpf (Bonn)

Laboratoire de math´ ematiques d’Orsay, Paris Sud

7 february 2019 The Mathematics of imaging, Paris

Why varifolds ?

◮ flexible: you can endow both discrete and continuous objects

with a varifold structure.

◮ encode order 1 information (tangent bundle): unoriented

• bjects.

◮ provide weak notion of curvatures. ◮ natural distances to compare varifolds.

Plan

A simple example What is a varifold ? Generalized curvature of a varifold Approximate curvature Numerical illustrations References Second fundamental form

We start with

◮ Γ ⊂ R2 a C2 closed curve, ◮ γ : [0, L] → R2 an injective (0 ∼ L) arc length

parametrization of Γ.

◮ unit tangent vector τ: for x = γ(t) ∈ Γ, τ(x) = γ′(t) and

θ(x) the angle between τ(x) and the horizontal.

◮ curvature vector κ: for x = γ(t) ∈ Γ, κ(x) = γ′′(t).

We start with

◮ Γ ⊂ R2 a C2 closed curve, ◮ γ : [0, L] → R2 an injective (0 ∼ L) arclength parametrization

• f Γ.

◮ unit tangent vector τ: for x = γ(t) ∈ Γ, τ(x) = γ′(t) and

θ(x) the angle between τ(x) and the horizontal.

◮ curvature vector κ: for x = γ(t) ∈ Γ, κ(x) = γ′′(t).

Let ϕ : R2 → R be C1: L d dtϕ(γ(t))γ′(t) dt =

• by parts
• ϕ(γ(t))γ′(t)

L

t=0

• =0

− L ϕ(γ(t)) γ′′(t)

κ(γ(t))

dt

We start with

◮ Γ ⊂ R2 a C2 closed curve, ◮ γ : [0, L] → R2 an injective (0 ∼ L) arclength parametrization

• f Γ.

◮ unit tangent vector τ: for x = γ(t) ∈ Γ, τ(x) = γ′(t) and

θ(x) the angle between τ(x) and the horizontal.

◮ curvature vector κ: for x = γ(t) ∈ Γ, κ(x) = γ′′(t).

Let ϕ : R2 → R be C1: L d dtϕ(γ(t))γ′(t) dt = −

• Γ

ϕ(x)κ(x)

We start with

◮ Γ ⊂ R2 a C2 closed curve, ◮ γ : [0, L] → R2 an injective (0 ∼ L) arclength parametrization

• f Γ.

◮ unit tangent vector τ: for x = γ(t) ∈ Γ, τ(x) = γ′(t) and

θ(x) the angle between τ(x) and the horizontal.

◮ curvature vector κ: for x = γ(t) ∈ Γ, κ(x) = γ′′(t).

Let ϕ : R2 → R be C1: L d dtϕ(γ(t))γ′(t) dt = −

• Γ

ϕ(x)κ(x) L d dtϕ(γ(t))γ′(t) dt = L

• ∇ϕ(γ(t)) · γ′(t)
• γ′(t) dt

=

• Γ

(∇ϕ(x) · τ(x)) τ =

• Γ

Πθ(x) (∇ϕ(x))

We start with

◮ Γ ⊂ R2 a C2 closed curve, ◮ γ : [0, L] → R2 an injective (0 ∼ L) arclength parametrization

• f Γ.

◮ unit tangent vector τ: for x = γ(t) ∈ Γ, τ(x) = γ′(t) and

θ(x) the angle between τ(x) and the horizontal.

◮ curvature vector κ: for x = γ(t) ∈ Γ, κ(x) = γ′′(t).

Let ϕ : R2 → R be C1:

✎ ✍ ☞ ✌

• Γ

ϕ(x)κ(x) = −

• Γ

Πθ(x) (∇ϕ(x))

We start with

◮ Γ ⊂ R2 a C2 closed curve, ◮ γ : [0, L] → R2 an injective (0 ∼ L) arclength parametrization

• f Γ.

◮ unit tangent vector τ: for x = γ(t) ∈ Γ, τ(x) = γ′(t) and

θ(x) the angle between τ(x) and the horizontal.

◮ curvature vector κ: for x = γ(t) ∈ Γ, κ(x) = γ′′(t).

