Optimal topological simplification of discrete functions on surfaces
OOptimal topological simplificationof discrete functions on surfaces
Ulrich Bauer Carsten Lange Max Wardetzky July 26, 2010
Abstract
We solve the problem of minimizing the number of critical points among allfunctions on a surface within a prescribed distance δ from a given input function.The result is achieved by establishing a connection between discrete Morse theoryand persistent homology. Our method completely removes homological noise withpersistence less than 2 δ , constructively proving the tightness of a lower bound onthe number of critical points given by the stability theorem of persistent homologyin dimension two for any input function. We also show that an optimal solution canbe computed in linear time after persistence pairs have been computed. Measured data and functions constructed from measured data su ff er from omnipresentnoise introduced during the measuring process. Separating relevant information fromnoise is therefore a widely considered problem.Taking a topological point of view, we regard noise as a source of critical points.Indeed, even arbitrarily small amounts of noise (with respect to the supremum norm)may give rise to an arbitrarily large number of critical points. We may hence interpretcritical points that can be eliminated by small perturbations as being caused by noise.Consequently, we consider the following optimization problem: Problem (Topological simplification on surfaces) . Given a function f on a surface anda real number δ ≥ , find a function f δ subject to (cid:107) f δ − f (cid:107) ∞ ≤ δ such that f δ has aminimum number of critical points. The class of functions and the notion of critical points we work with will be clarifiedlater; for now, we just want to mention that multiple saddles (such as a “monkey saddle”)are counted here with multiplicity.The
Bottleneck Stability Theorem [6], a fundamental result in the theory of persistenthomology [10, 34], provides a lower bound on the number of critical points:
Proposition (Stability Bound) . For any function f δ with (cid:107) f δ − f (cid:107) ∞ ≤ δ , the number ofcritical points of f δ is bounded from below by the number of critical points of f thathave persistence > δ . Institute for Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16–18, 37083 Göt-tingen, Germany. {bauer,wardetzky}@math.uni-goettingen.de Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 6, 14195 Berlin,Germany. [email protected] a r X i v : . [ c s . C G ] J u l learly the question about the tightness of this bound is of great importance forthe significance of the Bottleneck Stability Theorem. In the present article, we showconstructively that the bound given by the stability theorem is actually tight for functionson surfaces (see Theorem 16): Theorem (Tightness of the stability bound) . Given a function f on a surface and a realnumber δ ≥ , there exists a function f δ such that (cid:107) f δ − f (cid:107) ∞ ≤ δ and the number ofcritical points of f δ equals the number of critical points of f that have persistence > δ . A similar statement does not hold in higher dimensions or for non-manifold 2-complexes, see Section 6.5.
Discrete Morse theory [14, 15] provides equivalents of several core concepts of classicalMorse theory, like discrete Morse functions, discrete gradient vector fields, criticalpoints, and a cancelation theorem for the elimination of critical points from a vectorfield. Because of its simplicity, it not only maintains the intuition of the classical theorybut allows to go beyond it by providing explicit constructions that would become quitecomplicated in the smooth setting.
Persistent homology [10, 34] quantifies topological features of a function. It definesthe birth and death of homology groups at critical points, identifies pairs of these( persistence pairs ), and provides a measure of their significance ( persistence ).Whereas (discrete) Morse theory makes statements about the homotopy type of thesublevel sets of a function, persistence theory is concerned with their homology . Oursolution to the problem of topological simplification on surfaces relies on a combinationof both theories. In particular, we make contributions to the following problems:
Canceling a single pair of critical points of a function
Forman [14] describes asimple method for eliminating pairs of critical points in discrete vector fields . Modifyinga function according to the cancelation of a pair of critical points, however, is moredi ffi cult and requires additional e ff ort. We first observe that a discrete gradient vectorfield induces a partial order on the cells of the underlying complex, giving rise tothe notion of attracting and repelling sets (in analogy to the notion of stable andunstable manifolds in the classical theory). Building on these concepts, we describe acanonical method for eliminating a pair of critical points of a discrete Morse function .This complements Forman’s cancelation method for discrete gradient vector fields . Inparticular, it is applicable in any dimension (Section 4.1). An informal description indimension 1 is shown in Figure 1.To cancel a pair of critical points whose values di ff er by d , our method modifies thefunction by d in the supremum norm, which is the minimum required for cancelation(see Figure 1). To achieve this minimum, elimination of critical points has to take intoaccount the attracting and repelling sets of the canceled pair, containing cells of all dimensions; moreover, an arbitrary number of other critical points might have theirvalue changed. This is in contrast to previous related methods [12, 3] which operatejust on the 1-skeleton of the surface (and on the 1-skeleton of its dual) and does nota ff ect other critical points, but modifies the function by d and hence is not a minimalmodification in the supremum norm. Moreover, these methods do not extend to higherdimensions. 2 * † * † * (a) (b) (c) † * (d) (e) (f) Figure 1: Cancelation of critical points. (a) shows the graph of a function together withthe directions of its gradient vector field. The values of the repelling set (b) of the uppercritical point (marked with † ) and of the attracting set (c) of the lower critical point(marked with ∗ ) are cut o ff at the average value of the two critical points, creating aplateau (d). The old gradient directions are still consistent with the new function. Thegradient vector field can now be reversed along the path between the critical points,eliminating the pair (e). The resulting function has a plateau, but can be perturbedslightly to become non-degenerate (f). Degenerate functions
Morse theory, in any of its variations, fundamentally relies onthe assumption that the critical points of the function considered are non-degenerate.This condition not only prevents the theory from being applicable directly to arbitraryinput functions. The canonical function arising from canceling a pair of critical pointshas a plateau (see Figure 1) and hence is not a discrete Morse function. However, thereis a Morse function arbitrarily close to it. This necessitates a method to deal with suchdegenerate functions. To do so, we devise a symbolic perturbation scheme (Section 2.2)based on discrete gradient vector fields, allowing to treat the degenerate case in thesame way as the generic case by considering the larger class of pseudo-Morse functions ;in particular, we do not require the input function to be generic. Instead of derivinginformation about critical points from the function directly (which leads to ambiguitiesin degenerate cases), we work with an explicit gradient vector field consistent with thefunction, coinciding with the usual discrete gradient vector field in the generic case.Our scheme always allows to construct a Morse function arbitrarily close to a givenpseudo-Morse function and consistent with the given gradient vector field.A second symbolic perturbation scheme allows to relax the assumption that criticalcells have unique function values. It extends the first perturbation scheme by explicitlymaintaining a total order on the cells that is consistent with both the function and thegradient vector field.
Multiple cancelations
We establish a connection between persistence pairs and thecancelation of critical points by proving that for functions on surfaces, every persistencepair can eventually be canceled if a sequence of cancelations is performed according toa certain hierarchy on the persistence pairs (Section 3.3). The statement is no longer true3or manifolds in higher dimensions or non-manifold 2-complexes, where persistencepairs cannot always be canceled.
