Detection of particles and estimation of size distribution in process fluids

Rochester Institute of Technology

RIT Scholar Works

Theses

Thesis/Dissertation Collections

10-5-1992

Detection of particles and estimation of size

distribution in process fluids

Bo Li

Follow this and additional works at:

http://scholarworks.rit.edu/theses

MASTER'S THESIS

DETECTION OF PARTICLES AND ESTIMATION

OF SIZE DISTRIBUTION IN PROCESS FLUIDS

by

BoLi

B.S. QingHua University, China

(1986)

A thesis submitted in partial fulfillment of the

requirements for the degree of Master of Science

in the Center for Imaging Science

in the College of Imaging Arts and Sciences

of the Rochester Institute of Technology

September 1992

Signature of the Author:

_

Accepted by:

_

CENTER FOR IMAGING SCIENCE

COLLEGE OF IMAGING ARTS AND SCIENCES

ROCHESTER INSTITUTE OF TECHNOLOGY

ROCHESTER, NEW YORK

CERTIFICATE OF APPROVAL

DETECTION OF PARTICLES AND ESTIMATION

OF SIZE DISTRIBUTION IN PROCESS FLUIDS

The M.S. Degree Thesis of Bo Li

has been examined and approved by the

thesis committee as satisfactory for the

thesis requirement for the

Master of Science degree

Dr. E. Dougherty (Thesis Advisor)

Prof.

R.

Waag, University of.Rochester

Dr. Mehdi Vaez-Iravani, R.I.T.

THESIS RELEASE PERMISION FORM

ROCHESTER INSTITUTE OF TECHNOLOGY

COLLEGE OF IMAGING ARTS AND SCIENCES

CENTER FOR IMAGING SCIENCE

DETECTION OF PARTICLES AND ESTIMATION

OF SIZE DISTRIBUTION IN PROCESS FLUIDS

I. Bo Li. hereby grant pennission to the Wallace Memorial Library of R.I.T. to reproduce

my thesis in whole or in part. Any reproduction will not be for commercial use or profit.

Signature:_ _

~

_

Date:

¥_L..--_·_~

__

/_/f:_~_~

_

Abstract

Process fluids_maybe degraded

by

largeconcentrations of suspendedparticles,formations

of gel _slugs, or the infiltration of air bubbles. The scattering of ultrasound is dependent upon the relation of the wavelength and the

scatterers'

dimensions.

Relatively

large objects are_easily detected becauseoftheir_{strong scattering in} theshortwavelength limit

Acknowledgments

The project has been supported

by

Eastman Kodak

Company

(Kodak). Some previous

work, which was also supported

by

Kodak,

in ultrasonic visualization of process fluids

was carried out

by

K.

Udy

andProf. R.

Waag

at

University

ofRochester. The ultrasonic

imager,

ACUSON

128,

is an instrument of the ultrasonic research lab in

University

of

Rochester. All algorithms have beenperformedon the _{digital image processing}systemat

analytical

imaging

systems_{group in Eastman Kodak Company.}

I wouldliketo thankRobert

Kraus,

the ultrasoundproject

leader,

Steve

Clyde,

at

Kodak,

andMs. L. Hinkelman at

University

ofRochesterfortheir tremendous efforts and

help

in the

following

aspects:

*)

preparation of polystyrenelatex sphereparticles and processfluids.

*)

setting up a_{fluid circulating}system witha connected ultrasonictransducer.

*)

mountingthe transducertoan ultrasonicimagerand_recordingultrasonicimages

ofparticlesinprocessfluids.

I would also like to thank Prof. R.

Waag

for his advice in ultrasonic

theory

and

experiments.

Finally,

Iwishto_particularlythankR. Krausagainfor hisconsistent and great support and

Dedication

This thesis is dedicatedto_{my wife,}Julie andour

lovely

newborn, Jennifer. Shewas born

Tableof

Contents

Abstract

iv

Tableof

Contents

_vii

ListofFigures ix

ListofTables x

1.0 Introduction 1

1.1 Ultrasonic

Imaging

4

1.1.1

Scattering

Theory

₄

1.1.2 B-Scan ExperimentFundamentals 7

1.2 SpatialandTemporal

Differencing

and

Filtering

1 1

1.3Particle Recognition 13

1.3.1

Spatially Varying

Thresholdand

Binary

ImageMeasurements 13

1.3.2The MorphologicalSegmentationon

Touching

Particles 17

1.4Statistical RegressiDn 19

1.4.1 Least-Squarer Regression 19

1.4.2Least-Squares Regression for Normal Distribution 21

1.4.3Least-Squares Regression for Exponential Function 23

2.0 StatementofWork 24

2.1 Process Fluid PreparationandParticle Distribution Selection 24

2.2

Circulating

System,

Ultrasonic

Imager,

andImageAcquisition 27

2.3 Image

Analysis

and

Processing

_System 29 2.3.1 Image

Digitization,

ROI

Segmentation,

andNoise Suppression 29

2.3.2 Regional

Scatterers

ClassificationandSegmentation 31

2.3.3 ObservedParticle Area Distribution fromDispersion 36

2.4

Stability

ofObservationData Distribution 38

2.5 Observed Particle Distributionsfrom All Fluids 40

3.0 AnalysisofResults 44

3. 1 Regression andEstimation 44

3.1.1 Regression for Original Particle Distribution 44

3.1.2 Regression forObservedParticle Distribution 47

3.1.3 EstimationofSystem Parameter from Each Fluid 56

3. 1.4Invariance oftheSystem Parameter 60

3.2 EstimationofInput ParticleSize Distribution 63

4.0

Summary

andConclusions 65

5.0 Appendix The DerivationofPartialDerivatives 67

List

of

Figures

Fig. 1.1 _{The relationship between}acoustic wavelength

(X)

and particle radius

(r)

4

Fig. 1.2Schematicdecibelresponse of

P,/Pe

vs.

log(r)

7

Fig. 1.3 Anultrasonicimageofparticlesinfluid 9

Fig. 1.4Spatial

differencing

algorithm structurefor_a3x3_window 12

Fig. 1.5

Gray

level ROIandthespatialdifferential image 14

Fig. 1.6Thresholdarea andthresholdfunction t(v, g,)at

-6,

0,

6 dB 15

Fig. 1.7 The

binary

ultrasonicimageof

Fig

1.5 15

Fig. 1.8

(1)

Convexobject

(2)

Non-convexobject

(3)

Convex hullof

(2)

16

Fig. 1.9Anexample of

binary

watershed segmentationalgorithm 1 8

Fig. 1.10Curvesof errors

during

_minimizingeachpartialderivative 22

Fig. 2. 1 Size distributionoforiginal particlesvs.radius

(r)

26

Fig.2.2

Circulating

fluidconfiguration

[15]

27

Fig.2.3Thewhole ultrasonicimagedisplayedonAcuson 128 28

Fig.2.4

(a)

Differential Image (top),

(b)

Binary

Image

(bottom)

3 1

Fig. 2.5

(a)

Ultimateerosion setofparticlesin

binary

image

(Top)

(b)

Particles after watershed segmentation

(Bottom)

34

Fig. 2.6 Final Layoutofthe outputimage

(Top)

Original imagewindow

(Middle)

Differentialimage.

(Bottom)

Finalsegmented particles

(Upper-Right)

Statusoftheimager

(Lower-Right)

Statisticalresults 35

Fig. 2.7 Anormalizedobservationdata distribution 37

Fig._{2.8 Output distribution in different sampling}periods

(1)

20seconds.

(2)

30seconds.

(3)

40seconds.

(4)

All

(2)-(4)

38

Fig. 2.9 Observed distributions from

dispersion,

gel,and_{emulsion,respectively} 41

Fig. 3.1 (a). Original particledistribution vs.

log(r)

(b).

