Rochester Institute of Technology
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8-1-1991
Adaption and application of morphological
pseudoconvolutions to scanning tunneling and
atomic force microscopy
Andrew D. Weisman
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Recommended Citation
ADAPTATION AND APPLICATION OF MORPHOLOGICAL
PSEUDOCONVOLUTIONS TO SCANNING WNNELING AND ATOMIC FORCE
MICROSCOPY
by
Andrew D. Weisman
Rochester Institute of Technology
(1991)
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 Graphic
Arts
and Photography
of the Rochester Institute of Technology
August 1991
Signature of the Author
_
Coordinator,
M.
S. Degree Program
COLLEGE OF GRAPmC ARTS AND PHOTOGRAPHY
ROCHESTER INSTITIJTE OF TECHNOLOGY
ROCHESTER, NEW YORK
CERTIFICATE OF APPROVAL
M. S. DEGREE TIffiSIS
The M. S. Degree Thesis of Andrew D. Weisman has been examined and approved by the
thesis committee as satisfactory for the thesis requirement for the Master of Science degree
Dr. Edward
R.
Dougherty
Dr. Mehdi Vaez-Iravani
Dr. Howard A. Mizes
THESIS RELEASE PERMISSION
ROCHESTER INSTITUTE OF TECHNOLOGY
COLLEGE OF GRAPIDC ARTS AND PHOTOGRAPHY
)'
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II/le
1/
C1.IJ1!l;LJ;fI
be
0 -44
-1of
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!U.!t'c9/l
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+0
, 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.
ADAPTATION AND
APPLICATION
OF
MORPHOLOGICAL
PSEUDOCONVOLUTIONS TO
SCANNING
TUNNELING AND
ATOMIC FORCE
MICROSCOPY
by
Andrew
D. Weisman
Rochester Institute
ofTechnology
(1991)
A
thesissubmittedin
partialfulfillment
oftherequirementsfor
thedegree
ofMaster
ofScience
in
theCenter for
Imaging
Science
in
theCollege
ofGraphic Arts
andPhotography
ofthe
Rochester Institute
ofTechnology
ABSTRACT
A recently developed
class ofdigital
filters
known
as morphological pseudoconvolutionsare adapted and applied to
Scanning Tunneling Microscopy
(STM)
andAtomic
Force
Microscopy
(AFM)
images. These filters
are shown tooutperform,
both visually
andin
the
meansquare errorsense,
previously
introduced
Wiener
filtering
techniques.The
filters
are compared on typical
STM/AFM
images,
using both
modeled and actualdata.
The
technique
is
general,
andis
showntoperformvery
wellonmany
types ofSTM
andAFM
images.
Acknowledgements
I
wouldlike
toexpressmy
sincere gratitude toR. J.
Dwayne
Miller
andThe Center for
Photoinduced Charge
Transfer
for
theirsupportduring
this work, andfor
theinvaluable
opportunity
thatwas afforded me.I
wouldlike
to thankProfessor
Edward
R.
Dougherty
for
his
guidance,
expertise,
andenthusiasm
during
thiswork,
without whichI
would nothave had
the confidence orknowledge
tocarry
outthis thesis.I
would alsolike
to thankDr. Howard A. Mizes for his continuing
help
and expertguidance throughout the course of this work.
His
enthusiasmfor
science andlife in
general
has
helped
me greatly.I
wouldlike
to thankas wellDr. Mehdi Vaez-Iravani
for his
supportduring
my
education,
and
for accepting
tobe
onmy
defense
committee.I
wouldlike
to thankKong-Gay
Loh for his
help
and expertisein
Scanning
Tunneling
andAtomic
Force
Microscopy,
Lucius Seok-Hwan Doh for
valuablediscussions,
andMatthew
Lang
for
imaging
thepolyhexylthiophene.I
wouldespecially
like
to thankmy
wife,
Catherine,
andmy
family
for
giving
methemoralsupport
I
needed andfor
instilling
in
metheconfidencethatI
lacked
tobe
abletocarry
outThis
thesisis dedicated
tomy
parents,
David
andMorelyn
Weisman.
Contents
1
Introduction
1
1.1
Aims
of thestudy
1
1.2
Scanning
Tunneling
Microscopy
2
1.3
Atomic
Force
Microscopy
4
2
Post-processing
techniques previously
appliedto
Scanning Tunneling
and
Atomic
Force
Microscopy
11
2.1
Introduction
tolinear
system analysis11
2.2
Classical
filtering
techniques14
2.2.1
The
lowpass
filter
14
2.2.2
The
medianfilter
15
2.2.3
The
Wiener
filter
16
3
Morphological
pseudoconvolutions23
3.1
Motivation
behind
pseudoconvolutions23
3.2
Introduction
to mathematicalmorphology
24
3.3
Morphological
pseudoconvolutiondevelopment
26
4
Application
of morphological pseudoconvolutionsto
STM/AFM
images
and experimental results
31
4.1 Adaptation
ofmorphological pseudoconvolutionstoSTM/AFM
typedata
31
4.2
Development
ofcriteriafor evaluating
pseudoconvolutions32
4.2.1
Mean Square Error
-metric
for
comparison33
4.2.2
Development
of noise model33
4.2.3
Development
ofimage
model35
4.3
Comparison
of pseudoconvolutions withWiener
filtering
35
4.4
Results
and analysisfor
graphite model37
4.5
Analysis
on edges40
4.6
Empirical
results and analysis42
Conclusions/Recommendations
57
Table
of
Figures
Figure
1-1
7
Figure
1-2
8
Figure
1-3
9
Figure
1-4
10
Figure
2-1
20
Figure
2-2
21
Figure
2-3
22
Figure
3-1
29
Figure
3-2
30
Figure
4-1
45
Figure
4-2
46
Figure
4-3
47
Figure
4-4
48
Figure
4-5
49
Figure
4-6
50
Figure
4-7
51
Figure
4-8
52
Figure
4-9
53
Figure
4-10
54
Figure
4-11
55
Figure
4-12
56
Chapter 1
Introduction
1.1
Aims
ofthe
study
Both
Scanning Tunneling Microscopy (STM)
andAtomic Force
Microscopy
(AFM)
areimportant
toolsfor
surfaceimaging
atthesub-micronscale,
andit is
becoming
commontoenhance the
resulting
images
by
means ofdigital
postfilters,
mosttypically
lowpass
andWiener
filters.1*2Wiener
filtering
requires estimates ofboth
thedesired image
and thenoise,
andthesearedifficult
toobtainin STM
andAFM
systems.Moreover,
astypically
applied,
theWiener
filter
requiresthat theimage be
wide-sensestationary,
and whilethisassumption might
be
acceptablein
certaincircumstances,
in
othersit may
not.Finally,
application ofthe
Wiener methodology
presupposesthatwe wishtoemploy
alinear
filter,
and
in many
instances
suchanassumptionis
either unwarranted or not relevant.In
thisthesis,
arecently introduced
class of nonlinear spatial-domainfilters
known
asmorphological pseudoconvolutions3'4'5, which make
very few
assumptionsregarding
image
degradation,
are applied toSTM/AFM
data.6'7In
pseudoconvolutiondesign,
theshape oftypical
image
and noise structureis
takeninto
account.Because
thefilters
arenonlinear,
they
canbe
constructedso astokeep
thedistortion
oftheunderlying
image
toainformation loss is
critical,
is
that thefilters
contain a parameterthatallows onetoperformdesired degrees
offiltering,
and as the parameteris increased
one canactually
pass theobserved
image
without alteration.1.2
Scanning Tunneling Microscopy
Scanning Tunneling Microscopy
(STM)8is
a new
branch
ofmicroscopy
developed
in
the1980's
thatenables onetoresolve thesurface structure of aconducting
substratedown
totheatomic
level. The idea behind STM is
thatif
anatomically sharp conducting
tip
(i.e.
