approximation of set-valued functions (adaptation of classical approximation operators) || on sets

October 8, 2014 11:39 9in x 6in Approximation of Set-Valued Functions:. . . b1776-ch02

Chapter 2

On Sets

The sets considered throughout the book are compact. The purpose of thischapter is to provide preliminary information on sets, operations betweensets and parametrizations of sets. First we present some notation anddefinitions on compact sets in R

n and consider three set operations — theMinkowski sum, the metric average and the metric linear combination —and review some of their properties. Then, the notion of a parametrizationof sets and several ways for parametrizing sets are presented. The specificexamples of parametrizations given are for convex sets, for star-shaped sets,for sets in R and for general sets.

2.1. Sets and Operations Between Sets

2.1.1. Definitions and notation

We start with notation. We denote by K(Rn) the collection of all compactnon-empty subsets of R

n. All sets considered from now on are from K(Rn).The set-difference of two sets A and B is A \ B = {a ∈ A : a /∈ B}, theLebesgue measure of a measurable set A is µ(A). The Cartesian product oftwo sets A, B is A×B = {(a, b) : a ∈ A, b ∈ B}.

By Co(Rn) we denote the collection of all convex sets in K(Rn). Recallthat a set A is said to be convex if, for any x, y ∈ A and any t ∈ [0, 1],the point tx + (1− t)y is in A. The convex hull of a set A, i.e. the minimalconvex set containing A, is denoted by co(A); the closure of A by cl(A).The segment between two points a, b ∈ R

n is [a, b] = co({a, b}).Further we use the notation 〈·, ·〉 for the Euclidean inner product, | · |

for the Euclidean norm and Sn−1 for the unit sphere in Rn.

17

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18 Approximation of Set-Valued Functions

To measure the distance between two sets A, B ∈ K(Rn) we use in thisbook the Hausdorff metric,

haus(A, B) = max{

maxa∈A

dist(a, B), maxb∈B

dist(b, A)}

,

where

dist(a, B) = minb∈B|a− b|,

is the distance from a point a ∈ Rn to a set B.

It is well known that the spaces K(Rn) and Co(Rn) are complete metricspaces with respect to the Hausdorff metric.Note that for A and B convex sets in R, A = [a1, a2], B = [b1, b2],

haus(A, B) = max{|a1 − b1|, |a2 − b2|}. (2.1)

For A ∈ K(Rn), we use the notation ‖A‖ = maxa∈A |a| and denote by ∂A

the boundary of A. The set of projections of a ∈ Rn on a set B ∈ K(Rn)

is

ΠB(a) = {b ∈ B : |a− b| = dist(a, B)}.The projection of a set A ∈ K(Rn) on a set B ∈ K(Rn) is

ΠB(A) =⋃a∈A

ΠB(a).

In the following we present three operations between sets, which areused later in the construction of approximation operators for set-valuedfunctions.

2.1.2. Minkowski linear combination

Definition 2.1.1 A Minkowski linear combination of the sets A0, . . . , AN

with real coefficients λ1, . . . , λN is

N∑i=1

λiAi =

{N∑

i=1

λiai : ai ∈ Ai, i = 1, . . . , N

}.

In particular,

λA = {λa : a ∈ A}, A + B = {a + b, a ∈ A, b ∈ B}.The set A + B is called the Minkowski sum of A and B, and the set λA iscalled the product of A by a scalar λ.

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On Sets 19

A Minkowski convex combination or a Minkowski average of sets is aMinkowski linear combination with {λi}Ni=1 non-negative, summing up to 1.

While λ(A + B) = λA + λB for any sets A, B ∈ K(Rn) and λ ∈ R, thesecond distributive low (λ + µ)A = λA + µA holds only for A ∈ Co(Rn)and λ, µ ≥ 0.

For the convex hull the following identity holds

co(λA + µB) = λcoA + µcoB, λ, µ ∈ R. (2.2)

In Chapters 4 and 5 we study approximation operators based on Minkowskiconvex combinations.

