Image Quality and Information Content | 9 | Biomedical Image Analysis

ABSTRACT

Image Quality and Information Content

Several factors aect the quality and information content of biomedical images

acquired with the modalities described in Chapter A few considerations

in biomedical image acquisition and analysis that could have a bearing on

image quality are described in Section A good understanding of such

factors as well as appropriate characterization of the concomitant loss in

image quality are essential in order to design image processing techniques to

remove the degradation andor improve the quality of biomedical images The

characterization of information content is important for the same purposes as

above as well as in the analysis and design of image transmission and archival

systems

An inherent problem in characterizing quality lies in the fact that image

quality is typically judged by human observers in a subjective manner To

quantify the notion of image quality is a dicult proposition Similarly the

nature of the information conveyed by an image is dicult to quantify due

to its multifaceted characteristics in terms of statistical structural percep

tual semantic and diagnostic connotations However several measures have

been designed to characterize or quantify a few specic attributes of images

which may in turn be associated with various notions of quality as well as

information content The numerical values of such measures of a given image

before and after certain processes or the changes in the attributes due to cer

tain phenomena could then be used to assess variations in image quality and

information content We shall explore several such measures in this chapter

Diculties in Image Acquisition and Analysis

In Chapter we studied several imaging systems and procedures for the

acquisition of many dierent types of biomedical images The practical appli

cation of these techniques may pose certain diculties the investigator often

faces conditions that may impose limitations on the quality and information

content of the images acquired The following paragraphs illustrate a few

practical diculties that one might encounter in biomedical image acquisition

and analysis

Accessibility of the organ of interest Several organs of interest in

imagingbased investigation are situated well within the body encased in pro

tective and diculttoaccess regions for good reason For example the brain

is protected by the skull and the prostate is situated at the base of the blad

der near the pelvic outlet Several limitations are encountered in imaging

such organs special imaging devices and image processing techniques are re

quired to facilitate their visualization Visualization of the arteries in the

brain requires the injection of an Xray contrast agent and the subtraction

of a reference image see Section Special transrectal probes have been

designed for D ultrasonic imaging of the prostate Despite the use of

such special devices and techniques images obtained in applications as above

tend to be aected by severe artifacts

Variability of information Biological systems exhibit great ranges of in

herent variability within their dierent categories The intrinsic and natural

variability presented by biological entities within a given class far exceeds the

variability that we may observe in engineering physical and manufactured

samples The distinction between a normal pattern and an abnormal pat

tern is often clouded by signicant overlap between the ranges of the features

or variables that are used to characterize the two categories the problem

is compounded when multiple abnormalities need to be considered Imag

ing conditions and parameters could cause further ambiguities due to the

eects of subject positioning and projection For example most malignant

breast tumors are irregular and spiculated in shape whereas benign masses

are smooth and round or oval However some malignant tumors may present

smooth shapes and some benign masses may have rough shapes A tumor

may present a rough appearance in one view or projection but a smoother

prole in another Furthermore the notion of shape roughness is nonspe

cic and openended Overlapping patterns caused by ligaments ducts and

breast tissue that may lie in other planes but are integrated on to a single

image plane in the process of mammographic imaging could also aect the

appearance of tumors and masses in images The use of multiple views and

spot magnication imaging could help resolve some of these ambiguities but

at the cost of additional radiation dose to the subject

Physiological artifacts and interference Physiological systems are

dynamic and active Some activities such as breathing may be suspended

voluntarily by an adult subject in a reasonable state of health and well

being for brief periods of time to permit improved imaging However car

diac activity blood circulation and peristaltic movement are not under ones

volitional control The rhythmic contractile activity of the heart poses chal

lenges in imaging of the heart The pulsatile movement of blood through the

brain causes slight movements of the brain that could cause artifacts in an

giographic imaging see Section Dark shadows may appear in ultrasound

images next to bony regions due to signicant attenuation of the investigating

beam and hence the lack of echoes from tissues beyond the bony regions along

the path of beam propagation An analyst should pay attention to potential

physiological artifacts when interpreting biomedical images

Special techniques have been developed to overcome some of the limitations

mentioned above in cardiac imaging Electronic steering of the Xray beam

has been employed to reduce the scanning time required for CT projection

data acquisition in order to permit imaging of the heart see Figure

Stateoftheart multislice and helicalscan CT scanners acquire the required

data in intervals much shorter than the time taken by the initial models of

CT scanners Cardiac nuclear medicine imaging is performed by gating the

photoncounting process to a certain specic phase of the cardiac cycle by

using the electrocardiogram ECG as a reference see Figure and Sec

tion Although nuclear medicine imaging procedures take several min

utes the almostperiodic activity of the heart permits the cumulative imaging

of its musculature or chambers at particular positions repeatedly over several

cardiac cycles

Energy limitations In Xray mammography considering the fact that

the organ imaged is mainly composed of soft tissues a low kV p would be

desired in order to maximize image contrast However lowenergy Xray pho

tons are absorbed more readily than highenergy photons by the skin and

breast tissues thereby increasing the radiation dose to the patient A com

promise is required between these two considerations Similarly in TEM a

highkV electron beam would be desirable in order to minimize damage to

the specimen but a lowkV beam can provide improved contrast The practi

cal application of imaging techniques often requires the striking of a tradeo

between conicting considerations as above

Patient safety The protection of the subject or patient in a study from

electrical shock radiation hazard and other potentially dangerous conditions

is an unquestionable requirement of paramount importance Most organi

zations require ethical approval by specialized committees for experimental

procedures involving human or animal subjects with the aim of minimizing

the risk and discomfort to the subject and maximizing the benets to both the

subjects and the investigator The relative levels of potential risks involved

should be assessed when a choice is available between various procedures and

analyzed against their relative benets Patient safety concerns may preclude

the use of a procedure that may yield better images or results than others

or may require modications to a procedure that may lead to inferior im

ages Further image processing steps would then become essential in order to

improve image quality or otherwise compensate for the initial compromise

Characterization of Image Quality

Biomedical images are typically complex sources of several items of informa

tion Furthermore the notion of quality cannot be easily characterized with a

small number of features or attributes Because of these reasons researchers

have developed a rather large number of measures to represent quantitatively

several attributes of images related to impressions of quality Changes in

measures related to quality may be analyzed for several purposes such as

comparison of images generated by dierent medical imaging systems

comparison of images obtained using dierent imaging parameter set

tings of a given system

comparison of the results of several image enhancement algorithms

assessment of the eect of the passage of an image through a transmis

sion channel or medium and

assessment of images compressed by dierent data compression tech

niques at dierent rates of loss of data information or quality

Specially designed phantoms are often used to test medical imaging sys

tems for routine quality control Bijkerk et al

developed a phantom with gold disks of dierent diameter and thickness to

test mammography systems Because the signal contrast and location are

known from the design of the phantom the detection performance of trained

observers may be used to test and compare imaging systems

Ideally it is desirable to use numerical observers automatic tools to

measure and express image quality by means of numbers or gures of merit

FOMs that could be objectively compared see Furuie et al and Bar

rett for examples It is clear that not only are FOMs important but so

is the methodology for their comparison Kayargadde and Martens

discuss the relationships between image quality attributes in a psychometric

space and a perceptual space

Many algorithms have been proposed to explore various attributes of images

or imaging systems The attributes take into consideration either the whole

image or a chosen region to calculate FOMs and are labeled as being global

or local respectively Often the measured attribute is image denition

the clarity with which details are reproduced which is typically

expressed in terms of image sharpness This notion was rst mentioned by

Higgins and Jones in the realm of photography but is valid for image

evaluation in a broader context Rangayyan and Elkadiki present a

survey of dierent methods to measure sharpness in photographic and digital

images see Section Because quality is a subjective notion the results

obtained by algorithms such as those mentioned above need to be validated

against the evaluation of test images by human observers This could be done

by submitting the same set of images to human and numerical computer

evaluation and then comparing the results

Subjective and objective judgment should agree to some degree under dened

conditions in order for the numerical measures to be useful The following

sections describe some of the concepts and measures that are commonly used

in biomedical image analysis

Digitization of Images

The representation of natural scenes and objects as digital images for process

ing using computers requires two steps sampling and quantization Both of

these steps could potentially cause loss of quality and introduce artifacts

Sampling

Sampling is the process of representing a continuoustime or continuousspace