(a)

anditsnormaldistributionregression. 45

Fig. 3.2 Anormalized observationparticledistribution in dispersion 47

Fig. 3.3Observedparticledistributions in dispersionandtheirexponential regression 53

Fig. 3.4 Observedparticledistributions ingel andtheirexponential regression 54

Fig. 3.5Observedparticledistributions inemulsion andtheirexponential regression 55

Fig. 3.6Systemconfigurationand particledistributionatinputand output ₅₇

List

of

Tables

Table 1. Table 1. Measured ParticleRadius Probabilities 25

Table 2. DataofBlobs Before Segmentation 32

Table 3. DataofBlobs (i.e

Particles)

AfterSegmentation 33

Table 4. Probabilitiesof

log(r)

45

Table 5.q's obtainedfrom dispersion withdifferentconcentrationsandsignalgains 51 Table 6. q's obtainedfromgel withdifferent fluidconcentrations and signalgains 5 1 Table 7.q's obtainedfromemulsionwithdifferent fluidconcentrations andsignalgains52 Table8.q'swith minimumstandarddeviation

oq

in Table 5-7 respectively 52 Table9. Systemparameterc's obtainedfrom Table8

by

_applying

(36)

60

1.0 Introduction

Processfluids_{may be}degraded

by

largeconcentrations of suspendedparticles,formations

of gel _slugs, or the infiltration of air bubbles. Occurrences of these anomalies are

unpredictable and aredifficulttoobserveifthefluidis

flowing

through anopaque tubeor

is sensitivetolight. Onemethod of

detecting

undesirableprocess conditionsistodrawand

analyzefluidsamples.

However,

thismethod_{may disrupt}thesystem eachtimea sampleis

taken.

Also,

the results ofthe analysis _may not be representative of transient conditions

that _{may have} existed beforeor occurred afterthe sample was taken. A methodwhich is

noninvasive and which allows particle suspensions in process fluids to be monitored in

realtimeisthereforedesirable.

Mecocci hasproposedaPC-basedsystem tomonitor transparentfluid film [01]. Thesys

temis usedtoreplacehumanfilm

inspectors,

butthis system wasdesignedtodetect large

objects (size in cm) and assumed no particle _overlap

during

detection.

Also,

it did not

attempt to estimate particle distributions in process fluid. This needs statistical analysis

fromobservationdata.

In the past, acoustic waves have proven useful in the nondestructive evaluation of

materials [02]. Since these waves do not cause ionization and because

they

allow investigation inreal_time,without

disturbing

thephysical systemin_question,

they

arewell

suitedforthe_studyof_{fluids moving}throughflowtubes.

contami-nants also suggest the use of acoustic waves in

detecting

and_{characterizing} anomalous

conditions.

Process

fluids which contain particles have _scattering characteristics that

dependon particle_{size, concentration,}anddistribution.

Thus,

by

_observingchanges inthe

ultrasonicimages received, undesirable conditions_{may be determined in} the fluidon line

andinrealtime.

According

toacoustic _theory,

long

wavelength ultrasound is scattered_weakly

by

objects

and short wavelength ultrasound is scattered_strongly

by

objects. So particles are hardto

detect in theregion ofthe

long

wavelength limit and_{easily detected in} the region ofthe

short wavelength limit. A

long

wavelength limit

(LWL)

region is where the ultrasonic

wavelength is much greaterthan theparticle sizetobe

detected;

a shortwavelength limit

(SWL)

region is where the ultrasonic wavelengthismuch less than theparticlesize tobe

detected. Since the _scatteringcharacteristics of ultrasound are wellknown in the regions

ofboth LWLandSWL

[03],

weare_goingtoinvestigatetheproperties of particledistribu

tion in the ultrasonic image in the intermediate region where ultrasonic wavelength is

comparable withthe size ofparticles.

Hence,

_{it is very important}toinvestigate theinter

mediate region ofultrasound sothatultrasonicdetectioncan be applied

fully,

_extensively,

andthoroughly.

arerecorded

by

aVCR. A _variety of ultrasonic _{operating frequencies} areavailable, but 5

MHzisselectedformost oftheexperimental work.

After the ultrasonic images have been recorded, a sequence ofimage _processing tech

niques isused toextractinformation.These techniquesinclude

digitization,

classification,

and a number of morphological operations.

Following

the ultrasonic

imaging,

real-time

image processing, and automatic data gathering, an exponential regression model forthe

observed particledistribution is developedand astatisticalestimatoris derivedtoestimate

the variance oftheoriginal particle size distribution fromthe observeddata via a system

parameter._{While it is certainly true that the} system parameterdepends ontheparticles of

the ultrasonic

imaging

and _{image processing employed,} a

key

aspect ofthe _{study is} to

show _that,withinan acceptablelevel_{variability, the} systemparameteris invariantrelative

to the various process fluids of

interest,

_namely,

dispersion,

_gel, emulsion. Invariance is

statistically demonstrated

by

using fixed-effectsanalysis of varianceto showthat thenull

hypothesis of system-parameter_equalityoverthe threefluids is accepted atthe 0.05 level

of significance. Invariance

having

been

demonstrated,

thefinalvalue ofthe system param

eteristaken to be theaverageofthe threevalues fromthe different fluids. Both the theo

retical basis ofthis approach andexperimental methods are describedand theresults are

1.1

Ultrasonic

Imaging

1.1.1

Scattering

Theory

When sound travelsin a_medium, it is reflectedor scattered atthe interfaces ofdifferent

kinds of materialsitencounters._{Rayleigh scattering}

(long

wavelength

limit)

occurs when

the size of theobject ultrasound "illuminates" is much smallerthan the ultrasonic wave

length,

and specular reflection (short wavelength

limit)

occurs when the object size is

much greater than the wavelength. Fig. 1.1 illustrates the relationships between acoustic

wavelength

(X)

and particle radius (r).

Ultrasonic source

(

aperature=_a

)

Pe:emitted signal

Pr.returned signal

Sphereparticles

(

radius=_r

)

Ifweletk=

2nfk,

_{where k is}

called wave_number,then

kr 1 => Rayleigh

Scattering

kr\ => Specular Reflection

The amount of returned echo of _scattering or reflection is measured

by

the ratio

PfJPe,

where

Pe

is the power emitted

by

the transducerwith aperture a (Fig.

1.1)

and

Pr

is the

received echo power. That

is,

Pr

is thereturned powerafterit has propagatedthrough the

medium andbeen reflected at objectinterfaces.

The Kirchoff approximation

[03]

gives the result of

PrlPe

for reflection from a large

sphere whenkr 1:

pr

rV

k

=

~a?

(1)

where T is the reflectioncoefficient andR is the distance fromtheacoustic source to the

object.

Also,

a simplified approximation on particles like small rigid spheres under some addi

tionalassumptions is foundwhenkr 1:

Pr

25k4"6

The decibel _{is generally used as the unit of the ratio}

Pr

_{IPe. When} we focus only on the

decibelrelationships between

Pr

IPe

and particle radiusr

by

_takingthelogarithmofboth

sides,thenfunctions

(1)

and

(2)

become

log

U-

|

-log

{/)

=

21og

(r)

jfcr 1.

(3)

logffflO-logtr6)

=

61og(r)

kr\.

(4)

Therestrictionsof_{kr here carry} theabove approximations.Fig. 1.2gives thedecibelecho

response curves vs.

logir,

ofdifferentmaterials. Notethat the regionofkr near 1 cannot

be plottedbecauseapproximationswerederivedundereither(kr

1)

or(kr 1).

It can be seen that these curves are all linear with different slopes and offsets in the

Rayleigh scattering region and the specular reflection region (the linear curve is from

log(PrIPe)

vs.

log(r)

). The linear relation is obtained from the approximation ofeither

(kr

1)

or (k r 1). Butthe _theorydoes not give usa simple linearapproximation in

log(P/Pe)

Fig. 1.2 SchematicDecibel Responseof

P/Pe

vs.

log(r)

(1)

Gel-PSL

(2)

Dispersion-PSL

(3)

Emulsion-PSL

log(r)

1.1.2 B-Scan Experiment Fundamentals

The ultrasonic imager used in this experiment is based on the pulse-echo and B-scan

principle[04].