Let ϕ : R2 → R be C1:

✎ ✍ ☞ ✌

• Γ

ϕ(x)κ(x) = −

• Γ

Πθ(x) (∇ϕ(x))

◮ weak formulation of curvature, ◮ relies only on the knowledge of

• Γ

ψ(x, θ(x))

• ψ : R2 × R → R continuous

∀x ∈ R2, ω → ψ(x, ω) is π–periodic

• .

Our first varifold

C = ψ : R2 × R → R (x, ω) → ψ(x, ω)

• ψ continuous and π–periodic

w.r.t. ω

• .

The continuous linear form    VΓ : C → R ψ →

• Γ

ψ(x, θ(x)) is the 1–varifold naturally associated with Γ.

Our first varifold

C = ψ : R2 × R → R (x, ω) → ψ(x, ω)

• ψ continuous and π–periodic

w.r.t. ω

• .

The continuous linear form on C    VΓ : C → R ψ →

• Γ

ψ(x, θ(x)) is the 1–varifold naturally associated with Γ. With ψ(x, ω) = Πω∇ϕ(x),

✎ ✍ ☞ ✌

• Γ

ϕ(x)κ(x) = −VΓ(ψ) and

◮ Knowing VΓ is enough to recover the curvature κ. ◮ Conversely, it is possible to define a notion of generalized

curvature for any continuous linear form on C, that is for ANY 1–varifold in R2.

Plan

A simple example What is a varifold ? Generalized curvature of a varifold Approximate curvature Numerical illustrations References Second fundamental form

Varifolds ?

Generalized surface : couples a weighted spatial information and a non oriented direction information. Introduced by Almgren in the 60’ : weak notion of surface allowing good compactness properties. A d–varifold is a Radon measure in Rn × Gd,n.

The Grassmannian

Grassmannian of d–planes : Gd,n = {d–vector sub-spaces of Rn}. non-oriented d–planes. We identify P ∈ Gd,n with the orthogonal projector ΠP onto P, so that Gd,n can be seen as a compact subset of Mn(R) : Gd,n ≃   A ∈ Mn(R)

• A2 = A

AT = A Trace(A) = d    Distance on Gd,n : d(P, Q) = ΠP − ΠQ.

Radon measure

X = Rn, X = Rn × Gd,n. A Radon measure in X will equivalently be (thanks to Riesz theorem) :

◮ a Borel mesure on X that takes finite values on compact

sets.

◮ a positive linear form on Cc(X).

Weak star convergence: µi

∗

− ⇀ µ ⇔ ∀ϕ ∈ Cc(X),

• X

ϕ dµi →

• X

ϕ dµ . locally metrized for instance by the flat distance : ∆(µ, ν) = sup

• X

ϕ dµ −

• X

ϕ dν

• ϕ is 1–Lipschitz

supX |ϕ| ≤ 1

About ∆

• Condition sup |ϕ| ≤ 1 : for ε > 0 and µ = (1 + ε)δ0, ν = δ0,
• ϕ dµ −
• ϕ dν
• = ε |ϕ(0)| −

− − − − − →

ϕ(0)→+∞ +∞ .

• Condition ϕ 1–Lipschitz: for ε > 0 and µ = δε, ν = δ0,
• ϕ dµ −
• ϕ dν
• = |ϕ(ε) − ϕ(0)| = 2 .

with ϕ(ε) = 1 and ϕ(0) = −1.

• Localized version : B ⊂ Rn

✗ ✖ ✔ ✕

∆B(µ, ν) = sup   

• X

ϕ dµ −

• X

ϕ dν

• ϕ is 1–Lipschitz

supX |ϕ| ≤ 1 spt ϕ ⊂ B   

First examples

1–Varifold associated with

◮ a segment S ⊂ Rn whose direction is P ∈ G1,n :

V = H1

|S ⊗ δP , ◮ a union of segments

M = ∪8

i=1Si ,

Si of direction Pi ∈ G1,n: V =

8

• i=1

H1

|Si ⊗ δPi.

2–Varifold associated with a triangulated surface M =

• T∈T

T, where T has direction PT ∈ G2,n : V =

• T∈T

L2

|T ⊗ δPT .

Point cloud varifolds

d–Varifold associated with a point cloud in Rn, that is

◮ a finite set of points {xi}N i=1 ⊂ Rn, ◮ weighted by masses {mi}N i=1 ⊂ R∗ +, ◮ provided with directions {Pi}N i=1 ⊂ Gd,n.

V =

N

• i=1

miδxi ⊗ δPi.

