Homological Reconstruction and Simplification in R3

Dominique Attali∗

Gipsa-labSaint Martin d’Hères, France

Ulrich Bauer†

IST AustriaKlosterneuburg, Austria

Olivier Devillers‡

INRIA Sophia Antipolis –Méditerranée

Sophia Antipolis, France

Marc Glisse§

INRIA Saclay – Île-de-FranceOrsay, France

André Lieutier¶

Dassault SystèmeAix-en-Provence, France

ABSTRACTWe consider the problem of deciding whether the persistent homol-ogy group of a simplicial pair (K, L) can be realized as the homologyH∗(X) of some complex X with L ⊂ X ⊂ K. We show that this prob-lem is NP-complete even if K is embedded in R3.

As a consequence, we show that it is NP-hard to simplify leveland sublevel sets of scalar functions on S3 within a given toleranceconstraint. This problem has relevance to the visualization of med-ical images by isosurfaces. We also show an implication to thetheory of well groups of scalar functions: not every well group canbe realized by some level set, and deciding whether a well groupcan be realized is NP-hard.

Categories and Subject DescriptorsF.2.2 [Analysis of Algorithms and Problem Complexity]: Non-numerical Algorithms and Problems—Geometrical problems andcomputations

KeywordsNP-hard problems, homology, persistence

1. INTRODUCTIONIn this paper, we establish NP-completeness of a variety of related

problems that ask for an object in R3 with a prescribed homology.In the most basic setting, we have a point cloud in Rd that samples

a shape and want to retrieve information on the sampled shape.

∗http://www.gipsa-lab.grenoble-inp.fr/~dominique.attali/†http://pub.ist.ac.at/~bauer/‡http://www-sop.inria.fr/members/Olivier.Devillers/§http://geometrica.saclay.inria.fr/team/Marc.Glisse/¶mailto:Andre.LIEUTIER@3ds.com

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There exists a whole spectrum of possibilities regarding the type ofsought information. At the coarsest level, we can content ourselveswith the homology groups which record the “holes” of a givendimension, hereafter referred to as homological features (connectedcomponents, cycles, cavities and so on). At a finer level, we may beinterested in building an approximation of the shape, reflecting asaccurately as possible both its geometry and topology. The standardway is to construct a simplicial complex using the data points asvertices, such as for instance the α-complex, the Rips complex orthe Čech complex [12, 11]. All three constructions have in commonto depend upon a scale parameter α and to get bigger as α increases.In the ideal case, we expect the complex to have the right homologyfor some suitable value of α [19, 6, 7, 1]. Unfortunately, dependingon the sampling, it may happen that such a value of α does not exist.Nonetheless, we might still be able to infer the true homology of theshape hidden in the noisy data using persistent homology [14, 9, 8].Given two scale parameters α1 and α2, the persistent homologygroups record the homological features that persist from α1 to α2.Under very weak hypotheses, we know that the persistent homologyis precisely that of the sampled shape. The persistent homology canbe computed efficiently (i.e., in polynomial time).

A natural question is then to ask for a complex that carries thepersistent homology: given a complex K and a subcomplex L, canwe find a subcomplex of K that contains L and whose homologicalfeatures are precisely those common to L and K? Our answer isthat sometimes we cannot, and deciding whether we can is NP-complete. This answer was first given in the general case by Attaliand Lieutier [2], who posed the restriction to complexes embeddedin R3 as an open problem. We resolve this problem by provingNP-completeness even for complexes embedded in R3. Note thatin R2 a solution always exists and can be computed in polynomialtime. The above problem concentrates on building a complex whosehomology matches perfectly the persistent homology of L into K:all the homological noise has been removed. We call such an objecta homological reconstruction. However, when it does not exist, itis still relevant to look for a complex nested between L and K andwhose homology is as close as possible to the persistent homologyof L into K: as much noise as possible has been removed. We callsuch a complex a homological simplification and prove that findingone is also an NP-hard problem.

