Objective Determination of Branching Angles in Fractals

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J. Arrault, B. Pouligny

To cite this version:

J. Arrault, B. Pouligny. Objective Determination of Branching Angles in Fractals. Journal de Physique

I, EDP Sciences, 1996, 6 (3), pp.431-441. �10.1051/jp1:1996167�. �jpa-00247195�

J.

Phys.

I France 6

(1996)

431-441 _MARCH 1996, PAGE 431

Objective Determination of Branching Angles in Fractals

J.

Arrault (*)

and B.

Pouligny

Centre de recherche Paul-Pascal

(*"),

Avenue Schweitzer 33600 Pessac, France

(lteceived

31 A1Jg1Jst 1995, received in final form 21 November 1995,

accepted

28 November

1995)

PACS.42.30.Sy

Pattem

recognition

PACS.47.53.+n Fractals

Abstract. We propose an

objective

pattern

recognition procedure

to determme

branching

angles

in 2-dimensional fractals.

Recogmtion

_is

performed by correlating

the

object

with V- shaped identifiers. We define

a quantitative ^criterion to prove the existence of characteristic

branching angles

in such

figures.

The whole procedure is tested _{on a} mathematical

(snowflake)

fractal and then

applied

to _a few

physical examples.

A neat characteristic

angle

_is found in dendritic aggregates, m agreement with visual

impression.

However, the

procedure

does not find _any dominant

branching angle

_m basic Witten and Sander aggregates.

1. Introduction

From _a

qualitative point

^of _view, ^fractals _are

highly

ramified structures

featuring

structural details and branch

sphtting

^whatever the scale. A _more

rigorous

statement is that trie _mass

distribution in fractals exhibits a dilation invariance

iii.

Mass

scaling

is _a direct consequence of this property ^{and is} characterized

by

_a set of fractal dimensions

[2].

Figure 1,

_a to c, shows three

examples

of 2-dimensional fractals.

Figure

la is _{a so} called snowflake

(or Vicsek's)

_[3]

fractal,

which is built

through

_a

simple

mathematical construction rule.

Figure

16 shows _a classical Witten-Sander

(WS)

_[4]

_aggregate,

which is _a

computer

simulation of _a diffusion l1nlited

growth

_process. Here the

elementary growth step

^involves a

random walker

la partiale undergoing

Brownian

motion)

which

ultimately

colhdes the

already existing

cluster at _some site _on its

boundary.

The rule _is that the

partiale

remains bound to the cluster at this site. In other words _a lattice site

contiguous

to the cluster

boundary

is

occupied

_{as soon as} it is visited

by

_a random walker.

Figure

le shows _a "noise reduced" WS

aggregate (NRWS).

The rule here _is that for trie _same site to be

occupied

it bas to be visited

n times

in

_»

ii.

In trie

example shown,

_n

= 3000. This rule

obviously

favors sites with

large growth probability.

This leads to

typical

dendritic

shapes.

WS

aggregates

_are realizations of the most

general

_process of diffusion hmited

aggregation (DLA).

These structures have attracted _a considerable _interest in the recent years because many

growth phenomena

in very diiferent fields

(electro-chemistry,

fluid

mechanics,

bacteria

proliferation...) _pertain

_to DLA

through

the _same first

principles (the

so-called Stefan

problem

(*)

Author for correspondence

(e-mail: arrault©axpp.crpp.u-bordeaux.fr) (*")

CNRS

©

Les

Éditions

_de

_Physique

₁₉₉₆

la) (b)

ic) id)

Fig.

I.

Examples

of 2d fractals.

a)

Snowflake.

b) Regular

Witten and Sander aggregate.

c)

Noise

reduced Witten and Sander aggregate.

d)

Modified snowflake. The snowflake has 5

generations.

Trie WS and NRWS aggregates _were built

by

_an off-Iattice

procedure,

and contain 25000 and 9000

particles, respectively.

The

ordinary

snowflake

(a)

_is

generated

from _an initial _square, whose sides

are

cut in p = 3

equal

parts. This

gives

p~ ₌ 9 smaller _squares, from which

only

_{q =} 5 _ones

(the

central

and the _corner

ones)

_are

kept.