2.1.3. Metric average

The following average between two sets was introduced by Artstein.

Definition 2.1.2 For A, B ∈ K(Rn) and t ∈ [0, 1] the t-weighted metricaverage of A and B is

A⊕t B = {ta + (1− t)b : (a, b) ∈ Π(A, B)}, t ∈ [0, 1],

where

Π(A, B) = {(a, b) ∈ A×B : a ∈ ΠA(b) or b ∈ ΠB(a)}.

We call Π(A, B) the set of metric pairs of A, B ∈ K(Rn).The most important properties of the metric average are listed below.

For 0 ≤ t ≤ 1, 0 ≤ s ≤ 1

1. A⊕0 B = B, A⊕1 B = A, A⊕t B = B ⊕1−t A,2. A⊕t A = A,3. A ∩B ⊆ A⊕t B ⊆ tA + (1− t)B ⊆ co(A ∪B),4. A⊕t B = tA + (1− t)B, A, B ∈ Co(R),5. haus(A⊕t B, A⊕s B) = | t− s|haus(A, B).

In particular, it follows from Properties 1 and 5 that

haus(A⊕t B, A) = (1− t)haus(A, B),

haus(A⊕t B, B) = t haus(A, B).(2.3)

The fifth property is termed the metric property. Note that theanalogues of Properties 2 and 5 in the case of Minkowski averages aretrue only for convex sets, while with the metric average they are valid for

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general compact sets. The metric average is a geodesic average as definedin (1.10) in the metric space of X = K(Rn) endowed with the Hausdorffmetric. The Minkowski average is a geodesic average but only in Co(Rn).

In Chapter 6 we investigate approximation operators based on themetric average. The above properties of the metric average lead to theconvergence of some of them.

Remark 2.1.3 For A, B ∈ Co(Rn)

co(A⊕t B) ⊆ tA + (1− t)B, (2.4)

with equality for n = 1. For n > 1, A ⊕t B is not necessarily convex, andthen there is strict inclusion in (2.4).

We give below an example of two sets in R2 for which strict inclusion

in (2.4) holds.

Example 2.1.4 This example is illustrated by Fig. 2.1.5.In (a) two sets A = {(x, 0) : x ∈ [−1, 1]} and B = {(0, y) : y ∈ [−1, 1]}are shown.

Figure 2.1.5 The sets of Example 2.1.4.

Their Minkowski average with weight t is the rectangle

tA + (1− t)B = [−t, t]× [t− 1, 1− t].

Their metric average is a union of two segments

A⊕t B = tA ∪ (1− t)B.

These two averages for t = 12 are depicted in (b), the first in gray and the

second in black. In (c) co(A⊕1/2 B), in black, is compared with 12A + 1

2B

in gray.

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On Sets 21

Since the metric average is a non-associative binary operation, it cannotstraightforwardly be extended to three or more sets. Yet it can be extendedto a finite sequence of sets.

2.1.4. Metric linear combination

The metric linear combination is an operation on a finite number of orderedcompact sets, which extends the metric average. The operation is based onthe notion of a metric chain, which generalizes the notion of a metric pair.

Definition 2.1.6 Let {A0, A1, . . . , AN} be a finite sequence of compactsets. A vector (a0, a1, . . . , aN ) with ai ∈ Ai, i = 0, . . . , N is called a metricchain of {A0, . . . , AN} if there exists j, 0 ≤ j ≤ N such that

ai−1 ∈ ΠAi−1(ai), 1 ≤ i ≤ j and ai+1 ∈ ΠAi+1(ai), j ≤ i ≤ N − 1.

An illustration of such a metric chain is given in Fig. 2.1.7.

Figure 2.1.7 A metric chain.

Note that each element of each set Ai, i = 0, . . . , N generates at leastone metric chain. We denote by CH(A0, . . . , AN ) the collection of all metricchains of {A0, . . . , AN}. Note that CH(A0, . . . , AN ) depends on the orderof the sets. For N = 1, CH (A0, A1) = Π(A0, A1).