In the field of visualization and image analysis, another commonsetting consists in describing a shape through a continuous functionf : Rd → R instead of a point cloud in Rd. For instance, a medicalimage may be a collection of density measurements over a grid of

http://www.gipsa-lab.grenoble-inp.fr/~dominique.attali/http://pub.ist.ac.at/~bauer/http://www-sop.inria.fr/members/Olivier.Devillers/http://geometrica.saclay.inria.fr/team/Marc.Glisse/mailto:Andre.LIEUTIER@3ds.com

Figure 1: Example of a simplicial pair (K, L) having no homo-logical reconstruction. The simplices of L are colored red.

3D points and is best modeled as a continuous map over a certaindomain of R3. In the ideal case, the shape is a sublevel set of thefunction, f −1(−∞, t]. Unfortunately, noise can plague the data. Asthe parameter t increases, sublevel sets inflate and we can track theevolution of their homology. Features that appear and disappearquickly are considered topological noise, and we can consider thecommon features of two sublevel sets as those of a denoised sub-level set. The question now becomes: can we find another cleanerfunction, close enough to the original one, whose sublevel set hasthe denoised homology, i.e., a sublevel set reconstruction? Again,in R2 such a denoised function always exists and can be found inpolynomial time [3], while in R3 the problem does not always havea solution. The corresponding optimization problem asks for a sub-level set simplification, i.e., a function close to the original one thatminimizes the number of homological features of the sublevel set.We show that these two problems are the equivalents in the func-tional setting to the homological reconstruction and simplificationof simplicial pairs described above.

Often, one is also interested in the homology of a level set, f −1(t).We show how it can be related to the (persistent) homology ofsublevel sets, and consider the corresponding level set reconstruc-tion/simplification problems.

Further in this direction, Edelsbrunner et al. introduced the wellgroup [15, 4] as a denoised version of the homology group of a levelset. Again, we can ask whether one can find a realization of thewell group, i.e., a cleaner function whose level set has the samehomology as the well group?

We shall see in this paper that all of these related problems are NP-hard, as a consequence of the NP-completeness of the homologicalreconstruction problem.

1.1 Background and notationsWe are only concerned with topological spaces that are trian-

gulable by a finite simplicial complex, so simplicial and singularhomology are isomorphic and we make no distinction between thetwo. In particular, we use the simplicial versions of the Excision andMayer-Vietoris sequence theorems, which have less restrictive as-sumptions than their singular counterparts. If K is an abstract simpli-cial complex, we denote by |K| its geometric realization. Throughoutthis article, we consider homology with coefficients in an arbitraryfield F, so the homology groups are finite-dimensional F-vectorspaces and there is no torsion. Note that for simplicial complexes Kembedded in R3, this is in fact not a restriction, since due to theabsence of torsion in R3 the Betti numbers are independent of thechoice of coefficients [16, §3.3].

Given a topological space K , we write H∗(K) =⊕

i Hi(K)for the direct sum of homology groups in all dimensions, andβ(K) = ∑i≥0 βi(K) for the total Betti number. If (K ,L) is a pairof topological spaces L ⊂ K , the inclusion L ↪→ K induces a ho-momorphism H∗(L)→ H∗(K), which is denoted by H∗(L ↪→ K).

Figure 2: Example of a simplicial pair (left) having a homolog-ical reconstruction as a subspace (right), but not as a subcom-plex. The simplicial complex on the left contains three tetrahe-dra which share the inner blue edge and whose union forms atriangular bipyramid.

The rank of this map is the persistent Betti number of the inclusionL ↪→ K and is denoted by β(L ↪→ K) = rank H∗(L ↪→ K); theimage im H∗(L ↪→ K) is a persistent homology group. If (K, L)is a simplicial pair, that is, a pair of simplicial complexes suchthat L ⊂ K, then the persistent Betti number β(|L| ↪→ |K|) can becomputed in time cubic in the number of simplices in K [14].

A piecewise linear function on a topological space K is a contin-uous function f : K → R such that there exists a finite triangulationof K on which f is simplexwise linear. Note that a simplexwiselinear function must be linear on each simplex of the given triangu-lation, while a piecewise linear function is linear on each simplexof some arbitrary triangulation.

