J Math Imaging Vis DOI 10.1007/s10851-008-0096-z

Multidimensional Size Functions for Shape Comparison

S. Biasotti · A. Cerri · P. Frosini · D. Giorgi · C. Landi

© Springer Science+Business Media, LLC 2008

Abstract has proven to be a useful framework proof of their stability with respect to that distance. Experi- for shape analysis in the context of pattern recognition. Its ments are carried out to show the feasibility of the method. main tool is a shape descriptor called . Size Theory has been mostly developed in the 1-dimensional set- Keywords Multidimensional size function · ting, meaning that shapes are studied with respect to func- Multidimensional measuring function · Natural tions, defined on the studied objects, with values in R.The pseudo-distance potentialities of the k-dimensional setting, that is using func- tions with values in Rk, were not explored until now for lack of an efficient computational approach. In this paper we pro- 1 Introduction vide the theoretical results leading to a concise and complete shape descriptor also in the multidimensional case. This is In the shape analysis literature many methods have been de- possible because we prove that in Size Theory the compar- veloped, which describe shapes making use of properties ison of multidimensional size functions can be reduced to of real-valued functions defined on the studied object. Usu- the 1-dimensional case by a suitable change of variables. In- ally, the role of these functions is to quantitatively measure deed, a foliation in half-planes can be given, such that the re- the geometric properties of the shape while taking into ac- striction of a multidimensional size function to each of these count its . Because of the topological information half-planes turns out to be a classical size function in two conveyed by these descriptors, they often are better suited scalar variables. This leads to the definition of a new dis- to compare shapes non-rigidly related to each other. On the tance between multidimensional size functions, and to the other hand, the metrical information provided by the func- tion reveals specific instances of features. S. Biasotti · D. Giorgi The use of different functions defined on the same object IMATI, Consiglio Nazionale delle Ricerche, Via De Marini 6, can give insights of its shape from different perspectives. 16149 Genova, Italia Although many approaches rely on a fixed function for de- scribing shapes (e.g., the height function for contour trees A. Cerri · P. Frosini () ARCES, Università di Bologna, via Toffano 2/2, 40135 Bologna, [25], and the distance transform for the medial axis [4, 37, Italia 38]), the possibility of adopting different functions charac- e-mail: [email protected] terizes an increasing number of methods [3]. This strategy is currently being investigated also in graphics for the quite A. Cerri · P. Frosini Dipartimento di Matematica, Università di Bologna, P.zza di different purpose of detecting generators of the first homol- Porta S. Donato 5, 40126 Bologna, Italia ogy group (cf. [39]). Since the early 90s, Size Theory was proposed as a geo- C. Landi metrical/topological approach to shape description and com- Dipartimento di Scienze e Metodi dell’Ingegneria, Università di Modena e Reggio Emilia, via Amendola 2, Pad. Morselli, 42100 parison based on the use of classes of functions (cf., e.g., Reggio Emilia, Italia the papers [26, 27, 43, 44], and also the survey [29] and J Math Imaging Vis Sect. 8.4 of the textbook [34]), finding its roots in the classi- The use of the multidimensional size theory in real ap- cal Morse Theory ([28]). Particular classes of functions have plications needs answers to the following questions: How to been singled out as better suited than others to deal with spe- combinatorially represent multidimensional size functions? cific problems, such as obtaining invariance under groups How to compare multidimensional size functions in a way of transformations (cf., e.g., [16, 36, 42]), or working with that is resistant to perturbations? How to obtain a lower particular classes of objects (cf., e.g., [7, 33, 40, 41]). Nev- bound for the natural pseudo-distance based on multidimen- ertheless, the choice of the most appropriate functions for a sional size functions? Is the method computationally afford- particular application is not fixed a priori but can be changed able? The main aim of this paper is to answer these ques- up to the problem at hand. tions. Most of the research in this field was developed for func- tions with values in R. Retrospectively, we call this setting Outline All the basic results of the 1-dimensional Size 1-dimensional. The possibility of constructing an analogous Theory that we need also in the k-dimensional case are Rk theory based on functions with values in , therefore called recalled in Sect. 2. Before introducing the concept of k- k-dimensional or multidimensional, was investigated for the dimensional size function in Sect. 3, we discuss the moti- first time in [31]. The advantage of working with k-valued vations for this extension. functions is that shapes can be simultaneously investigated Our first result is the proof that in Size Theory the com- by k different 1-valued functions. In other words, k different parison of multidimensional size functions can be reduced functions cooperate to produce a single shape descriptor. to the 1-dimensional case by a suitable change of variables In this multidimensional framework, the comparison of (Theorem 3). The key idea is to show that a foliation in half- two objects in a dataset (e.g. 3D-models, images or sounds) planes can be given, such that the restriction of a multidi- is translated into the comparison of two suitable topologi- mensional size function to these half-planes turns out to be cal spaces M and N , endowed with two continuous func- a classical size function in two scalar variables. This reduc- tions ϕ : M → Rk, ψ : N → Rk. These functions are called tion scheme is presented in Sect. 4. k-dimensional measuring functions and are chosen accord- This approach implies that each size function, with re- ing to the application. In other words, they can be seen as de- spect to a k-dimensional measuring function, can be com- scriptors of the properties considered relevant for the com- pletely and compactly described by a parameterized fam- parison. In the same paper [31] the natural pseudo-distance ily of discrete descriptors (Remark 3). This follows by ap- d between the pairs (M, ϕ) , (N , ψ) was introduced, set- plying to each plane in our foliation the representation of ting d((M, ϕ),( N , ψ)) equal to the infimum of the change classical size functions by means of formal series of points of the measuring function, induced by composition with all and lines. An important consequence is the proof of the the homeomorphisms from M to N (if any). The natural stability of this new descriptor (and hence of the corre- pseudo-distance d is a measure of the dissimilarity between the studied shapes. Unfortunately, the direct computation of sponding k-dimensional size function) with respect to per- d is quite difficult, even for k = 1, although strong properties turbations of the measuring functions, also in higher dimen- can be proved in this case (cf. [17, 19, 20]). sions (Proposition 2), by using the stability result proved for To overcome this difficulty, the strategy proposed in [31] 1-dimensional size functions. Moreover, we prove stability for any k consists in obtaining lower bounds for d using of this descriptor also with respect to perturbations of the size groups. This strategy has been broadly devel- foliation leaves. oped in the case k = 1, using size functions instead of size As a further contribution, we show that a matching dis- homotopy groups ([14, 15, 18]). Size functions are shape tance between size functions, with respect to measuring Rk descriptors that analyze the variation of the number of con- functions taking values in , can easily be introduced (De- nected components in the lower-level sets of the studied finition 8). This matching distance provides a lower bound space with respect to the chosen measuring function. When for the natural pseudo-distance, also in the multidimensional k = 1, size functions can be efficiently employed in applica- case (Theorem 4). The higher discriminatory power of mul- tions since they admit a combinatorial representation as for- tidimensional size functions in comparison to 1-dimensional mal series of points and lines in R2 ([21, 30, 35]). Based on ones is proved in Proposition 4 and illustrated by a simple this concise representation, the matching distance between example. All these results, presented in Sect. 5, motivate the size functions was introduced in [35] and further studied in introduction of Multidimensional Size Theory in real appli- [14, 15]. The first useful property of the matching distance cations and open the way to their effective use. is its stability under perturbations of measuring functions The main issues related to the passage from the theoret- (with respect to the max-norm). Furthermore, the matching ical to the computational model are discussed in Sect. 6, distance between two size functions furnishes a lower bound where we also present experiments on small datasets. In this for the natural pseudo-distance between the corresponding way we demonstrate that all the properties proved in the the- size pairs. oretical setting actually hold in concrete applications. J Math Imaging Vis Before concluding the paper, we formally explore some links existing between multidimensional size functions and the concept of vineyard, recently introduced in [12].