signal on a discrete grid with samples that are separated by usually uniform

intervals The theory and practice of sampling D signals have been well

established In essence a bandlimited signal with the frequency of

its fastest component being f

Hz may be represented without loss by its

samples obtained at the Nyquist rate of f

Sampling may be modeled as the multiplication of the given continuous

time or analog signal with a periodic train of impulses The multiplication

of two signals in the time domain corresponds to the convolution of their

Fourier spectra The Fourier transform of a periodic train of impulses is

another periodic train of impulses with a period that is equal to the inverse

of the period in the time domain that is f

Hz Therefore the Fourier

spectrum of the sampled signal is periodic with a period equal to f

Hz A

sampled signal has innite bandwidth however the sampled signal contains

distinct or unique frequency components only up to f

If the signal as above is sampled at a rate lower than f

Hz an error known

as aliasing occurs where the frequency components above f

Hz appear at

lower frequencies It then becomes impossible to recover the original signal

from its sampled version

If sampled at a rate of at least f

Hz the original signal may be recovered

from its sampled version by lowpass ltering and extracting the baseband

component over the band f

Hz from the innite spectrum of the sampled

signal If an ideal rectangular lowpass lter were to be used the equivalent

operation in the time domain would be convolution with a sinc function which

is of innite duration This operation is known as interpolation Other

interpolating functions of nite duration need to be used in practice with

the equivalent lter extracting the baseband components without signicant

reduction in gain over the band f

In practice in order to prevent aliasing errors it is common to use an

antialiasing lter prior to the sampling of D signals with a passband that

is close to f

Hz with the prior knowledge that the signal contains no

signicant energy or information beyond f

Hz Analog spectrum

analyzers may be used to estimate the bandwidth and spectral content of a

given D analog signal prior to sampling

All of the concepts explained above apply to the sampling of D signals or

images However in most reallife applications of imaging and image process

ing it is not possible to estimate the frequency content of the images and

also not possible to apply antialiasing lters Adequate sampling frequen

cies need to be established for each type of image or application based upon

prior experience and knowledge Regardless even with the same type of im

ages dierent sampling frequencies may be suitable or adequate for dierent

applications

Figure illustrates the loss of quality associated with sampling an image

at lower and lower numbers of pixels

Biomedical images originally obtained on lm are usually digitized using

highresolution CCD cameras or laser scanners Several newer biomedical

imaging systems include devices for direct digital data acquisition In digital

imaging systems such as CT sampling is inherent in the measurement process

which is also performed in a domain that is dierent from the image domain

This adds a further level of complexity to the analysis of sampling Practical

experimentation and experience have helped in the development of guidelines

to assist in such applications

Quantization

Quantization is the process of representing the values of a sampled signal or

image using a nite set of allowed values In a digital representation using n

bits per sample and positive integers only there exist

possible quantized

levels spanning the range

If n bits are used to represent each

pixel there can exist values or gray levels to represent the values of the

image at each pixel in the range

It is necessary to map appropriately the range of variation of the given

analog signal such as the output of a chargecoupled device CCD detector

or a video device to the input dynamic range of the quantizer If the lowest

level or lower threshold of the quantizer is set too high in relation to the

range of the original signal the quantized output will have several samples

with the value zero corresponding to all signal values that are less than the

lower threshold Similarly if the highest level or higher threshold of the

quantizer is set too low the output will have several samples with the highest

a b

c d

FIGURE

Eect of sampling on the appearance and quality of an image a

pixels b pixels c pixels and d pixels All four

images have gray levels at bits per pixel

quantized level corresponding to all signal values that are greater than the

higher threshold Furthermore the decision levels of the quantizer should be

optimized in accordance with the probability density function PDF of the

original signal or image

The LloydMax quantization procedure to optimize a quan

tizer is derived as follows Let pr represent the PDF of the amplitude or

gray levels in the given image with the values of the continuous or analog

variable r varying within the range r

min

max

Let the range r

min

max

be divided into L parts demarcated by the decision levels R

with R

min

and R

max

see Figure Let the L output levels

of the quantizer represent the values Q

as indicated in

Figure

The meansquared error MSE in representing the analog signal by its

quantized values is given by

r Q

pr dr

Several procedures exist to determine the values of R

and Q

that minimize

the MSE A classical result indicates that the output level Q

should lie at the centroid of the part of the PDF between the decision levels

and R

given by

r pr dr

pr dr

which reduces to

if the PDF is uniform It also follows that the decision levels are then given

It is common to quantize images to bitspixel However CT images

represent a large dynamic range of Xray attenuation coecient normalized

into HU over the range for human tissues Small dierences

of the order of HU could indicate the distinction between normal tissue

and diseased tissue If the range of HU were to be quantized into

levels using an bit quantizer each quantized level would represent a change

HU which could lead to the loss of the distinction as above

in noise For this reason CT and several other medical images are quantized

using bitspixel

The use of an inadequate number of quantized gray levels leads to false

contours and poor representation of image intensities Figure illustrates

the loss of image quality as the number of bits per pixel is reduced from six

to one

FIGURE

Quantization of an image graylevel signal r with a Gaussian solid line or

uniform dashed line PDF The quantizer output levels are indicated by Q

and the decision levels represented by R

The quantized values in a digital image are commonly referred to as gray

levels with representing black and standing for white when bit quanti

zation is used Unfortunately this goes against the notion of a larger amount

of gray being darker than a smaller amount of gray However if the quantized

values represent optical density OD a larger value would represent a darker

region than a smaller value Table lists a few variables that bear dierent

relationships with the displayed pixel value

Array and matrix representation of images

Images are commonly represented as D functions of space fx y A digital

image fmn may be interpreted as a discretized version of fx y in a D

array or as a matrix see Section for details on matrix representation of

images and image processing operations The notational dierences between

the representation of an image as a function of space and as a matrix could

be a source of confusion

a b

c d

FIGURE

Eect of graylevel quantization on the appearance and quality of an image

a gray levels bits per pixel b gray levels bits per pixel c four

gray levels bits per pixel and d two gray levels bit per pixel All four

images have pixels Compare with the image in Figure a with

gray levels at bits per pixel

I m a g e Q u a l i t y a n d I n f o r m a t i o n C o n t e n t

TABLE

Relationships Between Tissue Type Tissue Density Xray Attenuation Coecient Hounseld

