Submanifold Reconstruction

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Submanifold Reconstruction

Jean-Daniel Boissonnat DataShape, INRIA

http://www-sop.inria.fr/datashape

Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 1 / 36

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Geometric data analysis

Images, text, speech, neural signals, GPS traces,...

Geometrisation: Data = points + distances between points Hypothesis: Data lie close to a structure of

“small” intrinsic dimension Problem: Infer the structure from the data

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Submanifolds of R

^d

A compact subsetM⊂R^d is a submanifold without boundary of (intrinsic) dimensionk<d, if anyp∈Mhas an open (topological) k-ball as a neighborhood inM

W

U R^m

φ R^N

M

Intuitively, a submanifold of dimensionkis a subset ofR^dthat looks locally like an open set of an affine space of dimensionk

Acurvea1-dimensional submanifold Asurfaceis a2-dimensional submanifold

More generally, manifolds are defined in an intrinsic way, independently of any embedding inR^d

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Triangulation of a submanifold

We call triangulation of a submanifoldM⊂R^d a simplicial complexMˆ such that

Mˆ is embedded inR^d its vertices are onM it is homeomorphic toM

Submanifold reconstruction

The problem is to construct a triangulationMˆ of some unknown submanifoldMgiven a finite set of pointsP⊂M

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Triangulation of a submanifold

We call triangulation of a submanifoldM⊂R^d a simplicial complexMˆ such that

Mˆ is embedded inR^d its vertices are onM it is homeomorphic toM

Submanifold reconstruction

The problem is to construct a triangulationMˆ of some unknown submanifoldMgiven a finite set of pointsP⊂M

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Issues in high-dimensional geometry

Dimensionality severely restricts our intuition and ability to visualize data

⇒need for automated and provably correct methods methods Complexity of data structures and algorithms rapidly grow as the dimensionality increases

⇒no subdivision of the ambient space is affordable

⇒data structures and algorithms should be sensitive to the intrinsic dimension(usually unknown) of the data

Inherent defects : sparsity, noise, outliers

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Issues in high-dimensional geometry

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Issues in high-dimensional geometry

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Looking for small and faithful simplicial complexes

Need to compromise Size of the complex

I can we havedimMˆ =dimM?

Efficiency of the construction algorithms and of the representations

I can we avoid the exponential dependence ond?

I can we minimize the number of simplices ?

Quality of the approximation

I Homotopy type & homology (Cech andαcomplexes, persistence)

I Homeomorphism (Delaunay-type complexes)

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Sampling and distance functions

[Niyogi et al.], [Chazal et al.]

Distance to a compactK: dK :x→infp∈K kx−pk

Geometric inference from noisy data

Pb: infering topological and geometric properties from point cloud data sets sampled

“around” unknown low-dimensional shapes.

Sc. challenges:

- dealing with noise

- well founded math. models

- algorithmic complexity issues (curse of dimensionality)

The distance function framework:

When the data C are close (Hausdorﬀ dist.) to the geometric structure K to infer...

• distance function d

K

: x → inf

p∈K

� x − p �

• Replace K and C by d

K

and d

C

• Stability results for the topology/geometry of the oﬀsets K

^r

= d

⁻_K¹

([0, r]) and C

^r

= d

⁻_C¹

([0, r])

Stability

If the data pointsCare close (Hausdorff) to the geometric structureK, the topology and the geometry of the offsetsKr=d⁻¹([0,r])and Cr=d⁻¹([0,r])are close

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Distance functions and triangulations

Computational Topology (Jeff Erickson) Examples of Cell Complexes

Corollary 15.1.For any points setPand radius�, the Aleksandrov-ˇCech complexACˇ_�(P)is homotopy- equivalent to the union of balls of radius�centered at points inP.

Aleksandrov-ˇCech complexes and unions of balls for two different radii. 2-simplices are yellow; 3-simplices are green.

15.1.2 Vietoris-Rips Complexes: Flags and Shadows

Theproximity graph N_�(P)is the geometric graph whose vertices are the pointsPand whose edges join all pairs of points at distance at most 2�; in other words,N_�(P)is the 1-skeleton of the Aleksandrov-ˇCech complex. TheVietoris-Rips complex VR_�(P)is theflag complexorclique complexof the proximity graphN_�(P). A set ofk+1 points inPdefines ak-simplex inV R_�(P)if and only if every pair defines an edge inN_�(P), or equivalently, if the set has diameter at most 2�. Again, the Vietoris-Rips complex is an abstractsimplicial complex.

The Vietoris-Rips complex was used by Leopold Vietoris[57]in the early days of homology theory as a means of creating finite simplicial models of metric spaces.²The complex was rediscovered by Eliayu Rips in the 1980s and popularized by Mikhail Gromov[35]as a means of building simplicial models for group actions. ‘Rips complexes’ are now a standard tool in geometric and combinatorial group theory.

