Stability of boundary measures
aa r X i v : . [ c s . C G ] J un a p p o r t (cid:13)(cid:13) d e r e c h e r c h e (cid:13) I SS N - I S RN I NR I A / RR -- -- F R + E N G Thème SYM
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Stability of boundary measures
Frédéric Chazal — David Cohen-Steiner — Quentin Mérigot
N° 6219
14 June 2007 nité de recherche INRIA Sophia Antipolis2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)
Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65
Stability of boundary measuresFrédéri Chazal , David Cohen-Steiner , Quentin MérigotThème SYM (cid:22) Systèmes symboliquesProjet Geometri aRapport de re her he n° 6219 (cid:22) 14 June 2007 (cid:22) 20 pagesAbstra t: We introdu e the boundary measure at s ale r of a ompa t subset of the n -dimensional Eu lidean spa e. We show how it an be omputed for point louds andsuggest these measures an be used for feature dete tion. The main ontribution of thiswork is the proof a quantitative stability theorem for boundary measures using tools of onvex analysis and geometri measure theory. As a orollary we obtain a stability resultfor Federer's urvature measures of a ompa t, allowing to ompute them from point- loudapproximations of the ompa t.Key-words: dimension dete tion, point louds, urvature measures, onvex fun tions,nearest neighbor.tabilité de mesures de bordRésumé : Nous introduisons la notion de mesure de bord d'é helle r d'un sous-ensemble ompa t de l'espa e eu lidien de dimension n . Nous montrons omment al uler es mesurespour un nuage de points et suggérons que es mesures peuvent être utilisées pour de ladéte tion de features. La prin ipale ontribution de e travail est la démonstration d'unthéorème quantitatif de stabilité des mesures de bord, utilisant des outils de l'analyse onvexeet de la théorie géométrique de la mesure. En orollaire, wous obtenons un résultat destabilité des mesures de ourbure d'un ompa t (notion introduite par Federer), permettantde les al uler à partir d'approximations du ompa t par des nuages de points.Mots- lés : déte tion de dimension, nuages de points, mesures de ourbure, fon tions onvexes, plus pro he voisin.oundary measures 3Introdu tionMotivations and previous work. The main goal of our work is to develop a frameworkfor features dete tion: (cid:28)nding the boundaries, sharp edges, orners of a ompa t set K ⊆ R n knowing only a possibly noisy point loud sample of it.This problem has been an area of a tive resear h in omputer s ien e for some years.Many of the urrently used methods for feature and dimension dete tion (see [DGGZ03℄and the referen es therein) rely on the omputation of a Voronoï diagram. The ost ofthis omputation is exponential in the dimension and annot be pra ti ally realized foran ambient dimension mu h greater than three. In low dimension, several methods havebeen invented for boundary dete tion (mostly to dete t holes), for example [FK06℄ (2D,graph-based), [BSK℄ (3D), and [RBBK06℄. Sharp edges dete tion has also been studied in[GWM01℄, and re ently in [DHOS07℄.The algorithms we develop have three main advantages: they are built on a strong math-emati al theory, are robust to noise and their ost depend only on the intrinsi dimensionof the sampled ompa t set. None of the existing methods for feature dete tion share thesethree desirable properties at the same time.Boundary measures and their stability. Given a s ale parameter r , we asso iate toea h ompa t subset K of R n a probability measure β K,r . This boundary measure of K ats ale r as we all it, gives for every Borel set A ⊆ R n the probability that the proje tion on K of a random point at distan e at most r of K lies in A (the proje tion on K , denoted by p K , maps almost any point in R n to its losest point in K ).Intuitively, the measure β K,r will be more on entrated on the features of K : for instan e,if K is a onvex polyhedron in R , β K,r will harge the edges more than the fa es, and theverti es even more (see example I). It should also be noti ed that this measure is loselyrelated to Federer's urvature measures (introdu ed in [Fed59℄).This arti le fo uses on the stability properties of the boundary measures, showing thatthey an be approximated from a noisy sample of K . The problem of extra ting geometri information from these boundary measures will be treated in an up oming work. The mainstability theorem an be stated as follow:Theorem (IV.1). If one endows the set of ompa t subsets of R n with the Hausdor(cid:27) dis-tan e, and the set of ompa tly supported probability measures on R n with the Wassersteindistan e, the map K β K,r is lo ally / -Hölder.In the sequel we will make this statement more pre ise by giving expli it onstants. Avery similar stability result for a generalization of Federer's urvature measures is dedu edfrom this theorem. We dedu e theorem IV.1 from the two theorems III.5 and II.3 below,whi h are also interesting in their own.Theorem (III.5). Let E be an open subset of R n with ( n − (cid:21)re ti(cid:28)able boundary, and f, g be two onvex fun tions su h that diam( ∇ f ( E ) ∪ ∇ g ( E )) k . Then there exists a onstant C ( n, E, k ) depending only on n and E su h that for k f − g k ∞ small enough, k∇ f − ∇ g k L ( E ) C ( n, E, k ) k f − g k / ∞ RR n° 6219 Chazal, Cohen-Steiner & MérigotTheorem (II.3). If K is a ompa t set of R n , for every positive r , ∂K r = { x ; d( x, K ) = r } is ( n − (cid:21)re ti(cid:28)able and H n − ( ∂K r ) N ( ∂K, r ) × ω n − (2 r ) Theorem III.5 is used to show that the map K p K ∈ L ( E ) (where p K is the proje tionon K ) is lo ally / -Hölder, whi h is the main ingredient for the stability result. TheoremII.3 improves upon [OP85℄, in whi h Oleksiv and Pesin prove the (cid:28)niteness of the measureof the level sets of the distan e fun tion to K . It is used here as a tool to show that K r ∆ K ′ r is small when K and K ′ are lose ( A ∆ B being the symmetri di(cid:27)eren e between A and B ,and K r being the set of points at distan e at most r from K ).Outline. In the (cid:28)rst se tion we give some examples of boundary measures and show howthey an be omputed e(cid:30) iently for point louds. The se ond and third se tions ontain theproofs of theorems II.3 and III.5 respe tively. In the fourth se tion we dedu e from thesetheorems the stability results for boundary and urvature measures.I De(cid:28)nition of boundary measuresSome examples of boundary measuresNotations. If K is a ompa t subset of R n , the distan e to K is de(cid:28)ned as d K ( x ) =min y ∈ K k x − y k . The r -tubular neighborhood or r -o(cid:27)set around a subset F ⊆ R n is theset of points at distan e at most r from F , and is denoted by F r .For x ∈ R n , the set of points y ∈ K that realizes this minimum is denoted by proj K ( x ) .One an show that K ( x ) = 1 i(cid:27) d K is di(cid:27)erentiable at x . Sin e d K is -Lips hitz, atheorem of Radema her ensures that both onditions are true for almost every point x ∈ R n .This allows us to de(cid:28)ne a fun tion p K ∈ L ( R n ) , alled the proje tion on K , whi h maps(almost) every point x ∈ R n to its only losest point in K . The s -dimensional Hausdor(cid:27)measure is denoted by H s ; in parti ular H n oin ides with the usual Lebesgue measure on R n .De(cid:28)nition I.1. The r -s ale boundary measure β K,r of a ompa t K of R n asso iates toany Borel set A ⊆ R n the probability that the proje tion of a random point at distan e lessthan r of K lies in A .If we denote by µ K,r the pushforward of the uniform measure on K r by the proje tion on K , ie. for all Borel set A ⊆ R n , µ K,r ( A ) = H n ( p − K ( A ) ∩ K r ) , then β K,r = H n ( K r ) − µ K,r .Examples. 1. If C = { x i ; 1 i N } is a (cid:19)point loud(cid:20), that is a (cid:28)nite set of points of R n , then β C,r is a sum of weighted Dira measures. Indeed, if
Vor C ( x i ) denotes theVoronoi ell of x i , that is the set of points loser to x i than to any other point of C ,we have µ C,r = n X i =1 H n (Vor C ( x i ) ∩ C r ) δ x i INRIAoundary measures 52. Let S be a unit-length segment in the plane with endpoints a and b . The set S r is theunion of a re tangle of dimension × r whose points proje ts on the segment and twohalf-disks of radius r whose points are proje ted on a and b . It follows that µ S,r = 2 r H (cid:12)(cid:12) S + π r δ a + π r δ b
3. Let P be a onvex solid polyhedron of R , { e j } be its edges and { v k } be its verti es.We denote by a ( e j ) the angle between the normals of the two fa es ontaining e i , andby K ( v k ) the solid angle formed by the normal one at v k . Then one an see that µ P,r = H (cid:12)(cid:12) P + r H (cid:12)(cid:12) ∂P + X j r a ( e j ) × H (cid:12)(cid:12) e j + X k r K ( v k ) δ v k
4. More generally, if K is a ompa t with positive rea h, in the sense that there existsa positive r su h that the proje tion on K is unique for any point in K r , there existBorel measures (Φ K,i ) i n on R n su h that µ K,r = n X i =0 r n − i ω n − i Φ K,i where ω i is the volume of the unit sphere in R i +1 . These measures Φ K,i are alled the urvature measures of the ompa t set K and have been introdu ed under this formby Federer in [Fed59℄, generalizing existing notions in the ase of onvex subsets and ompa t smooth submanifolds of R n (Minkowski's Quermassintegral and Weyl's tubeformula, f. [Wey39℄).The se ond and third examples show exa tly the kind of behaviour we want to exhibit(and so does (cid:28)gure I.1): the measure β K,r an be written as a sum of weighted Hausdor(cid:27)measures of various dimension, on entrated on the features of K : its boundary, its edgesand its orners. This remark together with the stability theorem for boundary measuresshows that they are a suitable tool to be used in robust feature extra tion algorithms. Inthe next paragraph we show how to ompute them e(cid:30) iently for point louds.The boundary measure of a point loudA fast method for omputing the boundary measures of point louds is of ru ial importan efor pra ti al appli ations. Indeed, most real-world data, either D (laser s ans) or higherdimensional is given in the form of an unstru tured point loud. Sin e omputing the Voronoïdiagram of a point loud has an exponential ost in the ambient dimension, we will be usinga probabilisti Monte-Carlo method to get an approximation of the boundary measures. Ina very general way, if µ is an absolutely ontinuous measure on R n , one an ompute p C µ as shown below. The three main steps of this algorithm (I, II, and III) are des ribed withmore detail in the following paragraphs.RR n° 6219 Chazal, Cohen-Steiner & MérigotInput: a point loud C = { x i } , a measure µ Output: an approximation of p C µ in the form P k ( i ) δ X i [I.℄ Choose N big enough to get a good approximation with high on(cid:28)den ewhile n N do[II.℄ Choose a random point X n with probability distribution µ [III.