The _same

procedure

_is

repeated

at

infinitum,

which _{grues a} self-similar

figure

whose fractal dimension _is equal to In

q/

In _p

= 1.464.... The modified snowflake is built in _a similar _way, with _p

= 5 and _q

= 13. This gives dF " 1.59...

[4]).

^This

problem, although

it _con be very

simply

stated

la Laplace equation

with _a

moving boundary)

has _no

analytical

solution

and,

_up to now, has resisted ail

attempts

^of numerical

resolution. WS

aggregates

are solutions of

Stephan's problem

^obtained

by

simulation.

Considerable effort has been devoted _to

characterizing

the structure of these

solutions,

with the ultimate

goal

of

elaborating

a

theory

in the form of _a construction rule

(as

for mathematical

fractals).

Such _a

theory

should of _course

reproduce

ail known

morphological

characteristics of the

aggregates. Reproducing

the value of the fractal dimension

(dF

'~

1.60) _[5,

_{fil is} not

enough:

this _is evident from

Figure

id. There is shown _a mathematical

fractal,

which _was

generated using

_a construction rule similar to that of the snowflake

(see _caption

of the

figure

for

details).

This

figure

has a fractal dimension

dF

"

In13/

In 5 _ci I.fi. It is obvions that WS

aggregates

and the structure shown _in

Figure

id have

completely

diiferent

morphologies, although they

hâve

nearly equal

fractal dimensions.

Diffusion limited

growth

_appears

everywhere directed,

_a

_property

which is

equivalent

to

N°3 OBJECTIVE DETERMINATION OF BRANCHING ANGLES IN FRACTALS 433

stating

that ranlification

(or branching) angles

_are ^smaller than 90°. The distribution of

branching angles

is _a basic

morphological property

^of a fractal

figure.

^This

property

^{is diiferent} front _nlass

scaling

in that it addresses the

anisotropy of tractai

_mass distribittion.

In

Figure la,

there is _an obvions dominant

angle equal

to

90°,

which is the direct consequence of the snowflake's construction rule.

Only

_very few attempts have been made to _measure

angles

in WS

aggregates, probably

because of the

dilliculty

of

defining angles

in a way which is both

unambiguous

and

objective.

^A look at

Figure

16 is suiiicient to convince oneself of trie

diiiiculty

of trie

problem.

Ossadnik

iii

used _a nonlinear

filtering procedure

to

identify

branch _stems in WS

aggregates

and calculated

angles

between stems from their inertia _tensors. Results tutu ont

dependent

_on trie branch "order"

(see

Ref.

iii ^fbr

_a definition of trie

order),

and range between 85 and 38°.

Recently,

^Arnéodo et ai.

[8,9]

^{have used} a wavelet transform

(see

Section 3 of this

paper for _a definition and

Figure

4d for

illustration)

and

by

~isitai

inspection

identified branches and measured

angles

between them. Trie

angle

distribution found

using

this

procedure

tutu ont to be

roughly independent

of the wavelet _size

Ii.e.

the resolution _in the

image)

and has _a

maximum at 9 _ct 36°. This

finding, together

with the

uncovering

of Fibonacci _sequences [9]

in the ramification of trie

aggregates,

supports the

conjecture

that there is _a hidden rive-fold

symnletry

[10] ^{in WS}

aggregates.

In this paper, we propose an

objecti~e procedure

to define and deternline characteristic

angles

in 2-dimensional fractal

figures.

At trie

present stage,

trie

procedure

is

fully numerical,

but it is based _on classical methods in

Optical

Pattern

Recognition _[11].

Trie paper is

organized

as follows: in Section

2,

_we

give

trie

principles

of trie

procedure,

then _we test it _{on a} mortel situation and _on the mathematical snowflake. As _we will see, the

branching points

^{and the}

characteristic

angles

in these

figures

_are

correctly

identified. The _saule

recognition procedure

is then

applied

to the WS and NRWS

aggregates.

In Section

3,

we elaborate _a modified version of the

procedure, namely

_a "Wavelet Transform"

version,

which anlounts to

reducing

the

resolution in the

object by bandpass filtering.

