Cone fields and topological sampling in manifolds with bounded curvature

Cone fields and top ological sampling in manifolds with b ounded curv ature Katharine T urner Octob er 25, 2018 Abstract A standard reconstruction problem is ho w to discov er a compact set from a noisy p oin t cloud that appro ximates it. A finite p oin t cloud is a compact set. This pap er prov es a reconstruction theorem whic h giv es a sufficient condition, as a b ound on the Hausdorff distance b et w een tw o compact sets, for when certain offsets of these tw o sets are homotopic in terms of the absence of µ -critical p oin ts in an annular region. W e reduce the problem of reconstructing a subset from a p oin t cloud to the existence of a deformation retraction from the offset of the subset to the subset itself. The am bien t space can b e any Riemannian manifold but we fo cus on am bien t manifolds which hav e nowhere negative curv ature (this includes Euclidean space). W e get an impro v emen t on previous b ounds for the case where the am bien t space is Euclidean whenever µ ≤ 0 . 945 ( µ ∈ (0 , 1) b y definition). In the pro cess, w e prov e stability theorems for µ -critical p oin ts when the ambien t space is a manifold. 1 In tro duction In mo dern science and engineering a common problem is understanding some shap e from a p oin t cloud sampled from that shap e. This p oin t cloud should b e thought of as some finite num ber of (p oten tially) noisy samples. T opology and geometry are considered very natural to ols in suc h data analysis (see e.g. [13] and [4]). One reason is b ecause top ological in v arian ts are often more stable under noise. W e will w ant to understand the homotopy type of a set - tw o ob jects are homotopy equiv alen t if there is a w a y to contin uously deform one ob ject in to another. Often we wish to kno w the exten t to which w e can, and ho w to, reconstruct shapes from noisy p oin t clouds. Naturally the more restrictions on the space the easier it is to reconstruct. The first area of fo cus w as the study of surfaces in R 3 , motiv ated by problems suc h as medical imaging, visualization and reverse engineering of physical ob jects. Algorithms with theoretical guaran ties exist for smo oth closed surfaces with sufficient dense samples. In [1] the concept of a lo cal feature size w as in tro duced. The lo cal feature size at p , denoted lfs( p ), is the distance from p to the medial axis of A . The sampling conditions for surface reconstruction were based on the concept of  -sampling. A p oin t cloud of A is an  -sample if for every p oin t p ∈ A there is some sample p oint at distance at most  lfs( p ) aw ay . The Co cone Algorithm pro duces a homeomorphic set from any 0 . 06-sampling of a smooth closed surface [1]. This pro cess has b een extended to smo oth surfaces with b oundaries [9]. Ho w ev er given an arbitrary compact set K , the b est we can hop e for is to find some nearby set that is homotopy equiv alent to an offset of K . F or instance from a p oin t cloud w e can not tell apart the original set and the same set with a sligh t thic k ening in places. 1 One of the simplest metho ds of reconstruction is to use the offset of a sampling. Giv en a set K , the r -offset of K , denoted K r , is defined to b e { x ∈ M : d ( x, K ) ≤ r } . This is top ologically the same as taking the α -shape of data p oin ts [11] or taking the Cˇ ec h complex [6]. This leads to the problem of finding theoretical guarantees as to when an offset of a sampling has the same top ology (i.e. homotopy type) as the underlying set. In other w ords w e w an t to find conditions on a p oin t cloud S of a compact set A so that S r is homotopy equiv alen t to A . W e will in fact find sufficient conditions for S r to deformation retract to A . Clearly this will only work if the point cloud is sufficien tly close. Usually “sufficien tly close” is interpreted as a b ound on the Hausdorff distance b et ween A and S (the Hausdorff distance b et ween A and S , denoted d H ( A, S ), is the smallest r ≥ 0 suc h that S ⊆ A r and A ⊆ S r ). Muc h of the earlier theory assumes that this Hausdorff distance is less than some measure of geometrical or top ological feature size of the shap e and sho w the output is correct. W e now survey some of this developmen t. The medial axis of a compact set A is the set of p oints p in the ambien t space for which there is more than one p oin t in A which is closest to p . In Figure 1 (a) and (b) the medial axes are the dashed lines. The reac h of A is the minim um distance b etw een p oints in A and p oints in the medial axis of A . It can b e thought of as the minim um lo cal feature size. The reach in (a) is 0 and the reac h in (b) is c . Sampling conditions based on reach include those found in [19] which consider smo oth manifolds in R n . Smooth submanifolds hav e p ositiv e reac h but a w edge, for instance, do es not. T o deal with a larger class of sets Chazal, Cohen-Steiner and Lieutier in [7] introduced the notion of µ -reac h. A p oin t is µ -critical when the norm of the gradient of the distance function at that p oin t is less than or equal to µ . In section 3 we elab orate on a geometric description. In brief, a p oin t p is a cos θ 1 -critical p oin t of the distance function to A if all the p oin ts in A that are closest to p cannot b e con tained in any cone emanating from p with an angle less than θ . In particular 0-critical p oin ts are critical p oin ts of the distance function and every p oin t on the medial axis is a µ -critical p oin t for some µ < 1. The µ -reach of a set is the supremum of r > 0 suc h that the r offset do es not con tain an y µ -critical p oin ts. In [7] and [2] sampling conditions are given in terms of the µ -reach. Another imp ortan t feature size is the weak feature siz e. The weak feature size is the infimum of the p ositiv e critical v alues of the distance function from A . This has several adv antages. Firstly , it can mean a significant impro v emen t on the b ounds suc h as in the case of “hairy” ob jects. Secondly , it can b e applied to a larger class of compact sets. Every semi-algebraic set has p ositive weak feature size [12]. This follows from the fact that the distance function from a semi-algebraic set has only finitely many critical v alues. Instead of making our b ounds in terms of µ -reac h, we will only require the absence of µ -critical p oin ts in an annular region of A along with a b ound on the weak feature size. V arious feature sizes are illustrated in Figures 1 and 2. In Figure 1, p has tw o p oin