℄ Finds its losest point x i in the loud C , add to n ( x i ) end whilereturn [ n ( x i ) /N ] i .Step I. The measure µ N = 1 /N P i N δ X i where ( X i ) is a sequen e of independentrandom variables whose law are µ is alled an empiri al measure. The question of whether(and at what speed) µ N onverge to µ as N grows to in(cid:28)nity is well-known to probabilistsand statisti ians. The results of this se tion are not original and an probably be improved,they are presented here only to give proof-of- on ept bounds for N .Theorem I.2 (Hoe(cid:27)ding's inequality). If ( Y i ) is a sequen e of independent [0 , -valuedrandom variables whose ommon law ν has a mean m ∈ R , and Y N = (1 /N ) P i N Y i then P ( (cid:12)(cid:12) Y N − m (cid:12)(cid:12) > ε ) − N ε ) In parti ular, let's onsider a family ( X i ) of independent random variables distributeda ording to the law p C µ . Then, for any -Lips hitz fun tion f : R n → R with k f k ∞ ,one an apply Hoe(cid:27)ding's inequality to the family of random variables Y i = f ( X i ) : P "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X i =1 f ( X i ) − Z f d µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > ε − N ε ) This kind of estimate also follows from Talagrand T ( λ ) -inequalities, in whi h ase thefa tor in the exponential is repla ed by λ . Bolley, Guillin and Villani use this fa t to getquantitative on entration inequalities for empiri al measures with non- ompa t support in[BGV07℄.We now let BL ( C ) be set of Lips hitz fun tions f on C whose Lips hitz onstant Lip f is at most and k f k ∞ . We let N (BL ( C ) , k . k ∞ , ε ) be the minimum number of ballsof radius at most r (with respe t to the k . k ∞ norm) needed to over BL ( C ) . PropositionI.3 gives a bound for this number. It follows from the de(cid:28)nition of the bounded-Lips hitzdistan e (see I.4) and from the union bound that P [ d bL (p C µ N , p C µ ) > ε ] N (BL ( C ) , k . k ∞ , ε/
4) exp( − N ε / Proposition I.3. For any ompa t metri spa e K , N (BL ( K ) , k . k ∞ , ε ) (cid:18) ε (cid:19) N ( K,ε/ INRIAoundary measures 7Proof. Let X = { x i } be an ε/ -dense family of points of K with X = N ( K, ε/ . It iseasily seen that for every -Lips hitz fun tions f, g on K , k f − g k ∞ k ( f − g ) | X k ∞ + ε/ .Then, one on ludes using that N (BL ( X ) , k . k ∞ , ε/ (4 /ε ) X .In (cid:28)ne one gets the following estimate on the bounded-Lips hitz distan e between theempiri al and the real measure: P [ d bL (p C µ N , p C µ ) > ε ] (cid:0) ln(16 /ε ) N ( C, ε/ − N ε / (cid:1) Sin e C is a point loud, the oarsest possible bound on N ( C, ε/ , namely C , showsthat omputing an ε -approximation of the measure p µ with high on(cid:28)den e (eg. ) anbe done with N = O ( C ln(1 /ε ) /ε ) .Step II. To simulate the uniform measure on K r one annot simply shoot points in abounding box of K r , keeping those that are a tually in K r sin e this has an exponential ostin the ambient dimension. Lu kily there is a simple algorithm to generate points a ordingto this law whi h relies on pi king a random point x i in the loud C and then a point X in B ( x i , r ) (cid:22) taking into a ount the overlap of the balls B ( x, r ) where x ∈ C :Input: a point loud C = { x i } , a s alar r Output: a random point in C r whose law is H n | K r repeatPi k a random point x i in the point loud C Pi k a random point X in the ball B ( x i , r ) Count the number k of points x j ∈ C at distan e at most r from X Pi k a random integer d between and k until d = 1 return X .Step III. The trivial algorithm for omputing the proje tion of a point on a point loudtakes exa tly n steps. Sin e generally N will an order of magnitude greater than n wemight improve the overall O ( n ) ost by maintaining a data stru ture whi h allows fastnearest-neighbour queries. This problem is notoriously di(cid:30) ult and until re ently most ofthe e(cid:30) ient algorithms in high dimension were only able to ompute approximate nearestneighbours. This amounts to repla ing p C by a map ˜p ε with the property that for all x , k ˜p ε ( x ) − p C ( x ) k (1 + ε )d C ( x ) . Unfortunately, the te hniques we develop in this paperdo not seem to apply dire tly to get quantitative loseness estimates for the measures ˜p ε µ and p K µ .It should be noted that for low entropy point louds, nearest neighbor queries an be donemore e(cid:30) iently. For instan e, a re ent arti le by Beygelzimer, Kakade and Langford ( f.[BKL06℄) introdu es a stru ture alled over trees whi h allows an exa t nearest neighbourquery with omplexity O ( c log n ) where c is related to the intrinsi dimension of the point loud, with an initialisation ost of O ( c n log n ) .RR n° 6219 Chazal, Cohen-Steiner & MérigotFigure I.1: Boundary measure for a sampled me hani al part.Wasserstein distan e and stabilitySin e our goal is to give a quantitative stability result for boundary measures, we needto put a metri on the spa e of probability measures on R n . The Wasserstein distan e,related to the Monge-Kantorovi h optimal transportation problem seemed intuitively (andlater happened to really be) appropriate for our purposes. A good referen e on this topi isCédri Villani's book [Vil03℄.De(cid:28)nition I.4. The set of measures (resp. probability measures) on R n is denoted by M ( R n ) (resp. M ( R n ) ). We endow M ( R n ) with the bounded Lips hitz distan e, ie. ∀ µ, ν ∈ M ( R n ) , d bL ( µ, ν ) = sup k ϕ k Lip (cid:12)(cid:12)(cid:12)(cid:12)Z ϕ d µ − Z ϕ d ν (cid:12)(cid:12)(cid:12)(cid:12) where the supremum is taken over all Lips hitz fun tions ϕ with k ϕ k Lip = Lip ϕ + k ϕ k ∞ ( Lip ϕ being the smallest onstant k su h that ϕ is k -Lips hitz).We put two distan es on M ( R n ) (whi h are in fa t identi , see below). The Fortet-Mourier distan e, whi h is almost the same as the bounded Lips hitz one: ∀ µ, ν ∈ M ( R n ) , d FM ( µ, ν ) = sup Lip ϕ (cid:12)(cid:12)(cid:12)(cid:12)Z ϕ d µ − Z ϕ d ν (cid:12)(cid:12)(cid:12)(cid:12) And the Wasserstein distan e: W ( µ, ν ) = inf { E (d( X, Y )) ; law( X ) = µ, law( Y ) = ν } where the in(cid:28)mum is taken over all random variables X and Y whose laws are µ and ν respe tively. INRIAoundary measures 9Notations. If µ and ν ∈ M ( R n ) are absolutely ontinuous with respe t to H n , ie. d µ = ϕ d H n and d ν = ψ d H n we denote by µ ∩ ν the measure de(cid:28)ned by d( µ ∩ ν ) = min( ϕ, ψ )d H n ,and µ ∆ ν = µ + ν − µ ∩ ν .Proposition I.5. If µ ∈ M ( R n ) is absolutely ontinuous with respe t to the Lebesgue mea-sure, and f, g : R n → R n are two fun tions in L ( µ ) , then d bL ( f µ, g µ ) k f − g k L ( µ ) If µ and ν are two absolutely ontinuous measures on R n , d bL ( f µ, g ν ) k f − g k L ( µ ∩ ν ) + mass( µ ∆ ν ) Proof. For any -Lips hitz fun tion ϕ on R n , (cid:12)(cid:12)(cid:12)(cid:12)Z ϕ d f µ − Z ϕ d g µ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z ϕ ◦ f d µ − Z ϕ ◦ g d µ (cid:12)(cid:12)(cid:12)(cid:12) Lip ϕ Z k f − g k d µ k f − g k L ( µ ) For the se ond inequality, let us (cid:28)rst remark that there exists two positive measures µ r and ν r su h that µ = µ ∩ ν + µ r and ν = µ ∩ ν + ν r . Then, d bL ( f µ, g ν ) d bL ( f µ, f µ ∩ ν ) + d bL ( f µ ∩ ν, g µ ∩ ν ) + d bL ( g µ, g µ ∩ ν ) Now let us bound one of the extreme terms of the sum, ∀ ϕ s.t k ϕ k ∞ , (cid:12)(cid:12)(cid:12)(cid:12)Z ϕ d f µ − Z ϕ d f µ ∩ ν (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z ϕ ◦ f d µ r (cid:12)(cid:12)(cid:12)(cid:12) mass( µ r ) One on ludes using that µ r + ν r = µ ∆ ν .Corollary I.6. If K and K ′ are two ompa t subsets of R n , d bL ( µ K,r , µ K ′ ,r ) k p K − p ′ K k L ( K r ∩ K ′ r ) + H n ( K r ∆ K ′ r ) Hen e to get a quantitative ontinuity estimate for the map K µ K,r one needs toshow that if K and K ′ are Hausdor(cid:27)- lose, K r ∆ K ′ r is small, and to evaluate the ontinuitymodulus of K p K ∈ L ( K r ∩ K ′ r ) . This is the purpose of the two following paragraphs.RR n° 62190 Chazal, Cohen-Steiner & MérigotII K r ∆ K ′ r is small when K and K ′ are loseIt is not hard to see that if d H ( K, K ′ ) is smaller than ε , then K r ∆ K ′ r is ontained in ( K r + ε \ K r − ε ) . The volume of this thi k tube around K an then be expressed as anintegral of the area of the hypersurfa es ∂K t .The next proposition gives a bound for the measure of the r -level set ∂K r of a ompa tset K ⊆ R n depending only on its overing number N ( K, r ) (ie. the minimal number of losed balls of radius r needed to over K ). In what follows, K r is the set of points of R n at distan e less than r of K , and ∂K r is the boundary of this set, ie. the r -level set of d K .In this paragraph, we prove the following theorem :Theorem. If K is a ompa t set of R n , for every positive r , ∂K r is H n − (cid:21)re ti(cid:28)able and H n − ( ∂K r ) N ( ∂K, r ) × ω n − (2 r ) This proposition improves over a result of (cid:28)niteness of the level sets of the distan efun tion to a ompa t set, proved by by Oleksiv and Pesin in [OP85℄. We begin by provingit in the spe ial ase of (cid:16) r -(cid:29)owers(cid:17). A r -(cid:29)ower F is the the boundary of the r -tube of a ompa t set ontained in a ball B ( x, r ) , ie. F = ∂K r where K ⊆ B ( x, r ) . The di(cid:27)eren ewith the general ase is that if K ⊆ B ( x, r ) , then K r is a star-shaped set with respe t to x .Thus we an de(cid:28)ne a ray-shooting appli ation s K : S n − → ∂K r whi h maps any v ∈ S n − to the interse tion of the ray emanating from x with dire tion v with ∂K r . x v s K ( v ) Figure II.2: Ray-shooting from the enter of a (cid:29)ower.Lemma II.1. Let K = { e } ⊆ B ( x, r ) and de(cid:28)ne s e as above. Then s e is r -Lips hitz (withrespe t to the sphere's inner metri ) and its Ja obian is at most (2 r ) n − .Proof. Solving the equation k x + tv − e k = r with t > gives s e ( v ) = x + (cid:18)q h v | x − e i + r − k x − e k − h v | x − e i (cid:19) v Denote by H v the orthogonal of the -plane P spanned by v and s e ( v ) − e . For ea hve tor w hosen in H v , a simple al ulation gives: s e ( v + tw ) = s e ( v ) + tw k s e ( v ) − x k + o ( t ) INRIAoundary measures 11Hen e the derivative of s e along H v is simply the multipli ation by k s e ( v ) − x k r .Now, we now onsider the ase of the -plane P . We denote by θ the angle between s e ( v ) − x and s e ( v ) − e and by w a ve tor tangent to v in the interse tion of the sphere with P . Then k (d s e ) v ( w ) kk w k = k s e ( v ) − x k| cos( θ ) | Now let us remark that k s e ( v ) − e k k s e ( v ) − x k | cos( θ ) | = |h s e ( v ) − e | s e ( v ) − x i| = 12 ( k x − s e ( v ) k + k s e ( v ) − e k − k x − e k ) > k x − s e ( v ) k Finally we have proved that k (d s e ) v k r . The result follows by integration.We denote by ω n ( r ) the n -Hausdor(cid:27) measure of the n -sphere of radius r .Corollary II.2. A r -(cid:29)ower in R n is a H n − (cid:21)re ti(cid:28)able set and its measure is at most ω n − (2 r ) .Proof. Let K ⊆ B ( x, r ) be