Bản chất của hình ảnh y sinh học (Phần 3)

When the values of a random process form a 2D function of space, wehave a noise image xy seeFigures3.2 and 3.3.. as above are useful in the analysis of the behavior of random processes,

the performance of the most sophisticated image processing algorithms is high.The removal of artifacts without causing any distortion or loss of the desiredinformation in the image of interest is often a signicant challenge Theenormity of the problem of noise removal and its importance are reected

by the placement of this chapter as the rst chapter on image processingtechniques in this book

This chapter starts with an introduction to the nature of the artifacts thatare commonly encountered in biomedical images Several illustrations of im-ages corrupted by various types of artifacts are provided Details of the design

of lters spanning a broad range of approaches, from linear space-domain andfrequency-domain xed lters, to the optimal Wiener lter, and further on

to nonlinear and adaptive lters, are then described The chapter concludeswith demonstrations of application of the lters described to a few biomedicalimages

(Note: A good background in signal and system analysis 1, 2, 3, 167] as well

as probability, random variables, and stochastic processes 3, 128, 168, 169,

170, 171, 172, 173] is required in order to follow the procedures and analysesdescribed in this chapter.)

3.1 Characterization of Artifacts

3.1.1 Random noise

The term random noise refers to an interference that arises from a randomprocess such as thermal noise in electronic devices and the counting of photons

A random process is characterized by the PDF representing the probabilities

of occurrence of all possible values of a random variable (See Papoulis 128]

152 Biomedical Image Analysis

or Bendat and Piersol 168] for background material on probability, randomvariables, and stochastic processes.)

Consider a random process that is characterized by the PDFp ( ) Theprocess could be a function of time as (t), or of space in 1D, 2D, or 3D

as (x), (x
y), or (x
y
z) it could also be a spatio-temporal function as(x
y
z
t) The argument of the PDF represents the value that the randomprocess can assume, which could be a voltage in the case of a function of time,

or a gray level in the case of a 2D or 3D image The use of the same symbolfor the function and the value it can assume when dealing with PDFs is usefulwhen dealing with several random processes

The mean  of the random process is given by the rst-order moment

of the PDF, dened as

 =E ] =

Z 1

;1

( ; )2p ( )d : (3.3)The square root of the variance gives the standard deviation (SD)  of theprocess Note that 2 = E 2];2 If the mean is zero, it follows that

2=E 2], that is, the variance and the MS values are the same

Observe the use of the same symbol to represent the random variable,the random process, and the random signal as a function of time or space.The subscript of the PDF or the statistical parameter derived indicates therandom process of concern The context of the discussion or expression shouldmake the meaning of the symbol clear

When the values of a random process form a time series or a function oftime, we have a random signal (or a stochastic process) (t) seeFigure 3.1

When one such time series is observed, it is important to note that the entityrepresents but one single realization of the random process An example of arandom function of time is the current generated by a CCD detector elementdue to thermal noise when no light is falling on the detector (known as thedark current) The statistical measures described above then have physicalmeaning: the mean represents the DC component, the MS value represents theaverage power, and the square root of the mean-squared value (the root mean-squared or RMS value) gives the average noise magnitude These measures

A time series composed of random noise samples with a Gaussian PDF having

= 0 and2= 0:01 MS value = 0:01 RMS = 0:1 See alsoFigures 3.2and3.3

When the values of a random process form a 2D function of space, wehave a noise image (x
y) seeFigures3.2 and 3.3 Several possibilities arise

in this situation: We may have a single random process that generates randomgray levels that are then placed at various locations in the (x
y) plane in somestructured or random sequence We may have an array of detectors with onedetector per pixel of a digital image the gray level generated by each detectormay then be viewed as a distinct random process that is independent of those

of the other detectors A TV image generated by such a camera in the presence

of no input image could be considered to be a noise process in (x
y
t), that

is, a function of space and time

A biomedical image of interestf(x
y) may also, for the sake of generality, beconsidered to be a realization of a random process f Such a representation

154 Biomedical Image Analysis

FIGURE 3.2

An image composed of random noise samples with a Gaussian PDF having

 = 0 and 2 = 0:01 MS value = 0:01 RMS = 0:1 The normalizedpixel values in the range ;0:5
0:5] were linearly mapped to the display range

