ref 27 improved minimal inter-quantile distance method for blind estimation of noise
TRANSCRIPT
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Improved minimal inter-quantile distance method for blind estimation
of noise variance in images
Vladimir V. Lukina, Sergey K. Abramov
a, Alexander A. Zelensky
a, Jaakko T. Astola
b,
Benoit Vozel
c
, Kacem Chehdi
c
aNational Aerospace University, 61070, Kharkov, Ukraine;
bTampere University of Technology, Institute of Signal Processing,
P.O. Box-553, FIN-33101, Tampere, FinlandcUniversity of Rennes 1 - TSI2M, 22 305 Lannion Cedex, BP 80518, France
ABSTRACT
Multichannel (multi and hyperspectral, dual and multipolarization, multitemporal) remote sensing (RS) is widely used in
different applications. Noise is one of the basic factors that deteriorates RS data quality and prevents retrieval of useful
information. Because of this, image pre-filtering is a typical stage of multichannel RS data pre-processing. Most efficient
modern filters and other image processing techniques employ a priori information on noise type and its statistical char-acteristics like variance. Thus, there is an obvious need in automatic (blind) techniques for determination of noise type
and its characteristics. Although several such techniques have been already developed, not all of them are able to per-
form appropriately in cases when considered images contain a large percentage of texture regions and other locally ac-
tive areas. Recently we have designed a method of blind determination of noise variance based on minimal inter-quantile
distance. However, it occurred that its accuracy could be further improved. In this paper we describe and analyze several
ways to do this. One opportunity deals with better approximation of inter-quantile distance curve. Another opportunity
concerns the use of image pre-segmentation before forming an initial set of local estimates of noise variance. Both ways
are studied for model data and test images. Numerical simulation results confirm improvement of estimate accuracy for
the proposed approach.
Keywords:inter-quantile distance, noise variance evaluation.
1. INTRODUCTION
An advantage of modern RS systems is that they are able to provide potential users by information valuable for such
important applications as meteorology, environment monitoring, pollution detection, agriculture, etc.1,2,3 To ensure wid-
er capabilities of remote sensing, modern RS systems are commonly provided by multichannel (multispectral, hyper-
spectral) operation facilities, and a general tendency is to increase the channel number. The examples of such systems
are AVIRIS4, CHRIS-Proba5, etc.
Meanwhile, increasing the number of channels (sub-bands) results in considerably increased complexity of different
stages and procedures of RS data processing: filtering, edge and object detection, and compression 6-9. Simultaneously
with demand to process data as quickly as possible, this explains a need in design of blind methods for different stages
of multichannel RS data processing.
In this paper, we consider a particular task of blind noise variance evaluation in images. Note that for multichannel RS
data this operation is to be carried out component-wise since statistical characteristics of noise in different component
(subband) images can vary a lot10,11. Besides, it is worth noting that in many practical situations noise is a dominating
a Correspondence to Lukin V.V.: e-mail [email protected] tel./fax +38 0573 151186c Work supported by the European Union. Co-financed by the ERDF and the Regional Council of Brittany, through the
European Interreg3b PIMHAI project.
Image and Signal Processing for Remote Sensing XIII, edited by Lorenzo Bruzzone,Proc. of SPIE Vol. 6748, 67481I, (2007) 0277-786X/07/$18 doi: 10.1117/12.738006
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factor degrading image quality. Finally, one should keep in mind that a priori knowledge of noise type and statistical
characteristics is required for the most effective algorithms of image denoising12-14, edge detection15,16, compression of
noisy images, etc.11,13,17 For all these applications it is desirable to provide high enough accuracy of noise variance eval-
uation18, otherwise the performance of image processing might inappropriately worsen.