According

to thepulse-echo_principle,atransducer sends an acoustic pulse

throughsometransmissionmedium.Thepulseis_subsequentlyreflected

by

discontinuities

in the medium and later received

by

the same transducer. The locations ofthe acoustic

interfaces in the fluid are determined from the times oftravel and the speed of sound. A

linear array of transducer elements can be used to implement the B-scan principle in

are then positioned on a

display

_according to their times of arrival and transducer posi

tions. With a lineararray, across-sectional image isobtained

by

_{sequentially pulsing} the

element inthe array.Echosignal amplitudes aredisplayed in araster.

Repeating

this pro cess _rapidlycomparedto themovements of echoes

being

imagedproduces a sequence of

B-scans which shows motion and _{may be} recorded on video tape.

Actually,

each ultra

sonic beamisgenerated

by

a_groupof elementswhich arefiredwith small timedifference

so that thebeamcouldfocus well at a certaindepth

([13],

[14]).

In the experimental system employed in this study, particles or other anomalies which

have acoustic impedance properties that differ from those ofthe fluid form acoustic dis

continuities. Itis fromthediscontinuitiesthat theultrasonic signalisreflectedor scattered.

The B-scanimagethencontainsinformationaboutallscatterersinthecross-section ofthe

medium through which the acoustic signals are sent.

Thus,

with thecombination ofthe

pulse-echo and B-scan principles, the task of

displaying

atwo-dimensional section ofa

medium and_monitoringthe contentsoftheprocess fluidas it flows is accomplished. An

ultrasonicimageof particles inprocessfluidis shownin

Fig

1.3.

Sinceparticles arerepresented

by

echoesintheultrasonic

image,

wefindthatnot all parti cles in fluids are resolved. Although particles are not _physically

touching

in

fluid,

their

sometime that some echoes are not independentin theultrasonic images. Some ofthese

unresolved echoes couldbesegmented

by

_{image processing}techniques.

Since theechoes ofparticles, not particles _themselves, are seen in theultrasonic

images,

we can _only process these echoes to obtain their information such as area,

location,

and

perimeters. But forsimplicity, we use theword particlesto representtheechoes ofparti

clesintheultrasonicimageinthispaper.For

instance,

aparticlearea meansthearea ofthe

particle's echo in

image,

not the actual one; and

touching

or multi-particles mean some

echoes of particleswhichcannotberesolved

by

the B-scanand theseechoes are not sepa

rateintheultrasonic image.

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It is importanttonotethataparticle mustbe centeredinthe ultrasonicbeam toobtain the

maximum echo ofreflection. Iftheparticle_{is only partially in}the

beam,

ormoves out ofit

altogether, reduction or complete loss of the return signal will occur. Some changes of

appearance ofthe scatterers in the sequential ultrasonic images can be explained

by

the

relative positions ofthe scatterersandthecenter oftheultrasonicbeam.

While moving particles i.i fluids can be detected

by

a B-scan imager and recorded on

video_{tape, image processing} techniquesare neededtoextractinformationfromthevideo

tape. The_{first step in} the_{image processing} sequenceis to_{find only moving}particles ina

digitized sequence of videoimages. Theultrasonicimagesconsist not_onlyof_movingpar

ticles, but also of a backgroundwhich changes in terms of signalgain ofthe imager and

slow variation of the loss properties of the _{fluids. It is necessary} to eliminate the back

ground inordertoimprove the signalto noise ratio ofthe imagedparticles. An approach

1.2 Spatial

and

Temporal

Differencing

_and

Filtering

Todetect a_movingtarget.Rauchand othershaveproposedtheuse of temporaldifferential

filters to detect changes in the image [05].

Unfortunately,

most algorithms designed to

detect changes caused

by

_moving targets also detect changes introduced

by

background

motion. Pattersonand others haveevaluated severalnewalgorithms andproposedthebest

solution _amongthem

[06]

The spatial differential algorithm is a nonlinear three-dimensional filter which is essen

tially

anadhocextensionofan algorithmfromthefieldofpattern_{recognition, the}Knear

est neighbor classifier[071. The spatial differential algorithm works on atime sequential

set of sub-windows ofthe spatialimage in the samefashion asthe spatialtemporal _adap

tivefilter (Fig. 1.4). Thealgorithmtakesthecenter pixelin awindowin thecurrentimage

frame and subtracts all the pixels in the window fromprevious frame. Theoutput ofthe

filter isthe magnitudeofthe smallestdifference. This hastheeffect of_{cancelling features}

in the images which are correlated both spatially andtemporally. Theoutput

image, Gn,

froman NxNspatial

differencing

filterwhenappliedto themxmimage

Fn

at_{time n,}is

G

=min

{

I

Fn>

tj

-Fn.lf

i+kJ+i I

}

(5)

where

i, j,

k, I,

n areintegors and

k,

h N

N'

~2'

2.

,

i,je mxm

(6)

A logical extension ofthis algorithmisto use boththe_precedingand

following

framesto

providedata forthealgorithm. This double-sided

differencing

subtractsthecenter pixelin

thewindow from allthepixelsinboththe_precedingand

following

framesand outputsthe

magnitude ofthe smallestdifference.Tests haveshownthat thisalgorithmexhibitsperfor

mance whichissuperiortoother algorithms_{[05]. This idea is going}tobeappliedtoobtain

differential images witheliminatedbackgroundand suppressed noise.

'n-1,

i-1,j-l

"n-l,i-l,j

Fn-l,i-l,j+l

'n-1,

i,j-l

Fn-l,i,j

Fn-1,

i,j+l

"n-1,

i+lj-l

"n-1,

i+l,j

Fn-1,

i+l,j+l

1.3

Particle Recognition

In this section, the methods of_recognizing particles in fluids are _going to be discussed.

Particle classificationis made on

binary

images obtained

by

_applying a _{spatially varying}

thresholdfunctionon _{gray level differential images.}

1.3.1

Spatially Varying

Thresholdand

Binary

Image Measurements

Afterthe differentialultrasonic imageshave been obtained, a_{spatially varying} threshold

function is _going to be applied to obtain

binary

images. Before we _{do that,} we need to

locate the

boundary

oftht fluidarea on the screen thatis theRegion

Of

Interest

(ROI)

to

be processed. It is importantto select a ROI sothat the _quantityofdata isreduced. Inthe

area offluid

(ROI),

a singlethresholddoesnotwork well becausethebackgroundofultra

sonic images isnotuniform,sothat thenoiseisnot_{evenly distributed}throughout theROI.

Fig. 1.5 shows theROIof

Fig

1.3 and adifferential image from it. We findthat theback

groundis brighteratthe _topandbottomthanit is atthecenteroftheflowchannel. Sothe

noise amplitudein thedifferentialimages isnot_{evenly distributed.}

Furthermore,

theback

ground is also affected

by

signal gain. So we have decided to use a threshold function

alongtheverticaldirectionoftheROIandthefunction isalso changedintermsofthe _sig

nal gain.

The thresholdfunction is defined as T - t(v,

g), where v isthe vertical coordinate ofthe

ROIand_{g is} thesignal gain(whichisone of-12, -6,

0,

6,

12 dB). Several functionscanbe

usedas the threshold function. For simplicity andfast_operation, the ROI is divided into

threerectangularranges and asinglethresholdvalueisappliedin each region. Fig. 1.6 illustrates

threeofthethreshold

functions

at-6 dB 0dB _and6

dB,

respectively.Wemake_upa

look-up

table for both the sizes oftheranges (heights_wl,w2, w3in Fig.

1.6)

andtheir threshold values under

each signal gain.

Thus,

thethresholdfunctiont(v, g)isassociatedwiththeROIvertical coordinate

and signal gains.

Fig

1.7 isa

binary

image resulting fromthethresholdfunctiononthedifferentialimagein

Fig

1.5.