As _we will

explain,

a donlinant

angle

is found

in the NRWS

aggregate

^whose value is not aifected

by

resolution.

Conversely, angle

values found in the WS

aggregate

are not robust with

respect

to _a resolution

change

_or statistical

fluctuations. These results _are sumnlarized and conlnlented in the final Section 4.

2. Bore Procedure

We start from the

simple

idea of

correlating

_a characteristic

shape,

that _we will _{name an}

"identifier",

^with trie

object

to be

investigated.

We calculate:

C(M,1, 9, Q)

=

p(M

_*

V(M,1, 9, Q) il)

Here

p(M)

is trie _mass

density

of trie

object

at

point M,

and trie

symbol

* stands for correlation

product.

In trie

practical

situation of _a finite size

(or

finite number of

generations) fractal, p(M)

is a

binary

function defined _{on a} 512 x 512 square

lattice,

which is

equal

to 1 at any

occupied

site and

equal

to 0 otherwise.

V is trie identifier. It is

just

_a

simple

V

(see Fig. 2a)

made of two

one-pixel-thick

_arms

of

length

1. 9 is the aperture

angle

^of the V and Q defines its orientation

relatively

to _a fixed direction. Because of

discretization,

the number of

pixels

in trie identifier

lits mass) depends

_on Q. This artifact leads _to

systematic

_errors which _we correct for

by dividing

the bore correlation

product (Eq. iii) by f V~(M)dM.

This definition of the

normahzing

factor is

general

and

applies

whatever trie definition of

V(M).

In trie

simple

_case of _a

binary function,

finis _is

just

^trie mass of trie identifier.

Kumerical calculations _are

performed

_{on an} Alliant VFX40

computer.

For _a given value of Q, trie fonction

C(M,1, 9, Q)

is

computed

_via trie Fourier transforms of _p and V

(for

this

"1 ^"

é~~~

~

~ ~~

8

~/

~ ^~'

~

,~

Î~

~~ l'j,[)

,~ ~

l_~ j '~_~

'Î

/ /~/

~/~ ~ '

(ai (b)

ioo 0.008

/=

°ce Q~

~i _{Ù 50}

~

jj~

~

ç_ L

'ij

0

n=

0 500 1000 5 5Q IQQ

7 ÎÎ ₆

(c) (dl

Fig. ^2. Illustration of the

branching angles

determination

procedure

_{using a}

simple

V-shaped ^iden- tifier

(a)

_{on a} test

sample (b)

made of _a collection of disordered different Vs. For

clarity

and

definition,

the identifier in

(a)

has been

magmfied

5 times. Apertures in 16) are

20°,40°, 60°,85°, 90°,100°

and

120°,

with 7 Vs _m each _case. The small

wiggles

_on most of the V _{arms are} due to discretization _on the 512 _* 512 square Iattice

(see text). (c)

contrast

histogram

obtained with _a 60° identifier. For instance,

p(j,1, 60°)

_is the number of sites _m

Figure

2b where the contrast i is

equal

to ~o within _a +à~

margin.

Ai _is chosen equal to

im/200,

where _{im is} the maximum contrast in the image.

(d)

different moments

(n=5,10,15)

of the contrast

histograms

_~ersus identifier aperture. ^{For each} order, the maximum of

El~~(1,

_Ù) has been normahzed to 100.

purpose, we used _a classical fast Fourier transform

(FFT) algorithm).

For each 9, _we collect 360 such

functions,

for Q ranging from 0 to 360° with _a 1°

step.

^{This demands about} hour of

computing

time. We calculate trie correlation

product

at each point for ail orientations of trie identifier and define trie contrast:

7(M,1,

_Ù)

=

Maxici~ Minici~

,

j2)

for 0 < Q < 27r. Trie relevance of this

quantity

is understandable

by looking

at the test

sample

shown _in

Figure

2b. This

sample

is _a collection of _{more or} less

entangled

V's ~vith 7 diiferent values of the

aperture angle. Clearly

C will take _on

large

values whenever trie identifier _is

brought

in coincidence ~vith _a V of

aperture

⁹ in the

figure.

