Title: The medial axis of closed bounded sets is Lipschitz stable with respect to the Hausdorff distance under ambient diffeomorphisms
Venue: NeurIPS 2023
Accepted: no
Keywords: Medial axis

Hausdorff distance

Lipschitz continuity
Abstract: We prove that the medial axis of closed sets is Hausdorff stable in the following sense: 
Let $\mathcal{S} \subseteq \mathbb{R}^d$ be a fixed closed set that contains a bounding sphere. Consider the space of $C^{1,1}$~diffeomorphisms of $\mathbb{R}^d$ to itself, which keep the bounding sphere invariant. 
The map from this space of diffeomorphisms (endowed with a Banach norm) to the space of closed subsets of $\mathbb{R}^d$ (endowed with the Hausdorff distance), mapping a diffeomorphism $F$ to the closure of the medial axis of $F(\mathcal{S})$, is Lipschitz.

This extends a previous stability result of Chazal and Soufflet on the stability of the medial axis of $C^2$~manifolds under $C^2$ ambient diffeomorphisms. 

1

Abstract

We prove that the medial axis of closed sets is Hausdorff stable in the following sense: Let S ⊆ R d be a fixed closed set that contains a bounding sphere. Consider the space of C 1 , 1 diffeomorphisms of R d to itself, which keep the bounding sphere invariant. The map from this space of diffeomorphisms (endowed with a Banach norm) to the space of closed subsets of R d (endowed with the Hausdorff distance), mapping a diffeomorphism F to the closure of the medial axis of F ( S ) , is Lipschitz.

This extends a previous stability result of Chazal and Soufflet on the stability of the medial axis of C 2 manifolds under C 2 ambient diffeomorphisms.

1 Introduction

In [19], Federer introduced the reach of a (closed) set S ⊂ R d as the infimum over all points in S of the distance from these points to the medial axis ax( S ) , the set of points in R d for which the closest point in S is not unique. Federer also introduced the reach at a point p ∈ S to be the distance from p to the medial axis of S . We now call this quantity the local feature size [3] and denote it by lfs( p ) .

Federer proved that the reach is stable under C 1 , 1 diffeomorphisms of the ambient space. Here, a C 1 , 1 map is a C 1 map whose derivative is Lipschitz, and a C 1 , 1 diffeomorphism is a C 1 , 1 bijective map whose inverse is also C 1 , 1 . Chazal and Soufflet [13] proved that the medial axis is stable with respect to the Hausdorff distance under ambient diffeomorphisms, but under stronger assumptions than the work of Federer, namely assuming that S is a C 2 manifold and the distortion is a C 2 diffeomorphism of the ambient space. Chazal and Soufflet based their work on earlier results by Blaschke [9], which were not as strong as Federer's.

In this paper we extend the stability result of the medial axis. More concretely, we generalize the result of Chazal and Soufflet [13] to arbitrary closed sets and C 1 , 1 diffeomorphisms of the ambient space; we show that the Hausdorff distance between the medial axes of the closed set and its image is bounded in terms of Lipschitz constants stemming from the diffeomorphism of the ambient space. Our result follows from the work of Federer [19] and in fact shortens the proof in [13] significantly.

Our bounds on the Hausdorff distance say nothing about the topology of the medial axis, which is 27 known to be highly unstable (see e.g. [5]), although it preserves the homotopy type (see [28]). 28

Contribution and related work Our work differs from the majority of the literature in three 29 essential ways: 30

Firstly, we make no assumptions on the set we consider apart from that it is closed. The stability of 31 the medial axis of (piecewise) smooth manifolds has been the object of intense study, see for example 32

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[13, 15-17, 24, 30, 37-40]. However, the manifold assumption is impossible to achieve in many applications - such as in the context of astrophysics, one of the main motivations of this paper.

Secondly, we achieve stability without pruning the medial axis. This contrasts with a large body of work, such as [6, 12, 16, 29]. Not having to prune the medial axis is a significant advantage. On the downside, we limit the changes of the considered set to those induced by ambient diffeomorphisms. Nevertheless, given the standard examples of the instability of the medial axis - see for example [5] -webelieve these limitations are near to the weakest assumptions necessary for Hausdorff stability.

Thirdly, our results hold for sets in arbitrary dimensions and are not sensitive to the dimension of the set itself. A large part of the related work only investigates sets of low dimensions or codimension one manifolds, although there are some notable exceptions such as [39], see also [17], and [12, 29].

Motivation The medial axis has many real world applications - among others, in robot motion planning [27], triangulation algorithms [4], graphics [35], and shape recognition, segmentation, and learning [10, 18, 25, 33, 41]. See also the overviews [32, 35]. The reach - the distance between a set and its medial axis - is a central concept in manifold learning [1, 2, 20-22, 34].