Tightness of the stability bound
While the stability bound can easily be seen tobe tight when only a single pair of critical points is canceled, we need to ensure thatrepeated cancelation does not violate the δ -tolerance constraint. Again, the situation isdi ff erent from previous work [12, 3], where simplification is treated separately for pairsof dimensions (0 ,
1) and (1 , ff erent dimensions at the same time (Section 4.2). We provide a constructive proof ofthe tightness of the stability bound (Theorem 16). The construction is well suited forproving our theorem; however, it has a suboptimal quadratic time complexity. E ffi cient solution We show that, after persistence pairs have been computed in time O (sort( n )) [10], an optimal solution to the topological simplification problem on surfacescan be computed in time O ( n ) using simple graph traversal methods (Section 5). Hence,we match the time complexity of [3]. Since Theorem 16 is already established, we canuse it to give a simple proof of correctness of the linear algorithm.This result is surprising in view of the fact that the topological simplification problemon surfaces is NP-hard when restricting to simplexwise linear functions on a triangu-lated surface. This follows from a recent result by Gray et al. [16], which states thatminimizing the number of extrema of a simplexwise linear function with interval con-straints for the vertex values is NP-hard. Their argument can easily be adapted to ourproblem setting, where all constraint intervals are assumed to have length 2 δ . Notethat the emphasis on simplexwise linear (i.e., linear on each simplex, as opposed tojust piecewise linear) functions is significant here: a multiple saddle can be split intoseveral non-degenerate saddles by an arbitrarily small (in L ∞ ) perturbation in the spaceof piecewise linear functions, but not in the subspace of simplexwise linear functions.This emphasizes the important role of discrete Morse theory in our problem: thehardness of the problem in the simplexwise linear setting arises from the possibilitythat the input contains multiple saddles, which is excluded by definition in discreteMorse theory. Going from simplexwise linear functions to discrete Morse functions(Section 2.3) can be interpreted as splitting multiple saddles. Energy minimization of simplified functions
The solution to the topological sim-plification problem is not unique in general: both the δ -constraint and the simplifieddiscrete gradient vector field impose a set of linear inequalities on the simplified func-tion, so the solution set is a convex polytope. This additionally allows to minimize asuitable convex energy functional. We employ this technique to remove artifacts fromthe initial solution and to improve the similarity to the input function (Section 6.4). Topological simplification of functions within a δ -tolerance constraint has been consid-ered before by Edelsbrunner et al. [12] and Attali et al. [3]. The problem consideredthere di ff ers from ours by a seemingly small but significant detail: in [12, 3] the criticalpoints of the input function f that are not eliminated are additionally assumed to retainthe same critical value for the output g . This restriction has serious consequences: whileit allows for the elimination of critical points of f with persistence ≤ δ , in certain casesnot all critical points with persistence ≤ δ can be eliminated; an example is given4n [12]. Hence, under this restriction it is not possible to match the stability bound.Moreover, this result does not provide any information about the tightness of the stabilitybound since it considers only a restricted set of functions.The methods presented in [12, 3] can be interpreted as variants of the carving method proposed by Soille [31] in the context of terrain simplification. There is anotherpopular method for removing extrema from terrains, called filling or flooding [19, 2, 9].A combination of both methods has been proposed in [32]. Our methods of cancelingcritical points from a function can be interpreted as a combination of carving andflooding in the realm of discrete Morse theory.Apart from the above mentioned works, persistent homology provides the basisof several other elegant methods for computation and simplification of multi-scalestructures derived from a function. For example, Edelsbrunner et al. [10] discuss simpli-fication of the persistent homology for filtrations of simplicial complexes. Edelsbrunneret al. [11] and Gyulassy et al. [17] consider simplification of cell decompositions (Morse-Smale complexes) resulting from a given Morse function. Unfortunately, a simplifiedMorse-Smale complex does not directly give rise to a simplified function. Indeed,simplifying a Morse-Smale complex is closely related to simplifying a discrete gradientvector field.The problem of constructing discrete gradient vector fields (as opposed to functions)that minimize the number of critical points without constraints is addressed by Lewineret al. [26] for surfaces and by Joswig and Pfetsch [20] for complexes of arbitrarydimension. King et al. [22] were the first to propose the combination of persistencewith discrete Morse theory to simplify the gradient vector field of an input function ona 3-dimensional simplicial complex. Their method has quadratic time complexity andproduces a simplified discrete gradient vector field but not a function. Moreover, it doesnot aim at optimality (in 3 dimensions, not every persistence pair can be canceled).Several statements of this article can also be transferred to the setting of piecewiselinear Morse-Smale complexes. For example, Theorem 13 can be used to show that thesuccessive simplification of a Morse-Smale complex on a surface proposed by Edels-brunner et al. [11] is always possible. This extends the Adjacency Lemma in [11],which shows a necessary but not su ffi cient condition for the successive cancelation ofpersistence pairs. Classical (smooth) Morse theory [28] relates the critical points of a generic smoothreal-valued function on a manifold to the global topology of that manifold. Forman[14, 15] carried over the main ideas of Morse theory to a combinatorial setting. Webriefly review some important notions and results here that are used throughout thisarticle, together with some extensions to Forman’s theory that provide important toolsfor our solution.A CW complex K is a topological space constructed inductively: starting witha discrete set K of 0-cells, the n -skeleton K n is formed by attaching n -cells (open n -dimensional balls) by continuous maps S n − → K n − from their boundary to the( n − K is denoted by K . Throughout this article, weconsider only finite CW complexes. Whenever a cell τ ∈ K is attached to a cell σ (i.e., σ ⊂ ∂τ , where ∂τ denotes the boundary of τ ), we call σ a face of τ ; a face ofcodimension 1 is called a facet . If all attaching maps are homeomorphisms, K is called5 φ Figure 2: Reversing a gradient vector field along the unique path from ∂ρ to φ producesa gradient vector field in which the 1-cell φ and the 2-cell ρ are no longer critical.a regular CW complex. A regular CW complex whose underlying space is a 2-manifoldis called a combinatorial surface . We refer to [27, 18] for details on CW complexes.
One of the central concepts of discrete Morse theory is that of a discrete vector field – apurely combinatorial analogue of a classical vector field.
Definition (discrete vector field, critical cell [14, 15]) . A discrete vector field V on aregular CW complex K is a set of pairs of cells ( σ, τ ) ∈ K × K, with σ a facet of τ , suchthat each cell of K is contained in at most one pair of V. A cell σ ∈ K is critical withrespect to V if σ is not contained in any pair of V. The dimension of a critical cell isalso called its index. A pair ( σ, τ ) in a discrete vector field V can be visualized as an arrow from σ to τ (as in Figure 2).In the following, we consider an important subclass of vector fields in which thearrows do not form closed paths. This can be made precise using the concept of V-paths . Definition ( V -path [15]) . Let V be a discrete vector field. A V-path Γ from a cell σ to acell σ r is a sequence ( σ , τ , σ , . . . , τ r − , σ r ) of cells such that for every ≤ i ≤ r − : σ i is a facet of τ i with ( σ i , τ i ) ∈ V and σ i + is a facet of τ i with ( σ i + , τ i ) (cid:60) V . Γ is closed if σ = σ r and nontrivial if r > . We call dim σ the dimension of Γ . By a V -path from ∂ρ to φ we mean a V -path from a facet of ρ to φ (see Figure 2 foran example). Definition (discrete gradient vector field [15]) . A discrete vector field V is a discretegradient vector field if it contains no nontrivial closed V-paths.
The main technique for reducing the number of critical points is that of reversing agradient vector field V along a V -path between two critical cells ρ and φ (see Figure 2for an example). It provides a discrete analogue of Morse’s cancelation theorem [30]: Theorem 1 (Forman [14]) . Let φ and ρ be two critical cells of a gradient vector field Vwith exactly one V-path Γ from ∂ρ to φ . Then there is a gradient vector field (cid:101) V obtainedby reversing V along the path Γ . The critical cells of (cid:101) V are exactly the critical cells of Vapart from { φ, ρ } . Moreover, V = (cid:101) V except along the path Γ . Gradient vector fields on combinatorial surfaces have additional properties that donot hold in higher dimensions. The following property is readily checked using the factthat a 1-cell is only attached to at most two 0-cells, and at most two 2-cells are attachedto a 1-cell: 6 emma 2.
Two V-paths of dimension 0 cannot branch at a common cell, and twoV-paths of dimension 1 cannot merge (except at their last cell).
Corollary 3.
Let ρ be a critical 1-cell of a discrete vector field V on a combinatorialsurface. Then there are at most two V-paths from ∂ρ to critical 0-cells, each startingat one of the two 0-cells in ∂ρ . Similarly, there are at most two V-paths from facets ofcritical 2-cells to ρ . As in smooth Morse theory, a discrete gradient vector field can be understood as thegradient of some non-degenerate function in the following sense:
Definition (discrete Morse function [14]) . A function f : K → R on the cells of aregular CW complex K is a discrete Morse function if there is a gradient vector field V f such that whenever σ is a facet of τ then ( σ, τ ) (cid:60) V f implies f ( σ ) < f ( τ ) and ( σ, τ ) ∈ V f implies f ( σ ) ≥ f ( τ ) . V f is called the gradient vector field of f . In contrast to simplexwise linear functions, which are determined by their functionvalues at the vertices, discrete Morse functions take values on cells of any dimension.The gradient vector field of a discrete Morse function encodes only the sign ofthe di ff erence between function values, not the di ff erence itself. Therefore a discretegradient vector field does not uniquely determine a discrete Morse function, but forevery discrete Morse function f there is exactly one gradient vector field V f .In order to be able to treat non-generic input functions, it is useful to consider amore general class of functions, which we call pseudo-Morse functions . Pseudo-Morsefunctions substitute the strict inequality in the definition of Morse functions by a weakone. Definition (pseudo-Morse function, consistency) . A function f : K → R on the cellsof a regular CW complex K is a discrete pseudo-Morse function if there is a gradientvector field V such that whenever σ is a facet of τ then ( σ, τ ) (cid:60) V implies f ( σ ) ≤ f ( τ ) and ( σ, τ ) ∈ V implies f ( σ ) ≥ f ( τ ) . In this case, we call f and V consistent . Note that a gradient vector field V consistent with a pseudo-Morse function f is notunique in general. The following lemma provides a useful characterization of discretepseudo-Morse functions. Lemma 4.