Regular varifolds

When M ⊂ Rn is a d–sub-manifold (or a d–rectifiable set) :

• 1. µ measure in Rn supported in M : µ = Hd

|M.

• 2. a family (νx)x∈M of probabilities in Gd,n : νx = δTxM.

Then define V = µ ⊗ νx Radon measure in Rn × Gd,n in the sense: for ψ ∈x Cc(Rn × Gd,n), V (ψ) =

• ψ dV =
• x∈Rn
• P∈Gd,n

ψ(x, P) dνx(P) dµ(x) =

• M

ψ(x, TxM) dHd(x)

Regular varifolds

When M ⊂ Rn is a d–sub-manifold (or a d–rectifiable set) :

• 1. µ measure in Rn supported in M : µ = Hd

|M.

• 2. a family (νx)x∈M of probabilities in Gd,n : νx = δTxM.

Then define V = µ ⊗ νx Radon measure in Rn × Gd,n in the sense: for ψ ∈ Cc(Rn × Gd,n), V (ψ) =

• ψ dV =
• x∈Rn
• P∈Gd,n

ψ(x, P) dνx(P) dµ(x) =

• M

ψ(x, TxM) dHd(x) Remember, for Γ: VΓ(ψ) =

• Γ

ψ(x, θ(x)) .

Disintegration

Mass of a varifold V : it’s the Radon measure V in Rn defined as V (A) = V (A × Gd,n) . Disintegration : a d–varifold V can be decomposed as V = µ ⊗ νx with µ = V where for V –a.e. x, νx is a probability measure in Gd,n.

More varifolds ...

Point cloud Volumic approx Rectifiable

• j

mjδxj ⊗ δPj

• K∈K

mKLn

|K ⊗ δPK

θ(x)Hd

|M ⊗ δTxM

V =

• j

mjδxj V =

• K∈K

mKLn

|K

V = θ(x)Hd

|M

Plan

A simple example What is a varifold ? Generalized curvature of a varifold Approximate curvature Numerical illustrations References Second fundamental form

Divergence theorem

• X ∈ C1

c(Rn, Rn),

• For P ∈ Gd,n, divPX =

n

• k=1

(ΠP∇Xk) · ek,

• M ⊂ Rn closed d–sub-manifold C2 with mean curvature

vector H,

✎ ✍ ☞ ✌

• M

divTxMX dHd = −

• M

H · X dHd. For V = Hd

|M ⊗ δTxM the d–varifold associated with M :

• Rn×Gd,n

divPX(x) dV(x, P) = −

• Rn H · X dV .

− → distributional definition of mean curvature.

First variation of a varifold

First variation of V δV : X ∈ C1

c(Rn, Rn) −

→

• Rn×Gd,n

divP X(x) dV (x, P) . δV is a distribution of order 1. V = Hd

|M ⊗ δTxM associated with M (C2 closed):

δV (X) = −

• M

X · H dHd thus

✞ ✝ ☎ ✆

δV = −H Hd

|M = −HV order 0.

When δV is of order 0,

• Riesz : δV is a vector mesure de Radon.
• Radon Nikodym : we decompose δV with respect to V :

✞ ✝ ☎ ✆

δV = −H V + (δV )sing, H ∈ L1(V ) generalized curvature.

Plan

A simple example What is a varifold ? Generalized curvature of a varifold Approximate curvature Numerical illustrations References Second fundamental form

Regularization of the first variation

ρ ∈ C∞

c (B1(0)) radial ≥ 0,

• ρ = 1 and ρε(x) = 1

εn ρ x ε

• .

Regularized first variation δV ∗ ρε(x) = 1 εn+1

• Bε(x)×Gd,n

∇S ρ y − x ε

• dV (y, S) .

Well-defined for any varifold. Case of a point cloud : V =

N

• i=1

miδ(xi,Pi),

✓ ✒ ✏ ✑

1 ε

N

• i=1

miρ′ |xi − x| ε ΠPi(xi − x) |xi − x| . − → explicit expression “easy” to implement numerically.

Approximate curvature

Radon-Nikodym derivative of δV ∗ ρε with respect to V ∗ ξε : δV ∗ ρε = −Hε(x, V ) V ∗ ξε with

✎ ✍ ☞ ✌

Hε(x, V ) = − δV ∗ ρε(x) V ∗ ξε(x) .

