Shape recognition and classification is a problem with a wide variety of applications. Several recent works have demonstrated that topological descriptors can be used as summaries of shapes and utilized to compute distances. In this abstract, we explore the use of a finite number of Euler Characteristic Curves (ECC) to reconstruct plane graphs. We highlight difficulties that occur when attempting to adopt approaches for reconstruction with persistence diagrams to reconstruction with ECCs. Furthermore, we highlight specific arrangements of vertices that create problems for reconstruction and present several observations about how they affect the ECC-based reconstruction. Finally, we show that plane graphs without degree two vertices can be reconstructed using a finite number of ECCs.
Shape comparison and classification is a common task in the field of computer science, with applications in graphics, geometry, machine learning, and several other research fields. The problem has been well-studied in, with several approaches described in the survey . One relatively new approach to the problem involves utilizing topological descriptors to represent and compare the shapes. In , Turner et al. proposed the use of the zero- and one-dimensional persistence diagrams from lower-star filtrations to compare triangulations of in , for . We call the mapping of a shape to to a parameterized set of diagrams the persistent homology transform (PHT). Their main result (Cor. of ) showed that the persistent homology transform (PHT) is injective for comparing triangulations of or embedded in (or triangulations of in ), and thus can be used to distinguish different shapes. Turner et al. also extend the idea of the PHT to the Euler Characteristic Curve (ECC) and describe the Euler Characteristic Transform (ECT), a topological summary that records changes in the Euler Characteristic across a height parameter, again from all directions. Finally, using experimental results, the authors show that the PHT and ECT performed well in clustering tasks. In , Crawford et al. extend this work by proposing the smooth Euler Characteristic Transform (SECT), a functional variant of the ECT with favorable properties for analysis. They show that features derived from the SECT of tumor shapes are better predictors of clinical outcomes of patients than other traditional features.
The proof of injectivity (i.e., that a shape can be reconstructed from the PHT or the ECC) uses an infinite set of a directions; however, using an infinite set of directions is infeasible for computational purposes. Thus, both [2, 7] use sampling a finite set of directions for the height filtrations in order to apply the technique to shape comparison. In , Belton et al. present an algorithm for reconstructing plane graphs using a quadratic (hence, finite) number of persistence diagrams. Simultaneous to that result, other researchers also attempted to give a finite number of directions sufficient to fully determine a shape. Both  and  give upper bounds on the number of directions needed to determine a hidden shape in . In order to do this, they make assumptions about the curvature and geometry of the input shape. In our work, by contrast, we restrict to plane graphs, but make no restrictions on curvature.
Here, we attempt to extend the work of  on the PHT to the ECT. However, difficulties arise when using ECCs because they do not encode information about every vertex from every direction, as a persistence diagram does when on-diagonal points are included. We show that, while the number of directions needed to give an ECT unique to the input graph is linear in the number of vertices of the graph, it is difficult to determine which directions generate the necessary ECCs. As we will see, the main difficulty lies with the presence of degree two vertices.
In this paper, we focus on a subset of finite simplicial complexes that are composed of only edges and vertices and are provided with a planar straight-line embedding in . We refer to these simplicial complexes as plane graphs. We refer the reader to  for a general background on persistent homology, and only present the necessary content here.
Let be a plane graph. In what follows, we assume that the vertices of have distinct - and -coordinates from one another. Furthermore, we assume that no three vertices are collinear.
Let be the unit sphere in . Consider
, i.e., a direction vector in; we define the lower-star filtration with respect to . Let be defined for a simplex by where is the inner (dot) product and measures height in the direction of unit vector . Intuitively, the height of with respect to is the maximum “height” of all vertices in . Then, for each , the subcomplex is composed of all simplices that lie entirely below or at the height , with respect to the direction . The lower-star filtration is sequence of subcomplexes , where increases from to ; notice that only changes when is the height of a vertex of .