The triangle inequality immediately implies the nesting relationshipACˇ_�(P)⊆V R_�(P)⊆ACˇ_2�(P) for any�, where⊆indicates containmentas abstract simplicial complexes. The upper radius 2�can be reduced to�3�/2 if the underlying metric space is Euclidean[21], but for arbitrary metric spaces, these bounds cannot be improved.

One big advantage of Vietoris-Rips complexes is that they determined entirely by their underlying proximity graphs; thus, they can be applied in contexts like sensor-network modeling where the underlying metric is unknown. In contrast, the Aleksandrov-ˇCech complex also depends on the metric of the ambient space that containsP; even if we assume that the underlying space is Euclidean, we need the lengths of the edges of the proximity complex to reconstruct the Aleksandrov-ˇCech complex.

On the other hand, there is no result like the Nerve Lemma for flag complexes. Indeed, it is easy to construct Vietoris-Rips complexes for pointsin the Euclidean planethat contain topological features of arbitrarily high dimension.

2Vietoris actually defined a slightly different complex. LetU={U1,U2, . . .}be a set of open sets that cover some topological spaceX. TheVietoris complexofUis the abstract simplicial complex whose vertices are points inX, and whose simplices are finite subsets ofXthat lie in some common setU_i. Thus, the Vietoris complex of an open cover is the dual of its Aleskandrov-ˇCech nerve. Dowker[25]proved that these two simplicial complexes have isomorphic homology groups.

2

Nerve theorem (Leray)

The nerve of the balls (Cech complex) and the union of balls have the same homotopy type (same result for theα-complex)

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Questions

+ The homotopy type of a compact setXcan be computed from the

˘Cech complex of a sample ofX + The same is true for theα-complex

– The ˘Cech and theα-complexes arehuge(O(n^d)andO(n^dd/2e)) and difficult to compute in high dimensions

– Both complexes arenotin general homeomorphic toX (i.e. nota triangulationofX)

– The ˘Cech complexcannot be realizedin general in the same space asX

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˘Cech and Rips complexes

The Rips complex is easier to compute but still very big, and less precise in approximating the topology

CHAPTER 2. SHAPE RECONSTRUCTION

ˇCech complex

The ˇCech complexC(P, α)is the abstract simplicial complex whosek-simplices correspond to subsets ofk+ 1points that can be enclosed in a ball of radiusα,

C(P, α) = {σ| ∅ �=σ⊂P,Rad(σ)≤α}.

Equivalently, ak-simplex{p₀, . . . , p_k}belongs to the ˇCech complex if and only if the k+ 1closed Euclidean balls B(p_i, α)have non-empty common intersection. Hence, the ˇCech complex is the nerve of the collection of balls{B(p, α)|p∈P}. Since balls are convex, the Nerve theorem implies that the ˇCech complexC(P, α)is homotopy equivalent to the union of these balls, that is,|C(P, α)| �P^α(see Figure2.2, left and right).

Rips complex

TheVietoris-Rips complexis a variant of the ˇCech complex which is easier to compute. The Vietoris-Rips complex,R(P, α)is the abstract simplicial complex whosek-simplices correspond to subsets ofk+ 1points inPwith diameter at most2α,

R(P, α) = {σ| ∅ �=σ⊂P,Diam(σ)≤2α}.

For simplicity, we refer toR(P, α)as the Rips complex. Recall that theflag complexof a graphG, denotedFlagG, is the maximal simplicial complex whose 1-skeleton isG. The Rips complex is an example of a flag complex. More precisely, this is the largest simplicial complex sharing with the ˇCech complex the same 1-skeleton,R(P, α) = Flag�

C(P, α)⁽¹⁾�

. Generally, R(P, α)andC(P, α)do not share the same topology. It follows that the Rips complexR(P, α) is generally not homotopy equivalent to theα-offsetP^α(see Figure2.2).

α ϑdα

Figure 2.2: Left: The ˇCech complex with parameterα. It comprises six triangles and is homotopy equivalent to a circle. Middle: Rips complex with parameterα. It contains two more triangles and is homeomorphic to a 2-sphere. Its shadow is a topological disk. Right: ˇCech complex with parameterϑ_dα. It contains all faces of the 5-simplex and is homeomorphic to a 5-ball.

22

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An example where no offset has the right topology !

1. Manifold + small noise assumption 2. Call persistent homology at rescue !

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The curses of Delaunay triangulations in higher dimensions

Complexity depends exponentially on the ambient dimension.

