Globally Optimal Smooth k-Vector Fields and Curvature Linesknoeppel/talks/globally... · felix...

GLOBALLY OPTIMAL SMOOTH k -VECTOR FIELDSAND CURVATURE LINES

Felix Knöppel

Institut für MathematikTechnische Universität Berlin

June 6, 2012

Joint work with Keenan Crane, Ulrich Pinkall, and Peter Schröder

FELIX KNÖPPEL (TU BERLIN) OPTIMAL SMOOTH k -VECTOR FIELDS JUNE 6, 2012 1 / 56

OUTLINE

1 INTRODUCTION

2 SMOOTH THEORYThe Canonical Bundle

The Degree of Complex Line Bundles

The Hopf Differential as 2-vector field

Smoothness Energies and Alignment

3 DISCRETIZATIONDiscrete Hermitian Lines Bundles

Connecting Topology

Discrete Energies

4 EXPERIMENTAL RESULTSSmoothest Sections

Optimal Curvature Direction Fields


FIGURE: Excerpt from a paper of Zorin and Hertzmann


FIGURE: Our method in compare


1 INTRODUCTION






Connecting Topology

Discrete Energies




Let f : M→ R3 ' ImH be an immersed oriented surface. Denote its Gauss map byN : M→ S2 ⊂ ImH.

the induced metric g is given by g = 〈df ,df〉define the shape operator S by df ◦S = dNthe the eigenvalues κ1,κ2 of S are called principal curvaturesGauss curvature K and mean curvature H of M are given by

K = detS, H = 12 trS

define J : TM→ TM by the relation

df ◦ J = N · df

J is complex structure: J2 =−Id


the induced metric is compatible with J, i.e. g (X ,Y) = g (JX ,JY)

further we can define a hermitian product g on TM by

g(X ,Y) = g (X ,Y) + ig (X ,JY)

If π : E→ M is a vector bundle over M, we denote its smooth sections by Γ(E):

Γ(E) = {ψ : M→ E | ψ smooth and π ◦ψ = idM} .

Denote the trivial bundle by C := M×C and set

C∞ (M) := Γ(C) – the set of smooth C–valued functions on MΩk (M) := Γ(C⊗Λk (M)) – the set of smooth C–valued k–forms on MΩk (E) := Γ(E⊗Λk (M)) – the set of smooth k–forms on M with values in E


DEFINITION (COMPLEX LINE BUNDLE)A complex line bundle is a vector bundle π : L→ M, where the fibers Lp = π−1 (p) are1–dimensional C–vector spaces and the trivialization maps are C–linear on the fibers.

For two complex line bundles L1,L2 the usual vector space constructions work:

L∗1 ,L1⊗ L2

are complex line bundles again.


DEFINITION (∗-OPERATOR)Define for a 1–form ω ∈ Ω1 (L):

∗ω := ω ◦ J.

DEFINITION (CANONICAL AND ANTI-CANONICAL BUNDLE)The canonical bundle K and anti-canonical bundle K̄ are given by

Kp := {ω : TpM→ C | ω R− linear , ∗ω = iω},K̄p := {ω : TpM→ C | ω R− linear , ∗ω =−iω}.


Let V and W be complex vector spaces with complex structures JV and JW resp. Then:

Any R – linear map A : V →W splits into a C–linear part B and a C–anti-linear part B̄

A = B + B̄, B = 12 (A− JW ◦A◦ JV ) , B̄ =12 (A + JW ◦A◦ JV ) .

Example:

the derivative d : C∞ (M)→ Ω1 (M) of functions splits in two parts:

d = ∂ + ∂̄ ,

where

∂ f = 12 (df − i ∗ df) and ∂̄ f =12 (df + i ∗ df)

∂̄ f ≡ 0 are the Cauchy-Riemann equations


DEFINITION (CONNECTION)A connection ∇ : Γ(L)→ Ω1 (L) is a linear map, that satisfies the Leibniz rule

∇(fψ) = (df)ψ + f (∇ψ)

for f ∈ C∞ (M) ,ψ ∈ Γ(L).

the connection is said to be a complex connection, if

∇(Jψ) = J∇ψ for ψ ∈ Γ(L)

decompose ∇ = ∂ + ∂̄ , where ∂ : Γ(L)→ Γ(KL) , ∂̄ : Γ(L)→ Γ(K̄ L)∂̄ is a holomorphic structure


1 INTRODUCTION






Connecting Topology

Discrete Energies




DEFINITION (CURVATURE TENSOR)

The curvature tensor R∇ ∈ Ω2 (End(L)) is given by

R∇ (X ,Y)ψ = ∇X ∇Y ψ−∇Y ∇X ψ−∇[X ,Y ]ψ, X ,Y ∈ TM.