2 Background on 1-dimensional Size Theory

In the 1-dimensional case, Size Theory aims to study the shape of objects based on capturing quantitative topological properties provided by some real ϕ de- fined on the space M representing the object. Actually, the term “1-dimensional” refers to the fact that ϕ takes values in R. Fig. 1 (a) The object to be studied is the curve M depicted by a solid In this section we review the basic results in 1-dimensio- line, and its features are investigated through the measuring function ϕ nal Size Theory. We confine ourselves to recall those re- distance from the point P .(b) The size function (M,ϕ) sults that in the following sections will be extended to the k-dimensional setting, where shapes will be studied with re- k spect to functions ϕ : M → R . When not explicitly stated, that cannot be joined under y.So(M,ϕ)(x, y) = 2for the different notations ϕ and ϕ will make self-evident if we a ≤ x

When this finite number, called multiplicity of r for (M,ϕ), In order to make this definition clear, we refer the reader is strictly positive, we call the line r a cornerline (or corner- M to Fig. 1. The object to be studied is the curve depicted by point at infinity) for the size function. a solid line, and the measuring function ϕ is distance from the point P . The size function  M is illustrated on the ( ,ϕ) Definition 4 For every point p = (x, y) with x

The finite number μ(p) is called multiplicity of p for has no cost. For more details on this distance we refer the (M,ϕ). Moreover, we call cornerpoint for (M,ϕ) any point reader to [14, 15]. p such that the number μ(p) is strictly positive. A key property of the matching distance is its stability with respect to perturbations of the measuring functions, as Cornerpoints and cornerlines allow us to compactly rep- stated by the following theorem ([14, 15]). resent size functions as formal series (cf. Fig. 2(a) and (b)), in virtue of the following theorem, asserting that the mul- Theorem 2 If (M,ϕ), (M,χ) are size pairs and tiplicities of cornerpoints and cornerlines are sufficient to maxP ∈M |ϕ(P) − χ(P)|≤η, then it holds that compute size functions’ values (cf. [15, 30]). For the sake of dmatch((M,ϕ),(M,χ)) ≤ η. simplicity, in this statement, lines with equation x = k are identified to points “at infinity” with coordinates (k, ∞). The second tool of Size Theory is the natural pseudo- distance (cf. [17Ð20, 31]). It is a measure of the similarity Theorem 1 For every (x,¯ y)¯ with x<¯ y<¯ ∞ we have between two size pairs not mediated by any shape descrip- tor. The idea underlying the definition of the natural pseudo- ¯ ¯ = (M,ϕ)(x,y) μ (x, y) . distance between two size pairs (M,ϕ) and (N ,ψ), with (x,y): xy¯ and homeomorphic, is to measure the maximum jump between the values taken by the measuring functions ϕ and Given two size functions 1 and 2, their comparison can ψ when M is homeomorphically deformed into N : be translated to the problem of comparing their multisets of cornerpoints. Hence, let us consider the multiset C (resp. d((M,ϕ),(N ,ψ))= inf max |ϕ(P) − ψ(f(P))|, 1 f P ∈M C2) of all cornerpoints for 1 (resp. 2) counted with their multiplicities, augmented by adding a countable infinity of where f varies among all the homeomorphisms between M points of the diagonal {(x, y) ∈ R2 : x = y}. We can com- and N . pare 1 and 2 using the matching distance The matching distance turns out to be less informative about the dissimilarity between the studied shapes than the = dmatch(1,2) min max δ(p,σ(p)) natural pseudo-distance. Indeed, it holds that σ p∈C1 ≤ M N where σ varies among all the bijections between C1 and C2 dmatch((M,ϕ),(N ,ψ)) d(( ,ϕ),( ,ψ)) (1) and (cf. [14, 15]).