ametal)
is brought
withinafew
angstroms of aconducting
substrate,
electrons willtunnelbetween
thetip
andthe substrate.The
currentdepends exponentially
on thetip-sampleseparation.
Specifically,
for
vacuumtunneling
the currentincreases
an order of magnitudewith
roughly
an angstrom changein
the tip-sample separation.This
allowsfor
anextremely
sensitive measurement oftheheight
ofthesample.In
addition,
only
the atomclosestto the end ofthe
tip
will contributeto thetunneling
current,
allowing
for
atomic resolutionlaterally.
The
basic setup
for
anSTM is
shownin
Figure 1-1.
The
tip
is
movedin both
thex andy directions
(along
the substratesurface)
andin
the zdirection (closer
andfarther away from
thesurface)
by
means ofa piezoelectrictube,
towhichthe
tip
is
affixed.Electrodes
are placed on thetube(which
is
in
general elongatedinto
the shape of atube),
and when a voltageis
applied,
the tube contracts or expands(giving
themotionin
thezdirection)
and can alsobe
madetobend
laterally
(for
scanning).resolution
in
both
the
lateral
andverticaldirections. The STM
canbe
operatedin
twobasic
modes: theconstant current mode andtheconstantheight
mode.The
constant current modeis
themostwidely
used ofthe two.It
givesthetopography
ofthesubstrate
for
image
dimensions
greaterthan10A.
It
can alsobe
usedtoimage
surfacesthat are not
atomically
flat.
The
tip
is lowered
towards thesurfaceuntilathresholdamount of current(on
the order ofInA)
is
achieved.The
tip
is
then scanned across thesurfacewhilea
feedback
loop
monitorsthecurrent,
attempting
tokeep
thecurrent constant.When
thecurrentincreases
ordecreases (i.e.
thetip
encounters avalley
or amountain),
thetip
is
lowered
or raisedtocompensate.For
example,
if
avalley
is
encountered whilescanning,
thecurrentwill
decrease
(because
thesurface andtip
willbe
farther apart)
andthefeedback
loop
willrespondby
sending
a voltagetothe tubesothat thetip
is lowered
untilthecurrentincreases
back
to thedesired level
again.This
idea is depicted in Figure 1-2. Since
thevoltage appliedto the tube
is
directly
proportional to thecurrentencountered,
whichis in
turnproportionalto therelative
height
ofthesurface,
wemay
plot an equispaced sampledarray
of the voltage(z)
appliedto the tubeas afunction
of position onthesubstrate(x
andy)
andthusdevelop
thetopography
ofthesurface.This may
be
displayed in
a number ofways,
including
gray
level
versusposition(thus
agray
scaleimage).
The
constantheight
mode ofoperation scans thetip
overthe surface whilekeeping
its
height
constantfrom
afixed
referenceheight.
The
current(which
will change whenmountainsand valleys are
encountered)
is
monitoredand canbe digitized
and plotted as afunction
of position.This may be
seenin Figure 1-3. Because
thereis
nolag
timefrom
at a much
faster
ratein
theconstantheight
mode(one
to twoorders ofmagnitudefaster).
This
canbe
usefulin reducing
scan-line noisein
theimage.
Because
thereis
an exponentialdrop-off
of thetunneling
current with thedistance
between
thetip
and thesurface,
however,
the
verticalrangein
theconstantheight
modeis
not aslarge
asin
the constantcurrent mode.
Also,
because
thetip
height
does
notchange,
oneis
atrisk
ofcrashing
thetip
into
the
surfaceif
ahigh
enough mountainis
encountered while scanning.Both
modesare capable of
giving
atomicresolution,
however.
1.3
Atomic Force
Microscopy
Atomic Force
Microscopy
(AFM)
is
another method of atomiclevel
imaging
that ,unlikeSTM,
does
not require thesubstrate tobe
a conductor.AFM
is
similartoSTM in
thatit
usesa
sharp
tip
thatis lowered
very
closeto thesurface whichis
scanned via a piezoelectrictube, but
instead
ofusing
thetunneling
currenttomonitorthe tip-sampledisplacement,
one usesthe
force
ofinteraction between
thetip
and sample to monitor this separation.By
monitoring
thechanging
forces
on thetip
whilescanning
thesurface,
imaging
may
be
accomplished.
The
force
onthetip
causesadisplacement
ofthecantilever.There
aretwobasic
modes of operationfor
theAFM:
the attractive and repulsive modes.9Only
therepulsive mode will
be
discussed
here,
asit is
the mostwidely
used ofthe two andis
capable ofgiving
atomic resolution.The
repulsivemode ofimaging9withtheAFM
is
performedby bringing
thetip
into
very
sample
(so
thatthey
arein
contact),
but
thevery
tip
ofthecantileverfeels
arepulsiveforce.
The
magnitude ofthisforce is
measured as thetip
movesalong
the sample.Different
schemes
for
detecting
thechangein
thisforce
whilescanning
have
been
documented
andimplemented,
but
only
onecommonly
usedtechnique
willbe
presentedhere.