ts in K closest to p mark ed by q and q 0 . Since the angle b et ween the geodesics from p to q and p to q 0 is α w e conclude that p is cos θ critical for all θ < α (or equiv alen tly cos θ > cos α ). There are no cos α -critical p oints whose distance to the cusp greater than l . Along the medial axis tra v eling tow ard the cusp w e hav e µ -critical p oints with µ tending to zero. This example show how a set can hav e a µ -reac h of 0 for all µ > 0 and yet hav e a p ositive weak feature size (which in this example is infinity). No w consider Figure 2. The w eak feature size is c . The radii of the tw o circles, a and b , are also critical v alues of the distance function. The µ -reac h is c for all µ ∈ [0 , 1). Now x is a cos β - critical p oin t. Since l > a we can say that there are no cos β critical p oin ts in the annular region 1 θ ∈ [0 , π / 2] 2 α p q q 0 l l K Figure 1: The medial axes is the dashed line. The weak feature size is ∞ . The closest p oints in K to p are marked by q and q 0 . Since ∠ ( q pq 0 ) = α we p is cos( α )-critical. There are no cos α -critical p oin ts whose distance to the cusp greater than l . K has a µ -reach of 0 for all µ > 0. b a c l β l x A Figure 2: The medial axes is the dashed line. The weak feature size is c . The radii of the tw o circles, a and b , are also critical v alues of the distance function. The µ -reach is c for all µ ∈ [0 , 1). x is a cos β -critical p oin t but there are no cos β critical p oin ts in the annular region { x : l < d ( x, A ) < b } . 3 { x : l < d ( x, A ) < b } . One limitation to any of these reconstruction theorems is the requiremen t of knowing geometric prop erties of the unkno wn ob ject w e are trying to reconstruct. A shift in p ersp ectiv e can ov ercome this limitation by considering the geometric prop erties of the p oin t cloud, which we do kno w, and can hence pro v e sufficien t conditions for an offset of the original set to deformation retract to an offset of the p oint cloud. W e know the p oint cloud and hence w e know the µ -critical v alues of its distance function. Theoretical guaran tees are given in [7] for when suitable homotopies exist by considering the µ -reach of an offset of the p oin t cloud. Unfortunately there is usually a significant n um b er of small critical v alues of the distance function to the p oin t cloud. This means the starting offset b eyond whic h µ -reach is considered is significant. Our approac h, which only considers the existence of µ -critical p oin ts in a annular region, thus gains a significant adv antage. W e note that previous reconstruction theories ha v e b een restricted to the case where the am bien t space is Euclidean. Another contribution of this pap er is to allo w the ambien t space to be an y manifold whose curv ature is b ounded from b elo w, th us answ ering an op en question ask ed in [7]. Although w e fo cus on the important case of non-negativ ely curved manifolds w e explore a paradigm of reconstruction whic h can b e applied to manifolds with curv ature b ounded from b elo w by some κ < 0 with analogous, alb eit messier, results. Examples of manifolds with no where negativ e curv ature are Stiefel and Grassmannian manifolds. These examples are imp ortan t b ecause there are many applications where data naturally lies on these manifolds such as in dynamic textures [10], face recognition [5], gait recognition [3] and affine shap e analysis and image analysis [20]. Ev en when restricted to the case where we use µ -reac h in Euclidean space w e still improv e on the previous results whenev er µ ≤ 0 . 945. The main theorem of this pap er is as follo ws. Theorem. L et µ ∈ (0 , 1) , r > 0 . L et M b e a smo oth manifold with nowher e ne gative curvatur e such that every p oint has an inje ctivity r adius gr e ater than r . L et L a c omp act subset with d H ( K, L ) < δ . Supp ose that ther e ar e no µ -critic al p oints in K [ r − δ,r − δ +2 δ /µ ] and (4 + µ 2 ) δ < µ 2 r . Then L r deformation r etr acts to K r − δ . By considering b oth the set A and its sampling p oin t cloud S in the role of K w e can reexpress this theorem as a sampling condition. It is a sampling condition as it gives a b ound on the Hausdorff distance b et ween the compact set we wish to reconstruct and its sampling p oin t cloud. Theorem. L et µ ∈ (0 , 1) , r > 0 . L et A b e a c omp act subset of a smo oth manifold M with nowher e ne gative curvatur e such that the inje ctivity r adius of every p oint in M is gr e ater than r and A r is homotopic to A . L et S b e a (p otential ly noisy) finite p oint cloud of A (i.e. a finite set of p oints). Supp ose that either (i) ther e ar e no µ -critic al p oints of the distanc e function fr om A in { x ∈ M : d ( x, A ) ∈ [ a, b ] } or (ii) ther e ar e no µ -critic al p oints of the distanc e function fr om S in { x ∈ M : d ( x, S ) ∈ [ a, b ] } . Then S r is homotopic to A whenever d H ( S, A ) ≤ min n r − a, bµ − rµ 4 − µ , µ 2 r 4+ µ 2 o . Note that if wfs( A ) ≥ a then there exists a deformation retraction from A a to A r for all 0 < r < a [14]. The new ingredient in our approac h is the study of cone fields whic h are generalizations of not necessarily contin uous unit vector fields where we attac h a closed ball in the unit tangent sphere, 4 a “cone”, to eac h p oint in the manifold. More precisely , at each p oin t x we chose a unit tangent v ector w x and an angle β x and then w e take the cone at x to b e the set of unit tangent v ectors whose angle with w x is at most β x . W e denote this cone at x by C ( w x , β x ) and call it acute if β x is acute. A cone field is a choice of cone at each p oin t. In section tw o we study cone fields, defining upp er and lo wer semicontin uous cone fields and sho w that acute lo w er semicontin uous cone fields admit smo oth vector fields. Of particular interest for our reconstruction theorem is the minimal acute r -spanning cone fields. The minimal acute r -spanning cone for K from the p oin t x , if it exists, is the cone C ( w x , β x ) where exp x { tv : t ∈ [0 , r ] , v ∈ C ( w x , β x } ⊇ K δ ∩ B ( x, r ) and β x is acute and minimal. W e can then define its complemen tary cone to b e C ( w x , π / 2 − β x ). W e will care whether the minimal acute r -spanning cone field (defined p oin t wise) for K δ exists ov er the ann ular region K [ r − δ,r + δ ] := { x ∈ M : d K ( x ) ∈ [ r − δ, r + δ ] } . In section tw o we show if the complemen tary cone field to the r -spanning cone field admits a smo oth vector field then the flow of this vector field pro duces a deformation retract