δ((x, y), (x ,y )) = min max{|x − x |, |y − y |}, Nevertheless, the computation of the natural pseudo- distance can be accomplished only in a few cases. There- y − x y − x fore inequality (1) turns out to be very useful as an esti- max , . mation from below of the natural pseudo-distance. More- 2 2 over, in [14]wehaveprovedthatdmatch gives the best lower Roughly speaking, the matching distance dmatch between bound for the natural pseudo-distance, in the sense that any two size functions is the minimum, over all the matchings other distance between size functions would furnish a worse between the cornerpoints of the two size functions, of the bound. maximum of the L∞-distances between two matched cor- It is worth mentioning here that, using tools of Alge- nerpoints. Since two size functions can have a different num- braic Topology, other lower bounds for the natural pseudo- ber of cornerpoints, these can be also matched to points of distance are obtained in [31] by means of size homotopy the diagonal, as illustrated in Fig. 2(c). Notice that the defi- groups. These are shape descriptors that generalize size nition of δ implies that matching two points of the diagonal functions by taking into account classes of loops in the lower J Math Imaging Vis level sets instead of connected components. However, size k-dimensional measuring functions allows us to produce just homotopy groups are not computationally feasible. Always one shape descriptor containing the information of k differ- from the perspective of , yet another ent measuring functions at the same time. shape descriptor, the Size Functor, has been developed in [5] A natural question arises, whether the results obtained us- for 1-dimensional measuring functions, based on . ing k-dimensional measuring functions were not obtainable More recently, similar ideas have independently led from k 1-dimensional measuring functions. Actually, a re- Edelsbrunner et al. to the definition of Persistent Homol- sult of this paper is not only the introduction of a procedure ogy (cf. [23, 24]), and Allili et al. to the definition of the to carry out computations in k-dimensional Size Theory, Morse Homology Descriptor (cf. [2]). but also furnishing the proof, illustrated by examples, that In particular, the concepts of size function and corner- k-dimensional size functions do not simply contain the sum point have been recently rediscovered in the framework of of the information contained in 1-dimensional size func- Persistent Homology (cf. [23, 24]). In the terminology of tions. Persistent Homology, size functions correspond to the di- mension zero persistent Betti number while formal series of 3.2 Definitions cornerpoints correspond to persistence diagrams. Also the theorem stating the stability of the matching distance has a For the present paper, M, N denote two non-empty com- counterpart in Persistent Homology (cf. [10, 11]). Moreover, pact and locally connected Hausdorff spaces. the first persistent homology group equals the Abelianiza- tion of the first size . In Multidimensional Size Theory [31], any pair (M, ϕ) , k where ϕ = (ϕ1,...,ϕk) : M → R is a continuous function, is called a size pair. The function ϕ is called a k-dimensional 3 k-dimensional Size Functions measuring function. The following relations and ≺ are k defined in R :forx = (x1,...,xk) and y = (y1,...,yk),we Multidimensional size functions are the extension of the no- say x y (resp. x ≺y) if and only if xi ≤ yi (resp. xi

(M,ϕ) (x, y) =  ϕ (s, t). (M,F ) 4 Reduction to the 1-dimensional Case (l,b) k Proof For every x = (x1,...,xk) ∈ R , with xi = sli + bi , In this section, we will show that there exists a parameter- ϕ i = 1,...,k, it holds that Mϕ x=MF ≤ s.This ized family of half-planes in Rk × Rk such that the restric- (l,b) is true because tion of (M,ϕ) to each of these planes can be seen as a par- ticular 1-dimensional size function. Mϕ x={P ∈ M : ϕi(P ) ≤ xi,i= 1,...,k}

= Rk ={P ∈ M : ϕi(P ) ≤ sli + bi,i= 1,...,k} Definition 7 For every unit vector l (l1,...,lk) of = = − such that li > 0fori 1,...,k , and for every vector b = ∈ M : ϕi(P ) bi ≤ = Rk k = P s, i 1,...,k (b1,...,bk) of such that = bi 0, we shall say that li i 1 the pair (l,b) is admissible. We shall denote the set of all ϕ k k = MF ≤ s. admissible pairs in R × R by Admk. Given an admissible (l, b) J Math Imaging Vis k Analogously, for every y = (y1,...,yk) ∈ R , with yi = functions corresponding to ϕ1,...,ϕk, considered indepen- tli + bi,i= 1,...,k, it holds that Mϕ y= dently. In Sect. 6 we shall check this point on a dataset of MF ϕ ≤ t. Therefore Remark 1 implies the claim.  shapes. (l, b)