When we observe a difference between and , we know that we have encountered a vertex. As in , we define a structure to encode what we know about this vertex in . Given , and a height , the filtration line at height is the line, denoted , perpendicular to direction and at height in direction . Given a finite set of vertices , the filtration lines of are the set of lines
Further, will contain lines if and only if no two vertices have the same height in direction . Our assumptions guarantee distinct vertex heights only for , , , and , referred to as the cardinal directions. In , every line in can be read off of the persistence diagram, as every simplex corresponds to either a birth or the death of a homology class. Next, we observe that we cannot witness all such lines for another topological descriptor, the Euler Characteristic Curve.
Euler Characteristic Curves
The Euler characteristic of a plane graph is . The Euler Characteristic Curve (ECC) is the piecewise step function of the Euler characteristic, whose domain is subcomplexes of a filtration defined by some parameterization of . In this paper, the parameter is the height of a lower-star filtration. Specifically, we define to be the function that maps a height to the Euler Characteristic of . Every change in the ECC corresponds to a filtration line from that direction, but not vice versa. For example, if an edge and vertex appear at the same height, then the ECC does not change. We now refine our definition of filtration lines:
This set corresponds to the subset of vertices in that are witnessed from through the ECC . As such, we refer to these lines as witnessed lines. We note that the only time that a vertex is not witnessed is if the vertex is included in the filtration at the same time as an edge because the vertex being added will be cancelled out by the inclusion of the edge. Furthermore, we note that lying on a filtration line from does not necessarily imply that is witnessed from , i.e., it could lie on a witness line for another vertex if they lie at the same height from .
3 Towards Vertex Reconstruction
We are interested in reconstructing a plane graph from ECCs from a finite number of directions. While three directions was sufficient for reconstructing vertices using persistence diagrams, ECCs contain strictly less information in each direction. We observe the existence of a linear number of directions that allows to fully reconstruct the vertices of a plane graph: [ECC Existence] Given a plane graph with , there exist directions that can be used to reconstruct all vertices in . The proof of this claim may be found in proofs. We note that while directions are sufficient, this bound is likely not tight.
Initially, attempting to use the techniques in  seems promising for plane graph reconstruction using ECCs, i.e., we can define a correspondence between three-way witness line intersections (from carefully chosen directions) and vertices. However, certain types of vertices introduce difficulties.
For example, consider deg2Trick. A degree two vertex is not witnessed by any of the witness lines from the cardinal directions and . However, we would like to generate a correspondence between three-way intersections of witness lines and non degree two vertices. If we use the technique described in Theorem of  to choose such a direction, that direction creates a witness line that causes a three-way intersection not corresponding to a vertex. In fact, when degree two vertices are introduced to the plane graph, several problems arise. We discuss these problems in detail in degTwoProb.
4 Degree Two Challenges
Degree two vertices introduce several complications in finding witness directions, because degree two vertices can have an arbitrarily small region on from which they can be witnessed.
For example, in deg2Colin the vertices , , and are nearly collinear. In order to witness , we must choose directions from within the red region, where a decrease in the ECC will be observed, or from the blue region, where an increase in the ECC will be observed. However, these these regions becomes arbitrarily small as , and approach collinear.
As mentioned earlier, degree two vertices can also introduce additional ambiguities when witnessing non-degree two vertices. Recall the example found in deg2Trick and the discussion in degTwoScary.
Despite these difficulties, several situations exist in which degree two vertices can be witnessed. The following propositions summarize these scenarios. Proofs are provided in proofs. For clarity, we discuss quadrants as though is located at the origin. However, note that the following propositions also apply to arrangements with similar orientations and angles.
[Same Quadrant] If and lie in the same quadrant, such as the vertices and in deg2scenarios, then will be witnessed in ECCs from every one of the cardinal directions.
[Neighboring Quadrants] If and lie in neighboring quadrants, such as vertices and in deg2scenarios, then will be witnessed in ECCs from exactly two of the four cardinal directions. [Degree Two Bounded Angle] If then will be witnessed in ECCs from at least two of the four cardinal directions. The above propositions show scenarios for which degree two vertices can be witnessed using cardinal directions. However, degree two vertices pose particular problems when the edges lie in non-neighboring quadrants, such as the edges and in deg2scenarios or and in deg2Colin. Then, when degree two vertices are not included in a plane graph , a constant number of ECCs can be used to determine the embeddings of the vertices.
