Resolution-limited measure and dimension

Resolution-Limited P. M.

Measure and Dimension*

Lapsa

Applied Research Laboratoq Pennsyloania State Unizjersity P.O. Box 30 State College, Pennsylvania

Transmitted

16804

by Melvin R. Scott

ABSTRACT

As extensions dimension

of Hausdorff

are introduced.

ize sets

as having

generated

one,

measure with

and

which

but

sets. Certain

1.

Hausdorff

must

be

simplifications

this

measured.

it possible fiirther

in the dimension

overcome

makes

and dimension,

the former

dimension,

have

dimension a set

Hausdorffs

fractIonal

must

measures Whereas

The enl1anc.e

were

real world which

problem

any

is exactlv

by explicitly

resulting

to attribute

a new measure

concepts

to character-

set, even

a computer-

an integer. limiting

development

non-integer the practical

and associated

intended

new

resolution

generally

parallels

dimensions usefulness

The

the

to real-world of the concepts.

INTRODUCTION Until

recently

in

thz

long

and

distinguished

history

of

lneasure

theory,

most science and engineering applications were thought to have been covered by Lebesgue measure. Within the last decade, however, intense interest has focused on sets not well characterized thereby, namely, those which can be said to exhibit fractional dirnensionality. Such sets display intricate patterns which may make them useful for modelin g complicated natural or artificial Ph enomena. One method of characterizing such intricacy is by means of Hausdorjf

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P. M. LAPSA

2

and Hausdorff dimension, where the latter in general can be a fraction. Although the concept of fractional dimension might appear contrary to intuition, the idea in fact has an extensive history, apparently originating in 1919 [2], undergoing substantial development in the subsequent decades, and residing in mainstream texts for perhaps half a century [l]. More recently,

measure

sets whose Hausdorff dimension exceeds their ordinary topological dimension have been named fractal.y, and study of the concept has been widely popularized, not only abstractly but also via computer-generated images [3]. Informally, the concept means that a set is inordinately cluttered in its details, regardless of how fine a scale is used to examine it. This property has been called erruticism, but the term connotes both irregularity and randomness, whereas neither property is central to the concept described. The classic example, which, though intricate, is not only quite regular but also permanently deterministic, is the Cantor ternary set, whose Hausdorff dimension turns out to be log 2/log3 E 0.6309. Therefore, a better term for the concept may be intricacy. Other examples, closely related to applications

in science

and engineering,

are white noise and Brownian motion, whose graphs have Hausdorff dimensions of 2 and 3/2, respectively. Since their topological dimensions can be no greater than I, these are clearly fractals, and, as is well known, exhibit ever finer details no matter how great a magnification scale is used. In fact, the former example is in a sense degenerate because its Hausdorff dimension is the dimension of the entire space containing the graph. In both cases, the Hausdorff dimension is related to intuitive impressions of the intricacy or “busyness” of the processes’ graphs, as white noise much more so than is Brownian motion. The present purpose is to introduce dimension, which will overcome practical

a new measure and associated limitations of the Hausdorff con-

cepts as applied to real-world sets. That is, in the real world, all sets-even spectacular computer-generated “fractals”-must have exactly integer Hausdorff dimension. The new concepts, however, permit the extension of fractional dimensionality concepts to these sets. For example, discrete sets such as computer graphs can have the same measured dimensionality as the underlying idealizations they simulate.

2.

HAUSDORFF

MEASURE

AND DIMENSION

To discuss these ideas more rigorously, for convenience the concepts leadings to the definition of Hausdorff measure and dimension will be given. It will be possible to build thereon and develop similar but more practically useful notions of measure and dimensionality. In the following discussion, A will be an M-dimensional set. A C R*‘.

Resolrction-Li,,lite~ Measure and Dimension DEFINITION.

The rlinnleter of A is IA1 =

where

(I( s, L/) is the distance

sup(d( s, y): between

s, y E

A}

x and y.