Let f : K → R be a function on the cells of a regular CW complex K and letV be a gradient vector field on K . Then f is a discrete pseudo-Morse function consistentwith V if and only if for every (cid:15) > there is a discrete Morse function f (cid:15) : K → R with (cid:107) f (cid:15) − f (cid:107) ∞ ≤ (cid:15) such that V is the gradient vector field of f (cid:15) .Proof. Assume that f is a pseudo-Morse function consistent with a gradient vector field V . There exists a discrete Morse function g whose gradient vector field V g is preciselygiven by V [14]. Let G be the maximum absolute value of g . Given (cid:15) >
0, for each cell σ define f (cid:15) ( σ ) : = f ( σ ) + (cid:15) g ( σ ) G . Then it is straightforward to check that f (cid:15) is a discreteMorse function with gradient vector field V and (cid:107) f (cid:15) − f (cid:107) ∞ ≤ (cid:15) .7n the other hand, assume that for every (cid:15) > f (cid:15) : K → R consistent with V and (cid:107) f (cid:15) − f (cid:107) ∞ ≤ (cid:15) . Choose (cid:15) such that for every φ, ρ ∈ K with f ( φ ) (cid:44) f ( ρ ) we have (cid:15) < | f ( φ ) − f ( ρ ) | . In this case, one easily verifies that f is apseudo-Morse function consistent with V . (cid:3) The previous lemma provides a symbolic perturbation scheme based on gradientvector fields in order to allow for non-generic (degenerate) input functions. Startingwith a pseudo-Morse function f , we can choose a consistent gradient vector field V ,which may not be unique. Lemma 4 asserts that there is a discrete Morse functions f (cid:15) arbitrarily close to f and consistent with V . Therefore we can work with f as if it were adiscrete Morse function with gradient vector field V . In particular, we use it to considercritical points associated to a pseudo-Morse function by choosing a consistent gradientvector field.This first symbolic perturbation scheme is not su ffi cient for all our purposes; thedefinition of persistence pairs in Section 3 not only requires a gradient vector field, butalso a total order on the critical cells, which again might not be uniquely defined bya pseudo-Morse function f and a consistent gradient vector field V . We now derive asecond perturbation scheme that meets these requirements.Since a gradient vector field imposes certain inequality constraints on the functionsconsistent with it, we can ask how these inequalities a ff ect the relation between thefunction values of any two cells. We observe that any discrete gradient vector field givesrise to a strict partial order on the set of cells: Definition (induced partial order) . Let V be a discrete gradient vector field and considerthe relation ← V defined on K such that whenever σ is a facet of τ then ( σ, τ ) (cid:60) V implies σ ← V τ and ( σ, τ ) ∈ V implies σ → V τ. Let ≺ V be the transitive closure of ← V . Then ≺ V is called the (strict) partial orderinduced by V. The interpretation of this partial order is that for any pseudo-Morse function f consistent with V and any two cells φ and ρ , the relation φ ≺ V ρ implies f ( φ ) ≤ f ( ρ ).The relation ← V is the covering relation of ≺ V , i.e., φ ← V ρ implies φ ≺ V ρ and thereis no ψ with φ ≺ V ψ ≺ V ρ . The covering relation of a partial order forms a directedacyclic graph called the Hasse diagram (with edges oriented as suggested by ← V ). TheHasse diagram H V of ≺ V is obtained from the Hasse diagram of the face lattice of K byinverting the orientation of all edges corresponding to pairs ( σ, τ ) ∈ V as described byChari [5]. H V has the property that φ ≺ V ρ if and only if there is a directed path from ρ to φ . Note that σ ← V τ implies f ( σ ) ≤ f ( τ ), i.e., both the arrow visualizing ( σ, τ ) ∈ V and the arrow symbolizing σ ← V τ point towards a (weakly) decreasing function valueof f .Assume we are given a pseudo-Morse function f consistent with a gradient vectorfield V . On the one hand we have the induced partial order ≺ V . On the other hand thefunction f canonically induces a strict partial order ≺ f given by φ ≺ f ρ ⇔ f ( φ ) < f ( ρ ).Since the two orders ≺ f and ≺ V are compatible by assumption (there are no two cells( φ, ρ ) with φ ≺ V ρ and φ (cid:31) f ρ ), we can merge them into a strict partial order ≺ f , V (thetransitive closure of ( ≺ f ∪ ≺ V ) ⊂ K × K ). A linear extension of this order is now a stricttotal order ≺ consistent with both f and V . Definition (consistent total order) . Let V be a discrete gradient vector field V consistentwith a discrete pseudo-Morse function f . Then a strict total order ≺ is called consistent with ( f , V ) if it is a linear extension of ≺ f and ≺ V . ≺ gives rise to a canonical function K → N , which is a discreteMorse function and consistent with V . If we use this function as the function g in theproof of Lemma 4 to construct f (cid:15) , then f (cid:15) is an injective discrete Morse function withgradient vector field V and the total order induced by f (cid:15) is ≺ again. We thus obtain asecond symbolic perturbation scheme for situations where a total order on the cells isrequired.We make use of this concept in the following definition. A classical object of studyin smooth Morse theory is the sublevel set { x ∈ M : f ( x ) ≤ t } of a function f : M → R on a manifold M . In the discrete theory, the analogous object is the level subcomplex ,and the equivalent construction using our second symbolic perturbation scheme is the order subcomplex : Definition (level subcomplex [14], order subcomplex) . Let f be a pseudo-Morse func-tion on a regular CW complex K . Let the carrier of a subset L ⊂ K be the smallestsubcomplex of K containing all of L. Then for t ∈ R , the level subcomplex is K ( t ) = carrier (cid:32) (cid:91) ρ ∈ K : f ( ρ ) ≤ t ρ (cid:33) . Similarly, let ≺ be a strict total order on the cells K of a regular CW complex K . Thenfor a cell σ ∈ K, the order subcomplex is K ( σ ) = carrier (cid:32) (cid:91) ρ ∈ K : ρ (cid:22) σ ρ (cid:33) . Like in the smooth theory, the homotopy type of level subcomplexes changes onlyat critical cells. The statement can trivially be rephrased for order subcomplexes:
Theorem 5 (Forman [14]) . Let V be a gradient vector field on K and let ≺ be a linearextension of ≺ V . If ρ and ψ are two cells such that ρ ≺ ψ and there is no critical cell φ with respect to V such that ρ ≺ φ (cid:22) ψ , then K ( ψ ) collapses to K ( ρ ) . The order subcomplexes provide a finer (cell-by-cell) filtration of the complex K than the level subcomplexes, in particular if f is degenerate. This turns out to be usefulwhen working with persistent homology in Section 3. In this section we discuss a canonical relationship between discrete and piecewise linear(PL) Morse theory. As it turns out, it is possible to translate statements from one settingto the other seamlessly. Similar constructions have been used by King et al. [22], Attaliet al. [3].Assume that K is a simplicial complex. Let f PL be a simplexwise linear function on K and let f be its restriction to the 0-skeleton of K . The function f inductively givesrise to a discrete pseudo-Morse function f in the following way. For each 0-cell α , let f ( α ) = f ( α ). For a cell τ with dim τ >
0, let f ( τ ) be the maximum value of f on anyfacet of τ . The function f can easily be seen to be pseudo-Morse since it is consistentwith the empty vector field V = ∅ (all cells are critical). Note that any level subcomplexof f coincides with the induced subcomplex of K on the corresponding sublevel setof f . This induced subcomplex, in turn, is homotopy equivalent to the correspondingsublevel set of f PL [25, 29]. This means that from a Morse-theoretic point of view, thePL function f PL and the pseudo-Morse function f are equivalent. We conclude:9 heorem 6. Let f PL be a simplexwise linear function on a simplicial complex K . Thenthere is a canonical pseudo-Morse function f on K such that for every t ∈ R the sublevelset { x ∈ K : f PL ( x ) ≤ t } is homotopy equivalent to the level subcomplex K ( t ) . Vice versa, we can interpret any discrete pseudo-Morse function f on a regular CWcomplex K as a simplexwise linear function f sd : | sd K| → R on the underlying space ofthe barycentric subdivision sd K . The barycentric subdivision of a regular CW complex K is the order complex of the face lattice, i.e., the abstract simplicial complex sd K whose vertices are the cells of K and whose simplices are the totally ordered subsetsof the face lattice. The underlying space | sd K| is homeomorphic to K . The function f sd is assumed to linearly interpolate the values of f at the vertices of | sd K| insideeach simplex of | sd( K ) | . Again, the sublevel sets of f sd are homotopy equivalent to thecorresponding level subcomplexes of f : Figure 3: Illustration to Theorem 7, showing the homotopy equivalence of the levelsubcomplex K ( t ) to the sublevel set { x ∈ | sd K| : f sd ( x ) ≤ t } for t =
5. From left to right:function f on K (5); barycentric subdivision sd K (5) = ∆ ( K (5)) = ∆ (7); vector fieldsdefining the collapse of ∆ (7) onto ∆ (6) and of ∆ (6) onto ∆ (5); sublevel set of f sd . Theorem 7.