from L r to K r − δ when d H ( K, L ) ≤ δ . Since the complemen tary cone of an upp er semicon tin uous cone field is low er semicontin uous, the problem of reconstruction is th us reduced to finding sufficient conditions for an acute r -spanning cone field of K δ to exist o v er K [ r − δ,r + δ ] and that this minimal r -spanning cone field is upp er semicon tin uous. W e find sufficien t conditions for the existence of an acute r -spanning cone field via the stability of µ -critical p oin ts. A stability result of µ -critical p oin ts when the ambien t space is Euclidean is pro v ed in [7]. W e prov e a generalization of this result for when the curv ature of the am bien t space is b ounded from b elo w. The k ey to the pro of is T op onogov’s theorem ab out triangle comparison. It is worth observing that although µ -reach is not stable under Hausdorff distance 2 w e do hav e some stabilit y of the absence of µ -critical p oin ts within of annular regions. The author thanks her advisor Shmuel W einberger for helpful conv ersations and the anonymous referees for their v ery helpful constructiv e criticism. 2 Cone fields One wa y to build a deformation retraction from Y to A is to construct a smo oth v ector field on Y − A suc h that the vectors alwa ys p oint tow ards A and nev er out of Y . More generally there ma y b e some lo cal condition such that if vectors in some smo oth vector field satisfy it then the flow of the vector field has some desirable prop ert y . This leads us to the definition of cone fields which giv e a ball of acceptable unit v ectors at each p oin t. W e will first rigorously define cone fields and then explore a sufficien t condition for them to admit a smo oth vector field. T o define cone fields w e must recall some differential geometry . A useful reference is [16]. Through- out ( M , g ) is a smo oth n -dimensional manifold without b oundary . The unit tangent bundle of a manifold ( M , g ), denoted by U T M , is the unit sphere bundle for the tangen t bundle T M . It is a fib er bundle ov er M whose fib er at eac h p oin t is the unit sphere in the tangent plane; U T M := a x ∈ M { v ∈ T x M : g x ( v , v ) = 1 } , 2 In particular, for any compact set K and any b ound on Hausdorff distance δ > 0 there is a compact set L with zero µ -reac h such that d H ( K, L ) < δ 5 where T x M denotes the tangen t space to M at x . Elemen ts of U T M are pairs ( x, v ), where x is some p oin t of the manifold and v is some tangen t direction (of unit length) to the manifold at x . The exp onential map at x is a map from the tangen t space T x M to M . F or an y v ∈ T x M , a tangen t v ector to M at x , there is a unique geo desic γ v satisfying γ v (0) = x with initial tangent v ector γ 0 v (0) = v . This uses the fact that geo desics trav el at a constant speed. The exp onen tial map at x is defined by exp x ( v ) = γ v (1). The injectivity radius at a p oin t x is the radius of the largest ball on which the exp onen tial map at x is a diffeomorphism. Normal co ordinates at a p oin t x are a lo cal co ordinate system in a neigh b orhoo d of x obtained by applying the exp onential map to the tangen t space at x . Consider the ( n − 1)-dimensional unit sphere, S n − 1 , lying inside R n with a metric induced from this em b edding. Denote by C ( w, β ) the closed ball in S n − 1 cen tered at w with radius β . W e can view C ( w, β ) as the intersection of S n − 1 with a particular infinite cone: C ( w, β ) = S n − 1 ∩ { v ∈ R n \{ 0 } : ∠ ( v , w ) ≤ β } . W e say C ( w , β ) is acute if β is acute. W e can equip the tangent bundle and the unit tangent bundle ov er a manifold with a Riemannian metric induced b y the Levi-Civita connection. Giv en a path γ we can consider the linear isomor- phism Γ( γ ) t s : T γ ( s ) M → T γ ( t ) M induced by parallel transp ort along γ . This map is an isometry and so it sends the unit sphere to the unit sphere. W e can then define a metric on U T M as follo ws. If γ is a geo desic, | s − t | small, v ∈ T γ ( s ) M , and w ∈ T γ ( t ) M w e define d U T M ( v , w ) 2 = ( t − s ) 2 + d U T γ ( t ) M ( w , Γ( γ ) t s ( v )) 2 . Here w e equip U T γ ( t ) M with the usual metric on S n − 1 . In the case where M = R n is Euclidean space then Γ( γ ) t s is just the iden tit y map (we can of iden tifying tangent spaces by translation of the base p oint) and the metric on U T M is the same as that on the pro duct space R n × S n − 1 . Let d K denote the distance function from K . The Hausdorff distance b et w een tw o compact sets K and L is denoted d H ( K, L ) and is defined by d H ( K, L ) := max { sup x ∈ K d L ( x ) , sup y ∈ L d K ( y ) } . Alternativ ely it is the smallest r ≥ 0 such that K ⊆ L r and L ⊆ K r . Denote by F the fibre bundle o v er M where each fibre ov er x ∈ M is the space of non-empty closed balls in the unit tangent s phere at x . F has a natural metric induced from the Hausdorff metric on compact subsets of U T M . A c one field ov er a subset U ⊆ M is a section of F restricted to U . A cone field is contin uous if it is contin uous as a section. As a set, w e can write a cone field ov er U as { ( x, C ( w x , β x )) : x ∈ U } where w x ∈ U T x M and β x ∈ [0 , π ] . One imp ortan t observ ation is that if we tak e the parallel transp ort of a cone we again hav e a cone. More precisely if Γ( γ ) t s is the linear isomorphism induced b y parallel transp ort along γ then Γ( γ ) t s ( C ( w γ ( s ) , β γ ( s ) )) = C (Γ( γ ) t s ( w γ ( s ) ) , β γ ( s ) ) . W e can consider v ectors inside the cone at a p oin t x . W e sa y a vector field X := { ( x, v x ) : x ∈ U, v x ∈ T x M} is sub or dinate to the cone field W = { ( x, C ( w x , β x )) } if v x alw a ys lies in C ( w x , β x ). 