DEFINITION. A countnhke cooer of A is a family of subsets of R”’ such that each point of A belongs to at least one member of the fk~~ily:

C = (C,,C,

,...I

suchthat

AC

;Ci. I

i=

DEFINITION. An mmer such that for any Ci E C,

DEFINITION. {C,,C,,...} is

The

of A is a countable

p-memm

of a countable

nz”{C} = i

cover

C = {C,, C,,

collection

of sets

. }

C =

/CL)”

k= I

Thus, it may be noted that /n’(C) has units of length, in”(C) has units of area, m”(C) has units of volume, and so forth, although the actual numerics do not agree with Lebesgue measure.

DEFINITION.

The p-dirwnsiond

)n$(A)

= i;f {rap(C):

E-wxz.srm

of A is

C isnn

&-coverof

A).

4

P. M. LAPSA DEFINITION.

The Hnt~.sdmjjf ( 1I - c1ittwnsionnl)

measure

of A is

,,I”( A) = sup [ ttQ’( A)} E>

where

the latter

monotonic

equality

results

0

from

the fact that

tttg( A) is necessarily

in E.

DEFINITION. The HnILsrkt~rlittlert.~iott of A, denoted unique real number rl, where 0 < (1 < M, such that

by EZU( A), is the

(1) if 0 < p < d, then ttzt’( A) = X, and (2) if p > d, then tut’( A) = 0. For characterizing the intricacy of a set, Hausdorff measure and its associated dimension provide a finer mechanism than Lebesgue measure and integer dimension. The classic example is the standard Cantor set, which has zero Lebesgue measure in one dimension and hence is not measurably larger than, say, a single point. When Hausdorff measure is used, however, Cantor set has unity measure i;l the fractional dimension log2/log3.

3.

A RESOLUTION

LIMITED

the

MEASURE

We now proceed to extend the above definitions of measure and dimension to give results which, while agreein, 0 with the previous concepts as limiting special cases, are more usef61 in real-world applications. The revision has two aspects. First, elegant though the Hausdorff construction of measure is, when applied to real-world sets or even computer-generated ones, the plain fact emerges that the theory must give exactly integer dimensions. In other words, by the very definition of the concept, r10 fractnls hoe ewr been fottnrl in nature. Second, Hausdorff measure does not agree numerically with Lebesgue measure, or, for that matter, with everyday experience. Although this may seem to be a matter of cosmetics, to calculate the volume of a unit cube as anything other than unity may well place an insuperable barrier between the mathematical legerdemain and everyday experience. The first difficulty can be treated by restricting the kinds of covers admissible

in computing

the

p-measure

of a set,

as will be shown.

The

second difficulty can be overcome

simply by resealing the previous definition

of p-measure.

ALTERNATIVE DEFINITION. Tlle p-rmmrm2 of a countable sets C ={C,,C,,...Iis

Y,,lC,l”

x=

scale

factor

r,,

is the

of

c

/-L”{C} =

where the p-volume,

collection

I

ratio of a ball’s

p-volume

to a cube’s

this scale factor obviously agrees with ordinary For integer dimensions, notions of length, area, and volume in case the covering sets are the largest possible for given diameter (C,I, i.e., in case the C, are balls. Since, however, the scale factor does not affect the calculation of dimension, it is omitted in the sequel; the interested reader can easily include it if so inclined. The restriction on admissible covers takes the form of limits on the minimum as well as maximum diameters of the sets included.

DEFINITION. An (E,,, s)-cocer {C,, C,,

of a set

A is a countable

cover

.} such that for any Ci E C,

DEFINITION. The p-dirnen.sionol

/_+:,,,E,(A)

(E,,, c)-tnmsure

= i;f(~P{C):Cisan(E~i.E)-coverof

of a set A is

A}.

C =

6

P. M. LAPSA

ARL

measure

is thus

defined

in a nxmner

H~usdo~~ff measure, and indeed nameIy, the case E(, = 0:

Again,

in the definition

the

the suprernun~

latter

quite

parallel

is ~1 special

may be replaced

to the definition case

of the

of

former,

by a limit:

The difference is that /_+‘kS,, Fj as ii function of E cannot be asymptotic to .5z= 0, i.e., cannot go to infinity there, hut rather is litnitecl by its \2ilue at Eft. When E,) # 0, AKL measure will differ in that it carmot admit infinite covers-lest the measure be infinite for all p, Howe\w, ;~ltlmugh unbounded sets are thus excluded, it slwuld be noted that in the important case of compact sets, ~I~lusdorff measure also can 1~ ~o~~ll~~ltec~bv using nothing but finite covers. This can be shown by first relating countablt arltitrary covers to countable

Pww. of A, with

Construct

open

covers,

C&en

as follows.

