Average and Expected Distortion of Voronoi Paths and Scapes
aa r X i v : . [ m a t h . M G ] D ec Average and Expected Distortion of VoronoiPaths and Scapes ∗ Herbert Edelsbrunner and Anton Nikitenko
IST Austria (Institute of Science and Technology Austria), Am Campus 1,3400 Klosterneuburg, Austria, [email protected] , [email protected] Abstract
We generalize the concept of the Voronoi path of a line to more general shapes and computethe distortion constant, which describes how it changes volume on average. Although initiallyasked for a Poisson point process, the distortion is a characteristic property of the space ratherthan the point process. In other words, the constant ratio of the perimeter of a circle and itspixelation—and the analogous ratios for spheres in three and higher dimensions— hold for allsmoothly embedded shapes on average.
Keywords and phrases:
Distortion, Voronoi tessellations, Delaunay mosaics, Voronoi paths,Grassmannians, mixed volume, Poisson point processes, average, expectation.
Given a locally finite set A ⊆ R d and a line segment, the Voronoi path of the line segment isthe dual of the Voronoi tessellation of A intersected with the segment. We generalize it tothe Voronoi scape : given A and a p -dimensional set Ω ⊆ R d , we define it to be the dual of theVoronoi tessellation A intersected with Ω. More formally, it consists of all Delaunay cells, γ , whose dual Voronoi cells, γ ∗ , have a non-empty intersection with Ω. We are interested inthe distortion , which is the ratio of the p -dimensional volume of the Voronoi scape over thevolume of Ω.Considering the Voronoi tessellation of a stationary Poisson point process and a linesegment in R , [2] proves that the expected distortion is π . Extending this work to d > p d/π + O (1 / √ d ). We remove theambiguity in this answer by proving that the expected distortion in R d is d !! / ( d − d isodd, and 2 d !! / ( π ( d − d is even, in which !! is the double factorial. Furthermore, wegeneralize the result to any dimension and prove that for p -dimensional sets the expecteddistortion is the binomial coefficient (cid:0) d/ p/ (cid:1) , in which non-integer parameters are understoodin the way the Gamma function extends the factorial: D p,d = (cid:18) d/ p/ (cid:19) = Γ( d + 1)Γ( p + 1) Γ( d − p + 1) = d !! p !! ( d − p )!! 2 π if d is even and p is odd , d !! p !! ( d − p )!! otherwise . (1) ∗ This project has received funding from the European Research Council (ERC) under the EuropeanUnion’s Horizon 2020 research and innovation programme, grant no. 788183, from the WittgensteinPrize, Austrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative ResearchCenter TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grantno. I 02979-N35.
Average and Expected Distortion
The binomial interpretation also provides the asymptotics for D p,d . Table 1 shows thevalues for small dimensions. More precisely, we prove that (1) is the average distortion forsufficiently regular p -dimensional sets and Voronoi tessellations. The claim for stationaryPoisson point processes follows because they are invariant under rotations and translations.The proof is based on a decomposition of R d × G r p,d that is related to the mixed complex [6]. As a byproduct, we get an expression for the volumes of the cells in the mixed complex;see Corollary 5.1. d = 1 2 3 4 5 6 7 8 9 10 p = 1 1 π
32 163 π
158 325 π π π
53 1 π
52 323 π
358 25615 π π
105 1 π
72 25615 π
638 51215 π
107 1 π
92 51221 π
59 1 π
10 1Table 1: The average resp. expected distortion in small dimensions. Note that even rows andcolumns form the Pascal triangle.
Outline.