0
255] See also Figure 3.3

When an image f(x
y) is observed in the presence of random noise , thedetected image g(x
y) may be treated as a realization of another randomprocess g In most cases, the noise is additive, and the observed image isexpressed as

g(x
y) =f(x
y) + (x
y): (3.4)Each of the random processes f, , and g is characterized by its own PDF

pf(f),p ( ), andpg(g), respectively

In most practical applications, the random processes representing an image

of interest and the noise a
ecting the image may be assumed to be cally independent processes Two random processes f and are said to bestatistically independent if their joint PDFpf
(f
) is equal to the product oftheir individual PDFs given aspf(f)p ( ) It then follows that the rst-ordermoment and second-order central moment of the processes in Equation 3.4are related as

statisti-E g] =g=f + =f =E f]
(3.5)

E (g;g)2] =2 g=2 f+2
(3.6)where  represents the mean and 2 represents the variance of the randomprocess indicated by the subscript, and it is assumed that = 0

Ensemble averages: When the PDFs of the random processes of cern are not known, it is common to approximate the statistical expectationoperation by averages computed using a collection or ensemble of sampleobservations of the random process Such averages are known as ensemble av-erages Suppose we haveM observations of the random processf as functions

con-of (x
y): f1(x
y)
f2(x
y)
:::
fM(x
y) seeFigure 3.4 We may estimate themean of the process at a particular spatial location (x1
y1) as

k =1 fk(x1
y1): (3.7)The autocorrelation function (ACF)f(x1
x1+
y1
y1+) of the randomprocessf is dened as

f(x1
x1+
y1
y1+) =E f(x1
y1)
f(x1+
y1+)]
(3.8)

156 Biomedical Image Analysiswhich may be estimated as

k =1 fk(x1
y1)fk(x1+
y1+)
(3.9)where and are spatial shift parameters If the imagef(x
y) is complex,one of the versions off(x
y) in the products above should be conjugated mostbiomedical images that are encountered in practice are real-valued functions,and this distinction is often ignored The ACF indicates how the values of

an image at a particular spatial location are statistically related to (or havecharacteristics in common with) the values of the same image at anothershifted location If the process is stationary, the ACF depends only upon theshift parameters, and may be expressed as f(
)

M

.

.

.

.

Ensemble and spatial averaging of images

The three equations above may be applied to signals that are functions

of time by replacing the spatial variables (x
y) with the temporal variable

t, replacing the shift parameter  with  to represent temporal delay, andmaking a few other related changes

we may compute averages by integrating over the spatial domain, to obtainspatial averages or spatial statistics see Figure 3.4 The spatial mean of theimagefk(x
y) is given by

f(k) = 1A

Z 1

;1

Z 1

;1

fk(x
y)dx dy
(3.12)whereA is a normalization factor, such as the actual area of the image Ob-serve that the spatial mean above is a single-valued entity (a scalar) For astationary process, the spatial ACF is given by

f(

k) =

Z 1

;1

Z 1

;1

fk(x
y)fk(x+
y+)dx dy: (3.13)

A suitable normalization factor, such as the total energy of the image which isequal tof(0
0)] may be included, if necessary The sample indexkbecomesirrelevant if only one observation is available In practice, the integrals change

to summations over the space of the digital image available

When we have a 2D image as a function of time, such as TV, video, roscopy, and cine-angiography signals, we have a spatio-temporal signal thatmay be expressed as f(x
y
t) seeFigure 3.5 We may then compute statis-tics over a single frame f(x
y
t1) at the instant of timet1, which are known

uo-as intraframe statistics We could also compute parameters through multipleframes over a certain period of time, which are called interframe statisticsthe signal over a specic period of time may then be treated as a 3D dataset.Random functions of time may thus be characterized in terms of ensembleand/or temporal statistics Random functions of space may be represented

158 Biomedical Image Analysis

f (x, y, t )

f (x, y, t)

.

.

.

.

temporal or interframe statistics

spatial or intraframe statistics 1

FIGURE 3.5

Spatial and temporal statistics of a video signal

as above are useful in the analysis of the behavior of random processes, and inmodeling, spectrum analysis, lter design, data compression, and data com-munication.