6 > 0, let C = {C,, C,,

} he if countable

(s/2)-cover

p-measure

a countable

open

&-cover

D = {D, , Ll,,

} such that each

D, is

the union of open

E’-balls

centered

at every point

x of C,.

Denoting

such

balls by S,(E’),

and its diameter

is

ID,) <

/c,/ + 2E’.

where

1

6

‘=

-

2

win

l/l’

!( -;,

IC,I”

+ g$

1

- /C,/ . I

ThUS,

d(D)

=

E jD,l” k= I

= rrL~‘{C) +

$

Therefore,

i;f { r)zj’( Ct} : D is a countdde

because

open c-cover of A) < ~2“I A]

W

6 was arbitrary.

Then it is a simple proposition.

matter

to

relate

to finite

covers

in the

following

P. M. LAPSA

8 PROPOSITION 1.

For corrz~mt A c

R"',

mJ'(A) = sup inf { III “( E} : E is clfitlite txowr

ofA}

&>O E

PROOF.

First we note that igf ( mI’( C) : C is an ewver

of A}

< inf ( m1’( E} : E is a finite c-cover of A} , E and therefore, mf’( A) < sq

inf ( rn/‘( E} : E is a finite c-cover of A} .

&>O E To establish

the opposite inequality, we can take advantage of the conquctness of A. Let D = {D,, D,, } d enote a countable oI>en .c-cover of A. Since A is compact, there exists a finite open sub-cover from D, sa?

E,j=(E,,E,,...,E,}. Since

we have iqf ( mI’( E} : E is a finite c-cover of A]

< i;f m

I’( D}

< tnJ’( A) by Lemma

1.

4.

SIMPLIFIED

CALCULATION

It was noted that, for B<) + 0, only finite covers need to be considered for ARL measure. Therefore, it is possible to simplify the calculation by using covers consisting of sets of eyunl dinmeter, say E. In terms of the definition in Section 3, such a uniform cover can be called an (E, &)-cover or, simply, an (E)-cover. This concept differs from the definition of E-cover, without parenthesis in Section 2 since the latter can, in general, be nonuniform. The three kinds of covers are related as follows:

C&o,&)-cover: (&)-cover

c-cover

&g < lCil G E;

= (8,

&)-cover:

/c,l = 8;

= (0, &)-cover:

It will also be useful to denote N(61,.E) A minimum nmnber N (&I,+) A lim NcEO,Ej.

the number

of sets involved:

of sets among all the (E,,, &)-covers

of A;

F’&,,f

This limit exists because NcE,,,6j cannot decrease as E becomes more restrictive by decreasing toward E(). Then the following lemma gives ARL measure in terms of uniform covers.

LEMMA2. Uniform Cowring For E,, # 0,

(i) First, it will be shown that only covers of minimal size are required in the definition of p[i,,, E,(A). This minimal size follows from the fact that N(Eg,Ej is an integer for each E. Thus there must be an E* > qj such that

10

P. M. LAPSA

N(Et).Ej is wmtant

for E E (se, E,];

k.,

by definition. Now with E E (E,,, E* 1, let C be any minimal-size of A; that is, C is of size NC,,,+ ). Its p-measure is bounded:

For C+ my ( E(), ~&cover

con~linin~

more sets, say N+ > AJ~~,~ + ) + I,

E/N+< p”{C+} But if E is sufficiently

then C+

( E(,, &)-cover

< &“N’.

near E,,, that is,

has larger ineasure:

>

E’J(l ++-j-k++

1)

Therefore, covers as large as C+ may be excluded from the (q,, ~&covers of A. (ii) Second, it will be shown that only uniform covers are required in order to form /.+‘l,,,(A). For C any (E,,, &)-cover of size Nc,tI+ ), let C’ be the cover

obtained

by replacing

each

p”{C’} Since

set in C by a similar

set of diameter

E. Then

=sp”(C) + iyFllf)(E” - 4).