Let f be a pseudo-Morse function on a simplicial complex K . Then finduces a simplexwise linear function f sd on | sd K| such that for every t ∈ R the levelsubcomplex K ( t ) is homotopy equivalent to the sublevel set { x ∈ | sd K| : f sd ( x ) ≤ t } .Proof. Let V be a discrete gradient vector field on K that is consistent with f and let ≺ be a total order consistent with ( f , V ). Let K ( t ) and K ( ρ ) denote the cells of the leveland order subcomplexes K ( t ) and K ( ρ ), respectively. Let ∆ ( U ) denote the inducedsubcomplex of sd K on a vertex set U (we identify a cell ρ ∈ K with the correspondingvertex { ρ } ∈ sd K ). The induced subcomplex ∆ ( K ( t )) is easily seen to be identical tosd K ( t ). Let F ( t ) = { φ ∈ K : f ( φ ) ≤ t } ⊂ K ( t ). We now show that ∆ ( K ( t )) collapsessimplicially onto ∆ ( F ( t )). See Figure 2.3 for an example.Let σ ∈ K ( t ) \ F ( t ) and let σ − denote its predecessor with respect to ≺ . We write ∆ ( ρ )for ∆ ( { φ ∈ K : φ ≺ ρ } ). We show that ∆ ( σ ) collapses onto ∆ ( σ − ). It follows from thedefinition of an order subcomplex that ( σ, τ ) ∈ V for a unique τ ∈ F ( t ). Consequently,for every simplex S ∈ ∆ ( σ ) with σ ∈ S and τ (cid:60) S the simplex T = S ∪ { τ } is alsocontained in ∆ ( σ ). Hence, these pairs ( S , T ) constitute a discrete gradient vector field W on ∆ ( σ ) such that exactly the simplices containing σ (the vertex star of σ ) are non-critical. This vector field W provides a simplicial collapse of ∆ ( σ ) onto ∆ ( σ − ) byapplying Theorem 5 with an arbitrary linear extension of ≺ W . By repeatedly applyingthis argument, we obtain that ∆ ( K ( t )) collapses onto ∆ ( F ( t )). This implies that theunderlying spaces are homotopy equivalent.Finally, let f sd be the simplexwise linear extension of f from the vertices of sd( K ) tothe whole complex. Recall that the induced subcomplex ∆ ( F ( t )) is homotopy equivalentto the sublevel set { x ∈ | sd K| : f sd ( x ) ≤ t } [25, 29]. The claim now follows. (cid:3) f ( σ ) as theminimum value of all cells that contain σ as a facet. This can be used to constructdiscrete pseudo-Morse functions from functions defined on cubical grids, such aspixel images, by interpreting each pixel as a 2-cell. The resulting level subcomplexescorrespond to the cubical complexes extracted from images as described by Kaczynskiet al. [21]. Vice versa, a pseudo-Morse function on a cubical complex can be interpretedas a function defined on a subdivided grid. This construction has been used in theexamples in Section 6.Note that starting with a PL function and constructing a pseudo-Morse functionconsistent with the empty vector field means that initially all cells are considered critical,which is a point worth discussing. King et al. [22] propose to construct an initial discretegradient vector field with critical cells corresponding to the critical vertices (in the PLsense, see [23, 13, 4]) of a (non-degenerate) input PL function instead. We omit such astep for two reasons. First, this step is unnecessary in our method and would not leadto di ff erent results. Second, the step can actually be interpreted as a special case ofthe topological simplification problem with δ =
0. In this case, the problem reduces tominimizing the number of critical points among all gradient vector fields consistent withthe input function. We discuss the simplification of a gradient vector field in Sections 3.3and 5.3.
The notions of persistent homology and persistence pairs were introduced in [10, 34, 6]in order to investigate the change of the homology groups in a filtration of a topologicalspace (a nested sequence of subspaces). This concept can naturally be applied to discretepseudo-Morse functions. The following definitions can be applied to cellular homologywith coe ffi cients in an arbitrary field F . We write H d ( K ) as a shorthand for the d th homology group H d ( K ; F ) of K and H ∗ ( K ) = (cid:76) d H d ( K ). Convention and Notation
Throughout Section 3 we consider a pseudo-Morse func-tion f consistent with a gradient vector field V on a regular CW complex K and a stricttotal order ≺ consistent with ( f , V ). As a consequence of Theorem 5, the homology groups of order subcomplexes changeonly at critical cells of V . Let σ and τ be critical cells such that σ ≺ τ and considerthe inclusion map i σ, τ : K ( σ ) (cid:44) → K ( τ ) between the order subcomplexes with regardto the total order ≺ . This map induces a homomorphism i σ, τ ∗ : H ∗ ( K ( σ )) → H ∗ ( K ( τ ))between homology groups. For every cell ρ , let ρ − denote its predecessor with regardto ≺ . Now consider the sequence H ∗ ( K ( σ − )) → H ∗ ( K ( σ )) → H ∗ ( K ( τ − )) → H ∗ ( K ( τ ))of induced homomorphisms. Here we allow for the cases σ = τ − and σ − = ∅ (if σ isthe first cell in ≺ , in which case H ∗ ( K ( σ − )) is the trivial group).11 efinition (birth, death, persistence pair [10]) . We say that a class h ∈ H ∗ ( K ( σ )) is born at (or created by ) a positive cell σ ifh (cid:60) im( i σ − , σ ∗ ) . Moreover, we say that a class h ∈ H ∗ ( K ( σ )) that is born at σ dies entering (or gets merged by ) a negative cell τ if there is a class ˜ h ∈ H ∗ ( K ( σ − )) such thati σ, τ − ∗ ( h ) (cid:60) im( i σ − , τ − ∗ ) but i σ, τ ∗ ( h ) = i σ − , τ ∗ (˜ h ) ∈ im( i σ − , τ ∗ ) . If there exists a class h that is born at σ and dies entering τ , then ( σ, τ ) is a persistencepair . The di ff erence f ( τ ) − f ( σ ) is called the persistence of ( σ, τ ) . Note that in this definition we always have dim τ = dim σ +
1. On combinatorialsurfaces, the only possible cases for (dim σ, dim τ ) are (0 ,
1) or (1 , For any closed combinatorial surface K , there is an associated dual complex K ∗ , acombinatorial surface homeomorphic to K whose i -cells correspond to (2 − i )-cells of K [18]. A discrete pseudo-Morse function f on K gives rise to a discrete pseudo-Morsefunction f ∗ on K ∗ via σ ∗ (cid:55)→ − f ( σ ) [14].Moreover, as shown by Cohen-Steiner et al. [7] and Attali et al. [3], the persistencepairs of dimension (1 ,
2) for K correspond to the persistence pairs of dimension (0 , K ∗ (with τ ∗ ≺ σ ∗ ⇔ σ ≺ τ ). The homology groups H ( K ( ρ i ))(generated by the connected components of K ( ρ i )), and hence the persistence pairs ofdimension (0 , K , also called the (primal)graph of K . Consequently, the persistence pairs of dimension (1 ,
2) are determined bythe 1-skeleton of K ∗ , called the dual graph . This means that all persistence pairs of asurface can be determined in terms of Morse functions on graphs.In order to treat surfaces with boundary, we employ the usual construction ofattaching an additional 2-cell (with function value ∞ ) to each boundary component.This way we obtain a closed surface having the same sequence of order subcomplexes(up to the additional cells) and hence the same persistence pairs as the original surface. Persistence pairs on surfaces carry a certain hierarchical structure that allows us toestablish a connection to the cancelation theorem of discrete Morse theory. The mainresult of this section is that persistence pairs on surfaces can always be canceledsequentially if the order of cancelations respects this hierarchy.
Definition (parent, child, persistence hierarchy) . On a combinatorial surface K , let ( σ, τ ) be a persistence pair with dim σ = , and let [ σ ] ∈ H ( K ( σ )) be the class createdby σ . Let ˜ σ be the unique cell creating the class [ ˜ σ ] ∈ H ( K ( τ )) into which [ σ ] getsmerged by τ , i.e., [ ˜ σ ] (cid:60) im( i ˜ σ − , τ ∗ ) and [ ˜ σ ] = i σ, τ ∗ ([ σ ]) . Then ˜ σ is called the parent of σ (in the persistence hierarchy ), and σ is called the child of ˜ σ . The transitive closure ofthe child relation is called descendant . Let ( σ, τ ) and ( ˜ σ, ˜ τ ) be two persistence pairs. If either dim σ = dim ˜ σ = σ is the parent of σ or dim τ = dim ˜ τ = τ ∗ is the parent of τ ∗ (with regard to thepersistence hierarchy on the dual complex), then we also call the pair ( ˜ σ, ˜ τ ) the parent of ( σ, τ ) and ( σ, τ ) the child of ( ˜ σ, ˜ τ ). The following definition and lemma justify thisnomenclature: 12 ττ ˜ σ σ ˆ σ ˆ τ Figure 4: The persistence hierarchy. Both ( σ, τ ) and ( ˆ σ, ˆ τ ) are children of, and hencenested in, ( ˜ σ, ˜ τ ). Only ( σ, τ ) needs to be canceled before ( ˜ σ, ˜ τ ) can be canceled. Definition (nested pairs) . On a combinatorial surface K , let ( σ, τ ) and ( ˜ σ, ˜ τ ) be twopersistence pairs. We say that ( σ, τ ) is nested in ( ˜ σ, ˜ τ ) if ˜ σ ≺ σ ≺ τ ≺ ˜ τ . Lemma 8.