Units HU Optical Density OD and Gray Level The Xray Attenuation

Coecient was Measured at a Photon Energy of keV

Tissue Density Xray Hounseld Optical Gray level Appearance

type gmcm

attenuation cm

units density brightness in image

lung lower low high low dark

liver medium medium medium gray

bone higher high low high white

An M N matrix has M rows and N columns its height is M and width

is N numbering of the elements starts with at the topleft corner and

ends with MN at the lowerright corner of the image A function of space

fx y that has been converted into a digital representation fmn is typi

cally placed in the rst quadrant in the Cartesian coordinate system Then an

MN will have a width ofM and height of N indexing of the elements starts

with at the origin at the bottomleft corner and ends with M N

at the upperright corner of the image Figure illustrates the distinction

between these two types of representation of an image Observe that the size

of a matrix is expressed as rows columns whereas the size of an image is

usually expressed as width height

FIGURE

Array and matrix representation of an image

Optical Density

The value of a picture element or cell commonly known as a pixel or

occasionally as a pel in an image may be expressed in terms of a physical

attribute such as temperature density or Xray attenuation coecient the

intensity of light reected from the body at the location corresponding to the

pixel or the transmittance at the corresponding location on a lm rendition

of the image The last one of the options listed above is popular in medical

imaging due to the common use of lm as the medium for acquisition and

display of images The OD at a spot on a lm is dened as

OD log

where I

is the intensity of the light input and I

is the intensity of the light

transmitted through the lm at the spot of interest see Figure A perfectly

clear spot will transmit all of the light that is input and will have OD

a dark spot that reduces the intensity of the input light by a factor of

will have OD Xray lms in particular those used in mammography are

capable of representing gray levels from OD to OD

FIGURE

Measurement of the optical density at a spot on a lm or transparency using

a laser microdensitometer

Dynamic Range

The dynamic range of an imaging system or a variable is its range or gamut of

operation usually limited to the portion of linear response and is expressed

as the maximum minus the minimum value of the variable or parameter of

interest The dynamic range of an image is usually expressed as the dierence

between the maximum and minimum values present in the image Xray

lms for mammography typically possess a dynamic range of OD

Modern CRT monitors provide dynamic range of the order of cdm

in luminance or in sampled gray levels

Figure compares the characteristic curves of two devices Device A

has a larger slope or gamma see Section than Device B and hence

can provide higher contrast dened in Section Device B has a larger

latitude or breadth of exposure and optical density over which it can operate

than Device A Plots of lm density versus the log of Xray exposure are

known as HurterDrield or HD curves

FIGURE

Characteristic response curves of two hypothetical imaging devices

The lower levels of response of a lm or electronic display device are af

fected by a background level that could include the base level of the medium

or operation of the device as well as noise The response of a device typically

begins with a nonlinear toe region before it reaches its linear range of oper

ation Another nonlinear region referred to as the shoulder region leads to

the saturation level of the device It is desirable to operate within the linear

range of a given device

Air in the lungs and bowels as well as fat in various organs including the

breast tend to extend the dynamic range of images toward the lower end of

the density scale Bone calcications in the breast and in tumors as well

as metallic implants such as screws in bones and surgical clips contribute

to highdensity areas in images Mammograms are expected to possess a

dynamic range of OD CT images may have a dynamic range of about

to HU Metallic implants could have HU values beyond the

operating range of CT systems and lead to saturated areas in images the

Xray beam is eectively stopped by heavymetal implants

Contrast

Contrast is dened in a few dierent ways but is essentially the dierence

between the parameter imaged in a region of interest ROI and that in a

suitably dened background If the image parameter is expressed in OD

contrast is dened as

where f

and b

represent the foreground ROI and background OD re

spectively Figure illustrates the notion of contrast using circular ROIs

FIGURE

Illustration of the notion of contrast comparing a foreground region f with

its background b

When the image parameter has not been normalized the measure of con

trast will require normalization If for example f and b represent the average

light intensities emitted or reected from the foreground ROI and the back

ground respectively contrast may be dened as

f b

or as

f b

Due to the use of a reference background the measures dened above are

often referred to as simultaneous contrast It should be observed that the

contrast of a region or an object depends not only upon its own intensity but

also upon that of its background Furthermore the measure is not simply a

dierence but a ratio The human visual system HVS has bandpass lter

characteristics which lead to responses that are proportional to dierences

between illumination levels rather than to absolute illumination levels

Example The two squares in Figure are of the same value in the

scale but are placed on two dierent background regions of value

on the left and on the right The lighter background on the left makes

the inner square region appear darker than the corresponding inner square

on the right This eect could be explained by the measure of simultaneous

contrast the contrast of the inner square on the left using the denition in

Equation is

whereas that for the inner square on the right is

The values of C

and C

using the denition in Equation are respectively

and the advantage of this formulation is that the values of

contrast are limited to the range The negative contrast value for

the inner square on the left indicates that it is darker than the background

whereas it is the opposite for that on the right By covering the background

regions and viewing only the two inner squares simultaneously it will be seen

that the gray levels of the latter are indeed the same

Just noticeable di erence The concept of justnoticeable dierence

JND is important in analyzing contrast visibility and the quality of medical

images JND is determined as follows For a given background level

b as in Equation the value of an object in the foreground f is increased

gradually from the same level as b to a level when the object is just perceived

The value fbb at the level of minimal perception of the object is the JND

for the background level b The experiment should ideally be repeated many

times for the same observer and also repeated for several observers Exper

iments have shown that the JND is almost constant at approximately

or over a wide range of background intensity this is known as Webers

law

Example The ve bars in Figure have intensity values of from left to

right and The bars are placed on a background of

The contrast of the rst bar to the left according to Equation is

This contrast value is slightly greater than the nominal JND the object should

be barely perceptible to most observers The contrast values of the remaining

four bars are more than adequate for clear perception

Example Calcications appear as bright spots in mammograms A cal

cication that appears against fat and lowdensity tissue may possess high

FIGURE

Illustration of the eect of the background on the perception of an object

simultaneous contrast The two inner squares have the same gray level of

but are placed on dierent background levels of on the left and

on the right

FIGURE

Illustration of the notion of justnoticeable dierence The ve bars have

intensity values of from left to right and and are

placed on a background of The rst bar is barely noticeable the contrast

of the bars increases from left to right

contrast and be easily visible On the other hand a similar calcication that

appears against a background of highdensity breast tissue or a calcication

that is present within a highdensity tumor could possess low contrast and

be dicult to detect Figure shows a part of a mammogram with several

calcications appearing against dierent background tissue patterns and den

sity The various calcications in this image present dierent levels of contrast

and visibility

Small calcications and masses situated amidst highdensity breast tissue

could present low contrast close to the JND in a mammogram Such features

present signicant challenges in a breast cancer screening situation Enhance

ment of the contrast and visibility of such features could assist in improving

the accuracy of detecting early breast cancer see Sections

and

Histogram

The dynamic range of the gray levels in an image provides global information

on the extent or spread of intensity levels across the image However the dy

namic range does not provide any information on the existence of intermediate

gray levels in the image The histogram of an image provides information on

the spread of gray levels over the complete dynamic range of the image across

all pixels in the image

Consider an image fmn of size M N pixels with gray levels l

L The histogram of the image may be dened as

fmn l l L

where the discrete unit impulse function or delta function is dened as

if k

otherwise

The histogram value P

l provides the number of pixels in the image f

that possess the gray level l The sum of all the entries in a histogram equals

the total number of pixels in the image

l MN

The area under the function P

l when multiplied with an appropriate scal

ing factor provides the total intensity density or brightness of the image

depending upon the physical parameter represented by the pixel values

FIGURE

Part of a mammogram with several calcications associated with malignant

breast disease The density of the background aects the contrast and visi

bility of the calcications The image has pixels at a resolution of

m the true width of the image is about mm

A histogram may be normalized by dividing its entries by the total number

of pixels in the image Then with the assumption that the total number of

pixels is large and that the image is a typical representative of its class or the