the ompa t set generating the (cid:29)ower ∂K r . As above, for anyve tor v ∈ S n − , we denote by s the interse tion of the ray { x + tv ; t > } with ∂K r . Sin e K r is a star-shaped set around x , s is a bije tion from S n − to ∂K r .Now let ( y k ) be a dense sequen e in K , and denote by s k the proje tion from S n − tothe (cid:29)ower ∂ ( ∪ i k { y i } ) r de(cid:28)ned as above. Then ( s k ) onverges simply to p on S n − . Indeed,if we (cid:28)x v ∈ S n − and ε > , the segment joining x and s ( v ) trun ated at a distan e ε of s ( v ) is a ompa t set ontained in int K r . It is overed by the union ∪ i B ( y i , r ) , so that for N big enough it is also overed by ∪ k N B ( y k , r ) . For those N , k s k ( x ) − s ( x ) k ε .Finally, ∂K r is the image of the sphere by p , whi h is r -Lips hitz as a simple limit of r -Lips hitz fun tions.We now dedu e a general bound on the measure of the tube boundary ∂K r around ageneral ompa t set K by overing it with a family of (cid:29)owers:Theorem II.3. If K is a ompa t set of R n , for every positive r , ∂K r is a H n − -re ti(cid:28)ablesubset of R n and moreover, H n − ( ∂K r ) N ( ∂K, r ) × ω n − (2 r ) Proof. It is easy to see that ∂K r ⊆ ∂ ( ∂K r ) . Thus, if we let ( x i ) be an optimal overingof ∂K by open balls of radius r , and denote by K i the ( ompa t) interse tion of ∂K with B ( x i , r ) , the boundary ∂K r is ontained in the union ∪ i ∂K ri . Hen e its Hausdor(cid:27) measuredoes not ex eed the sum P i H n − ( ∂K ri ) . One on ludes by applying the pre eding lemma.RR n° 62192 Chazal, Cohen-Steiner & MérigotRemark II.4. 1. The bound in the theorem is tight, as one an he k taking K = B (0 , r ) .2. Let us noti e that for some onstant C ( n ) , N ( B (0 , , r ) C ( n ) r − n . From thisand the above bound it follows that H n − ( ∂K r ) (1 + C ( n ) × (diam( K ) /r ) n ) ω n − (2 r ) C ′ ( n ) × (1 + diam( K ) n r ) for some universal onstant C ′ ( n ) depending only on the ambient dimension n . Thislast inequality was the one proved in [OP85℄.To on lude we use a weak formulation of the o-area formula, a standard result ofgeometri measure theory ([DG54℄, [Fed59℄), whi h reads Z R n |∇ x f | d H n ( x ) = Z R H n − ( f − ( y ))d H ( y ) whenever f : R n → R is a Lips hitz map. From this formula and the previous estimationfollows thatCorollary II.5. For any ompa t sets K, K ′ ⊆ R n , with d H ( K, K ′ ) ε , H n ( K r ∆ K ′ r ) Z r + εr − ε H n − ( ∂K t )d t N ( K, r − ε ) ω n − (2 r + 2 ε ) × ε III The map K p K is lo ally / -HölderWe now study the ontinuity modulus of the map K p K ∈ L ( E ) , where E is a suitableopen set. We remind the reader of two well-known fa ts of onvex analysis (see for instan e[Cla83℄):1. If f : Ω ⊆ R n → R is a lo ally onvex fun tion, its subdi(cid:27)erential at a point x ,denoted by ∂ x f is the set of ve tors v of R n su h that for all h ∈ R n small enough, f ( x + h ) > f ( x ) + h h | v i . Then f admits a derivative at x i(cid:27) ∂ x f = { v } is a singleton,in whi h ase ∇ x f = v .2. A lo ally onvex fun tion has a derivative almost everywhere.Lemma III.1. The fun tion v K : R n → R , x
7→ k x k − d K ( x ) is onvex with gradient ∇ v K = 2 p K almost everywhere.Proof. By de(cid:28)nition, v K ( x ) = sup y ∈ K k x k − k x − y k = sup y ∈ K v K,y ( x ) with v K,y ( x ) =2 h x | y i − k y k . Hen e v K is onvex as a supremum of a(cid:30)ne fun tions. Be ause v K,p K ( x ) and v K take the same value at x , ∂ x v K,p K ( x ) = { p K ( x ) } ⊆ ∂v K . Sin e v K is di(cid:27)erentiablealmost everywhere, equality must be true almost everywhere whi h on ludes the proof.INRIAoundary measures 13This lemma shows that k p K − p K ′ k L ( E ) = 1 / k∇ v K − ∇ v K ′ k L ( E ) . Our estimation ofthe ontinuity modulus of the map K p K will follow from a general theorem whi h assertsthat if ϕ and ψ are two uniformly lose onvex fun tions with bounded gradients then ∇ ϕ and ∇ ψ are L - lose. The next proposition below is the -dimensional version of this result,from whi h we then dedu e the general theorem.Proposition III.2. If I is an interval, and ϕ : I → R and ψ : I → R are two onvexfun tions su h that diam( ϕ ′ ( I ) ∪ ψ ′ ( I )) k , then letting δ = k ϕ − ψ k L ∞ ( I ) , Z I | ϕ ′ − ψ ′ | π (length( I ) + k + δ / ) δ / Lemma III.3. Let f : I → R be a nonde reasing fun tion with diam ϕ ( I ) k . Then, if F isthe ompleted graph of f , ie. the set of points ( x, y ) ∈ I × R su h that lim x − ϕ y lim x + ϕ ,then H n ( F r ) π (length( I ) + k + r ) × r .Proof. Let γ : [0 , → F be a ontinuous parametrization of F , in reasing with respe t to thelexi ographi order on R . Then, for any in reasing sequen e ( t i ) ∈ [0 , and ( x i , y i ) = γ ( t i ) , X i k γ ( t i +1 ) − γ ( t i ) k X i x i +1 − x i + y i +1 − y i length( I ) + k Hen e length( F ) length( I ) + k . Thus we an hoose a -Lips hitz parametrization of F , ˜ γ : [0 , length( I ) + k ] → F . Then for any positive r , the set X = { ˜ γ ( i × r ) ; 0 i N } with N the upper integer part of (length( I ) + k ) /r , is su h that any point of F is at distan eat most r of X . Hen e F r is ontained in X r , implying that H n ( F r ) N π (3 r/ π (length( I ) + k + r ) r .Proof of proposition III.2. Let I = [ a, b ] and J = [ c, c + k ] be su h that ϕ ′ ( I ) ∪ ψ ′ ( I ) ⊆ J .Without loss of generality we will suppose that ψ ′ ( a ) = ϕ ′ ( a ) = c and ψ ′ ( b ) = ϕ ′ ( b ) = c + k .With this assumption, the ompleted graphs Φ and Ψ of ϕ ′ and ψ ′ de(cid:28)ned as above are twore ti(cid:28)able urves joining ( a, c ) and ( b, c + k ) . We let V be the set of points ( x, y ) ∈ R lyingbetween those graphs; the quantity we want to bound is R I | ϕ ′ − ψ ′ | = H ( V ) .Let δ = k ϕ − ψ k L ∞ ( I ) . For any point p = ( x, y ) in V , and any δ ′ > δ , the losed disk D = B ( p, p δ ′ /π ) of volume δ ′ entered at p annot be ontained in V . Indeed if it were,then the di(cid:27)eren e κ = ϕ − ψ would in rease too mu h around p : sin e κ ′ has a onstantsign on this segment, | κ ( x + 2 δ ′ /π ) − κ ( x − δ ′ /π ) | = Z x +2 δ ′ /πx − δ ′ /π | κ ′ | > H ( D ) = 2 δ ′ > δ This ontradi ts k κ k ∞ = δ . Hen e, D must interse ts ∂V implying that V must be ontainedin ( ∂V ) √ δ ′ /π for any δ ′ > δ . Sin e ∂V = Φ ∪ Ψ , the previous lemma gives H ( V ) H (cid:16) Φ √ δ ′ /π (cid:17) + H (cid:16) Ψ √ δ ′ /π (cid:17) π (length( I ) + k + p δ ′ /π ) p δ ′ /π Letting δ ′ onverge to δ on ludes the proof.RR n° 62194 Chazal, Cohen-Steiner & MérigotA generalization of this proposition in arbitrary dimension will follow from an argument oming from integral geometry, ie. we will integrate the inequality of proposition III.2 overthe set of lines of R n to get a bound on k∇ ϕ − ∇ ψ k L ( E ) .We let L n be the set of oriented a(cid:30)ne lines in R n seen as the submanifold of R n madeof points ( u, p ) ∈ R n × R n with u ∈ S n − and x in the hyperplane { u } ⊥ , and endowed withthe indu ed Riemannian metri . The orresponding measure d L n is invariant under rigidmotions. We let D nu be the set of oriented lines with a (cid:28)xed dire tion u .The usual Crofton formula ( f. [Mor88℄ for instan e) states that for any H n − (cid:21)re ti(cid:28)ablesubset S of R n , with β n the volume of the unit n -ball, H n − ( S ) = 12 β n − Z ℓ ∈L n ℓ ∩ S )d ℓ (III.1)where X is the ardinality of X . We will also use the following Crofton-like formula: if K is a H n (cid:21)re ti(cid:28)able subset of R n , H n ( K ) = 1 ω n − Z ℓ ∈L n H ( ℓ ∩ K )d ℓ (III.2)whi h follows from the Fubini theorem (remember ω n − is the volume of the ( n − (cid:21)sphere).Lemma III.4. Let X : E → R n be a L -ve tor (cid:28)eld on an open subset E ⊆ R n . Z E k X k = n ω n − Z ℓ ∈L n Z y ∈ ℓ ∩ E |h X ( y ) | u ( ℓ ) i| d y d ℓ Sket h of proof. The family of ve tor (cid:28)elds of the form P i X i χ Ω i , where the Ω i are a (cid:28)nitenumber of disjoint open subsets of R n and X i are onstant ve tors, is L -dense in the spa e L ( R n , R n ) . Using this fa t and the ontinuity of the two sides of the equality, it is enoughto prove this equality for X = x k X k χ E where x is a onstant unit ve tor and E a boundedopen set of R n .In that ase, one has Z ℓ ∈D nu Z y ∈ ℓ |h X ( y ) | u i| d y d ℓ = k X k |h x | u i| Z ℓ ∈D nu length( E ∩ ℓ )d ℓ = k X k L ( E ) |h x | u i| By a Fubini-like theorem one has Z ℓ ∈L n Z y ∈ ℓ |h X ( y ) | u ( ℓ ) i| d y d ℓ = Z u ∈S n − Z ℓ ∈D nu Z y ∈ ℓ |h X ( y ) | u ( ℓ ) i| d y d ℓ d u = k X k L ( E ) Z u ∈S n − |h x | u i| d u INRIAoundary measures 15The last integral does, in fa t, not depend on x and its value an be easily omputed: Z u ∈S n − |h x | u i| d u = 2 ω n − Z t (1 − t ) n − d t = 2 n ω n − Theorem III.5. Let E be an open subset of R n with ( n − (cid:21)re ti(cid:28)able boundary, and f, g be two lo ally onvex fun tions on E su h that diam( ∇ f ( E ) ∪ ∇ g ( E )) k . Then, letting δ = k f − g k L ∞ ( E ) k∇ f − ∇ g k L ( E ) C ( n )( H n ( E ) + ( k + δ / ) H n − ( ∂E )) δ / with C ( n ) πn as soon as n > (in fa t, C ( n ) = O ( √ n ) ).Proof of the theorem. The -dimensional ase follows from proposition III.2: in that ase, E is a ountable union of intervals on whi h f and g satisfy exa tly the hypothesis of theproposition. Summing the inequalities gives the result with C (1) = 6 π .The general ase will follow from this one with the use of integral geometry. If we set X = ∇ f − ∇ g , f ℓ = f | ℓ ∩ E and g ℓ = g | ℓ ∩ E . Lemma III.4 gives, letting D ( n ) = n/ (2 ω n − ) , Z E k∇ f − ∇ g k = D ( n ) Z ℓ ∈L n Z y ∈ ℓ ∩ E |h∇ f − ∇ g | u ( ℓ ) i| d y d ℓ = D ( n ) Z ℓ ∈L n Z y ∈ ℓ ∩ E | f ′ ℓ − g ′ ℓ | d y d ℓ The fun tions f ℓ and g ℓ satisfy the hypothesis of the one-dimensional ase, so that forea h hoi e of ℓ , and with δ = k f − g k L ∞ ( E ) , Z y ∈ ℓ ∩ E | f ′ ℓ − g ′ ℓ | d y πD ( n )( H ( E ∩ ℓ ) + ( k + δ / ) H ( ∂E ∩ ℓ )) δ / It follows by integration on L n that Z E k∇ f − ∇ g k πD ( n ) (cid:18)Z L n H ( E ∩ ℓ )d L n + ( k + δ / ) Z L n H ( ∂E ∩ ℓ )d L n (cid:19) δ / The formula III.1 and III.2 show that the (cid:28)rst integral is equal (up to a onstant) to thevolume of E and the se ond to the ( n − -measure of ∂E . This proves the theorem with C ( n ) = 6 πD ( n )( ω n − + 2 β n − ) . To get the bound on C ( n ) one uses the formula ω n − = nβ n and β n +1 β n as soon as n > .Multiplying f and g by the same positive fa tor t and optimizing the result in t yields abetter, homogeneous, bound :RR n° 62196 Chazal, Cohen-Steiner & MérigotCorollary III.6. Under the same hypothesis as in theorem III.5, one gets the followingbound, with δ = k f − g k L ∞ ( E ) : k∇ f − ∇ g k L ( E ) C ( n )[( H n ( E ) H n − ( ∂E ) diam( ∇ f ( E ) ∪ ∇ g ( E ))) / + H n − ( ∂E ) δ / ] δ / Remark III.7. To get an homogeneous bound as in this orollary, one ould also optimizethe one-dimensional bound of proposition III.2 before integrating on the set of a(cid:30)ne lines of R n as in the proof of theorem