A

"branching point"

_in trie

figure

is defined _{as a}

position

of the identified vortex where C _varies very much _{as a} functioii of

N°3 OBJECTIVE DETERMINATION OF BRANCHING ANGLES IN FRACT~~LS 435

Q, namely

where _~ takes _{on a}

"large"

value.

Here, quotation

marks _mean that _we need _a

quantitative

criterion to decide how

large

_~ has to be. This is trie

subject

of trie

following

discussion.

In

Figure 2d,

it is obvious that

"branching points"

are

just

trie V vortices. These

high

contrast

points (sites) represent

a very small _mass fraction in the

system.

^{Most of the}

points

have small contrast. This _means that in _a

histogram

of ail contrasts

(the

set of _j values for ail the

occupied

sites _in trie

image) corresponding

to a

particular identifier,

_say

p(~, i,9),

trie

branching points participate only

in trie

histogram

tait. This _is

clearly

visible in

Figure 2c,

where 60° vortices

give

_rise to a small

bump

in the tait of trie

p(~, i,60°) graph.

Trie

large bump

near ~ = 0 is due to trie many

points

in the

figure

which bave _{lo1A~ contrasts.}

Dur

goal

is to find trie dominant

branching angles

_in trie

figure.

One way ^to do _so is to count trie number of

points having

_a contrast

higher

than _some threshold value j for each identifier

aperture (9)

^{and then} to

plot

this number _~ersits 9. This _{amounts to}

weighing

each

histogram

tait _{as a} function of 9. However this

procedure

has the drawback of

introducing

_an

arbitrary

parameter, j, ^{in the} final result. A

slightly

^diiferent but smarter way is to calculate _a

high

order moment of _p,

E(") ii, 9)

₌

~"p(i,1, 9)d~, (3)

and to

plot E(~l

_~ersits 9. This is shown in

Figure 2d,

for different values of _n. As the _moment order

increases,

the characteristic V

angles 20,40, 60, 85, 90,100

^and

120°, progressively

sho1A~

up as

pronounced peaks.

As _n

increases,

the 1A~idth of each

peak progressively

decreases clown to about

1°,

which is the

practical

limit set

by

discretization.

We viii take the ezistence

of

these

peaks for large

_{n as oitr}

operationai dejinition of

dominant

branching angles

in the

pictitre.

In

Figure 2d,

the

large

order moments tend to _an

asymptotic limit,

which tutu out to be the exact distribution of

branching angles

in the test

figure.

The existence of this

asymptotic

limit _means that trie result of _our

analysis

_can be stated

independently

of the value of _n, which

in tutu _means that the _way in which the tails of the

histograms

_are

"weighed"

_is of

marginal

importance.

Note that the coincidence of

E(Ù(1,9)

with the distribution of

branching angles (in

trie

n - +cc

limit),

which is visible _in trie test

example,

_may not be _a

general property.

Dur

point

is

just

that if trie distribution bas _a few well defined

peaks,

these will

clearly

_appear in

E(Ù ii, 9)

for

sulliciently large

_n.

We _now

proceed

to

applying

_our

procedure

to trie fractals shown in

Figure

1.

Looking

at

Figure la,

it is obvious that the number of _resonances

(or branching points)

will

depend

_on

the value of trie identifier size 1.

Choosing

the value of t amounts to

selecting

_a number of

generations,

_say

N,

in the

figure.

IV

=

1, 2, 3,... correspond

to

Lli equal

to

1,3,9,.

,

i-e-

3~

IL

is the overall size of trie

figure).

From trie snowflake's construction rule

[3],

we

expect

^trie number of identified

branching points

to scale

roughly

_as

5~,

with _{x given}

by

3~

=

Lli. By

"roughly",

_{we mean} that this result is exact

only

for

integer

^{values of} x. The results shown _m

Figure

3a and 5b

correspond

to

1equal

to

L/20.

This choice _is rather

arbitrary.