DEFINITION (DEGREE)Let M be a closed oriented surface. The degree of a complex line bundle L over M is

given by

deg(L) := i2π

∫MR∇,

where R∇ is the curvature tensor of any complex connection ∇ on M.


The group of line bundles

L = {L→ M | L is complex line bundle}/Isomorphism

is an abelian group with respect to⊗ with C as identity and L−1 = L∗.

The map

deg : L → Z

is a group isomorphism.

Hence we know te degree of the powers of the canonical bundle over a Riemann surface:

deg(K n) =−nχ (M) .


1 INTRODUCTION






Connecting Topology

Discrete Energies




A k–vector field is the field of the k–th roots of a section ψ of the bundle TMk :

If X is a local basis section of TM then ψ = z X k and the corresponding k–vector field isgiven by the set–valued map

p 7→ {w Xp | w ∈ C,wk = z (p)}.

A k–direction field is a k–vector field, such that |z|= 1.

The Hopf–differential Q is the trace–free part of the shape operator:

S = Q +H · I.

Let End± (TM) := {T ∈ End (TM) | T ◦ J =±J ◦T}. Q ∈ End− (TM).


Let g be the hermitian form corresponding to the metric g, then we have the followingisomorphism:

End− (TM) 3 T 7→ g(.,T (.)) ∈ K 2.

Let X ∈ TpM be an eigendirection of Q wrt. say κ1−κ22 , then g(X ,QX) =κ1−κ2

2 .

Identify K 2 with (TM)2, then Hopf–differential at p takes the form

κ1−κ22 X

2.

Hence, away from umbilical points, the corresponding 2–vector field consists of two

vectors pointing towards the larger principal curvature direction.


1 INTRODUCTION






Connecting Topology

Discrete Energies




The Dirichlet energy ED is given by

ED (ψ) = 12∫

M|∇ψ|2 ,

where the norm on Ω1 (L)

|ωp|2 = |ω(X)|2 + |ω(Y)|2

for an orthonormal basis X ,Y ∈ TpM.

It decomposes into a holomorphic energy EH and an anti–holomorphic energy EA:

ED (ψ) = EH (ψ) + EA (ψ) ,

where

EH (ψ) = 12∫

M

∣∣∂̄ψ∣∣2 , EA (ψ) = 12 ∫M|∂ψ|2


We define our smoothness energy as

Es = (1 + s)EH + (1− s)EA = ED + s (EH−EA) , s ∈ [−1,1] .

THEOREM (RELATION OF EH AND EA)The holomorphic and the anti–holomorphic energy of a section ψ are related as

EA (ψ)−EH (ψ) = 12∫

MK |ψ|2 ,

where K denotes the curvature of the line bundle.


Let ψ ∈ Γ(L) a section of L and write it as product of an R–valued function u ≥ 0 anda direction field φ

ψ = uφ .

Differentiation yields

∇ψ = (du)φ + u∇φ = (du)φ + uωJφ .

We obtain

Es (ψ) = 12∫

M|du|2 +

(|ω|2− s2K

)︸︷︷︸=:Vs

|u|2 = 12〈〈(−∆ + Vs)u,u〉〉.

For every direction field φ we get a Schrödinger operator S with potential Vs.


Vs =∣∣ω∣∣2− s2K


Opposite, take a direction field φ and multiply it with a function u ≥ 0: ψ := uφ .The energy of the section is the energy of u wrt. the operator S :

E (ψ) = 〈〈S u,u〉〉.

Moreover:

‖ψ‖2 = ‖u‖2.


DEFINITIONWe define the energy of a direction field φ ∈ L as the ground state of the correspondingSchrödinger operator S .

E (ψ) = 〈〈S u,u〉〉, ‖ψ‖2 = ‖u‖2.

Forget about φ :Minimizing over all u ≥ 0 with norm 1 gives the ground state.Minimizing over the unit sphere in the line bundle L gives the smallest ground state

wrt. to all underlying directions fields.

Normalizing this minimizer gives the smoothest direction field.


Let ψ0 be an arbitrary section of the hermitian line bundle L. Define an alignment energyEl by

El (ψ) =−∫

M〈ψ,ψ0〉.