Let ( σ, τ ) be a descendant of ( ˜ σ, ˜ τ ) in the persistence hierarchy. Then ( σ, τ ) is nested in ( ˜ σ, ˜ τ ) .Proof. Without loss of generality, assume dim σ =
0; otherwise, by duality, the argu-ment can be applied to ( τ ∗ , σ ∗ ) instead of ( σ, τ ).By definition of the persistence hierarchy, [ σ ] gets merged into the class [ ˜ σ ] ∈ H ( K ( τ )) created by ˜ σ . This implies that ˜ σ ≺ σ . It also implies that the class createdby ˜ σ has not been merged by any cell of K ( τ ), hence τ ≺ ˜ τ . (cid:3) We now turn our attention to the sequential cancelation of persistence pairs. Notethat the cancelation theorem (Theorem 1) applies to vector fields, which only provide a partial order on the cells, while the notion of persistence is based on a total order . Aftercanceling a persistence pair, the new vector field is no longer consistent with the initialtotal order. It is important to keep in mind that we only talk about persistence pairsof the initial total order ≺ , which is consistent with ( f , V ); we do not consider a newtotal order after applying a cancelation (which would complicate things considerably).Applying several cancelations results in a sequence of simplified vector fields: Definition (persistence cancelation sequence) . A persistence cancelation sequence isa sequence of gradient vector fields ( V , V , . . . , V n ) with V = V, where each V i isconstructed from V i − by canceling a persistence pair ( σ i , τ i ) using Theorem 1.A persistence cancelation sequence is called nested if in this construction every pair ( σ i , τ i ) nested in another pair ( σ j , τ j ) is canceled first, i.e., σ j ≺ σ i ≺ τ i ≺ τ j ⇒ i < j.A persistence cancelation sequence is called a δ -persistence cancelation sequence ifexactly those persistence pairs are canceled that have persistence ≤ δ . A persistence pair ( σ, τ ) can be canceled from a vector field as soon as all descen-dants have been canceled (compare also to Edelsbrunner et al. [11] for the existencepart of the following statement in a special case):
Lemma 9.
On a combinatorial surface K , let ( V , V , . . . , V i ) be a persistence cance-lation sequence. Assume that a persistence pair ( σ, τ ) has not been canceled in thesequence but that every descendant of ( σ, τ ) has been canceled. Then there exists aV i -path from ∂τ to σ and this path is unique.Assume further that every persistence pair nested in ( σ, τ ) has been canceled. Ifthere is a unique V i -path from ∂τ to another cell ˜ σ (cid:44) σ that is critical in V i then wehave σ (cid:31) ˜ σ . ffi cient but not necessary. The proof ofLemma 9 relies on a few auxiliary lemmas and is given after Lemma 12. Lemma 10.
Let ( V , V , . . . , V i ) be a persistence cancelation sequence and let ( σ, τ ) bea persistence pair with dim σ = that has not been canceled in the sequence. Let C bethe connected component of the subcomplex K ( τ − ) containing σ , and let C denote thecells of C . Then every ( φ, ρ ) ∈ V i with dim φ = satisfies φ ∈ C ⇔ ρ ∈ C.Proof.
The claim is shown by induction. The base case follows from consistency ofthe total order ≺ with ( f , V ). Consider the cancelation of a persistence pair ( σ i , τ i ).If dim σ i (cid:44)
0, the tuples in V i of dimensions (0 ,
1) stay unchanged and the claimimmediately follows from the induction hypothesis. Now assume dim σ i =
0. We showthat the claim holds for every ( φ, ρ ) ∈ V i \ V i − .The cells in V i \ V i − are τ i and the cells on the V i − -path ( φ , ρ , φ , . . . , ρ r − , φ r )from φ ∈ ∂τ i to φ r = σ i . By the induction hypothesis we have φ k ∈ C ⇔ ρ k ∈ C .Because C is a subcomplex, we also have ρ k − ∈ C ⇒ φ k ∈ C (with ρ − = τ i ). Moreover,if σ i ∈ C then σ i is a descendant of σ and by Lemma 8 ( σ i , τ i ) is nested in ( σ, τ ),implying that σ i and τ i are in the same connected component of K ( τ − ). Hence we alsohave σ i ∈ C ⇒ τ i ∈ C . Consequently, either all or none of the cells in V i \ V i − arecontained in C and the claim immediately follows. (cid:3) We also require the notion of the restriction of a vector field to a subcomplex:
Definition (restriction of a vector field to a subcomplex) . Let V be a discrete vectorfield on K and let (cid:101) K be a subcomplex of K with cells (cid:101) K. The restriction of V to (cid:101) K is (cid:101) V = V ∩ (cid:0)(cid:101) K × (cid:101) K (cid:1) , i.e., the pairs of cells in V that are both in (cid:101) K. As a direct consequence of this definition, the restriction of a vector field V onto asubcomplex may have critical cells that are not critical in V : Lemma 11.
Let (cid:101)
V be the restriction of a discrete vector field V on K to a subcomplex (cid:101) K .The critical d-cells of (cid:101) V are exactly the critical d-cells of V that are contained in (cid:101) K ifand only if each pair ( σ, τ ) ∈ V with dim σ = d satisfies σ ∈ (cid:101) K ⇔ τ ∈ (cid:101) K. Moreover, we use the following fact:
Lemma 12.
Let V be a discrete gradient vector field V on K with only one critical0-cell σ . Then there is a V-path from every -cell ˜ σ to σ .Proof. Each V -path of dimension 0 ending at a non-critical cell ˜ σ (cid:44) σ , ( ˜ σ, ˜ τ ) ∈ V , canbe extended by ˜ τ and the unique 0-cell ˆ σ ∈ ∂ ˜ τ , ˆ σ (cid:44) ˜ σ . Since K is finite and V does notcontain nontrivial closed paths, the extension will eventually end up at σ . (cid:3) Proof of Lemma 9.
Without loss of generality, assume dim σ =
0; otherwise, by duality,the argument can be applied to ( τ ∗ , σ ∗ ) instead of ( σ, τ ).Let C be the connected component of the subcomplex K ( τ − ) created by σ . Apartfrom σ , every 0-cell in C that is critical in V is a descendant of σ . By assumption, alldescendants of σ have been canceled, and hence σ is the only 0-cell in C that is criticalin V i . By Lemmas 10 and 11, σ is also the only critical 0-cell in the restriction of V i to C . By Lemma 12, there is a V i -path to σ from every 0-cell in C , in particular fromexactly one of the two 0-cells in ∂τ since ∂τ ∩ C contains exactly one cell. By Lemma 2,this path is unique.Now assume that every persistence pair nested in ( σ, τ ) has been canceled and thereis a unique V i -path from ∂τ to another cell ˜ σ (cid:44) σ that is critical in V i . By assumption, ˜ σ
14s not a descendant of σ , meaning that ˜ σ and σ are in di ff erent connected componentsof K ( τ − ). Moreover, ˜ σ creates the component (cid:101) C (cid:44) C , because otherwise we would havean uncanceled pair ( ˜ σ, ˜ τ ) nested in ( σ, τ ). Since τ is paired with σ and merges (cid:101) C and C ,we know that σ is a descendant of ˜ σ and σ (cid:31) ˜ σ . (cid:3) As a consequence of Lemma 9, we can construct a sequence of cancelations toeliminate all persistence pairs below a certain persistence threshold:
Theorem 13.
Let f be a pseudo-Morse function on a combinatorial surface K andlet δ ≥ . Then there exists a nested δ -persistence cancelation sequence.Proof. If the subsequence ( V , V , . . . , V i − ) satisfies the assumptions of Lemma 9 forsome persistence pair ( σ i , τ i ), we can use Theorem 1 to construct V i from V i − . Acanonical choice satisfying these assumptions is given by canceling the persistence pairs( σ i , τ i ) with persistence ≤ δ according to the order ≺ on the negative cells, i.e., τ i ≺ τ i + for every i . The claim follows by induction. (cid:3) Cohen-Steiner et al. [6] studied properties of persistence diagrams , which are a repre-sentation of the value pairs ( f ( σ ) , f ( τ )) corresponding to the persistence pairs ( σ, τ ) ofa function f . Here we use R = R ∪ {−∞ , ∞} . Definition (Persistence diagram [6]) . The persistence diagram D ( f ) ⊂ R of a pseudo-Morse function f is the multiset consisting of ( f ( σ ) , f ( τ )) for all persistence pairs ( σ, τ ) of f , together with all points on the diagonal counted with (countably) infinite multiplic-ity. An unpaired positive cell σ is represented by ( f ( σ ) , ∞ ) . The main result of [6] is the
Bottleneck Stability Theorem for persistence diagrams:if two functions are close then their persistence diagrams are also close. Due to thecorrespondence between piecewise linear functions and discrete pseudo-Morse functions(Section 2.3), the statement reads as follows in the language of discrete Morse theory:
Definition (Bottleneck distance) . Let X and Y be two multisets of R . The bottleneckdistance is d B ( X , Y ) : = inf γ sup x ∈ X (cid:107) x − γ ( x ) (cid:107) ∞ , where γ ranges over all bijections fromX to Y. Here we assume ( a , ∞ ) − ( b , ∞ ) = ( a − b , a , ∞ ) − ( b , c ) = ( a − b , ∞ ), and (cid:107) ( a , ∞ ) (cid:107) ∞ = ∞ for a , b , c ∈ R . Theorem 14 (Cohen-Steiner et al. [6]) . Let f , g : K → R be two discrete pseudo-Morsefunctions. Then the respective persistence diagrams satisfy d B ( D ( f ) , D ( g )) ≤ (cid:107) f − g (cid:107) ∞ . Note that the bottleneck distance provides a metric on the persistence diagramsof pseudo-Morse functions on K , in particular, d B ( D ( f ) , D ( g )) = D ( f ) = D ( g ). Therefore, in contrast to the persistence pairs , the persistence diagram ofa discrete pseudo-Morse function f is well-defined; in particular, it is independent ofthe total order ≺ chosen and even independent of the gradient vector field V consistentwith f . Theorem 14 provides a lower bound on the number of persistence pairs amongall pseudo-Morse functions f δ with (cid:107) f δ − f (cid:107) ∞ ≤ δ : Corollary 15 (Stability Bound) . For any pseudo-Morse function f δ with (cid:107) f δ − f (cid:107) ∞ ≤ δ ,the number of persistence pairs of f δ is bounded from below by the number of persistencepairs of f that have persistence > δ . roof. Let D and D δ be the persistence diagrams of f and f δ , respectively. By Theo-rem 14 we have d B ( D , D δ ) ≤ δ . This means that there is a bijection γ between D and D δ with (cid:107) p − γ ( p ) (cid:107) ∞ ≤ δ for all p ∈ D . Let p = ( p ∗ , p † ) = ( f ( σ ) , f ( τ )) ∈ D represent apersistence pair ( σ, τ ) of f with persistence p † − p ∗ > δ . Letting q = ( q ∗ , q † ) : = γ ( p ),this implies that p ∗ + δ ≥ q ∗ and p † − δ ≤ q † . Together with p † − p ∗ > δ , thisyields q † − q ∗ >
0. Hence there must be a persistence pair of f δ corresponding to eachpersistence pair of f with persistence > δ . (cid:3) We are interested in functions that achieve the lower bound of Corollary 15:
Definition (Perfect δ -simplification) . Let f be a pseudo-Morse function on a combina-torial surface K . A perfect δ -simplification of f is a pseudo-Morse function f δ suchthat (cid:107) f δ − f (cid:107) ∞ ≤ δ and the number of persistence pairs of f δ equals the number ofpersistence pairs of f that have persistence > δ . In this section, we prove the following central result:
Theorem 16.