process that generates images of its kind the normalized histogram may be

taken to represent the PDF p

l of the imagegenerating process

It follows that

Example The histogram of the image in Figure is shown in Figure

It is seen that most of the pixels in the image lie in the narrow range of

out of the available range of The eective dynamic range of the image

may be taken to be rather than This agrees with the dull

and lowcontrast appearance of the image The full available range of gray

levels has not been utilized in the image which could be due to poor lighting

and image acquisition conditions or due to the nature of the object being

imaged

The gray level of the large blank background in the image in Figure is

in the range the peak in the histogram corresponds to the general

background range The relatively bright areas of the myocyte itself have gray

levels in the range The histogram of the myocyte image is almost

unimodal that is it has only one major peak The peak happens to represent

the background in the image rather than the object of interest

Example Figure a shows the histogram of the image in Figure

b The discrete spikes are due to noise in the image The histogram of the

image after smoothing using the mean lter and rounding the results to

integers is shown in part b of the gure The histogram of the ltered image

is bimodal with two main peaks spanning the gray level ranges and

representing the collagen bers and background respectively Most

of the pixels corresponding to the collagen bers in crosssection have gray

levels below about most of the brighter background pixels have values

greater than

Example Figure shows a part of a mammogram with a tumor The

normalized histogram of the image is shown in Figure It is seen that the

histogram has two large peaks in the range representing the background

in the image with no breast tissue Although the image has bright areas the

number of pixels occupying the high gray levels in the range is

insignicant

Example Figure shows a CT image of a twoyearold male patient

with neuroblastoma see Section for details The histogram of the image

is shown in Figure a The histogram of the entire CT study of the

patient including sectional images is shown in Figure b Observe

FIGURE

Histogram of the image of the ventricular myocyte in Figure The size of

the image is pixels Entropy H bits

FIGURE

a Histogram of the image of the collagen bers in Figure b H

bits b Histogram of the image after the application of the mean

lter and rounding the results to integers H bits

FIGURE

Part of a mammogram with a malignant tumor the relatively bright region

along the upperleft edge of the image The size of the image is

pixels The pixel resolution of m the width of the image is about

mm Image courtesy of Foothills Hospital Calgary

FIGURE

Normalized histogram of the mammogram in Figure Entropy H

bits

that the unit of the pixel variable in the histograms is HU however the gray

level values in the image have been scaled for display in Figure and do

not directly correspond to the HU values The histograms are multimodal

indicating the presence of several types of tissue in the CT images The peaks

in the histogram in Figure a in the range HU correspond to liver

and other abdominal organs and tissues The small peak in the range

HU in the same histogram corresponds to calcied parts of the tumor

The histogram of the full volume includes a small peak in the range

HU corresponding to bone not shown in Figure b Histograms of

this nature provide information useful in diagnosis as well as in the follow up

of the eect of therapy Methods for the analysis of histograms for application

in neuroblastoma are described in Section

Entropy

The distribution of gray levels over the full available range is represented

by the histogram The histogram provides quantitative information on the

FIGURE

CT image of a patient with neuroblastoma Only one sectional image out of a

total of images in the study is shown The size of the image is

pixels The tumor which appears as a large circular region on the left

hand side of the image includes calcied tissues that appear as bright regions

The HU range of has been linearly mapped to the display range

of see also Figures and Image courtesy of Alberta Childrens

Hospital Calgary

FIGURE

a Histogram of the CT section image in Figure b Histogram of the

entire CT study of the patient with sectional images The histograms are

displayed for the range HU only

probability of occurrence of each gray level in the image However it is often

desirable to express in a single quantity the manner in which the values of a

histogram or PDF vary over the full available range Entropy is a statistical

measure of information that is commonly used for this purpose

The various pixels in an image may be considered to be symbols produced by

a discrete information source with the gray levels as its states Let us consider

the occurrence of L gray levels in an image with the probability of occurrence

of the l

gray level being pl l L Let us also treat the gray

level of a pixel as a random variable A measure of information conveyed by an

event a pixel or a gray level may be related to the statistical uncertainty of

the event giving rise to the information rather than the semantic or structural

content of the signal or image Given the unlimited scope of applications of

imaging and the contextdependent meaning conveyed by images a statistical

approach as above is appropriate to serve the general purpose of analysis of

the information content of images

A measure of information hp should be a function of pl satisfying the

following criteria

hp should be continuous for p

hp for p a totally unexpected event conveys maximal infor

mation when it does indeed occur

hp for p an event that is certain to occur does not convey

any information

if p

an event with a lower probability of occurrence

conveys more information when it does occur than an event with a higher

probability of occurrence

If two statistically independent image processes or pixels f and g are

considered the joint information of the two sources is given by the sum

of their individual measures of information h

These requirements are met by hp logp

