 # How Many Directions Determine a Shape and other Sufficiency Results for Two Topological Transforms

In this paper we consider two topological transforms based on Euler calculus: the persistent homology transform (PHT) and the Euler characteristic transform (ECT). Both of these transforms are of interest for their mathematical properties as well as their applications to science and engineering, because they provide a way of summarizing shapes in a topological, yet quantitative, way. Both transforms take a shape, viewed as a tame subset M of R^d, and associates to each direction v in S^d-1 a shape summary obtained by scanning M in the direction v. These shape summaries are either persistence diagrams or piecewise constant integer valued functions called Euler curves. By using an inversion theorem of Schapira, we show that both transforms are injective on the space of shapes---each shape has a unique transform. By making use of a stratified space structure on the sphere, induced by hyperplane divisions, we prove additional uniqueness results in terms of distributions on the space of Euler curves. Finally, our main result provides the first (to our knowledge) finite bound required to specify any shape in certain uncountable families of shapes, bounded below by curvature. This result is perhaps best appreciated in terms of shattering number or the perspective that any point in these particular moduli spaces of shapes is indexed using a tree of finite depth.

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## 1. Introduction

In this paper we consider two topological transforms that are of theoretical and practical interest: the Euler Characteristic Transform (ECT) and the Persistent Homology Transform (PHT). At a high level both of these transforms take a shape, viewed as a subset of , and associates to each direction a shape summary obtained by scanning in the direction . This process of scanning has a Morse-theoretic and persistent-topological flavor—we study the topology of the sublevel sets of the height function as the height varies. These evolving sublevel sets are summarized using either the Euler Curve, which is the assignment of heights to Euler characteristic of the corresponding sublevel set, or the persistence diagram, which pairs critical values of in a computational way. Consequently, a more precise summary of the ECT and the PHT is that they associate to any sufficiently tame subset a map from the sphere to the space of Euler curves and persistence diagrams, respectively.

Before introducing the mathematical properties of these transforms, as well as the results proved in this paper, we would like to point out some of the applications of interest. In both data science and computational geometry, quantifying differences in shape is a difficult problem. Part of the problem is structural: most statistical analysis operates on scalar-valued quantities, but it is often very hard to summarize a shape with a single number. Nevertheless, both Euler curves and persistence diagrams have metrics on them that are good for detecting qualitative and quantitative differences in shape. For example, in

 the persistent homology transform was introduced and applied to the analysis of heel bones from primates. By quantifying the shapes of these heel bones and clustering using certain metrics on persistence diagrams, the authors of that paper were able to automatically differentiate species of primates using quantified differences in their particular phylogenetic expression of heel bone shape. A variant on the Euler characteristic transform was also used in  to discriminate between tumor types. Here the applicability of topological methods is particularly compelling: agressive tumors tend to grow “roots” that invade nearby tissues and these protrusions are often well-detected using critical values of the Euler characteristic transform.

A question of interest to both theoretical and applied mathematicians is the extent to which any shape summary is lossy. Topological invariants such as Euler characteristic and homology are in and of themselves obviously lossy; there are non-homeomorphic spaces that appear to be identical when using these particular lenses for viewing topology. The surprising fact developed in [13, 14] and carried further in this paper and independently in , is that by considering the Euler curve for every possible direction one can completely recover a shape. Said differently, the Euler characteristic transform is a sufficient statistic for compact, definable subsets of , see Theorem 3.1. Moreover, since homology determines Euler characteristic via an alternating sum of Betti numbers, we obtain injectivity results for the PHT as well, see Theorem 4.1.

The choice of our class of shapes is an important variable throughout the paper. We use “constructible sets” at first, these are compact subsets of that can be constructed with finitely many geometric and logical operations. The theory for carving out this collection of sets comes from logic and the study of o-minimal/definable structures, as they provide a way of banishing shapes are are considered “too wild.” Every constructible set can be triangulated, for example. Moreover, the world of definable sets and maps is the natural setting for Schapira’s inversion result , which is the engine that drives many of the results in our paper, specifically Theorems 3.1 and 4.1, and independently Theorem 5 and Corollary 6 in .

An additional reason to consider o-minimal sets is that they are naturally stratified into manifold pieces. This is important because it implies that the constructions that underly the ECT and PHT are also stratified, and that both of these two transforms can be described in terms of constructible sheaves. The upshot of this observation, which is developed in Section 5, is that the ECT of a particular shape should be determined, in some sense, by finitely many directions, certainly if is known in advance. Indeed, any o-minimal set induces a stratification of the sphere of directions, and whenever we restrict to a particular stratum the variation of the transform can be described by homeomorphisms of the real line. To make precise how to relate the transform for two different directions in a fixed stratum, we specialize to the case where is a geometric simplicial complex. This allows us to provide an explicit formula for the Euler characteristic transform in addition to providing explicit linear relations for two directions in the same stratum. This is the content of Lemma 5.3 and Proposition 5.1, as well as its persistent homology variation in Proposition 5.2.