A
simplescheme often referredtoas the"beam-bounce"10techniqueis
illustrated
in
Figure
1-4. The
substrateis
attachedtoa piezoelectric tube sothatit
may
be
movedin both
thevertical and
lateral directions
asin STM
(notice
thatit is
actually
the substrate thatis
"scanned"
here,
while thetip
remains stationary).A
small metallic rodis
bent into
acantileversothat theapparatus consistsofa
tip
connectedtoanarm,
and a minutemirroris
placed onthe
backside
ofthetip. The
arm ofthetip
is
somewhatflexible
andis
attachedtoa
rigid
body
so thatit may be displaced in
thez(vertical)
direction
freely
whenforces
areencountered.
A
laser beam coming from
an oblique angleis
focussed
down
ontothemirrorand
from
thereis
reflected onto aposition sensitivedetector, typically
consisting
oftwophotodiodes separated
by
a10
micronspace.By
placing
thedetector
at arelatively large
distance
from
thecantilever,
the amount ofdeflection
is
magnified and thus smallmovements
may
be
measured quite easily.The
sampleis
moveduntilit
contactsthetip
andthe
tip
bends backwards.
As
thesampleis
scanned,
bumps
andvalleysin
thesurface willcause the
tip
tomove,
and thus thelaser beam
willbe displaced
on thedetector.
A
feedback
loop
monitorsthisdisplacement,
and attempts tokeep
thebeam
centered onthedetector
by
adjusting
thesample'sverticalheight
by
sending
voltagesto the tubesothatit
eithercontracts or expands.
This
voltageis
sampled and recorded as afunction
oflateral
positionon the
substrate,
and thusa surfacetopography
may
be
developed
anddisplayed
As
illustrated,
theresulting images
for both
STM
andAFM
aredigital images
thatmay
be
stored
in
a computer.Digital
image processing
techniquesmay
thusbe
appropriatefor
filtering
out noiseincurred
during
theimaging
process.Classical
techniquesincluding
mean,
median,
andWiener
filters have
been
applied1'2with some
success,
however
moreeffective
techniques
aredesired,
asprocessing
images
wherethe
input is
not alwaysknown
height
(z)
STM
Tip
Atomic Surface
Tip
Path
Atomic Surface
Tip
Path
Atomic Surface
Figure 1-3:
Set-up
for
Scanning
Tunneling
Microscope
operatedin
Position
Sensitive
Detector
Cantilever
Mirror
Sample
Piezoelectric Tube
Figure
1-4: Basic set-up for Atomic Force Microscope using
beam-bounce
technique.
Chapter
2
Post-processing
techniques
previously
applied
to
Scanning Tunneling
and
Atomic Force
Microscopy
2.1
Introduction
to
linear
systemanalysis
Traditionally,
theModulation Transfer Function
(MTF)
of animaging
systemis
studiedbefore image
processing
is
performed togainknowledge
of whatthe systemdoes
to theinput
to
achievethe output one observes.If
theMTF
is
known
andthesystemis found
tobe
linear
and shiftinvariant,
onemay
performvariouslinear
filtering
techniques
toattempttocorrect
for
the effects ofthe system.A. filter *P
is
linear if for any
images
f
andg
andscalars a and
b,
*F(af
+bg)
=axP(f)
+b<P(g);
otherwiseit
is
nonlinear.Linearity
represents a restriction on
filter
design.
Linear
techniques such asdeconvolution
andWiener
filtering,
however,
canbe
effectivefor
filtering
certaintypesofimages.
System
linearity
means that given aninput
afi(x,y)
+bf2(x,
y),
where a andb
are real constants, the outputmay be
written asagi(x, y)
+bg2(x,
y).In
an optical radiationdetector,
for
example,
f
is
theintensity
ofthelight incident
on thedetector,
andg
is
thecurrent generated.
For
anSTM
operatedin
constantheight
mode,
f
is
theheight
ofthesample
defined
by
a contour of constantFermi
level
chargedensity
and referred to anarbitrary
zero,
andg
is
the position of thetip,
referred to the same zero.The
optical radiationdetector
canbecome
nonlinear atlow light levels
and atsaturation,
where theresponse
may level
off,
but
it
may
have
alinear
regionbetween
these twoextremes.A
source ofnonlinearity in
theSTM is
its
finite
responsetime
to changesin
the surfacetopography.
This
becomes
much more pronounced asthescanning
speedincreases.
System
shiftinvariance
meansthatmoving
theinput
laterally
before
imaging
is
equivalentto
taking
theimage
and thenshifting
the output.Shift
invariance may
be
expressedmathematically
asf(x
-a,
y
-b)
> g(xa,
y
-b),
where a andb
are real constants.Shift
invariance is
a more strictdefinition
thanit
may
appear.A
systemthatmagnifies aninput
f(x,
y)
into
an outputf(x/k,
y/k),
such as an opticalmicroscope,
is
not shiftinvariant
because
aninput
off(x
-a,
y
-b)
is
not givenby
f(x/k
-a,
y/k-b),
but
ratherby
f(x/k
-a/k,
y/k-b/k). The STM does
notmagnify in
this manner,but it is
not clearthat theSTM is
shiftinvariant
because
anonlinearresponseofthescanning
piezoelectricmay
causethe
sensitivity
tobe
positiondependent.
Although
few
imaging
systemssatisfy linear
and shiftinvariance
conditions,
many
aremodeledas such
in
ordertoutilizelinear
filtering
techniques.To
usethesetechniques,
the systemMTF
mustbe
studied.To
gainknowledge
ofthesystemMTF,
onenormally
takesan
image
ofaknown
object,
and then comparesthe outputimage
to thisoriginal object.The known
objectis
typically
onethatcontains mostfrequencies (such
as astep
oradelta
function)
sothat a system analysisoveralarge
frequency
rangemay be done.
Assuming
that the
MTF is
consistent underdifferent
imaging
conditions,
an algorithmmay be
developed
to
correctfor
its
effecttosome extentIn STM/AFM
images, however,
this processis
very
difficult
toperform,
because
a"known"
input is
nearly
impossible
toconsistently
create ontheatomic scale.One is
alsofaced
withthe
problems ofthermaldrift
overtimewhichinhibits
reproducibility
ofresults,
different MTFs for
different
scanspeeds,
different
scanareas,
anddifferent
substratesproducing very
different
noise effects.All
ofthesefactors
makethedevelopment
of a setlinear
algorithmvirtually
impossible.
What
is
generally
done,
therefore,
is
touse somepost-filtering techniques,
such aslowpass
filtering
andmedianfiltering. Wiener
filtering,
which requires
linear
systemanalysis,
has
alsobeen
employeddespite
theseproblems.1'2There is
a needfor
apost-filtering
technique thatis
robust andthatdoes
not makemany
assumptionsaboutthe
imaging
process.Filtering
ofdata
ofthistype,
wherethegoal oftheimaging
processis
toinvestigate
the unknown, mustbe
done
in
a prudent manner.One
does
not wishtoremovedesired information from
theimage
orintroduce
artifacts.Here,
artifacts are structures
resulting from
thefiltering
processthatmay
be
misinterpreted.As
wewill
see,
filtering
techniquesbased in
thefrequency
domain
canintroduce
artifacts ofvarious types.
A
numberof examplesin
this thesisareintroduced
via onedimensional
signalsfor clarity
ofpresentationand ease ofcomputation.
It
shouldbe
clearthata signalmay
be
thoughtofas aone-dimensional
image,
orequivalently,
a crosssectionof animage may be
consideredto
be
asignal.2.2
Classical
filtering
techniques
2.2.1
The
lowpass
filter
In
thepresentsection,
threeclassicalfiltering
techniqueswillbe
briefly
discussed,
paying
special attentionto the
Wiener filter. Lowpass
filters
operateby
passing only frequencies
lying
below
a certain cutofffrequency.
Filtering
is
accomplishedin
one oftwoways:(1)
theFourier
transformis
applied,
the transformedimage is
truncatedin
frequency,
and then theinverse
Fourier
transformis
applied,
or(2)
thefirst
methodis
approximatedby
a convolutionin
thespatialdomain.
Digitally,
convolutionis
a moving-window operationthat replaces each gray-value
in
theimage
by
a weighted average ofitself
and someneighborhood of
surrounding gray
values.Lowpass
filters
arelinear
filters.
A
basic
lowpass filter is
themoving
mean,
which replaces eachgray
value with theunweightedaverage ofthewindowedvalues.
The underlying
assumptionsfor lowpass
filtering
are that the noise consists ofhigher
frequencies
than theimage,
and that thehigher
spatialfrequencies in
theimage do
notcontain
important
information.
These
may
ormay
notbe
reasonable assumptions.For
example,
the systemmay
giverise
to
1/f
noise,
whichis
alow
frequency
noise.This
is
oftenthe case
for
STM.2-11'12'13Also,
theimage may
containdifferent desired
edge ortexture
information
that willbe
lost
uponlowpass filtering.
An
example of textureinformation
in STM
is
animage
containing
periodic atomic structure.An
edgeis
madeup
of an
infinite
number offrequencies,
and whenhigher frequencies
aretruncated,
edgesharpness
is lost.
Thus,
if
quantitativeinformation
is
desired
from
thedata,
such as widthmeasurements of a polymer
strand,
theability
toextractthisinformation
willbe hampered
upon
lowpass
filtering.
Also,
whilelowpass
filtering
does decrease
theamplitude of pointnoise,
it
also spreads point noiseinto neighboring
pixels anddoes
not eliminateit.
For
instance,
Figure
2-1
depicts
the effect of athree-point
mean on point noise.Notice how
thespreading
effect ofthemean causestheoutputtoappeartohave
structure(an
artifact).2.2.2
The
medianfilter
The
medianis
a nonlinearfilter.
It
replacesthegray
valueat a pixelby
themedian ofitself
and
its
neighborsin
alocal
window.It completely
eliminates point noisein
flat
regions(Fig.
2-1).
Also,
whereas the meanblurs
edges,
themedian preserves edges(Fig.
2-2).
Because
the medianfilter
preservesedgeinformation,
onemay
say
thatedges arein
theinvariant
class ofthemedianfilter. An
image
f
is
saidtobe in
theinvariant
class of afilter
*F
if
*F(f)
=f.
While
the medianfilter does have
certaindesirable
properties,
its
noise-suppression
efficiency
is less
thanthe mean's, andtheeffect ofits
useis
tosometimes givea
blotchy
output,
aswell astoflatten higher background
variationthatmay
be
containedin
theuncorrupted
image.
Specifically,
both
mean and medianfilters have detrimental
effectson simple
highly
varying
textureinformation
thatwemay
wishto
preserve.This may be
seenin
Figure
2-3. We
notethatJustusson14 provides a concise account ofthestatisticaleffects of median
filters,
and that a number of approacheshave
been
developed for
designing
median-likefilters
thateither enhanceedges,
possessincreased
invariant
classes,
or result
in less
degradation
oftheunderlying
image.15'16'17'18'192.2.3
The
Wiener
filter
The
Wienerfilter
is
arelatively
optimalfilter,
which meansthatit is
themost efficientfilter
of a certain
kind
with respecttoa specific criterion of goodness.Moreover,
it
depends
onjoint
statisticalknowledge
concerning
theimage
and noise.The
criterion under whichtheWiener
filter is
optimalis
themean square error(MSE). Given
thatwehave
n observationsXi,
X2,....,
X
andwishtouse theseobservationstoestimatethevalue of another variableY,
theoptimalMS filter
is
givenby
theestimation ruleg
thatminimizes(with
respecttoMSE)
the expectedvalueE[IY
g(Xi,X2,...,Xn)|2].(1)
It is
well-known that the optimalfilter
is
givenby
the conditional expectationE[YIXi,
X2,...,
Xn].
If
we restrict theform
ofg
tobe
alinear
combination of theobservations,
g(Xi,X2,...,
Xn)
=aiXi
+a2X2
+ - +anXn,
(2)
we obtain a
relatively
optimalfilter known
as the optimallinear
filter. The
weightsai,
a2,...,
an
thatminimizeMSE
arefound
by
solving
a systemoflinear
equationsknown
astheYule-
Walker
equations.20In
noise
filtering
theobservationsconsist of windowedgray
values and
Y
is
the
gray
valuewe wish restored.As
it
stands,
theoptimallinear filter
is
spatially
variant,
whichmeansthateachpixelmustbe filtered
by
its
own particular set of weights.For
thefilter
tobe spatially
invariant,
which means we can
filter
by
a single weightedsum,
we must make an assumption on theclass of
images
underconsideration; namely,
as a randomprocess,
thecollection of(noisy)
images
mustbe
wide-sense stationary.This
meansthatsecond-order statistics mustonly
depend
upon the(vector)
difference
between
pixelgray
values.The
assumption ofstationarity in
real-worldimages is
quite strong.Under
wide-sensestationarity,
theoptimallinear filter
canbe
expressedby
means oftheFourier transform,
andin
this contextit
is
calledthe
Wiener
filter.