When a source generates a number of gray levels with dierent probabilities

a measure of average information or entropy is dened as the expected value

of information contained in each possible level

plhpl

Using log

in place of h we obtain the commonly used denition of entropy

pl log

pl bits

Because the gray levels are considered as individual entities in this denition

that is no neighboring elements are taken into account the result is known

as the zerothorder entropy

The entropies of the images in Figures and with the corresponding

histogram or PDF in Figures and are and bits respectively

Observe that the histogram in Figure has a broader spread than that in

Figure which accounts for the correspondingly higher entropy

Dierentiating the function in Equation with respect to pl it can

be shown that the maximum possible entropy occurs when all the gray levels

occur with the same probability equal to

that is when the various gray

levels are equally likely

max

log

If the number of gray levels in an image is

then H

max

is K bits the

maximum possible entropy of an image with bit pixels is bits

It should be observed that entropy characterizes the statistical information

content of a source based upon the PDF of the constituent events which are

treated as random variables When an image is characterized by its entropy

it is important to recognize that the measure is not sensitive to the pictorial

structural semantic or applicationspecic diagnostic information in the

image Entropy does not account for the spatial distribution of the gray levels

in a given image Regardless the entropy of an image is an important measure

because it gives a summarized measure of the statistical information content of

an image an imagegenerating source or an information source characterized

by a PDF as well as the lower bound on the noisefree transmission rate and

storage capacity requirements

Properties of entropy A few important properties of entropy

are as follows

with H

only for p or p no information is conveyed

by events that do not occur or occur with certainty

The joint information H

conveyed by n events with probabil

ities of occurrence p

is governed by H

logn

with equality if and only if p

for i n

Considering two images or sources f and g with PDFs p

and p

where l

and l

represent gray levels in the range L the average

joint information or joint entropy is

log

If the two sources are statistically independent the joint PDF p

reduces to p

Joint entropy is governed by the condition

with equality if and only if f and g are statistically

independent

The conditional entropy of an image f given that another image g has

been observed is

f jg

log

f jg

log

f jg

where p

f jg

is the conditional PDF of f given g Then H

f jg

with equality if and only if f and g are statistically

independent Note The conditional PDF of f given g is dened as

f jg

if p

otherwise

Higher order entropy The formulation of entropy as a measure of in

formation is based upon the premise that the various pixels in an image may

be considered to be symbols produced by a discrete information source with

the gray levels as its states From the discussion above it follows that the

denition of entropy in Equation assumes that the successive pixels pro

duced by the source are statistically independent While governed by the

limit H

max

K bits the entropy of a realworld image with K bits per

pixel encountered in practice could be considerably lower due to the fact

that neighboring pixels in most real images are not independent of one an

other Due to this reason it is desirable to consider sequences of pixels to

estimate the true entropy or information content of a given image

Let pfl

g represent the probability of occurrence of the sequence fl

g of gray levels in the image f The n

order entropy of f is dened

pfl

g log

pfl

where the summation is over all possible sequences fl

g with n pix

els Note Some variations exist in the literature regarding the denition of

higherorder entropy In the denition given above n refers to the number

of neighboring or additional elements considered not counting the initial or

zeroth element this is consistent with the denition of the zerothorder en

tropy in Equation H

is a monotonically decreasing function of n and

approaches the true entropy of the source as n

Mutual information A measure that is important in the analysis of

transmission of images over a communication system as well as in the analysis

of storage in and retrieval from an archival system with potential loss of

information is mutual information dened as

f jg

gjf

This measure represents the information received or retrieved with the follow

ing explanation H

is the information input to the transmission or archival

system in the form of the image f H

f jg

is the information about f given that

the received or retrieved image g has been observed In this analysis g is

taken to be known but f is considered to be unknown although g is expected

to be a good representation of f Then if g is completely correlated with f

we have H

f jg

and I

f jg

this represents the case where there is no

loss or distortion in image transmission and reception or in image storage and

retrieval If g is independent of f H

f jg

and I

f jg

this represents

the situation where there is complete loss of information in the transmission

or archival process

Entropy and mutual information are important concepts that are useful in

the design and analysis of image archival coding and communication systems

this topic is discussed in Chapter

Blur and Spread Functions

Several components of image acquisition systems cause blurring due to intrin

sic and practical limitations The simplest visualization of blurring is provided

by using a single ideal point to represent the object being imaged see Fig

ure a Mathematically an ideal point is represented by the continuous

unit impulse function or the Dirac delta function x y dened as

x y

undened atx y

otherwise

and

x y dx dy

Note The D Dirac delta function x is dened in terms of its action within

an integral as

fx x x

if a x

otherwise

where fx is a function that is continuous at x

This is known as the sifting

property of the delta function because the value of the function fx at the location