III.5. The bound obtained this way is always stri tly betterthan the ones of both theorem III.5 and orollary III.6, but involves an integral term Z ℓ ∈L n p H ( ℓ ∩ ∂E ) H ( ℓ ∩ E )d ℓ whose intuitive meaning is not quite lear.Applying theorem III.5 to the fun tions v K and v K ′ introdu ed at the begining of thispart and using lemma III.1, one easily gets :Corollary III.8. If E is an open set of R n with re ti(cid:28)able boundary, K and K ′ two ompa tsubsets of R n then, with R K = k d K k L ∞ ( E ) and ε = d H ( K, K ′ ) , k p K − p K ′ k L ( E ) C ( n )[ H n ( E ) + (diam( K ) + ε + (2 R K + ε ) / ε / ) H n − ( ∂E )] × (2 R K + ε ) / ε / In parti ular, if d H ( K, K ′ ) is smaller than min( R K , diam( K ) , diam( K ) /R K ) , there is an-other onstant C ( n ) depending only on n su h that k p K − p K ′ k L ( E ) C ( n )[ H n ( E ) + diam( K ) H n − ( ∂E )] p R K d H ( K, K ′ ) Remarks III.9. 1. This theorem gives in parti ular a quantitative version of the onti-nuity theorem . of [ Fed59 ] : if ( K n ) is a sequen e of ompa t subsets of R n with reach( K n ) > r > , onverging to a ompa t set K , then reach( K ) > r and p K n onverges to p K uniformly on ea h ompa t set ontained in { x ∈ R n ; d K ( x ) < r } .However we have to stress that the result we have proved is more general sin e itdoes not make any assumption on the regularity of K n (cid:22) at the expense of uniform onvergen e.2. The se ond term of the bound involving H n − ( ∂E ) is ne essary. Indeed, let us supposethat a bound k p K − p K ′ k L ( E ) C ( K ) H n ( E ) √ ε were true around K for any open set E . Now let K be the union of two parallel hyperplane at distan e R interse ted witha big sphere entered at a point x of their medial hyperplane M . Let E ε be a ball ofradius ε tangent to M at x and K ε be the translation by ε of K along the ommonnormal of the hyperplanes su h that the medial hyperplane of K ε tou hes the ball E ε on the opposite of x . Then, for ε small enough, k p K − p K ′ k L ( E ε ) ≃ R × H n ( E ε ) ,whi h learly ex eeds the assumed bound for a small enough ε . INRIAoundary measures 173. A ording to this theorem, the map K p K ∈ L ( E ) is lo ally / -Hölder. Thefollowing example shows that this result annot be improved even around a very simple ompa t set. ℓ R Figure III.3: A sequen e of (cid:19)knife blades(cid:20) onverging to a segment.Let S and S ′ be two opposite sides of a re tangle E , ie. two segments of length L andat distan e R . We now de(cid:28)ne a Hausdor(cid:27) approximation of S : for any positive integer N , divide S in N small segments s i of ommon length ℓ , and let C i be the unique ir le with enter in S ′ whi h ontains the two endpoints of s i . We now let S N be theunion of the ir le ar s of C i omprised between the two endpoints of s i .Then it is not very hard to see that if R ε = R + ε is the ommon radius of all the C i , R ε = R + ( ℓ/ , ie. d H ( S, S N ) = p R + ( ℓ/ − R Rℓ / . Then the L -distan ebetween the proje tions on S and S N is at least Ω( ℓ ) (be ause almost half of the pointsin E proje ts on the orners of S N , see the shaded area in (cid:28)g. III.3). Hen e, k p S − p S N k L ( E ) = Ω( ℓ ) = Ω(d H ( S, S N ) / ) Repla ing L ( E ) with L ( µ ) where µ has bounded variationAs we have seen before, a orollary of the previous result is that if µ = H n | E , the map K p K µ is lo ally / -Hölder. This result an be generalized when µ = u H n where u ∈ L ( R n ) has bounded variation. We re all some fa ts about the theory of fun tions withbounded variation, taken from [AFP00℄. If Ω ⊆ R n is an open set and u ∈ L (Ω) , thevariation of u in Ω is V( u, Ω) = sup (cid:26)Z Ω u div ϕ ; ϕ ∈ C c (Ω) , k ϕ k ∞ (cid:27) A fun tion u ∈ L (Ω) has bounded variation if V ( u, Ω) < + ∞ . The set of fun tions ofbounded variation on Ω is denoted by BV(Ω) . We also mention that if u is Lips hitz on Ω ,then V( u, Ω) = k∇ u k L (Ω) . Finally, we let V( u ) be the total variation of u in R n .Theorem III.10. Let µ ∈ M ( R n ) be a measure with density u ∈ BV( R n ) with respe t tothe Lebesgue measure, and K be a ompa t subset of R n . We suppose that supp( u ) ⊆ K R .Then, if d H ( K, K ′ ) is small enough, d bL (p K µ, p K ′ µ ) C ( n ) (cid:16) k u k L ( K R ) + diam( K ) V ( u ) (cid:17) √ R × d H ( K, K ′ ) / RR n° 62198 Chazal, Cohen-Steiner & MérigotProof. We begin with the additional assumption that u has lass C ∞ . The fun tion u an be written as an integral over t ∈ R of the hara teristi fun tions of its superlevel sets E t = { u > t } , ie. u ( x ) = R ∞ χ E t ( x )d t . Fubini's theorem then ensures that for any Lips hitzfun tion f de(cid:28)ned on R n with k f k Lip , p K ′ µ ( f ) = Z R n f ◦ p K ′ ( x ) u ( x )d x = Z R Z R n f ◦ p K ′ ( x ) χ { u > t } ( x )d x d t By Sard's theorem, for almost any t , ∂E t = u − ( t ) is a ( n − -re ti(cid:28)able subsetof R n . Thus, for those t the previous orollary implies, for ε = d H ( K, K ′ ) ε =min( R, diam( K ) , diam( K ) /R K ) , Z E t | f ◦ p K ( x ) − f ◦ p K ′ ( x ) | d x k p K − p K ′ k L ( E t ) C ( n )[ H n ( E t ) + diam( K ) H n − ( ∂E t )] √ Rε Putting this inequality into the last equality gives | p K µ ( f ) − p K ′ µ ( f ) | C ( n ) (cid:18)Z R H n ( E t ) + diam( K ) H n − ( ∂E t )d t (cid:19) √ Rε Using Fubini's theorem again and the oarea formula one (cid:28)nally gets that | p K µ ( f ) − p K ′ µ ( f ) | C ( n ) (cid:16) k u k L ( K R ) + diam( K ) V( u ) (cid:17) √ Rε.