Assume that noise type has been already pre-determined. This can be done, e.g., by means of the already designed meth-ods6,19-21. Then, the basic task is to evaluate the main parameters of the determined dominating factor, desirably in auto-
matic mode. Quite many blind techniques for evaluation of additive or multiplicative noise variance have been already
proposed22-32. A practical choice depends upon several factors. The most important factor is a priority of requirements to
the technique. So, let us remind the basic requirements. First, a technique should be applicable to images with different
structure, i.e., irrespectively, what is a percentage of pixels belonging to image homogeneous regions (IHR). Many exist-
ing techniques perform well enough if an image is not too textural. But they fail if the percentage of pixels that belong to
IHR is less than 3040% 22,24,26,28. Because of this, considerable efforts have been spent by us for designing the blind
methods able to operate well for highly textural images25,27,29,30. Second, a blind technique has to be able to perform
properly (with providing appropriate accuracy) in cases of both spatially uncorrelated and correlated noise, desirably
even if spatial correlation properties of noise are a priori unknown. While the methods 27,29 are more resilient to absence
of information on noise spatial correlation properties, the method 25 produces biased estimates of noise variance in case
of spatially correlated noise. The third requirement is to provide high accuracy of estimation for a wide range of possible
values of noise variance. As a marginal case, a method should produce near-zero estimates for noise-free images. In thissense, the methods27,29,31,32 are characterized by the best accuracy. Moreover, the methods31,32 are able to evaluate fluc-
tuative (additive or multiplicative) noise variance in cases of simultaneous presence of impulse noise. Finally, the fourth
requirement is to perform quickly enough. In this sense, the methods31,32 and the corresponding algorithms are slow
and require intensive computations. The method27 is faster but the technique29 is one of the most efficient. However, the
performance of the latter technique can be further improved.
This paper deals with considering modifications that can be done for the technique 29 in order to improve its accuracy.
The paper is organized as follows. Some common stages of blind evaluation of noise variance are described in Section
2. A model for local variance estimate distribution is also given. Section 3 describes the proposed approximation of
inter-quantile curve and its performance is studied for model data. Then, in Section 4, we test this modification for a set
of test images artificially corrupted by noise. Performance comparison for several methods is provided. Section 5 con-
tains description of another modification that deals with exploiting segmentation maps. Then, the conclusions follow.
2. COMMON STEPS IN BLIND NOISE VARIANCE EVALUATION AND LOCAL ESTIMATES
DISTRIBUTION
In general, one can mention two main approaches to blind evaluation of noise variance. As said earlier, estimation in
spectral domain 12,25 suffers from problems that arise in case of spatially correlated noise. In turn, methods operating in
spatial domain better cope with this frequently met phenomenon. Thus, let us concentrate on the latter approach. Ac-
cording to it, an image under interest at the first step is divided into bN overlapping or non-overlapping blocks of a
rather small size that tessellate this image. Typically a block size is 5x5 or 7x7 pixels, the latter is preferable for spatially
correlated noise. At the second step, the local meanlI and the local variance estimates
2 , 1,...,l b
l N = are calculated
for each l-th block. If the considered dominant noise is multiplicative, the only difference is that one has to calculate
local relative variance estimate as 2 2 2/l l lI = . Without loosing generality, let us further suppose that noise is pure
additive and we deal with a set of estimates 2 , 1,...,l bl N = .
Then, at the third stage, the obtained set of such estimates is to be processed in some way. At this stage there is a variety
of possible variants. However, almost all of the designed methods are based on the following fact. The local estimates
calculated for blocks that belong to IHRs (normal estimates) do not differ too much from the true value of noise variance2
tr and they form a mode of local estimates distribution. Other local estimates obtained in image heterogeneous (lo-
cally active) regions like edge and detail neighborhoods, textural regions, etc, are usually considerably larger than the
true value and such estimates produce a heavy one-sided tail of local estimates distribution. The shape and parameters
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of such tail depend upon image properties, noise variance, block size, etc. Examples of histograms of such distributions
have been provided in our previous papers7,23,29
. Moreover, it has been shown that normal estimates have practically
Gaussian distribution if block size is large enough (e.g., 7x7)23,30.
Thus, as in our earlier papers, let us use the following model for probability density function (PDF) of local estimates
( ) ( ) ( ) ( )2, 1 0,Gaussian Uniformx p m p M = + , (1)
where ( )2,Gaussian m denotes the Gaussian PDF with the mean m (2
trm = or 2=m for additive and multiplicative
noise, respectively) and variance 2 ; do not mix 2 with 2tr or2
), ( )0,Uniform is the uniform PDF within the
limits from 0 toM, p is the parameter characterizing the percentage of normal local estimates. The variance 2 in (1)
is determined by the used block size, noise PDF and variance, and spatial correlation properties 23,30. Other parameters of
the model (1) are the following:Mdescribes the range of possible variation of local estimates obtained in image hetero-
geneous regions (obviously, a local estimate is non-negative). For modeling different practical situations, it is possible to
vary the ratio M / m. For given block size and noise PDF, 2 is approximately proportional to 2m . The PDF
( )0,Uniform relates to variance estimates obtained in heterogeneous blocks. We used uniform PDF although other
appropriate model PDFs can be used without considerable influence on accuracy of noise variance estimates.
