## Introduction

Shape comparison is probably one of the most challenging issues in Computer Vision and Pattern Recognition. In recent years many papers have been devoted to this subject and new mathematical techniques have been developed to deal with this problem. In the early 90’s, Size Theory was proposed as a geometrical/topological approach to shape comparison. The main idea is to translate the comparison of two datasets (e.g. 3D-models, images or sounds) into the comparison of two suitable topological spaces

and , endowed with two continuous functions , . These functions are called*-dimensional measuring functions*and are chosen according to the application. In other words, they can be seen as descriptors of the properties considered relevant for the comparison. In [19] the definition of the natural pseudo-distance between the pairs , was introduced, setting equal to the infimum of the change of the measuring function, induced by composition with all the homeomorfisms from to . Unfortunately, the study of is quite difficult, even for , although strong properties can be proved in this case (cf. [9, 11]). Size Theory tackles this problem by introducing some mathematical tools that allow us to easily obtain lower bounds for , such as

*size homotopy groups*and

*size functions*(cf. [19] and [10]). The idea is to study the pairs , where is defined by setting for . The -th size homotopy group describes the non-trivial equivalence classes of -dimensional loops in that remain homotopically non-trivial also in . Size functions count the number of connected components in that meet . It turns out that is a set whose cardinality is equal to the value taken by the size function at . From the homological point of view, an analogous approach, named

*Size Functor*, has been developed in [2] for -dimensional measuring functions.

More recently, similar ideas have independently led Edelsbrunner
et al. to the definition of *Persistent Homology* (cf.
[12, 13]), and Allili et al. to the definition of
the *Morse Homology Descriptor* (cf. [1]).

From the beginning of the 90’s, size functions have been studied and applied in the case of -dimensional measuring functions (cf., e.g., [8, 14, 15, 16, 18, 20, 21, 22, 23, 24]). The multidimensional case presented more severe difficulties, since a concise, complete and stable description of multidimensional size functions was not available before this work.

In [3], Carlsson and Zomorodian examine the completeness problem by studying Multidimensional Persistent Homology. In that paper, it is claimed that multidimensional persistence has an essentially different character from its -dimensional version. Indeed, their approach does not seem to lead to a concise, complete and stable descriptor in the multidimensional case, whereas it does in classical Persistent Homology (see [4]).

The first result of this paper is the proof that in Size Theory
the comparison of multidimensional size functions can be reduced
to the -dimensional case by a suitable change of variables
(Theorem 1). The key idea is to show that a
foliation in half-planes can be given, such that the restriction
of a multidimensional size function to these half-planes turns out
to be a classical size function in two scalar variables. Our
approach implies that each size function, with respect to a
-dimensional measuring function, can be completely and
compactly described by a parameterized family of discrete
descriptors (Remark 2). This follows from the
results proved in [17] about the representation of
classical size functions by means of formal series of points and
lines, applied to each plane in our foliation. An important
consequence is that we can easily prove the stability of this new
descriptor (and hence of the corresponding -dimensional size
function) also in higher dimensions (Proposition 2),
by using a recent result of stability proved for -dimensional
size functions (cf. [6, 7]) and analogous to the
result obtained in [4] for Persistence Homology. As a
final contribution, we show that a matching distance between size
functions, with respect to measuring functions taking values in
, can easily be introduced (Definition 4).
This matching distance provides a lower bound for the natural
pseudo-distance, also in the multidimensional case (Theorem
2). All these results open the way to the use of
Multidimensional Size Theory in real applications.

Outline. In Section 1 we give the
definition of -dimensional size function. In Section
2 the foliation we use is presented, and the
reduction to the -dimensional case is proved. Section
3 shows the stability of our computational method,
implying a lower bound for the natural pseudo-distance.
Additionally, a new distance between multidimensional size
functions is introduced. In Section 4 the
effectiveness of the multidimensional approach is tested on an
example. Section 5 examines some links between
multidimensional size functions and the concept of
*vineyard*, recently introduced in [5]. Section
6 concludes the paper, presenting some discussion
and future work.