6 W e will call a vector field strictly sub or dinate if the v ector at each p oin t lies in the interior of the cone. Define the c omplementary c one of C ( w , β ) to b e C ( w, π / 2 − β ). Given a cone field where the cone at eac h p oin t is acute we can construct the c omplementary c one field p oin twise. F rom the triangle inequalit y on the unit tangen t sphere w e obtain the follo wing useful lemma. Lemma 2.1. L et v b e a ve ctor in some acute c one C and v 0 a ve ctor strictly inside the c omple- mentary c one to C . Then ∠ ( v , v 0 ) < π / 2 . W e will show the existence of smo oth vector fields which are sub ordinate to particular cone fields. W e will define a class of cone fields for whic h the existence of sub ordinate Lipsc hitz v ector fields is guaran teed. Let W = { ( x, C ( w x , β x )) } b e a cone field ov er U with β x > 0 for all x ∈ U . Completely analogous to real-v alued functions we say W is upp er semic ontinuous if for ev ery x ∈ U and every  > 0 there is a δ > 0 such that for every unit sp eed geo desic γ with γ (0) = x we hav e Γ( γ ) t 0 ( C ( w x , β x +  )) ⊇ C ( w γ ( t ) , β γ ( t ) ) for all t ∈ (0 , δ ). W e sa y W is lower semic ontinuous if for every x ∈ U and every  ∈ (0 , β x ) there is a δ > 0 such that for every unit sp eed geo desic γ with γ (0) = x we hav e Γ( γ ) t 0 ( C ( w x , β x −  )) ⊆ C ( w γ ( t ) , β γ ( t ) ) for all t ∈ (0 , δ ). Analogous the real-v alued function case, one can pro v e that a cone field is contin uous if and only if it is b oth upp er and low er semicontin uous. There is a useful relationship b et ween upp er and lo w er semicon tin uous cone fields which is analogous to the fact that the negativ e of a upp er contin uous function is lo w er semicon tin uous and vice versa. Lemma 2.2. An acute c one field is upp er semic ontinuous if and only if its c omplementary c one field is lower semic ontinuous. Pr o of. The pro of follows from the observ ation that Γ( γ ) t 0 ( C ( w x , β x +  )) ⊇ C ( w γ ( t ) , β γ ( t ) ) if and only if ∠ (Γ( γ ) t 0 ( w x ) , w γ ( t ) ) ≤ β x +  − β γ ( t ) = ( π / 2 − β γ ( t ) ) − (( π / 2 − β x ) −  ) if and only if Γ( γ ) t 0 ( C ( w x , ( π / 2 − β x ) −  )) ⊆ C ( w γ ( t ) , π / 2 − β γ ( t ) ) . Prop osition 2.3. L et W = { ( x, C ( w x , β x )) } b e a lower semic ontinuous c one field over U ⊂ M with β x > 0 for al l x ∈ U . Then ther e exists a smo oth unit ve ctor field strictly sub or dinate to W . Pr o of. Since W is assumed to b e low er semicontin uous, for each x there exists a δ x > 0 such that for ev ery unit sp eed geo desic γ with γ (0) = x we hav e Γ( γ ) t 0 ( C ( w x , β x / 2)) ⊆ C ( w γ ( t ) , β γ ( t ) ) for all t ∈ (0 , δ ). This means that the v ector field ov er B ( x, δ x ) constructed by parallel transp ort of w x is strictly sub ordinate to W | B ( x,δ x ) . Let X x denote this vector field. By construction X x is a smo oth unit vector field on B ( x, δ x ). 7 The set of op en balls { B ( x, δ x ) : x ∈ U } cov er U and since M is paracompact there is a lo cally finite sub co v er { B ( x i , δ i ) } and a partition of unit y { ρ i } sub ordinate to this cov er. This means that there is a collection of functions ρ i : U → [0 , 1] of smo oth functions such that supp( ρ i ) ⊆ B ( x i , δ i ) for eac h i and for eac h y ∈ U w e hav e P i ρ i ( y ) = 1. Let X = P i ρ i X x i . This is defined ov er all of U as the X x i are all defined o v er the supp ort of the corresp onding ρ i . It is smo oth b ecause the X x i and the ρ i are all smo oth. A t each p oin t y ∈ U the vector X ( y ), can b e written as the sum P j a j v j where ∠ ( v j , w y ) < β y < π / 2 and the a j ∈ [0 , 1] for eac h j and P j a j = 1. This implies that ∠ ( P j a j v j , w y ) < β y and that X ( y ) is nonzero. Th us we can conclude that X ( y ) k X ( y ) k ∈ W ( y ). Since X is nowhere v anishing we can rescale the vectors at each p oin t to construct a smo oth v ector field ˆ X of unit v ectors. This vector field ˆ X is strictly sub ordinate to W . W e are interested in cone fields that reflect lo cal geometric prop erties of the distance function to a set. W e call γ a se gment from x / ∈ K to K is if γ is a distance achieving path from x to K . If M is Euclidean then these segments are straigh t lines. Observe that on an arbitrary manifold there can b e more than one segment connecting x to the same y ∈ K . W e say C ( w , β ) is an r -sp anning c one for K if { exp x ( tv ) : t ∈ [0 , r ] , v ∈ C ( w , β ) } ⊇ K ∩ B ( x, r ) 6 = ∅ . W e say C ( w , β ) is a minimal if whenever C ( w 0 , β 0 ) is also an r -spanning cone then β ≤ β 0 . W e can see that when an acute minimal r -spanning cone exists then it is unique. Lemma 2.4. L et 0 < δ < r / 2 , and let K , L b e c omp act subsets of a manifold M with d H ( K, L ) ≤ δ . Supp ose that ther e exists an acute r -sp anning c one field W over K [ r − δ,r + δ ] for K δ . L et W 0 b e the c omplementary c one field to W . If X is a smo oth ve ctor field strictly sub or dinate to W 0 , then X induc es a deformation r etr action fr om L r to K r − δ . Pr o of. Since X is a smo oth vector field it has a unique smo oth in tegral flow (a standard result, for example [15]). The idea is to follow this flow from each p oint in L r un til it reac hes K r − δ . Denote the acute r -spanning cone field W by { ( x, C ( w x , β x )) } . F rom Lemma 2.1 we know that any v ector sitting strictly inside C ( w x , π 2 − β x ) forms an acute angle when paired with angle vector in C ( w x , β x ). Let X b e a v ector field strictly sub ordinate to W 0 . Let x ∈ L r ∩ K [ r − δ,r + δ ] , and let v b e the vector at x in X. Now x ∈ B ( y , r ) for some y ∈ L . Let γ y x denote a geo desic from x to y of length at most r , and γ 0 y x (0) its tangen t v ector at x . By construction, γ 0 y x (0) ∈ C ( w x , β x ) and hence it forms an acute angle with v . This means that their images form an acute angle in the normal co ordinates giv en b y the exp onen tial map at x . No w consider the normal co ordinates given by the exp onen tial map at y . In these co ordinates γ y x is a radial straigh t line emitted from the origin. Gauss’s lemma (see [16]) tells us that the angle b et ween γ v and γ y x in the normal co ordinates at y is acute if and only if the angle b et ween them in the normal co ordinates based at x is acute. W e hav e already shown that this second angle is acute. This means that in the normal co ordinates given by the exp onen tial map at y , γ v m ust remain inside B (0 , r ) for some p ositive amoun t of time. Since this is true all x w e know that the integral flo w do es not leav e L r . 8 The in tegral flow of X is alwa ys trav eling tow ards K as it lies in W 0 . F or each x ∈ K [ r − δ,r + δ ] , let λ x b e the rate at whic h the integral flo w of X at x is trav eling tow ards K . Since λ x > 0 for all x ∈ K [ r − δ,r + δ ] and K [ r − δ,r + δ ] is compact there is some λ > 0 which forms a lo wer b ound on how fast the in tegral flo w of X trav els tow ards X . Construct the deformation retraction from L r to K r − δ b y following each p oin t along the flow of X until it reaches K r − δ and then remaining stationary . The uniform low er b ound on how fast the in tegral flow of X trav els tow ards K com bined with the observ ation that every p oint in L r is at most 2 δ from K r − δ , te lls us that in a finite amoun t of time every p oin t in L r will b e sen t to one in K r − δ . 