3.1.2 Examples of noise PDFs

As we have already seen, several types of noise sources are encountered inbiomedical imaging Depending upon the characteristics of the noise sourceand the phenomena involved in the generation of the signal and noise values,

we encounter a few di
erent types of PDFs, some of which are described inthe following paragraphs 3, 128, 173]

Gaussian: The most commonly encountered and used noise PDF is theGaussian or normal PDF, expressed as 3, 128]

A Gaussian PDF is completely specied by its mean x and variance 2 x

= 3
= 1 See also Figures 3.2and 3.3

When we have two jointly normal random processesxandy, the bivariatenormal PDF is given by

px
y(x
y) = p 1

4 2(1; 2)xy
exp

= E (x;x)(y;y)]

If = 0, the two processes are uncorrelated The bivariate normal PDF thenreduces to a product of two univariate Gaussians, which implies that the twoprocesses are statistically independent

160 Biomedical Image Analysis

In practice, an image is typically a
ected by a series of independent sources

of additive noise the net noise PDF may then be assumed to be a Gaussian

Uniform: All possible values of a uniformly distributed random processhave equal probability of occurrence The PDF of such a random process overthe range (a
b) is a rectangle of height(b ; 1 a )over the range (a
b) The mean ofthe process is (a+2b), and the variance is (b; a ) 2

PDFs corresponding to random processes with values spread over the ranges(;10
10) and (;5
5) The quantization of gray levels in an image to a nitenumber of integers leads to an error or noise that is uniformly distributed

Poisson: The counting of discrete random events such as the number ofphotons emitted by a source or detected by a sensor in a given interval oftime leads to a random variable with a Poisson PDF The discrete nature

of the packets of energy (that is, photons) and the statistical randomness intheir emission and detection contribute to uncertainty, which is reected as

162 Biomedical Image Analysisquantum noise, photon noise, mottle, or Poisson noise in images Shot noise

in electronic devices may also be modeled as Poisson noise

One of the formulations of the Poisson PDF is as follows: The probabilitythatk photons are detected in a certain interval is given by

P(k) = exp(;) k

Here, is the mean of the process, which represents the average number ofphotons counted in the specied interval over many trials The values ofP(k)for all (integer) k is the Poisson PDF The variance of the Poisson PDF isequal to its mean

The Poisson PDF tends toward the Gaussian PDF for large mean values.Figure 3.8 shows two Poisson PDFs along with the Gaussians for the sameparameters it is seen that the Poisson and Gaussian PDFs for =2 = 20match each other well

−8 −6 −4 −2 0 2 4 6 8 10 0.1

Two Laplacian PDFs with= 0
2= 1 (solid) and= 0
2= 4 (dashed)

Rayleigh: The Rayleigh PDF is given by the function

px(x) = 2b(x;a) exp



;

(x;a)2b



u(x;a)
(3.19)whereu(x) is the unit step function such that

164 Biomedical Image AnalysisFigure 3.10 shows a Rayleigh PDF with a = 1 and b = 4 The RayleighPDF has been used to model speckle noise 175].

It should, however, be noted that the phase of the interfering waveform will notusually be known Furthermore, the interfering waveform may not be an exactsinusoid this is indicated by the presence of harmonics of the fundamental

50Hzor 60Hzcomponent Notch and comb lters may be used to removepower-line artifact 31] It is not common to encounter power-line interference

image cast by such labels should be well removed from the image of the tient on the acquired image, although, occasionally, they may get connected

pa-or even overlap Such items interfere in image analysis when the procedure

is applied to the entire image Preprocessing techniques are then required torecognize and remove such artifacts seeSection 5.9for examples

Surgical implants such as staples, pins, and screws create diculties andartifacts in X-ray, MR, CT, and ultrasound imaging The advantage withsuch artifacts is that the precise composition and geometry of the implantsare known by design and the manufacturers' specications Methods maythen be designed to remove each specic artifact

3.1.4 Physiological interference

As we have already noted, the human body is a complex conglomeration ofseveral systems and processes Several physiological processes could be active

at a given instant of time, each one a
ecting the system or process of interest

in diverse ways A patient or experimental subject may not be able to exercisecontrol on all of his or her physiological processes and systems The e
ect ofsystems or processes other than those of interest on the image being acquiredmay be termed as physiological interference several examples are listed below

 E
ect of breathing on a chest X-ray image

 E
ect of breathing, peristalsis, and movement of material through thegastro-intestinal system on CT images of the abdomen

 E
ect of cardiovascular activity on CT images of the chest

 E
ect of pulsatile movement of arteries in subtraction angiography.Physiological interference may not be characterized by any specic wave-form, pattern, or spectral content, and is typically dynamic and nonstationary(varying with the level of the activity of relevance and hence with time see

lters will usually not be e
ective in removing physiological interference

166 Biomedical Image AnalysisNormal anatomical details such as the ribs in chest X-ray images and theskull in brain imaging may also be considered to be artifacts when other details

in such images are of primary interest Methods may need to be developed toremove their e
ects before the details of interest may be analyzed