With this notion of metric chains we can define a set-operation.

Definition 2.1.8 A metric linear combination of a finite sequence ofsets A0, . . . , AN with coefficients λ0, . . . , λN ∈ R, is

N⊕i=0

λiAi =

{N∑

i=0

λiai : (a0, . . . , aN ) ∈ CH (A0, . . . , AN )

}. (2.5)

In particular⊕N

i=0 1·Ai is called a metric sum and is denoted by⊕N

i=0 Ai.

In the special case N = 1 and λ0, λ1 ∈ [0, 1], λ0 + λ1 = 1, the metriclinear combination is the metric average. The following are four important

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properties of the metric linear combination which can be easily derived fromthe definition:

1.N⊕

i=0

λiAi =N⊕

i=0

λN−iAN−i,

2.N⊕

i=0

λiA =( N∑

i=0

λi

)A,

3.N⊕

i=0

λAi = λ

( N⊕i=0

1 ·Ai

),

4.( N∑

i=0

λi

)( N⋂i=0

Ai

)⊂

N⊕i=0

λiAi ⊂N∑

i=0

λiAi.

Remark 2.1.9 The metric sum of two sets A0 ⊕A1 =⊕1

i=0 Ai is com-mutative by the first property, and is not associative in view of (2.5).Similarly one can define the metric difference between two sets byA0 � A1 =

⊕1i=0 λiAi with λ0 = 1, λ1 = −1. Then it follows from the

second property, that

A�A = {0}. (2.6)

Yet the operation A�B is not the inverse operation of the metric sum asis demonstrated by the following example: A = [0, 1], B = {0, 1}.

A�B = [−1/2, 1/2], but (A�B)⊕B = [−1/2, 1/2] ∪ {3/2} = A.

For∑N

i=0 λi = 1, from the second property we get

N⊕i=0

λiA = A. (2.7)

The analogue of this property does not hold for Minkowski linearcombinations with some negative coefficients, even for convex sets. Thisis one reason why only positive operators based on Minkowski sum areconsidered for set-valued functions with convex images. The metric linearcombination allows the adaption of non-positive approximation operatorsto SVFs. In Chapter 7 we study such operators and also positive ones.

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On Sets 23

2.2. Parametrizations of Sets

Some classes of sets can be described by a collection of single-valuedfunctions. In this section we bring several examples of such classes, withtheir corresponding parametrizations.

Definition 2.2.1 Let A be a collection of sets in K(Rn). We call a familyof functions G with values in R

m a parametrization of A if

1. there exists a bijection T : A → G,2. all g ∈ G have common domain of definition D.

We denote for every A ∈ A its image by gA = TA ∈ G.We term the functions in G parametrizing functions and the elementsof D parameters.

The parametrization is called canonical if in addition the family G isclosed under convex combinations.

2.2.1. Induced metrics and operations

A parametrization G induces a metric on A. For A, B ∈ A this metric isgiven by

dG(A, B) := supξ∈D|gA(ξ)− gB(ξ)| (2.8)

and is termed hereafter induced metric.Moreover, a canonical parametrization also induces a set operation

analogous to a convex combination of numbers. For A, B ∈ A, this operationis defined by

tA � (1− t)B := T−1(tgA + (1− t)gB), t ∈ [0, 1]

and is termed hereafter induced convex combination.This binary operation is easily extended to an induced convex combi-

nation of a finite number of sets,

k⊎i=1

tiAi = T−1

(k∑

i=1

tigAi

), (2.9)

for t1, . . . , tk ∈ [0, 1], satisfying∑k

i=1 ti = 1.

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Every induced convex combination satisfies the metric property, withrespect to the induced metric, namely

dG(tA � (1− t)B, sA � (1− s)B) = |t− s|dG(A, B).