5.1 Main Results In the following, we shall use the symbol F ϕ in the (l, b) sense of Theorem 3. In the sequel, we shall denote by d ( ϕ , match (M,F ) (l, b)  ) the matching distance between the 1-dimensional (N ,F ψ ) Remark 2 By applying the parameterization described in (l, b) Definition 7, we can change the problem of computing and size functions  ϕ and  ψ . (M,F ) (N ,F ) describing k-dimensional size functions into the one of com- (l,b) (l, b) As an easy corollary of Theorem 3 and Remark 3 we have puting and describing families of 1-dimensional size func- that the set of 1-dimensional size functions  ϕ ,as (M,F ) tions. We point out that the same technique can be straight- (l, b) forwardly applied to the ranks of size homotopy groups (l,b) varies in Admk, completely characterizes (M,ϕ) . and to the ranks of multidimensional persistent homology groups, since our approach allows us to exploit the theory Corollary 1 Let us consider the size pairs (M, ϕ) , (N , ψ). ≡ developed in dimension 1 essentially without any change. Then, the identity (M,ϕ) (N ,ψ) holds if and only if = dmatch( M ϕ , ψ ) 0, for every admissible pair ( ,F ) (N ,F ) Remark 3 Theorem 3 allows us to represent each multidi- (l,b) (l,b) (l, b) . mensional size function as a parameterized family of formal series of points and lines, on the basis of the description in- The next result proves the stability of d with respect troduced in [21, 30, 35] for the 1-dimensional case and re- match to the chosen measuring function, i.e. that small enough called in Sect. 2. Indeed, we can associate a formal series changes in ϕ with respect to the max-norm induce small σ with each admissible pair (l,b), with σ describ- (l,b) (l,b) changes of  ϕ with respect to the matching distance. (M,F ) ing the 1-dimensional size function  ϕ .Thefam- (M,F ) (l,b) (l, b) { : ∈ } ily σ(l, b) (l,b) Admk is a new complete descriptor for Proposition 2 If (M, ϕ) , (M, χ) are size pairs and (M,ϕ) , in the sense that two multidimensional size func- maxP ∈M ϕ(P) −χ(P) ∞ ≤ , then for each admissible tions coincide if and only if the corresponding parameter- pair (l, b) , it holds that ized families of formal series coincide.

d ( ϕ , χ ) match (M,F ) (M,F ) (l, b) (l, b)  5 Lower Bounds for the k-dimensional Natural ≤ . Pseudo-Distance mini=1,...,k li

ϕ Proof From Theorem 2, taking η = max ∈M |F (P ) − As recalled in Sect. 2, 1-dimensional size functions can P (l, b) be compared by means of a distance, called matching dis- F χ (P )|, we obtain that (l, b) tance. This distance is based on the observation that each 1-dimensional size function can be represented by a formal dmatch( ϕ , χ ) (M,F ) (M,F ) series of cornerpoints. The matching distance is computed (l,b) (l,b) by finding an optimal matching between the multisets of ≤ max |F ϕ (P ) − F χ (P )|. (l, b) (l, b) cornerpoints describing two size functions. Key properties P ∈M of the matching distance are its stability under perturbations Let us now fix P ∈ M. Then, denoting by ιˆ the index for of the measuring functions and the possibility of computing ϕi (P )−bi ϕ which maxi is attained, by the definition of F li (l, b) lower bounds for the natural pseudo-distance. In this section and F χ we have that we show that these results are still valid also in the multi- (l, b) dimensional case. Moreover, we prove that the multidimen- F ϕ (P ) − F χ (P ) sional matching distance provides a better lower bound for (l, b) (l, b) the natural pseudo-distance than the 1-dimensional match- ϕ (P ) − b χ (P ) − b ing distance. We conclude the section with an example = max i i − max i i showing that the information contained in size functions cor- i li i li responding to measuring functions ϕ = (ϕ ,...,ϕ ) is not ϕιˆ(P ) − bιˆ χi(P ) − bi 1 k = − max simply the sum of that contained in the 1-dimensional size lιˆ i li J Math Imaging Vis

ϕˆ(P ) − bˆ χˆ(P ) − bˆ (l − lˆ)ϕˆ(P ) − l bˆ + lˆb ≤ ι i − ι ι ιˆ ι ι ιˆ ι ι ιˆ = lˆ lˆ ι ι lιˆlιˆ − − ∞ ϕιˆ(P ) χιˆ(P ) ϕ(P) χ(P) (l − lˆ)ϕˆ(P ) + lˆ(b − bˆ) + bˆ(lˆ − l ) = ≤ . = ιˆ ι ι ι ιˆ ι ι ι ιˆ lιˆ mini=1,...,k li lιˆlιˆ χ − ϕ ≤ | − || |+ | − |+| || − | In the same way, we obtain F (P ) F (P ) lιˆ lιˆ ϕιˆ(P ) lιˆ bιˆ bιˆ bιˆ lιˆ lιˆ (l,b) (l,b) ≤ ϕ(P)−χ(P) ∞ lˆl . Therefore, if max ∈M ϕ(P) −χ(P) ∞ ι ιˆ mini=1,...,k li P ≤ , + + ≤ ( ϕ(P) ∞ l ∞ b ∞) − ϕ(P) −χ(P) ∞ lιˆ(lιˆ ) max |F ϕ (P ) − F χ (P )|≤ max ∈M (l, b) (l, b) ∈M P P mini=1,...,k li ( ϕ(P) ∞ + l ∞ + b ∞) ≤ .  mini=1,...,k{li(li − )} ≤ .  mini=1,...,k li Analogously, we can prove that F ϕ (P ) − F ϕ (P ) ≤ (l ,b ) (l, b) Analogously, it is possible to prove that dmatch is stable + + ( ϕ(P) ∞ l ∞ b ∞) . Therefore, with respect to the choice of the half-planes in the folia- mini=1,...,k{li (li −)} tion. Indeed, the next proposition states that small enough | ϕ − ϕ | changes in (l,b) with respect to the max-norm induce small max F (P ) F (P ) P ∈M (l,b) (l ,b ) changes of  ϕ with respect to the matching distance. (M,F ) (l, b) max ∈M ϕ(P) ∞ + l ∞ + b ∞ ≤  · P . mini=1,...,k{li(li − )} Proposition 3 If (M, ϕ) is a size pair, (l,b) ∈ Admk and  is a real number with 0 ≤ 