3.1.5 Other types of noise and artifact

Systematic errors are caused by several factors such as geometric tion, miscalibration, nonlinear response of detectors, sampling, and quantiza-tion 3] Such errors may be modeled from a knowledge of the correspondingparameters, which may be determined from specications, measured experi-mentally, or derived mathematically

distor-A few other types of artifact that cannot be easily categorized into thegroups discussed above are the following:

 Punctate or shot noise due to dust on the screen, lm, or examinationtable

 Scratches on lm that could appear as intense line segments

 Shot noise due to inactive elements in a detector array

 Salt-and-pepper noise due to impulsive noise, leading to black or whitepixels at the extreme ends of the pixel-value range

 Film-grain noise due to scanning of lms with high resolution

 Punctate noise in chest X-ray or mammographic images caused by metic powder or deodorant (which could masquerade as microcalcica-tions)

cos- Superimposed images of clothing accessories such as pins, hooks, tons, and jewelry

but-3.1.6 Stationary versus nonstationary processes

Random processes may be characterized in terms of their temporal/spatialand/or ensemble statistics A random process is said to be stationary in thestrict sense or strongly stationary if its statistics are not a
ected by a shift inthe origin of time or space In most practical applications, only the rst-orderand second-order averages are used A random process is said to be weaklystationary or stationary in the wide sense if its mean is a constant and itsACF depends only upon the di
erence (or shift) in time or space Then, wehave f(x1
y1) =f and f(x1
x1+
y1
y1+) =f(
) The ACF isnow a function of the shift parametersand only the PSD of the processdoes not vary with space

statistics that vary with time or space The statistics of most images vary overspace indeed, such variations are the source of pictorial information Mostbiomedical systems are dynamic systems and produce nonstationary signalsand images However, a physical or physiological system has limitations in therate at which it can change its characteristics This limitation facilitates thebreaking of a signal into segments of short duration (typically a few tens ofmilliseconds), over which the statistics of interest may be assumed to remainconstant 31] The signal is then referred to as a quasistationary process Tech-niques designed for stationary signals may then be extended and applied tononstationary signals Analysis of signals by this approach is known as short-time analysis 31, 176] On the same token, the characteristics of the features

in an image vary over relatively large scales of space statistical parameterswithin small regions of space, within an object, or within an organ of a giventype may be assumed to remain constant The image may then be assumed

to be block-wise stationary, which permits sectioned or block-by-block cessing or moving-window processing using techniques designed for stationaryprocesses 177, 178] Figure 3.11illustrates the notion of computing statisticswithin a moving window

pro-Certain systems, such as the cardiac system, normally perform rhythmic erations Considering the dynamics of the cardiac system, it is obvious thatthe system is nonstationary However, various phases of the cardiac cycle |

op-as well op-as the related components of the op-associated electrocardiogram (ECG),phonocardiogram (PCG), and carotid pulse signals | repeat over time in

an almost-periodic manner A given phase of the process or signal possessesstatistics that vary from those of the other phases however, the statistics

of a specic phase repeat cyclically For example, the statistics of the PCGsignal vary within the duration of a cardiac cycle, especially when murmursare present, but repeat themselves at regular intervals over successive cardiaccycles Such signals are referred to as cyclo-stationary signals 31] The cycli-cal repetition of the process facilitates synchronized ensemble averaging (see

of the signal over many cycles

The cyclical nature of cardiac activity may be exploited for synchronizedaveraging to reduce noise and improve the SNR of the ECG and PCG 31] The

168 Biomedical Image Analysis

same technique may also be extended to imaging the heart: In gated pool imaging, nuclear medicine images of the heart are acquired in severalparts over short intervals of time Images acquired at the same phases of thecardiac cycle | determined by using the ECG signal as a reference, trigger, or

blood-\gating" signal | are accumulated over several cardiac cycles A sequence ofsuch gated and averaged frames over a full cardiac cycle may then be played

as a video or a movie to visualize the time-varying size and contents of theleft ventricle (SeeSection 3.10for illustration of gated blood-pool imaging.)