Here 〈., .〉 denotes the real part of the hermitian product.

Form for t ∈ [0,1] the energy

Es,t (ψ) = (1− t)Es (ψ) + tEl (ψ) .

minimize over the unit sphere in L


We write the energies Es,El as

Es(ψ) = 12〈〈Asψ,ψ〉〉, El (ψ) =−〈〈ψ0,ψ〉〉.

Minimizing Es,t (ψ) = (1− t)Es (ψ) + tEl (ψ) over all ψ with 〈〈ψ,ψ〉〉= 1 leads to

(As−λ I)ψ = t1−t ψ0.

λ is a monotone function of tλ t→0−−→ λ1 and λ

t→1−−→−∞Method:

solve (As−λ I) ψ̃ = ψ0 for λ ∈ (−∞,λ1)ψ = ψ̃|ψ̃| gives the minimizer


1 INTRODUCTION






Connecting Topology

Discrete Energies




A discrete surface M is a cell decomposition (V ,E,F) of a compact oriented surface(without boundary). V ,E,F denote the set of vertices, oriented edges and faces resp.

DEFINITION (DISCRETE HERMITIAN LINE BUNDLE)A hermitian line bundle L (with connection) over M is then given by

1 1–dimensional hermitian vector spaces Li for each vertex i ∈ V .2 unitary maps Ce = Cij : Li → Lj for each oriented edge e = (i, j) ∈ E .3 C is a discrete connection, i.e. Cji ◦Cij = idLi .

Example: A discrete tangent bundles from a smooth surface.

triangulate smooth surface by geodesics and obtain discrete surface M

construct discrete connection C using parallel transport of the Levi–Civita

connection


Let γ = (γ1, . . . ,γn) be a loop on M. We define the monodromy hγ around γ by

hγ · idL0 = Cγn ◦ · · · ◦Cγ1 .

For a face ϕ we set hϕ := h∂ϕ .

DEFINITION (DISCRETE HERMITIAN LINE BUNDLE*)4 hϕ 6=−1 for all ϕ ∈ F .

We define the curvature 2-form σϕ : F → (−π,π) by the condition hϕ = eiσϕ .

THEOREMLet M be a discrete surface. Then there is an integer n ∈ Z such that∫

Mσ = 2πn.


DEFINITION (DEGREE OF A DISCRETE LINE BUNDLE)Let L be a discrete hermitian line bundle over the discrete surface M. Then the degree

deg(L) of the hermitian line bundle L is given by

deg(L) = 12π

∫M

σ .

A discrete hermitian line bundle isomorphism is a map U that assigns to each vertex i a

unitary vector space isomorphism between the corresponding fibers such that

C̃ij ◦Ui = Uj ◦Cij .

THEOREMTwo Hermitian line bundles with connection are isomorphic if and only if they have the

same monodromy around every closed loop.


COROLLARYThe degree of a discrete hermitian line bundle only depends on the isomorphism class

of the bundle.

Let ψ be any section of a hermitian line bundle L over M such that |ψi |= 1 for all i ∈ V .Set

Cij (ψi) = zijψj .

We say that ψ is smooth, ifze 6=−1 for all e ∈ E.

Then there are unique real numbers δe ∈ (−π,π) such that

ze = eiδe .


DEFINITION (INDEX)Let ψ be a smooth section of a hermitian line bundle L over M. The index of a ψ atϕ ∈ F is defined as

2π indexϕ (ψ) = σϕ −

(∑

e∈∂ϕδe

).

Hence, from the definitions we get immediately

deg(L) = ∑ϕ∈F

indexϕ (ψ) .


1 INTRODUCTION






Connecting Topology

Discrete Energies




The degree of the canonical bundle comes with the combinatoric:

1 Euler characteristic of a M is given by χ (M) = |V |− |E|+ |F |2 further we know that

deg(K) =−χ (M)

We want to preserve that equation!

Again think of a geodesic triangulation of a smooth surface: If αi denote the exteriorangles of a geodesic triangle ϕ , then Gauss–Bonnet gives∫

ϕK = 2π−

3

∑i=1

αi = σϕ + 2πn,

for some n ∈ {−1,0,1}.


Since per definition σϕ ∈ (−π,π) we get∣∣∣∣∫ϕ K∣∣∣∣< π ⇒ deg(K−1)= χ (M) .

If we are interested in higher powers of the canonical bundle K n, we have to assume that

the triangulation is finer: For n 6= 0 demand that∣∣∣∣∫ϕ K∣∣∣∣< π|n| .