all NCfi(,+ ) sets in C’ are of the same

size E, i.e., C’ is an (E)-cover

of

A, i;f

{ /_4,“{Cj : C 15 ‘. an ( &)-cover

< i;f

of A, of size NC,<,+,}

(PI’{ C} : C is an ( E,,, E)-cover

of A,

ofsize

NC,<,+,}

Q,,+,(&” - 42

+

= P&,6)(A) + qEc,+)(&” - 4) for E E (qr,

E*).

l_i;rr+

On the other

irgf { /..kjJ{C} : C IS an ( 8) -cover

hand,

.>IJI+

i;f

2

and the lemma

Thus.

since

any (&)-cover

is also an (E,,, &)-cover,

{ PI’{ C) : C is an ( &)-cover

lim E’E,,+

follows.

i;f

{ k”“(C}:

of A, of size NCt‘(,+ ,}

of A, of size NC,,,+ ,}

C is an ( E(,, &)-cover

of A}

n

As a consequence, it is possible for ARL measure to simplify one of the most awkward practical difficulties of Hausdorff measure: the optimization over all c-covers, having sets of possibly unequal diameters, to find the required infimum. This is shown in the next proposition, which requires only

P. M. LAPSA

12 the determination of nrry minimal-size all sets of diameter E.

PROOF. consisting

By Lemma

(8 I-cover, i.e., a uniform cover having

2, it suffices to find the infimum over uniform

of NCEl,+ t sets, each of which has diameter

covers

E. But all such covers

have the same p-measure,

p”{ c) = NC,,,. )F”. Hence,

: C is any ( &)-cover

of A, of size NC,,,+ ,}

II

A further simplification is derived in the Appendix for the case where ARL measure is based on cov’ers restricted to consist of balls, i.e., ARL round meast~re. In that wse, it is shown that there always exists a cover which actually realizes the measure. This result would be i!~ll~(~ssible for Hausdorff measure, which, in general, can only he al~l~roached as the limit of greatest lower bounds. 5.