Indeed,

dG(tA � (1− t)B, sA � (1− s)B)

= supξ∈D|(tgA(ξ) + (1− t)gB(ξ))− (sgA(ξ) + (1− s)gB(ξ))|

= supξ∈D|(t− s)(gA(ξ) + (s− t)gB(ξ))| = |t− s| sup

ξ∈D|gA(ξ)− gB(ξ)|

= |t− s|dG(A, B).

Canonical parametrizations facilitate the adaptation of approximationoperators defined on real-valued functions to set-valued functions, as isshown in Chapter 4. The rest of Section 2.2 is devoted to examples ofparametrizations.

2.2.2. Convex sets by support functions

A well-known parametrization of convex compact sets is by their supportfunctions. We recall the definition and some important properties of supportfunctions.

The support function δ∗(A, ·) : Sn−1 → R for A ∈ K(Rn) is

δ∗(A, ξ) = maxa∈A〈ξ, a〉, ξ ∈ Sn−1. (2.10)

There is a one-to-one correspondence between a convex compact set andits support function. For a non-convex set the support function determinesits convex hull.

In this parametrization G is the set of all support functions of sets inA = Co(Rn), and D = Sn−1.

We list below several important properties of support functions.For A, B ∈ Co(Rn) and ξ, ξ ∈ Sn−1,

1. δ∗(A + B, ·) = δ∗(A, ·) + δ∗(B, ·),2. δ∗(λA, ·) = λδ∗(A, ·), λ ≥ 0,

3. A ⊆ B ⇐⇒ δ∗(A, ·) ≤ δ∗(B, ·),

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On Sets 25

4. haus(A, B) = maxξ∈Sn−1 |δ∗(A, ξ)− δ∗(B, ξ)|,5. |δ∗(A, ξ)− δ∗(A, ξ)| ≤ (supa∈A |a|)|ξ − ξ|.

Remark 2.2.2

(i) There are three important consequences of the above properties ofsupport functions:

— This parametrization is canonical by Properties 1 and 2.— The induced metric is the Hausdorff metric by Property 4.— The induced convex combination is the Minkowski convex combi-

nation by Properties 1 and 2.

(ii) Definition (2.10) is often extended to ξ ∈ Rn, and δ∗(A, ξ) in this case

becomes homogenous and sublinear in ξ; namely, it satisfies

δ∗(A, λξ) = λδ∗(A, ξ), λ ≥ 0,(2.11)

δ∗(A, ξ + ξ) ≤ δ∗(A, ξ) + δ∗(A, ξ).

In fact, the set G is the restriction to Sn−1 of all functions defined onR

n which satisfy (2.11).

For general compact sets the existence of a canonical parametrizationis an open question, except for compact sets in R considered in the nextsection.

2.2.3. Parametrization of sets in R

We consider compact sets in R which are finite unions of compact intervalsof positive measure. We denote this class of sets by A. Thus a set A ∈ A isgiven by

A =N⋃

n=1

[an, bn], N <∞, (2.12)

where an < bn, n = 1, . . . , N , bn < an+1 for n = 1, . . . , N − 1.For the class AN of all sets as in A with fixed N , there is a

parametrization in terms of the boundary points.For a given set A ∈ AN the corresponding parametrizing function is

gA : {1, . . . , N} → R2, gA(n) = (an, bn) ∈ R

2, n ∈ {1, . . . , N}.

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It is clear that this parametrization of AN is canonical. For the class A thecollection {gA : A ∈ A} is not a parametrization, since the domain of gA

depends on the number of segments in A.Now we introduce a parametrization of A which is canonical.

For A ∈ A given by (2.12), the corresponding function gA : [0, 1]→ R isdefined by

gA(ξ) = min{

a ≥ a1 :µ([a1, a] ∩A)

µ(A)= ξ

}, ξ ∈ [0, 1], (2.13)

where µ(·) is the Lebesgue measure of R, or equivalently

gA(ξ) =

{a1, ξ = 0,

ai + µ(A)(ξ − λi−1), λi−1 < ξ ≤ λi, i = 1, . . . , N,

(2.14)λ0 = 0, λi =

1µ(A)

i∑j=1

µ([aj , bj ]), i = 1, . . . , N.