Robustness issues become very tricky

Higher dimensional Delaunay triangulations are not thick even if the vertices are well-spaced

The restricted Delaunay triangulation is no longer a good approximation of the manifold even under strong sampling conditions (ford>2)

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3D Delaunay Triangulations are not thick even if the vertices are well-spaced

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Each square face can be circumscribed by an empty sphere This remains true if the grid points are slightly perturbed therefore creating thin simplices

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Badly-shaped simplices

Badly-shaped simplices lead to bad geometric approximations

which in turn may lead to topological defects inDel|M(P) [Oudot]

see also [Cairns], [Whitehead], [Munkres], [Whitney]

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Tangent space approximation

Mis a smoothk-dimensional manifold (k>2) embedded inR^d

Bad news [Oudot 2005]

The Delaunay triangulation restricted toMmay be a bad approximation of the manifold even if the sample is dense

u w v p0 c0

t=∆

x y z t

p

t=∆+δ/2 c

x y z t

Figure 3: Left: tetrahedron [u, v, w, p0] and its dual Voronoi edge. Right: after perturbation ofS.

c0is the center of a Delaunay ball of radius greater than 1, we can assume thatδis small enough for the points ofL0to remain onS. Letc= (¹₂,¹₂,^δ₂,∆+^δ₂) be at the top of the bump. Since the points ofL0are located in hyperplanet=∆in the vicinity of [u, v, w, p0],cis equidistant to u, v, w, p0, and closer to these points than to any other point ofL0. This implies that the open ball Bc=B(c,!c−u!) contains no point ofL0and hasu, v, w, p0on its bounding sphere. Hence,Bcis a Delaunay ball circumscribing [u, v, w, p0], andcbelongs to the Voronoi edge dual to [u, v, w, p0].

Moreover, sinceu, v, wand (0,0,0,∆) are cocircular,∂Bcpasses also through (0,0,0,∆).

We deformSfurther by creating another small bump, at point (0,0,0,∆) this time, so as to move this point byδinto thet-dimension, outward the hypercube. Letp= (0,0,0,∆+δ) be the top of the bump — see Figure 3 (right). A quick computation shows that!c−p!=!c−u!, which implies thatp∈∂Bc. Here again, by choosingδsufficiently small, we can make sure that the radius of curvature of the bump is at least^∆₂, which means that the reach of the deformed hypersurface is still^∆₂=¹_µ. We can also make sure that the bump ofpis disjoint from the bump of c since

!c−p!>√¹2, and that the points ofL0\ {p0}remain³onS. It follows thatBcis empty of points ofL, whereLis defined byL=L0∪{p} \ {p0}. Since∂Bccontainsu, v, w, p,Bcis a Delaunay ball circumscribing [u, v, w, p]. Equivalently,cbelongs to the Voronoi edgeedual to [u, v, w, p]. Note also thatLis an (ε−δ)-sparse (2ε+δ)-sample ofS.

Since [u, v, w, p] is included in hyperplanez= 0, its dual Voronoi edgeeis aligned with (0,0,1,0), as illustrated in Figure 3 (right). This edge is incident to four Voronoi 2-faces, which are dual to the four facets of [u, v, w, p]. These 2-faces can be seen as extrusions, into thez-dimension (0,0,1,0), of the edges of the Voronoi diagram of{u, v, w, p}inside hyperplanez= 0. Among these Voronoi edges, two lie above the planet=∆+^δ₂, and two lie below. As a result, inR⁴, two Voronoi 2-faces incident toelie above hyperplanet=∆+^δ₂. These two Voronoi 2-faces do not intersect S, except atcand possibly at the bump ofp. Now, the circumradii of the facets of [u, v, w, p] are at most!c−u!=^√^1+δ√2²< µrch(S), thus, inside hyperplanez= 0, Amenta and Bern’s normal lemma [1, Lemma 7] states that the edges of the Voronoi diagram of{u, v, w, p}make angles of at most arcsin^µ

√3

1−µ<^π₃with vector (0,0,0,1). As a consequence, any Voronoi 2-facefincident toe

3They lie at leastεaway fromp0, and hence at leastε−δaway from (0,0,0,∆).

9

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Thickness and tangent space approximation

Lemma [Whitney 1957]

Ifσis a j-simplex whose vertices all lie within a distanceη from a hyperplaneH⊂R^d, then

sin∠(aff(σ),H)≤ 2jη D(σ) Corollary

Ifσis a j-simplex,j≤k, vert(σ)⊂M, ∆(σ)≤δrch(M)

∀p∈σ, sin∠(aff(σ),Tp)≤ δ Θ(σ)

(η≤_{2 rch(M)}^∆(σ)² by the Chord Lemma)

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The assumptions

Mis a differentiable submanifold ofpositive reachofR^d The dimensionkofMissmall

P is anε-netofM, i.e.