By considering a “generic” class of embedded simplicial complexes, we leverage the explicit formula for the Euler characteristic transform to produce a new measure-theoretic perspective on shapes. In Theorem 6.1 we prove that by considering the pushforward of the Lebesgue measure on into the space of Euler curves (or persistence diagrams) one can uniquely determine a generic shape up to rotation and reflection, i.e. an element of . The importance of this result for applications is that one can faithfully compare two un-aligned shapes simply by studying their associated distribution of Euler curves.

Finally, we extend the themes of stratification theory and injectivity results to answer our titular question, which is perhaps more accurately worded as “How many directions are needed to infer a shape?” Here the idea is that there is a shape hidden from view, perhaps cloaked by our sphere of directions, and we would like to learn as much about the shape as possible. Our mode of interrogation is that we can specify a direction and an oracle will tell us the Euler curve of the shape when viewed in that direction. Of course, by our earlier injectivity result, Theorem 3.1, we know that if we query all possible direction, then we can uniquely determine any constructible shape. However, a natural question of theoretical and practical importance is whether finitely many queries suffice. The main result of our paper, Theorem 7.1, shows that if we impose some a priori assumptions on our (uncountable) set of possible shapes, then finitely many queries indeed do suffice. However, unlike the popular parlor game “Twenty Questions,” the number of questions/directions we might need to infer our hidden shape is only bounded by

 Δ(d,δ,k)=⎛⎝(d−1)k(2δsin(δ))d−1+1⎞⎠(1+2δ)d+O(dkδd−1)2d.

The parameters above reflect a priori geometric assumptions that we make about our hidden shape. Specifically, we assume our shape is well modeled as a geometric simplicial complex embedded in , with a lower bound on the “curvature” at every vertex, and a uniform upper bound on the number of “critical values” of the ECT in any direction. We call this class of shapes . Moreover, the two terms in the above sum are best understood as a two step interrogation procedure. The first term counts the number of directions that we’d sample the ECT at for any shape. The second term in the above sum counts the maximum number of questions we might ask, adapted to the answers given in the first round of questions. We note that there is perhaps an interesting way of interpreting our result as a shattering number for the set of shapes . Every question/direction “shatters” the set of possible shapes into two pieces and our results prove that we can uniquely specify any element of after at most number of shatters.

## 2. Background on Euler Calculus

Often in geometry and topology we require that our sets and maps have certain tameness properties. These tameness properties are exhibited by the following categories: piecewise linear, semialgebraic, and subanalytic sets and mappings. Logicians have generalized and abstracted these categories into the notion of an o-minimal structure, which we now define.

###### Definition 2.1.

An o-minimal structure specifies for each , a collection of subsets of closed under intersection and complement. These collections are related to each other by the following rules:

1. If , then and are both in ; and

2. If , then where is axis-aligned projection.

We further require that contains all algebraic sets and that contain no more and no less than all finite unions of points and open intervals in . Elements of are called tame or definable sets. A definable map is one whose graph is definable.

###### Example 2.1.

The collection of semialgebraic sets, sets defined in terms of polynomial inequalities, is an example of an o-minimal structure. Subanalytic sets provide another, larger example of an o-minimal structure.

Tame/definable sets play the role of measurable sets for an integration theory based on Euler characteristic called The Euler Calculus, see  for an expository review. The guarantee that definable sets can be measured by Euler characteristic is by virtue of the following theorem.

###### Theorem 2.1 (Triangulation Theorem ).

Any tame set admits a definable bijection with a subcollection of open simplices in the geometric realization of a finite Euclidean simplicial complex. Moreover, this bijection can be made to respect a partition of a tame set into tame subsets.

In view of the Triangulation Theorem, one can define the Euler characteristic of a tame set in terms of an alternating count of the number of simplices used in a definable triangulation.

###### Definition 2.2.

If is tame and is a definable bijection with a collection of open simplices, then the definable Euler characteristic of is

 χ(X):=∑i(−1)dimσi

where denotes the dimension of the open simplex . We understand that since this corresponds to the empty sum.

As one might expect, the definable Euler characteristic does not depend on a particular choice of triangulation [16, pp.70-71], as it is a definable homeomorphism invariant. However, as the next example shows, the definable Euler characteristic is not a homotopy invariant.