From (2.14) it is easy to infer that the parametrization G consists of allpiecewise linear functions defined on [0, 1], which are left continuous witha constant non-negative slope. Moreover, (2.14) implies that gA for A ∈ Ahas a constant slope µ(A), and discontinuity points λ1, . . . , λN−1. Also, forg ∈ G, the set T−1(g) is the closure of the image of g.

The next simple example illustrates the relation between A and gA.

Example 2.2.3 The set A is given by [a1, b1] ∪ [a2, b2] witha1 < b1 < a2 < b2. Let λ = b1−a1

b2−a2+b1−a1. Then for ξ ∈ [0, 1]

gA(ξ) =

a1 +b1 − a1

λξ, 0 ≤ ξ ≤ λ,

a2 +b2 − a2

1− λ(ξ − λ), λ < ξ ≤ 1.

Note that µ(A) = b2 − a2 + b1 − a1 and that b1−a1λ = b2−a2

1−λ = µ(A).The graph of gA is depicted in Fig. 2.2.4.

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On Sets 27

Figure 2.2.4 The graph of gA in Example 2.2.3.

Since G is closed under convex combinations, the parametrization iscanonical. The induced convex combination is a subset of the Minkowskiconvex combination and, in addition to the metric property, also has themeasure property, namely

µ(tA � (1− t)B) = tµ(A) + (1− t)µ(B). (2.15)

The equality in the above formula is easily obtained by considering theslope of tgA + (1− t)gB .

Next we show that the induced metric bounds the Hausdorff metric.

Proposition 2.2.5 For A, B ∈ Ahaus(A, B) ≤ dG(A, B) = sup

ξ∈[0,1]|gA(ξ)− gB(ξ)|.

Proof Since for any ξ ∈ [0, 1], gA(ξ) ∈ A, gB(ξ) ∈ B, we get

|gA(ξ)− gB(ξ)| ≥ max{dist(gA(ξ), B), dist(gB(ξ), A)}.Since any set C ∈ A is the closure of the image of gC , taking thesupremum over ξ ∈ [0, 1] in the above inequality we get the claim of theproposition. �

It is interesting to note that tA � (1− t)B = tA + (1− t)B forA, B ∈ Co(R) and that on Co(R) the induced metric coincides with theHausdorff metric.

2.2.4. Star-shaped sets by radial functions

We recall that a set A is star-shaped if there exists a point c ∈ A such thatfor all a ∈ A the segment [c, a] is contained in A. The point c is called a

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center of A. The set of all centers of A is called the kernel of A and is aconvex set. We denote it by Ker(A).

The mapping of a compact star-shaped set A to the pair (cA, ρA(·)),where cA is the center of mass of Ker(A) and where

ρA(ξ) = max{β ∈ R : cA + βξ ∈ A}, ξ ∈ Sn−1,

is bijective.The function gA : Sn−1 → R

n+1

gA(ξ) = (cA, ρA(ξ)),

defines a parametrization. This parametrization is not canonical in general.Yet in the class of all compact star-shaped sets which are centrallysymmetric this parametrization is canonical. Indeed, if a∗ ∈ A is the centerof symmetry of A, then it is the center of mass of both A and Ker(A).We note that for t ∈ [0, 1] the set tA � (1− t)B is centrally symmetric withcenter of symmetry tcA + (1− t)cB .The induced metric in this case can be defined as

dG(A, B) = |cA − cB |+ supξ∈Sn−1

|ρA(ξ)− ρB(ξ)|.

This metric bounds the Hausdorff metric since for a = cA + ρA(ξ) ξ andb = cB + ρB(ξ) ξ, |a− b| ≤ |cA − cb|+ |ρA(ξ)− ρB(ξ)|.