I ∀x∈M, ∃ p∈P, kx−pk ≤εrch(M)

I ∀p,q∈P, kp−qk ≥η ε¯ εis small enough

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The tangential Delaunay complex

[B. & Ghosh 2010]

p T_p

M

1 Construct the star ofp∈P in the Delaunay triangulationDelTp(P) of Prestricted toTp

2 DelTM(P) =S

p∈P star(p)

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+ DelTM(P)⊂Del(P)

+ star(p),DelTp(P)and thereforeDelTM(P)can be computed without computingDel(P)

– DelTM(P)isnotnecessarily a triangulated manifold

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Construction of Del

Tp

(P)

Given ad-flatH⊂R,Vor(P)∩His aweightedVoronoi diagram inH

pi

pj x

p0 i

p⁰j

H

kx−pik²≤ kx−pjk²

⇔ kx−p⁰_ik²+kpi−p⁰_ik²≤ kx−p⁰_jk²+kpj−p⁰_jk²

Corollary: construction of DelT_p

Ψ_p(pi) = (p⁰_i,−kpi−p⁰_ik²) (weighted point)

1 project P ontoTpwhich requiresO(Dn)time

2 constructstar(Ψp(pi))inDel(Ψp(pi))⊂Tpi 3 star(pi)≈star(Ψ_p(pi)) (isomorphic)

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Construction of Del

Tp

(P)

Given ad-flatH⊂R,Vor(P)∩His aweightedVoronoi diagram inH

pi

pj x

p0 i

p⁰j

H

kx−pik²≤ kx−pjk²

⇔ kx−p⁰_ik²+kpi−p⁰_ik²≤ kx−p⁰_jk²+kpj−p⁰_jk²

Corollary: construction of DelT_p

Ψ_p(pi) = (p⁰_i,−kpi−p⁰_ik²) (weighted point)

1 project P ontoTpwhich requiresO(Dn)time

2 constructstar(Ψp(pi))inDel(Ψp(pi))⊂Tpi 3 star(pi)≈star(Ψ_p(pi)) (isomorphic)

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Inconsistencies in the tangential complex

A simplex isnotin the star of all its vertices

τ ∈star(pi) ⇔ Tp_i∩Vor(τ)6=∅ ⇔ B(cp_i(τ)∩P=∅ τ 6∈star(pj) ⇔ Tp_j∩Vor(τ) =∅ ⇔ B(cp_j(τ)∩P3p

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Inconsistency (k + 1)-trigger

Inconsistent simplex

τis said to beinconsistentiff

∃pi,pj∈τs. t.Vor(τ)∩Tpi�=∅andVor(τ)∩Tpj=∅

p_i

p_j τ

B_p_j(τ)

B_p_i(τ)

p T_p_i

∈Vor(τ)

∈aﬀ(Vor(τ))

cpi(τ)

T_p_j

c_p_j(τ) M

i_φ

Arijit Ghosh PhD defense

Bpi(τ): ball circumscribingτ centered onTpi,cpi its center Inconsistency : Bp_i(τ)∩P=∅ and Bp_j(τ)∩P6=∅ pl ∈Bij, first point hit by(1−λ)Bpi+λBpj,λ:0→1 Triggerτ^†:(k+1)-simplexτ ?pl ∈Del((P))

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Inconsistency (k + 1)-triggers are flakes

Inconsistent simplex

τis said to beinconsistentiff

∃pi,pj∈τs. t.Vor(τ)∩Tpi�=∅andVor(τ)∩Tpj=∅

p_i

p_j τ

B_p_j(τ)

B_p_i(τ)

p T_p_i

∈Vor(τ)

∈aﬀ(Vor(τ))

cpi(τ)

T_p_j

c_p_j(τ) M

i_φ

Arijit Ghosh PhD defense

Ifτ is small and thick, then

Tpi ≈Tpj ≈aff(τ) (sample density)

kcpi−cpjksmall ⇒ Bij:=Bpi(τ)\Bpj(τ)6=∅ is small (τthick)

the triggerτ^†=τ ?pl isnotthick

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Bound on the diameter of the simplices of Del

TM

(P)

(i) Vor(p)∩Tp⊆B(p, α0rch(M))whereα₀ ≈ε (ii) ∀σ ∈star(p),Rp(σ)≤αrch(M)

(iii) ∀σ ∈DelTM(P),∆(σ)≤2αrch(M).

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Proof of (i)

x⁰ x x⁰⁰ p

x∈Vor(p)∩Tp,kp−xk=αrch(M)

x⁰be the point ofMclosest tox and x⁰⁰= Π_p(x⁰) kp−x⁰k ≤ kp−xk+kx−x⁰k ≤2kp−xk

⇒ kx⁰−x⁰⁰k ≤ ^kp−x_2rch(_M⁰^k²₎ ≤2α²rch(M) (Chord Lemma)

kx⁰−x⁰⁰k=kx−x⁰k cosφ, whereφ=∠(Tx⁰,Tp)andcosφ≥1−8α²

⇒ kx−x⁰k ≤ 2α²rch(M)

1−8α² assumingα≤

√2 4 P is anε-dense sample : ∃q∈P,kx⁰−qk ≤εrch(M)

kx−pk=αrch(M)≤ kx−qk ≤ kx−x⁰k+kx⁰−qk ≤

2α² 1−8α² +ε

rch(M)