Section 2 prepares the proof of our main result by computing the first and secondmoments of the p -dimensional volume of the projection of a unit p -cube in R d . Section 3studies the space of point-direction pairs. Section 4 introduces a mild regularity conditionfor Voronoi tessellations. Section 5 computes the volume of the cells in the mixed complex.Section 6 proves that D p,d is the average distortion for p -dimensional shapes in R d . Section7 applies the result to the Poisson case. Section 8 concludes the paper. We need some preliminary computations. Let G r p,d be the linear Grassmannian manifold ,whose points are the p -planes that pass through the origin of R d . Given a p -dimensionalunit cube, E ⊆ R d , and a p -plane, L ∈ G r p,d , we write E | L for the projection of the cubeonto the plane, and k E | L k p for its p -dimensional volume. The j -th projection moment is theaverage j -th power of the volume of the projection. We express this moment as an integralover the Grassmannian equipped with the uniform probability measure in (2) and convertit to two equivalent expressions involving the angle to a fixed plane in (3) and (4): m ( j ) p,d = Z L ∈ G r p,d k E | L k jp d L, (2)= Z L ∈ G r p,d cos j ϕ ( L, L ) d L (3)= Z F ∈ S t p,d k F | L k jp d F. (4) . Edelsbrunner and A. Nikitenko 3 To explain (3) and (4), we fix the plane L ∈ G r p,d containing E . The angle between two p -planes, ϕ ( L, L ) ∈ [0 , π ], is defined as the arc-cosine of the ratio of k B | L k p over k B k p for any compact set with non-empty interior, B ⊆ L . The angle is symmetric, so we caninstead consider the integrand in (3) as the projection of a unit p -cube in a random p -planeonto L . Formally, we write S t p,d for the Stiefel manifold of orthonormal p -frames in R d , weidentify a frame with the unit p -cube it spans, and we integrate using the uniform probabilitymeasure of S t p,d to arrive at (4).By construction, the 0-th projection moment is equal to 1, independent of p and d .We compute the 1-st and 2-nd projection moments, which curiously both have intuitivegeometric interpretations. ◮ Lemma 2.1 (Projection Moments) . Let d ≥ and ≤ p ≤ d . Then m (1) p,d = Γ( p +12 ) Γ( d − p +12 )Γ( ) Γ( d +12 ) , (5) m (2) p,d = 1 / (cid:18) dp (cid:19) = p ! ( d − p )! d ! . (6) Proof.
The 1-st projection moment appears in the classic Crofton formula of integral geo-metry, which says that the volume of a convex body is proportional to the average volumeof its projections. The constant of proportionality given in (5) can be found in [9, Formula(5.8)]. We use (4) together with a generalization of the Pythagorean theorem to computethe 2-nd moment. By Pythagoras, the squared length of a line segment is the sum of squaredlengths of its projections onto the coordinate axes. The Cauchy–Binet formula [4, §4.6] canbe used to generalize this to the squared volume of a p -dimensional parallelepiped in R d .Let P be such a parallelepiped, and write P i for its projection onto the i -th coordinate p -plane (in which the numbering is arbitrary). There are (cid:0) dp (cid:1) coordinate p -planes, and theCauchy–Binet formula asserts k P k p = X ( dp ) i =1 k P i k p . (7)Letting P = F ∈ S t p,d be the uniformly random unit p -cube, we can take the expectationon both sides of (7). We get 1 on the left-hand side and the sum of (cid:0) dp (cid:1) identical terms onthe right-hand side. Hence, the average squared p -dimensional volume of the projection is1 / (cid:0) dp (cid:1) , as claimed. (cid:4) We set D p,d = m (1) p,d / m (2) p,d and leave it to the reader to verify that this agrees with (1),where D p,d is given in terms of Gamma functions as well as double factorials. We use the Delaunay mosaic to tile the space of point-direction pairs , R d × G r p,d . Given aDelaunay mosaic of a set A ⊆ R d , denoted Del( A ), consider a p -dimensional cell, γ ∈ Del( A ),and its dual ( d − p )-dimensional Voronoi cell, γ ∗ ∈ Vor( A ). We define the p -tile of γ to consistof all pairs ( x, L ) ∈ R d × G r p,d such that L + x has a non-empty intersection with γ ∗ , and x lies in the projection of γ onto L + x : J ( γ, γ ∗ ) = { ( x, L ) ∈ R d × G r p,d | x ∈ γ | L + x and ( L + x ) ∩ γ ∗ = ∅} . (8)The tiles decompose the space R d × G r p,d in the sense that they cover the space of point-direction pairs and their interiors are pairwise disjoint. Since the detailed analysis of theboundaries is irrelevant for the current work, we only prove a weaker statement. Average and Expected Distortion ◮ Lemma 3.1 (Uniqueness of Tile) . Let A ⊆ R d be locally finite with conv A = R d , and let ≤ p ≤ d . Then for almost every point-direction pair, ( x, L ) ∈ R d × G r p,d , there exists aunique p -tile, J ( γ, γ ∗ ) , that contains ( x, L ) . Proof.