0

Threshold

Area inROIat6 dB

t

(v,

_g)

wl

V

wl

w>2

vv3

w2 w3

6dB

OdB

-6dB

Fig

1.6 Thresholdarea andthreshold_{function t(v,} g)at_-6,0and6 dB

Fig. 1.7 The

binary

ultrasonicimageof

Fig

1.5

Afterthe

binary

image has beenobtained,measurementof

binary

objects needstobereal

ized. We use blobs to represent _any continuous object

(echo)

areas in the

binary

image

(Fig. 1.7). Some important measurement parameters are the area

(i.e.,

pixel _number) of

each

blob,

number_of

blobs

_inthe

image,

_andthe

boundary

ofeachblob. Wehaveselected

an algorithm which issupported

by

hardware

toaccomplishthemeasurements.

An object doesn't always representan independentparticle. Itis only an isolated

binary

blob in the

image.

Sometimes,

ifparticles are _touching, or _{overlapping, the} object looks

likemulti-particlesinthe

binary

image.

Thus,

wefindthatparticlesdetected

by

ultrasound

can be classified into two categories: independentparticles andmulti-particles

touching

each other.These twocategories are_goingtobeprocessedin differentways. Todoso,we

apply

binary

morphologicalimage_processing

[8, 9,

17].

Since anindependentparticle shouldhavean_essentiallyconvex shape and a multi-particle

object a nonconvex shape in the

binary

image,

the two categories ofparticlescan be dis

tinguished

by

the difference between the perimeters of an object and its convex hull.

Mathematically,

a convex areais described asan areawhichcannotbecutthrough

by

_any

tangentofit. Fig. 1.8 illustratestheseconcepts.

(1)

(2)

(3)

1.3.2 The Morphological

Segmentation

on

Touching

Particles

Afterallblobs areclassifiedinto independentparticles and

touching

ones, analgorithmis

usedtosegment

touching

particlesinordertoobtaintheareaofeachindividualparticlein

the

binary

image. The particular algorithm we _apply is the

binary

watershed, which is

composed of a number of morphological operations. Its action is to segment _touching

(overlapping)

binary

particles in such a _way as to maintainthe

integrity

ofeach particle.

There aretwo main steps when _applyingthe watershed algorithm. The first isto findthe

ultimate erosion set ofthe

touching

particle sothat the pseudo-center ofeach particle in

the

touching

areacanbe addressed.Thesecondisto_{build up}the_{divide lines among}these

pseudo-centers inside the

boundary

of _touching blob. These divide lines segment the

touching

particles. (Fordetailsofthe

binary

_watershed,see

[10],

[12]).

Fig.

1.9(a)

shows an example of_overlappingparticles andFig.

1.9(b)

illustrates the ulti

mate erosion set in

determining

all pseudo-centersin _touchingparticles. Fig.

1.9(c)

gives

the procedures of _{setting up} watershed divide lines in separating the

touching

particles.

These figures arefrom [10].

Afterthe watershed segmentation algorithm is appliedon the

binary

image,

all

touching

particles are _properly segmented. Then we measure the area ofeach individual particle.

Those areas compose the observed particle sizedistribution which is going to be statisti

callyanalyzed.Themethodsare_going tobe discussed innextSection.

(a)

A

touching

blob

(b)

Ultimateerosionset of

(a)

(c)

Proceduresof

building

_upthedivide lines

1.4

Statistical Estimation

1.4.1

Least-Squares Regression

The principle ofleast-squares errors is to minimize (with respect to some criterion) the

vertical distances oftheobservationdata setto thefittedmodel. Theresults are the

least-squares estimatorsforthemodel parameters.

Using

theproperties ofthese estimators,itis possibletomakeinferences_regardingtheparametersinthe_underlyingmodel [11].

Intheclassicaldeterministic setting, adependentvariable_{y is}consideredto beafunction

y(x) of an independent variable x

if,

given _{x, there} exists an exact _{corresponding} value y = _{y(x). The independentvariablex will continue tobe deterministic but thedependent}

variable

Y,

called a response variable,is consideredto be random.

Consequently,

given_x,

thebestthatcanbe done istoarrive at some prediction ofY. Inpractice, toaccomplish the

estimation,observations will be made atnpoints,x\,X2, ....xn, each

x,-being

knownas a

regressor orpredictor variable. The resultis adata set _consisting of n points ofthe form

(xi,

yi),each

y,-being

a sample value oftherandomvariableFix,-.

If_yi = _yfxf), i=

0, 1,

..., n,is theobservationdata set and

ffri)

is used to estimatey(xj) in

thesamedomainof_x,wedefine

e(xi)=_y(xi)-f(xi)

(7)

as theerroroftheestimation at point jc,-. The least-squares estimates are selected so as to

minimizethe sum ofthesquares ofthese residuals.The sumis calledthe sum of squares error

(SSE)

andis given

by

SSE=

Xe/

=

S

(?(*i>

-/<*/)

)2

<8>

i=l i=l

Minimization ofSSE results in a sample regressionmodel

f(x)

that best fits theobserva

tionaldataset_y(xi),

i=l, 2,

...n.

To accomplish the minimization of

SSE,

partial derivatives ofSSE are investigated. Let

the estimation

f(x{)

be a function with m parameters: _Pj , P2 >Pm-

Then/fx,)

can be expressed in anotherform:/(x,-

)

=

f(X( \p\

,P2> >Pm ) ^e parameters determine the

function

f(x)

atthepoint_x,,i=

1, 2,

..., n. All

(X,-,

y,jareknownfrom observationdataset,

only pi ,P2, .-,pmaretobeestimated.

Treating

SSEas afunctionof variables_Pi ,pi, ...,pm,minimization canbe accomplished

by

taking

partialderivativesand_settingthemequaltozero.

-SSE

= _-2

Y

[

(y

(x.)

-f(xi))^-f(xi) = 0 7=1,2, ...,m.

(9)

1.4.2

Least-Squares

Regression forNormal Distribution

If

f(x)

isa_verticallyscaled

Gaussian,

then therearetwoparametersin

(*-n)2

fix)

=/(*|u.,o) = _-jL-e 22

(10)

V27XO

f(x)

is a_probability

density

_functionwith mean(i andvariance a2.

By

putting

(10)

into

(9),

the two partial derivatives of normal distribution are (see

Appendix for derivation):

^SSE=

^xT(l)-^xT(0)

(11)

-SSE

=

*xT(2)-^xT(l)+l

r^-i|xr(0)

(12)

where

T(w)

=

[ (/(x,)

-v,)/^.^] w=0,l,2.

(13)

.= i

It is not apparenttoresolve themathematical solutions of[Iand afromthese two partial

derivative

functions; however,

our goalistomake them closetozero

by

_adjusting|iando

so that the minimization can be practically reached under this approximation. Several

numeral approachesare availablein [16].

3000

2500

6 2000 ^1500 w

w ₁₀₀₀

CO "0 500

80 100

regression index i

0 20 40 60 80 100

regression index i

Squares Sum of Error

20 40 60 80 100

regression index i

Fig

1.10Curvesof errors

during

_{minimizing SSE.}

(a)

partialderivativeon[i

(top)

(b)

partialderivativeona

(middle)

(c)

CurveofSSE

(bottom)

Thepracticalrealizationoftheminimizationis done

by

the

following

steps:

1)

settheinitialvalues of u, anda_tof(x)fromsome ofthe

(x,-,

y,J.

2)

compute thetwopartialderivativesandtheirsigns.

3)

adjusttheseparameterstodecreasetheabsolutevalues ofthe twopartialderivatives

achieved.

Inthe_step

(3),

since alltwopartialderivatives are in directproportionto u and_a,respec

tively

(see

Appendix),

the values of the partial derivatives can be increased

(decreased)

gradually

by

adding

(reducing)

u. ando,respectively.Whentheupdated partialderivatives

are_relativelycloseenough_{to zero, the}numeralminimization approximationisconsidered

to be reached and these (i and o are used as the parameters for/fx).

Fig

1.10 giveserror

curves ofthesepartialderivativesofSSE.