Let f be a discrete pseudo-Morse function on a combinatorial surface.Then there exists a perfect δ -simplification of f . The proof of Theorem 16 is constructive and hence leads to an algorithm. Thecorresponding construction is outlined in Section 4.1. Unfortunately, the resultingalgorithm has a running time that is quadratic in the input size. We present an e ffi cient algorithm in Section 5. The proof of its correctness becomes easier once Theorem 16 isestablished. This is the reason why we present two separate constructions. Corollary 17 (Tightness of the stability bound) . Given a discrete pseudo-Morse functionf on a surface and δ ≥ , there exists a discrete pseudo-Morse f δ consistent with agradient vector field V δ such that (cid:107) f δ − f (cid:107) ∞ ≤ δ and the number of critical points of V δ equals the number of critical points of f that have persistence > δ . Using Lemma 4, the result can also be stated for (non-degenerate) discrete Morsefunctions (in a slightly di ff erent form, because only critical points with persistence < δ can be eliminated within a tolerance of δ in the set of discrete Morse functions): Corollary 18.
Given a discrete Morse function f on a surface and δ > , there exists adiscrete Morse function f δ such that (cid:107) f δ − f (cid:107) ∞ < δ and the number of critical points off δ equals the number of critical points of f that have persistence ≥ δ . Convention and Notation
Throughout this section we consider a given pseudo-Morsefunction f consistent with a gradient vector field V on a combinatorial surface K , a stricttotal order ≺ consistent with ( f , V ), and a nested 2 δ -persistence cancelation sequence( V , . . . , V n ) with V = V . Moreover, we let ≺ j : = ≺ V j denote the partial order inducedby V j . 16 .1 The plateau function For every V i in the cancelation sequence, we inductively define a pseudo-Morse function f i consistent with V i , see Figure 1 for an illustration. By assumption we start with apseudo-Morse function f : = f consistent with V : = V . Suppose that we have con-structed a pseudo-Morse function f i − consistent with V i − . Let ( σ, τ ) be the persistencepair that is canceled in the construction of V i from V i − using Theorem 1. We define thecorresponding plateau function f i as follows: m i = f ( σ ) + f ( τ )2 and f i ( ρ ) : = m i if ρ (cid:23) i − σ and f i − ( ρ ) < m i or ρ (cid:22) i − τ and f i − ( ρ ) > m i , f i − ( ρ ) otherwise . This means that the attracting set { ρ : ρ (cid:23) i − σ } of σ is raised to at least the value m i ,and analogously the repelling set { ρ : ρ (cid:22) i − τ } of τ is lowered. Hence, f i creates a local plateau at the value m i . The following lemma is a direct consequence of the way weconstruct f i from f i − and the fact that f i is constant along the path from ∂τ to σ . It canbe proven using a straightforward induction argument. Lemma 19.
The plateau function f i is consistent with both V i − and V i . Note that the construction of the plateau function does not depend on the propertiesof combinatorial surfaces but can be applied to regular CW complexes of arbitrarydimensions . Moreover, it does not depend on the cancelation persistence pairs: wheneverwe have a pseudo-Morse function f consistent with a gradient vector field V and (cid:101) V isconstructed from V by a cancelation using Theorem 1, we can obtain a plateau function˜ f that is consistent with both V and (cid:101) V . It remains to show that the plateau construction above is admissible, i.e., that all of thefunctions f i satisfy the δ -constraint. Lemma 20.
Each plateau function f i satisfies (cid:107) f i − f (cid:107) ∞ ≤ δ .Proof. We show the statement by induction. The base case is trivial since f = f .Let ( σ, τ ) be the persistence pair that is canceled when constructing V i from V i − .We show that the δ -constraint is neither violated by increasing the value of any cell ρ inthe attracting set of σ in V i − , nor by decreasing the value of any cell in the repellingset of τ . Since f i ( ρ ) = f i − ( ρ ) for all cells ρ not treated in these two cases, the claimfollows.We first show | f i ( ρ ) − f ( ρ ) | ≤ δ for any cell ρ (cid:23) i − σ with f i − ( ρ ) < m i . By theinduction hypothesis we have a lower bound f i − ( ρ ) ≥ f ( ρ ) − δ . By construction of f i ,the value of ρ is increased: f i ( ρ ) = m i > f i − ( ρ ). Therefore, the lower bound remainsvalid after step i : f i ( ρ ) > f i − ( ρ ) ≥ f ( ρ ) − δ. To show the upper bound f i ( ρ ) ≤ f ( ρ ) + δ , we first use f ( τ ) − f ( σ ) ≤ δ to obtain f i ( ρ ) = m i = f ( σ ) + f ( τ )2 ≤ f ( σ ) + ( f ( σ ) + δ )2 = f ( σ ) + δ. This is almost the desired inequality except that the right hand side contains f ( σ ) insteadof f ( ρ ). To finish the proof, it therefore su ffi ces to show that f ( σ ) ≤ f ( ρ ). This, in turn,17 ν ψφ φ ψ Figure 5: Example illustrating Lemma 21. Left: gradient vector field W (before reversingthe path from ∂ν to µ ). Right: gradient vector field (cid:101) W (after path reversal). Note that wehave the new relation φ ≺ (cid:101) W ψ (corresponding in this example to a (cid:101) W -path from ψ to φ ).In the example, the conclusion φ (cid:22) W ν and µ (cid:22) W ψ of Lemma 21 is reflected by the two W -paths from ∂ν to φ and from ψ to µ , respectively.is a consequence of the facts that, according to Lemma 22, σ ≺ i − ρ implies σ ≺ ρ , andthat ≺ is consistent with ( f , V ).It remains to show that | f i ( ρ ) − f ( ρ ) | ≤ δ for any cell ρ (cid:22) i − τ with f i − ( ρ ) > m i . Theproof of this statement is analogous to the above. (cid:3) Before proving Lemma 22, we first investigate how the reversal of a gradient vectorfield may change the induced partial order (see Figure 5 for an example):
Lemma 21.
Let µ, ν, φ, ψ be (not necessarily disjont) cells of a regular CW complex K ,and let W and (cid:101) W be two gradient vector fields. Assume that the cells µ, ν are criticalin W and that (cid:101)
W is constructed by reversing W along the unique W-path from ∂ν to µ .Assume further that φ ⊀ W ψ and φ ≺ (cid:101) W ψ . Then φ (cid:22) W ν and µ (cid:22) W ψ .Proof. By definition of the induced partial order, φ ≺ (cid:101) W ψ implies that there exists asequence ( ρ , . . . , ρ k ) with ρ = φ , ρ k = ψ and ρ i ← (cid:101) W ρ i + for all 1 ≤ i ≤ k −
1. Hereeither ρ i is a facet of ρ i + or ρ i + is a facet of ρ i , and we therefore also have either ρ i ← W ρ i + or ρ i → W ρ i + . But since φ ⊀ W ψ , there exists a smallest index j suchthat ρ j → W ρ j + . Since the relations ← W and ← (cid:101) W di ff er only along the W -path from ∂ν to µ (including ν ), it follows that the cells ρ j and ρ j + are contained in this W -path.Hence we have ρ j (cid:22) W ν . Moreover, by the choice of j we have φ = ρ (cid:22) W ρ j . Thereforewe conclude that φ (cid:22) W ν . By an analogous argument one also shows that µ (cid:22) W ψ . (cid:3) Lemma 22.