The motivation of this paper is twofold: Firstly, we tackle the following challenge from the processing of images collected with optical devices which use lenses - such as cameras or telescopes. A shape extracted from such an image may be imprecise due to the imperfection of the lenses. Our result implies that the medial axis of such a shape is stable under these imperfections. As a consequence, the outcome of any shape recognition or shape segmentation algorithm based on the medial axis will

be stable.

In addition to the disciplines listed above, such stability is sought after in astrophysics, in particular for shape analysis and automated shape identification in observational astronomy. Observational astronomers are interested in reconstructing objects like stars or galaxies, and their place in the universe from data gathered by telescopes. They can deduce the distance from the object to the observer thanks to so-called standard candles or red shift [14, 23, 31]. However, the image gets distorted due to optical effects - either through gravitational lensing ([7]) or lensing inside the telescope itself ([36]).

Such a distortion can be modeled as a diffeomorphism of the ambient space. At the same time, this problem cannot be tackled using the result by Chazal and Soufflet [13], since the observed objects might not be smooth - for example due to interactions with shock waves or jets. In addition, with our method astrophysicists can not only reconstruct objects in space (3D), but also in spacetime (4D).

The second motivation is more formal in nature: The stability of the medial axis is instrumental in establishing its computability. Indeed, when proving properties of algorithms based on the medial axis, authors generally assume the real RAM model. 1 However, as was recently argued in [29], the medial axis needs to be stable in order to be computable in more realistic models of computation.

There is a more practical component to this formal question: It is not a priori clear if using possibly noisy real world data or the output of other computer programs as input for these algorithms yields answers that are close to the ground truth. To be able to prove that the output is correct, we need (numerical) stability of the medial axis.

Outline After revisiting preliminaries and known results in Section 2, we state the main stability 74 result in Section 3. In Section 4 we reformulate this result in terms of norms on Banach spaces. This 75 also exhibits the fact that the stability of the medial axis is Lipschitz in the following sense: We think 76 of the set S as fixed and consider the map from the space of diffeomorphisms (endowed with a norm 77 which makes it a Banach space) to the space of closed subsets of R d (endowed with the Hausdorff 78 distance), mapping each diffeomorphism F : R d → R d to the closure of the medial axis of F ( S ) . 79

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The Lipschitz constant then only depends on the diameter of the bounding sphere of the set

S

.

We only include proof sketches of the two main theorems in this article. The full proofs of the 81 theorems and of the supporting lemmas, can be found in the supplementary material. 82

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2 Preliminaries: Sets of positive reach and the closest point projection

In this section we recall some definitions and results concerning the medial axis and sets of positive 84 reach. Essentially, we need three ingredients from the literature to prove our main theorem: the 85 notions related to the closest point projection, the properties of the generalized normal and tangent 86 spaces, and Federer's result on the stability of the reach under ambient diffeomorphisms. 87

We write d ( · , · ) for the Euclidean distance between two points, and the distance between a point and 88 a set. That is, for any closed set S and point p , 89

We denote the Hausdorff distance between two sets A,B ⊆ R d by d H ( A,B ) : 90

We write B ( c, r ) , resp. S ( c, r ) , to denote balls, resp. spheres, with centre c and radius r . Lastly, |·| denotes the Euclidean norm, and ∥·∥ an operator norm.

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The closest point projection and related notions The projection of points in the ambient space 93 R d to the (set of) closest point(s) of the set S ⊆ R d is denoted by π S , and illustrated in Figure 1. 94

S

P4

πS(P3)

∈πs(p3)

)s)

P3 = πs(p3)

P2

Ts(p1)!

(d)s 3

ax(S)

P1

The medial axis of S is the set of all points p ∈ R d where the set π S ( p ) consists of more than one 95 point: 96

Here, # π S ( p ) denotes the cardinality of the set π S ( p ) . 97

For a point p ∈ S , the local feature size of p is the distance from p to the medial axis of the set S : 98

Finally, the reach of the set S is the infimum of the local feature size over all its points: 99

Throughout this paper we assume that S ⊆ R d is a closed set. We shall further assume that the set 100 S as well as its medial axis are bounded, and that the bounding sphere of S is contained in S itself. 101

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More specifically, we assume that there exists a closed ball B of positive radius such that S ⊆ B , and ∂B ⊆ S . We call ∂B the bounding sphere of S .

The addition of the bounding sphere ∂B to the set S is necessary to obtain the desired bound on the Hausdorff distance between the two medial axes of the set S and its image under the ambient diffeomorphism. Indeed, consider the following example, illustrated in Figure 2.