We

just

choose the value of1 far from the

large

and small cut-off

lengths

in the

figure,

1-e- in _a size domam where ail the

figures

_are dilation _invariant

(this

_is trie basic

pi~operty

^of

fractals).

Thus _we expect trie conclusion of the

analysis

in

terms of dominant

branching angles

to be

independent

of the value of the identifier _size.

We checked this

by trying

^diiferent values

of1,

_{as we} will

explain.

Dur results _ai-e

displayed

in

Figure

3a to c. In trie first _case

(ai, ^E(~°l (9)

^and

E(~51(9)

feature

a very ^neat

peak

at

90°,

which is the result _one expects from the snowflake construction rule.

This _may be taken _{as a} further test of trie method. Trie

p(~) histogram

for 9

= 90° features _a

ioo

)i

j~

50 i~i

~

~

r]j

~i

'~ ii

r

~ ._ ^~

[

r

Ù

50 100 20 40 60 80100 20 40 60 80100

Îo ~--Z. 6 11 6 11 6

~~

(Î

~

ÎÎ

~

(a) (b) (c)

Fig.

3. Moments

El~~(t,Ù)

of the contrast

histogram

_{as a} function of identifier aperture

angle.

a)

Mathematical snowflake.

b)

WS aggregate.

c)

Noise red1Jced WS aggregate. In

la),

the different

moments shown

(n

₌ 5,10,

là)

have ail _{a maximum} at ₌ 90°.

distinct

bump

at

high

contrasts _as in

Figure

2c. This

bump

is made of 25

branching points.

If trie _size of trie identifier is reduced to

L/40,

_we still find _a

unique

maximum in

E(~51(9)

at 9

= 90° and 125

branching points.

Notice that trie ratio

25/125

=

ils

is

equal

to

3~~F,

with

dF

_" In

SI ln3,

_as

expected.

The other

fairly

clear result is with the noise reduced

aggregate

(Fig. 3c)

where _a dominant

angle

near 70°

progressively

shows _{up as n} increases. This result is obtained with

=

L/20 (where

L is trie

largest

linear size of trie

object).

Trie result with

=

L/40 (not

shown _in trie

figure)

is

quite

similar.

Trie bare WS

aggregate (Fig. lb)

is

obviously

_{a more}

complex

_case. Several

peaks

show up in the

E(~) ii, 9) diagrams (Fig. 3b),

but with _no dear

asymptotic

convergence as n increases.

Notice that

Figure

16 is

just

_one realization of the statistical WS

aggregate.

^We

repeated

the

angle analysis

_on 10 diiferent realizations. Each of them resulted in _a multitude of

peaks,

but at

positions depending

_on the

particular

reahzation. In other

words,

trie

peaks

shown

in

Figure

3b _are not stable with

respect

to the statistical fluctuations. As _we will _{see in} the

following paragraph,

these _are also not robust with

respect

to trie resolution set in trie

analysis procedure.

3. Wavelet Transform Version

At this

stage

we

just blindly applied

trie V-identifier

procedure

to fractals. We found _{a quan-} titative

(numerical)

criterion that

yields angle values,

not _a

geometrical

^definition of

angles

_in

a

complex figure.

In fact there is _no way of

correctly defining angles

_in

fractals,

because of trie existence

(by definition)

of structural details at all scales. One _way of

circumventing

this

dilliculty

is to observe the fractal Db

ject

at finite

spatial resolution,

in other words to _smear Dut all details smaller than _an

arbitrary length.

This _can be achieved

by low-pass filtering,

more

generally by band-pass filtering.

In

Figure

4 the _mass densities of the diiferent Db

jects

were convoluted

by

_a Mexican

hat,

a

function

equal

to the second derivative of _a Gaussian function:

wal~)

"

j12 ₎₂ ~~Pl~ )2

_,

¹⁴⁾

N°3 OBJECTIVE DETERMINATION OF BRANCHING ANGLES IN FRACTALS 437

~~ ^~ ~'

~~ =, ~~

~

m~ ii j _j)j[

~ @~~ /~

~~)

^~/

)

~ /~

~--~J -~~~J~ (

(~) (b) (c)

x5

Fig.