1.4.3 Least-SquaresRegression for Exponential Function

Ifthe function to be regressed is an exponential

density

function,

_only oneparameter is

goingtobe determined. Let

f(x)

= ae-ax

a>0

(14)

whereaneedstobe found

by

theregression.FromthepartialderivativeexpressionofSSE

in

(9),

thepartialderivativeof ais (see Appendix for

derivation)

3-SSE=-xT(0)-2xT(l)

(15)

da a

T(w),

w=

0,

1,

is defined inEquation (13). Themethod ofminimizing SSE isthe same as

theone we statedintheprevious section.

2.0

Statement

of

Work

Since the approximations of

long

and shortwavelength limits have beengiven as linear response curves of

log(P^Pe)

vs.

log(r)

(see Eq.

(3), (4)

andFig.

1.2),

the_scatteringchar

acteristics of particles _may be well understoodin these tworegions. The experiment has beendesigned to findthe _scattering characteristics of particles intheintermediateregion

wheretheultrasound wavelengthisnearthe object size(radius).

To study the characteristics in this region with a _variety ofindustrial process

fluids,

we

haveselected a particle distributionwitha radius mean

(|ir)

close to theultrasound wave length

(X,)

and have measured the observed particle size distribution obtained from the computer.

By

_analyzingthedistributionofobserved

data,

we are not_only

finding

the_way

of _estimating the original distribution but also

determining

the parameters which can describethe_scattering characteristicsinthisintermediateregion.

2.1 Process Fluid

Preparation

and

Particle Radius

Distribution

Selection

In the film

industry,

_{many kinds} ofprocess fluids can be selected so that the actions of ultrasound on themcan be understood. Some industrial fluids have beenmadein orderto covercertain _variety offluids in real circumstances. Polystyrene latex

(PSL)

is selected

hereas theparticle materialin fluids.

Dispersion,

gel,and emulsion are used asthefluids.

The particle concentration sequence is

5.0, 10.0,

15.0,

20.0

(ppm)

in the three

fluids,

incrementtoexaminethe effectsof signal gain onthe ultrasoundimage.

Since 5 MHzis chosen as besttrade-offbetweenpenetration and resolution inultrasound

detection forthis _{experimentation, it is used as the}

frequency

_{in the experiment and all}

velocities ofultrasound in the fluids are near 1500 m/sec. The particle mean should be

near48 micronsinorderto

keep

kr=\. This isbecause

k=2nfk _and

X

=

vlf

then

r=I/k=_v/

(2nf)

= ₁₅₀₀

/(

_2*3.14*5*IO6

)

=_47.8micron

(16)

(17)

(18)

where

X

isultrasound_wavelength,risparticle radiusinmeters(IO6_micron),v_{is velocity}

of sound

(m/sec)

and/is the

frequency

ofsound (Hz).

Table 1 gives themeasured probabilities

(percentage)

ofthepolystyrenelatex atthecenter

value of each radius

(r)

measurement _{step (increment). These} values are measured

by

a

device called Full Range Analyzer

(FRA)

before particles are added to the fluids. All

pointsin Table 1 are illustrated in Fig. 2.1.

r

(microns)

26.17 31.12 37.00 44.00 52.33 62.23 74.00 88.00 104.65 124.45

% 0.00 1.97 10.03 25.74 31.48 20.71 8.03 1.88 0.16 0.00

Table 1: Measured Particle Radius

Probabilities

Particle Distribution

vs log(r)

1,5 1.6 1.7 1.8 1.9 2 2.1

log(r)

(r:

particle radius)

2.2

Circulating

System,

Ultrasonic

Imager,

and

Image Acquisition

Tosimulatethe

industrial

process,afluid

circulating

system(Fig.

2.2)

is designedto_pump

the test

fluids

through a custom designed flow through cell for ultrasonic imaging. The

inside diameter

(ID)

ofthe cell is the same as theID ofthe _connecting

hoses,

which are

typicalforprocess use.The attachmentofthe transducer to thecellhasbeen accomplished

in such a _way as to

maximizing

_{transmission of ultrasound into circulating fluidwhile at}

the same time

eliminating

artifacts. Details ofthis _assemblyare not coveredinthe thesis.

Stirrer

cdb

Reservoir

Pump

Chamber.

Transducer

3*_Imager

Row-D

Fig. 2.2

Circulating

Fluid Configuration

[15]

When the systemhas been set_{up, the}detectedparticles

(actually,

they

arethe amplitudes

of returned _echoes) can be displayed on the monitor of the acoustic imager and be

recorded

by

aVCRsimultaneously.Theacousticimagerusedinexperimentsisadiagnos

tic acousticimagercalledAcuson 128 which hasbeen widelyemployed inmedical

diag

nosis.

Fig

2.3 isone oftheimagesobtainedinthisexperiment. Inthe

image,

therearebothxand

yaxis scalesfortheobservation window and a small arrowhead ontheleftsideindicates

thefocusoftheacousticbeamssothat thevaliddepthofultrasound canbe foundvisually.

The current status and settings ofthe imager are shown at the upper-right comer ofthe

image,

such as

date,

_time,

frequency,

signal_power,signal gain

(dB),

etc.

_

:

-TCR

00:53:39:86

>-h.

y

#5-37 18P L55S MP?ri= _fc* 628

384 B 8/3/*

s<r*=

2Z

/V

Ui

2.3 Image Analysis

and

Processing

System

Theacousticimagerconverts thereturned ultrasonic echoestoa video signal whichcanbe

displayedonthemonitor andbeabletobeprocessed_{simultaneously if}theimageprocess

ing

system is connectedto the acousticimager.Forthis experiment, werecordedthe sig

nal on videotapeforoff-line_processingand analysis.

2.3.1 Image

Digitization,

ROI

Segmentation,

andNoise Suppression

So farasthedigital imageacquisitionis concerned,themethodforimageacquisitionis to

digitizethevideo signal _usinga real time imagedigitization board from DATA CUBE Co.

Itsamples a videoimageframe

by

aformatof

512x512x8,

whichmeans512 dotsperscan

line,

512 lines per frame and 8 bits _{(256 grey}

levels)

per pixel.

(Actually,

because ofthe

TV scanning

format,

only 482 linescan beused anddisplayed so wehavetoconsiderthe

image as 512X482 pixels rather than 521X512 pixels.) The digitized images are put in

framememories/buffers(called ROI _store) for furtherprocessing. Oursystemhas 2image

hardware banks andeach bank has 8 frame buffers foreight512X512X8 images.

By

tak

ing

advantage of this _{image processing} system, we can process the video signal in real

timeandtheimage fromeach_{processing step} canbe storedinthe framebuffers.

From the acoustic image in Fig.

2.3,

it is obvious that we need process _onlypart of the

whole

image,

theobservationchannel window.Theultrasonicstatus window atthe

upper-rightcorner,

however,

is meaningfulto ustoo.We callthesewindows Regions Of Interest

measuring the coordinates of all

ROIs,

we cutthese ROIs fromtheoriginal image and re

arrangethem andthe resultantdatainan outputimage. The layout ofthe final image can

beseenlater.

Since all detected particles are in the fluid channel observation window, only this area

shouldbeprocessed.Theideaof_{processing ROIs in}awindowof animage is very impor

tant in _processing sequential images (orvideoimage). Because the frame rate is 30 per

second, which means theframeinterval is 1/30 second,thereis not much time toprocess

strictlysequentialimages ina normal playback speed.

Saving

processtimeon eachframe

is important.

Using

_{hardware processing is}the

key

issue. Current

technology

puts alotof

algorithmsin thehardware.

The averageperiod of each experimentis about a_minute, which includes 30X60frames.

We donot restrict ourselvestoprocess_{exactly frame}

by

frame. We considerit is sufficient

to process each frame in less thana secondin this study. _{The processing}rate dependson

theoperationsinthealgorithm andhardwareconfigurationofthe system.