Let the set S consist of two points in the plane, S = { p, q } ⊆ R 2 . The medial axis of S is then the bisector line of p and q . After a generic perturbation F of p and q -that is, not a translation and not a perturbation in the direction ± ( p -q ) -the bisector line ax( F ( S )) of the perturbed points intersects the bisector ax( S ) of the original pair. The Hausdorff distance between these two non-parallel lines is infinite, and thus unboundable.

Figure 2: In black the set S and its medial axis, in light blue the perturbed set and its medial axis. The Hausdorff distance between ax( S ) and ax( F ( S )) is infinite.

At the same time, the addition of the bounding sphere ∂B to the considered set S is not a restriction. Indeed,

Remark 2.1 The medial axes of S and S \ ∂B coincide in the interior of the ball B sufficiently far away from its boundary ∂B . More precisely:

Any point x ∈ ax( S ) , such that π S ( x ) ∩ ∂B = ∅ , lies on the medial axis ax( S \ ∂B ) .

Conversely, if a point x lies on the medial axis ax( S \ ∂B ) , and any (and thus every) point q ∈ π S\ ∂B ( x ) satisfies d ( x, q ) < d ( x, ∂B ) , then x ∈ ax( S ) .

Thus, the medial axis is locally stable if the ambient diffeomorphism is close to the identity. 2

A recurring strategy in this article is to start at a point p on the set S , move away from this point in a 'normal' direction, and see if by projecting using the closest point projection π S we get back to p . To this end, we define the projection range .

Definition 2.2 (Projection range) Let p ∈ S be a point and v ∈ R d a vector. The projection range d ( p, v, π S ) in direction v is the maximal distance one can travel from p along v such that the closest point projection yields only the point p :

Since π S ( p ) = { p } , the projection range is canonically non-negative. Furthermore, the directions for which the range is positive are key to our study, because of the following property:

Lemma 2.3 (Theorem 4.8 (6) of [19]) Consider a point p ∈ S and a vector v ∈ R d . If

We call these special directions v back projection vectors :

Definition 2.4 (Unit back projection vectors) For a point p ∈ S , UBP( p, S ) is the set of unit vectors with a positive projection range:

We further define

Thanks to Lemma 2.3, the following map is well-defined: 134

The generalized tangent and normal space Back projection vectors are intricately related to the generalized tangent and normal spaces.

Definition 2.5 (Definitions 4.3 and 4.4 of [19]) Let p ∈ S . The generalized tangent space Tan( p, S ) is the set of vectors u ∈ R d , such that either u = 0 or, for every ε > 0 there exists a point q ∈ S with

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The generalized normal space Nor( p, S ) consists of vectors v ∈ R d such that ⟨ v, u ⟩ ≤ 0 for all 140 u ∈ Tan( p, S ) . Vectors contained in the generalized tangent, resp. normal, space are called tangent , 141 resp. normal , to S at p . 142

The generalized tangent and normal spaces are illustrated in Figure 3. 143

p4 + Nor(p4,S)

: p3 + Nor(p3, S)

S

P4 + Tan(p4, S)

P4

P3

p3 + Tan(p3,S)

R2

Tan(p2,S) = IR2

Nor(p2,S) = {0}

P1

p1 + Tan(p1,S)

p1 + Nor(p1,S)

Stability of the reach under ambient diffeomorphisms Our last ingredient is the following result by Federer.

Theorem 2.6 (Stability of the reach under ambient diffeomorphisms, Theorem 4.19 of [19]) Pick two constants 0 < t < rch( S ) and s > 0 . If the map

is injective and continuously differentiable, and the maps F , F -1 , and DF are Lipschitz continuous with Lipschitz constants Lip( F ) , Lip( F -1 ) , Lip( DF ) , respectively, then the reach rch( F ( S )) of the image of the set S under the map F is lower-bounded by

3 Stability of the medial axis under ambient diffeomorphisms

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In this section we present the main result of this paper, Theorem 3.9. This theorem extends earlier 152 work by Chazal and Soufflet [13]. Its proof relies on Federer's result on the stability of the reach, 153 Theorem 2.6. To give a more geometrical interpretation we introduce the concept of a weakly tangent 154 sphere and ball, and a maximal empty weakly tangent ball. 155

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Definition 3.1 (Weakly tangent sphere and ball) Let p ∈ S . A sphere is called weakly tangent to S at p if it contains the point p and its centre lies in the (translated) generalized normal space Nor( p, S ) + p . In other words, spheres weakly tangent to S at p are spheres with centres p + v and radii | v | , for a vector v ∈ Nor( p, S ) .