A RESOLUTION-LIMITED

DIMENSION

With the definition of measure thus extended, the next step is to develop an associated notion of dimension. For Hausdorff dimension, the choice was clear-cut: if too small a dimension p is assumed, then Hausdorff measure 111J’( A) becomes infinite. For ARL measure, the choice cannot be so dramatic because the resolution of the admissible covers is limited to some E,). What can be done, however, is to judge p according to whether or not ,u/L,,, has a

~~~~~~~~~c?~ fo incfmse near the resolution limit E(). If an assumed ~~ilnensi(~n p yields SLI& ;Itendency, approl~riately defined, then the assumed value will be judged too small. This basic idea needs only slight technical el~~l~or~lti~}n. It will not do to judge the behavior of pCI’6C,) ( A) by its derivative ~lp/i.~,)( A)/c!E,) because at imy particular value of .9(), the derivative may not represent the overall tendency. Indeed, the ~~I?l~roI~~~~tecriterion emerges when we examine the behavior of y,‘LS,,( A) in relation to the limiting minimum number of covering sets,

NC,,,+,.

PKOOF. L,et gl, be @yen. By the proof such that for 6 E (et), &*I,

N(fit,. Let

ct; E ( LS,,. E* 1. Then

Cl

=

of Lemma

2, there

exists

E,

> E,),

y.,,+).

also for F E (E;,

E* ],

Hence.

or, by definition,

Ncc,;+1 = That

is, NCgI,+, is right-continuous.

N,,,,+,’

By the proof

of Proposition

2,

Since EJ is continuous, p(k,,)( A) is right-continuous. Likewise, it is continuous wherever N( E,,+ f is c[~~~tinu~)l~s (i.e., constant, since Nt,,, + ) is integer-valued). a As argued at the beginning of this section, it is now clear that an overall tendency of &i,,,( A) to increase cannot he judged by its derivative. Indeed,

P. M. LAPSA

14

=

illf{F : A%\,,) < N(,<), ,)

By right-continltity,

A is so small ant 1, unless Inininral valrie of 1,

In terms

in relation

of E,, :tnd Z,), it is possilde

to

F,, that

to define

A\,,,+ ) already

has

the

the ilrcretnents

For ;I given E,, > 0, the incremental tendency - Ap/:.,,)( A)/A 8,) may 1~ positive, zero, or negative; this indicates that if 8,) were decreased, /.L[:~,~(A) would tend to increase, remain the sane, or decrease, respectively. If the measure tends to increase, the :tssund chnension p will be jrltlgetl to0 Andogous to the‘ Hnu~dorJrf small, and conversely if it tends to decrease. which defines Hausclorff dimension, it is possible to estaldisl~ for ?2tirde,’

~l~OlGdTIOS

3.

GiWJl E,, > 0, q- xc,,,+ I > 1, fh

.srccl1tllrff

p* =

h;,.,,+,

( i

1 log

A

log

%,,+I

z,, -

if&(, - > 0.

i SC,i

I

0

ant1 for }’ < p*,

ifg,,

= 0.

tl,cJ-c!c3iet.Yp* > 0,

16

P. M. LAPSA

and

A4:,,,(Al <

(),

2% c‘(1 p > p”, the proof is sidar. (ii) In the case gg = 0, the minimtim required nwnber of covering sets does not increase even for arbitrarily small F,,. Therefore, the set A consists of a finite number of points, say N. Thus for go small, For

which

Hence,

gives for any positive

p* is less than

value

any positive

1’

value: II

F’* = 0.

DEE’INITION. The nsymptotically r~sol~~tion-lit?litecl tlirnension at resolution level co is the value p*( cl,) defined by Proposition

6.

APPLICATION

As mentioned concept is that

TO SAMPLED

of a set 3.

A

SETS

in Section 3, a practical drawback no real-world or computer-generated

of the Hausdorff measure set can actually have

Resohttion-Lirlliterl

Measure

and Dimension

17

non-integer dimension. That ARL measure overcomes this difficulty by use of the resolution limit E,, will be demonstrated by means of sampled sets, which are typically used as computer representations of unachievable ideal sets. For example, the graph of a function is represented by a finite number of samples. In the sequel, A, will denote a sampled set, i.e., a set of samples of an original set A, such that the sampling accuracy is 6. By this it is meant that if x E A, then there exists r~ E A,, such that 11x - ~11 < 6. Since A is bounded, A, can be finite; let N,,,;,, be the number of points contained in A,. Whereas, the sampled set A, clearly has zero Hausdorff dimension, it will be demonstrated that, for suitable resolution limit co, its ARL dimension approximates that of the original set A. Of course, cc, must not be so small that each point of A, requires a separate covering set, i.e., NcEII+) = N,,, ,, an d co = 0, which would imply that, for any p > 0,