j Remark 6 Another interesting choice for applications could and hence min = l ·d ( ϕ , ) = i 1,...,k i match (M,F ) N ψ j j ( ,F ) be the weighted mean pseudo-distance computed on the (l ,b ) (lj ,bj ) √1 j j · d ( ϕ , ) = d ( M , finite subset A ={(l , b ) : j = 1,...,h}⊆Admk, and k match (M,F ) N ψ match ( ,ϕj ) j j ( ,F ) h j · j · (l ,b ) (lj ,bj ) defined as j=1 w mini=1,...,k li dmatch( M ϕ , ( ,F ) (N ,ψj )). From the definition of Dmatch((M,ϕ) , N ),the (lj ,bj ) ( ,ψ) j claim immediately follows.   ψ ) (assuming that w are real numbers with (N ,F ) (lj ,bj ) j = h j = 5.2 An Example w > 0, for every j 1,...,h, and j=1 w 1): In- deed, it takes into account the information conveyed from As a consequence of Proposition 4, the multidimensional each leaf (lj , bj ) ∈ A. This weighted mean pseudo-distance size function associated with ϕ contains at least as much in- between size functions gives a lower bound for the natural formation about the studied shape as the whole set of the 1- pseudo-distance that is at most as good as that of Remark 5. dimensional size functions associated with ϕ1,...,ϕk.Ac- tually, the size function of ϕ can be strictly more informa- The following result proves that the lower bound for the tive than the set of the size functions of ϕ1,...,ϕk. This fact natural pseudo-distance provided by Dmatch((M,ϕ) ,(N ,ψ) ) will be evident from the experiments illustrated in Sect. 6, is better than the ones obtained from the 1-dimensional but can also be easily checked in the following example. = R3 Q =[− ]×[− ]×[− ] matching distances dmatch((M,ϕi ),(N ,ψi )),fori In consider the set 1, 1 1, 1 1, 1 1,...,k. and the sphere S of equation x2 + y2 + z2 = 1. Let also  = J Math Imaging Vis Fig. 3 The topological spaces M and N and the size functions

 ϕ , ψ associated (M,F ) (N ,F ) (l,b) (l, b) with the half-plane π ,for √ √ (l,b) = 2 2 = l ( 2 , 2 ) and b (0, 0)

√ 3 2 (1,2) : R → R be the continuous function, defined as (0, 2) onto the point of coordinates (0,√1), computed with (x,y,z) = (|x|, |z|). In this setting, consider the size pairs respect to the max-norm. The points (0, 2) and (0, 1) are M N M = Q N = S ( , ϕ) and ( , ψ) where ∂ , , and ϕ and cornerpoints for the size functions  ϕ and  ψ , (M,F ) (N ,F ) ψ are respectively the restrictions of  to M and N .In (l,b) (l, b) respectively. order to compare the size functions (M,ϕ) and (N ,ψ) ,we ≡ We conclude by observing that (M,ϕ1) (N ,ψ1) and are interested in studying the foliation in half-planes π(l, b) , (M,ϕ ) ≡ (N ,ψ ). In other words, the multidimensional where l = (cos θ,sin θ) with θ ∈ (0, π ), and b = (a, −a) 2 2 2 size functions, with respect to ϕ, ψ , are able to discriminate with a ∈ R. Any such half-plane is represented by the cube and the sphere, while both the 1-dimensional size ⎧ functions, with respect to ϕ ,ϕ and ψ ,ψ , cannot do that. ⎪x1 = s cos θ + a, 1 2 1 2 ⎨ This higher discriminatory power of multidimensional size x2 = s sin θ − a, ⎪ = + functions motivates their definition and use. ⎩⎪y1 t cos θ a, y2 = t sin θ − a with s,t ∈ R, s