4.

a)

Scheme of the Mexican haï wavelet.

b)

to

f): Isotropic

wavelet components ^{of the test}

sample 16),

^the snowflake

(c),

the WS aggregate

Id),

the _noise reduced WS aggregate

(e). If)

is the

V.shaped

identifier. The wavelet size parameter a was chosen

equal

to 5 lattice sizes, which is about

1/8

of the V-identifier _arm

Iength.

The quantifies

represented

in

b)

to

f)

_are

positive values,

with _a 16-Ievel

coding

from white

(0)

to black _max

(Ta()

For

clarity, (a)

and

(f)

_are

magnified

_views

(x

10 and _x 5,

respectively).

where _r is the 2-dimensional coordinate. The parameter a defines the width of the function

(the

full width at half maximum is

equal

to 1.33

a).

The Fourier transform of

Wa(r)

reads:

where _q is the 2-dimensional

spatial frequency. Clearly,

convolution

by Wa(r)

_{acts as a} band- pass filter

(with

smooth

edges).

In the _context of

signal analysis,

the Mexican hat is known _as

a very

simple example

of _a wavelet. This wavelet is

"isotropic",

because

Wa(r depends only

on the modulus of _r. The convolution

product

~ai~

"

Pi~1

~

~ai~ (fil

may be called _an

isotropic

wavelet

component

^of _p. The whole set of wavelet components of p, for _a

ranging

from 0 to

infinity,

is the so-called Wavelet Transform

(WT)

of _p.

Isotropic

WT of ld and 2d fractals has

recently

been the

subject

of considerable research

[12,13]

and shown to be _a

powerful

tool to visualize structural

hierarchy

and to characterize

mass

scaling (~).

There is _a

great flexibility

in the choice of _a

particular

function _{as a} ~avelet.

Here the Mexican hat _was selected for _reasons of smoothness and numerical

simplicity (~).

Figure

4f shows _an

isotropic

wavelet

component

of the

V-shaped identifier,

which _we will denote

~â:

(=V3Wa

_1?1

(

itself is _an

anisotropic

wavelet

lit depends

_on bath the nlodulus and the direction of

ri,

which _we used _as the

angle

identifier to

analyse Ta

in the different _cases

(Fig. 4b, 4c, 4d, 4e).

We choose

ali

=

1/8

in this

study.

Here the choice of _a fixes the resolution _in the

image

to be

processed.

i has to be

larger

thon this resolution limit for the identifier to be resolved _{as a}

V-shape.

Notice that

Ca

=

Ta

@

Wa (8)

"

[p

^@

wl~~j

^{@ v 19)}

where

WÎ~I

= l§~a l§

Wa

is

just

_an other

isotropic

wavelet whose size is still _on the order of _a.

Equation (9)

_means that the _same if _as before _is used to

investigate

_a finite resolution _version of the

image.

This choice makes the

procedure

closest to the visual _one used in reference

[8].

In the _same

spirit

_as in Section

2,

_we ^define in each _{case a contrast ~a, a contrast}

histogram

pa and the

corresponding

moments

EÎ"1

The results _are

displayed

in

Figure

Sa to Sd. for

=

L/20.

Notice that the results obtained with the test

sample (Fig. Sa)

and with the sno1A~flake

(Fig. 5b)

_{are very} similar to those shown in

Figure

2d and 3a. Peak

positions

and widths

ai-e identical. There _are small fluctuations between the

peak heights

in

Figure

Sa which _are not present ⁱⁿ

Figure

2d. These fluctuations _are due _to the fact that there _are diiferent

length

scales in the test

sample.

For instance the branches of small

angle (20°)

Vs _are close

enough

for their wavelet

components

to interfere. This

implies

that the ~vavelet i~ersions of Vs of diiferent

aperture angles

_are

qualitatively

diiferent. A

complication

of the _same nature _occurs between

neighboring Vs,

which also interfere at finite resolution.

These interferences do not

greatly

^affect the behavior of the

high

order moments in the

s1nlple figures, namely

the _test

sample,

the snowflake and the _noise reduced

aggregate (Fig. 5d).