This proves the theorem in the ase of Lips hitz fun tions. To on lude the proof in thegeneral ase, one has to approximate the bounded variation fun tion u by a sequen e of C ∞ fun tions ( u n ) su h that both k u − u n k L ( K R ) and | V( u ) − V( u n ) | onverge to zero, whi his possible by theorem 3.9 in [AFP00℄.Remark III.11. Taking u = χ E where E is a suitable open set shows that theorem III.8 analso be re overed from III.10.IV Stability of boundary and urvature measuresWe ombine the results of orollaries I.6, II.5 and III.8 to getTheorem IV.1. If K and K ′ are two ompa t sets with ε = d H ( K, K ′ ) smaller than min(diam K, r, r / diam K ) , then d bL ( µ K,r , µ K ′ ,r ) C ( n ) N ( K, r − ε ) r n [ r + diam( K )] r εr In parti ular, if for a given bounded Lips hitz fun tion f on R n , one de(cid:28)nes ϕ K,f ( r ) = µ K,r ( f ) , the map K ϕ K,f ∈ C ([ r min , r max ]) with < r min < r max is lo ally / -Hölder.INRIAoundary measures 19In what follows we suppose that ( r i ) is a sequen e of n distin t numbers < r < ... < r n .For any ompa t set K and f ∈ C ( R n ) , we let h Φ ( r ) K,i ( f ) i i be the solutions of the linearsystem ∀ i s.t i n, n X j =0 ω n − j Φ ( r ) K,j ( f ) r n − ji = µ K,r i ( f ) Sin e the system is linear in ( µ K,r i ( f )) and these values depends ontinuously on f , the map f Φ ( r ) K,i ( f ) is also linear and ontinuous, ie. Φ ( r ) K,i is a signed measure on R n . It is also tobe noti ed that if K has positive rea h with reach( K ) > r n , the Φ ( r ) K,i oin ide with the usual urvature measures of K . In that ase, the following result gives a way to approximate the(usual) urvature measures of K from a Hausdor(cid:27)-approximation of it even if its rea h isarbitrary small.Corollary IV.2. There exist a onstant C depending on K and ( r ) su h that for any ompa t subset K ′ of R n lose enough to K , ∀ i, d bL (cid:16) Φ ( r ) K ′ ,i , Φ ( r ) K,i (cid:17) C d H ( K, K ′ ) / Referen es[AFP00℄ L. Ambrosio, N. Fus o, and D. Pallara. Fun tions of bounded variation and freedis ontinuity problems. Oxford Mathemati al Monographs, 2000.[BGV07℄ F. Bolley, A. Guillin, and C. Villani. Quantitative Con entration Inequalitiesfor Empiri al Measures on Non- ompa t Spa es. Probability Theory and RelatedFields, 137(3):541(cid:21)593, 2007.[BKL06℄ A. Beygelzimer, S. Kakade, and J. Langford. Cover trees for nearest neighbor.Pro eedings of the 23rd international onferen e on Ma hine learning, pages 97(cid:21)104, 2006.[BSK℄ G.H. Bendels, R. S hnabel, and R. Klein. Dete ting Holes in Point Set Surfa es.[Cla83℄ F.H. Clarke. Optimization and nonsmooth analysis. Wiley New York, 1983.[DG54℄ E. De Giorgi. Su una teoria generale della misura (r- 1)-dimensionale in unospazio adr dimensioni. Annali di Matemati a Pura ed Appli ata, 36(1):191(cid:21)213,1954.[DGGZ03℄ T.K. Dey, J. Giesen, S. Goswami, and W. Zhao. Shape Dimension and Approx-imation from Samples. Dis rete and Computational Geometry, 29(3):419(cid:21)434,2003.RR n° 62190 Chazal, Cohen-Steiner & Mérigot[DHOS07℄ J. Daniels, L. K. Ha, T. O hotta, and C. T. Silva. Robust smooth feature ex-tra tion from point louds. In Shape Modeling International, 2007. Pro eedings,2007.[Fed59℄ H. Federer. Curvature Measures. Transa tions of the Ameri an Mathemati alSo iety, 93(3):418(cid:21)491, 1959.[FK06℄ S. Funke and C. Klein. Hole dete tion or: how mu h geometry hides in onne -tivity? Pro eedings of the twenty-se ond annual symposium on Computationalgeometry, pages 377(cid:21)385, 2006.[GWM01℄ S. Gumhold, X. Wang, and R. Ma Leod. Feature extra tion from point louds.Pro . 10th International Meshing Roundtable, pages 293(cid:21)305, 2001.[Mor88℄ F. Morgan. Geometri Measure Theory: A Beginner's Guide. A ademi Press,1988.[OP85℄ I.Y. Oleksiv and NI Pesin. Finiteness of Hausdor(cid:27) measure of level sets ofbounded subsets of Eu lidean spa e. Mathemati al Notes, 37(3):237(cid:21)242, 1985.[RBBK06℄ G. Rosman, A. M. Bronstein, M. M. Bronstein, and R. Kimmel. Topologi ally onstrained isometri embedding. In Pro . Conf. on Ma hine Learning and Pat-tern Re ognition (MLPR), 2006.[Vil03℄ C. Villani. Topi s in Optimal Transportation. Ameri an Mathemati al So iety,2003.[Wey39℄ H. Weyl. On the Volume of Tubes. Ameri an Journal of Mathemati s, 61(2):461(cid:21)472, 1939. INRIA nité de recherche INRIA Sophia Antipolis2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)
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