Upon

digitization,

noise suppression is required in

designing

the motion detection

algorithm. Fromthe videotape, we see that particles are _{moving in} a nonuniformback

groundandthe background becomes strongerwhen the signal gain is increased.

Keeping

track of these _moving particles from the background is important in further processing.

Sec. 1.2 and Sec. 1.3.1 proposed an algorithm which has proven tobe successful in both

motiondetectionand noisesuppression. Theresult of_applyingthisalgorithm_{is illustrated}

Fig. 2.4

(a)

Differential Image (top),

(b)

Binary

Image

(bottom)

2.3.2 RegionalScatterersClassificationandSegmentation

We apply a_{spatially varying}threshold functionon thepure particleimagetoget a

binary

image. _{This image is going} to be measured to get the perimeters of all blobs. Because

these blobs areeither independentor _touching, we use the

following

procedures to _seg

mentthosemultiparticles

first,

andthenobtain allindividualparticlesthereafter:

1)

Findallisolated blobs intheimageandlabelthem

2)

Measuretheimageto getperimetersand equivalent convexperimetersof eachblob.

3) Classify

independentand

touching

particles

by

theratio oftwoperimeters.

4)

Findthearea of each

independent

_particle.

5)

Segment

touching

particles

by

watershed algorithm andthencountthese segmented

particleareasrespectively.

Blob# _P

ecp ecp/p area

(pixels)

0 8.000 7.410 0.9268 4

1 6.000 1.000 0.1667 2

2 29.69 28.14 0.9478 53

3 19.02 18.05 0.9488 22

4 15.85 15.95 1.0070 14

5 15.39 15.24 0.9900 13

6 149.6 101.6 0.6791 504

7 22.95 22.53 0.9815 29

8 11.39 9.650 0.8472 7

9 15.39 15.24 0.9900 13

10 10.78 9.830 0.9116 7

11 15.56 14.48 0.9304 16

12 16.00 16.00 1.0000 15

13 30.52 28.31 0.9274 41

14 36.21 35.54 0.9815 67

15 28.12 28.42 1.0110 50

Table 2: Dataofblobs

(i.e.particles)

beforesegmentation

p: perimeterofblob

ecp: equivalent convex-hullperimeter

Blob# _{P ecp ecp/p} area

(pixels)

0 8.000 7.410 0.9268 4

1 6.000 1.000 0.1667 2

2 29.69 28.14 0.9478 53

3 28.57 27.91 0.9771 42

4 47.15 44.02 0.9336 118

5 19.02 18.05 0.9488 22

6 15.85 15.95 1.0070 14

7 15.39 15.24 0.9900 13

8 53.08 50.74 0.9559 127

9 59.39 57.25 0.964 191

10 22.95 22.53 0.9815 29

11 11.39 9.650 0.8472 7

12 15.39 15.24 0.9900 13

13 10.78 9.830 0.9116 7

14 15.56 14.48 0.9304 16

15 16.00 16.00 1.0000 15

16 30.52 28.31 0.9274 41

17 36.21 35.54 0.9815 67

18 28.12 28.42 1.0110 50

Table 3: Dataofblobs

(i.e.particles)

after segmentation

p: perimeterofblob

ecp: equivalent convex-hullperimeter

area: segmented particle areainpixels

Pixel

connectivity

_{determines the area of each blob.}

Searching

the blob

boundary

brings

out the perimeter of each

blob.

_{The equivalent convex hull is the} smallest convex blob

which contains the original one. ECP stands for Equivalent Convex Perimeter ofa blob

(Fig. 1.8). Table2containsdata obtainedfrom the

binary

imagein

Fig

2.4(b).

*

<b

m

V U o

Fig

2.5

(a)

Pseudo centersofparticlesin

binary

image

(Top)

(b)

Particlesafterwatershedsegmentation

(Bottom)

Because theECPwill neverbe greaterthan the realperimeter, themaximum value ofthe

ratiois 1 (somevalues are_slightly greaterthan 1 because oftheeffects ofdigitizationof

scaled X/Y aspect ratio). From the values in Table

2,

a significant _gap can _{be found}

between independentparticlesand

touching

ones(bold ones).Thisis_whywe can_classify

been

found

after a

large

number of perimeters ofblobs in bothcategorieshave been inves tigated. When amultiparticle _{blob is classified, the} morphological watershed segmenta

tion algorithm is applied on the blob to segment the

touching

particles. Fig.

2.5(a)~(b)

illustrate

two ofthosesteps ofthesegmentation.

. >'W i

itX "

7X.

>jn v/ >*>*

Frame No. 0

Particle Count

18

Total Area

855

Thresh: Var.

Fig

2.6 Final Layoutoftheoutputimage

(Top)

Original imagewindow

(Middle)

Differentialimage.

(Bottom)

Finalsegmented particles

(Upper-Right)

Status oftheimager

(Lower-Right)

Statisticalresults

Afterall

touching

particles are_properlysegmented, the

binary

imageshould bemeasured

again toget the finalparticleareaintheimage. Table 3 shows theparticledata after _seg

mentation. Blobs

#3, #4,

#8and#9 in

Fig 2.5(b)

arethose particles segmentedfrom blob

#6 in

Fig

2.4(b).A finalimage withmeasurementresultsis shownin

Fig

2.6.

When the _{image processing is complete,} all particles in the ultrasonic image have been

measured. Becauseof_{applying hardware operations and}optimal algorithms, the process

ing

time for each image frame is about 0.7-0.8 _second, which is considered "real

time"

processing in

dealing

with process fluids. Thenext_{step is} toperform a statistical analysis

ontheobservation data.

2.3.3Observed Particle Area Distribution from Dispersion

When each individual particle areahas been obtained, we accumulate all particles in all

sampledimage frames and sortthemin terms of area sothat theparticle area distribution

canbe determined. It is importantto

keep

in mindthat thex-axis_along theparticledistri

bution should not be the actualdiscretepixel numberof particle. Sincewe took the loga

rithm of particle area (in square _microns) as the x-axis of the original

distribution, i.e.,

/og(area)

=log(nr

2),

_{we should}

keep

the_{identical meaning}ofthex-axis fortheobserva

tion

distribution,

i.e.,

/og(area)

=

log(px),

_where

pxis thepixelnumber

(area)

oftheparti

cle observed.

Also,

thecount of particles undereach area shouldbenormalized sothatthe

y-axis representsthepercentages

(probabilities)

ofparticlesundereach area.

particles on the x-axis. Itcomes fromall recognized particles in 59 imagessampled from

40-second

periodsoftheexperimentfor dispersion.

Dispersion, 5 ppm, 6dB

0.15

density

0.05

0.25 0.5 0.75 1 1.25 1.5 1.75 2

log(x) (x: area in pixel)

Fig. 2.7 A Normalized Observation Data Distribution

2.4

Stability

of

Observation

Data

Distribution

Before examining theexperimentaldata fromvariousfluids andsignalgains, thedistribu

tion _stabilityoftheobservationdata shouldbe investigated.Sincetheparticle numberin a

singleimageframe is toosmalltoestimatethedistributionofparticles in

fluid,

we needto

run through a period of experimentationtocollectenoughparticles,notjustthoseinasin

gleframe. What is a sufficient_quantityof particles fromwhichastableoutputdistribution

can beobtained? Theeffect of a series of selected_samplingperiodsfromeach experiment

willbe studied here.

(1)

(3)

12/3/91 Dispersion OdB 1.2ml

loq(r) <r: particle radius)

12/3/91 Disprsion OdB 7.2ml

200 150 o100 u 50 0 loc 12/

v_

u y C o 200 150 Sioo SO 0

v__

A K V U 0 E o

0.2 0.40.60.8 1 1.2 <r (r: particle radi

3/91 Dispersion OdB 7. J3)

2ml

0.20.40.60.8 1 1.2 log(r) (r: particle radi

12/3/91 Dispersion OdB 7.