a b

FIGURE

a An ideal point source b A Gaussianshaped point spread function

of the delta function is sifted or selected from all of its values The expression

may be extended to all x as

f x d

which may also be interpreted as resolving the arbitrary signal fx into a weighted

combination of mutually orthogonal delta functions A common denition of the

delta function is in terms of its integrated strength as

x dx

with the conditions

undened atx

otherwise

The delta function is also dened as the limiting condition of several ordinary func

tions one of which is

x lim

exp

jxj

The delta function may be visualized as the limit of a function with a sharp peak

of undened value whose integral over the full extent of the independent variable is

maintained as unity while its temporal or spatial extent is compressed toward zero

The image obtained when the input is a point or impulse function is known

as the impulse response or point spread function PSF see Figure b

Assuming the imaging system to be linear and shiftinvariant or position

invariant or spaceinvariant abbreviated as LSI the image gx y of an ob

ject fx y is given by the D convolution integral

gx y

hx y f d d

h fx y d d

hx y fx y

where hx y is the PSF and are temporary variables of integration and

represents D convolution

Note For details on the theory of linear systems and convolution refer to

Lathi Oppenheim et al Oppenheim and Schafer and Gonzalez and

Woods In extending the concepts of LSI system theory from timedomain

signals to the space domain of images it should be observed that causality is

not a matter of concern in most applications of image processing

Some examples of the cause of blurring are

Focal spot The physical spot on the anode target that generates

X rays is not an ideal dimensionless point but has nite physical di

mensions and an area of the order of mm

Several straightline

paths would then be possible from the Xray source through a given

point in the object being imaged and on to the lm The image so

formed will include not only the main radiographic shadow the um

bra but also an associated blur the penumbra as illustrated in

Figure The penumbra causes blurring of the image

Thickness of screen or crystal The screen used in screenlm X

ray imaging and the scintillation crystal used in gammaray imaging

generate visible light when struck by X or gamma rays Due to the

nite thickness of the screen or crystal a point source of light within the

detector will be sensed over a wider region on the lm see Figure

or by several PMTs see Figure the thicker the crystal or screen

the worse the blurring eect caused as above

Scattering Although it is common to assume straightline propagation

of X or gamma rays through the body or object being imaged this is not

always the case in reality X gamma and ultrasound rays do indeed

get scattered within the body and within the detector The eect of

rays that are scattered to a direction that is signicantly dierent from

the original path will likely be perceived as background noise However

scattering to a smaller extent may cause unsharp edges and blurring in

a manner similar to those described above

Point line and edge spread functions In practice it is often not

possible or convenient to obtain an image of an ideal point a microscopic hole

in a sheet of metal may not allow adequate Xray photons to pass through and

create a useful image an innitesimally small drop of a radiopharmaceutical

may not emit sucient gammaray photons to record an appreciable image on

a gamma camera However it is possible to construct phantoms to represent

ideal lines or edges For use in Xray imaging a line phantom may be created

FIGURE

The eect of a nite focal spot Xraygenerating portion of the target on

the sharpness of the image of an object

by cutting a narrow slit in a sheet of metal In SPECT imaging it is common

to use a thin plastic tube with diameter of the order of mm and lled with

a radiopharmaceutical to create a line source Given that the spatial

resolution of a typical SPECT system is of the order of several mm such a

phantom may be assumed to represent an ideal straight line with no thickness

An image obtained of such a source is known as the line spread function LSF

of the system Because any crosssection of an ideal straight line is a point

or impulse function the reconstruction of a crosssection of a line phantom

provides the PSF of the system Observe also that the integration of an ideal

point results in a straight line along the path of integration see Figure

In cases where the construction of a line source is not possible or appropri

ate one may prepare a phantom representing an ideal edge Such a phantom

is easy to prepare for planar Xray imaging one needs to simply image the

ideal and straight edge of a sheet or slab made of a material with a higher

attenuation coecient than that of the background or table upon which it is

placed when imaging In the case of CT imaging a D cube or parallelepiped

with its sides and edges milled to be perfect planes and straight lines re

spectively may be used as the test object A prole of the image of such

a phantom across the ideal edge provides the edge spread function ESF of

the system see Figure see also Section The derivative of an edge

along the direction perpendicular to the edge is an ideal straight line see Fig

FIGURE

The relationship between point impulse function line and edge step im

ages The height of each function represents its strength

ure Therefore the derivative of the ESF gives the LSF of the system

Then the PSF may be estimated from the LSF as described above

FIGURE

Blurring of an ideal sharp edge into an unsharp edge by an imaging system

In practice due to the presence of noise and artifacts it would be desirable

to average several measurements of the LSF which could be performed along

the length of the line or edge If the imaging system is anisotropic the LSF

should be obtained for several orientations of the line source If the blur of

the system varies with the distance between the detector and the source as is

the case in nuclear medicine imaging with a gamma camera one should also

measure the LSF at several distances

The mathematical relationships between the PSF LSF and ESF may be

expressed as follows Consider integration of the D delta function

along the x axis as follows

x y

x y dx

x dx

The last integral above is equal to unity the separability property of the D

impulse function as x y x y has been used above Observe that