The aforementioned peculiarities of local estimates distribution have been put into basis of an idea that our task is to get
some accurate and robust estimate of this distribution mode and accept it as a final estimate of noise variance. The re-
cent reseach has been concentrated on analysis and design of such estimation techniques including the use of the sample
myriad23, bootstrap based approach27 and inter-quantile distance minimization29. This last modification has resulted in
obtaining quite accurate estimates even if the percentage p of local estimates calculated in homogeneous image blocks
is small, up top=0.1.
3. THE DESIGNED MODIFIED INTER-QUANTILE METHOD
Let us first briefly consider the approaches27,29 based on quantile and inter-quantile estimates. Assume that as an esti-
mate of noise variance we can use some quantile of the set of estimates
2
, 1,...,l bl N=
. The best (optimal) quantileindex
optn is such for which its bias and variance are minimal. However, it is a priori unknown and, in fact, it depends
upon bN and27,29. To prove the existence of such quantile, consider several sets of parameters (Cases) of the model
(1) presented in Table 1. These sets mimic different possible relationships between the parameters of the model (1). In
particular, the Cases 5 and 6 are the most unfavourable for majority of techniques of blind noise variance estimation
since for themp is rather small (this corresponds to highly textured images).
Table 1. The considered sets of parameters of the
model (1)
Model
Parameters p m 2 M / m
Case 1 0.9 100 800 100Case 2 0.8 300 7200 100
Case 3 0.65 100 800 100
Case 4 0.5 200 3200 100
Case 5 0.15 300 7200 100
Case 6 0.1 400 12800 100
Case 7 0.65 400 12800 10
0
200
400
600800
1000
1200
1400
1600
1800
2000
0 10 20 30 40 50 60
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Fig. 1. The aggregate errors for the quantile estimate as the
functions of quantile index %n for different Cases
%n
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Let us characterize the accuracy of estimation techniques by the following three quantitative parameters: the bias
m m = , the variance ( )22 m m = and the aggregate error 2 2 = + . Here m denotes the used estimate of
noise variance; is the expectation for an ensemble of realizations. We have also calculated the parameter rel as
( / ) 100%rel m = . Simulations have been carried out for 10000 realizations.
Detailed analysis of behavior of the bias, variance and aggregate error on% 100 / bn n N= (n denotes quantile index) is
presented in the paper27
. Therefore, here we represent only the dependences of the aggregate error on %n (Fig. 1).
They all have global minima for % /100 / 2optn p , and the estimates of noise variance obtained for the corresponding
quantile % /100opt b opt n N n= are practically unbiased.
The values%opt
n or, respectively,opt
n are to be estimated. An efficient way to do this is to apply minimal inter-quantile
distance (range) IQR approach29. Assume that by sorting an original set of local variance estimates 2 , 1,...,l bl N = in
ascending order we have obtained a sample bt NtX ,...,1,)(
= where )(tX denotes the t-th order statistic. Then it can be
expected that for PDF (1) the difference )()( tst XX + for fixed s (where s is an even integer) is the smallest in theneighbourhood of distribution mode. This assumption occurred to be true 29, and Fig. 2,a illustrates this for two different
values ofs: 2.0/;1.0/ == bb NsNs forCase 2. As seen, for both values of bNs / the plots have global minima for40/)2/(100 + bNst , i.e. for 2/4.0/)2/( pNst b + . And according to the plot of in Fig. 2 the quantile that has
the smallest forCase 2 is just .40% =n
Then, by finding such test ( sNt b = ,...,1 ) for which)()( tst XX + is the smallest one can get an estimate of distribution
mode and, respectively, noise variance. However, two questions arise: 1) how to produce a distribution mode estimate X
(i.e., m in the model (1)) knowing the interval [ ])()( ; stt estest XX + , and 2) what is the optimal or, at least, reasonable choice ofs? In the paper29, we have considered the simplest way to obtain X:
( / 2) estt sX X+
= wheres is even.