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Bound on the diameter of inconsistency triggers

τ an inconsistentk-simplex,τ^†a trigger,θ=maxp∈τ∠(aff(τ),Tp) Lemma sinθ≤ _Θ(τ))^∆(τ)_rch(_M₎ and R(τ^†)≤ ^R(τ)_cos_θ

T_pi

cpi

c(τ) pi

τ R(τ)

ω pl

Proof

d(pl,Tp)≤ _2rch(^∆²^(τ)_M₎ (Chord Lemma)

sin∠(aff(τ),Tp)≤ ²

∆2(τ) 2rch(M)

Θ(τ) ∆(τ) = _Θ(τ)^∆(τ)_rch(M) (Whitney’s angle bound)

Rpi(τ) =kpi−cpik ≤ ^R(τ)_cos_θ and R(τ^†)≤ ki(τ^†)−pik ≤ _cos^R(τ)_θ

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Bound on the thickness of inconsistency triggers

Lemma Θ(τ^†)≤ _2(k+1)rch(^∆(τ^†⁾_M₎

1+ _Θ(τ)²

T_pi

cpi

c(τ) pi

R(τ) τ ω pl

ProofLetq∈τ.

D(pl, τ^†) = kpl−qk sin∠(pl−q,aff(τ))

≤ ∆(τ^†) (sin∠(pl−q,Tq) +sin∠(Tq,aff(τ)))

≤ ∆(τ^†)

∆(τ^†)

2rch(M) + ∆(τ) Θ(τ)rch(M)

(Chord + previous Lemmas)

≤ ∆²(τ^†) 2rch(M)

1+ 2 Θ(τ)

Hence, ifτ is thick,τ^†cannot be so : we say thatτ^†is aflake

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Reconstruction of smooth submanifolds

1 For each vertexv, compute the starstar(p)ofpinDelp(P)

2 Remove inconsistencies among the stars by weighting the points

3 Stitch the stars to obtain a triangulation of P

v

Figure 3. A two-dimensional link triangulation, represented as a collection of two-dimensional stars.

Star flipping is a variant of star splaying that adds two more ideas. First, the representation and the algorithm are recursive on the dimensionality. For example, in a three-dimensional triangulation, the star of a vertexvis represented byv’s link, which is a two- dimensional triangulation. This two-dimensional triangulation is represented by a set of two-dimensional stars, as illustrated in Fig- ure 3. These stars are not required to agree with each other either.

Each two-dimensional star is represented by a one-dimensional link triangulation (recall Figure 2). The one-dimensional triangulations are calledlink rings, and unlike their higher-dimensional counter- parts, they are always internally consistent.

Second, the workhorse of star flipping is the classic flip algorithm, at every level of the recursion. To make a star locally convex, star flipping tries to apply classic flipping within the link triangulation. Only if classic flipping gets stuck before restoring local convexity to a star does star flipping call itself recursively.

Star flipping, described in Section 5, seems likely to run faster than star splaying if the input triangulation is close to Delaunay, because it takes better advantage of the input triangulation.

3 Stars, Rays, and Cones

Star splaying is founded on several observations about the relation- ships between stars, rays, polyhedral cones, convex hulls, and De- launay triangulations.

Consider the convex hullHof a setV of vertices inE^d⁺¹. (Com- putingHis a standard way to compute a Delaunay triangulation in E^d; see below.) Suppose thatV isgeneric: nod+2 points ofV lie on a common hyperplane. ThenH is a simplicial polytope—every facet of H is ad-simplex. Let∂H denote the boundary triangulation of H. For consistency, facets ared-simplices and ridgesare (d−1)-simplices throughout this paper, whether inE^d⁺¹or inE^d.

Imagine wishing to compute not all ofH, but just the star of one vertex vof H—specifically,v’s star in∂H, leaving outH proper.

See Figure 4(a). Define the set of rays that originate atvand pass through other vertices ofV, namelyR = {vw" :w ∈ V\{v}}. LetHv

be the convex hull of the raysR, illustrated in Figure 4(b). Hv is a polyhedral cone with vertexvandH ⊂ Hv. The star ofvwraps around the tip of Hv like a paper shell around an ice cream cone.

The star is combinatorially equivalent to the cone’s boundary: the face lattice for the proper faces ofHvis isomorphic to the face lattice for the star ofv. In the isomorphism, the rays on the boundary of Hv are in one-to-one correspondence with the edges inv’s star and the vertices inv’s link.

Lethbe a hyperplane that separatesvfrom all the other vertices inV, illustrated in Figure 4(c). The cross-sectionP= Hv∩h=H∩h is a d-polytope, namely the convex hull of the intersection points

{"r∩h : r ∈ R}. The face lattice ofP’s boundary is isomorphic to

the face lattice ofv’s link.