###### Example 2.2.

The definable Euler characteristic of the open unit interval is . Note that is contractible to a point, which has definable Euler characteristic . The reader might notice that these computations coincide with the compactly-supported Euler characteristic, which can be defined in terms of the ranks of compactly-supported cohomology or Borel-Moore homology groups.

###### Remark 2.1.

We will often drop the prefix “definable” and simply refer to “the Euler characteristic” of a definable set.

Like its compactly-supported version, the definable Euler characteristic satisfies the inclusion-exclusion rule:

###### Proposition 2.1 ( p.71).

For tame subsets we have

 χ(A∪B)+χ(A∩B)=χ(A)+χ(B)

Consequently, the definable Euler characteristic specifies a valuation on any o-minimal structure, i.e. it serves the role of a measure without the requirement that sets be assigned a non-negative value. This allows us to develop an integration theory called Euler calculus that is well-defined for so-called constructible functions, which we now define.

###### Definition 2.3.

A constructible function is an integer-valued function on a tame set with the property that every level set is tame. The set of constructible functions with domain , denoted , is closed under pointwise addition and multiplication, thereby making into a ring.

###### Definition 2.4.

The Euler integral of a constructible function is the sum of the Euler characteristics of each of its level-sets, i.e.

 ∫ϕdχ:=∞∑n=−∞χ(ϕ−1(n))

Like any good calculus, there is an accompanying suite of canonical operations in this theory—pullback, pushforward, convolution, etc.

###### Definition 2.5.

Let be a tame mapping between between definable sets. Let be a constructible function on . The pullback of along is defined pointwise by

 f∗ϕY(x)=ϕY(f(x)).

The pullback operation defines a ring homomorphism .

The dual operation of pushing forward a constructible function along a tame map is given by integrating along the fibers.

###### Definition 2.6.

The pushforward of a constructible function along a tame map is given by

 f∗ϕX(y)=∫f−1(y)ϕXdχ.

This defines a group homomorphism .

Putting these two operations together allows one to define our first topological transform: the Radon transform.

###### Definition 2.7.

Suppose is a locally closed definable subset of the product of two definable sets. Let and denote the projections from the product onto the indicated factors. The Radon transform with respect to is the group homomorphism that takes a constructible function on , , pulls it back to the product space , multiplies by the indicator function before pushing forward to . In equational form, the Radon transform is

 RS(ϕ):=πY∗[(π∗Xϕ)1S].

The following inversion theorem of Schapira  gives a topological criterion for the invertibility of the transform in terms of the subset .

###### Theorem 2.2 ( Theorem 3.1).

If and have fibers and in satisfying

1. for all , and

2. for all ,

then for all ,

 (RS′∘RS)ϕ=(χ1−χ2)ϕ+χ2(∫Xϕdχ)1X.

In the next section we show how to use Schapira’s result to deduce the injectivity properties of the Euler Characteristic and Persistent Homology Transforms.

## 3. Injectivity of the Euler Characteristic Transform

In the previous section we introduced background on definable sets, constructible functions and the Radon transform. We now specialize this material to the study of persistent-type topological transforms on definable subsets of . What makes these transforms “persistent” is that they study the evolution of topological invariants with respect to a real parameter. In this section we begin with the simpler of these two invariants, the Euler characteristic.

###### Definition 3.1.

The Euler Characteristic Transform takes a constructible function on and returns a constructible function on the whose value at a direction and real parameter is the Euler integral of the restriction of to the half space . In equational form, we have

 ECT:CF(Rd)→CF(Sd−1×R)whereECT(ϕ)(v,t):=∫x⋅v≤tϕdχ.

For many applications of interest, it suffices to restrict this transform to the collection of compact definable subsets of , which we call constructible sets , where we identify a definable subset with its associated constructible indicator function . When we fix and let vary, we refer to as the Euler curve for the direction . This allows us to equivalently view the Euler Characteristic Transform for a fixed as a map from the sphere to the space of Euler curves (constructible functions on ).

 ECT(M):Sd−1→CF(R)whereECT(M)(v)(t):=χ(M∩{x∣x⋅v≤t})
###### Remark 3.1.

The restriction to constructible sets CS permits us to ignore the difference between ordinary and compactly-supported Euler characteristic. This is because the intersection of a compact definable set and the closed half-space is compact and definable, and these two versions of Euler characteristic agree on compact sets. This restriction is not strictly necessary, but would require a reworking of certain aspects of persistent homology (reviewed in Section 4), which is beyond the scope of this paper.

###### Remark 3.2 (Constructibility).

The reader may wish to pause to consider why the ECT associates to a definable set , viewed as the constructible function , a constructible function on . This is again by virtue of o-minimality and its good behavior with respect to products, polynomial inequalities, and projections. In particular, one can associate to definable another definable set

 XM:={(x,v,t)∈M×Sd−1×R∣x⋅v≤t}

One can then project onto the last two coordinates to obtain a map

 π:XM→Sd−1×Rwhereπ−1(v,t)=Mv,t.