X(t+s)
-X(t)
0
20
40
60
80
100
0 10 20 30 40 50 60 70 80 90 100
100(t+s/2)/Nb
s/Nb=0.1
s/Nb=0.2
Fig. 2. The plots of
)()(
)2/(
tst
XXst=+
+
vs bNst /)2/(100+
(Case 2)
The way to selects was two-stage. More in detail, the estimation procedure29
(algorithm AIQR) was the following:
1) calculate )()( tst XX + ( sNt b = ,...,1 ) for [ ]21.0 bNs = and find suchinit
estt for which )()( tst XX + is the smallest;
2) calculate [ ]2
067.0444.1 binit
est
qopt Nts += , if [ ]2
1,1 => bqopt
bqopt NsNs where [ ]
2 denotes rounding off to a nearest
even integer;
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3) calculate)()( tst XX
qopt
+ (
qopt
b sNt = ,...,1 ), findfin
estt for which
)()( tst XXqopt
+ is the smallest and obtain the final
estimate as( / 2)fin qoptestt sX
+.
This algorithm has exploited the fact that optimals depends upon parameters of the model (1). This is clearly seen from
analysis of plots presented in Fig. 3. As seen, optimal ratios for which the aggregate errors are the smallest take place for
/opt bs N p .
0
50
100
150
200
250
300
0 10 20 30 40 50 60 70 80 90 100s%
IQR Case 1Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Fig. 3. The plots of IQR as the functions of bNs /100 for the considered Cases.
One of the main disadvantage of AIQR estimator is a rather large estimation variance29
. Our research has shown that the
reason of such behavior of the AIQR is rather large variance of the estimate initestt obtained for [ ]21.0 bNs = at the first
step of the algorithm. Note thatinit
estt is directly involved in calculation ofqopts at the second step of the algorithm and
the calculated qopts is used at the third step. In other words, errors at the first step propagate to the next ones. Then,
one can expect that if the estimates initestt will be more accurate, the final estimates of noise variance will be more accu-
rate as well.
One observation that follows from examples in Fig. 2 is that the curves )2/( st+ are noisy. To suppress this noise
(to reduce the curve fluctuations and, hence, to decrease the variances of initial and the final estimates) one should usesome preprocessing procedure. We propose to use a square (second order polynomial) LMS regression in the neighbor-
hood of the curve minimum (rough quantile number estimation / 2initestt s+ ). Then the coordinate of approximation curve
minimum fltestt is used to obtain corrected estimate( )fltesttX .
X(t+s)-X(t)
0
20
40
60
80
10 0
0 10 20 30 40 50 60 70 80 90 100
100(t+s/2)/Nb
s/Nb=0.1
s/Nb=0.2
Fig. 4. The plots of )()()2/( tst XXst =+ + vs bNst /)2/(100 + and their approximations (Case 2)The examples of regression curves for / 0.1
bs N = and / 0.2
bs N = are shown in Fig.4 (thick solid lines). One particu-
lar question is how to determine a range of the values tin which approximation is to be done? Our investigations have
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shown that optimal range for LMS approximation can be determined as 0.1( / 2);1.9( / 2)init init
est est t s t s + + . Such choice
for majority of Cases provides the best accuracy of resulting estimates of inter-quantile curve global minimum. The
method based on such approach of inter-quantile range filtering is further referred as IQRF.
Consider now the performance of the designed algorithm. First, let us compare the accuracy of initial estimates for
IQR29 and IQRF. The data for different values ofp are given in Table 2. We have controlled such additional statistical
parameters: the mean%
n of % 100( / 2) /est bn t s N = + and its variance%
2
n . Besides, we present the results for optimal
values of parameter % 100 /opt opt
bs s N= whereopts is the optimal value ofs that produces the smallest aggregate error.
Below for all considered statistical parameters we use subscripts to denote their relation to the corresponding method.
As seen, the obtained values of the optimal quantile%IQR
n are characterized by better accuracy smaller variances
%
2
IQRn and 2IQR . Bias values for both estimators are close to zero. As the result, the values
rel
IQR for the new estimator
IQRF are smaller than for the IQR estimator.