In the

latter _{case, we recover} the dominant

angle

around

70°,

~vhich makes this result robust _{~ersus a} resolution

change.

For the snowflake and NRiVS

aggregate,

we also checked for

self-similarity,

as ~ve dia with the bai-e

procedure.

The results with

= L

/20

and

= L

/40

_are

essentially

the

same in terms of characteristic

angles.

^The

proliferation

of

branching points

in the snowflake goes as

i~~F,

_as

expected.

The situation _is very diiferent with the basic WS

aggregate,

^{because the}

peaks

found in

Figure

âc

ii

= L

/20)

_are uncorrelated to those found with the bare

procedure (Fig. 3b).

The

same is true ~vith

=

L/40.

The _set of

angle

values _is net robust with

respect

to _a

change

_in

the _size of the identifier. As _a

practical conclusion,

_we state that _our

procedure

identifies _no

significant angle

_in

regular

iVS

aggregates.

(~)

Studying

self similarity in _a signal requires

eliminating

_any regular

(non

self

similar)

background from the

signal.

This task is achieved

by bandpass,

not

by Iowpass filtering.

See for instance [14]

(~) On the other

hand,

the

optical

WT version ~vhich _was

developed

in

our

laboratory [là,16]

is based

on true

(sharp edges) bandpass filtering,

because this made the optical hardware

simplest.

N°3 OBJECTIVE DETERMINATION OF BRANCHING ANGLES IN FRACTALS 439

100 100

-~

-~ ii

ii

/= /= jj

~

_{Qi i'}

fi 50 fi 50 _i~ 1~(

~~ ~~

©ù ©ù ~j

i_i

_i~j

0 0

ÎÎ

^~~

~

~°°

)j

^{m--m- 50 100}

6

la) ₁₆₎

IÙÙ ~j

IÙÙ rj

Il

L~

Îj j~

/~

j~ ^~

ç~

jj ^~

50

~~

~ 5Ù Î

~ ~~

~Î_

~~ /~~~

~

~~

~zQ

o o

Î[m__m_20

_{40 60} 80100

1[1__1-

_{zo 40 60 80100}

3° 6 ~° 6

(c) _~d)

Fig.

5. Moments

EÎ~(1,Ù)

of ihe contrast

histogram

in the wavelet

procedure.

Here the

object

and the identifier _are

isotropic

wavelet components ^of the bare _mass

density (p)

and of the bare V.

a)

Test

sample. b)

Mathematical sno~vflake.

c)

WS aggregate.

d)

Noise-reduced WS aggregate.

4. Conclusion

We defined _an

objective procedure

to find

significant angles

_in 2-dimensional

pictures.

The

procedure

is based _{on a} classical pattern

recognition by

_means of _a

V-shaped identifier,

follo1A~ed

by

calculation of

high

order moments in the contrast distribution. The

procedure

_can be

applied

directly

to the

picture,

_or to _an

isotropic

wai,elet

component

^of it.

The nlethod _was

successfully

tested _on

s1nlple exanlples,

such _{as a} collection of

s1nlple

V-

shapes

_{or a} mathematical

snowflake,

and sho1A>n to

produce angle

values in full agreement ^with the

expected

_ones.

ive then

applied

the

procedure

to

physical

fractals with _{no a}

priori knowledge

of the branch- mg

angles

therein. In _a noise reduced WS

aggregate

we found _a

conspicuous signal

at 9

r-

70°,

which _{was moreover} found

independent (robust)

of the

particular

version of the

procedure.

Conversely,

we found _no definite robust

signal

in the basic WS

aggregate.

^{This result should}

not be taken _{as a} failure to detect _a weak

signal.

Dur statement that there is _no dominant

angle

in WS

aggregate

is _a

significant result,

which is _to be understood in the context of pattern

recognition.

This however ares _{net mean} that _we

definitely

rule out the idea of _a dominant

(36°) _[17j angle

_in such

aggregates.

^In

fact,

_a look _at

Figure

4 shows that the

isotropic

WT components of the fractal

figure

_may be viewed _{as a} collection of

spots,

say, "atoms". In the

snowflake,

these atoms _are

obviously

ordered in _{a square} lattice. In the noise reduced

aggregate they

are

dosely packed

in well defined _rows. There _{is no} such _a well defined local order in the basic WS

aggregate

^and

accordingly

no obvious _way of

defining

local directions in the

picture.