• Choice of ρ, ξ. For V associated with M smooth, the leading

term in the expansion of |C Hε − H| around a point is proportional to 1 (sρ′(s) + dCξ(s))sd−1 ds = 0

• by PI
• ☛

✡ ✟ ✠

ξ(s) = −sρ′(s) dC = −sρ′(s) n .

Convergence

Let V = θHd

|M ⊗ δTxM be a rectifiable d–varifold s.t.

δV = −HV + (δV )sing. is a measure.

• Consistency C = Cρ,ξ > 0 constant, for Hd–a.e. x ∈ M,

C Hε(x, V ) − − − →

ε→0 H(x) = − δV

V (x)

• Stability :

◮ x ∈ sptV and zi −

− − →

i→∞ 0

◮ (Vi)i sequence of d–varifolds weak–∗ converges to V with a

localized flat distance around x controlled by di ↓ 0. Then, for εi ↓ 0 satisfying

✎ ✍ ☞ ✌

di + |zi − x| ε2

i

− − − →

i→∞ 0 ,

|Hεi(zi, Vi) − Hεi(x, V )| = Oi→∞ di + |zi − x| ε2

i

Case of a point cloud varifold

Let V =

N

• i=1

miδ(xi,Pi),

✬ ✫ ✩ ✪

Hε(x, V ) = − 1 ε

N

• i=1

miρ′ |xi − x| ε ΠPi(xi − x) |xi − x|

N

• i=1

miξ |xi − x| ε

• .

Plan

A simple example What is a varifold ? Generalized curvature of a varifold Approximate curvature Numerical illustrations References Second fundamental form

Mean curvature

Code C++ using nanoflann library.

Figure: Intensity of mean curvature from blue (zero) to red through white ε = 0.007 for a diameter 1

Mean curvature

Code C++ using nanoflann library.

Figure: Intensity of mean curvature from blue (zero) to red through white ε = 0.007 for a diameter 1

Gaussian curvature

Figure: Gaussian curvature, negative (blue), zero (white), positive (red)

Sharp features

Figure: |k1| + |k2| from blue (zero) to red (high) through white

(a) (b) (c) (d)

Figure: Left: Gaussian curvature, Right: |k1| + |k2|, Top: without noise, Bottom: with white noise

0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2

100 200 300 400 600 700 1000 1500 2000

(a)

0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2

100 200 300 400 600 1000 1500 3000

(b)

0.8 0.4 0.0 0.4 0.8 0.8 0.4 0.0 0.4 0.8

(c)

(a) Step 1 (b) Step 101 (c) Step 13 (d) Step 25 (e) Step 37 (f) Step 49 (g) Step 61 (h) Step 73 (i) Step 85 (j) Step 97

Figure: Evolution of a tetrahedron whose edges are fixed, discretized with N = 6052 points and for a time–step τ = 0.005.

(a) Step 1 (b) Step 2701 (c) Step 401 (d) Step 801 (e) Step 1201 (f) Step 1601 (g) Step 2001 (h) Step 2401 (i) Step 2701 (j) Step 2701

Figure: Evolution of a cube whose edges are fixed, discretized with N = 18600 points and for a time–step τ = 0.01.

Thanks for your attention !

Plan

A simple example What is a varifold ? Generalized curvature of a varifold Approximate curvature Numerical illustrations References Second fundamental form

` A propos des varifolds

• W. K. Allard.

On the first variation of a varifold. Annals of Mathematics, 95:417–491, 1972.

• J. Almgren.

Theory of Varifolds. Mimeographed lecture notes, 1965. K.A. Brakke. The motion of a surface by its mean curvature. Mathematical notes (20), Princeton University Press, 1978.

• N. Charon, A. Trouv´

e. The varifold representation of nonoriented shapes for diffeomorphic registration. SIAM Journal on Imaging Sciences, 6(4):2547–2580, 2013.

J.E. Hutchinson. Second Fundamental Form for Varifolds and the Existence of Surfaces Minimising Curvature. Indiana Univ. Math. J., (35)1, 1986.

• U. Menne.

The Concept of Varifold. Notices Amer. Math. Soc., 64(10):1148–1152, 2017.

• L. Simon.

Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, volume 3, 1983.

` A propos de courbure discr` ete

• N. Amenta, Y.-J. Kil.

Defining Point-set Surfaces. ACM SIGGRAPH 2004 Papers, 264–270, 2004.

• F. Chazal, D. Cohen-Steiner, A. Lieutier, B. Thibert.