3 Stabilit y of µ -critical p oin ts W e wan t to study the gradient v ector fields for distance functions from compact subsets of a general manifold ( M , g ). This can b e though t of as the ob vious generalization of the gradient vector fields for distance functions from compact subsets of Euclidean space (as studied in [7] and [17]). Recall that γ is a segment from x / ∈ K to K is if γ is a distance achieving path from x to K . The p oint x is a called a critic al p oint of the distance function from K if, for all non-zero v ∈ T x M , there exists a segment γ from x to K suc h that ∠ ( γ 0 (0) , v ) ≤ π / 2. Equiv alently , if Γ( x ) := { y ∈ K : d K ( x ) = d ( x, y ) } , then x is a critical p oin t if and only if 0 lies in the conv ex hull of exp − 1 x Γ( x ) ∩ B (0 , d K ( x )). W e need to construct the gradient vector field so that it v anishes at critical p oin ts of the distance function. F or all non-critical p oints w e can consider the minimal spanning cone C ( w x , β x ) for Γ( x ) from x of length d K ( x ). W e set ∇ K ( x ) := − cos( β x ) w x whenev er x is not critical. Observ e that ∇ K a ( x ) = ∇ K ( x ) whenever d K ( x ) > a . F or µ ∈ R , w e call x µ -critic al if k∇ K ( x ) k ≤ µ . A p oin t is 0-critical exactly when it is a critical p oin t for the distance function. It is easy to v erify that these definitions agree with those giv en in [7] when M is Euclidean. W e will wan t to prov e a generalization of the stabilit y result in [7] where the am bien t space is a manifold with curv ature bounded from below in a suitable neigh b orhoo d of the compact subset under study . By appropriate scaling it is sufficient to consider the cases where the curv ature is b ounded from b elo w b y 1, 0 or − 1. Lemma 3.1. L et K ⊂ M b e a c omp act subset of a manifold ( M , g ) for which the curvatur e on K 2 α is b ounde d fr om b elow by κ . L et x ∈ K α \ K b e a µ -critic al p oint of d K . Then for any y ∈ K 2 α we have cos d K ( y ) ≥ cos d ( y , x ) cos d K ( x ) − sin d ( y , x ) sin d K ( x ) µ if κ = 1 d K ( y ) 2 ≤ d K ( x ) 2 + d ( x, y ) 2 + 2 d ( y , x ) d K ( x ) µ if κ = 0 cosh d K ( y ) ≤ cosh d ( y , x ) cosh d K ( x ) + sinh d ( y , x ) sinh d K ( x ) µ if κ = − 1 . Pr o of. Let θ ∈ (0 , π / 2] such that cos θ = µ . Let Γ( x ) = { z ∈ K : d K ( z ) = d ( z , x ) } and set d Γ( x ) := { z / k z k : z ∈ exp − 1 x (Γ( x )) ∩ B (0 , d K ( x )) } . Fix y ∈ K 2 α and c ho ose v ∈ T x M suc h that exp x ( d ( x, y ) v ) = y . 9 W e wan t to show that there is some z ∈ K and length ac hieving geo desics γ y x and γ z x suc h that ∠ ( y , x, z ) ≤ π − θ where ∠ ( y , x, z ) is the angle b et ween γ y x and γ z x . Supp ose not. This means that no p oin t of d Γ( x ) lies in C ( v , π − θ ). Geometrically this means that d Γ( x ) must lie in the interior of C ( − v , θ ) whic h is the complemen t of C ( v , π − θ ) in the sphere. Ho w ev er this implies that the minimal spanning cone for Γ( x ) from x of length d K ( x ) lies strictly inside C ( − v , θ ) and hence k∇ K ( x ) k > cos θ = µ . This contradicts the assumption that x is a µ -critical p oint (i.e. k∇ K ( x ) k ≤ µ ). Th us by contradiction, there is some p oin t z ∈ K and length ac hieving geo desics γ y x and γ z x suc h that ∠ ( y , x, z ) ≤ π − θ . Let 4 x,y ,z b e the geo desic triangle with γ y x and γ z x suc h that ∠ ( y , x, z ) ≤ π − θ which we hav e just sho wn must exist. Let ˜ 4 ˜ x, ˜ y , ˜ z b e the corresp onding triangle in M ( κ ), the manifold with constant curv ature κ , where the length of the sides are preserved. T op onogo v’s theorem is a triangle com- parison theorem which quantifies the assertion that a pair of geo desics emanating from the same p oin t spread apart more slowly in a region of high curv ature than they would in a region of low cur- v ature. The details of this theorem and its pro of can b e found in [8]. By taking the contrapositive of T op onogov’s theorem we know ∠ ( ˜ y , ˜ x, ˜ z ) ≤ ∠ ( y , x, z ) ≤ π − θ and hence cos ∠ ( ˜ y , ˜ x, ˜ z ) ≥ − µ. W e finally substitute d K ( y ) ≤ d ( ˜ y , ˜ z ), d K ( x ) = d ( ˜ x, ˜ z ) and cos ∠ ( ˜ y , ˜ x, ˜ z ) ≥ − µ in to the spherical, Euclidean and h yp erbolic cosine rules resp ectiv ely to obtain the desired inequalities. Our stability result will arise from comparing t wo opp osing inequalities - one from the previous lemma alongside one coming from the following lemma whic h is Lemma 4.1 in [18]. It is easy to c hec k the definition of k∇ K ( x ) k coincides with k∇ x f k when f = d K and X is a manifold. 3 Lemma 3.2 (Lemma 4.1 in [18]) . L et X b e a metric sp ac e. Supp ose f : X → R is a lo c al ly Lipschitz map, x ∈ X , and f ( x ) = 0 . F or µ, r > 0 , assume that the b al l B ( x, r ) is c omplete and that k∇ z f k ≥ µ for e ach z with d ( z , x ) < r and f ( z ) ≥ 0 . Then for e ach 0 < C < µ ther e is a p oint z ∈ X with d ( z , x ) ≤ r and f ( z ) = C r . The follo wing proposition is a generalization of critical p oin t stabilit y theorem in [7] where the am bien t space can no w b e any manifold with non-negative curv ature. Prop osition 3.3. L et K, L b e c omp act subsets of M with d H ( K, L ) ≤ δ . L et x b e a µ -critic al p oint of d K . If C ≥ µ + 2 s δ d K ( x ) and K d K ( x )+4 δ / ( C − µ ) has nowher e ne gative curvatur e, then ther e exists a C -critic al p oint y of d L with d L ( y ) ≥ d L ( x ) and y ∈ B ( x, 4 δ / ( C − µ )) . Pr o of. W e w an t to sho w that there is some y suc h that k∇ L ( y ) k ≤ C and d L ( y ) ≥ d L ( x ) and d ( x, y ) ≤ 4 δ / ( C − µ ). Supp ose not. Then there is some ˜ µ > C such that k∇ L ( y ) k ≥ ˜ µ whenever d L ( y ) ≥ d L ( x ) and d ( x, y ) ≤ 4 δ / ( C − µ ). If C ≥ µ + 2 p δ /d K ( x ) then d K ( x ) − 4 δ / ( C − µ ) 2 ≥ 0 and hence we can construct K 0 := K d K ( x ) − 4 δ / ( C − µ ) 2 and L 0 := L d K ( x ) − 4 δ / ( C − µ ) 2 . 