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Medianalgorithmtotheinterpolationof2-Ddigitaldata
19
expectedvalueofE{ε(m,n)}=0,thenthesumofsquaresoffixedcoefficientsin(1)
willbeameasureofnoiseamplificationinpolynomialinterpolationalgorithm[2]:
σ
2
y
=
∑∑
α
2
k
,
l
(
p
,
q
)
σ
2
=
σ
2
∑∑
α
2
k
,
l
(
p
,
q
)
(2)
k
l
k
l
Theothercoefficientdetermininginterpolationquality(notonlyfordisturbed
pictures)mentionedintheintroductionwasinterpolationerror.Twoerrorindices
areusedinthiswork[3]:NAE(NormalizedAbsoluteError)andNMSE
(NormalizedMeanSquareError):
NAE
=
2
m
∑∑
M
=
−
1
1
2
n
N
=
1
−
1
X
d
(
m
,
n
)
−
Y
(
m
,
n
)
(3)
2
m
∑∑
M
=
−
1
1
2
n
N
=
1
−
1
X
d
(
m
,
n
)
NMSE
=
2
m
∑∑
M
=
−
1
1
2
n
2
N
m
=
∑∑
M
1
−
=
1
−
1
[
1
X
2
n
N
d
=
−
1
(
1
[
m
X
,
d
n
)
(
m
−
,
Y
n
(
)]
m
2
,
n
)]
2
Intheaboverelations,Y(m,n)isofthesameimportanceasin(1),whileXd(m,
n)isahighresolutionoriginalpictureconnectedwithalowresolutionoriginal
pictureX(m,n)bymeansofthefollowingrelation
1:
X
(
m
,
n
)
=
X
d
(
2
l
m
−
1
,
2
l
n
−
1
)
:
(5)
1
≤
m
≤
M
,
1
≤
n
≤
N
Puttingtogethersoachievedqualityindicescouldconstructabasisforchoosing
anoptimalalgorithmwithrespecttosmallnoiseamplificationcoefficientand
smallinterpolationerror.However,theseindicesdonotconsidercertainalgorithm
properties,visibleonlywhensubjectivelyassessingenlargedpicture.
Amplifiednoisecanbringhardlypredictableeffectsonoutputpicture.Asan
example,seeFig.1bwhereresultsoftripleinterpolationofapicturedisturbedwith
pulsenoiseareshown.
Onthepresentedphoto,a“whippingrain”effectcanbeclearlyvisible.This
effectoccurrencecanbejustifiedbyspecificamplitudeperformanceof
interpolationfilter[5].Errorsintroducedbyinterpolationfiltersshownherefora
chosenpolynomialinterpolationalgorithmonly,areinducingtosearchother,more
efficientsolutions.
(4)
1Itisassumedinfurtherexperiments,thatahighresolutionpictureisknownaswellasa
lowresolutionpictureareknown.Suchassumptionhasbeenmadeinordertoenableto
verifyresultsobtained