3.1.7 Covariance and cross-correlation

When two random processesf andgneed to be compared, we could computethe covariance between them as

fg=E (f;f)(g;g)] =

Z 1

;1

Z 1

;1(f;f)(g;g)pf
g(f
g)df dg
(3.21)where pf
g(f
g) is the joint PDF of the two processes, and the image coor-dinates have been omitted for the sake of compact notation The covarianceparameter may be normalized to get the correlation coecient, dened as

fg= fg

width of images, as well as for the detection of objects by template matching.

3.1.8 Signal-dependent noise

Noise may be categorized as being independent of the signal of interest if nostatistical parameter of any order of the noise process is a function of thesignal Although it is common to assume that the noise present in a signal

or image is statistically independent of the true signal (or image) of interest,several cases exist in biomedical imaging where this assumption is not valid,and the noise is functionally related to or dependent upon the signal Thefollowing paragraphs provide brief notes on a few types of signal-dependentnoise encountered in biomedical imaging

Poisson noise: Imaging systems that operate in low-light conditions, or

in low-dose radiation conditions such as nuclear medicine imaging, are oftena
ected by photon noise that can be modeled as a Poisson process 179, 180]

value)go(m
n) under the conditions of a Poisson process is given by

P(go(m
n)jf(m
n) ) = (m
n)]go ( m
n ) exp ; (m
n)]

go(m
n)!
(3.24)where f(m
n) is the undegraded pixel value (the observation in the absence

of any noise), and is a proportionality factor Because the mean of thedegraded imagego is given by

E go(m
n)] = f(m
n)]
(3.25)images corrupted with Poisson noise are usually normalized as

g(m
n) =go(m
n) (3.26)

It has been shown 179] that, in this case, Poisson noise may be modeled asstationary noise uncorrelated with the signal and added to the signal as inEquation 3.4, with zero mean and variance given by

2(m
n) = E f(m
n)] = E g(m
n)]: (3.27)

170 Biomedical Image Analysis

Film-grain noise: The granular structure of lm due to the silver-halidegrains used contributes noise to the recorded image, which is known as lm-grain noise When images recorded on photographic lm are digitized in order

to be processed by a digital computer, lm-grain noise is a signicant source

of degradation of the information According to Froehlich et al 181], themodel for an image corrupted by lm-grain noise is given by

g(m
n) =f(m
n) +F f(m
n)] 1(m
n) + 2(m
n)
(3.28)where  is a proportionality factor, F ] is a mathematical function, and

1(m
n) and 2(m
n) are samples from two random processes independent ofthe signal This model may be taken to represent a general imaging situationthat includes signal-independent noise as well as signal-dependent noise, andthe noise could be additive or multiplicative Observe that the model reduces

to the simple signal-independent additive noise model in Equation 3.4 if= 0.Froehlich et al 181] modeled lm-grain noise withF f(m
n)] = f(m
n)]p,usingp= 0:5 The two noise processes 1and 2were assumed to be Gaussian-distributed, uncorrelated, zero-mean random processes According to thismodel, the noise that corrupts the image has two components: one that issignal-dependent through the factorp

f(m
n) 1(m
n), and another that

is signal-independent given by 2(m
n) Film-grain noise may be modeled

as additive noise as in Equation 3.4, with (m
n) = p

coher-g(m
n) =f(m
n) 1(m
n)
(3.30)where 1(m
n) is a stationary noise process that is assumed to be uncorrelatedwith the image If the mean of the noise process 1 is not equal to one, thenoisy image may be normalized by dividing by 1such that, in the normalizedimage, the multiplicative noise has its mean equal to one Depending uponthe specic application, the distribution of the noise may be assumed to beexponential 175, 183, 186, 187], Gaussian 184], or Rayleigh 175]

The multiplicative model in Equation 3.30 may be converted to the additivemodel as in Equation 3.4 with (m
n) being zero-mean additive noise having

applying an appropriate transformation to the whole image, the noise can bemade signal-independent One of the transformations proposed is 189, 190]

T g(m
n)] =p

g(m
n)
(3.32)where is an appropriate normalizing constant It has been shown that thenoise in the transformed image is additive, has a Gaussian distribution, isunbiased, and has a standard deviation that no longer depends on the signalbut is given by  2 :

3.2 Synchronized or Multiframe Averaging

In certain applications of imaging, if the object being imaged can remainfree from motion or change of any kind (internal or external) over a longperiod of time compared to the time required to record an image, it becomespossible to acquire several frames of images of the object in precisely the samestate or condition Then, the frames may be averaged to reduce noise this isknown as multiframe averaging The method may be extended to the imaging

of dynamic systems whose movements follow a rhythm or cycle with phasesthat can be determined by another signal, such as the cardiac system whosephases of contraction are indicated by the ECG signal Then, several imageframes may be acquired at the same phase of the rhythmic movement oversuccessive cycles, and averaged to reduce noise Such a process is known assynchronized averaging The process may be repeated or triggered at everyphase of interest (Note: A process as above in nuclear medicine imaging may

be viewed simply as counting the photons emitted over a long period of time

in total, albeit in a succession of short intervals gated to a particular phase

of contraction of the heart Ignoring the last step of division by the number

of frames to obtain the average, the process simply accumulates the photoncounts over the frames acquired.)