=  ϕ (s, t), (M,F ) When dealing with Computer Graphics and Computer Vi- (l, b) sion applications, a relevant problem is to find a suitable dis- s s t t cretization of both the space M and the function ϕ.Asclas-  N (x1,x2,y1,y2) =  N √ , √ , √ , √ ( ,ψ) ( ,ψ) 2 2 2 2 sical discrete models we mention the simplicial complexes =  (s, t). and the digital spaces. In this Section we highlight how our (N ,F ψ ) (l, b) theory is able to go well in both model representations. Given a model |X| represented by the geometric realiza- In this case, by Theorem 4 and Remark 5 (applied for A con- tion of a X of arbitrary dimension n, and taining just the admissible pair that we have chosen), a lower a piecewise linear map ϕ :|X|→Rk, first defined on the bound for the natural pseudo-distance d((M, ϕ),( N , ψ)) is vertices of X and then linearly extended over all the other given by √ √ simplices by using barycentric coordinates, we consider the √ size pair (|X|, ϕ) . Triangulations and tetrahedralizations are 2 = 2 − dmatch( ϕ , ψ ) ( 2 1) (M,F ) (N ,F ) examples of geometric realizations of simplicial complexes 2 (l,b) (l, b) 2 √ for n = 2 and n = 3, respectively. 2 Digital spaces are mainly relevant in image processing. = 1 − . 2 A digital space is often represented by a graph structure based on the local adjacency relations of the digital points Indeed, the matching distance dmatch( ϕ , ψ ) (M,F ) (N ,F ) (l,b) (l, b) (e.g., pixels, voxels, etc.) [22]. In our experiments on digital is equal to the cost of moving the point of coordinates spaces, the shape of interest is represented by the non-zero J Math Imaging Vis points of a binary 3D image, but our reasonings are immedi- complexity for evaluating the k-dimensional size function ately applicable to 2D and higher dimensions. Moreover, the over h half-planes is O(h· (n log n + m · α(2m + n, n))). procedure would be analogous if the shape were represented In the same way, as stated in Definition 8, the computa- by all the points of the image (zero and non-zero). To obtain tion of the k-dimensional matching distance easily follows a size pair from this data, we take a geometric realization |I| from the computation of the 1-dimensional matching dis- of a graph that encodes the non-zero elements of the image tance dmatch together with the computation of the minimum as nodes and the neighborhood adjacency among the digital of the k factors li ,fori = 1,...,k, over the set A of half- points as edges. Depending on the number of neighbors that planes. Since the computational complexity for computing may be adjacent to a point, the connectivity (i.e. the number the 1-dimensional matching distance in a single half-plane of edges) of the graph depends on the number of neighbors is O(p2.5), with p the number of cornerpoints taken into that are admitted to be adjacent to a point in a 2D image account for the comparison, it follows that computing the k- (i.e., 8-, 6- or 4-neighborhoods) or 3D image (e.g., 6-, 18- or dimensional matching distance between two k-dimensional 26-neighborhoods). Then, the size pair is the pair (|I|, ϕ) , size functions requires O(h· (p2.5 + k)) operations. where ϕ :|I|→Rk is a piecewise linear function first de- fined on the nodes of the graph and then linearly extended 6.2 Experimental Results to the edges. Once the size pair has been identified, the proposed re- We present some experimental results on a set of 8 human duction of k-dimensional size functions to the 1-dimensional models represented by triangle meshes and 6 different ob- case allows us to use the existing framework for computing jects represented by voxelized digital models. Our goal is to 1-dimensional size functions [13], based on a discrete struc- check the stability of the proposed framework and to vali- ture. date its potential for shape comparison. The algorithm in [13] takes as input a size graph,i.e.a To this aim, we have to define a multidimensional mea- pair (G, f ) where G is a graph and f : V(G)→ R is a suring function to describe the shapes, and to choose and function labeling each vertex of G by a real number. For the discretize a foliation, for both the cases of meshes and digi- simplicial case, our input is given by the 1-skeleton of the tal objects. complex X, with the nodes labeled by the values of the re- Let us begin with triangle meshes. We define a 2- striction of ϕ on the vertices of the complex. For the case of dimensional measuring function extending to triangle digital spaces, the size graph corresponds to the graph used meshes and to the multidimensional case the reasonings in to encode the binary image. Similarly to the simplicial case, [36], where complete families of invariant 1-dimensional the node labels given in input correspond to the restriction measuring functions are introduced. For a given triangle { } of ϕ to the vertices of the graph. mesh of vertices P1,...,Pn we compute the barycenter = 1 n The output of the algorithm is the multiset of corner- B n i=1 Pi , and normalize the model so that it is con- points and cornerlines that completely determines the dis- tained in a unit sphere. We then define a vector crete size function of the size graph. We recall the definition n = (P − B)||P − B|| of discrete size function of a size graph ([32]): For every v = i 1 i i . n 2 = ||Pi − B|| a

Fig. 4 2-dimensional size functions restricted to five leaves of the foliation, for the first four models

7 Links Between Dimension 0 Vineyards and polating between f0 and f1. These authors assume that the Multidimensional Size Functions topological space is homeomorphic to the body of a simpli- f tame In a recent paper [12], Cohen-Steiner et al. have introduced cial complex, and that the measuring functions t are . the concept of vineyard, that is a 1-parameter family of per- We shall do the same in this section. We recall that dimen- sistence diagrams associated with the homotopy ft , inter- sion p persistence diagrams are a concise representation of J Math Imaging Vis

Fig. 5 2-dimensional size functions restricted to five leaves of the foliation, for other four models

x,y x,y x,y the function rank Hp , where Hp denotes the dimension equivalent to the knowledge of the function rank Hp , com- p persistent homology group computed at point (x, y) (cf. puted with respect to the function ft . We are interested in the = x,y [12]). Therefore, the information described by vineyards is case p 0. Since, by definition, for x

Table 1 Distances between models: the 2-dimensional matching distance between the 2-dimensional size functions associated with ϕ(1,−1) (top value) and the maximum between the 1-dimensional matching distances associated to the 1-dimensional measuring functions ϕ1 and ϕ−1 (bottom). The 2-dimensional matching distance provides a better lower bound for the natural pseudo-distance. Notice also that the values related to the 1- dimensional matching distance are each other very close, up to the considered number of digits, thus revealing a lower discriminatory power than that provided by the 2-dimensional matching distance

0.0000 0.0181 0.1411 0.1470 0.1325 0.1287 0.1171 0.1187 0.0000 0.0003 0.0025 0.0026 0.0023 0.0022 0.0020 0.0021

0.0181 0.0000 0.1419 0.1478 0.1304 0.1265 0.1171 0.1187 0.0003 0.0000 0.0026 0.0026 0.0023 0.0022 0.0020 0.0021

0.1411 0.1419 0.0000 0.0137 0.1583 0.1370 0.1127 0.1017 0.0025 0.0025 0.0000 0.0002 0.0028 0.0024 0.0020 0.0018

0.1470 0.1478 0.0137 0.0000 0.1533 0.1381 0.1137 0.1021 0.0026 0.0026 0.0002 0.0000 0.0027 0.0024 0.0020 0.0018