3 k∇ x f k is the nonnegativ e num b er max { 0 , lim sup y → x f ( y ) − f ( x ) d ( y,x ) } . That this is k∇ K ( x ) k follows from our geometric construction of ∇ K ( x ) and from the cosine rule. 10 By construction d H ( K 0 , L 0 ) ≤ d H ( K, L ) ≤ δ and d K 0 ( x ) = 4 δ / ( C − µ ) 2 . Using f = d L 0 − d L 0 ( x ) and r = 4 δ / ( C − µ ) in Lemma 3.2 w e know that there exists a p oin t y ∈ B ( x, 4 δ / ( C − µ )) suc h that f ( y ) = C 4 δ / ( C − µ ) whic h means d L 0 ( y ) = d L 0 ( x ) + C 4 δ / ( C − µ ). Using d H ( K 0 , L 0 ) ≤ δ we can sho w that d K 0 ( y ) ≥ d L 0 ( y ) − δ = d L 0 ( x ) + C 4 δ / ( C − µ ) − δ ≥ d K 0 ( x ) + C 4 δ / ( C − µ ) − 2 δ. Since d K 0 ( x ) = 4 δ / ( C − µ ) 2 w e conclude that d K 0 ( y ) ≥ 4 δ + C 4 δ ( C − µ ) − 2 δ ( C − µ ) 2 ( C − µ ) 2 . (1) A t the same time, Lemma 3.1 implies that d K 0 ( y ) 2 ≤ d K 0 ( x ) 2 + d ( x, y ) 2 + 2 d ( y , x ) d K 0 ( x ) µ and hence d K 0 ( y ) 2 ≤ 16 δ 2 + 16 δ 2 ( C − µ ) 2 + 32 δ 2 µ ( C − µ ) ( C − µ ) 4 (2) By com bining (1) and (2) and p erforming some algebraic manipulation w e obtain 1 + ( C − µ ) 2 + 2( C − µ ) µ ≥ (1 + C ( C − µ ) − ( C − µ ) 2 / 2) 2 . (3) Ho w ev er algebraic manipulation of (3) implies 0 ≥ ( C − µ ) 2 / 4 + C µ whic h is a con tradiction. One of the problems with working with the µ -reac h is that it is not stable under Hausdorff distance. Indeed by the creation of an arbitrarily small cusp w e know that for any compact subset K , and an y δ > 0, there exists some compact subset L with d H ( K, L ) < δ whose µ -reach is zero for all µ > 0. How ever by only considering µ -critical p oints in an annular regions we can ha v e s tabilit y results. Corollary 3.4. L et K, L b e c omp act subsets of a manifold with non-ne gative se ctional curvatur e such that d H ( K, L ) ≤ δ . Supp ose that ther e ar e no C -critic al p oints for d L in the annular r e gion L [ a,b ] . If C ≥ µ + 2 r δ a + δ then ther e ar e no µ -critic al p oints for d K in the annular r e gion K [ a + δ,b − 4 δ/ ( C − µ ) − δ ] . Pr o of. If x is a µ -critical p oin t with d K ( x ) ∈ [ a + δ, b − 4 δ / ( C − µ ) − δ ] then b y Prop osition 3.3 there exists some C -critical p oin t y with d L ( y ) ∈ [ d L ( x ) , d L ( x ) + 4 δ / ( C − µ )] ⊂ [ d K ( x ) − δ, d K ( x ) + 4 δ / ( C − µ ) + δ ] ⊂ [ a, b ] whic h is a con tradiction. Analogous stability results should hold for the cases when κ = 1 , − 1 . How ever, w e will only later require the case when µ = 0 and so to significantly simplify calculations we restrict to this case. Prop osition 3.5. L et K , L b e c omp act subsets of M with d H ( K, L ) ≤ δ . L et x b e a critic al p oint of d K . Supp ose that the se ctional curvatur e of K 2 d K ( x ) is b ounde d fr om b elow by κ = − 1 . Then for al l C > 0 ther e exists a C -critic al p oint y of d L with d L ( y ) ≥ d L ( x ) and d ( x, y ) ≤ 4 δ /C whenever 9 δ ≤ 2 tanh( d K ( x )) C 2 . 11 Pr o of. Let C ∈ (0 , 1). W e wan t to show that there is some p oin t y such that k∇ L ( y ) k ≤ C and d L ( y ) ≥ d L ( x ) and d ( x, y ) ≤ 4 δ /C . Suppose not. Then there is some ˜ µ > C such that k∇ L ( y ) k ≥ ˜ µ whenev er d L ( y ) ≥ d L ( x ) and d ( x, y ) ≤ 4 δ /C Using f = d L − d L ( x ) and r = 4 δ /C in Lemma 3.2 w e know that there exists a p oin t y ∈ B ( x, 4 δ /C ) suc h that f ( y ) = 4 δ ; i.e. d L ( y ) = d L ( x ) + 4 δ . F rom d H ( K, L ) ≤ δ we kno w that d K ( y ) ≥ d K ( x ) + 2 δ. A t the same time, Lemma 3.1 implies that cosh d K ( y ) ≤ cosh d ( y , x ) cosh d K ( x ) and hence cosh( d K ( x )+ 2 δ ) ≤ cosh(4 δ /C ) cosh d K ( x ) . Using the hyperb olic cosh sum form ula and dividing through b y cosh d K ( x ), this can b e rewritten as cosh(4 δ /C ) − cosh(2 δ ) ≥ tanh( d K ( x )) sinh(2 δ ) . Our assumption that 9 δ ≤ 2 tanh( d K ( x )) C 2 implies that 4 δ /C ≤ 8 tanh( d K ( x )) C / 9 < 1. Now cosh( t ) − cosh(2 δ ) < cosh( t ) − 1 < 9 16 t 2 whenev er t ∈ (0 , 1) and sinh( t ) > t for all t . This means that w e can conclude that 9 16  4 δ C  2 > tanh( d K ( x ))2 δ and hence that 9 δ > 2 tanh( d K ( x )) C 2 . This contradicts our assumption of δ implying that there m ust exist a suitably nearb y C -critical p oint. 4 Reconstruction theorem Our reconstruction pro of will inv olve finding sufficient conditions for the existence of useful cone fields. W e will use the stabilit y of µ -critical p oin ts to show the existence of acute minimal spanning cones of A δ from p oin ts in an annular region. Lemma 4.1. L et µ ∈ (0 , 1) , r > δ > 0 and M b e a manifold with nowher e ne gative curva- tur e. L et K ⊂ M b e a c omp act subset and x ∈ K r + δ . If ther e ar e no µ -critic al p oints of d K in K [ d K ( x ) ,d K ( x )+2( r − d K ( x )+ δ ) /µ ] and δ ≤ d K ( x ) − 4 − µ 2 4 + µ 2 r then ther e is an acute r -sp anning c one for K δ fr om x . Pr o of. Supp ose, by wa y of contradiction, that 0 is in the conv ex hull of (exp − 1 x K δ ) ∩ B (0 , r ). Set ˆ K := (exp − 1 x K δ ) ∩ B (0 , r ). Define the map ϕ : B (0 , r ) → ∂ B  0 , r + ( d K ( x ) − δ ) 2  z 7→ r + ( d K ( x ) − δ ) 2 z k z k . and set ˆ L to b e ϕ ( ˆ K ). This construction is illustrated in Figure 2. By construction d ˆ L ( x ) = 1 2 ( r + ( d K ( x ) − δ )) and d H (exp x ˆ L, exp x ˆ K ) ≤ 1 2 ( r − ( d K ( x ) − δ )) . 12 d K ( x ) − δ 1 2 ( d K ( x ) − δ + r ) r x y ∈ ˆ K ϕ ( y ) exp − 1 x ( K δ ) ϕ ( ˆ K ) Figure 3: Construction of ˆ L = ϕ ( ˆ K ) Set L = exp x ˆ L ∪ { y ∈ K : d ( y , x ) ≥ r } . By construction K = exp x ˆ K ∪ { y ∈ K : d ( y , x ) ≥ r } . Since taking the union with b oth sets by the same set can only decrease the Hausdorff distance, d H ( K, L ) ≤ d H (exp x ˆ K , exp x ˆ L ). Also, by construction, d L ( x ) = d ˆ L ( x ) = 1 2 ( r + ( d K ( x ) − δ )). Since 0 is in the conv ex hull of ˆ K (by assumption), there exists z 1 , . . . , z m ∈ ˆ K and a 1 , . . . a m > 0 suc h that Σ m i =1 a i z i = 0. Ho wev er this gives m X i =1 a i 2 k z i k r + d K ( x ) − δ ϕ ( z i ) = 0 and hence 0 is in the conv ex hull of ˆ L . Since all the p oints in ˆ L are equidistant from 0, and hence all the p oin ts in exp x ˆ L are equidistant to x , w e conclude that x is a 0-critical p oin t of d exp x ˆ L . Adding p oin ts further aw ay from x do es not affect this criticalit y and so x is a 0-critical p oin t of d L . Our condition that δ ≤ d K ( x ) − r (4 − µ 2 ) / (4 + µ 2 ) can b e rewritten as 1 2 ( r − ( d K ( x ) − δ )) < 1 2 ( r + ( d K ( x ) − δ )) µ 2 4 . This implies that d H ( K, L ) < d L ( x ) µ 2 / 4, or in other w ords µ ≥ 0 + 2 p d H ( K, L ) /d L ( x ). This enables us to apply Prop osition 3.3 to say that there exists a µ -critical p oin t y of d K with d K ( y ) ∈ [ d K ( x ) , d K ( x ) + 4 d H ( K, L ) /µ ] = [ d K ( x ) , d K ( x ) + 2( r − d A ( x ) + δ ) /µ ] . This con tradicts our assumption ab out the absence of µ -critical p oin ts in that ann ular region. The pro cess can b e applied for the case when κ = − 1 using Prop osition 3.5 instead of Prop osition 3.3. This leads to the following lemma. 