Take any point-direction pair, ( x, L ). Assume without loss of generality that x = 0is the origin and L = R p is a coordinate p -plane in R d . Map each point a ∈ A to the point a ′ = a | L ∈ R p , and let a ′′ = −k a − a ′ k ∈ R be its weight . The weighted points define aweighted Voronoi tessellation and the corresponding weighted Delaunay mosaic; see e.g. [1].The mosaic is generically a simplicial complex and generally a polyhedral complex, whichis geometrically realized in R p by drawing each cell, γ , as the convex hull of the points thatgenerate the p -dimensional Voronoi cells sharing γ ∗ . Consider the cells in Vor( A ) that havea non-empty intersection with L , write V L ( A ) for the collection of dual cells in Del( A ), andobserve that V L ( A ) is the Voronoi scape of L .As proved in [10], the weighted Voronoi tessellation is the intersection of L with Vor( A )and, by duality, the weighted Delaunay mosaic is the orthogonal projection of V L ( A ) to L .If L and Del( A ) are in general position, then all Delaunay cells in V L ( A ) project injectivelyto L , and the cells of dimension less than p form a set of zero measure. If Del( A ) covers R d ,then the weighted Delaunay mosaic covers R p . Hence, for almost all point-direction pairs,( x, L ), there is a unique Delaunay p -cell γ , such that ( x, L ) ∈ J ( γ, γ ∗ ), as claimed. (cid:4) The proof of the lemma gives some insight into the motivation for choosing this particulartiling of the space of point-direction pairs. We now compute the measure of a tile. ◮ Lemma 3.2 (Volume of Tile) . The measure of J = J ( γ, γ ∗ ) is k J k = k γ k p k γ ∗ k d − p / (cid:0) dp (cid:1) . Proof.
The measure of the tile is the integral of 1 over its pairs. Setting x = y + z , in which y ∈ L and z ∈ L ⊥ , the integral is k J k = Z L ∈ G r p,d Z y ∈ L y ∈ γ | L Z z ⊥ L ( L + z ) ∩ γ ∗ = ∅ d z d y d L (9)= k γ k p k γ ∗ k d − p Z L ∈ G r p,d cos ϕ ( L, γ ) d L, (10)where we get (10) by noticing that the innermost integral in (9) is the ( d − p )-dimensionalvolume of the projection of γ ∗ to L ⊥ , which is t k γ ∗ k d − p with t = cos ϕ ( L, γ ), and the middleintegral is the p -dimensional volume of the projection of γ to L , which is t k γ k p . Using (3)and (6), we see that integral in (10) is m (2) p,d . (cid:4) We take a closer look at the projection of a tile to R d . Let ( x, L ) be a point-direction pairin J = J ( γ, γ ∗ ) with dim γ = p . There are points u ∈ γ and v ∈ γ ∗ such that x = u | L + x and v = ( L + x ) ∩ γ ∗ . Because of the right angle between the direction and the projection, wehave k x − u k + k x − v k = k u − v k , so x lies on the smallest sphere that passes through u and v . Indeed, u and v define a ( d − z = z ( γ, γ ∗ ) = aff γ ∩ aff γ ∗ andobserve that the sphere defined by u and v also passes through z . Let R = R ( γ, γ ∗ ) be themaximum distance between a point of γ and a point of γ ∗ and note that R is the radiusof every largest empty sphere that passes through the vertices of γ . Since the diameter ofthe sphere spanned by u and v is k u − v k ≤ R , it follows that the ball with center z andradius R contains this sphere. This implies an upper bound on the size of the projectionof the tile to R d , which we state formally for later reference. . Edelsbrunner and A. Nikitenko 5 ◮ Lemma 3.3 (Projection of Tile) . The projection of J = J ( γ, γ ∗ ) to R d is contained in theclosed ball with center z = z ( γ, γ ∗ ) and radius R = R ( γ, γ ∗ ) . It follows that the volume of the projection of the tile to R d is at most R d times thevolume of the unit