Table 2. Accuracy of the IQR and novel IQRF estimators for the different p ( m =10; 2 =8; m =100)
IQR estimator IQRF estimatorp
%
opts %IQRn %2
IQRn
IQR 2
IQR IQR
rel
IQR %opts %IQRFn %
2
IQRFn
IQRF 2
IQRF IQRF
rel
IQRF
0.90 88.0 45.02 0.098 0.013 0.015 0.015 1.23 89.5 45.00 0.178 -0.0069 0.014 0.014 1.18
0.75 73.4 37.73 0.253 0.019 0.019 0.019 1.38 74.5 37.82 0.206 -0.0069 0.018 0.018 1.34
0.65 63.7 32.80 0.338 0.022 0.022 0.023 1.52 64.5 32.91 0.287 -0.0068 0.021 0.021 1.45
0.50 48.2 25.44 0.420 0.023 0.030 0.031 1.76 49.5 25.58 0.333 -0.0010 0.027 0.027 1.64
0.25 23.7 13.23 0.436 0.022 0.063 0.063 2.51 24.6 13.24 0.354 -0.0089 0.058 0.056 2.37
0.15 13.7 8.32 0.373 0.023 0.114 0.115 3.39 14.6 8.29 0.302 -0.0085 0.098 0.098 3.13
0.10 9.1 5.87 0.326 0.023 0.185 0.186 4.31 9.2 5.82 0.244 -0.0204 0.157 0.158 3.97
Based on IQRF estimation, an automatic inter-quantile range filtered (AIQRF) estimator exploiting the same idea as
AIQR estimator has been proposed
29
. As well as the AIQR estimator, the AIQRF consists of two stages obtaining theinitial estimate and the final one. More in details, it is the following:
1) apply to the sample at hand the IQRF estimator with parameter [ ]2
1.0 bNs = and obtain the initial estimation of
minimal inter-quantile range left boundary 2I flt Iest est t t s= ;
2) calculate quasi-optimal inter-quntile range2
1.835 0.079qopt I est b
s t N = + , if [ ]21, 1qopt qopt
b bs N s N> = ;
3) apply to the sample the IQRF estimator with qopts s= , find flt IIestt and obtain the final estimate as ( )flt IIesttX .Table 3. Accuracy of the AIQR, IQRF and AIQRF estimators for the different Cases
AIQR estimator IQRF estimator (with optimals) AIQRF estimatorCase
AIQR 2
AIQR AIQR rel
AIQR IQRF 2
IQRF IQRF rel
IQRF AIQRF 2
AIQRF AIQRF rel
AIQRF
1 -0.09 6.27 6.28 2.51 0.04 1.44 1.44 1.20 -0.13 1.59 1.61 1.272 -0.39 56.70 56.85 2.51 -0.40 14.78 14.95 1.29 -0.50 16.55 16.80 1.37
3 -0.17 7.53 7.56 2.75 -0.17 2.14 2.17 1.47 -0.18 2.24 2.28 1.51
4 -0.43 35.76 35.95 3.00 -0.30 11.39 11.48 1.69 -0.39 11.81 11.95 1.73
5 -3.59 185.97 198.82 4.70 -0.13 90.55 90.56 3.17 -1.60 95.18 97.75 3.30
6 0.50 294.65 294.90 4.29 -0.85 259.13 259.86 4.03 -0.97 253.41 254.36 3.99
7 -0.56 105.05 105.36 2.57 -0.31 32.88 32.98 1.44 -0.48 34.36 34.59 1.47
Simulation results for the old estimator AIQR29
and the designed estimators for different Cases are presented in Table 3.
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As seen, the proposed modified estimator AIQRF outperforms previously designed algorithm AIQR and provides com-
parable results with IQRF in case of optimal selection ofs. The relative aggregate error relBM for AIQRF is considerably
(by approximately two times) reduced in comparison to AIQR.
4. ANALYSIS OF THE PROPOSED METHOD PERFORMANCE FOR TEST IMAGES
Consider now the accuracy of the proposed method for a set of conventional test images (Peppers, Barbara, Baboon,
Goldhill). Additive noise (Gaussian, i.i.d.) with zero mean and different variances2tr (50, 100, 400) has been added to
these images. Estimates of noise variance have been obtained for a large number of noise realizations and then statistical
characteristics of estimates have been calculated. Also, we have processed noise-free images for which an estimate is
characterized by the only parameter its bias . The obtained results are given in Table 4.