## 1 Definition of k-dimensional size function

For the present paper, , denote two
non-empty compact and locally connected Hausdorff spaces.

In Multidimensional Size Theory [19], any pair
, where
is a
continuous function, is called a *size pair*. The function
is called a *-dimensional measuring function*. The
following relations and are defined in :
for and , we say
(resp. ) if and only if
(resp. ) for every index . Moreover,
is endowed with the usual -norm:
. In
this framework, if and are homeomorphic, the
size pairs and can
be compared by means of the *natural pseudo-distance* ,
defined as

where varies among all the homeomorphisms between and . The term pseudo-distance means that can vanish even if the size pairs do not coincide. Here, and in what follows, and are identified.

Now we introduce the -dimensional analogue of size function for a size pair . We shall use the following notations: will be the open set , while . For every -tuple , let be the set .

###### Definition 1.

For every -tuple , we shall say that
two points are
-*connected* if and only if a
connected subset of
exists, containing and .

###### Definition 2.

We shall call *(-dimensional) size function* associated
with the size pair the function
, defined
by setting equal to the number
of equivalence classes in which the set
is divided by the
-connectedness relation.

###### Remark 1.

In other words, counts the connected components in containing at least one point of .

## 2 Reduction to the 1-dimensional case

In this section, we will show that there exists a parameterized family of half-planes in such that the restriction of to each of these planes can be seen as a particular 1-dimensional size function.

###### Definition 3.

For every unit vector

of such that for every , and for every vector of such that , we shall say that the pair is*admissible*. We shall denote the set of all admissible pairs in by . Given an admissible pair , we define the half-plane of by the following parametric equations:

for , with .

###### Proposition 1.

For every there exists one and only one admissible pair such that .

###### Proof.

The claim immediately follows by taking, for ,

Then, , , with

∎

Now we can prove the reduction to the -dimensional case.

###### Theorem 1.

Let be an admissible pair, and be defined by setting

Then, for every the following equality holds:

###### Proof.

For every , with , it holds that . This is true because

Analogously, for every , with , it holds that . Therefore Remark 1 implies the claim. ∎

In the following, we shall use the symbol in the sense of Theorem 1.

###### Remark 2.

Theorem 1 allows us to represent each multidimensional size function as a parameterized family of formal series of points and lines, on the basis of the description introduced in [17] for the -dimensional case. Indeed, we can associate a formal series with each admissible pair , with describing the -dimensional size function . The family is a new complete descriptor for , in the sense that two multidimensional size functions coincide if and only if the corresponding parameterized families of formal series coincide.

## 3 Lower bounds for the k-dimensional natural pseudo-distance

In [6, 7], it has been shown that -dimensional
size functions can be compared by means of a distance, called
*matching distance*. This distance is based on the
observation that each

-dimensional size function is the sum of characteristic functions of triangles. The matching distance is computed by finding an optimal matching between the sets of triangles describing two size functions. For a formal definition we refer to

[6, 7] (see also [4] for the analogue of the matching distance in Persistent Homology). In the sequel, we shall denote by the matching distance between the -dimensional size functions and .###### Corollary 1.

Let us consider the size pairs , . Then, the identity holds if and only if , for every admissible pair .

The next result proves that small enough changes in with respect to the -norm induce small changes of with respect to the matching distance.

###### Proposition 2.

If , are size pairs and , then for each admissible pair , it holds that

###### Proof.

###### Remark 3.

Analogously, it is easy to show that small enough changes in with respect to the -norm induce small changes of with respect to the matching distance.

Now we are able to prove our next result, showing that a lower bound exists for the multidimensional natural pseudo-distance.