5 A Special Case
If a plane graph contains no degree two vertices, the graph can be reconstructed using a finite number of ECCs. Let denote a plane graph with vertex and edge sets and respectively. Recall from  that three way filtration line intersections from carefully chosen directions correspond to a vertex location for plane graphs using persistence diagrams. We show that this result still holds for reconstructing plane graphs using ECCs, if they do not contain degree two vertices. The proofs of the following lemmas and theorem can be found in proofs.
First, we provide a lemma that yields insight into how non-degree two vertices are witnessed. [Linear Witness Lines] Let be a plane graph in with vertices such that for all , and denote . Let be a line in such that any line parallel to intersects at most one vertex in . Let be chosen perpindicular to . Then,
By generalizing the results of vertFiltLine, we introduce the the following Lemma to generate potential vertex locations in , where is the number of vertices. [Witness Line Intersections] Recall the cardinal directions . If for all , then
Utilizing these horizontal and vertical witness lines, we are able to pick two additional directions to generate three-way filtration line intersections using a technique similar to the one described in Theorem of . Then, the following theorem holds as well. [ECC Vertex Reconstruction] Let be a plane graph with vertices and edges . If for all , then the locations of all vertices can be determined using six ECCs in time. The proof of noDeg2VertexReconstruct is found in proofs, but note that the result follows using similar arguments to those found in Theorem of .
6 Discussion and Future Work
We have shown that, for any known plane graph , we can choose a linear number of directions to fully describe using only ECCs from those directions. However, when is unknown, determining such a set is difficult. We emphasize that although there is an infinite number of directions in which the vertices of a plane graph can be witnessed by an ECC, the presence of degree two vertices can restrict these directions to an arbitrarily small subset of .
Our ultimate goal is to further develop the theory on determining the minimal set of directions necessary to reconstruct shapes. We are currently investigating upper bounds on the number of directions needed to reconstruct a plane graph from ECCs. Additionally, we are exploring what assumptions we can place on the underlying shape in order to overcome the challenges of degree two vertices. For example, we observe that if the number of vertices is known, then the intersection of filtration lines determines the location of all vertices. Another simplifying assumption is that minimum angle between any three vertices, , is known. Then, we can avoid some of the issues described in degTwoProb by employing pairs of directions whose difference in angle is less than . Finally, we would like to extend our work to more general shapes embedded in .
This material is based upon work supported by the National Science Foundation under Grant No. CCF 1618605 (authors BTF and SM) and Grant No. DBI 1661530; BTF and AS acknowledge the support of NIH and NSF under Grant No. NSF-DMS 1664858.
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-  Lorin Crawford, Anthea Monod, Andrew X. Chen, Sayan Mukherjee, and Raúl Rabadán. Functional data analysis using a topological summary statistic: The smooth Euler characteristic transform. arXiv:1611.06818, 2016.
-  Justin Curry, Sayan Mukherjee, and Katharine Turner. How many directions determine a shape and other sufficiency results for two topological transforms. arXiv:1805.09782, 2018.
-  Herbert Edelsbrunner and John Harer. Computational Topology: An Introduction. American Mathematical Society, 2010.
-  Robert Ghrist, Rachel Levanger, and Huy Mai. Persistent homology and Euler integral transforms. arXiv:1804.04740, 2018.
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Appendix A Proofs
Proof of nDirExist
Let be a vertex in . First, we show that each vertex is witnessed from an infinite number of directions . If , is witnessed from any direction for which it lies on a unique witness line (So, for all but directions). If with edge for some , then is observed from an the infinite set of directions from which appears after in the lower-star filtration, and lies on a unique witness line. If with edges and for , then is observed from any direction from which and appear before in the filtration and lies on a unique witness line. Thus, each vertex is witnessed from an infinite number of directions.