2. RECONSTRUCTION AND SIMPLIFICA-TION OF SIMPLICIAL PAIRS

In this section, we consider a simplicial pair and define the homo-logical reconstruction problem and the homological simplificationproblem. We prove that both problems are NP-hard when the sim-plicial pair is embedded in R3. We start with a simple lemma:

Lemma 1. Consider a triple of topological spaces L ⊂ X ⊂ Kwith finite Betti numbers. Then

β(X) ≥ β(L ↪→ K).Proof. This is a consequence of the fact that whenever we con-

sider two linear maps j : U → V and i : V → W between finitedimensional vector spaces, then dim V ≥ rank j ≥ rank i ◦ j.

This property suggests the following definition:

Definition 1. Consider a triple of topological spaces L ⊂ X ⊂ Kwith finite Betti numbers. Then X is called a homological recon-struction of the pair (K ,L) if β(X) = β(L ↪→ K).

We will often omit “homological” since there is no ambiguity inthis paper. An equivalent condition for X being a reconstructionis that H∗(L ↪→ X) is surjective and H∗(X ↪→ K) is injective, asdefined in [2]. Not every pair (K ,L) admits a reconstruction; asimple counterexample is shown in Fig. 1. The use of topologicalspaces in the definition (as opposed to simplicial complexes) ismotivated by the following observation. Let (K, L) be a simplicialpair. Then there might be a reconstruction of (|K|, |L|), but not as asubcomplex of K. An example is shown in Fig. 2.

2.1 Homological reconstruction is NP-hardWe now focus our attention on spaces that are geometric realiza-

tions of finite simplicial complexes embedded in R3.

Truei Falsei

A

B C

DE

FG

H

Figure 3: Variable (top) and clause (bottom) gadgets for thereduction of homological reconstruction to 3-SAT.

Theorem 1. The homological reconstruction problem is NP-hard: Given as input a simplicial pair (K, L) embedded in R3, decidewhether there exists a reconstruction X of (|K|, |L|). The problem isNP-complete if X is required to be a subcomplex of K.

This section is devoted to the proof of Theorem 1 by reductionfrom 3-SAT. Recall that a Boolean formula Φ is in 3-CNF if it isa conjunction of several clauses, each of which is a disjunctionof three literals, a literal being either a variable or its negation.Given a 3-CNF formula Φ, we construct a simplicial pair (KΦ, LΦ)embedded in R3 and prove that (|KΦ|, |LΦ|) has a reconstruction (asa subcomplex of KΦ) if and only if Φ has a satisfying assignment(see Lemmas 2 and 3 below).

For this, we associate to the 3-CNF formula Φ a simplicial pair(KΦ, LΦ) with trivial persistent homology. Equivalently, any recon-struction X of (|KΦ|, |LΦ|) has trivial homology, i.e.,

βd(X) = βd(|LΦ| ↪→ |KΦ|) =1 if d = 0,0 otherwise.

This means that X has a single connected component, no loops,and no cavities. X has to fill all loops or cavities in LΦ and has toconnect the different connected components of LΦ by adding to LΦportions of KΦ without creating any new loops or cavities. In thefigures, red represents the subcomplex LΦ.

The variable gadget.The variable gadget is a simplicial pair (Vi,Wi) as depicted in

Fig. 3, top. The simplicial complex Vi contains 4 edges forming acycle. The two red edges belong to Wi, but not the two blue ones.One of the blue edges will be called Truei and the other one willbe called Falsei. The key property of this construction is that anyreconstruction of the pair (|Vi|, |Wi|) cannot contain both edges Trueiand Falsei, for otherwise they would create a 1-cycle together withthe red ones. This property will allow us to match the presence ofthe edge Truei to a true assignment of the variable vi.

The clause gadget.The clause gadget is a simplicial pair (C j,D j) as depicted in

Fig. 3, bottom. The simplicial complex D j contains a cycle ABCDE.

The cycle is closed with two surfaces in C j (thereafter referred toas the lower hemisphere and the disk) thereby creating a cavity.Furthermore, the complex D j contains an arc that ends inside thedisk. Whenever we fill the cycle ABCDE with the disk, this connectsthe two endpoints of the arc, thus creating a new cycle, which weclose twice in C j by a left hemisphere and a right hemisphere.Consider one bold edge in the interior of each hemisphere, which iswhere the clause gadget will connect to the variable gadgets.