2.2.5. General sets by signed distance functions

For any compact set A ∈ K(Rn) the function

sd(A, ξ) =

dist(ξ, ∂A), ξ ∈ A

−dist(ξ, ∂A), ξ /∈ A, ξ ∈ R

n

is called the signed distance function of A.The set A is easily determined by A = {ξ : sd(A, ξ) ≥ 0}. Thus the

mapping T (A) = sd(A, ·) from K(Rn) into the set G of all signed distancefunctions of sets in K(Rn) is bijective, and the domain of definition of allfunctions in G is R

n, hence G is a parametrization of K(Rn).

Remark 2.2.6 The parametrization by signed distance functions is not acanonical parametrization because it is not closed under convex combina-tions. Yet, one can define a weak linear combination with real coefficients

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On Sets 29

λ1, . . . , λN by the set{

x :N∑

i=1

λisd(Ai, x) ≥ 0

}. (2.16)

In case λi ∈ [0, 1], i = 1, . . . , N and∑N

i=1 λi = 1, (2.16) is termed weakconvex combination.

For sets A, B with a “large” intersection A ∩ B and a “small”symmetric difference (A \B) ∪ (B \A), this weak convex combination is“geometrically reasonable”, which explains the ubiquitous use of thisoperation in algorithms for reconstruction of objects from their parallelcross-sections. On the other hand, when A and B are disjoint, their averagewith t1 = t2 = 1/2 is empty, which is not acceptable.

It is interesting to note that the weak convex combination of twointersecting convex sets is convex, because the sign distance function ofa convex set is concave.

The metric induced by the signed distance function bounds theHausdorff metric from above. Indeed, if we assume that for two compactsets A = B the Hausdorff distance is achieved at a point a ∈ A, namelyhaus(A, B) = dist(a, B), then a ∈ ∂A \B and

dist(a, B) = |sd(B, a)| = |sd(B, a)− sd(A, a)| ≤ ‖sd(B, ·)− sd(A, ·)‖∞.

To realize that there is no equality between these two metrics, considerthe two sets A = {x : |x| ≤ 1} and B = A \ {x : |x| ≤ ε}, with ε < 1/2.Indeed haus(A, B) = ε, while dG(A, B) = sd(A, 0)− sd(B, 0) = 1 + ε.

2.3. Bibliographical Notes

Minkowski linear combinations are well known and used mostly in convexanalysis [82,84] and in mathematical morphology [6, 86].

Definitions and properties of the metric spaces K(Rn) and Co(Rn)endowed with the Hausdorff metric can be found in [82–84], in particularthe completeness of these two spaces. More about convex sets and convexhulls of sets can also be found in these books.

The metric average is introduced in [5], where its metric property isproved. Some additional properties of this average are studied in [10, 42],and an algorithm for its computation in R

1 is given in the latter. The metricaverage operation is further extended to metric linear combinations of

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compact sets in [44]. Algorithms for the computation of the metric averageof planar polygons are developed in [55,66]. The examples calculated thereindicate that in many cases the geometry of the metric average is not “inbetween” the geometry of the two averaged sets, even when the polygonsare convex.

The concept of parametrization of sets is closely related to theconcept of embedding a collection of sets in a vector space. For a surveyof embeddings see Chapter 3 of [6]. In particular, the parametrizationof convex compact sets by support functions which was introduced byHormander in [54], see also [82, 84], is equivalent to the embedding ofCo(Rn) in the normed vector space of pairs of convex compact setsdefined in [81]. This parametrization is used for approximations of set-valued functions and their Aumann integrals in [9, 16, 34, 87]. It is alsoapplied in approximation of control systems and differential inclusions, e.g.,in [15,39,51,90,95].

The parametrizing radial functions of star-shaped sets are studied, e.g.,in [89]. Kernels of star-shaped sets are studied in [60].

The signed distance function introduced in [65] is the basis of variousmethods for reconstruction of objects from cross-sections, see e.g. [27, 52]and references therein.

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approximation of set-valued functions (adaptation of classical approximation operators) || on sets

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