0.1325 0.1304 0.1583 0.1533 0.0000 0.0921 0.0588 0.1000 0.0023 0.0023 0.0028 0.0027 0.0000 0.0014 0.0016 0.0017

0.1287 0.1265 0.1370 0.1381 0.0921 0.0000 0.1069 0.1048 0.0022 0.0022 0.0024 0.0024 0.0014 0.0000 0.0019 0.0018

0.1171 0.1171 0.1127 0.1137 0.0588 0.1069 0.0000 0.0350 0.0020 0.0020 0.0020 0.0020 0.0016 0.0019 0.0000 0.0006

0.1187 0.1187 0.1017 0.1021 0.1000 0.1048 0.0350 0.0000 0.0021 0.0021 0.0018 0.0018 0.0017 0.0018 0.0006 0.0000

M  cides with the value taken by the size function (M,ft )(x, y), p. 203). Moreover, n Kn is a subset of ϕ y and it follows that, for x0, P and Q are ϕ L(x,y) of equivalence classes of ϕ x quotiented with   (y1 + ε,...,yk + ε)-connected in M, then they are also respect to the ϕ y -connectedness relation. ϕ y-connected. Note that, for x ≺y, L(x, y) simply coincides with Proof For every positive integer number n,letKn be the (M,ϕ) (x,y). M + 1 + 1  connected component of ϕ (y1 n ,...,yk n ) con- taining P and Q. Since connected components are closed Proof of Lemma 2 First of all we observe that the func- M  M (x,(y + ε,...,y + ε)) sets and is compact, each Kn is compact. The set n Kn tion ( ,ϕ) 1 k is nonincreasing in the is the intersection of a family of connected compact Haus- variable ε, and hence the value limε→0+ (M,ϕ) (x,(y 1 + + dorff subspaces with the property that Kn+1 ⊆ Kn for every ε,...,yk ε)) is defined. The statement of the lemma is n, and hence it is connected (cf. Theorem 28.2in[45], trivial if limε→0+ (M,ϕ) (x,(y 1 + ε,...,yk + ε)) =+∞, J Math Imaging Vis Table 2 Space and time requirements for the computation of the size function of some 3D images of different dimensions. |V | and |E| represent the number of vertices and edges of the size graphs of the models. Avg. time is the average time required to compute the size function on a single half-plane of the foliation, while Total time refers to the computation of the size functions on 9 half-planes. Analogously, Avg. |C| is the average number of cornerpoints of the size function on a single half-plane of the foliation, and Total |C| is the sum of the number of cornerpoints of the size functions on 9 half-planes. These results are obtained using a AMD Athlon 3500, with 2 GB RAM

Model |V ||E| Avg. time Total time Avg. |C| Total |C| 19 538 187 005 0.035 s 0.315 s 180 1623

18 779 193 181 0.041 s 0.369 s 84 756

11 504 97 757 0.015 s 0.135 s 16 142

29 061 289 461 0.056 s 0.504 s 15 133

114 277 1 233 063 0.212 s 1.908 s 28 254

5472 45 689 0.006 s 0.054 s 14 122

+ x,¯ y¯ ¯ since, for every ε>0, the inequality (M,ϕ) (x,(y1 where H0 (t) denotes the dimension 0 persistent homology + ≤ ε,...,yk ε)) L(x,y) holds by definition, and hence group computed at point (x,¯ y)¯ with respect to ft¯. the equality L(x, y) =+∞ immediately follows. Let us + + + = x,¯ y¯ now assume that limε→0 (M,ϕ) (x,(y1 ε,...,yk ε)) Proof We know that rank H (t)¯ is equal to the number of +∞ P ={ } 0 r< . In this case a finite set P1,...,Pr of equivalence classes of Mf¯ ≤¯x quotiented with respect M  t points in ϕ x exists such that, for every small enough to the f¯ ≤¯y-connectedness relation. On the other hand, ∈ M   + +  t ε>0, every P ϕ x is ϕ (y1 ε,...,yk ε) - Lemma 2 states that lim → +  M× (x,¯ t,¯ −t,¯ y¯ + ε, t¯ + ∈ P M = ε 0 ( I,χ) connected to a point Pj in . Furthermore, for i j ε, −t¯ + ε) is equal to the number of equivalence classes ∈ P  + +  the points Pi,Pj are not ϕ (y1 ε,...,yk ε) - of M × Iχ (x,¯ t,¯ −t)¯  quotiented, with respect to the   connected and hence not ϕ y -connected either. From χ (y,¯ t,¯ −t)¯ -connectedness relation. By definition of χ, ∈ M    Lemma 1 it follows that every P ϕ x is ϕ y) - this last number equals the number of equivalence classes connected to a point Pj ∈ P in M. Therefore L(x, y) = r = of Mft¯ ≤¯x quotiented, with respect to the ft¯ ≤¯y- lim → +  M (x,(y + ε,...,y + ε)).  ε 0 ( ,ϕ) 1 k connectedness relation. This concludes our proof.  Theorem 5 For t ∈ I =[0, 1], consider the family of size However, although these two links exist, the concept pairs (M,ft ) where ft is a homotopy between f0 : M → R 3 of multidimensional size function has quite different pur- and f1 : M → R. Define χ : M × I → R by χ(P,t) = (f (P ), t, −t). Then, for every t¯ ∈ I and x,¯ y¯ ∈ R with x¯ ≤ poses than that of vineyard. First of all, vineyards are t R3 y¯, it holds that based on a 1-parameter parallel foliation of , while the study of multidimensional size functions depends on a x,¯ y¯ ¯ − + ⊆ Rk ×Rk rank H0 (t) (2k 2)-parameter non-parallel foliation of . ¯ ¯ ¯ ¯ In fact, multidimensional size functions are associated with = lim (M×I,χ) (x,¯ t,−t,y¯ + ε, t + ε, −t + ε), ε→0+ k-dimensional measuring functions, instead of with a homo- J Math Imaging Vis Table 3 Multidimensional and 1-dimensional matching distance be- • Choice of the planes inside the foliation. The compari- tween an airplane and another airplane (first column) or a chair model (second column). The first three rows refer to the multidimensional son technique expressed by Remark 5 requires the choice matching distance using 3 different half-planes of the foliation, the of a finite set of foliation leaves, on which we compute the fourth and fifth row show the average and the maximum value of the reduction from multidimensional to 1-dimensional size multidimensional matching distance using 9 planes, and the last three functions. It would be interesting to determine a method rows represent the 1-dimensional matching distance using the single components of ϕ to make this choice optimal. • Existence of size pairs having assigned k-dimensional size functions. At this time we do not know if any link ex- ists between the 1-dimensional size functions associated