The key property of this clause gadget is that at least one of the3 bold edges must be present in any reconstruction X of the pair(|C j|, |D j|). Indeed, the cycle ABCDE in X must be filled up. If itis filled by the lower hemisphere, we are done. If it is filled by thedisk, we have a new cycle EFGH in X which in turn must be killedeither by the left or by the right hemisphere. In any case, X containsat least one of the hemispheres and thus one of the three bold edges.

Correspondence with a formula.Given a 3-CNF formula Φ with n clauses c1, . . . , cn and m vari-

ables v1 . . . , vm, we construct a 2-dimensional pair (KΦ, LΦ) as fol-lows. For each variable vi we take a copy (Vi,Wi) of the variablegadget. For each clause c j, we take a copy (C j,D j) of the clausegadget; for each literal evi of c j, we identify one of the bold edgesof C j to Falsei if e is a negation and Truei otherwise. See Fig. 4 foran example.

First notice that β2(LΦ) = 0 (i.e., LΦ has no cavities). Second,we can assume that β0(KΦ) = 1 (i.e., KΦ is connected). Indeed,if KΦ is disconnected, it means that the 3-SAT problem (and thereconstruction problem) can be decomposed into 2 independentsubproblems with disjoint sets of variables, which can be solvedseparately. Last, β1(LΦ ↪→ KΦ) = 0 (i.e., the cycles in LΦ areboundaries in KΦ). Indeed, the only 1-cycles in LΦ are the 1-cyclesABCDE in each D j, and they are filled in KΦ. This means that weare looking for a reconstruction with trivial homology.

From a reconstruction to a satisfying assignment.Let X be a homological reconstruction of the pair (|KΦ|, |LΦ|). We

do not assume that X is the geometric realization of some subcom-plex of KΦ. Assign to each variable vi the value true if the edgeTruei is entirely contained inX, and false otherwise. For each clausegadget (C j,D j), at least one bold edge is contained in X. If this edgecorresponds to a positive literal vi, this means that Truei is in X, vi istrue and the clause is satisfied. If the edge corresponds to a negativeliteral ¬vi, this implies that Falsei is in X. Truei is thus not in X, sovi was assigned false and the clause is satisfied. We have thus shownthat the assignment of the variables makes the formula evaluate totrue:

Lemma 2. If (|KΦ|, |LΦ|) has a homological reconstruction, thenΦ has a satisfying assignment.

From a satisfying assignment to a reconstruction.Given a satisfying assignment for the formula Φ, we construct a

subcomplex X ⊂ KΦ such that |X| is a reconstruction of (|KΦ|, |LΦ|).We start with X = LΦ and add to X a selected set of simplicesfrom KΦ. For each clause c j, we pick one literal that evaluates totrue and close the cycle in the clause gadget complex D j corre-spondingly. If the literal corresponds to the bold edge of the lowerhemisphere, we add this hemisphere. Otherwise, we add the diskand the hemisphere that contains the bold edge corresponding to theselected literal.

The only 2-cycles in KΦ are in the clause gadgets. As we did notcreate any 2-cycle in X, it follows that β2(X) = 0. By construction,filling the clause gadgets never introduced both Truei and Falsei

(¬t ∨ u∨ v)

(t ∨¬v∨¬w)

(¬u∨¬v∨w)

t

u

v

w

Figure 4: Embedding of the clause gadget with aligned hemispheres (top), and the simplicial pair (KΦ, LΦ) generated in the reductionfrom the 3-SAT instance (¬t ∧ u ∧ v) ∨ (t ∧ ¬v ∧ ¬w) ∨ (¬u ∧ ¬v ∧ w) (bottom left), with parallel projection (bottom right) orthogonalto the alignment axis of the variable gadgets.

in X. Indeed, it could only introduce Truei if vi was assigned thevalue true and Falsei if it was assigned the value false. Therefore,the 1-cycle in the variable gadgets do not appear in X. Also, foreach clause gadget, we filled the ABCDE 1-cycle, and whenever wecreated an extra EFGH 1-cycle by adding the disk, we immediatelyfilled it with the left or right hemisphere. Now we only need tocheck that the construction did not create any “non-local” 1-cycles.Since for each clause we have only used one of the literals whichevaluate to true, the only contact a clause gadget in X has with therest of X is through a single bold edge, and the clause gadget canbe collapsed to that edge. After collapsing all clause gadgets, allthat remains are disconnected variable gadgets with at most 3 edgeseach, and so β1(X) = 0. We finally add to X just enough edgesfrom KΦ so that it becomes connected, without creating any extracycles in the process. This is possible since we assumed that KΦ isconnected. Thus we have β0(X) = 1. We conclude:

Lemma 3. If Φ has a satisfying assignment, then there is a sub-complex X of KΦ such that |X| is a homological reconstruction of(|KΦ|, |LΦ|).