Table 4. Accuracy of the AIQR and proposed AIQRF techniques
for different test images and variances of additive noise
AIQR estimator AIQRF estimator
Image
2tr
%AIQRn
%
2
AIQRn
AIQR 2
AIQR AIQR
rel
AIQR %AIQRFn
%
2
AIQRFn AIQRF
2
AIQRF AIQRF
rel
AIQRF
0 16.99 9.79 18.90 10.44
50 28.87 2.52 12.80 1.98 165.69 25.74 31.53 0.26 14.87 0.36 221.4 29.76
100 32.19 1.79 13.10 3.50 175.14 13.23 34.92 0.12 16.49 0.61 272.4 16.51Peppers
400 37.46 2.41 6.21 36.33 74.83 2.16 40.30 0.13 16.62 5.43 281.5 4.20
0 7.72 4.24 8.95 4.56
50 19.13 0.74 6.99 0.94 49.81 14.12 20.89 0.08 8.83 0.26 78.30 17.70
100 21.80 1.23 7.68 4.78 63.74 7.98 23.55 0.10 10.67 0.96 114.8 10.72Barbara
400 28.63 1.96 12.42 57.11 211.31 3.63 31.55 0.21 27.11 11.64 746.5 6.83
0 6.87 28.33 8.66 32.36
50 11.27 0.87 36.44 9.90 1337 73.13 12.78 0.21 41.27 3.80 1707 82.63
100 13.95 0.79 43.25 16.73 1888 43.45 15.63 0.26 49.92 6.70 2499 49.99Baboon
400 22.31 1.71 75.63 125.84 5845 19.11 25.16 0.40 98.55 36.47 9748 24.68
0 5.35 4.54 5.42 4.61 50 21.92 2.46 21.15 5.43 452.79 42.56 24.76 0.48 25.01 1.31 626.7 50.07
100 26.87 1.96 25.93 7.09 679.43 26.07 29.95 0.34 31.47 1.97 992.1 31.50GoldHill
400 36.49 2.03 35.04 44.19 1272 8.92 39.65 0.20 48.31 7.27 2341 12.10
Let us analyze the obtained data. As seen, for noise free images both considered methods produce noise variance esti-
mates that are not equal to zero (bias valuesAIQR
andAIQRF
are positive). There are several reasons behind this fact.
First, real life images practically always contain some noise (even those ones commonly used in experiments) and/or
they do not have ideally homogeneous regions. Second, local estimate distribution for noise-free images does not obey
the model (1) on which we have relied in design of our methods for blind evaluation of noise variance. One interesting
observation is that bothAIQR
andAIQRF
are larger for images that contain more textured regions. Since%AIQR
n and
%AIQRF
n can serve as some estimates of the percentagep of IHRs, it follows that the images Baboon and GoldHill are the
most textural. This is in good agreement with visual inspection of the considered test images.
For noisy images we have got, in some sense, quite surprising results. Recall that for simulation data for the model (1)rel
BM for AIQRF was considerably smaller than for AIQR (Table 3 in Section 3). However, for the data in Table 4 the
situation is the opposite for practically all test images and additive noise variance values. Analysis shows that this is due
to the fact thatAIQR
is smaller thanAIQRF
. Note that at the same time2
AIQR is considerably larger than2
AIQRF . In
other words, due to using polynomial approximation we have reduced estimate variance but increased its bias.
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To understand why this has happened, we have, first, returned to the model (1) again. Its analysis shows that approxima-
tion of normal local estimate distribution by Gaussian ( )2,Gaussian m is not absolutely adequate. In fact, this distribu-tion is slightly asymmetric23 and its approximation by the second-order polynomial produces slightly biased estimates of
distribution mode. Another reason could be that another term in the model (1), ( )0,Uniform , can not be well enough
approximated by uniform distribution for practical distributions of local estimates for real life images. Thus, we decided
to somehow additionally reduce this negative influence of abnormal estimates obtained for blocks placed in image het-
erogeneous regions.