A ball is called weakly tangent to S at p if its boundary sphere is weakly tangent to S at p .

Remark 3.2 Using the definition of Nor( p, S ) , a weakly tangent ball can also be defined as follows: A ball B ( c, r ) is weakly tangent at p if and only if its centre c and radius r satisfy

̸

We remark:

Lemma 3.3 Let p ∈ S and v ∈ R d , and suppose that for some λ > 0 we have π S ( p + λv ) = { p } . Then, for all λ ′ ≥ λ , we have π S ( p + λ ′ v ) = { p } and for all λ ′ > λ , that p / ∈ π S ( p + λ ′ v ) .

S

P2

P1

ax(S)

Lemma 3.3 essentially tells us that a family of weakly tangent balls { B ( p + λv, λ | v | ) } λ ≥ 0 contains at most one which is maximal with respect to inclusion among those whose interior is disjoint from the set S . Two such families are illustrated in Figure 4.

We call such balls maximal empty . For the purpose of this article, we define maximal empty balls in terms of unit back projection vectors (Definition 2.4). To see that each maximal empty ball is indeed weakly tangent, we emphasise:

Lemma 3.4 If ( p, v ) ∈ BP( S ) , then ( p, v ) ∈ Nor( S ) . That is, BP( S ) ⊆ Nor( S ) . In particular, for any pair ( p, u ) ∈ UBP( S ) and radius λ ≥ 0 , the ball B ( p + λu, λ ) is weakly tangent to S .

Remark 3.5 For general closed sets, the converse of Lemma 3.4, that is, Nor( S ) ⊆ BP( S ) , is not true. One counter-example is the graph of the function x ↦→ | x | 3 / 2 at the origin. However, the inclusion Nor( S ) ⊆ BP( S ) holds for sets of positive reach, thanks to Theorem 4.8 (12) of [19] (recalled in the supplementary material).

Definition 3.6 (Maximal empty weakly tangent ball) Let ( p, u ) ∈ UBP( S ) . A weakly tangent ball B ( p + λu, λ ) is called maximal empty to S if λ = d ( p, u, π S ) , or, equivalently, if π ax , S ( p, u ) = p + λu .

(Maximal empty) weakly tangent balls satisfy the following properties. Let ( p, u ) ∈ UBP( S ) .

For any radius 0 < λ ≤ d ( p, u, π S ) , the interior of the ball B ( p + λu, λ ) is disjoint from the set S . This follows directly from Definition 3.6 and Lemma 3.3.

The centres of maximal empty weakly tangent balls lie on the closure of the medial axis of S . This is due to Lemma 2.3 and the definition of the map π ax , S (equation (1)).

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Assumption 3.8

We assume that the set S has a bounding sphere of radius r , which we denote by S ( r ) .

We consider a C 1 diffeomorphism F : R d → R d such that the Lipschitz constants of F and F -1 are bounded by L F , and the Lipschitz constants of the differentials DF and DF -1 are bounded by L DF . We call such a diffeomorphism a C 1 , 1 diffeomorphism.

We further assume that the map F leaves the bounding sphere S ( r ) invariant, that is, F ( S ( r )) = S ( r ) .

We pick a point c ∈ ax( S ) , a point p ∈ π S ( c ) , and write ρ = | c -p | . Observe that since S ∩ ax( S ) = ∅ , ρ is positive. By Lemma 3.7, the ball B ( c, ρ ) is a maximal empty weakly tangent ball to S at p . Moreover, we define u = c -p | c -p | and note that u ∈ UBP( p, S ) .

We denote the tangent hyperplane to the boundary sphere of B ( c, ρ ) at p by p + T . The hyperplane T is the orthocomplement of the vector u : T = u ⊥ .

We work with the unit vector at F ( p ) that points inside the image of the ball B ( c, ρ ) and is orthogonal to the hyperplane D p F ( T ) . We denote this vector by u ′ .

Theorem 3.9 Under the above assumptions, there exists a maximal empty weakly tangent ball B ( c ′ , ρ ′ ) to the set F ( S ) , whose boundary sphere has an internal normal u ′ . In particular, the ball B ( c ′ , ρ ′ ) is tangent to the affine hyperplane F ( p ) + D p F ( T ) . Its radius ρ ′ is bounded by ρ ′ ∈ [ ρ ( L F ) 3 + ρL DF ( L F ) 2 , ( L F ) 3 ρ 1 -ρL DF ( L F ) 2 ] . Assume, moreover, that the distortions of both F and DF are bounded, that is, for all x ∈ R d ,

then the map π ax , S satisfies 213

The following lemma moreover tells us, that each point on the medial axis is a centre of a maximal empty weakly tangent ball.