172 Biomedical Image AnalysisSynchronized averaging is a useful technique in the acquisition of severalbiomedical signals 31] Observe that averaging as above is a form of ensembleaveraging.

Let us represent a single image frame in a situation as above as

gi(x
y) =f(x
y) + i(x
y)
(3.33)wheregi(x
y) is theithobserved frame of the imagef(x
y), and i(x
y) is thenoise in the same frame Let us assume that the noise process is independent

of the signal source Observe that the desired (original) image f(x
y) isinvariant from one frame to another It follows that g 2 i ( x
y ) =2

i =1 gi(x
y): (3.34)

If the mean of the noise process is zero, we haveP Mi=1 i(x
y)!0 asM ! 1(in practice, as the number of frames averaged increases to a large number).Then, it follows that 8]

E g(x
y)] =f(x
y)
(3.35)and

g 2 ( x
y )= 1M 2( x
y ): (3.36)Thus, the variance at every pixel in the averaged image is reduced by a factor

of M 1 from that in a single frame the SNR is improved by the factorp

M.The most important requirement in this procedure is that the frames be-ing averaged be mutually synchronized, aligned, or registered Any motion,change, or displacement between the frames will lead to smearing and distor-tion

Example: Figure 3.12 (a) shows a test image with several geometricalobjects placed at random Images (b) and (c) show two examples of eight noisyframes of the test image that were obtained by adding Gaussian-distributedrandom noise samples The results of averaging two, four, and eight noisyframes including the two in (b) and (c)] are shown in parts (d), (e), and(f), respectively It is seen that averaging using increasing numbers of frames

of the noisy image leads to a reduction of the noise the decreasing trend inthe RMS values of the processed images (given in the caption of Figure 3.12)conrms the expected e
ect of averaging

SeeSection 3.9for an illustration of the application of multiframe averaging

in confocal microscopy See also Section 3.10for details on gated blood-poolimaging in nuclear medicine

(c) (d)

FIGURE 3.12

(a) \Shapes": a 128
128 test image with various geometrical objects placed

at random (b) Image in (a) with Gaussian noise added, with= 0
2= 0:01(normalized), RMS error = 19:32 (c) Second version of noisy image, RMSerror = 19:54 Result of multiframe averaging using (d) the two frames in (b)and (c), RMS error = 15:30 (e) four frames, RMS error = 12:51 (f) eightframes, RMS error = 10:99

174 Biomedical Image Analysis

3.3 Space-domain Local-statistics-based Filters

Consider the practical situation when we are given a single, noisy observation

of an image of nite size We do not have access to an ensemble of images toperform multiframe (synchronized) averaging, and spatial statistics computedover the entire image frame will lead to scalar values that do not assist in re-moving the noise and obtaining a cleaner image Furthermore, we should alsoaccommodate for nonstationarity of the image In such situations, moving-window ltering using windows of small size such as 3
3, 5
5, or 7
7pixels becomes a valuable option rectangular windows as well as windows ofother shapes may also be considered where appropriate Various statisticalparameters of the pixels within such a moving window may be computed, withthe result being applied to the pixel in the output image at the same locationwhere the window is placed (centered) on the input image see Figure 3.13.Observe that only the pixel values in the given (input) image are used inthe ltering process the output is stored in a separate array Figure 3.14

illustrates a few di
erent neighborhood shapes that are commonly used inmoving-window image ltering 192]