The central observation is that these three problems are essen- tially equivalent: computing the star or link of vin the boundary of the (d+1)-dimensional convex hullH, computing the (d+1)-

v

H

v

H _v

(a) (b)

H _v

v h

P

(c)

Figure 4. (a) The star ofvin a convex polyhedronH. (b) The convex hull Hv of rays, a polyhedral cone whose boundary is combinatorially equivalent tov’s star. (c) A cross-section of the cone is the convex hull of the points where the rays intersect the cross-sectional hyperplane.

dimensional convex hull Hv of rays, and computing the d-dimen- sional convex hullP of points. The wealth of ideas computational geometers employ for the last problem apply immediately to the first problem.

One way to compute H is to use a d-dimensional convex hull algorithm to compute the star of each vertex in V individually.

Naively applied, this method is strikingly slow—itsbest-caserun- ning time is inΘ(n²), wherenis the number of vertices inV. But if the candidates for inclusion in the link of each vertex could be pruned to a small number—say, a constant number of vertices in each link—then the method becomes strikingly attractive, as it en- tails just a linear number of constant-time convex hull computa- tions.

At first glance, one inconvenience of this method appears to be finding a corner-cutting hyperplaneh. This step is not only unnec- essary; it is unwise, because computing the intersections of the rays inRwithhintroduces avoidable roundofferrors. Many algorithms for computing convex hulls of point sets in E^d can be adapted to compute convex hulls of rays originating at a common pointv in E^d⁺¹, simply by replacing the orientation tests ond+1 points inE^d with orientation tests ond+2 points in E^d⁺¹ (whereinvis always one of those points).

The standard incremental insertion method for updating a convex hull is the beneath-beyond method of Kallay [17], which adds one new vertex (or ray) at a time and maintains after each addition the convex hull of the vertices (or rays) processed so far. LetC be the convex hull at a fixed moment in time between vertex insertions.

Let f be a facet ofC. A point pis said to bebeyond f ifpandC lie on opposite sides of the affine hull of f. The beneath-beyond algorithm adds a vertex w and transformsC into conv(C ∪w) by finding and deleting every facet ofCthatwis beyond, then creating new facets that attachwto every ridge ofCthat adjoins exactly one surviving facet.

This idea works whetherCis a convex hull of points or a convex hull of rays with a common originv. The latter circumstance is illustrated in Figure 5. There is one important algorithmic difference between these two circumstances: a ray can lie beyondeveryfacet of the cone C, meaning that the convex hull of the rays is E^d⁺¹.

v

Figure 3. A two-dimensional link triangulation, represented as a collection of two-dimensional stars.

Star flipping is a variant of star splaying that adds two more ideas. First, the representation and the algorithm are recursive on the dimensionality. For example, in a three-dimensional triangula- tion, the star of a vertex v is represented by v’s link, which is a two- dimensional triangulation. This two-dimensional triangulation is represented by a set of two-dimensional stars, as illustrated in Fig- ure 3. These stars are not required to agree with each other either.

Each two-dimensional star is represented by a one-dimensional link triangulation (recall Figure 2). The one-dimensional triangulations are called link rings, and unlike their higher-dimensional counter- parts, they are always internally consistent.

Second, the workhorse of star flipping is the classic flip algo- rithm, at every level of the recursion. To make a star locally con- vex, star flipping tries to apply classic flipping within the link trian- gulation. Only if classic flipping gets stuck before restoring local convexity to a star does star flipping call itself recursively.

Star flipping, described in Section 5, seems likely to run faster than star splaying if the input triangulation is close to Delaunay, because it takes better advantage of the input triangulation.

3 Stars, Rays, and Cones

Star splaying is founded on several observations about the relation- ships between stars, rays, polyhedral cones, convex hulls, and De- launay triangulations.

Consider the convex hull H of a set V of vertices in E

^d⁺¹

. (Com- puting H is a standard way to compute a Delaunay triangulation in E

^d

; see below.) Suppose that V is generic: no d + 2 points of V lie on a common hyperplane. Then H is a simplicial polytope—every facet of H is a d-simplex. Let ∂H denote the boundary triangula- tion of H. For consistency, facets are d-simplices and ridges are (d − 1)-simplices throughout this paper, whether in E

^d⁺¹

or in E

^d

.

Imagine wishing to compute not all of H, but just the star of one vertex v of H—specifically, v’s star in ∂H, leaving out H proper.

See Figure 4(a). Define the set of rays that originate at v and pass through other vertices of V , namely R = { vw " : w ∈ V \{ v }} . Let H

v

be the convex hull of the rays R, illustrated in Figure 4(b). H

v

is a polyhedral cone with vertex v and H ⊂ H

v

. The star of v wraps around the tip of H

v

like a paper shell around an ice cream cone.

The star is combinatorially equivalent to the cone’s boundary: the face lattice for the proper faces of H

v

is isomorphic to the face lat- tice for the star of v. In the isomorphism, the rays on the boundary of H

v

are in one-to-one correspondence with the edges in v’s star and the vertices in v’s link.