Lemma 3.7 (Surjectivity on ax( S ) ) For any point x ∈ ax( S ) and p ∈ π S ( x ) , there exists a vector u ∈ UBP( p, S ) such that π ax , S ( p, u ) = x . In other words, B ( x, | x -p | ) is a maximally empty weakly tangent ball. Moreover, we have that

We are now almost ready to state our main theorem. Before phrasing the result, we walk the reader through the assumptions and fix the notation on the way. The assumptions are illustrated in Figure 5.

S(r)

F(S(r) = S(r)

S\S(r)

F(S\S(r))

B(c, p)

F(B(c,p))

ax(SS(r)

ax(F(S(S(r)))

u

p

()

:p+T

F(p) + DpF(T)

Thus, the Hausdorff distance between the medial axes of S and its image F ( S ) is bounded by 214

The bound | F ( x ) -x | ≤ ε 1 is really necessary, because we want our theorem to accommodate for rotations and translations, which rotate and translate the medial axis without changing distances and hence have Lipschitz constant 1. We further stress that if the diffeomorphism F is close to the identity, its Lipschitz constant satisfies L F ≥ 1 , because by assumption F leaves the bounding sphere S ( r ) invariant, and L DF is close to zero.

Sketch of the proof of Theorem 3.9 The idea of the proof is depicted in Figure 5. Thanks to Federer's result (Theorem 2.6), we know that the reach of the maximal empty weakly tangent ball B ( c, ρ ) does not change too much under the ambient diffeomorphism F . This gives a lower bound on the radius of every maximal empty weakly tangent ball of the image of this ball - the set F ( B ( c, ρ )) . We show that in the interior of F ( B ( c, ρ )) , the radii of the maximal empty weakly tangent balls of F ( B ( c, ρ )) are close to ρ . One of these balls is also empty weakly tangent to F ( S ) at F ( p ) , though not necessarily maximal. We denote its centre by c ′ . Since we can apply the same argument for the map F -1 , we find an upper and lower bound on the radius of the maximal weakly tangent ball B ( c ′ , ρ ′ ) of F ( S ) at F ( p ) that is also weakly tangent to F ( B ( c, ρ )) , or equivalently tangent to D p F ( T ) .

While this bound on the difference of the radii is essentially a bound on the distance || c -p | - | c ′ -F ( p ) || between the points c -p and c ′ -F ( p ) , the bound ε 2 on ∥ DF -1 ∥ allows one to bound the angle between the vectors c -p and c ′ -F ( p ) . With the assumption (2) we can then derive a bound the distance between the points c and c ′ . Finally, thanks to [19, Theorem 4.8 (6)] (Lemma 2.3) this induces a bound on the Hausdorff distance between the (closure of the) two medial axes ax( S ) and ax( F ( S )) . □

It was a surprise to the authors that no assumption on the set (apart from closedness) needed to be made, and that the techniques used were that simple and well established; they go back to Federer [19]. In fact, the authors at first envisioned a far more elaborate argument assuming the set had positive µ -reach [11].

4 Quantifying C 1 , 1 diffeomorphisms as deviations from identity

In this section we reformulate the main result in terms of norms on Banach spaces. This reformulation offers a more theoretical insight, and we believe the reformulated bounds are easier to work with in certain applications. Indeed, in the context of practical numerical computations, a bound on the Lipschitz constant of an operator - or, at least, a modulus of continuity - allows to control the condition number. This control is particularly useful when we calculate with objects such as the medial axis,whose (numerical) stability is often problematic in practice.

As we will see below, for this reformulation we somewhat strengthen our assumptions.

We decompose a diffeomorphism F into the identity map 1 R d on R d , and a displacement field φ : F = 1 R d + φ . For the choice of the displacement field, we restrict ourselves to the vector space U of all C 1 , 1 maps φ from R d to R d whose restriction to the exterior R d \ B ( r ) of a certain bounding ball B ( r ) equals 0 . 3

A natural norm associated to U is one that makes it a Banach space. A typical choice, inherited from general Banach spaces of C 1 , 1 functions, would be for example, for φ ∈ U ,

Here we used the following notation:

∥ φ ∥ ∞ = sup x ∈ R d | φ ( x ) | denotes the sup norm on x ↦→| φ ( x ) | , where | · | is the Euclidean norm in R d ,

∥ Dφ ∥ ∞ = sup x ∈ R d ∥ Dφ ( x ) ∥ denotes the sup norm on x ↦→∥ Dφ ( x ) ∥ , where ∥ Dφ ( x ) ∥ is the operator norm induced by the Euclidean norm on R d .