Let ( V , . . . , V n ) be a nested persistence cancelation sequence and let ( σ, τ ) be a persistence pair of ≺ with σ and τ critical cells of V i . Then for any ρ ∈ K,(a) ρ (cid:31) i σ implies ρ (cid:31) σ , and(b) ρ ≺ i τ implies ρ ≺ τ .Proof. We only present the proof of part (a), which is done again by induction: we showthat ρ (cid:31) i σ implies ρ (cid:31) σ for all 0 ≤ i ≤ n . Part (b) can be shown analogously.The base case i = (cid:31) is a linear extension of (cid:31) . Assume that ρ (cid:31) i σ . If ρ (cid:31) i − σ , then the claim follows directly from the induction hypothesis.Hence we assume that ρ (cid:7) i − σ . Let ( ˜ σ, ˜ τ ) be the persistence pair that is canceledwhen constructing V i from V i − ; this implies ˜ σ ≺ i − ˜ τ . From Lemma 21 with ( W , (cid:101) W ) = ( V i − , V i ) and ( µ, ν, φ, ψ ) = ( ˜ σ, ˜ τ, σ, ρ ), we infer that σ (cid:22) i − ˜ τ and ˜ σ (cid:22) i − ρ . This hastwo consequences:(i) σ ≺ i − ˜ τ (since σ is critical in V i while ˜ τ is not), and(ii) ˜ σ (cid:22) ρ (by the induction hypothesis).18o finish the proof of the claim, by (ii) it su ffi ces to show that σ ≺ ˜ σ . We proceedby case analysis on the dimensions of ˜ σ and σ . Since these two cells are positive byassumption, they have dimension less than 2. Case 1 (dim σ =
1, dim ˜ σ = V i − -pathfrom the 1-cell ˜ τ to the 0-cell ˜ σ does not change the attracting set of any critical 1-cell(and in particular σ ), contradicting ρ (cid:7) i − σ and ρ (cid:31) i σ . Case 2 (dim σ =
0, dim ˜ σ = τ ≺ ˜ τ . If additionally ˜ σ ≺ σ , thiscontradicts the assumption that the cancelation sequence is nested and ( σ, τ ) is canceledafter ( ˜ σ, ˜ τ ). Therefore τ ≺ ˜ τ implies σ ≺ ˜ σ .Now assume τ (cid:31) ˜ τ . This means that σ creates a connected component that is notyet merged in K (˜ τ ). Since σ ≺ i − ˜ τ by (i), there is a sequence ( ρ , . . . , ρ k ) with ρ = σ , ρ k = ˜ τ , and ρ j ← V i − ρ j + for all 1 ≤ j ≤ k −
1. For each ρ j we trivially have ρ j ≺ i − ˜ τ and hence ρ j ≺ ˜ τ by the induction hypothesis, implying that ρ j ∈ K (˜ τ ). Moreover, sinceeither ρ j is a facet of ρ j + or ρ j + is a facet of ρ j , we know that all ρ j , and in particular σ and ˜ τ , are in the same connected component of K (˜ τ ). In an analogous way one showsthat ˜ σ and ˜ τ , and hence σ and ˜ σ , are in one and the same connected component. Sincewe know that σ created that component, it follows that σ ≺ ˜ σ . Case 3 (dim σ = dim ˜ σ ∈ { , } ): The relation σ ≺ i − ˜ τ from (i) above implies theexistence of a V i − -path from ˜ τ to σ . We will show by contradiction that this path mustbe unique. To see this, assume that there are two V i − -paths from ˜ τ to σ . Without loss ofgenerality, assume that dim σ = dim ˜ σ = τ = σ ∗ , ˜ τ ∗ instead of ˜ τ, σ . By Corollary 3, each ofthe 0-cells in ∂ ˜ τ must belong to exactly one of the two V i − -paths from ˜ τ to σ . Now bya similar argument as in Case 2 above, we obtain that each cell of these two V i − -pathsis contained in the same connected component of K (˜ τ − ) as σ . But since ˜ τ is a negative1-cell, the two 0-cells in its boundary belong to di ff erent connected components of K (˜ τ − ), a contradiction.Hence, there is a unique V i − -path from ˜ τ to σ . Lemma 9 asserts that ˜ σ is the largestcell (with regard to ≺ ) with a unique V i − -path from ˜ τ to ˜ σ . Since σ (cid:44) ˜ σ , we obtain σ ≺ ˜ σ . (cid:3) Proof of Theorem 16.
According to Theorem 13 there exists a nested 2 δ -persistencecancelation sequence ( V , V , . . . , V n ) for the pseudo-Morse function f . Let f n be theplateau function corresponding to V n . Since f n is consistent with V n by Lemma 19 and (cid:107) f n − f (cid:107) ∞ ≤ δ by Lemma 20, it is a perfect δ -simplification. (cid:3) ffi cient algorithm The definition of the plateau function in the previous section canonically leads to analgorithm that runs in time quadratic in the input size. In this section we present amethod for computing a perfect δ -simplification in time dominated by the computationof persistence pairs, i.e., O (sort( n )), where n = | K | is the number of cells of K . Apartfrom this computation, all steps of our algorithm take linear time O ( n ). We stress thatpre- and post-processing steps, like conversion from and to PL functions, also requireonly linear time O ( n ).The algorithm can be summarized as follows. First, persistence pairs are computedusing a variant of Kruskal’s algorithm for minimum spanning trees. Next, the persistencepairs are used to construct a simplified gradient vector field by a graph traversal of boththe primal and dual 1-skeleton. In a third step, the simplified vector is used to compute19he simplified function by a graph traversal on the Hasse diagram of the partial orderinduced by the simplified vector field. Assume we are given a pseudo-Morse function f consistent with a discrete gradientvector field V as input. We write φ (cid:39) V ρ if neither φ ≺ V ρ nor φ (cid:31) V ρ , and similarlyfor (cid:39) f . Let ≺ T be an arbitrary total order on K . We define the order ≺ as the lexicographicorder given by ≺ f , ≺ V , and ≺ T : we have φ ≺ ρ if and only if either(a) φ ≺ f ρ ,(b) φ (cid:39) f ρ and φ ≺ V ρ , or(c) φ (cid:39) f ρ and φ (cid:39) V ρ and φ ≺ T ρ .Now assume that f is constructed from data given as a PL or piecewise constant functionas explained in Section 2.3. Then V is the empty vector field (all cells are critical),meaning that φ ≺ V ρ if and only if φ is a face of τ . If now the order ≺ T is chosen suchthat the cells are sorted by dimension, then φ ≺ V ρ implies φ ≺ T ρ . The definition nowsimplifies to: φ ≺ ρ if and only if either(a) φ ≺ f ρ or(b) φ (cid:39) f ρ and φ ≺ T ρ . Recall that the persistence pairs of dimension (0 ,
1) are determined solely by the 1-skeleton G of K . Therefore, persistence pairs can be computed by applying a variant ofKruskal’s algorithm [24] for finding a minimum spanning tree to both the primal and thedual 1-skeleton [10, 3]. Let G be the 1-skeleton of K and M ( G ) the minimum spanningtree of G (using the total order ≺ for determining the edge weights, which impliesuniqueness of M ( G )). Kruskal’s algorithm for computing M ( G ) initializes a graph T with the vertices of G , sweeps over the edges of G in order ≺ , adds to T every edge of G that does not create a 1-cycle, and returns the final graph T . Note that the set of edges of M ( G ) consists of all negative 1-cells together with all 1-cells τ with ( σ, τ ) ∈ V for some σ ; all other 1-cells create a cycle in T . When encountering a negative 1-cell, we computethe persistence of the corresponding (0 ,
1) pair by storing for each connected componentof the intermediate graph T the 0-cell that created it. Clearly we obtain all dimension(0 ,
1) persistence pairs this way. Simultaneously, we construct the subgraph M δ ( G ) of M ( G ) not containing the negative 1-cells with persistence > δ . In an analogous way,for the dual 1-skeleton G ∗ we can compute the minimum spanning tree M ( G ∗ ) andobtain the subgraph M δ ( G ∗ ) together with all (1 ,
2) persistence pairs.Kruskal’s algorithm has a time complexity of O (sort( n )), yielding a complexityof O ( n log n ) for comparison-based sorting. Assuming that the function values arerepresented by a small ( O (log n )) word size, Attali et al. [3] point out that persistencepairs on a graph can be computed in linear time O ( n ) on a RAM using radix sort togetherwith a linear-time algorithm for minimum spanning trees. We now explain how to construct a simplified gradient vector field V δ . To this end,we traverse (using depth-first search) each of the connected components of the primalgraph M δ ( G ) (constructed in the previous section) from the 0-cell that created the20omponent. During this traversal, whenever we encounter an edge (1-cell) ψ thatconnects a previously visited vertex (0-cell) ρ with an unvisited vertex φ , we add ( φ, ψ )to the gradient vector field V δ . This construction takes O ( n ) time.We perform an analogous traversal for the dual graph M δ ( G ∗ ). Again, whenever weencounter an edge ψ ∗ that connects a visited vertex ρ ∗ with an unvisited vertex φ ∗ (with ψ a 1-cell and ρ, φ ψ, φ ) to the gradient vectorfield V δ . Note that the final V δ results from both the primal and dual traversals and is avector field on K . Theorem 23.