ball in R d . Since we assume the uniform probability measure on G r p,d ,the same upper bound holds for the measure of the tile itself. Taking the union of progressively more tiles, we eventually cover all of R d × G r p,d . However,at each step some of the points miss some of the directions, and which directions are covereddepends on the mosaic. For what follows, we require a mild regularity condition for thistiling. For a set Ω ⊆ R d we call a tile a boundary tile of Ω if its projection to R d contains atleast one point inside and at least one point outside Ω. ◮ Definition 4.1 (Mixed Regularity) . Let A ⊆ R d be locally finite. We say that A has theproperty of mixed regularity if, for any p , the total measure of the boundary p -tiles of a d -ball of radius R centered at the origin is o ( R d ).Note that conv A = R d is necessary for A to have the mixed regularity property. Indeed,if conv A does not cover R d , then there exists an unbounded Voronoi cell and thus a tile withinfinite measure. Motivated by the analysis in Section 3, we give some sufficient conditionsfor a set A ⊆ R d to satisfy the mixed regularity property. ◮ Lemma 4.2 (Sufficient Conditions) . A locally finite set A ⊆ R d has the mixed regularityproperty if one of the following holds: The radii of all circumspheres of top-dimensional Delaunay cells are bounded. Each ball in R d of radius greater than R contains a point of A . There is a function g ( R ) = o ( R ) such that every ball of radius g ( R ) that intersects the d -ball of radius R centered at the origin contains at least one point of A . Conditions 1 and 2 are equivalent, while the last one is weaker. We finish the sectionwith an application to Poisson point processes. ◮ Lemma 4.3 (Mixed Regularity in Expectation) . A stationary Poisson point process A ⊆ R d satisfies the mixed regularity property in expectation; that is: the total expected measure ofthe boundary tiles of a d -ball of radius R centered at the origin is o ( R d ) . Proof.
For each boundary tile, its Delaunay cell (which is almost surely a simplex) is aface of a top-dimensional Delaunay simplex whose circumsphere intersects the boundary ofthe ball. In [7, Appendix A] it is proved that the total number of such spheres is o ( R d ).A minor modification of that proof suffices to show that the total volume of these balls is o ( R d ). Together with Lemma 3.3, this implies the claimed bound. (cid:4) Call k γ k p k γ ∗ k d − p the mixed volume of a p -cell γ ∈ Del( A ) and its dual ( d − p )-cell γ ∗ ∈ Vor( A ). We note that this concept is related to a particular decomposition of R d , as wenow explain. Given A ⊆ R d , the d -dimensional cells of the mixed complex defined in [6] aretranslates of the products γ × γ ∗ . The d -dimensional volume of this cell is k γ × γ ∗ k d = k γ k p k γ ∗ k d − p / d . As proved in [6], the cells in the mixed complex have pairwise disjoint Average and Expected Distortion interiors and they cover R d . Assuming the mixed regularity property, this implies that,up to a lower order term, the cells for p = 0 cover a fraction of 1 / d of the ball of radius R centered at the origin. By symmetry, this is also true for p = d . We continue with ageneralization of these bounds to dimension p between 0 and d . ◮ Corollary 5.1 (Mixed Volumes) . Let A ⊆ R d have the mixed regularity property. For any ≤ p ≤ d , the sum of the mixed volumes over all p -dimensional Delaunay cells containedin a ball of radius R is ν d (cid:0) dp (cid:1) R d + o ( R d ) , in which ν d is the volume of the unit ball in R d . Proof.