with the planes π(l, b) , apart from continuity. A question naturally arises about the conditions of existence of size pairs having an assigned continuous family of size func- θ = π , φ = π 0.1337 0.7593 4 4 tions on the planes of our foliation. θ = π , φ = π 0.1336 0.7592 4 12 • Reduction moves for the multidimensional size graph. θ = π , φ = π 0.2198 0.4741 12 4 In the discrete framework, the computation of 1-dimen- Average over 9 half-planes 0.1359 0.5449 sional size functions is usually speeded up by applying Maximum over 9 half-planes 0.2198 0.7593 some reduction moves to the size graph. Similar moves ϕx 0.0782 0.2543 can be introduced in the multidimensional setting ([8]) ϕy 0.1151 0.3481 but so far we do not know when this reduction is worth its ϕz 0.0712 0.0963 computational cost.

Acknowledgements The authors thank Bianca Falcidieno and topy between 1-dimensional measuring functions. Further- Michela Spagnuolo (IMATI-CNR) for their support and the helpful more, [12] does not aim to identify distances for the com- discussions. This work has been supported by the CNR project parison of vineyards, while we are interested in quantitative DG.RSTL.050.004, the Italian National Project SHALOM, funded by methods for comparing multidimensional size functions. the Italian Ministry of Research (contract number RBIN04HWR8), and partially developed in the CNR research activity (ICT.P10.009.002) and within the activity of ARCES “E. De Castro”, University of Bologna, under the auspices of INdAM-GNSAGA. 8 Conclusions and Future Work This paper is dedicated to the memory of Marco Gori.

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In: Intelligent Ro- She authored more than 50 reviewed bots and Computer Vision X: Algorithms and Techniques, Boston, papers, published in scientific jour- MA. Proc. of SPIE, vol. 1607, pp. 122Ð133 (1991) nals and conferences, has been serv- 28. Frosini, P.: Connections between size functions and critical points. ing on the programme committee of several conferences, and has been Math. Methods Appl. Sci. 19, 555Ð569 (1996) involving in national and international projects. 29. Frosini, P., Landi, C.: Size theory as a topological tool for com- puter vision. Pattern Recognit. Image Anal. 9, 596Ð603 (1999) A. Cerri received his Ph.D. degree 30. Frosini, P., Landi, C.: Size functions and formal series. Appl. Al- in Mathematics from the University gebra Eng. Commun. Comput. 12, 327Ð349 (2001) of Bologna in 2007. He is currently a 31. Frosini, P., Mulazzani, M.: Size homotopy groups for computa- post-doc student at the University of tion of natural size distances. Bull. Belg. Math. Soc. 6, 455Ð464 Bologna. His research interests in- (1999) clude geometrical-topological meth- 32. Frosini, P., Pittore, M.: New methods for reducing size graphs. Int. ods for shape comparison and appli- J. Comput. Math. 70, 505Ð517 (1999) cations in Pattern Recognition. 33. Handouyaya, M., Ziou, D., Wang, S.: Sign language recognition using moment-based size functions. In: Proc. of Vision Interface 99, Trois-Rivières, Canada, 19Ð21 May 1999, pp. 210Ð216 (1999) 34. Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational Ho- mology, Applied Mathematical Sciences, vol. 157. Springer, New York (2004) J Math Imaging Vis P. Frosini received the Ph.D. degree C. Landi graduated in Mathemat- in Mathematics from the University ics in 1994 at the University of of Florence in 1991. Since 1993, he Bologna (Italy), and received the has been a researcher in the Fac- Ph.D. degree in Mathematics from ulty of Engineering at the University the University of Pisa (Italy). She of Bologna. He is a member of the participates in research on pattern Advanced Research Center on Elec- recognition with the Vision Mathe- tronic Systems for Information and matics Group of the University of Communication Technologies at the Bologna, where she works mainly University of Bologna. His research on shape analysis and comparison. interests Since 2005 she is assistant profes- sor at the Department of Sciences and Methods for Engineering of the University of Modena and Reggio Emilia (Italy). D. Giorgi gottheLaureaDegree cum laude in Mathematics at the University of Bologna in 2002. In 2006 she got a Doctoral Degree in Computational Mathematics at the University of Padua; her research theme concerned the retrieval of im- ages and 3D digital shapes. Since March 2006 she has been a research fellow at CNR-IMATI in Genoa. Her current research interests are related to computational topology techniques for 3D shape analysis, description and matching. She has also participated in many national and international research projects.