Notice that all the fibers are definable and vary over a definable base, which implies that this is a definable map. This observation provides an alternative definition of the Euler characteristic transform: The Euler Characteristic Transform is simply the pushforward of the indicator function along the definable map , which defines a constructible function on .

We now use Schapira’s inversion theorem, recalled as Theorem 2.2, to prove our first injectivity result.

###### Theorem 3.1.

Let be the set of constructible sets, i.e. compact definable sets. The map is injective. Equivalently, if and are two constructible sets give the same association of directions to Euler curves, then they are the same set. Said symbolically,

 ECT(M)=ECT(M′):Sd−1→CF(R)⇒M=M′
###### Proof.

Let . Let be the hyperplane defined by . By the inclusion-exclusion property of the definable Euler characteristic

 χ(M∩W) = χ({x∈M:x⋅v=t}) = χ({x∈M:x⋅v≤t}∩{x∈M:x⋅(−v)≤−t}) = χ({x∈M:x⋅v≤t})+χ({x∈M:x⋅(−v)≤−t})−χ(M) = ECT(M)(v,t)+ECT(M)(−v,−t)−ECT(M)(v,∞)

This means that from we can deduce the for all hyperplanes .

Let be the subset of where when is in the hyperplane . For simplicity, we denote the projection to by and the projection to by . For this choice of the Radon transform of the indicator function for at is

 (RS1M)(W) = (π2)∗[(π∗11M)1S](W) = ∫(x,W)∈S(π∗11M)dχ = ∫x∈M∩W(π∗11M)dχ = χ(M∩W)

This implies that from we can derive .

Similarly let be the subset of where when is in the hyperplane . For all , is the set of hyperplanes that go through and hence is and . For all , is the set of hyperplanes that go through and and hence is and . Applying Theorem 2.2 yields

 (RS′∘RS)(1M)=(−1)d−11M+12(1+(−1)d−2)χ(M)1Rd.

Note that if then , since determines the Euler characteristic of every slice. Moreover, if , then , which by inspecting the inversion formula above further implies that and hence . ∎

## 4. Injectivity of the Persistent Homology Transform

The primary transform of interest for this paper is the persistent homology transform, which was first introduced in  and was initially defined for an embedded simplicial complex in . The reader is encouraged to consult  (and  for a related precursor) for a more complete treatment of the persistent homology transform, but we will briefly outline how this transform can be defined for any constructible set .

As already noted, given a direction and a value , the sublevel set is the intersection of the constructible set with a closed half-space. This intersection has various topological summaries, one of them being the (definable) Euler Characteristic . One can also consider the (cellular) homology with field coefficients , which is defined in each degree

. These are vector spaces that summarize topological content of any topological space

, which will always be in this paper spaces of the form . In low degrees the interpretation of these homology vector spaces for a general space are as follows: is a vector space with basis given by connected components, is a vector space spanned by “holes” or closed loops that are not the boundaries of embedded disks, is a vector space spanned by “voids” or closed two-dimensional (possibly self-intersecting) surfaces that are not the boundaries of an embedded three dimensional space. The higher homologies are understood by analogy: these are vector spaces spanned by closed (i.e. without their own boundary) -dimensional subspaces of that are themselves not the boundaries of -dimensional spaces. The dimension of is also called the Betti number .

The proof that ordinary Euler characteristic is a topological invariant is best understood via homology. Indeed, the Betti numbers determine the Euler characteristic via an alternating sum:

 χ(X)=β0(X)−β1(X)+β2(X)−β3(X)+⋯

However, one feature that homology enjoys that the Euler characteristic does not is functoriality, which is the property that any continuous transformation of spaces

induces a linear transformation of homology vector spaces

for each degree . This is the key feature that defines sublevel set persistent homology.

###### Definition 4.1.

Let be a compact definable set and let be the restriction of the inner product to points . The sublevel set persistent homology group in degree between and is

 PHk(M,hv)(s,t)=imιs→tk:Hk(Mv,s)→Hk(Mv,t).

The remarkable feature of persistent homology is that one can encode the persistent homology groups for every pair of values using a finite number of points in the extended plane. This is done via the persistence diagram.

###### Definition 4.2.

Let be the part of the extended plane that is above the diagonal, i.e . The persistence diagram in degree associated to when filtered by sublevel sets of is the unique finite multi-set of points with the property that for every pair of values the following equality holds

 dimPHk(M,hv)(s,t):=rankιs→tk=card{PHk(M,hv)∩[−∞,s]×(t,∞]}.
###### Remark 4.1 (O-Minimality and Persistence).