Assuming
theimaging
processis
modeledby
alinear
observation system withuncorrelated, additive,
stationary
noise,
alsoknown
as whitenoise,
theWiener filter
is
given
by
U(kX,
ky)
=G(kX, ky)[U(kX,
ky)
+N(kX, ky)],
(3)
where
U,
N,
andU
aretheFourier
transformsoftheuncorruptedimage,
thenoise,
andtheoptimal
linear
estimate oftheuncorruptedimage,
respectively,
and whereG,
also calledtheWiener
filter,
is
givenby
r<v i -l
H*(kx, kv)S(kT,
ky)
0(kX'ky)"IH(kX,ky)l2SU(kX,ky)
+Sn(kX,ky)
'W
where
Su
andSn
are the power spectra of the uncorruptedimage
u and noisen,
respectively,
and whereH
is
thefrequency
response ofthesystem.21If
we assumethere
is
noblurring
due
to thesystem,
thenH
=1
andtheWiener filter
takes theform
nn , >.
Su(kx,
ky)
G(kx,
ky)
-Su(kx>
ky)
+Sn(kxj ky)
.(5)
This
filter
is
calledtheWiener smoothing
filter.
Note
thatit
reduces to1
whenthesignalpower spectrum
is
much greater than thenoise spectrum and reducesto0
when thenoisedominates
theimage.
The
Wiener
filter
requiresan estimate ofthe power spectrum ofboth
the noise and theimage. While
thismay
be
a reasonable requirementfor
generallarge
scaleimaging
systemsthattakecertain classes of
images,
it is
notfor
eithertheSTM
ortheAFM. In
additiontothe
Wiener
filter's
requirement ofsystemlinearity
and shiftinvariance,
it
also requiresstationarity
oftheimage. This
means that theimage
musthave
somebasic
frequency
or texture thatdoes
not change acrosstheimage. In
typicalSTM/AFM
atomic scaleimages,
images
ofuniform crystalline surfaces canbe
described
asstationary,
whileimages
ofabsorbed molecules onthesesame surfaces are not stationary.
Edges
separatethemoleculeandthe
surface,
andstationarity
does
nothold. The Wiener filter
willhave
thetendency
toblur
edges, and thusinhibit
ourability
tomake quantitative measurements.Blurring
the
edgesof an
image
alsohas
theeffect ofmaking
theimage aesthetically
undesirabletoview,
as our eyes respond more
strongly
to
edgeinformation
thanto
many
othertypes
ofinformation.22
The Wiener filter
is
optimized with respecttoMSE,
whichis strictly
alinear
metricderiving
from inner
product spaces.It is
wellknown
that our eyes respondin
a non-linearmanner22,
and giventhatit is
theeyes ofthe scientistthatareprobing
andanalyzing
theimage,
it is
notnecessarily
appropriate to use alinear
metric to optimize animage for
human
viewing.While
theWiener filter
does have drawbacks
relativetofiltering
STM/AFM
data,
it is
usedsomewhat
successfully
under the conditions that theimage is slowly varying
andhomogeneous,
there are no edgesin
theimage,
andthenoiseis
nottoogreat,
is
consistent,
and
is
known
to agood extent.These
situations,
however,
do
not alwaysexist,
andit
is
then thata nonlinear
filter
can performbetter.
X
6
s
Figure
2-1
:Effect
ofthree
point meanand
medianfilters
on pointnoise,
wheref(x)
is
the
input. Note
how
the
meanfilter
spreadsthe
noise while
the
medianfilter
eliminatesthis
noise.X
s
Figure
2-2: Effect
ofthree
point meanand
medianfilters
on adigitized
step,
where
f(x)
is
the
input. Note how
the
meanblurs
the
step
whilethe
medianpreserves
the
step.
e
Figure
2-3:
Effect
ofthree
point mean and medianfilters
on simplevariation,
where
f(x)
is
the
input. Bom have detrimental
effectshere.
Chapter 3
Morphological
pseudoconvolutions
3.1
Motivation behind
pseudoconvolutions
Both
mean and medianfilters
have
the side effect ofdegrading high-frequency
textureinformation in
theunderlying image. It
wouldthereforebe
highly
desirable
torecoverthisuseful
information
after mean ormedianfiltering
has
been
performed.If
wecould utilizethe
desirable
white noise suppressionproperties ofthemeanfilter but
also preserve edgeandtexture
information,
for
example, we wouldhave
an excellent white noisefilter. If it
werepossible aswell tousethepoint noisesuppression and edge preservation properties
ofthe median
filter,
but
also preservetextureinformation,
we wouldthenhave
auseful pairof
filters
to
workwith.In
themicroscopy
field,
given that wedo
notalwaysknow
what we arelooking
for,
wewould
like
to seeless
information
lost
due
tofiltering.
A
goodfilter
should notintroduce
artifacts, andshould
have
a controllable parameter whichwould allow all oftheoriginalinput
tobe
restored after a certainpoint,
thusallowing
theimage
tobe filtered
aslittle
asdesired. The information
shouldbe
restoredin
a manner suchthat thenoiseis last
tocomeback,
and that all ofthedesired
information is in
the
invariant
class ofthe
filter.
Increasing
a
filter's invariant
classis
theunderlying idea behind
a new set offilters
known
aspseudoconvolutions3'4'5
thatarise
in
mathematical morphology.3.2
Introduction
to
mathematical
morphology
Mathematical morphology is
abranch
ofimage
processing utilizing
primitiveshapes,
known
asstructuring
elements,
that are"fit"
into
animage.
The degree
to which astructuring
eitherfits
ordoes
notfit
determines
thedegree
towhich theimage is
oris
notfiltered.
Here,
only
twobasic
morphologicaloperations arepresented,
referring
thereaderto the
literature
(see Giardina
andDougherty23,
Serra24,
orHaralick,
Sternberg,
andZhuang25). Morphological
pseudoconvolutionswillthenbe introduced.
A filter
is
saidtobe
morphologicalif
it
is both
translationinvariant
andincreasing. A filter
*P is
translationinvariant
if
*P(fx
+y)
=*F(f)x
+y,
wherefx
is
the translationoff
by
x andfx
+y
is
theoffsetting
offx
by
y.A filter
is
increasing
if,
given thatAB,
Y(A)XP(B),
whereAB
meansthatall pointsin
the setA
arebelow
all pointsin
thesetB,
orthe
domain
ofA is
a subset ofthedomain
ofB,
andfor
allj
in
thedomain
ofA,
A(j)
<B(j)-
Notice
thatboth
themoving
mean and median are morphologicalfilters
by
definition.
The
twomostfundamental
morphologicaloperationsareerosionanddilation.
Erosion
off
by
the
structuring
elementeis defined
by
(f
6
e)[j]
=max{k:ej
+k
f
},
(6)
where
ej,
thespatialtranslationof eby
j,
is defined
by
ej(i)
=e(i-j). The
erosion off
by
eat
j
is found
by
translating
etoj
anddetermining
themaximum amount e canbe "pushed
up"
and still remain
beneath f. An
example of erosionis
givenin
Figure 3-1.