Let B ( R ) be the ball with radius R centered at the origin. Set B p ( R ) = B ( R ) × G r p,d ,let M p ( R ) be the smallest collection of p -tiles whose union contains B p ( R ), and let ∂M p ( R )be the boundary tiles of B ( R ). Clearly, M p ( R ) \ ∂M p ( R ) ⊆ B p ( R ) ⊆ M p ( R ) . (11)We note, that if a tile, J = J ( γ, γ ∗ ), contains a point inside the ball, then either γ is insidethe ball, or J is a boundary tile. Indeed, for every point x ∈ γ \ B ( R ), there is a direction L , such that L + x intersects γ ∗ , hence ( x, L ) ∈ J ( γ, γ ∗ ). By Lemma 3.2, the measure ofthis tile is k γ k p k γ ∗ k d − p / (cid:0) dp (cid:1) and, by the mixed regularity property, the measures of the tilescorresponding to Delaunay cells inside the ball sum up to k B ( R ) k d (1 + o (1)). Multiplyingby (cid:0) dp (cid:1) concludes the proof. (cid:4) Recall that the Voronoi scape of a set Ω ⊆ R d and a locally finite set A ⊆ R d consists of allDelaunay cells whose dual Voronoi cells have a non-empty intersection with Ω: V Ω ( A ) = { γ ∈ Del( A ) | Ω ∩ γ ∗ = ∅} . (12)We are ready to prove the main result of the paper. ◮ Theorem 6.1 (Voronoi Scapes of Flat Shapes) . Let A ⊆ R d have the mixed regularityproperty, and let Ω be a p -dimensional rectifiable set contained a p -dimensional plane in R d .The average volume of V Ω ( A ) over all congruent copies of Ω inside the d -ball of radius R centered at the origin is k Ω k p ( D p,d + o (1)) as R goes to infinity. Proof.
By the Crofton formula, the total measure of the ( d − p )-planes that intersect Ω is k Ω k p m (1) p,d [9, Formula (5.7)]. Moving Ω instead of the planes, we see that the total measure ofcongruent copies of Ω that have a non-empty intersection with a given ( p − d )-dimensionalpolytope γ ∗ is k γ ∗ k d − p k Ω k p m (1) p,d . Here we use that Ω is flat, which implies that almostevery congruent copy intersects γ ∗ in at most one point. A p -cell γ ∈ Del( A ) belongs to theVoronoi scape of a congruent copy Ω ′ of Ω iff Ω ′ ∩ γ ∗ is not empty, and we just computedthe total measure of such copies. The total contribution of γ to the p -volume of the Voronoiscapes of the congruent copies of Ω is thus k γ k p k γ ∗ k d − p k Ω k p m (1) p,d . We get the final resultby dividing the total contribution of the Delaunay cells inside the ball—which we computeusing Corollary 5.1—by the total measure of the congruent copies inside the ball: k B ( R ) k d k Ω k p m (1) p,d / m (2) p,d k B ( R ) k d (1 + o (1)) = k Ω k p ( D p,d + o (1)) , (13)in which we use (5) and (6) to get the expression on the right. (cid:4) . Edelsbrunner and A. Nikitenko 7 The proof of Theorem 6.1 makes use of the fact that Ω intersects almost every ( d − p )-polytope in at most one point, which is not true for general p -dimensional shapes. Wetherefore refrain from making any claim for this more general case, and only give a limitingtheorem for the Poisson point process shortly. We finish with stating the answer to the original question that motivated the work reportedin this paper. We showed in Section 4 that the stationary Poisson point process has themixed regularity property in expectation, which allows us to repeat all results while addingthe expectation to all quantities. By the isometry invariance of the process, for any setΩ, the expected volume of V Ω ( A ) does not depend on the position of Ω. Exchanging theexpectation and the average inside the ball of radius R centered at the origin and letting R go to infinity, we arrive at probabilistic versions of Theorem 6.1. ◮ Theorem 7.1 (Expected Volume for Flat Shapes) . Let A ⊆ R d be a stationary Poisson pointprocess, and let Ω be a rectifiable set in a p -dimensional plane in R d . Then the expected p -dimensional volume of the Voronoi scape of Ω is D p,d k Ω k p . For a non-flat p -dimensional set, we can get artifacts in the Voronoi scape. Considerfor example the unit circle in R . It is not difficult to prove that a sufficient small positiveconstant bounding the gaps between the points in A from above implies a Voronoi scapethat has the homotopy