The existence of the persistence diagram is a non-trivial result and depends on certain tameness properties of the maps . Finite dimensionality of all of the homology vector spaces suffices, but in the o-minimal setting things are even better behaved because the dimension of the persistent homology groups can only change finitely times. To see this we can modify the construction described in Remark 3.2

 π:XM,v→RwhereXM,v:={(x,t)∈M×R∣x⋅v≤t}

and is projection onto the second factor. Sublevel sets of are now encoded as fibers of the map and The Trivialization Theorem [16, p.7] implies that the topological type of the fiber of this map can only change finitely many times.

To consider persistence diagrams associated to different directions , we need to consider the set of all persistence diagrams.

###### Definition 4.3.

Persistence diagram space, written Dgm, is the set of all possible finite multi-sets of .

###### Remark 4.2.

We have used the term “space” with the implication that there is a topology on the set of all persistence diagrams. Indeed this topology comes from various choices of metrics on the set of persistence diagrams, the most notable one being the bottleneck distance. This distance is somewhat complicated to define and requires the addition of a countable union of copies of the diagonal to each persistence diagram. We refer the reader to  for the details on how the bottleneck distance is defined and the original proof that the persistence diagram is stable. Moreover, the bottleneck distance can be viewed as the -Wasserstein distance on the the space of persistence diagrams with . For statistical analysis it can be better to use other -Wasserstein distances, such as in  where the -Wasserstein distance was used. For more details about the geometry of the space of persistence diagrams under -Wasserstein metrics see .

###### Definition 4.4.

The persistent homology transform of a constructible set is the map that sends a direction to the set of persistent diagrams gotten by filtering in the direction of :

 PHT(M):v↦(PH0(M,hv),PH1(M,hv),…,PHd−1(M,hv))

Letting the set vary gives us the map

 PHT:CS(Rd)→C0(Sd−1,Dgmd)

Where is the set of continuous functions from to , the latter being equipped with some Wasserstein -distance.

Before moving on with the remainder of the paper, we offer a sheaf-theoretic interpretation of the persistent homology transform, which is not necessary for the remainder of the paper. The reader that is uninterested in sheaves can safely ignore the following remark.

###### Remark 4.3 (Sheaf-Theoretic Definition of the PHT).

Extending Remark 3.2, we know that associated to any constructible set is a space

 XM:={(x,v,t)∈M×Sd−1×R∣x⋅v≤t}

and a map whose fiber over is the sublevel set . The derived persistent homology transform is the right derived pushforward of the constant sheaf on along the map , written . The associated cohomology sheaves of this derived pushforward, called the Leray sheaves in , has stalk value at the cohomology of the sub-level set . If we restrict the sheaf to the subspace , then one obtains a constructible sheaf that is equivalent to the persistent (co)homology of the filtration of viewed in the direction of . The persistence diagram in degree is simply the expression of this restricted sheaf in terms of a direct sum of indecomposable sheaves.

We now give a persistent analog of the classical result that homology determines the Euler characteristic.

###### Proposition 4.1.

The Persistent Homology Transform () determines the Euler Characteristic Transform (), i.e. we have the following commutative diagram of maps.

###### Proof.

The persistence diagram determines the homology of the sublevel set by the cardinality of the intersection of with the half-open quadrant . This is equal to , which is . Consequently we can associate to the integer valued function

 χ(PHT(M))(v,t)=n∑k=0(−1)kdimPHk(M,hv)(t,t)=n∑k=0(−1)kβk(Mv,t)=χ(Mv,t),

which is transparently . ∎

###### Remark 4.4 (Grothendieck Group Interpretation).

Continuing Remark 4.3, the reader familiar with the Grothendieck group of constructible sheaves (see  for a clear and concise treatment) will note that Proposition 4.1 is precisely the statement that the image of the (derived) persistent homology transform in is the Euler characteristic transform.

We can combine Proposition 4.1 with Theorem 3.1 to obtain a generalization of an injectivity result proved in  for simplicial complexes in and .

###### Theorem 4.1.

Let be the set of constructible sets, i.e. compact definable subsets of . The persistent homology transform is injective.

###### Proof.

If two constructible sets and have , then they have , which by Theorem 3.1 implies that . ∎

###### Remark 4.5 (Co-Discovery).

In the final stages of preparing this article, a pre-print  by Rob Ghrist, Rachel Levanger, and Huy Mai appeared independently proving Theorem 4.1. The authors of that paper and this paper want to make clear that these results were independently discovered.