Owing
to theshape ofthe
structuring
element,
erosion reduces maximum noise(noise
spikes)
whilepreserving
theunderlying
image. Dilation
off
by
eis defined similarly
by
(f
e)[j]
=min{k:ej
+k
f
},
(7)
wheree
is
the
reflection of e through the origindefined
by
e(j)
= -e(-j).Geometrically,
dilation
markstheorigin position at whichthe "flipped"structuring
element wouldhit
thesignal
if
it
wereraised abovethesignaland thenlowered.
This is illustrated
in
Figure
3-1,
wherethe
input is dilated
by
thestructuring
element e.Here,
dilation
reduces minimumnoise
(dropouts in
thesignal)
but
preserves theunderlying
signal.It is
interesting
torecognizethat themovement of an
STM
tip
over a samplemay
be
thoughtof as adilation
ofthesample
by
astructuring
element of the same shape asthetip.
The tip,
like
astructuring
element,
willfit
into
some places on the samplebut
notinto
others.The structuring
element usedin
Figure 3-1 is
employedin
the pseudoconvolutionsintroduced
in
Refs.
3, 4,
5.
Here,
thestructuring
elementis defined in
onedimension
by
e[-l]
=e[l]
=-X, and
e[0]
=0,
whereX
is
afilter
parameterchosenby
the userandtheargumentof e
is
thepixelindex. As
one can seefrom
the
previousexample,
thegreaterX,
the
less
work the erosion/dilationdoes,
or themore theinput
signalis
preserved.As
X
approaches
infinity,
nofiltering
is
done,
and asX
approacheszero,
thedata is
flattened.
One
wouldthereforelike
to adjustX
so that thestructuring
elementjust
"fits"into
thesignal,
and notinto
thenoise.Since
theshape ofthestructuring
element plays as much of a role asits
size,
one wouldlike
the shape tobe
of a type thatwill notfit into
thenoise,
but
willfit into
the typeofimage
one wishestopreserve.The structuring
element shownin Figure 3-1 is
notideal
for
thetype of
images
found in
STM,
asit
tends tofit
slightly
into
the noisefound in
theseimages,
andbecause
its
shapeis
not characteristicofthe typeofstructuresfound
in STM.
The underlying
principle ofpseudoconvolutions,however,
willprovevery
useful.3.3
Morphological
pseudoconvolutiondevelopment
Pseudoconvolutions
aredefined in
thefollowing
manner:Given
afilter
(here
wewilltake
*F
tobe
themean ormedian),
thecorresponding
pseudofilter*'
is
givenby:
*F'(f)
=min{
f
e,
max[f0
e,
^(f)] },
(8)
where
f
is
theimage
and eis
thepreviously defined structuring
element.In
words,
thepseudofilter
is
computedby
taking
the
maximum oftheimage filtered
by
*
andtheerodedimage,
andthen theminimum ofthisresult andthedilated image.
In
effect weemploy
theoriginal
filter,
and thenbring
back
moreinformation
ofhigher
variation, the
amount ofwhichcorrespondsto thechoice of
A,.
In particular,
alllocal
variation smallerthanlambda
is
preserved.Figure 3-2
gives an example ofthis
process on a signal.As
mentionedpreviously,
the mainimport
of morphological pseudoconvolutionsis
theirability
to preserve certain classes of structure that would otherwisebe
lost
under theoriginal
filter.
In
particular,
oneproperty
thatmorphological pseudoconvolutions possessis
that theinvariant
class of structure ofthe originalfilter *P
willbe
a subclass oftheinvariant
class of thecorresponding
pseudofilter,
orInvOP)
InvOF').
For
example,
since steps are
invariant
under medianfiltering,
steps are alsoinvariant
underpseudomedians,
but fast
varying
texturemay
alsobe
invariant
under pseudomedians.Another
desirable property
of pseudoconvolutionsis
thatif
the originalfilter is
morphological,
then thecorresponding
pseudofilterwillalsobe
morphological.This
is
ahighly
desirable property
indeed,
asthebasic
propertiesintrinsic
tomorphologicalfilters,
namely
translationinvariance
andincreasing,
areimportant (and
oftenassumed)
whenfiltering
is
performed.Notice
alsothatif
X
=0,
thentheoriginalfiltering
operation(mean
ormedian)
is
theresult.Conversely,
asX
tends towardinfinity,
no matter whatfilter
has
been
chosen,
theresultwill
be
theoriginal unfilteredimage.
A
properX
mustthereforebe
chosentobring
back
anacceptableamount of
image
whilenotbringing
back
toomuchnoise.Pseudoconvolutions
areespecially
effective on noisethatis
notnecessarily
uniformacrossthe
image,
but
occursin
bursts.
This is because they
act somewhatadaptively
acrosstheimage;
thatis,
they
treat some parts oftheimage
thatmay
contain more orless
noisedifferently
thanother parts.This
is
very
usefulin
STM/AFM
imaging
because
the typeofnoise processesthatoccurtend to
be
highly
non-uniform.If
thestructuring
elementfits in
one region ofthe
image,
nofiltering
willbe done.
If,
however,
in
a noisier region oftheimage
thesamestructuring
elementdoes
notfit,
filtering
willbe
performed.This
is
seenin
Figure
3-1.
Pseudoconvolutions
are generalin
the sensethatany filter may be
usedin
conjunctionwiththem, (for
example wecouldhave
a pseudoWienerfilter)
but here only
pseudomeans andpseudomedians are studied.
The decision
touseeitherthepseudomeanorpseudomedianis
made
based
ontheimage
and noisecontent,
following
thesame guidelines aswhethertousethemean or median.
If,
for
example,
theimage
is
slowly varying
andhas
whitenoise,
the
pseudomean shouldbe
used;if, however,
there aredesired
edges and/or undesiredsparse point noise
in
theimage,
the pseudomedianis
preferable(see
Refs.
3,
4 for
astatistical analysis
regarding
thischoice).0>
8 o
<u
<+H~
<D
3
?* T3
Figure
3-1: Example
of erosion anddilation
on a signalby
the
structuring
element
e,
wheref(x)
is
the
input. The
erosion eliminates maximum noiseand
the
dilation
eliminates minimum noise.X
<4*
Input
V
t
t
t
Q
t<D
X
e
#
^_G
s
t
-*-t
t
Output
Figure
3-2: Example
of athree
point pseudomean ona
signalby
the
structuring
element e.