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· We write Lip( Dφ ) for the Lipschitz semi-norm of Dφ . The Lipschitz semi-norms of φ and Dφ are defined as

̸

̸

The norm defined in (4) makes U into a Banach space, since every Cauchy sequence in U has a limit in U . In addition, any function φ ∈ U satisfies:

since the restriction of φ to R d \ B ( r ) is 0 . This in turn yields that Lip( Dφ ) ≤ ∥ φ ∥ C 1 , 1 ≤ max(1 , r, r 2 ) Lip( Dφ ) . Thus, in U , the norm φ ↦→ Lip( Dφ ) is equivalent to the norm φ ↦→∥ φ ∥ C 1 , 1 .

We can now state slightly less general version of Theorem 3.9 in terms of the Banach space ( U , φ ↦→ Lip( Dφ ) ) .

Theorem 4.1 Let S ⊆ R d be bounded by the ball B ( r ) of radius r > 0 , such that S ( r ) = ∂B ( r ) ⊆ S . Let further F be a C 1 , 1 diffeomorphism from R d to itself that leaves the set R d \ B ( r ) invariant, and define two displacement fields φ, ˜ φ ∈ U such that F = 1 R d + φ and

Define ε = max(Lip( Dφ ) , Lip( D ˜ φ )) .

If rε ≤ 1 / 4 , the Hausdorff distance between the medial axes of the set S and its image F ( S ) is bounded by d H (ax( S ) , ax( F ( S ))) ≤ ( 1 + √ 50 ) r 2 ε + O ( r 3 ε 2 ) . In particular, d H (ax( S ) , ax( F ( S )))= O ( r 2 ε ) .

Sketch of the proof Essentially, the proof consists of rewriting Theorem 3.9 in terms of the language developed in this section. □

Remark 4.2 Observe that the bound O ( r 2 ε ) is consistent with a scaling by factor λ : S ↦→ λ S , F ( · ) ↦→ λF ( · /λ ) . Under such a scaling, the radius r is multiplied by λ , while the Lipschitz constant Lip( Dφ ) - and therefore ε - is divided by λ . Furthermore, the Hausdorff distance d H (ax( S ) , ax( F ( S ))) increases by a factor λ . By considering a diffeomorphism that translates the set S \ S ( r ) while keeping the bounding sphere S ( r ) fixed, we see that this bound is asymptotically optimal.

5 Conclusion and future work

We proved the Hausdorff stability of the medial axis of a closed set without any further assumption on it (as explained in Remark 2.1, the existence of the bounding sphere serves to formulate the main result in a clean way).

With regard to applications, our result is the first step towards providing a provably correct image recognition in particular in the context of astrophysics. The next step is to produce physics-informed models for the medial axis as occurring in astronomical data.

On the mathematical side, we conclude with a conjecture generalizing our result. We believe that our result generalizes to compact Riemannian manifolds with bounded curvature.

Conjecture 5.1 Let M be a compact Riemannian manifold with bounded sectional curvature 4 and S a closed subset of M . Then the medial axis (also called cut locus [26]) of S in M is Lipschitz stable under diffeomorphisms of M .

and

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Peer reviews:
comments: SUMMARY:
This paper is an extension of a result from Chazal and Soufflet, which states that the Hausdorff distance of a set to its medial axis is lipschitz-bound under ambient deformations. The authors extend the result from C2 sets and C2 deformations to arbitrary closed sets and C1,1 diffeomorphisms which preserve a bounded sphere in the set.

STRENGTHS:
Medial axis play a central role in vision, and 3D geometry and investigating may lead to novel approaches and algorithms.

WEAKNESSES:
Unfortunately, I do not believe it is within my capacity to evaluate the full correctness of the theorems presented in this paper, as such I feel uncomfortable to recommend its acceptance. I am a first-time reviewer hence will reassess my thoughts given other reviewers' input, but in a sense, placing the proofs in the supplementary is a somewhat odd choice to me, as the paper is a completely theoretical paper with its crux being the proofs themselves. Additionally, I find the theorem somewhat esoteric, being both a slight extension of Chazal and Soufflet, and requiring the ambient deformation preserve a bounded sphere to only yield a lipschitz bound. Given the above, I find this paper more suiting to a computational geometry/mathematical journal than NeurIPS.

QUESTIONS:
Can you describe some applied experiment (in, e.g., learning, vision, geometry) you would perform to show the argued practical impact of this result?