Let be the set of directions that witness . We can choose any three directions from and generate a unique three-way intersection at . Now, we need to show that a set of directions exist for each of the vertices such that no three-way intersections exist at locations where a vertex is not located. In order to do this, we give the vertices some arbitrary ordering . Then, select vertices in ascending order. For the first, any three directions in will give a single three-way intersection of witness lines. For each successive vertex , there exist up to witness lines. More importantly, the number of three-way witness line intersections is finite. Thus, there exist three directions in such that none of the witness lines created by these directions intersect existing intersections. Since the - and -coordinates of a vertex can be determined using a three-way line intersection, we can see that there exist a set of directions which generates exactly three-way intersections of witness lines, revealing the location of all vertices.
Proof of deg2SameQuad
If and lie in the same quadrant, then and will appear before from exactly one of the two -axis parallel directions directions or and before in exactly one of the -axis parallel directions or . Let and be the directions that witness and before . and will witness by seeing a decrease in the Euler Characteristic at the time that is first included in the filtration. Then, and will witness before or . Since no other edges with as an endpoint exist, there will be an increase in and at the time that is first included in the filtration. Then, is witnessed from every cardinal direction, as required. ∎
Proof of deg2NeighborQuad
Recall that no two vertices share - or -coordinates, then any witness line from a cardinal direction will be unique. Let be the cardinal direction for which and and the cardinal direction chosen such that and . Then, there is no change in Euler Characteristic at from either or , since is added at the same time as or , respectively. Now, let and be the remaining two cardinal directions, where is the direction from which we include before or . Direction witnesses because no edges are included at height from that direction. Direction witnesses because both and are added along with . Thus, is witnessed from exactly two of the four cardinal directions. ∎
Proof of deg2Angle
(Degree Two Bounded Angle)
If , then and must lie in neighboring quadrants or the same quadrant, since neither can lie on the boundary of a quadrant by assumption. If they are in the same quadrant, deg2SameQuad tells us that they must be seen from all four cardinal directions. If they are in neighboring quadrants, deg2NeighborQuad tells us that we can witness with ECCs from exactly two of the four cardinal directions. ∎
Proof of vertFiltLine
(Linear Witness Lines)
We show that each vertex is seen by at least one of or . Let be a vertex with . Then, will correspond to for any arbitrary direction because will always increase by at least one at time . As such, will be observed by both and .
Let be a vertex with and for some . Then, if is chosen such that , will not result in a change in . However, was chosen such that no two vertices will be observed at the same time. As a result, no edge in can be parallel to . Then, if then and an increase in is seen at time . This implies that is observed by or but not both.
Finally, if is a vertex with , then we must consider two cases. If, for , there exists exactly one edge such that , then there must exist at least two additional edges that will result in a decrease in at time . As such, will be observed by at least one of the ECCs resulting from or . On the other hand, if, for , there exists either zero edges or more than one edge that appear before in the height filtration from , then will either increase (in the case where no edges appear before ) or decrease (in the case where two or more edges appear before ). Then, all non-degree two vertices result in a change in or and, as such, , as required. ∎
Proof of noDeg2SeenVertices
(Witness Line Intersections)
By vertFiltLine, if is chosen such that no two vertices are intersected by a line perpendicular to , then will result in filtration lines. Recall that no two vertices in share an - or -coordinate. Then, by vertFiltLine, and , as required. ∎
Proof of noDeg2VertexReconstruct
(ECC Vertex Reconstruction)
Using noDeg2SeenVertices we construct horizontal and vertical lines corresponding to vertices using four ECCs and we denote them and respectively. Then, we must identify an additional two directions which will, together, generate an additional unique witness lines and exactly three-way filtration line intersections. We choose these final directions and using the method described in Theorem 5 of . We observe that, by Lemma 4 of , no two vertices will be intersected by any single line perpindicular to . Then, since each vertex will be witnessed by at least one of the ECCs from or by vertFiltLine, these two directions will yield distinct filtration lines each of which will intersect exactly one two-way intersection between lines of and . Then, Lemma 3 of  implies that these three-way intersections are the locations of the vertices in . The running time follows from the proof of Theorem 5 in . ∎