13 Lemma 4.2. L et µ ∈ (0 , 1) , r > δ > 0 and M b e a manifold with curvatur e b ounde d fr om b elow by κ = − 1 . L et K ⊂ M b e a c omp act subset and x ∈ K r + δ . If ther e ar e no µ -critic al p oints of d K in K [ d K ( x ) ,d K ( x )+4 δ /µ ] and 9( r + δ − d K ( x )) ≤ 4 tanh  1 2 ( r − δ + d K ( x )  µ 2 ther e is an acute r -sp anning c one for K δ fr om x . Theorem 4.3. L et µ ∈ (0 , 1) , r > 0 . L et M b e a smo oth manifold with nowher e ne gative curvatur e such that every p oint has an inje ctivity r adius gr e ater than r . L et K , L ⊂ M b e c omp act with d H ( K, L ) < δ . Supp ose that ther e ar e no µ -critic al p oints in K [ r − δ,r − δ +2 δ /µ ] and (4 + µ 2 ) δ < µ 2 r . Then L r deformation r etr acts to K r − δ . Pr o of. Supp ose that there are no µ critical point of d K in K [ r − δ,r + δ +2 δ /µ ] . F or eac h x ∈ K [ r − δ, r + δ ] w e ha v e d K ( x ) ∈ [ r − δ, r + δ ] and hence [ d K ( x ) , d K ( x ) + 2( r − d K ( x ) + δ ) /µ ] ⊆ [ r − δ, r + δ + 2 δ /µ ] . Lemma 4.1 tells us that there exists an acute r -spanning cone field W = ( x, C ( w x , β x )) ov er K [ r − δ,r + δ ] for K δ . By Lemma 2.4 it is sufficien t to show that there exists a smo oth vector field strictly sub ordinate to the complemen tary cone W 0 of W . Using Lemma 2.2 and Prop osition 2.3 the theorem will follow if w e can sho w W is upp er semicontin uous. Let x ∈ K [ r − δ,r + δ ] and  > 0. Since C ( w x , β x ) is the minimal spanning cone for K δ ∩ B ( x, r ) of length r and d K δ ( x ) ≥ r − 2 δ we hav e K δ ∩ B ( x, r ) ⊆ { exp x ( tv ) : v ∈ C ( w x , β x ) , t ∈ [ r − 2 δ, r ] } . This implies that there exists an α 0 > 0 such that for all α < α 0 w e ha v e  K δ ∩ B ( x, r )  α ⊆ { exp x ( tv ) : v ∈ C ( w x , β x + / 2) , t ∈ [ r − δ − α, r + α ] } . (4) Define the sequence of compact sets A n , n ∈ N , n ≥ 1 as follows. Let y ∈ A n if and only if y ∈ K δ ∩ B ( x, 1 /n ) and there do es not exist a path γ : [0 , 1] → K δ with γ (0) = y , γ (1) ∈ K δ ∩ B ( x, r ) suc h that d K δ ∩ B ( x,r ) ( γ ( t )) is strictly decreasing. The A n are compact b ecause K δ is closed. The A n are decreasing for the inclusion and ∩ n A n = ∅ . This implies that for some n , A n = ∅ . Set ˜ r := min { 1 /n, α } . F or this ˜ r (using equation (4) for the second inclusion) w e ha v e K δ ∩ B ( x, r + ˜ r ) ⊆  K δ ∩ B ( X, r )  ˜ r ⊆ { exp x ( tv ) : v ∈ C ( w x , β x + / 2) , t ∈ [ r − δ − ˜ r , r + ˜ r ] } . (5) By recalling that d K δ ( x ) ≥ r − 2 δ we can refine (5) to state K δ ∩ B ( x, r + ˜ r ) ⊆ { exp x ( tv ) : v ∈ C ( w x , β x + / 2) , t ∈ [ r − 2 δ, r + ˜ r ] } . (6) W e may assume that r + ˜ r is less than the injectivit y radius by taking ˜ r > 0 small enough. Let Γ y x denote the isometry b etw een the tangent plane at x to that at y induced by parallel transp ort. 14 along the geo desic from x to y . This is well defined when the distance b et ween x and y is less than the injectivit y radius. The function F : B ( x, ˜ r ) × T x M → M defined b y ( y , u ) 7→ exp y ◦ Γ y x ( u ) is contin uous in b oth y and u . This implies that for eac h pair of compact sets A ⊆ T x M and L ⊆ M such that int( F ( x, A )) ⊃ L there is a η > 0 such that F ( y , A ) ⊇ L whenever d ( x, y ) < η . By taking A = { tv : v ∈ C ( w x , β x + / 2) , t ∈ [ r − 2 δ, r + ˜ r / 2] } and L = exp x ( { tv : v ∈ C ( w x , β x +  ) , t ∈ [ r − 2 δ − ˜ r , r + ˜ r ] } ) we can conclude that there is a η > 0 such that exp x ( { tv : v ∈ C ( w x , β x + / 2) , t ∈ [ r − 2 δ, r + ˜ r ] } ) ⊆ exp y Γ y x ( { tv : v ∈ C ( w x , β x +  ) , t ∈ [ r − 2 δ − ˜ r , r + 2 ˜ r ] } (7) whenev er d ( x, y ) < η . W e ma y assume that η < ˜ r . Com bining (6) with the triangle inequality w e kno w that for d ( x, y ) < η , K δ ∩ B ( y , r ) ⊆ exp x { ( tv ) : v ∈ C ( w x , β x + / 2) , t ∈ [ r − 2 δ, r + ˜ r ] } . W e then use (7) to obtain K δ ∩ B ( y , r ) ⊆ exp y Γ y x ( { tv : v ∈ C ( w x , β x +  ) , t ∈ [ r − 2 δ − ˜ r , r + 2 ˜ r ] } . No w Γ y x is an isometry and so the in tersection of b oth sides with B ( y , r ) pro duces the con tainmen t K δ ∩ B ( y , r ) ⊆ exp y Γ y x ( { tv : v ∈ C ( w x , β x +  ) , t ∈ [0 , r ] } whenev er d ( x, y ) < η . C ( w y , β y ) is defined to b e the minimal spanning cone of K δ ∩ B ( y , r ) of length r and so exp y { tv : v ∈ C ( w y , β y ) , t ∈ [0 , r ] } ⊆ exp y Γ y x { tv : v ∈ C ( w x , β x +  ) , t ∈ [0 , r ] } . F rom the assumption that r is less than the injectivit y radius w e conclude C ( w y , β y ) ⊆ Γ y x C ( w x , β x +  ). By doing the same process but using Lemma 4.2 instead of Lemma 4.1 w e get the analogous theorem for when the am bien t space has its sectional curv ature b ounded b elo w b y − 1. Theorem 4.4. L et µ ∈ (0 , 1) , r > 0 . L et M b e a smo oth manifold whose se ctional curvatur e b ounde d b elow by − 1 and with an inje ctivity r adius gr e ater than r . L et K , L ⊂ M b e c omp act with d H ( K, L ) < δ . Supp ose that ther e ar e no µ -critic al p oints in K [ r − δ,r − δ +4 δ /µ ] and 9 δ < 2 tanh( r − δ ) µ 2 . Then L r deformations r etr acts to K r − δ . 5 Applications to p oin t cloud data In this section we now consider the situation where w e hav e some unknown compact set A whic h w e are w an ting to understand and we can sample A to generate a (p otentially noisy) p oin t cloud of A which we will denote by S . Historically geometric conditions hav e b een giv en on A for when offsets of the S and A are homotopic. This is b ecause A is often assumed to hav e nice geometric prop erties whereas S , as a p oint cloud, has man y critical p oin ts of its distance function nearby . The corresp onding theorem pro duced using Theorem 4.3 and Theorem 4.4 with K = A and L = S is as follo ws. 15 Corollary 5.1. L et µ ∈ (0 , 1) , r > 0 . L et M b e a smo oth manifold with se ctional curvatur e b ounde d by κ and whose inje ctivity r adius is gr e ater than r . L et A b e a c omp act subset of M and S b e a (p otential ly noisy) p oint cloud of A . Supp ose that ther e ar e no µ -critic al p oints in A [ a,b ] . Then S r is homotopic to A r − d H ( S,A ) whenever d H ( S, A ) ≤ min  r − a, bµ − r µ 4 − µ  , and d H ( S, A ) < µ 2 r 4 + µ 2 if κ = 0 or d H ( S, A ) ≤ min  r − a, bµ − r µ 4 − µ  , and 9 d H ( S, A ) < 2 tanh( r − d H ( S, A )) µ 2 if κ = − 1 . F urthermor e if A r − d H ( S,A ) deformation r etr acts to A then S r deformation r etr acts to A . Pr o of. In order to apply Theorem 4.3 or Theorem 4.4 we need to make sure that [ a, b ] ⊃ [ r − d H ( S, A ) , r − d H ( S, A ) + 4 d H ( S, A ) /µ ] and also that d H ( S, A ) < µ 2 r 4+ µ 2 or 9 d H ( S, A ) < 2 tanh( r − d H ( S, A )) µ 2 resp ectiv ely . Of general in terest is finding the homotop y t yp e of A rather than A r . How ev er, a sufficient condition for A b to deformation retract to A a (0 < a < b ) is that there are no 0-critical points in A b \ A a [14]. It w ould b e impossible from a p oin t cloud to distinguish A from A a for sufficien tly small a > 0. F urthermore, there are many shap es, suc h as hairy ob jects, for whic h many offsets ha v e a deformation retract ev en if there are small 0-critical v alues. W e now wan t to present a paradigm for finding sufficient conditions on p oin t cloud data for recon- structing any compact subset, lying in any Riemannian manifold, which has p ositiv e weak feature size. The first observ ation we need is that for Corollary 5.1 it is sufficien t to hav e low er b ounds on the sectional curv ature and the injectivity radius only for the p oints in A 6 r and A 3 r resp ectiv ely . This is b ecause no p oin ts outside this region are used in any of the pro ofs. Since A is compact there is some r > 0 such that the injectivity radius of ev ery p oint in A 3 r is greater than r . Reduce r if necessary to ensure that r < wfs( A ) where wfs( A ) is the weak feature size of A whic h w e hav e assumed is p ositive. A 3 r is compact so there is some finite lo w er b ound on the sectional curv ature for p oin ts in A 3 r . By rescaling the metric on the ambien t manifold if necessary (and with it scaling r ) we can assume that the lo wer b ound on sectional curv ature is 0 or − 1. This means w e can apply Corollary 5.1. It is clear a suitable µ and b ound on d H ( A, S ) in the Corollary m ust exist. Because r < wfs( A ) we can further state that the S r deformation retracts to A . This paradigm of reconstruction pro cesses shows that what the ambien t manifold is do es not p ose a theoretical barrier to the existence of reconstruction pro ofs. The homological feature size of a set A is the infim um of the distances α > 0 such that A α has a differen t homology to A . If we were only interested in reconstructing a set with the same homology as the original set it would b e sufficient to do the ab ov e reconstruction pro cess with replacing the w eak feature size w ith the homological feature size. After applying Corrollary 5.1 to show S r is homotopic to A r w e observe that since the homological feature size of A is greater than r then A r is homotopic to A . An alternativ e approach, as p oin ted out in [7], is to consider geometric prop erties of S (or in their case offsets of S ) itself rather than A . W e can tak e his approac h b ecause S is a compact set and w e do not require any smo oth structure. This means we can also conclude another corollary with K = S and L = A . 16 Corollary 5.2. L et µ ∈ (0 , 1) , r > 0 . L et M b e a smo oth manifold with se ctional curvatur e b ounde d by κ whose inje ctivity r adius is gr e ater than r . L et A b e a c omp act subset of M and S b e a (p otential ly noisy) p oint cloud of A . Supp ose that ther e ar e no µ -critic al p oints in S [ a,b ] . Then S r deformation r etr acts to A r whenever d H ( S, A ) ≤ min  r − a, bµ − r µ 4 − µ  , and d H ( S, A ) < µ 2 r 4 + µ 2 if κ = 0 or d H ( S, A ) ≤ min  r − a, bµ − r µ 4 − µ  , and d H ( S, A ) < 2 9 tanh( r − d H ( S, A )) µ 2 if κ = − 1 . F urthermor e if A r is homotopic to A then S r is homotopic to A . When the am bient space is Euclidean, it is reasonable to w ant to compare our reconstruction pro cess to previous ones in the literature. Since these hav e b een quan tified in terms of µ -reac h we can first compare the required Hausdorff b ounds on δ := d H ( A, S ) where A is a compact set with µ -reach r µ > 0 and S is a p oin t cloud. If we consider the limiting case when r − δ + 4 δ /C < r µ w e can apply our reconstruction theorem (in the case of κ = 0) once δ /r µ < µ 2 / (4 + 4 µ ) . Notably this is an impro v emen t on the b ounds presen ted in [7], which is δ /r µ < µ 2 / (5 µ 2 + 12), for all µ and an impro v emen t on the b ounds in [2], where it is δ r µ < − 3 µ + 3 µ 2 − 3 + p − 8 µ 2 + 4 µ 3 + 18 µ + 2 µ 4 + 9 + µ 6 − 4 µ 5 7 µ 2 + 22 µ + µ 4 − 4 µ 3 + 1 , for µ < 0 . 945. One adv antage of the approach of this pap er is not having any requiremen ts ab out the absence of µ -critical p oin ts very close to A . A sev ere limitation of restricting to set with p ositiv e µ -reach is the inabilit y to cop e with sets that hav e cusps. A t cusps the the µ -reach is zero for all v alues of µ > 0. The metho d used to ov ercome the shortfalls of µ -reach is to consider offsets of the compact set. F or a compact set K , there are no µ -critical p oin ts of d K in K [ a,b ] if and only if the µ -reac h of K a is at least b − a . This means we can compare different reconstruction theorems in terms of a lac k of µ critical p oin ts in an annular region. Let us assume that there are no µ -critical p oin ts of d K in K [ a,b ] . F or our reconstruction pro cess we need δ < min  µ ( b − a ) 4 , µ 2 b 4 + 4 µ  . Here w e w ould use r = b 4+ µ 2 4+4 µ . In comparison, for the reconstructions in [7] would need δ < ( b − a ) µ 2 5 µ 2 + 12 and the the reconstructions in [2] w e w ould need δ < ( b − a ) − 3 µ + 3 µ 2 − 3 + p − 8 µ 2 + 4 µ 3 + 18 µ + 2 µ 4 + 9 + µ 6 − 4 µ 5 7 µ 2 + 22 µ + µ 4 − 4 µ 3 + 1 whic h is significan tly w orse when b − a is small in comparison to b . One p ossible future direction is to use these results to find suitable sampling conditions for when a compact set can b e reconstructed. In particular, probabilistic results would b e interesting. 17 6 Index of Notation M is a smo oth Riemanian manifold which forms the ambien t space. A is a compact subset of M whic h w e desire to reconstruct. S is a noisy p oin t cloud sample of A . δ is a b ound on the Hausdorff distance b et ween tw o compacts sets. U T M is the unit tangent bundle of M . T x M is the tangen t plane to M at the p oin t x . γ is a geo desic on M (usually unit sp eed and alwa ys constant sp eed). x, y , z are p oin ts in M . exp x is the exp onen tial map from the tangent plane at x to M . Γ( γ ) is the isometry b et w een tangen t planes induced b y parallel transp ort along γ . w , v are unit tangent vectors. β , θ are angles. W e mainly care ab out acute angles. C ( w, β ) is a cone. It is a ball in the unit tangent sphere at a p oin t in M with cen ter w and radius β . W is a cone field. Also denoted by { ( x, C ( w x , β x )) } . W 0 denotes the complemen tary cone field to W when W is an acute cone field. F or W ab o ve it is { ( x, C ( w x , π / 2 − β x )) } . X is a vector field. K, L are compact subsets of M . d K is the distance function from K . K a is the a -offset of K . That is { x ∈ M : d K ( x ) ≤ a } . K [ a,b ] is the [ a, b ] ann ulus of K . 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