LIMITATIONS:
Yes
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-05

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe medial axis of a closed set $\\mathcal{S} \\subset \\mathbb{R}^d$ is defined to be the set of points in $\\mathbb{R}^d$ which do not have a unique closest point on $\\mathcal{S}.$ The authors develop a notion of stability for such sets with respect to ambient diffeomorphisms of $\\mathbb{R}^d.$ The main result proves stability with respect to $C^{1,1}$ diffeomorphisms under additional assumptions about the set $\\mathcal{S}$ (for instance, that $\\mathcal{S}$ is bounded, See Assumption 3.8 for a full list.) This result is considered a generalization of an earlier result, which makes stronger smoothness assumptions (namely $C^2$) on both the set $\\mathcal{S}$ and the class of ambient diffeomorphisms. The authors argue in the early parts of the paper that this extra generality is needed for (unspecified) applications in astrophysics.\n\nSTRENGTHS:\nThe expository style in the early parts of the paper is inviting, where it does a good job of illustrating some basic notions, including familiar ones such as the medial axis and less-familiar ones like the generalized tangent space.\n\nWEAKNESSES:\nThere are a few different criticisms one can make of this paper:\n\n  1. The topic is niche for a NeuRIPS audience.\n  2. The main result is technical and difficult for non-experts to verify.\n  3. The main result, as described in the introduction, is a marginal improvement over the current state of the art in reference [13], in the sense that one gains only less restrictive assumptions about the regularity of the functions and shape that appear.\n  4. The connection to applications is tenuous at best, and no experiments are provided.\n\nWith regards to 1 and 2 above, let me draw attention to the statement of the main result in Theorem 3.9 and the preceding assumptions it requires, which take nearly a page to write down even with many prerequisite definitions that appear before it. One would at least hope based on the promises of the introduction that a simple definition of \"stability\" would be available for use in the statement of the theorem. With regards to 3 and 4, there is very little given to convince the reader that the $C^2$ results are insufficient for applications. \n\nI would not necessarily suggest that a paper with one or more of these deficiencies be excluded from NeuRIPS. However, given that all four issues are present, it seems better to focus on resolving some of them, or to send the paper to another venue (eg. a pure mathematics journal) where some of these criteria are judged to be less important.\n\nQUESTIONS:\nIt would be very helpful for readers to point out where to find your main result early on in the introduction. That would be Theorem 3.9, right?\n\nlines 112: I don't understand why the assumption $\\mathcal{S}$ and its medial axis being bounded is not a further restriction needed to state your result. There is nothing in Remark 2.1 that addresses the case of an unbounded set $\\mathcal{S},$ nor anywhere else in the paper. Moreover, these assumptions do appear in the statement before Theorem 3.9. Thus, it seems incorrect for you to state your theorem in the introduction simply for \"closed sets\" without further qualification.\n\nLIMITATIONS:\nn/a", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 1, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

comments: SUMMARY:
The authors prove that the medial axis of closed set is Hausdorff stable without any further assumption on it. In this proof, the authors achieve stability without pruning the medial axis which is a significant advantage. Meanwhile, the results hold for sets in arbitrary dimensions.

STRENGTHS:
In originality, this work holds for sets in arbitrary dimentions and removes the limitation of manifold assumption when in proof, and it does not need to prune the medial axis which is a significant advantage.

The quality and clarity is good enough, it is easy to understand the motivation, outline and contribution.

The proof in this paper implies that the medial axis of an imprecise shape is stable. The medial axis plays an important role in the field of computational geometry, computer vision and graphics.

WEAKNESSES:
The result of this work shows the numerical stability of medial axis, but there is little analysis about the impact from noise size and quantity in real world data.

QUESTIONS:
1. In real world data, the noise quantity and size maybe different, does the result in this work mean that the stability of medial axis of different noisy data is always guaranteed?

2. What does the meaning of `considered set` in line 37 and 112? It seems that the word `considered` is not necessary there.

3. There are some standard examples of the instability of the medial axis mentioned in line 38. Could you give more explanation why these kinds of instability exist? Is the instability essential or numerical?

LIMITATIONS:
From line 36 to 39, the authors give the limitation about the work, but it could be more clear if there is more explanation. I also asked the questions about it in section Question.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-06