Chapter 4
Application
of
morphological
pseudoconvolutions
to
STM/AFM images
and experimental results
4.1
Adaptation
of morphologicalpseudoconvolutions
to
STM/AFM
type
data
Upon
first using
pseudoconvolutions,
improvement
overpreviously
used methods wasfound
in many
images.
Because
ofthe sharpness of thestructuring
element,
however,
some
line
noisein STM images
resulting from
the1/f
noise wasbrought
back
along
withuseful
information,
the reasonbeing
that thesharp
peak of thestructuring
elementincreasingly
fits into
thelines
asX
grows.White
noise wasalso noticeable asX
increased,
for
thesame reason.The structuring
elementintroduced in Refs.
3, 4,
5
was usedtobring
outhighly
varying
texture
information
and preserve edgesin
the presence of point noise.STM
andAFM
images
do
notgenerally have
edges or texture assharp
as thatdiscussed in Ref. 3.
Because
most structureshave
a rounded shape(atoms,
absorbedmolecules, etc.), the
structuring
elementis
chosentobe
a paraboloid of revolution sampledin
a5x5
mask(see
Figure 4-1).
This
allows thestructuring
element tofit
very
wellinto
thedesired
information,
while notpassing
thenoise.With
its
newshape,
thestructuring
element probes morefully
thedata,
andthus
point noiseand white noise are not allowedthrough.
The
desired
image, however,
is
restored(with
relatively
smallX)
andthereis
strong
improvement
overthemean and median.Also,
withthe
newfilter
mask(using
25
pointsas opposedto5
in Ref.
3),
line
noiseis
not allowedthrough,
because
thestructuring
element nowdoes
notfit into
thinlines.
Although
rigor
will notbe
providedhere,
the same properties asin Ref.
3
hold
truefor
pseudoconvolutions with the new
structuring
element.As
X
approachesinfinity,
for
example,
thefilter becomes very sharp
and thus all ofthe originaldata
passes.This
process,
however,
is
notas abrupt aswiththepreviousstructuring
element4.2
Development
of criteriafor
evaluating
pseudoconvolutionsThe
usual methodfor evaluating
theeffectivenessofafilter in
a certain environmentis
tofirst
create noise andideal image
modelsappropriateto thatenvironment.The
noise modelis
addedto theimage
modelandthisnoisy image
is
filtered
by
thefilter
in
question.The
filtered image
is
thencomparedto theideal image.
4.2.1
Mean
Square Error
- metricfor
comparisonThe
standardbasis
ofcomparisonis
globalMSE.
The
globalMSE
is found
by
taking
thesquared
difference
between
each pixelin
theideal image
andthecorresponding
pixelin
thefiltered
image,
summing
all ofthese
squareddifferences,
anddividing
thesumby
the totalnumberof pixels.
As
mentionedbefore,
thisis
notnecessarily
thebest
metricfor
theeye.We
will useit, however,
for lack
of abetter
one,
and alsotofollow
standard practices.One
mustkeep
in
mind,
however,
thatedges,
whichhave
astrong
influence
on oureyes22and on quantitative
analysis,
willnothave
astrong
influence
ontheoverallMSE
because
they
do
not makeup
much ofthe totalimage
area,
andthus theblurring
ofthemwill notmake much of an overallnumerical
difference. In
ordertoillustrate
thispoint,
theMSE
ofa
filtered
edgeis included
in
theanalysis.4.2.2
Development
of noise modelOften,
thenoise of animaging
systemis
estimatedby
collecting
severalimages,
averaging
them,
andsubtracting
this averagefrom
one oftheimages.
Such
an approach presentsstatisticalpitfalls.
In
particular, there areproblemsin applying
thismethod toSTM
andAFM images.
Thermal
and piezoelectricdrifts
cause offsetsin
subsequentimages
andmake apixel
by
pixelcomparisondifficult. For
example,if
theimage
consists ofa numberof
bumps,
and thebumps do
not alignbetween
images,
then the misalignment of thebumps
willappear as noise.Another
possiblemethod,
usefulif
theimage
information
is
at alower
frequency
than thenoise,
is
tosubtractthemoving
average oftheimage from
theimage
itself. One
advantageof pseudoconvolutions over
linear
filters,
however,
is
theirability
to preservestep
information.
If
thisaveraging
methodis
used,
step
information
in
theoriginalimage
wouldappear as noise
in
thefrequency
domain.
This
is because
steps wouldbe
averagedout,
asthey
consist ofhigh
frequency
information.
Because
of theseproblems,
the source ofnoise
in
STM images
willbe
considered,
andthen thisnoise willbe
simulated.The
noisein
STM images
arisesfrom
a number of sources.These
sourcesinclude
noisegenerated
by
thefeedback
circuit,
inadequate
damping
ofvibrations,
variationin
the tunneljunction
characteristics,
and piezoelectric andthermaldrift
overtime.The
magnitude ofeach ofthese noise sources
depends
on theconditionsunder which thesampleis imaged.
For
example,
mobile moleculesentering
thetunneling
gap
willin
general cause spikesin
the
tunneling
current andthussome responsein
thetip
tosamplespacing,
whilein
vacuumthisnoise source will
for
themost partbe
absent.The
size ofthescan andtherate at whichthescan
is
takenwilldetermine
thespatialfrequency
ofthenoise.As
an extremeexample,
thermal
drifts
canbe
eliminatedby
scanning
thesamplesufficiently fast.
Many
physical processes possess a1/f
noisespectrum,
whichhas been
observedin
theSTM
current spectrumby
a number ofresearchers.2'11'12'13However,
sincetheimage is
collected
in
time as the probeis
scannedalong
the xaxis,
there willbe
no correlationbetween
adjacent points ondifferent
scanlines,
andthenoise spectrumalong
they
axis willbe
whitenoise.1The
noise spectrumis
thereforeobtainedby
generating
white noisein
thespatial
domain,
taking
theFourier transform, multiplying
each point ofthe
Fourier
transform
by
l/kx,
wherekx
is
thewave numberin
thescandirection,
andthencalculating
the
inverse
Fourier
transform.4.2.3
Development
ofimage
modelBecause
STM
andAFM
image features
are not always observablewithothertechniques,
it
is
not always clear what anideal image
shouldlook like. The
graphite surfacehas
been
extensively
studiedin
air and sources of noisehave been discussed
by
several researchgroups.26
An ideal image
consist of atriangularlattice.
However,
asymmetrictipsoftencause a
distortion
oftheseatoms.27Even
though the
image
is
distorted,
thesymmetry
ofthe graphite
lattice
constrains the structure tohave 3-fold
symmetry,
and since themicroscope can resolve
only
thelowest Fourier
components,
animage is
wellapproximated
by
a summation ofthreecosinewavesoriented at120
degree
angles.27This
sum
is
therefore
taken torepresenttheideal image.