(8-connected) or integer

distance 1

(h) 1x5 bar (g) 5x1 bar

(f) cross (e) 5x5 square

lter-176 Biomedical Image Analysis

If we were to select the pixels in a small neighborhood around the pixel to beprocessed, the following assumptions may be made:

 the image component is relatively constant that is, the image is tationary and

quasis- the only variations in the neighborhood are due to noise

Further assumptions regarding the noise process that are typically made arethat it is additive, is independent of the image, and has zero mean Then, if wewere to take the mean of the pixels in the neighborhood, the result will tendtoward the true pixel value in the original, uncorrupted image In essence,

a spatial collection of pixels around the pixel being processed is substitutedfor an ensemble of pixels at the same location from multiple frames in theaveraging process that is, the image-generating process is assumed to beergodic

It is common to use a 3
3 or 8-connected neighborhood as inFigure 3.14

(a)for mean ltering Then, the output of the lterg(m
n) is given by

2 4

The mean lter can suppress Gaussian and uniformly distributed noise fectively in relatively homogeneous areas of an image However, the operationleads to blurring at the edges of the objects in the image, and also to the loss

ef-of ne details and texture Regardless, mean ltering is commonly employed

proximations will have to be made: the most common procedure rank-ordersthe pixels in a neighborhood containing an odd number of pixels, and thepixel value at the middle of the list is selected as the median The procedurealso permits the application of order-statistic lters 193]: theith element in

a rank-ordered list of values is known as theith order statistic The median

lter is an order-statistic lter of orderN=2 whereN is the size of the lterthat is, the number of values used to derive the output

The median lter is a nonlinear lter Its success in ltering depends uponthe number of the samples used to derive the output, as well as the spatialconguration of the neighborhood used to select the samples

The median lter provides better noise removal than the mean lter withoutblurring, especially when the noise has a long-tailed PDF (resulting in outliers)and in the case of salt-and-pepper noise However, the median lter couldresult in the clipping of corners and distortion of the shape of sharp-edgedobjects median ltering with large neighborhoods could also result in thecomplete elimination of small objects Neighborhoods that are not square

in shape are often used for median ltering in order to limit the clipping ofcorners and other types of distortion of shape seeFigure 3.14

Examples: Figure 3.15 (a)shows a 1D test signal with a rectangular pulsepart (b) of the same gure shows the test signal degraded with impulse (shot)noise The results of ltering the noisy signal using the mean and median with

lter lengthN = 3 are shown in plots (c) and (d), respectively, of Figure 3.15.The mean lter has blurred the edges of the pulse it has also created artifacts

in the form of small hills and valleys The median lter has removed the noisewithout distorting the signal

of the same gure shows the test signal degraded with uniformly distributednoise The results of ltering the noisy signal using the mean and median with

lter lengthN = 5 are shown in plots (c) and (d), respectively, of Figure 3.16.The mean lter has reduced the noise level, but has also blurred the edges ofthe pulses in addition, the strength of the rst, short pulse has been reduced.The median lter has removed the noise to some extent without distortingthe edges of the long pulse however, the short pulse has been obliterated

178 Biomedical Image Analysis

10 20 30 40 50 60 70 80 0

uni-180 Biomedical Image Analysis

Figure 3.17shows the original test image \Shapes", the test image degraded

by the addition of Gaussian-distributed random noise with = 0 and 2 =

0:01 (normalized), and the results of ltering the noisy image with the 3
3and 5
5 mean and median lters The RMS errors of the noisy and lteredimages with respect to the test image are given in the gure caption All ofthe lters except the 3
3 median have led to an increase in the RMS error.The blurring e
ect of the mean lter is readily seen in the results Closeobservation of the result of 3
3 median ltering Figure 3.17 (d)] shows thatthe lter has resulted in distortion of the shapes, in particular, clipping ofthe corners of the objects The 5
5 median lter has led to the completeremoval of small objects see Figure 3.17 (f) Observe that the results of the

3
3 mean and 5
5 median lters have similar RMS error values however,the blurring e
ect in the former case, and the distortion of shape information

as well as the loss of small objects in the latter case need to be consideredcarefully

dis-tributed A comparable set with speckle noise is shown in Figure 3.19 though the lters have reduced the noise to some extent, the distortions in-troduced have led to increased RMS errors for all of the results

of pixels a
ected by noise being 0:05 and 0:1, respectively The 3
3 median

lter has given good results in both cases with the lowest RMS error and theleast distortion The 5
5 median lter has led to signicant shape distortionand the loss of a few small features

its degraded versions with Gaussian, Poisson, and speckle noise It is evidentthat the signal-dependent Poisson noise and speckle noise have a
ected thehistogram in a di
erent manner compared to the signal-independent Gaussiannoise

by Gaussian noise Although the RMS errors of the ltered images are lowcompared to that of the noisy image, the lters have introduced a mottledappearance and ne texture in the smooth regions of the original image Fig-

ure 3.24shows the case with Poisson noise, where the 5
5 lters have providedvisually good results, regardless of the RMS errors