We are now in a position to prove Theorem 1.

Proof of Theorem 1. First we show that the homological recon-struction problem is NP-hard. We proceed by reduction from 3-SAT.Let (K, L) = (KΦ, LΦ) be a simplicial pair defined by a 3-SAT in-stance Φ. We show that the following propositions are equivalent:

(a) (|K|, |L|) has a reconstruction;(b) Φ has a satisfying assignment;

(c) there exists a subcomplex X ⊂ K such that |X| is a reconstructionof (|K|, |L|).

The implication (a) =⇒ (b) is shown in Lemma 2; (b) =⇒ (c) isshown in Lemma 3. Finally, (c) =⇒ (a) is trivial.

The pair (K, L) can be constructed from Φ in time polynomialin the size of Φ. Together with the equivalence (a) ⇐⇒ (b), thisestablishes NP-hardness of the homological reconstruction problem.

The equivalence (b) ⇐⇒ (c) also yields NP-hardness of the ho-mological reconstruction problem restricted to subcomplexes X ⊂ K.Moreover, given a subcomplex X as a polynomial size certificate,we can decide in polynomial time whether |X| is a reconstruc-tion of (|K|, |L|). Thus the problem is also in NP and hence NP-complete.

Embedding.Later, we have to consider not only an embedding of KΦ, but also

a triangulation of its complement. The following fact will be useful:

Lemma 4. There is a triangulation of S3 with size polynomial inthe size of KΦ and having KΦ as a subcomplex.

Proof. First, referring to Fig. 4, it is clear that KΦ can be embed-ded in R3. Indeed, we can align the clause gadgets and the variablegadgets along two lines parallel to the coordinate axes and makeeach clause gadget look like a small body with three long tenta-cles (colored green in Fig. 4) that connect to the variable gadgets.Due to the way the variable and clause gadgets are aligned in theconstruction along skew axes, the tentacles do not intersect in theirinterior.

We can subdivide the space by first projecting KΦ onto a planeorthogonal to the line carrying the variable gadgets. We get a polyg-onal region whose complement can easily be triangulated insidea bounding box without adding any new vertex and thus adding alinear number of edges. Extending each triangle in the direction ofthe projection, we get a collection of tubes, one for each triangle.The tubes can easily be triangulated while respecting KΦ to obtain apolynomial size triangulation of a bounding box of the construction,which can trivially be extended to a polynomial size triangulationof S3.

We want to remark that a similar construction can be realizedeven if we restrict edges and faces of L and K to be edges and facesof a 3D grid (see Fig. 5). This means that a variant of Theorem 1can also be shown for cubical complexes arising from 3D imagedata.

Corollary 1. The homological simplification problem is NP-hard: Given as input a simplicial pair (K, L) embedded in R3, find acomplex X minimizing β(X) subject to L ⊂ X ⊂ K.

Proof. We use a reduction from the homological reconstructionproblem. To determine if a reconstruction exists, we can first finda complex X minimizing β(X) subject to L ⊂ X ⊂ K. We thenonly need to check if its Betti number matches the lower boundβ(L ↪→ K).

3. RECONSTRUCTION AND SIMPLIFI-CATION OF LEVEL AND SUBLEVELSETS

In this section, we consider a real-valued simplexwise linear func-tion defined on a simplicial complex embedded in R3 and establishthe NP-hardness of problems that ask for a nearby function witha simplified sublevel set (Section 3.1) and a simplified level set(Section 3.3).