Let h be a hyperplane that separates v from all the other vertices in V , illustrated in Figure 4(c). The cross-section P = H

v

∩ h = H ∩ h is a d-polytope, namely the convex hull of the intersection points

{ " r ∩ h : r ∈ R } . The face lattice of P’s boundary is isomorphic to

the face lattice of v’s link.

The central observation is that these three problems are essen- tially equivalent: computing the star or link of v in the boundary of the (d + 1)-dimensional convex hull H, computing the (d + 1)-

v

H

v

H _v

(a) (b)

H _v

v h

P

(c)

Figure 4. (a) The star of v in a convex polyhedron H. (b) The convex hull Hv of rays, a polyhedral cone whose boundary is combinatorially equivalent to v’s star. (c) A cross-section of the cone is the convex hull of the points where the rays intersect the cross-sectional hyperplane.

dimensional convex hull H

v

of rays, and computing the d-dimen- sional convex hull P of points. The wealth of ideas computational geometers employ for the last problem apply immediately to the first problem.

One way to compute H is to use a d-dimensional convex hull algorithm to compute the star of each vertex in V individually.

Naively applied, this method is strikingly slow—its best-case run- ning time is in Θ(n

²

), where n is the number of vertices in V . But if the candidates for inclusion in the link of each vertex could be pruned to a small number—say, a constant number of vertices in each link—then the method becomes strikingly attractive, as it en- tails just a linear number of constant-time convex hull computa- tions.

At first glance, one inconvenience of this method appears to be finding a corner-cutting hyperplane h. This step is not only unnec- essary; it is unwise, because computing the intersections of the rays in R with h introduces avoidable roundoff errors. Many algorithms for computing convex hulls of point sets in E

^d

can be adapted to compute convex hulls of rays originating at a common point v in E

^d⁺¹

, simply by replacing the orientation tests on d + 1 points in E

^d

with orientation tests on d + 2 points in E

^d⁺¹

(wherein v is always one of those points).

The standard incremental insertion method for updating a convex hull is the beneath-beyond method of Kallay [17], which adds one new vertex (or ray) at a time and maintains after each addition the convex hull of the vertices (or rays) processed so far. Let C be the convex hull at a fixed moment in time between vertex insertions.

Let f be a facet of C. A point p is said to be beyond f if p and C lie on opposite sides of the affine hull of f . The beneath-beyond algorithm adds a vertex w and transforms C into conv(C ∪ w) by finding and deleting every facet of C that w is beyond, then creating new facets that attach w to every ridge of C that adjoins exactly one surviving facet.

This idea works whether C is a convex hull of points or a convex hull of rays with a common origin v. The latter circumstance is il- lustrated in Figure 5. There is one important algorithmic difference between these two circumstances: a ray can lie beyond every facet of the cone C, meaning that the convex hull of the rays is E

^d⁺¹

.

v

Figure 3.A two-dimensional link triangulation, represented as a collection of two-dimensional stars.

Star flipping is a variant of star splaying that adds two more ideas. First, the representation and the algorithm are recursive on the dimensionality. For example, in a three-dimensional triangulation, the star of a vertexvis represented byv’s link, which is a two- dimensional triangulation. This two-dimensional triangulation is represented by a set of two-dimensional stars, as illustrated in Fig- ure 3. These stars are not required to agree with each other either.

Each two-dimensional star is represented by a one-dimensional link triangulation (recall Figure 2). The one-dimensional triangulations are calledlink rings, and unlike their higher-dimensional counter- parts, they are always internally consistent.

Second, the workhorse of star flipping is the classic flip algorithm, at every level of the recursion. To make a star locally convex, star flipping tries to apply classic flipping within the link triangulation. Only if classic flipping gets stuck before restoring local convexity to a star does star flipping call itself recursively.

Star flipping, described in Section 5, seems likely to run faster than star splaying if the input triangulation is close to Delaunay, because it takes better advantage of the input triangulation.

3 Stars, Rays, and Cones

Star splaying is founded on several observations about the relation- ships between stars, rays, polyhedral cones, convex hulls, and De- launay triangulations.

Consider the convex hullHof a setV of vertices inE^d⁺¹. (Com- putingHis a standard way to compute a Delaunay triangulation in E^d; see below.) Suppose thatV isgeneric: nod+2 points ofV lie on a common hyperplane. ThenHis a simplicial polytope—every facet of H is ad-simplex. Let ∂H denote the boundary triangulation of H. For consistency, facets ared-simplices and ridges are (d−1)-simplices throughout this paper, whether inE^d⁺¹or inE^d.

Imagine wishing to compute not all ofH, but just the star of one vertexv of H—specifically, v’s star in ∂H, leaving out H proper.