I-$,,,(45) = 0,

Thus, ARL dimension p* would be zero as well. However, the point is that for a larger E,), if the assumed dimension ,Q is too big, then the incremental measure slope - A /_$,,( A, >/A F will get very large and thereby indicate that fact. The following propositions develop this result.

PHOWSITION4. Ifp exist.s E,) > 0, such that

< HD(A),

then gioen m-bitt-wily large b > 0, there

PROOF. Let the lengths of the successive intervals over which NC,,,+) is constant be denoted by A,(&,)), where k = 1,2,. If the change in measure over such an interval is AI, &L,,,(A), then the slope is

P. M. LAPSA

18

Let ~6 be an endpoint of any such interval. If r_’ < HE)(A), then Hausdorff measure is infinite. Thus, for sufficiently small E, the measure of A in terms of E-covers is large:

m;( A) > ba; + p&J A). Let ~;j be an interval endpoint that is sufficiently small in this sense, and that is also less than Ed:. AHL measure at &I; is at least as large as ml,,(A) because

its covers are restricted

more:

Also,

where the summation

Let

IY~ be the normalized

Since Ck AR&* = E; -

Thus,

is over the intewals

the average

( ---sk) exceeds

b.

between

.s;t and E;;. Therefore,

interval length

8;; < E;),

of the (-sk)

exceeds

b, which implies that at least one I

LEMMA 4.

Gicen

E,, > 0 nnd A c R.“, for snrnpling

nccumcy

6 less

than &(,,

PKOOF. Let x be any point in A. Then there is solne point L/ in A, which is within 6 of x. For E > &,,, if C is any (&)-cover of A,, there is :I set B, E C which contains y. Since (B, 1= E, B, cm be contained in a ball of diiuneter 2~. Furthermore, a ball of dialneter 4~ and with the salne center will cover a distance 6 outside B,, and hence will cover to a distance 6 ;lw;ly frown y. Thus, it will cover s as well. Let C’ be the cover obtained by replacing each B, by such a ball of dialneter 4~; then C’ covers not only A,, but A ~1swell. Each ball in C’ can be covered by a finite number of balls of dianieter E. Let K,, , which depends on dilnension M, be that nulnber, and let C” be the (&)-cover comprised of all such balls. Then, for each set B, E C. there are K,,, balls of dialneter E in C”. Thus,

p”{ C”} = K,, El.“{c} Since C” is an (&)-cover

for A,

/-$I.,,,< A) G d’{c”}~ and the lelnlna follows.

n

PHOPOSITION 5. Zf p < HD(A), tI Zen there exists n sampling accuracy 6, .ruch that the sampled set A, has nrhitmdy large ARL mmwre slope for some E(, > 0. Thnt is, @en h > 0,

PROOF. As in the proof of Proposition 4, let E’ > 0 be an arbitrary interval-endpoint; ARL lneasure at E,\ is @A,;,(A). Since p < HD(A), E;;

20

P. M. LAPSA

can be chosen

so that p&.;; is as large as desired.

Specifically,

P&;;,(A) > (&I + /-&,( A))& By Lemma

Following

4, if S < .YR, then

the proof of Proposition No,,,+ ),

4, let sk be the measure

slope over the

k th interval of constant

It follows that /-&,(A,)

= /-&,j(Afi)

+

C(-“I,)‘%% k

Therefore,

and at least one ( -sL)

must exceed

h.

7.

EXAMPLES

AND DISCUSSION

ILLUSTRATIVE

n

The operation of these concepts can be demonstrated by means of the simple sets in Figure 1. The ARL measure of the set 1 in Figure la is illustrated in Figure 2 for three values of assumed dimension p. The actual dimension is of course 1, and indeed at a given resolution level E() the incremental tendency, shown by the dashed lines, is negative for p too small ( p = i) and p osi. ‘t’ive for p too large (p = 1,s). The benefit of using the

Resolution-Lit,lited

Measure nnd Dimension

21

(a)

.

.

.

.

.

.

.

.

.

.

.

(b) FK:. 1.

(a) The Unit Interval 1. (1)) Sampletl Version of 1.

2.5

Y

/

2.

1.5

1 .

.5

0.

1

0.

.2

.4

.6

eo FK:. 2.

AKL Meas~w

of Z for p = 0.5, 1.0, 1.5

.8

1.

22

P. M. LAPSA

incremental measure slope rather than the derivative is clear, for the latter would be positive ahnost everywhere, eveu if p were too small. Figure 3 likewise displays ARL measure for values of p spaced more closely, viz., 0.1 apart.