type of the circle. In contrast, there is no such constant that wouldimply the topology type of the circle [8]. Indeed, the circle may cross a Voronoi edge twice,in which case the Voronoi scape has an edge sticking out, like a spine of a cactus. This edgeand similar artifacts are caused by multiple local intersections between Ω and ( d − p )-cellsin the Voronoi tessellation. This is the only apparent obstacle to extending Theorem 6.1:the measure of congruent copies of a Voronoi cell, γ ∗ , intersecting Ω diverges slightly from k γ ∗ k d − p times the number of intersection points. For a smoothly embedded p -manifold andany fixed direction of γ ∗ , the divergence happens only around tangent ( d − p )-planes in thisdirection, which vanishes as we tighten the notion of locality, e.g. by increasing the intensityof the Poisson point process. This implies an extension of Theorem 7.1 in the limit. ◮ Theorem 7.2 (Expected Volume for Smooth Shapes) . Let A ⊆ R d be a stationary Poissonpoint process with intensity ρ > , and let Ω be a compact and smoothly embedded p -manifoldin R d . Then the expected p -dimensional volume of the Voronoi scape of Ω goes to D p,d k Ω k p as ρ goes to infinity. The main contribution of this paper is a complete analysis of the average and expecteddistortion of p -dimensional Voronoi scapes in R d , for 0 ≤ p ≤ d . For p = 1, these scapesare known as Voronoi paths, for which the expected distortion has been studied but wasknown only in R ; see [2]. A useful insight from our analysis is that the expected distortionfor a stationary Poisson point process is the average distortion of a general locally finitepoint set. We make crucial use of this insight in the proof of our results. On the otherhand, the properties that make a mosaic a Delaunay mosaic are not used other than in thequantification of the mixed regularity property for locally finite sets. Indeed, we only needa pair of dual complexes in which dual cells are orthogonal to each other, a property that Average and Expected Distortion holds also for the generalizations of Voronoi tessellations and Delaunay mosaics to pointswith real weights; see e.g. [1].As a consequence of the expected distortion for planes, we get the expected p -dimensionalvolume of the Voronoi scape of a smoothly embedded p -manifold in R d . Can these resultsbe extended to other measures, such as notions of curvature, for example?
The proof ofTheorem 6.1 suggests that this extension would require a detailed analysis of the Croftonformula. Insights in this direction could be helpful in using the Voronoi scape to measureotherwise difficult to measure shapes [3].
Acknowledgements
The authors thank Ranita Biswas and Tatiana Ezubova for the collaboration on an experimental projectthat motivated the work reported in this paper.
References F. Aurenhammer.
Power diagrams: properties, algorithms and applications.
SIAM J. Comput. (1987), 78–96. F. Baccelli, K. Tchoumatchenko and S. Zuyev.
Markov paths on the Poisson–Delaunaygraph with applications to routing in mobile networks.
Adv. Appl. Probab. (2000), 1–18. R. Biswas, H. Edelsbrunner, T. Ezubova and A. Nikitenko.
Image analysis with isotropicrandom background. Manuscript, IST Austria, Klosterneuburg, Austria, 2020. J.G. Broida and S.G. Williamson.
A Comprehensive Introduction to Linear Algebra.
Addison-Wesley, Reading, Massachusetts, 1989. P.M.M. de Castro and O. Devillers.
Expected length of the Voronoi path in a high dimensionalPoisson–Delaunay triangulation.
Discrete Comput. Geom. (2018), 200–219. H. Edelsbrunner.
Deformable smooth surface design.
Discrete Comput. Geom. (1999), 87–115. H. Edelsbrunner, A. Nikitenko and M. Reitzner.
Expected sizes of Poisson–Delaunay mosaicsand their discrete Morse functions.
Adv. Appl. Prob. (2017), 745–767. H. Edelsbrunner and N.R. Shah.
Triangulating topological spaces.
Internat. J. Comput. Geom.Appl. (1997), 365–378. R. Schneider and W. Weil.
Stochastic and Integral Geometry.
Springer, Berlin, Germany, 2008. R. Sibson.
A vector identity for Dirichlet tessellations.
Math. Proc. Cambridge87