The gradient vector field V δ is identical to the final vector field V n of a δ -persistence cancelation sequence ( V , . . . , V n ) .Proof. First observe that if ( σ, τ ) ∈ V n and dim σ =
0, then both σ and τ are cellsof M δ ( G ) since all non-critical cells of V n either are non-critical in V as well or havepersistence ≤ δ (with respect to f and ≺ ). Moreover, the 0-cells creating a connectedcomponent of M δ ( G ) are the only critical 0-cells of V n (by definition) and of V δ (byconstruction). Since M δ ( G ) is a tree, the pairs ( σ, τ ) ∈ V n with dim σ = M δ ( G ∗ ), the statementfollows. (cid:3) Finally, we construct a function f δ (di ff erent from the plateau function defined inSection 4.1) that is consistent with the simplified gradient vector field V δ . Consider theHasse diagram H : = H V δ of the strict partial order ≺ V δ as described in Section 2.2. Wevisit the vertices K of H in a linear extension of ≺ V δ . The problem of finding a linearextension of a partial order is also called topological sorting and can be solved usingdepth-first search on H [8]. At each visited cell σ , we define f δ ( σ ) as the minimumvalue that satisfies the lower bound f δ ( σ ) ≥ f ( σ ) − δ and renders f δ consistent with V δ ,i.e., f δ ( σ ) = max (cid:16) f ( σ ) − δ, max ρ ← V δ σ f δ ( ρ ) (cid:17) . The construction of f δ also takes O ( n ) time. Theorem 24.
The function f δ constructed using the above algorithm is a perfect δ -simplification of f .Proof. By construction f δ is consistent with V δ . At the same time, by Theorem 23, V δ is the final vector field of a 2 δ -persistence cancelation sequence. Therefore, bythe definition of a perfect δ -simplification, it only remains to show that the constraint (cid:107) f δ − f (cid:107) ∞ ≤ δ is satisfied. The lower bound f δ ≥ f − δ is satisfied by construction. Itremains to show the upper bound f δ ≤ f + δ .Observe that the set of all perfect δ -simplifications consistent with V δ is defined bya set of linear inequalities: the upper and lower bounds on the function values givenby f ± δ , and the inequalities that define consistency with V δ . Therefore, the set of δ -simplifications is a convex polyhedron P ⊂ R n with n = | K | . The polyhedron P isbounded since it is a subset of the product of intervals (cid:81) σ ∈ K [ f ( σ ) − δ, f ( σ ) + δ ]. FromTheorem 16, we know that P is not empty. We now show that f δ is contained in P .21igure 6: Visualization of simplification artifacts. Function values indicated by graylevels. Left: Original function. Middle: Function obtained by the algorithm of Section 5.Note the bright path joining the two spots. Right: Function obtained after constraintenergy minimization according to Section 6.4. While the simplified topological structureis maintained, the visual appearance is closer to the original function.First, consider the (unbounded) convex polyhedron ˜ P defined by the lower bound f δ ≥ f − δ and the inequalities induced by V δ . By construction, f δ is contained in ˜ P .Moreover, again by construction, f δ minimizes the function value of any cell amongall functions in ˜ P . In other words, for any function ˜ f in P ⊂ ˜ P , we have ˜ f ≥ f δ . Thisimplies the upper bound f δ ≤ f + δ . (cid:3) We implemented the algorithm of Section 5 in C ++ . For a complex with over 4 millioncells (the cubical complex for a 1025 × δ -simplification on a 2.4GHz Intel Core 2Duo laptop. The method described in Section 5 assigns to each cell the smallest possible value. Asa consequence, the output function di ff ers from the input function f even if the inputfunction is already a perfect δ -simplification. Moreover, the method is not symmetricin the sense that we obtain an output function which maximizes the values if we applythe algorithm to the function − f on the dual complex and return the negative of thesimplified function. Since both the minimal and maximal solutions are points of a convexpolyhedron as explained in Section 5.5, we can take the component-wise arithmeticmean to obtain another perfect δ -simplification.With this modification, if the input function f is already a perfect δ -simplification,then the minimal solution is given by f − δ , while the maximal solution equals f + δ , sothe arithmetic mean of both solutions returns f again as desired.22
000 2000 3000 40001000200030004000
Figure 7: Top: Topographic map of elevation data set “Puget Sound” [1], showingthe region around Tacoma. Contour lines shown every 500 meters. Elevation data isconverted from a 512 ×
512 grid into a pseudo-Morse function on 1050625 cells. 33120critical cells have persistence > δ =
500 meters. The function has 1 minimum, 3 saddles, and 3maxima. 23 .3 Flooding and carving artifacts
Since the methods presented in the present article can be seen as combinations of the carving and flooding approaches, they also inherit some characteristics that may notalways be desirable in practical applications (see Figure 6).Carving methods [31, 12, 3]) cancel a pair of critical cells by changing only therepelling or attracting set of the 1-cell (saddle). This results in a noticeable thin pathbeing carved in the function. On the other hand, modifying extrema, i.e., loweringmaxima and raising minima, produces regions with constant function value; this iscalled filling or flooding [19, 9]). Although this e ff ect is less disturbing, it might appearunnatural in certain applications. In the next section, we propose a way to remedy bothkinds of artifacts. As mentioned in Section 5.5, the set of perfect δ -simplifications consistent with thesimplified gradient vector field V δ is a convex polyhedron P . Hence, the presentedmethod can be combined with energy minimization methods, since the polyhedron P can be used as the feasible region for an arbitrary convex optimization problem. Forexample, we used the interior point solver Ipopt [33] to minimize (a discretization of)the Dirichlet energy of the di ff erence f δ − f in order to obtain a function f δ that looksas similar as possible to the input function f (see Figures 6 and 7). Alternatively, weminimized the Dirichlet energy of the simplified function itself in order to obtain smoothcontour lines. The example of Figure 8 shows that a perfect δ -simplification may not exist on a non-manifold 2-dimensional cell complex. For the sake of simplicity, the example is givenfor a non-regular CW complex; it is straightforward to rephrase this example using aregular CW complex by subdividing the cells. The complex consists of two 0-cells ζ and γ with f ( ζ ) = f ( γ ) =
0, three 1-cells a , b , and c with f ( a ) = f ( b ) =
2, and f ( c ) = A and B with f ( A ) = f ( B ) =
3. Note that the complex is notmanifold since it is not locally euclidean at the 1-cell b . The persistence pairs are ( a , A ),( b , B ), and ( γ, c ). To obtain a perfect δ -simplification for δ = .
5, one would need to set f δ ( b ) = f δ ( B ) = . f δ ( a ) = f δ ( A ) = .
5. The corresponding simplified gradientvector field would be V δ = { ( a , A ) , ( b , B ) } . But since b is a facet of A , we must have f δ ( b ) ≤ f δ ( A ). Hence, we cannot cancel both ( a , A ) and ( b , B ) at the same time. Thisconstellation also appears in [10] under the name conflict of type (1,2) . A (2) a (1) c (0) γ (0) ζ (0) B (3) b (2) Figure 8: A discrete Morse function on a 2-complex that does not have a perfect δ -simplification. The function values of the cells are indicated in brackets.24ince such a 2-complex can also appear as a level subcomplex of an n -manifoldCW complex for n ≥
3, the example also shows that a perfect δ -simplification does notalways exists for functions on manifolds. As a concluding remark, we want to mention that the same constructions and proofspresented in this article can also be adapted to the problem of minimizing the number oflocal extrema of a pseudo-Morse function within a δ -tolerance on any d -dimensionalmanifold CW complex. Problem (Extrema simplification on manifolds) . Given a pseudo-Morse function f ona regular manifold CW complex and a real number δ ≥ , find a function f δ subject to (cid:107) f δ − f (cid:107) ∞ ≤ δ such that f δ has a minimum number of local extrema. Theorem 25.
Given a pseudo-Morse function f on a finite regular closed manifold CWcomplex and a real number δ ≥ , there exists a pseudo-Morse function f δ such that (cid:107) f δ − f (cid:107) ∞ ≤ δ and the number of local extrema of f δ equals the number of local extremaof f that have persistence > δ . This number is minimal. Note that in the case d = c i denote the number of critical cells ofdimension i . Since the Euler characteristic χ = c − c + c is a topological invariantand we have c + c + c = c + c ) − χ , the number of critical points is minimal ifand only if the number of extrema is minimal. Acknowledgements
The “Puget Sound” data set used in Figure 7 is taken from theLarge Geometric Models Archive of the Georgia Insitute of Technology. The originalelevation data is obtained from The United States Geological Survey (USGS), madeavailable by The University of Washington.
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