## 5. Stratified space structure of the ECT and PHT

One of the essential observations of this paper is that for constructible sets , the topological summaries provided by the ECT or PHT exhibit similarly tame or constructible behavior. To be precise, these transforms associate to every constructible set a stratification of the sphere , where the Euler curves or persistence diagrams associated to two directions and in the same stratum are related in a controlled way. We outline the high-level reasons this must be true before providing explict relationships for constructible sets that are piecewise linear. First, we recall what we mean by a stratified space structure. We note that there are many notions of a stratification of a space , perhaps the most famous being the one due to Whitney . The following definition was communicated to the authors by Robert MacPherson.

###### Definition 5.1.

A paracompact, Hausdorff space is stratified by a collection of subspaces , called strata, if

1. the strata partition ;

2. each stratum is a connected topological manifold of some dimension;

3. if , then ;

4. for any and any pair of points there is a stratum-preserving homeomorphism taking to .

A stratified map is a map of stratified spaces that sends strata to strata and which restricts to a fiber bundle with stratifiable fiber over each stratum in the codomain. A map is stratifiable if there are stratifications of and making a stratified map.

###### Example 5.1.

One can check directly that every Whitney stratification has the properties listed above. In particular, the fourth property comes from constructing a controlled vector field that flows one point to another.

As observed in Remarks 3.2 and 4.1, both the ECT and PHT can be viewed as auxiliary definable constructions associated to an o-minimal set . Since every o-minimal/definable set can be Whitney stratified , these auxiliary constructions are stratified as well.

###### Lemma 5.1.

For a general o-minimal set , the PHT or the ECT will induce a stratification of as well as a stratification of . Moreover, in the induced stratification of the sphere , a necessary condition for two directions and to be in the same stratum is that there is a stratum-preserving homeomorphism of the real line that induces an order preserving bijection between

 d−1⋃k=0⋃i∈PHk(M,hv){bki(v)}∪{dki(v)}→d−1⋃k=0⋃i∈PHk(M,hw){bki(w)}∪{dki(w)}.

These two sets being the union of all the birth times and death times of all the points in each of the corresponding persistence diagrams in all degrees , associated to filtering by and .

###### Proof.

The space

 XM:={(x,v,t)∈M×Sd−1×R∣x⋅v≤t}

is an element of , and thus admits a decomposition into strata satisfying the conditions listed in Definition 5.1. The projection map onto the section two factors is a continuous definable map and hence restricts to a stratifiable map . Note that the fiber of this map over a direction is the space .

We can also project further from onto and this map will be stratifiable as well. The fibers of this further projection will be stratified real lines, which are (possibly) refinements of the stratification given by taking all the critical values of and . Consequently, since stratifications imply local triviality of the stratified structure of the real line induced by the birth and death times of the persistence diagrams in each degree since each of these imply some topological change in the fiber of . ∎

There is sense in which Lemma 5.1 is unsatisfactory because it does not provide an explicit stratification, but just guarantees the existence of one, provided by the abstract properties of o-minimality. To provide more explicit relationships between Euler curves or persistence diagrams associated to vectors in the same stratum, we specialize to sets that are geometric simplicial complexes. We now recall some of the basic definitions.

###### Definition 5.2.

A geometric -simplex is the convex hull of affinely independent points and is denoted . We call a face of if .

###### Example 5.2.

For example, consider three points that determine a unique plane containing them. The -simplex is the vertex . The -simplex is the edge between the vertices and . Note that and are faces of . The -simplex is the triangle bordered by the edges , and , which are also faces of .

###### Remark 5.1 (Orientations).

Traditionally, we view the order of vertices in a -simplex as indicating an equivalence class of orientations, where if is a permutation then . However, for the purposes of this paper we can ignore orientation.

###### Definition 5.3.

A finite geometric simplicial complex is a finite set of geometric simplices such that

• Every face of a simplex in is also in ;

• If two simplices are in then their intersection is either empty or a face of both and .

###### Remark 5.2 (Re-Triangulation).

Strictly speaking we only care about the embedded image of the finite simplicial complex. We consider different triangulations as equivalent; our uniqueness results will always be up to re-triangulation.

For a pair of points let be the hyperplane that is orthogonal to the vector . This hyperplane divides the sphere into two halves depending on whether or . More generally, for a set of points we can define

 W(X):=⋃i≠jW({xi,xj})

as the union of the hyperplanes determined by each pair of points.

###### Definition 5.4.

Let be a finite set of points in and let be the union of hyperplanes determined by each pair of points. The hyperplane arrangement induces a stratification of , which we call the hyperplane division of by . The connected components of are the top ()-dimensional strata. We denote the union of these strata by . A stratum of consists of a single connected component of .