J3) 2ml 200 ulSO c ol00 V 50

v^

A it K c o 200 150 O100 u 50

L

A fl E o

0.2 0.40. 60. B 1 1.2 0.2 0.4 0.6 0.8 1 1.2 log(r) (t: particle radius)

(2)

(4)

Fig. 2.8 OutputDistributionin Different

Sampling

Periods

(1)

20 Seconds.

(2)

30 Seconds.

(3)

40 Seconds.

(4)

All

(2)-(4)

We selected four periods

(10, 20,

30,

40 in seconds) as the processing periods in the

approximately

50-second-long

experiments to test the fluctuation oftheobserved particle

whatthefluidconcentrations andthesignal gains_are,thenormalized observationdata dis tribution doesn'tchangemuch. Sowe can _say that20seconds is

long

enoughtoobtain a stable output

distribution

_{from the three kinds offluids underthe currentframe}

sampling

rate.

2.5

Observed Particle

Distributions

from

All

Fluids

Three kinds oftypical_{industrial fluids}

(dispersion,

gel andemulsion) are selectedforthe

experiment.

Each_processingprocedure goes asbelow:

1)

set_upthegain and _samplingperiod(20-40_seconds)foreach fluid.

2)

playbackthevideotape and start_samplinganddigitizationsimultaneously.

3)

thresholdimageand segment

touching

particles.

4)

measure

binary

imageof sampledframetogeteachparticle area.

5)

accumulate all particles and sort theminterms of areainall sampledframes.

6)

take thelogarithmof particle area as x-axis oftheobserveddistribution.

7)

normalizetheparticle count at each area

by

thetotalnumber ofdetectedparticles.

8)

plotthe_probabilitycurve asthe outputdistributionvs. log(px).

Allthe fluidswithdifferentconcentrations (5-20_ppm)have beenprocessed andtheparti

cle distributions have been obtained.

Fig

2.9(a)~(c)

illustrates some of the results. Our

goalis tofindthe_{relationship between}theoriginaland observeddistributionssothatesti

density

Dispersion, 5 ppm. 6dB

0.2

0.15

0.1

0.05

0 '

0.25 0.5 0.75 1 1.25 1.5 1.75 2

log(x) (x: area in pixel)

0.2

Dispersion, 15 ppm. 6dB

0.15

0.1

0.05

n "v/^e-Ay.

0.25 0.5 0.75 1 1.25 1.5 1.75 2

log(x) (x: area in pixel)

Fig

2.9

(a)

Twoobservedparticledistributionsobtainedfromdispersion.

Gel, 10 ppm, OdB

0.25 0.5 0.75 1 1.25 1.5 1.75 2

log(x) (x: area in pixel)

Gel, 20 ppm, OdB

density

0.25 0.5 0.75 1 1.25 1.5 1.75 2

log<x) <x: area in pixel)

density

Emulsion, 10 ppm. OdB

0.2

0.15

0.1

0.05

0 .*..'...rfS.^fc-JtaHv'

... 0.25 0.5 0.75 1 1.25 1.5 1.75 2

log(x) (x: area in pixel)

density

Emulsion, 15 ppm. OdB

0.2

0..15

0.1

0.05

fi ".,VV.--ws.fl.

0.25 0.5 0.75 1 1.25 1.5 1.75 2

log(x) (x: area in pixel)

Fig

2.9

(a)

Two observed particledistributionsobtained fromemulsion(continued).

3.0

Analysis

of

Results

Having

obtainedthe

discrete distributions

_{oftheoriginal data (Fig.}

2.1)

_andobservation

data (Fig. 2.9). We need_{to analyze the two distributions and discover}theirrelationship.

We dothis inthree steps.

First,

fromparticles ineach ofthe three

fluids,

_apply nonlinear

regression to find functional expressions

fitting

the original and observed distributions.

Second,

findthe_{relationshipbetweenthe twosimulation expressionsinordertodetermine}

some parameterswhichreflect the ultrasonic systemin theregion ofkr=l.

Third,

test the

stability of the system parameter obtained from different concentrations and fluids

by

applyinganalysis of variance.

3.1 Regression

and

Estimation

3.1.1. Regression forOriginal ParticleDistribution

The radius probabilities (relative

frequencies)

forthe original particle distribution in all

fluidsare givenin Table 1, andthedistributioncurveisplottedin

Fig

2.1. Thecurvedoes

not appear to represent a normal distribution

(ND),

but

taking

the logarithm ofparticle

radii

(

log(r) )

yieldsthedistributionofTable 4and

Fig

3.

1(a),

andthisnew_{distribution is}

fit well

by

a normaldistribution (Fig.

3.1(b))

obtained

by

_{numerally rmnimizing} the sum

of squares of errors (see Appendix I). Several numerical approaches are availablein [11].

The curve of regression possesses mean 1.7056 and standard deviation _{0.0918. Because}

distribution

risa

Log

Normal

Distribution

_(LND).

Particle Distribution vs log(r)

1.5 1.6 1.7 1.8 1.9 2 2.1 log(r) (r: particle radius)

Normal Distribution Regression

1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 log(r) (r: particle radius)

Fig

3. 1 (a). Original Particle Distributionvs.

log(r) (top)

(b).

(a)

andIts Normal Distribution Regression,

(bottom)

log(r)

1.418 1.493 1.568 1.643 1.719 1.794 1.869 1.944 2.020 2.095

% 0.00 1.97 10.03 25.74 31.48 20.71 8.03 1.88 0.16 0.00

Table4: Probabilitiesof

log(r)

In orderto

keep

the

distribution's

unitsidenticalatboth input and_output,wenow take a spe

ciallinearcombinationof

log(r),

i.e.,

x =

2log(

_r

)

+

log(

tx

)

=

log(

7tr2

)

=

log(

_a

)

(19)

where

"a"

is the cross-_{sectional area of}

the particle sphere. So we have

\i.iog(a)

=

3.91,

_and

log(a)

~ 0-1836

whichare usedin future. Thereason_{for using}

log(a)

ratherthan

log(r)

inesti

mationis that theultrasonic waveis faced

by

thatarea and what we haverecognizedfromthe

ultrasonic imageis thedigitalarea of a particlein_pixels,whichis therelative areaofthereal

particleunderthecurrent512x482x8image digitization format.

Since the distribution of

log(r)

is _normal, so is the distribution of

log(a),

which is a linear

function oflog(r). This is because ifx is arandom variable, x~N(p.

, a

)

, andy = mx+n,

wheremand nare constants. Then

(J.y

=

mE[x

]

+n =

m\ix+ n

(20)

cy2

=

m2

E[

(x

-\yx

)2

]

=m2a2

(21)

and_y= _mx+nis

3.1.2

Regression

for

Observed Particle

_Distribution

The original particle

distribution

have

_{beendescribed}

by

themean andstandarddeviation of a

fitted

normal

distribution,

_{but it is importanttodiscoverafitted function fortheobser}

vation data so that the

relationship

_{between input} and output distributions can be

described.

This is a

key

aspectofthe thesis: toregress theobserved distribution

by

a con

tinuous mathematical

function

which was not hitherto known. It is the

discovery

ofthe

mathematical expression that allows estimation of the original distribution

by

using the

parameters in the observed

distribution

and system as a whole. We have selected a

log-normal distribution as theinput.What is theobserved distribution aftertheoriginal distri

bution hasgone through thesystem?

Dispersion, 5 ppm, 6dB

0.15

density

0.05

0.25 0.5 0.75 1 1.25 1.5 1.75 2 log(x) <x: area in pixel)

Fig

3.2 Anormalized observationdistribution in dispersion (points

joined)

We haveapplied polynomial regression.

Any

polynomial canbeexpressed as

n-1

f(x)

= ^a,*'

(22)

i=0

which hasnundetermined_{coefficients,}

a,-, i=

0, 1,

...n-1. These

a,-can bedetermined

by

nindependent functionsfromtheobservationdataset

(

_XiJiXj)

),

i =

0,

1,

... n-1.In matrix

form

F= A*X

(23)

whereF=

[/7*0

)

/(*!