But what is the discrete tangent bundle?


Idea: Smooth the discrete surface!

Problems:

It is a hard task to smooth a surface.

The tangent spaces at the vertices depends on the smoothing.


Better idea: Take an intrinsic point of view.

neighborhood of vertex is isometric to a cone (rotationally symmetric)

smooth this preserving rotational symmetry

parallel transport of Levi–Civita connection along geodesics on smoothed surface

is independent of smoothing


Denote the angle sum of all the angles αj incident to the vertex i by α :

α = ∑αj .Then the curvature at vertex i is given by Ki = 2π−α .

We set si :=2πα and define new angles α̃j = siαj at the vertex, then

∑ α̃j = 2π.


Subdivide the surface M combinatorically:

The monodromy σ̃ieϕ over the subtriangle {i,e,ϕ} is then given by

σ̃ieϕ = si2 αi + αe + αϕ =12 (si − 1)αi .


The monodromy σϕ over the complete face ϕ is the sum of the monodromies over the 6subtriangles of ϕ .

DEFINITION (k -SMOOTH DISCRETE SURFACES)A discrete surface is called k–smooth, if∣∣σϕ ∣∣< πk .On a k–smooth mesh we have that for the curvature form σ̃ of TMk

∑ϕ∈F

σ̃ϕ = ∑ϕ∈F

kσϕ = 2kπχ (M) .

THEOREM (DISCRETE POINCARÉ–HOPF)On a k-smooth closed discrete surface the index sum of every k-direction field equals

kχ (M).


1 INTRODUCTION






Connecting Topology

Discrete Energies




Suppose we have an affine map f : C→ C mapping a non–degenerate triangle toanother. Decompose it:

df (v) = λ v + µ v̄.

For a discrete function f : M→ C, i.e. f ∈ Γ(C), we find that

ED = EH + EA,

where

EH = ∑ϕ∈F

∣∣µϕ ∣∣2 Aϕ , EA = ∑ϕ∈F

∣∣λϕ ∣∣2 Aϕ .How can we define that energies on curved bundles?


Strategy: Recall that we are working on the a subdivision of M.

1 use the connection C to identify the fiber of a face with the fibers of its vertices

2 interpolate linearly and apply the same construction as for functions

Let φ be a smooth global basis section, then ψ = uφ and we end up with a positivedefinite matrix As ∈ R2n given by

Es (u) = (1 + s)EH (u) + (1− s)EA (u) = 12〈〈u,Asu〉〉R2n .

Finding the minima of that energy is simply searching for eigenvectors corresponding to

the smallest eigenvalue.


1 INTRODUCTION






Connecting Topology

Discrete Energies




FIGURE: Holomorphic, harmonic and anti–holomorphic 1–direction fields


FIGURE: Anti–holomorphic (left) and holomorphic (right) 1–direction fields


FIGURE: Anti–holomorphic (left) and holomorphic (right) 2–direction fields


0.0

2.0

4.0

- 1 . 0 0.0 1.0

0.0

0.1

0.2

0.3000

- 1 . 0 0.0 1.0

FIGURE: First 40 eigenvalues over the interval [−1,1] for k = 2 for a round sphere and for agenus 3 surface.


FIGURE: Smooth section of TM3 and TM4


1 INTRODUCTION






Connecting Topology

Discrete Energies




We know the integral of Q in a distributional sense: If we chose the basis sections X on

the tangent bundle on an edge e to align with e then this takes the form

Qe =−αe le

2X 2,

where αe denotes the dihedral angle at e and le its length.

Again we use the connection to transport the section (via faces) to the adjacent vertices.


FIGURE: Discrete Hopf direction fields


FIGURE: Smoothed discrete Hopf direction fields.


FIGURE: Smooth line and cross fields on a rounded cube.


FIGURE: Smooth cross fields: Input for remeshing algorithms.


FIGURE: Principal curvature line fields on differently triangulated ellipsoids.


IntroductionSmooth TheoryThe Canonical BundleThe Degree of Complex Line BundlesThe Hopf Differential as 2-vector fieldSmoothness Energies and Alignment

DiscretizationDiscrete Hermitian Lines BundlesConnecting TopologyDiscrete Energies

Experimental ResultsSmoothest SectionsOptimal Curvature Direction Fields

Globally Optimal Smooth k-Vector Fields and Curvature Linesknoeppel/talks/globally... · felix...

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