Stability of Curvature Measures.

• Comput. Graph. Forum, 28(5), 2009.
• U. Clarenz,M. Rumpf, A. Telea.

Robust Feature Detection and Local Classification for Surfaces based on Moment Analysis. IEEE Transactions on Visualization and Computer Graphics, 10(5):516–524, 2004.

• D. Cohen-Steiner, J.-M. Morvan.

Second fundamental measure of geometric sets and local approximation of curvatures. Journal of Differential Geometry, 74(3):363–394, 2006.

• D. Levin.

The approximation power of moving least-squares. Mathematics of Computation, 67:1517–1531, 1998.

• Q. M´

erigot. Robust Voronoi-based Curvature and Feature Estimation. 2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling, 1–12, 2009.

• U. Pinkall, K. Polthier.

Computing discrete minimal surfaces and their conjugates.

• Experiment. Math., 2(1):15–36, 1993.
• M. Wardetzky.

Convergence of the Cotangent Formula: An Overview. Discrete Differential Geometry, Birkh¨ auser Basel, 275–286, 2008.

• B. Buet, G.P. Leonardi, S. Masnou.

A Varifold Approach to Surface Approximation. ARMA, 2017.

• B. Buet, G.P. Leonardi, S. Masnou.

Weak and Approximate Curvtaures of a Measure: a Varifold Perspective. arxiv, 2019.

Second fundamental form

d n

N

l=1 mlρ′

|xl0 −xl|

ε

Pl(xl0 −xl)

|xl0 −xl| · 1 2

• (Pl−Pl0)jkei+(Pl−Pl0)ikej−(Pl−Pl0)ijek
• N

l=1 mlξ

|xl0 −xl|

ε

• .

Link with the Cotangent formula

◮ Let T = (F, E, V) be a triangulation in R3, where V ⊂ R3 is

the set of vertices, E ⊂ V × V is the set of edges and F is the set of triangle faces.

◮ 2–varifold

VT =

• F∈F

H2

|F ⊗ δPF , ◮ The nodal function ϕv, v ∈ V, associated with T is defined by

ϕv(v) = 1, ϕv(w) = 0 for w ∈ V, w = v and ϕv affine on each face F ∈ F. δVT ( ϕx) = −1 2

• v∈V(x)

(cot αxv + cot βxv) (v − x).

Plan

A simple example What is a varifold ? Generalized curvature of a varifold Approximate curvature Numerical illustrations References Second fundamental form

Second fundamental form

Back to the divergence theorem:

• M ⊂ Rn C2 closed;
• P(x) = (Pjk(x))jk ∈ Mn(R) orthogonal projection onto TxM ;
• ϕ ∈ C1

c(Ω × Mn(R)),

X(x) := ϕ(x, P(x))ei. Divergence theorem 0 =

• M

divP (PX) leads to a weak formulation of the second fundamental form through Aijk = (P(x)∇Pjk(x))i : −

• M

(P(x)∇xϕ)i dHd =

• M

j,k

(P(x)∇Pjk(x))i

• =:Aijk

Djkϕ +

• q

(P(x)∇Piq(x))q

• Aqiq

ϕ

• dHd

Second fundamental form

Back to the divergence theorem:

• M ⊂ Rn C2 closed;
• P(x) = (Pjk(x))jk ∈ Mn(R) orthogonal projection onto TxM ;
• ϕ ∈ C1

c(Ω × Mn(R)),

X(x) := ϕ(x)Pjk(x)ei. Divergence theorem 0 =

• M

divP (PX) leads to a weak formulation of the second fundamental form through Aijk = (P(x)∇Pjk(x))i : −

• M

(P(x)∇ϕ)i dHd =

• M
• Aijkϕ + Pjk(x)
• q

Aqiqϕ

• dHd

It is then possible to define, for i, j, k = 1 . . . n, δijkV : X ∈ C1

c(Rn, Rn) −

→

• Rn×Gd,n

SjkdivSX(x) dV (x, S) · ei . When those distributions are Radon measures, we define βijk s.t. δijkV = −βijkV + (δijkV )sing . And for V –a.e. x, we can define Aijk as the pointwise solution

• f the linear system with n3 equations :

✓ ✒ ✏ ✑

Aijk + cjk

n

• q=1

Aqiq = βijk with cjk(x) =

• Gd,n

S dνx(S) and V = V ⊗ νx.