)

...f(xn.x)], A=

[a0

ax _...anA

]

and

/

X = 1 1 x x 0 1 n-1 n-1 x x 0 1 1 x

\

n-1

\

n-1 n-1 / Thus

A = F*

X"1

iff X isreversible,whichmeansthatall

xt

areindependentobservation data.

Any

analyticfunctiong(x)canbe expressed

by

aTaylor'sseries at point x0:

g(x) =

g(x0)+g'(x0)(x-xQ)+-^-(x-xQ) +... + ^(x-x0) +...

When

x0

=

0,

8(x) =

g(0)+g'(0)x+8-^x2

+ ...+

8-^xn+...

(25)

*J! _n!

g(n)(0)

withcoefficients

-: .

Witha polynomial regressionfortheobservation

data,

wefoundtheaveraged coefficients

a{ from several sets ofobserved data to _{be very} close to the coefficients of the Taylor's

series ofpe'qx

at_xq=

0,

_whichis

-qx

P* =

P 1 +

(^)

+

ffl

+ +

2! n!

(26)

Equation

(26)

reminds us of a standard exponential

density

functionwhich is

ae-, x>=0

fix)

0 1

0,

xo

However,

because we decided to ignore single-pixel particle in observation data as

they

are_{easily disrupted}

by

noise,thesmallest particle areais2pixels, whichrepresents a par

ticle arearange from 1.5to2.5 pixels in theobservation data.

Moreover,

the exponential

functionisobtained after

taking

thelogarithmof pixelarea, sotheregression

density

tionoftheactual

distribution

hasa rangefrom

log(l.5)

to

infinity,

i.e.

pe'**

x>=log(l.5)=_0.\16

fix)

;

log(l.5)

= _0.176

0 x<

Any density

function shouldhave

\f(x)

dx = 1

, sothat

\peqxdx

=

J

pe-qxdx= ₁

(27)

log (1.5)

By

resolving

(27),

we get/?= _1.5

qq.

_So

, 1.5

V7*,

*>=

log(l.5)

=0.176

/W=

(28)

L

0,

x<log(\.5) =0.176

The regressed exponenti?l distribution _{function has only} one _{parameter, q} to be deter

mined. We apply the method ofleast-squares to _{find q for}the output distributionestima

tion.

Table 5-7 givethe values ofa, the mean

\iq

and the standardderivation

aq

obtainedin

dispersion,

gel and _emulsion, respectively. No matter the fluid concentration and signal

gain, the values ofa are quite stable. We select those a'swhich

bring

rninimum standard

some observeddatacurves andtheirexponentialfunctionregressioncurves, whichfitthe

former _very well. In the next _section, we are _going to investigate the system parameter

fromwhich_{the mathematical}

relationship

betweenobserveddataandtheoriginaldistribu

tioncanbeestablished.

Gain concentrations

(ppm)

mean stddev

(dB)

5.0 10.0 15.0 20.0

Hg

ag

-12 3.2867 3.0030 2.4140 2.6348 2.8421 0.3917

-6 3.2117 2.4358 2.6328 2.6842 2.7411 0.3315

0 2.9896 2.7344 2.5688 2.6920 2.7462 0.1768

6 3.0173 2.8642 2.7682 2.8278 2.8694 0.1063

12 2.5331 2.6749 2.8099 2.6655 2.6709 0.1131

Table 5: a's obtainedfrom dispersionwithdifferentconcentrations and signal gains

Gain concentrations

(ppm)

mean stddev

(dB)

5.0 10.0 15.0 20.0

^g

Gg

-12 2.2018 2.4423 1.8742 2.0633 2.1454 0.2392

-6 2.4669 1.9552 1.8185 2.3457 2.1466 0.3090

0 2.7597 2.6867 2.7788 2.8808 2.7765 0.0801

6 2.6583 2.6892 2.4874 2.7428 2.6444 0.1104

12 2.8775 2.4472 2.6116 3.3080 2.8111 0.3757

Table 6:a's obtainedfromgel withdifferentfluidconcentrations and signal gains

Gain _{concentrations}

(ppm)

mean stddev

(dB)

5.0 10.0 15.0 20.0

^g

ag

-12

3.2553

_{1.5552 2.6592}

3.3493 2.7048 0.8250

-6 2.9636 _{2.8747 2.6573 2.4615 2.7393 0.2255}

0 2.9873 2.9209 2.9521 2.8808 2.9353 0.0453

6 3.2199 3.1886 2.8424 2.9659 3.0542 0.1809

12 3.2084 2.9489 2.7243 2.7984 2.9200 0.2138

Table 7: a's obtainedfromemulsionwithdifferentfluidconcentrations andsignal gains

concentrations

(ppm)

mean stddev

Fluids 5.0 10.0 15.0 20.0

^g

g

Dispersion 3.0173 2.8642 2.7682 2.8278 2.8694 0.1063

Gel 2.7597 2.6867 2.7788 2.8808 2.7765 0.0801

Emulsion 2.9873 2.9209 2.9521 2.8808 2.9353 0.0453

Exp. Regression of Dispersion <q=3.0173)

0.2

0.15

y(x)

0.1

0.05

5 ppm, 6dB

0.25 0.5 0.75 1 1.25 1.5 1.75 2 log(x) (x: area in pixel)

Exp. Regression of Dispersion (q=2.7682) 0.2

0.15

<x) 0.1

ytx

0.05

15 ppm, 6dB

0.25 0.5 0.75 1 1.25 1.5 1.75 2 logCx) <x: area in pixel)

Fig

3.3 Observed distributions in dispersionandtheirexponential regressions

Exp. Regression of Gel <q=2.4867)

10 ppm, OdB

0.25 0.5 0.75 1 1.25 1.5 1.75 2

log(x> (x: area in pixel)

Exp. Regression of Gel (q=2.6808)

y(x) 20 ppm, OdB

0.25 0.5 0.75 1 1.25 1.5 1.75 2

log(x) (x: area in pixel)

Exp. Regression of Emulsion (q=2.9209)

y<x) ₁₀

ppm, OdB

0.25 0.5 0.75 1 1.25 1.5 1.75 2

log(x) (x: area in pixel)

Exp. Regression of Emulsion <q=2.9521)

y(x) 15 ppm, OdB

0.25 0.5 0.75 1 1.25 1.5 1.75 2 log(x) (x: area in pixel)

Fig

3.5 Observed distributions inemulsion and theirexponential regressions

3.1.3

Estimation

of

System Parameter

_{from Each Fluid}

As the regression of both the original and the observed distribution have been accom

plished, we nextanalyzebothregressionssothat theestimation rulebetweenthemcan be

established.Recallthe regressionfunctionoforiginaldata:

(*-n)2

fix)

=/(jc|u,o) = _-jL-e

2a*

(29)

/27ta

where x=

log(%r),

_{ris particle radiusin}

micron and_p.,a are mean and standarddeviation

of

\og(nr),

respectively.

Let's rewrite the exponential regression distribution as a function of t for the observed

data:

g(t)=pe-*= _1.5iqe-*

(30)

where t =

log(px)

_>

log(l.5)

andpxis particle area inpixels obtained fromthe processed

ultrasonicimage. Itisnotdifficulttorealize_thatflx)and_git)arebothexponential butthe

formerisquadratic tox andthe lateris linearto t. Sincetheoriginal particles are detected

by

theultrasonic imagerand recognized

by

theimage _processing system,we can useFig.

Fluids

Log

Normal Distribution

fix)

Acoustic Imager Digital Image

Processing

System Observation Data Statistics Exponential Distribution git)

Fig.3.6 SystemConfigurationandParticleDistribution atInputandOutput

Weneedtogive ahypothesisofthe system responsethatcan explaintheoutputdata distribu

tion.

By

_considering the two regression

functions,

we postulate that the system response to

log(px)

is

t= _c

(

_x- _[i

f

_+b

(3D

where c andbaretheparameters ofthe wholesystem.

By

replacing

(

x p.)2

by

(t

-b)

/c

infix),

fix)