Given a real-valued function f , we write Ft for the t-level setf −1(t), F≤t for the (closed) t-sublevel set f −1((−∞, t]), and F

¬tt

¬uu

¬vv

¬ww

Variable gadget

TrueFalse

Clause gadget

(¬t ∨ u∨ v) (t ∨¬v∨¬w) (¬u∨¬v∨w)

Figure 5: Example of 3-SAT reduction using a 3D grid embedding.

propositions are equivalent to propositions (a)–(c) in the proof ofTheorem 1:

(d) f has a simplexwise linear sublevel set (0, 1)-reconstruction g.

(e) f has a sublevel set (0, 1)-reconstruction g.

To show (c) =⇒ (d), we define a simplexwise linear function g on| sd K| by its values on the vertices of sd K (the simplices of K); seeFigure 6:

g : σ 7→−2 if σ ∈ L,−1 if σ ∈ X \ L,1 if σ ∈ K \ X.

(2)

We have ‖g − f ‖∞ = 1. By Proposition 1, the sublevel set G≤0 ishomotopy equivalent to |X| and hence is a reconstruction of the pair

(|K|, |L|) ' (F≤1, F≤−1).Finally, (d) =⇒ (e) is trivial and (e) =⇒ (a) follows directly withG≤0 as a reconstruction of (F≤1, F≤−1) ' (|K|, |L|).

The function f can be constructed from Φ in polynomial time.Together with the equivalence (b) ⇐⇒ (e), this establishes NP-hardness of the sublevel set reconstruction problem.

The equivalence (b) ⇐⇒ (d) also yields NP-hardness of thesublevel set reconstruction problem restricted to simplexwise lin-ear functions | sd K| → R. Moreover given a simplexwise linearfunction g as a polynomial size certificate, by computing and com-paring β(G≤0) and β(F≤−1 ↪→ F≤1) we can decide in polynomialtime whether g is a sublevel set (0, 1)-reconstruction of f . Thus theproblem is also in NP and hence NP-complete.

Corollary 2. The sublevel set simplification problem is NP-hard: Given as input a simplexwise linear function f on a simplicialcomplex embedded in R3 and parameters t and δ, find a simplexwiselinear function g minimizing β(G≤t) subject to ‖g − f ‖∞ ≤ δ.

Proof. We use a reduction from the sublevel set reconstructionproblem. To determine if f has a sublevel set (t, δ)-reconstruction,we can first find a simplexwise linear minimizer of β(G≤t). We thenonly need to check if β(G≤t) matches the lower bound

β(F≤t−δ ↪→ F≤t+δ),which can be done in time polynomial in the size of K.

3.2 Betti numbers of level and sublevel setsThe Betti numbers of level and sublevel sets are related by the

following formula:

Lemma 5. Let f be a piecewise linear function on Sn, n > 1, andlet t be in the interior of the image of f , t ∈ int(im f ). Then

βd(Ft) = βd(F≤t) + βn−d−1(F

Definition 3. Let f , g be piecewise linear functions on Sn andconsider real parameters t and δ with t ± δ ∈ int(im f ). The functiong is called a level set (t, δ)-reconstruction of f if ‖g − f ‖∞ ≤ δ and

β(Gt) = β(F≤t−δ ↪→ F≤t+δ) + β(F

Proof. To determine if f has a level set (t, δ)-reconstruction, wecan first find a minimizer of β(Gt). We then only need to check ifβ(Gt) matches the lower bound

β(F≤t−δ ↪→ F≤t+δ) + β(F

Acknowledgements.This work was initiated during the 10th McGill–INRIA Workshop

on Computational Geometry at the Bellairs Research Institute. Theauthors wish to thank all the participants for creating a pleasant andstimulating atmosphere, in particular Nina Amenta for discussionsleading to a first version of Theorem 1. Some of the authors werepartially supported by the GIGA ANR grant (contract ANR-09-BLAN-0331-01) and the European project CG-Learning (contract255827).

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IntroductionBackground and notations

Reconstruction and simplification of simplicial pairsHomological reconstruction is NP-hard

Reconstruction and simplification of level and sublevel setsSublevel set reconstruction is NP-hardBetti numbers of level and sublevel setsLevel set reconstruction is NP-hard

Realizations of well groupsRealizability of well groups is NP-hard

Conclusion