###### Theorem 2.

Let and be two size pairs, with , homeomorphic. Setting , where varies among all the homeomorphisms between and , it holds that

###### Proof.

For any homeomorphism between and , it holds that . Moreover, by applying Proposition 2 with and , and observing that , we have

for every admissible . Since this is true for each homeomorphism between and , the claim immediately follows. ∎

###### Remark 4.

We observe that the left side of the inequality in Theorem 2 defines a distance between multidimensional size functions associated with homeomorphic spaces. When the spaces are not assumed to be homeomorphic, it still verifies all the properties of a distance, except for the fact that it may take the value . In other words, it defines an extended distance.

###### Definition 4.

Let and be two size
pairs. We shall call *multidimensional matching distance* the
extended distance defined by setting

###### Remark 5.

If we choose a non-empty subset and we substitute with in Definition 4, we obtain an (extended) pseudo-distance between multidimensional size functions. If is finite, this pseudo-distance appears to be particularly suitable for applications, from a computational point of view.

## 4 An example

In consider the set and the sphere of equation . Let also be the continuous function, defined as . In this setting, consider the size pairs and where , , and and are respectively the restrictions of to and . In order to compare the size functions and , we are interested in studying the foliation in half-planes , where with , and with . Any such half-plane is represented by

with , . Figure 1 shows the size functions and , for and , i.e. and . With this choice, we obtain that and . Therefore, Theorem 1 implies that, for every

In this case, by Theorem 2 and Remark 5 (applied for containing just the admissible pair that we have chosen), a lower bound for the natural pseudo-distance is given by

Indeed, the matching distance
is equal to the cost of moving the point of coordinates
onto the point of coordinates , computed
with respect to the -norm. The points and
are representative of the characteristic triangles of the
size functions
and ,
respectively.

We conclude by observing that
and
.
In other words, the multidimensional size functions, with respect
to , are able to discriminate the cube and the
sphere, while both the -dimensional size functions, with
respect to and , cannot do
that. The higher discriminatory power of multidimensional size
functions motivates their definition and use.

## 5 Links between dimension vineyards and multidimensional size functions

In a recent paper [5], Cohen-Steiner et al. have introduced the concept of vineyard, that is a -parameter family of persistence diagrams associated with the homotopy

, interpolating between

and . These authors assume that the topological space is homeomorphic to the body of a simplicial complex, and that the measuring functions are*tame*. We shall do the same in this section. We recall that dimension persistence diagrams are a concise representation of the function , where denotes the dimension persistent homology group computed at point (cf. [5]). Therefore, the information described by vineyards is equivalent to the knowledge of the function , computed with respect to the function . We are interested in the case . Since, by definition, for , coincides with the value taken by the size function , it follows that, for , dimension vineyards contain the same information as the -parameter family of size functions . Anyway, another interesting link exists between dimension vineyards and multidimensional size functions. This link is expressed by the following theorem. In order to prove it, we need the next two lemmas. The former states that the relation of -connectedness passes to the limit.

###### Lemma 1.

Assume that is a size pair and . If, for every , and are -connected in , then they are also -connected.

###### Proof.

For every positive integer number , let be the connected component of containing and . Since connected components are closed sets and is compact, each is compact. The set is the intersection of a family of connected compact Hausdorff subspaces with the property that for every , and hence it is connected (cf. Theorem in [25] p. ). Moreover, is a subset of and contains both and . Therefore, and are -connected. ∎

The following lemma allows us to study the behavior of multidimensional size functions near (where they have not been defined because of instability problems when the measuring functions are not assumed to be tame).

###### Lemma 2.

Let be a size pair. If then is equal to the number of equivalence classes of quotiented with respect to the -connectedness relation.

Note that, for , simply coincides with .

###### Proof of Lemma 2.

First of all we observe that the function is nonincreasing in the variable , and hence the value is defined. The statement of the theorem is trivial if , since, for every , the inequality holds by definition, and hence the equality immediately follows. Let us now assume that . In this case a finite set of points in exists such that, for every small enough , every is -connected to a point in . Furthermore, for the points are not -connected and hence not -connected either. From Lemma 1 it follows that every is -connected to a point in . Therefore . ∎

###### Theorem 3.

For , consider the family of size pairs where is a homotopy between