It
is displayed in Figure
4-2,
along
witha cross section of the
data.
4.3
Comparison
of pseudoconvolutions withWiener
filtering
Pseudoconvolutions
are compared with theWiener filter
methoddiscussed
by
Park
andQuate1
both qualitatively
and relativeto
MSE. In
theirmethod,
thespectrumis
chosen as1/f
in
thekx
(scan)
direction,
andindependent
offrequency
(white)
in
theky
direction. It
should
be
notedthatthey
employ
aform
ofthe
Wiener
filter
thatdemands
uncorrectednoise,
whereas1/f
noiseis
astrongly
correlated noise.The
powerspectrum estimateofthe(uncorrupted)
image is
taken tobe
that
of theinput
(noisy) image,
and aweighting
parameteradjuststheamplitude ofthe noisesothat
from
eqn.(5)
thefilter
reads:G(kx'ky)=l
+[(a/kx)/Su(kx,ky)]
>(9)
where a
is
the parameterin
question,
and wehave
divided
throughby
Su
to show thedirect influence
ofthe
noise spectrum andthesignal spectrum onthefilter.
In
practice,
ais
adjusted until an acceptable amount of
filtering
has been
accomplished.Several important
points shouldbe
made clearhere.
First,
for
reasonspreviously
discussed,
a graphitelattice is
chosenfor
theimage
model.As
canbe
seenin
Figure
4-2,
this model
is
notonly slowly
varying,
but is
alsoessentially
wide-sense stationary.The
model
is
therefore
an excellent environmentfor Wiener
filtering.
Second,
thel/fx
noisehas
been
chosenandimplemented,
and aWiener filter
thatcontains asits
noise estimatethevery
same noise presentin
theimage has
thenbeen
chosen.In
a realimaging
environment,
the noisewillnot
be
givenin
this manner andwillmostlikely
change,
thereby
making
aperfect noise estimate unobtainable.
In sharp
contrast to the presentWiener
filter,
pseudoconvolutionswill remain
symmetric,
andthuswillfilter equally in both
thex andy
directions.
Lastly,
thenumericalbasis for
comparisonis
MSE,
whichis
notonly
alinear
metric,
but
is
precisely
the
goodness measuretheWiener filter
minimizes.The
presence of1/f
noise causeshorizontal lines
to appearin
the
input image.
This
is
refered to as scan-line noise.
Three
noise conditions aresimulated,
in
whichonly
theamplitudeof
the
scan-line noiseis
adjusted.For
noiseA,
the
amplitudeofthe
scanlines is
half
ofthe amplitudeoftheimage,
for
B
the amplitudeof the scanlines is
approximately
thatofthe
image,
andthe amplitudeof noiseC is
1.5
times
that oftheimage.
4.4
Results
and
analysis
for
graphite
modelBecause
the simulated graphiteimage has
no edges and the simulated noise contains nopoint
noise,
the appropriatefilter is
the
pseudomean.Also,
aandX
have
been
adjustedtominimize
MSE in
theWiener filter
andthepseudomean,
respectively.One
shouldkeep
in
mind thatthemotivation
behind
using
pseudoconvolutionsis
notsimply
tominimizeMSE
as
in
theWiener
case,
but
tobring
back
usefulinformation
without a greatdeal
of noisegain.
We
expect a plot ofMSE
versusX
tostay relatively
constant over a range wherethestructuring
elementfits into
theimage
and not the noise.For larger
X,
the
structuring
element
fits into
the noise andMSE
willincrease
morerapidly
withX.
If MSE
staysconstant or goes
up
slowly
while we acquire moreinformation,
thisis fine.
If MSE
actually
goesdown
withincreasing
X
for
aregion,
thisis
all thebetter.
A
cross sectionthrough themaxima ofFigure 4-2
givenby
cos(47ty/3a)
+2cos(2rcy/3a),
wherea
is
the nearest neighbor carbon atomspacing,
showstwovalleys,
one shallow andone
deep.
The
difference
in
thedepth
of the valleysis due
to thehigher
frequency
component;
however,
the shallower ofthe twois
moredifficult
to recover afteradding
noise.
A
gauge of afilter's
effectivenessis
the
restoration ofthe
smallvalley
uponfiltering.
This is
exacdy
the type ofinformation
thatmay
be
important
in
orderto gaingreater
knowledge
of a real sample.Figures
4-3(a)
and4-3(b)
showthe
plotsfor
noisetypes
A, B,
andC,
for
theWiener
filter
and
pseudomean,
respectively.A
5x5
mean was usedfor
the pseudomean.MSE
is
normalized
by dividing by
theMSE
oftheunfilterednoisy
image.
This is
thenmultipliedby
100
toshowthepercentage oftheunfilteredMSE.
Referring
to thecurvefor
noiseA,
theminimumMSE
is
almostidentical
for both
filters.
Note
the
drop
of about3%
MSE
in
the
pseudomeanfrom
themean atX
=30.
The
drop
in
MSE
from
the originalnoisy
image is
approximately
67%
for
both.
The fact
that thepseudomean performs aswell as the
Wiener
filter in
thisenvironment relativetoMSE
is
remarkable.
Figures
4-4(a), 4-4(b)
and4-4(c)
show the unfilteredimage
with noiseA,
theoptimally Wiener
filtered
image,
andtheoptimal pseudomeanfiltered
image,
respectively.Even
though thedrop
in
MSE
is
thesamein
both
cases, the
Wiener
filter has introduced
anasymmetry
into
the
previously
six-foldrotationally
symmetricgraphiteimage. The
graphiteimage
arisesfrom
the sum of six equal amplitudeFourier
coefficientslocated
on thevertices ofahexagon.27
However,
Wiener
filtering
decreases
theamplitudeoftheFourier
coefficients with a
larger
kx
relative to those with a smallerkx.
The
same generalphenomenoncan arise
from
multipletips,27and gives riseto the triangulararray
of ellipsesin
Figure
4-4(b). The
effect ofWiener
filtering
can alsobe
seenin
thecross section.For
acrosssectionperpendiculartothe
scanning
direction,
theamplitudeis
given ascos(4rcy/3a)
+ 2cos(27cy/3a).
The first
term,
whichis
ofhigher
spatialfrequency,
is decreased
in
the
filtering
and this eliminates the shallow valley.In
Figure
4-4(c), however,
theimage
filtered
by
the pseudomean moreclosely
resembles the originalimage.
The
6 fold
rotational
symmetry
is
maintained and the cross section shows that the smallvalley is
pre