See Figure 4(a). Define the set of rays that originate atvand pass through other vertices ofV, namelyR = {vw" :w ∈ V\{v}}. Let Hv

be the convex hull of the raysR, illustrated in Figure 4(b). Hv is a polyhedral cone with vertexvandH ⊂ Hv. The star ofvwraps around the tip of Hv like a paper shell around an ice cream cone.

The star is combinatorially equivalent to the cone’s boundary: the face lattice for the proper faces ofHvis isomorphic to the face lattice for the star ofv. In the isomorphism, the rays on the boundary of Hv are in one-to-one correspondence with the edges inv’s star and the vertices inv’s link.

Lethbe a hyperplane that separatesvfrom all the other vertices inV, illustrated in Figure 4(c). The cross-sectionP=Hv∩h=H∩h is a d-polytope, namely the convex hull of the intersection points

{"r∩h : r ∈ R}. The face lattice ofP’s boundary is isomorphic to

the face lattice ofv’s link.

The central observation is that these three problems are essen- tially equivalent: computing the star or link ofvin the boundary of the (d+1)-dimensional convex hullH, computing the (d+1)-

v

H

v

H _v

(a) (b)

H _v

v h

P

(c)

Figure 4. (a) The star ofvin a convex polyhedron H. (b) The convex hull Hv of rays, a polyhedral cone whose boundary is combinatorially equivalent tov’s star. (c) A cross-section of the cone is the convex hull of the points where the rays intersect the cross-sectional hyperplane.

dimensional convex hull Hv of rays, and computing thed-dimensional convex hullPof points. The wealth of ideas computational geometers employ for the last problem apply immediately to the first problem.

One way to compute H is to use a d-dimensional convex hull algorithm to compute the star of each vertex in V individually.

Naively applied, this method is strikingly slow—itsbest-caserun- ning time is inΘ(n²), where nis the number of vertices inV. But if the candidates for inclusion in the link of each vertex could be pruned to a small number—say, a constant number of vertices in each link—then the method becomes strikingly attractive, as it en- tails just a linear number of constant-time convex hull computa- tions.

At first glance, one inconvenience of this method appears to be finding a corner-cutting hyperplaneh. This step is not only unnec- essary; it is unwise, because computing the intersections of the rays inRwithhintroduces avoidable roundofferrors. Many algorithms for computing convex hulls of point sets inE^d can be adapted to compute convex hulls of rays originating at a common pointv in E^d⁺¹, simply by replacing the orientation tests ond+1 points inE^d with orientation tests ond+2 points inE^d⁺¹ (whereinvis always one of those points).

The standard incremental insertion method for updating a convex hull is the beneath-beyond method of Kallay [17], which adds one new vertex (or ray) at a time and maintains after each addition the convex hull of the vertices (or rays) processed so far. LetC be the convex hull at a fixed moment in time between vertex insertions.

Let f be a facet ofC. A pointpis said to bebeyond f if pandC lie on opposite sides of the affine hull of f. The beneath-beyond algorithm adds a vertex wand transformsC into conv(C∪w) by finding and deleting every facet ofCthatwis beyond, then creating new facets that attachwto every ridge ofCthat adjoins exactly one surviving facet.

This idea works whetherCis a convex hull of points or a convex hull of rays with a common originv. The latter circumstance is illustrated in Figure 5. There is one important algorithmic difference between these two circumstances: a ray can lie beyondeveryfacet of the cone C, meaning that the convex hull of the rays is E^d⁺¹.

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Algorithm hypotheses

Known quantities inred

M=a differentiable submanifold of positive reach of dim. k⊂R^d P=an(ε, δ)-sample ofM

ε≤ε0

ε/δ≤η₀

we can estimate the tangent spaceTp at anyp∈P

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Removing inconsistencies by removing flakes

Another application of Moser-Tardos algorithmic LLL

Input: P,{Tp,p∈P},w˜0,Θ₀

Initialize all weights to0and computeDelTM( ˆP)

whilethere areΘ₀-flakes or inconsistencies inDelTM( ˆP)do whilethere is aΘ₀-flakeσinDelTM( ˆP) do

resampleσ, i.e. reweight the vertices ofσ updateDelTM( ˆP)

ifthere is an inconsistent simplexσinDelTM( ˆP) then compute a trigger simplexσ^†associated toσ resample the flakeσ⊂σ^†

updateDelTM( ˆP)

Output: A weighting scheme on P andDelTM( ˆP) DelTM( ˆP)isΘ₀-thick and has no inconsistency

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Summary

Termination

I If ^η^¯₂ ≥ρ¯≥f(Θ₀), the algorithm terminates and returns a complexMˆ that has no inconsistent configurations Complexity

I Nod-dimensional data structure⇒linear ind

I exponential ink

Approximation

I Mˆ is a PL simplicialk-manifold

I Mˆ ⊂tub(M, ε)

I Mˆ is homeomorphic toM

Submanifold Reconstruction