###### Remark 5.3 (Lower Dimensional Strata).

Other strata of the hyperplane division can be described precisely. Intuitively -dimensional strata correspond to (components of) complements of pair-wise intersections of distinct planes inside of , -dimensional strata correspond to complements of triple-intersections of planes in what is remaining, and so on. For our purposes, we only need the top-dimensional strata.

###### Lemma 5.2.

Let . If are in the same stratum of , then the order of is the same as the order of the .

###### Proof.

The proof is a result of the observation that and lie in the same hemisphere of and so if and only if . ∎

###### Definition 5.5.

For each vertex , the star of , denote is the set of simplices containing . Given a function we can define the lower star of with respect to , denoted , as the subset of simplicies whose vertices have function values smaller than or equal to . Both stars and lower stars are generally not simplicial complexes as they are not closed under the face relation.

###### Remark 5.4 (Topological Interpretation of the Star).

Although a geometric simplex is defined here in terms of convex hulls, which are closed when viewed as topological spaces, they are better viewed as topological spaces via their interiors. For example, if is the simplicial complex consisting of the 1-simplex along with its two faces and , then according to the above definition . If we identify each geometric simplex with its interior, then the star is the “open star,” which in this case is the half-open interval . Below we will give a combinatorial formula for computing the Euler characteristic of the star, which in this case is . Note that if one is used to thinking in terms of Euler characteristic of the underlying space, then one must use compactly-supported Euler characteristic of the open star in order for these viewpoints to cohere.

For a finite geometric simplicial complex with vertex set , the height function is the piece-wise linear extension of the restriction of to the set of vertices . When then all the function values of over are unique. This implies that each simplex belongs to a unique lower star, namely to the vertex with the highest function value.

Note that the sublevel set is homotopic to , this holds due to a deformation retraction. Since both and are also compact we conclude that they have the same definable Euler characteristic or Euler characteristic with compact support. We will use the notation

 K(x,v)={Δ∈St(x)⊆K∣x⋅v=maxy∈Δ{y⋅v}}

to denote the lower star of in the filtration by the height function in direction .

###### Lemma 5.3.

Let be a finite simplicial complex with vertex set . Let be a stratum of . Then for fixed , the lower stars are all the same for all . We will sometimes denote the lower star by to highlight this consistency. Moreover, for all we have the following formula for the Euler characteristic transform:

 ECT(K,v)=∑x∈X1{[v⋅x,∞)}χ(K(x,U))
###### Proof.

We can see that the agree for all for all by the observation that the vertices of appear in the same order in each of the . This is the subset of cells that are added at the height value .

Note that the change in the Euler characteristic of the sublevel sets of as the height value passes is

 χ(K(x,U))=∑Δ∈K(x,U)(−1)dimΔ.

Here we use the notation for the (compactly-supported) Euler characteristic of this set of simplicies. Consequently, we can write the Euler characteristic curve for in direction as

 ECT(K,v)=∑x∈X1{[v⋅x,∞)}χ(K(x,U))

Note that many of the elements in this sum are zero. ∎

###### Example 5.3.

As a simple example, let be and embedded geometric 2-simplex along with all of its faces and suppose is a direction in which, filtering by projection to , the vertices appear in the order , then , then . Note that , which has Euler characteristic 1; , which has Euler characteristic ; and , which has Euler characteristic . Note that the combinatorial formula we provided is equivalent to thinking of the compactly supported Euler characteristic of the topological spaces , , and , respectively.

We can observe that these Euler characteristic curves over are essentially linear in

. It is just that the linear interpolation occurs inside the dot products within the summand and not on the function level.

###### Proposition 5.1.

Let be a finite simplicial complex with vertex set . If lie in the same stratum of , then given the Euler curve for we can deduce the Euler curve for .

###### Proof.

Let be the stratum of containing and . Since , we know that for each vertex that is the only vertex of that appears at height . From the formula proved in Lemma 5.3 we have that we can deduce as the change in the Euler characteristic curve at . We can then plug each of these values into the formula

 ECT(K,w)=∑x∈X1{[w⋅x,∞)}χ(K(x,U))

to construct the Euler curve for the direction . ∎

Proposition 5.1 generalizes readily to the persistent homology transform.

###### Proposition 5.2.

Let be a finite simplicial complex with vertex set . If lie in the same stratum of , then the th persistence diagram associated to filtering by determines the th persistence diagram for filtering by .

###### Proof.

Let be the stratum of containing and . Since lies inside , every vertex of appears at a distinct height. From this we can deduce which pairs of vertices correspond to the birth-death coordinates of . Recall that the birth coordinate of a point in a persistence diagram corresponds to the largest value of when restricted to a simplex that generates new homology in degree . By our hypothesis that , the largest value is obtained at a vertex. The persistence algorithm pairs this generating simplex with another simplex that “kills” this class. The death coordinate of a point in the persistence diagram corresponds to the largest value of on this killing simplex, which by our assumption occurs at a vertex. Said succinctly, for a point in we know there are unique with and .