The key advantage of ARL measure and dimension becomes evident when these concepts are applied to sets that appear substantially different at different resolution levels. For example, a discretely sampled graph of a function, as might be produced by a computer, will appear vastly different above and below the sampling resolution. As an illustration, the set 1 is shown sampled with interval 0.1 in Figure lb; the cm-responding ARL measure is illustrated in Figures 4 and 5 for various values of 11. To demonstrate non-integer dimension, ARL measure of the standard Cantor set K is plotted in Figure 6. The assumed values of 11 range from 0.75 to 1.15, where 5 = log2/log 3 is the actual dimension. Finally, Figure 7 shows ARL measure of K sampled at a spacing &; these curves are quite similar to those of Figure 6, except that they inevitably collapse to zero when resolution level E,) shrinks below the sampling spacing.

2.5

2.

1.5 6%) 1 .

. 5

0.

0.

‘

.2

.4

.6

.8

1.

23

FK:. 4.

Several

ARL Mraswe

features,

noting in Figures

of I,, , fhr p = 0.5, 1.0, 1.5.

which

are generally

true of ARL

measure,

are worth

2 through 7.

The unsampled set’s measure can never be exceeded by the sampled set’s measure because any cover of the former will also be a cover of the latter. For resolution level &Cl well above the sampling interval, the sampled and unsampled sets measure about the same. As resolution level approaches zero, so does ARL measure for any 11 > O-in agreement with Hausdorff measure, which must give exactly zero for any uniformly sampled set. Consequently, as resolution level approaches zero, so does ARL dimension of the sampled set-in agreement with Hausdorff dimension. It may be concluded the ARL measure and dimension meet the objectives desired. The Hausdorff concepts are extended consistently, in a way that dimension&y of sampled sets and other real-world sets can be characterized an additional in meaningful fashion. The resolution level E() constitutes

24

P M. LAPSA

l 0.

.2

.6

.4

.8

1.

co FIN:. 5.

ARL Measrire of I,, , for More Closely Spaced Values of 11.

parameter for the measure, actually simpler.

but by virtue of it, the resulting computations

are

REFERENCES 1.

P. K. Halmos,

2.

1950. F. Hausdorff,

3.

B.

B.

Measure

Theq,

Dimension

Mandelbrot,

D. VanNostrand

and aussere

The

Fractnl

Mass, Mnth.

Ger,r,wtnj

Company,

Inc.,

Atlrl. 79:15i-179

of Nnturc,

W.

H.

New York, (1919).

Freeman

and

Company, New York, 1977. APPENDIX:

RELATION

If the covers

TO ROUND

that define

ARL

MEASURE

measure

are restricted

to consist

of balls,

then the result can be called ARL round mmsure and be denoted by p&J A). The restriction implies that it is simpler to compute since the choice of covering sets is so limited. Also, it serves as an upper bound to the

Resollltion-Litlliteri

25

Measure and Dir,Ensior:

.5

0.

0.

.2

.4

.6

.8

1.

eo FIN:. 6. ARL Measureof K.

unrestricted

measure,

It turns out that ARLR measure is amenable to the great simplification of having covers which actually realize it, rather than only approach it in the limit. The following lemma and proposition

develop this result.

LEMMA 5. Let A C R”’ be bounded. Then there exists a minimal unijkm ball cover (4given size N,

c* = {B

I,...,

BN).

26

P. M. LAPSA

0.

I-

0.

.2

.4

.6

.8

1.

CO

PROOF.

(i) Let n be the diameter of A and x0 be any point in A. Then A is covered by B,,$n), the closed ball with center at x,) and radius a. However, if x, is a ball center more than 3n away from s,), then a radius of more than 2n would be required for a ball B, ,
(iii) Let f(X) = r* be the imppin, 0 that associates with each X the corresponding r * , the infimum of the covering radii. Thus, it maps as follows,

Now given E > 0, consider any set of centers near X, say

X’ =

{s, + 8,).

) s,y + S,.},

where jakl < E. Then, to cover A, it suffices to increase the balls’ radii by E. Thus, f(X’) Conversely,

a similar argument


+ E.

shows that

f(X)


If(X)

-f(X’)I

+

E.

Therefore, =GE,

and f is continuous. (iv) Since f is a continuous mapping on a closed and bounded domain, it assumes its minimum at some point X*. Then C,%.( r*) is the minimal uniform cover of A. H

PROPOSITION6. For hounded realizes ARLR measure: p”{C*}

= E196,, i;f{p”{C}:

A c

R", there is nn (s,,)-cmer

C 19 ‘. on ( ET,,, E)-hnkmer

C * which

ofA}

= P&,l( A)

PROOF. By the proof of Lemma 2, for E sufficiently close to E() only covers of size NcES,+ , need to be considered. By Lemma 5, there exists a minimal uniform ball-cover of size NC,,,+ ). Let the common radius of the balls be r*. If eO = r*, that cover is the desired (.sJ-cover. If q, > r*, each ball may be replaced with a ball of radius E,,. w