5. METHOD BASED ON USING PRE-SEGMENTATION MAPS
Suppose that we have managed somehow to carry out discrimination between image homogeneous and heterogeneous
regions and, after this, have obtained a set of local variance estimates 2 , 1,...,lh bh
l N = only for blocks that belong to the
determined IHRs. Obviously, in this case a number of such blocksbh
N occurs smaller than the maximally possible
number of blocks that can tessellate an image. Then, a question is how to perform discrimination between image homo-
geneous and heterogeneous regions without knowing noise type and variance.
There exist many image segmentation methods able to indicate and localize quite large image homogeneous regions
characterized by compactness of pixels that are referred to the corresponding areas33
. Many methods of image segmen-
tation require a priori information on noise type and variance34,35 and they can not be directly applied in our case since
we do not know noise variance a priori. Fortunately, there are image segmentation techniques that do not require know-
ing noise type and statistical characteristics a priori36
. Among them, we retain a recent unsupervised one, based on a
variational classification method of observed pixels, following a preliminary optimized histogram transformation by
gravitational clustering. In the first stage, the original histogram is greatly reduced to highlight only pertinent modes of
the observed image. This is achieved by progressively decreasing the dispersion of the initial modes with regard to their
relative centres of gravity. Then, the best thresholds and the best modes are obtained by alternate optimization of some
energy of multi-thresholding. This energy measures the quality of a map of homogeneous regions as a function of the
intra-area variance of the detected regions. An area is decided to be uniform (homogeneous) if the dispersion of its grey
levels is sufficiently low. In the second stage, a supervised variational classification method to which it is sufficient to
give the previously obtained set of representative modes of the classes and which takes into account an a priori homo-
geneity constraint is applied to obtain the final result.
Let us give an example of segmentation method operation. Fig. 5,a presents the original (noise-free) test image Baboon.
Its noisy version is given in Fig. 5,b (additive noise variance is equal to 100). Image segmentation result obtained by the
method36 is represented in Fig. 5,c. Pixels that belong to the same region (segmentation class) are shown by the same
intensity in gray scale representation. For getting discrimination maps, we have applied the following post-processing of
segmented image. Within 5x5 scanning window, it has been tested to how many segmentation classes the pixels belong.
It has been done using the following rule for all scanning window positions:
a) if all scanning window pixels belong to the same segmentation class, then this scanning window position corre-sponds to homogeneous region (and the corresponding pixel of discrimination map 128
ijDM = );
b) if scanning window pixels belong to two different segmentation classes, then this scanning window positioncorresponds to edge/detail neighborhood (and the corresponding pixel of discrimination map 0
ijDM = );
c) if scanning window pixels belong to three or more segmentation classes, then it is supposed that this scanningwindow position corresponds to texture (and the corresponding pixel of discrimination map 255ijDM = ).
The obtained discrimination map is presented in Fig. 5,d. Pixels that belong to detected homogeneous regions are
shown by gray color. As seen, the proposed two-stage procedure of image processing (segmentation+discrimination)
determines homogeneous and quasi-homogeneous areas quite well. And the percentage of pixels that belong to these
areas for the test image Baboon is rather small. Note that image segmentation and, thus, discriminations results depend
upon noise level.
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rE 4 : ? T
:
4
4
- .
f r -
The obtained discrimination map ijDM can be further used for blind evaluation of noise variance. We propose to
calculate local estimates 2 , 1,...,lh bh
l N = only for those scanning window (block) positions for which 128ij
DM = . As
seen, segmentation method localizes image homogeneous (or, at least, quasi-homogeneous) regions shown by gray color
well enough.
