Ref 27 Improved Minimal Inter-quantile Distance Method for Blind Estimation of Noise

7/29/2019 Ref 27 Improved Minimal Inter-quantile Distance Method for Blind Estimation of Noise

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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

Proc. of SPIE Vol. 6748 67481I-1

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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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Ref 27 Improved Minimal Inter-quantile Distance Method for Blind Estimation of Noise