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work proves that the medial axis of closed sets is Hausdorff stable, this extends existing stability result on the stability of the medial axis of C^2 manifolds under C2 ambient diffeomorphisms. The contributions are\n\n1. This work makes no assumptions of the set except the closedness. The stability of the medial axis of smooth manifolds has been intensively studied in the literature, this work omits the manifold assumption.\n\n2. The stability is achieved without pruning the medial axis. Large body of works have to prune the medial axis.\n\n3. The stability results hold for sets in arbitrary dimensions and are insensitive to the dimension of the set itself.\n\nThis theoretical result plays a fundamental role in many fields, the generalization is important  to many practical applications.\n\nSTRENGTHS:\nThe theoretic results are much general, it doesn't require the manifold assumption, it doesn't need to prune the medial axis, the results hold for any dimensions. \n\nThe work is clearly represented. All the key concepts are explained thoroughly, the lemmas, theorems, corollaries are explained in detail, and rigorously formulated. The proofs are step by step, clean and easy to follow.\n\nWEAKNESSES:\nThe theoretical results are elegant and convincing. It will be helpful to give some numerical experimental results.\n\nQUESTIONS:\n1. The stability result is established with respect to smooth diffemorphisms of the ambient space, is the bounding sphere preserving condition intrinsically essential or technically necessary ? To what extent, one can remove this restriction ?\n\n2. Suppose the set S is a C2 surface, if S is deformed to generate a curvature singularity, the surface becomes C1 at the singularity, the medial axis may change drastically. Please explain why the ambient diffeomorphism C^{1,1} can avoid this situation.\n\n3. For the conjecture, the cut locus is closely related to the sign of the Gaussian curvature on the surface. Small perturbation changing the sign of the curvature may generate conjugate points suddenly. From this point of view, it seems the stability of cut locus may be hard to achieve. Please explain your insights for the conjecture.\n\nLIMITATIONS:\nThe current stability result assumes the diffeomorphisms is a small perturbation of the identity, and it preserves the bounding sphere. This constraint seems to be artificial and inconvenient for practical applications. Maybe this requirement can be weakened or the bounding sphere is pushed to infinity.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-07"}

comments: SUMMARY:
This paper proves the Hausdorff stability of the medial axis of closed bounded sets. This is a mathematics paper. The authors set up a foundation of their problem, then applied Theorem 2.6 (from [19]) to complete their proof. The end result is quite beautiful in fact. The authors also show that the results in [13] is a special case of their result.

STRENGTHS:
- The paper is written well. Despite not having a mathematics background, I am able to read and understand the majority part of the proof. (Nitpick: there are small typos, for example, some \pi_{S}(p_4) are annotated incorrectly in Figure 1.)
- The authors proved a difficult result (as an indication, [13] is a special case of the result). The proof seems to be correct to me.

WEAKNESSES:
- I have a hard time understanding how this result can be used in machine learning / computer vision / computational geometry applications. Yet the motivation is explained in ln45 - ln73. However, I still do not see how this result can be applied. For the benefit of the readers, I think applications need to be demonstrated, otherwise Neurips might not be the right audience.

QUESTIONS:
I'd like to understand how this result can be applied in applications that are of interest to Neurips audience.

LIMITATIONS:
N/A
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-26

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors analyze the stability of the medial axis of a set S, when S is perturbed by a map that is lipschitz with lipschitz derivatives.\n\nThis stability result is of interest in numerous applications in machine learning, such as astrophysics.\n\nThe author's results improve upon an existing result by Chazal and Soufflet in a few ways:\n1. The authors remove an assumption that the set S must be a piecewise smooth manifold; here they only require S to be closed and bounded.\n2. They do not require pruning the medial axis.\n3. Their result holds in high dimensions.\n\nSTRENGTHS:\nI think the result is significant, and of interest to the neurips community. Compared to Chazal and Soufflet, I think another significant aspect of this result is that this result is quantitative whereas Chazal and Soufflet's result is only qualitative.\n\nWEAKNESSES:\nI have some questions about how this paper's results compare to existing results, as well as about several aspects of the result (see below). These may not be considered weaknesses if the authors can address them.\n\nQUESTIONS:\nI have the following questions regarding this result:\n\n1) How significant is removing the manifold assumption? Even if a set is not smooth, can it not be made smooth via some infinitesimally small perturbation (e.g. a gaussian convolution)? Can the authors elaborate on the strength of their result in the context of an application? E.g. for the astrophysics image example, can we not simply smooth the image via a infinitesimal smoothing operation, and then apply Chazal and Soufflet?\n\n2) Is there any existing quantitative bounds that the authors can compare to (even if assumptions differ)? If there are, how does the rate in Theorem 3.9 (line 212) compare to existing rates?\n\n3) To double check, on line 212, as L_F -> 1, L_{DF}-> 0, epsilon_1 and epsilon_2 -> 0, C_L(r, L_F, L_{DF},eps_1,eps_2) -> 0, is that correct?\n\n4) Can the authors give an interpretation of rch(S) defined on line 99? In particular, for a non-smooth S, can rch(S) be 0?\n\n5) if rch(S)=0, the result due to Federer in Theorme 2.6 becomes vacuous. However, the result in Theorem 3.9 does not seem to depend on rch(S) at all, even though it crucially uses Federer's result. Can the authors explain why this is?\n\nLIMITATIONS:\nn/a", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-31"}