a b
c d
Fig. 5. The results of image region discrimination: a) the original (noise-free) image;
b) the noisy image; c) the segmentation result; d) the obtained discrimination map
Then, the method based on image pre-segmentation and homogeneous region detection that we have called AIQRF-
HRD contains the following stages:
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1) perform image pre-segmentation, find blocks that fully belong to the pre-determined IHRs, and obtain for thema set of local variance estimates 2 , 1,...,
lh bhl N = ;
2) make up sorting of the estimates 2 , 1,...,lh bh
l N = in ascending order with obtaining ( ) , 1,...,tbh
X t N= ;
3) calculate )()( tst XX + ( 1,...,= bh
t N s ) for [ ]2
1.0 bNs = and find suchinit
estt for which
)()( tst XX + is the
smallest;4) approximate )()( tst XX + curve (as the function of index t) in the neighborhood
0.1( / 2);1.9( / 2)init init est est t s t s + + of 2init
estt s+ and find its minimum
flt I
estt analytically as described in Section 3;
use this minimum to calculate the first stage quantile index 2I flt I
est est t t s= ;
5) calculate quasi-optimal inter-quantile range2
1.835 0.079 = + qopt I
est bhs t N , if [ ]
21, 1> =
qopt qopt
bh bhs N s N ;
6) repeat one time the steps 3 and 4 for qopts s= and determine the second stage inter-quantile curve minimumflt II
estt and quantile index 2
II flt II qopt
est est t t s= ;
7) obtain the final estimate as ( / 2)II qoptestt sX + .Consider now the performance of this method for the test images. The obtained simulation data are presented in Table 5
and they can be easily compared to results given for other techniques in Table 4. This comparison shows the following.First, for noise-free images, estimates have become more accurate (the values AIQRFH are smaller than AIQR and
AIQRF for the corresponding images). Besides, the values
%AIQRFHn are larger than
%AIQRFn for all considered images
and noise variance values. This indirectly shows that the majority of obtained estimates 2 , 1,...,lh bh
l N = can be consid-
ered normal (histogram analysis of these estimates is presented in our paper30 and it shows that this conclusion is true).
In case of noisy images, estimates of noise variance have become more accurate than for the technique AIQRF. Bias
absolute values for the AIQRF-HRD have considerably decreased in comparison to the method AIQRF. There is no
obvious tendency to be observed in comparing variance values for both techniques. However, comparing the relative
aggregate errors for the corresponding situations (the same image and the same noise variance), we can state that accu-
racy of AIQRF-HRD is sufficiently better.
Table 5. Accuracy of the proposed AIQRF technique with homogenous region detection (AIQRF-HRD)for different test images and variances of additive noise
AIQRF-HRD estimator
Image
2tr
%AIQRFHn
%
2
AIQRFHn
AIQRFH
2
AIQRFH AIQRFHrel
AIQRFH
0 31.34 3.92
50 45.01 1.50 3.73 0.76 14.64 7.65
100 45.92 1.34 2.08 2.60 6.90 2.63Barbara
400 45.38 17.41 -5.28 396.54 424.43 5.15
0 24.37 25.24
50 36.58 1.71 31.84 2.91 1016 63.75
100 40.46 1.39 34.16 5.87 1172 34.23B
aboon
400 45.31 1.50 30.21 63.22 976.02 7.81
0 26.97 3.39
50 40.22 14.57 14.67 9.96 225.13 30.01
100 43.18 11.59 14.22 25.59 227.92 15.10GoldHill
400 44.05 47.63 -6.98 989.99 1038 8.05
Concluding our analysis, we would like to mention that the method AIQRF-HRD is rather fast and computationally
efficient. Segmentation of images that contain less than 106 pixels can be performed at modern computers faster than in
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one second. Other used operations do not require too much efforts as well. It is quite easy to determine blocks that be-
long to pre-determined image homogeneous regions and to calculate for them 2 , 1,...,lh bh
l N = . Sorting in modern DSP
applications is a widely used operation and fast algorithms for its implementation exist 15. Finding a global minimum of)()( tst XX + , curve approximations by second-order polynomials and determination of a coordinate of approximation
curve minimum are trivial algorithmic operations as well.
6. CONCLUSIONS
In this paper, we first briefly discuss a practical need in design and application of blind methods for noise variance eval-
uation in multichannel remote sensing. Note that such need can also arise in processing of color photo and medical im-
ages. The situations in which the existing methods can fail are considered. The main requirements to blind methods of
noise variance evaluation are listed. Then, we briefly describe the model for distribution of local estimates of noise vari-
ance for techniques operating in spatial domain. We revisit the approach based on finding minimal inter-quantile dis-
tance and discuss the ways how its performance can be improved. The first modification deals with approximation of
inter-quantile distance curve by second-order polynomials. It is shown that for model data this modification improves
inter-quantile method performance. Problems in processing real life images are considered. Based on this analysis, an-
other modification that assumes image pre-segmentation is introduced. It is demonstrated via simulations that the intro-
duced modifications, in aggregate, produce considerable improvement of the inter-quantile method accuracy. Some
aspects of algorithm implementation are discussed as well showing that an algorithm can be rather efficient.
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