These local directions _are what _{one may} term

"branches",

and the collection of these branches is the fractal

skeleton,

at the scale of the wavelet used. We believe that the

recognition

of

significant angles

in _{a non}

elementary

fractal first demands

elaborating

_a

skeleton,

which is _a

highly

nonlinear

operation

and then

applying

the

recognition procedure

to the skeleton. We

are now

currently investigating

diiferent _ways of

building

_a skeleton: this _may be

simply

^the collection of crestlines _{in one} WT

component

or in _{a more}

sophisticated

version be deduced from the

proliferation

of "atoms" when the wavelet scale is decreased.

Acknowledgments

This research was

supported by

the Direction des Recherches

Études

_et

Techniques

under

contract

(DRET n°92/097).

We would like to thank J-F-

Muzy

and A. Améodo for

interesting

discussions. We _are very

grateful

to M. Tabard for his technical assistance.

References

Ill

^Feder

J., Fractals, (Plenum Press,

New

York, 1988).

[2]

Halsey T.C.,

Jensen

M.H.,

Kadanoif

L.P.,

Procaccia I. and Shraiman

B-I-, Phys.

Re~. A 33

(1986)

1141-1151.

[3] Vicsek

T.,

Fractal Growth

Phenomena,

Second Edition

(World

Scientific

Publishing,

Sin- gapore,

1992).

[4] Witten T.A. and Sander

L.M., Phys.

Re~. B 27

(1983)

5686-5697.

[Si

Argoul F.,

Arnéodo

A.,

Grasseau G. and

Swinney Hi., Phys.

Re~. Lett. 61

(1988)

2558- 256l.

[fil Li

G.,

^Sander L.M- and Meakin

P., Phys.

Re~. Lett. 63

(1989)

1322.

[7] Ossadnik P.,

Phys.

Re~. A 45

(1992)

1058-1066.

[8] Arnéodo

A., Argoul F., Muzy

J-F- and Tabard

M., Physica

A 188

(1992)

217-242.

[9] Arnéodo

A., Argoul F., Muzy

J-F- and Tabard

M., Phys.

Lett. A 171

(1992)

31-3G.

[loi

Procaccia I. and Zeitak

R., Phys.

Re~. Lett. 60

(1988)

2511-2514.

[Il]

Vander

Lugt A.,

_in

Optical

and

Electro-Optical

Information

Processing,

J.

Tipped

et

ai.,

Eds

(MIT

_press,

Cambridge, Mass., 1965).

[12] Argoul F.,

Arnéodo

A., Elezgaray J.,

Grasseau G. and Murenzi

R., Phys.

Re~. A 41

(1990)

5537-5560.

[13] Muzy J-F-, Bacry

E. and Arnéodo

A.,

International Journal

of Bifiircation

and Chaos 4

(1994)

245-302.

[14] Muzy J-F-, Bacry

E. and Arnéodo

A., Phys.

Re~. E 47

(1993)

875-884.

N°3 OBJECTIVE DETERMINATION OF BRANCHING ANGLES IN FRACTALS 441

ils]

Arnéodo

A., Argoul F., Muzy J-F-, Pouligny

B. and

Freysz E.,

The

optical

wavelet

transform,

"Wavelet and Their

Applications",

M.S.

Ruskai,

G.

Beyklin,

R.

Coifman,

I.

Daubechies,

S.

Mallat,

Y.

Meyer

and L.

Raphael,

Eds.

(Jones

and

Bartlett, Boston, 1992)

241-273.

[16] Pouligny B.,

Gabriel

G., Muzy J-F-,

Arnéodo

A., Argoul

A. and

Freysz E.,

J.

Appt. Cryst.

24

(1991)

526-530.

[17]

^Arnéodo

A., Argoul F., Bacry E., Muzy

J-F- and Tabard

M., Phys.

Re~. Lett. 68

(1992)

3456-3459.