Since the vertices of appear in the same order for as for we know that the birth and death coordinates in occur at the heights of the same set of vertices. That is to say that the off diagonal points of are . ∎

## 6. Uniqueness of the Distributions of Euler curves up to O(d) actions

A practical challenge in comparing two “close” shapes using either of the two transforms that we have discussed—the ECT or the PHT—is that the shapes should be first aligned or registered in similar poses in order to have some guarantee that the resulting transforms are close. For example, if one wanted to compare simplicial versions of a lion and a tiger, we would need to first embed them in such a way that they are facing the same direction. Rephrase, for centered shapes we would like to make them as close as possible by using actions of the special orthogonal group . If we wish to optimize also allowing reflections we would like to make them as close as possible by using actions of the orthogonal group . In general, aligning or registering shapes is a challenging problem .

In this section we show how studying distributions on the space on Euler curves—or persistence diagrams, since homology determines Euler characteristic—can bypass this process. These distributions are naturally invariant of actions as Lebesgue measure on the sphere is invariant; acting on a shape and acting on the space of directions using the same element of produces the same Euler curve. The key development of this section, Theorem 6.1, is the proof that for “generic” shapes the distribution of Euler curves uniquely determines that shape up to an action. Moreover our proof of this theorem is constructive; If we know the actual transforms of two generic shapes and we recognize they produce the same distribution of Euler curves, then we construct an element of that relates them. However, the deeper implication is that knowing the distribution of Euler curves for a generic shape is a sufficient statistic for shape comparison. Continuing the aforementioned example, we can compare an arbitrarily embedded tiger and lion without ever aligning them.

We now specify what we mean by a “generic shape.”

###### Definition 6.1.

A geometric simplicial complex in with vertex set is generic if

• the Euler curves for the height functions are distinct for all , and

• the vertex set is in general position.

Before proving this section’s main result, we give a more detailed characterization of the image of the Euler characteristic transform for a generic simplicial complex .

###### Proposition 6.1.

If is a generic finite geometric simplicial complex embedded in , then determines a stratified, in fact piece-wise linear, embedding of the sphere into the space of Euler curves.

###### Proof.

Let be the hyperplane division of determined by the vertices of . Let be a stratum in . Recall that the sets are the same for all which we denoted by . Furthermore, we proved in Lemma 5.3 that for

 ECT(K,v)=∑x∈X1{[v⋅x,∞)}χ(K(x,U)).

The above formula implies that we have a proper injective immersion from the -dimensional stratum to the image of when restricted to . For lower-dimensional strata of induced by the hyperplance division, two or more vertices have the same heights, which by an obvious extension of the above formula implies a proper injective immersion over each of those strata. ∎

Using the previous result we can bound from below the number of “critical values” on a generic complex when filtered in a generic direction. Since critical points and critical values are usually understood in a differentiable setting, and the shapes are dealing with are only piece-wise differentiable, we make precise what we mean by this intuitive term.

###### Definition 6.2.

Let be an embedded geometric complex. A function has an Euler critical value at if the Euler characteristic of the sub-level set changes at . Similarly, one can say has a homological critical value at if there is a such that for all small enough , the map induced by inclusion is not an isomorphism. For the purposes of this paper we will just use the term critical value when there is no chance for confusion.

###### Remark 6.1 (Homological Critical Value).

The proper definition of a homological critical value is much more subtle and we refer the reader to  for a comparison and contrast of two candidate definitions along with several interesting examples.

We now state a lower bound on the number of critical values of a generic complex when filtered in a generic direction.

###### Corollary 6.1.

If is a generic finite geometric simplicial complex embedded in , then for almost all the height function has at least Euler critical values, i.e. values for which the Euler characteristic of the sub-level set changes.

###### Proof.

Note that Proposition 6.1 implies a lower bound on the number of vertices such that , because if there were fewer than critical points then the Euler curves over a stratum would be specified completely by varying fewer than values, which would contradict the dimension bound just proved. Note that since this argument can be repeated for each top-dimensional stratum , which together form an open and dense subset of . This proves the claim. ∎

The following is the main result of this section and is our formal statement about the uniqueness of a distribution over diagrams or curves up to an action by .

###### Theorem 6.1.

Let and be generic geometric simplicial complexes in . Let be the Lebesgue measure on . If (that is the pushforward of the measures are the same), then there is some such that , that is to say that is some combination of rotations and reflections of .

###### Proof.

First we describe the proof at a high-level. Since and are continuous and injective onto the same subset of constructible functions we can define a homeomorphism