All of the lters have performed reasonably well in the presence of specklenoise, as illustrated in Figure 3.25,in terms of the reduction of RMS error.However, the visual quality of the images is poor

in ltering salt-and-pepper noise Although the RMS values of the results

of the mean lters are lower than those of the noisy images, visual tion of the results indicates the undesirable e
ects of blurring and mottledappearance

inspec-The RMS error (or the MSE) is commonly used to compare the results ofvarious image processing operations however, the examples presented above

The class of order-statistic lters 193] is large, and includes several nonlinear

lters that are useful in ltering di
erent types of noise in images The rststep in order-statistic ltering is to rank-order, from the minimum to themaximum, the pixel values in an appropriate neighborhood of the pixel beingprocessed The ith entry in the list is the output of the ith order-statistic

lter A few order-statistic lters of particular interest are the following:

 Min lter: the rst entry in the rank-ordered list, useful in removinghigh-valued impulse noise (isolated bright spots or \salt" noise)

 Max lter: the last entry in the rank-ordered list, useful in removinglow-valued impulse noise (isolated dark spots or \pepper" noise)

 Min/Max lter: sequential application of the Min and Max lters, useful

in removing salt-and-pepper noise

 Median lter: the entry in the middle of the list The median lter isthe most popular and commonly used lter among the order-statistic

lters see Section 3.3.2 for detailed discussion and illustration of themedian lter

 -trimmed mean lter: the mean of a reduced list where the rstandthe last  of the list is rejected, with 0   < 0:5 Outliers, that ispixels with values very di
erent from the rest of the pixels in the list,are rejected by the trimming process A value close to 0:5 forrejectsthe entire list except the median or a few values close to it, and theoutput is close to or equal to that of the median lter The mean ofthe trimmed list provides a compromise between the generic mean andmedian lters

 L-lters: a weighted combination of all of the elements in the ordered list The use of appropriate weights can provide outputs equal

rank-182 Biomedical Image Analysis

FIGURE 3.17

(a) Shapes test image (b) Image in (a) with Gaussian noise added, with

 = 0
2 = 0:01 (normalized), RMS error = 19:56 Result of ltering thenoisy image in (b) using: (c) 3
3 mean, RMS error = 22:62 (d) 3
3 median,RMS error = 15:40 (e) 5
5 mean, RMS error = 28:08 (f) 5
5 median,RMS error = 22:35

184 Biomedical Image Analysis

FIGURE 3.19

(a) Shapes test image (b) Image in (a) with speckle noise, with= 0
2=

0:04 (normalized), RMS error = 12:28 Result of ltering the noisy image in(b) using: (c) 3
3 mean, RMS error = 20:30 (d) 3
3 median, RMS error

= 15:66 (e) 5
5 mean, RMS error = 26:32 (f) 5
5 median, RMS error =

24:56

in (b) using: (c) 3
3 mean, RMS error = 24:85 (d) 3
3 median, RMS error

= 14:59 (e) 5
5 mean, RMS error = 28:24 (f) 5
5 median, RMS error =

23:14

186 Biomedical Image Analysis

= 15:42 (e) 5
5 mean, RMS error = 31:25 (f) 5
5 median, RMS error =

23:32

in order to show the important details The probability values of gray els 0 and 255 have been clipped in some of the histograms Each histogramrepresents the gray-level range of 0
255].

lev-188 Biomedical Image Analysis

(a) (b)

FIGURE 3.24

(a) Peppers test image (b) Image in (a) with Poisson noise, RMS error =

10:94 Result of ltering the noisy image in (b) using: (c) 3
3 mean, RMSerror = 11:22 (d) 3
3 median, RMS error = 8:56 (e) 5
5 mean, RMS error

= 15:36 (f) 5
5 median, RMS error = 10:83

190 Biomedical Image Analysis

FIGURE 3.25

(a) Peppers test image (b) Image in (a) with speckle noise, with= 0
2=

0:04 (normalized), RMS error = 26:08 Result of ltering the noisy image in(b) using: (c) 3
3 mean, RMS error = 13:68 (d) 3
3 median, RMS error

= 15:73 (e) 5
5 mean, RMS error = 16:01 (f) 5
5 median, RMS error =

14:66