although y has been expressed as a function of y only it represents a D

function of x y that is independent of x in the present case Considering

y over the entire D x y space it becomes evident that it is a line function

that is placed on the x axis The line function is thus given by an integral of

the impulse function see Figure

The output of an LSI system when the input is the line image f

x y

y that is the LSF which we shall denote here as h

x y is given by

x y

h f

x y d d

h y d d

h y d

hx y dx

In the equations above hx y is the PSF of the system and the sifting

property of the delta function has been used The nal equation above shows

that the LSF is the integral in this case along the x axis of the PSF This

result also follows simply from the linearity of the LSI system and that of

the operation of integration given that hx y is the output due to x y as

the input if the input is an integral of the delta function the output will be

the corresponding integral of hx y Observe that in the present example

x y is independent of x

Let us now consider the Fourier transform of h

x y Given that h

x y is

independent of x in the present illustration we may write it as a D function

y correspondingly its Fourier transform will be a D function which we

shall express as H

v Then we have

y expj vy dy

dx hx y expj ux vyj

Hu vj

H v

where Hu v is the D Fourier transform of hx y see Sections and

This shows that the Fourier transform of the LSF gives the values of

the Fourier transform of the PSF along a line in the D Fourier plane in this

case along the v axis

In a manner similar to the discussion above let us consider integrating the

line function as follows

x y

x d

The resulting function has the property

x f

x y

if y

which represents an edge or unit step function that is parallel to the x axis

see Figure Thus the edge or step function is obtained by integrating

the line function It follows that the ESF is given by

Conversely the LSF is the derivative of the ESF

Thus the ESF may be used to obtain the LSF which may further be used

to obtain the PSF and MTF as already explained Observe the use of the

generalized delta function to derive the discontinuous line and edge functions

in this section

In addition to the procedures and relationships described above based upon

the Fourier slice theorem see Section and Figure it can be shown

that the Fourier transform of a prole of the LSF is equal to the radial prole

of the Fourier transform of the PSF at the angle of placement of the line

source If the imaging system may be assumed to be isotropic in the plane of

the line source a single radial prole is adequate to reconstruct the complete

D Fourier transform of the PSF Then an inverse D Fourier transform

provides the PSF This method which is essentially the Fourier method of

reconstruction from projections described in Section was used by Hon et

al and Boulfelfel to estimate the PSF of a SPECT system

Example of application In the work of Boulfelfel a line source

was prepared using a plastic tube of internal radius mm lled with mCi

milli Curie of

Tc The phantom was imaged using a gamma camera at

various sourcetocollimator distances using an energy window of width of

keV centered at keV Figure shows a sample image of the line

source Figure shows a sample prole of the LSF and the averaged prole

obtained by averaging the rows of the LSF image

FIGURE

Nuclear medicine planar image of a line source obtained using a gamma

camera The size of the image is pixels with an eective width of

mm The pixel size is mm

It is common practice to characterize an LSF or PSF with its full width

at half the maximum FWHM value Boulfelfel observed that the FWHM

of the LSF of the gamma cameras studied varied between cm and cm

depending upon the radiopharmaceutical used the sourcetocollimator dis

FIGURE

Sample prole dotted line and averaged prole solid line obtained from

the image in Figure Either prole may be taken to represent the LSF

of the gamma camera

tance and the intervening medium The LSF was used to estimate the PSF

as explained above The FWHM of the PSF of the SPECT system studied

was observed to vary between cm and cm

See Section for illustrations of the ESF and LSF of a CT imaging

system See Chapter for descriptions of methods for deblurring images

Resolution

The spatial resolution of an imaging system or an image may be expressed in

terms of the following

The sampling interval in for example mm or m

The width of a prole of the PSF usually FWHM in mm

The size of the laser spot used to obtain the digital image by scanning

an original lm or the size of the solidstate detector used to obtain the

digital image in m

The smallest visible object or separation between objects in the image

in mm or m

The nest grid pattern that remains visible in the image in lpmm

The typical resolution limits of a few imaging systems are

Xray lm lpmm

screenlm combination lpmm

mammography up to lpmm

CT lpmm

CT lpmm or m

SPECT lpmm

The Fourier Transform and Spectral Content

The Fourier transform is a linear reversible transform that maps an image

from the space domain to the frequency domain Converting an image from

the spatial to the frequency Fourier domain helps in assessing the spectral

content and energy distribution over frequency bands Sharp edges in the

image domain are associated with large proportions of highfrequency con

tent Oriented patterns in the space domain correspond to increased energy

in bands of frequency in the spectral domain with the corresponding ori

entation Simple geometric patterns such as rectangles and circles map to

recognizable functions in the frequency domain such as the sinc and Bessel

functions respectively Transforming an image to the frequency domain as

sists in the application of frequencydomain lters to remove noise enhance

the image or extract certain components that are better separated in the

frequency domain than in the space domain

The D Fourier transform of an image fx y denoted by F u v is given

F u v

fx y expj ux vy dx dy

The variables u and v represent frequency in the horizontal and vertical direc

tions respectively The frequency variable in image analysis is often referred

to as spatial frequency to avoid confusion with temporal frequency we will

however not use this terminology in this book Recall that the complex expo

nential is a combination of the D sine and cosine functions and is separable

expj ux vy

expj ux expj vy

cos ux j sin ux cos vy j sin vy

Images are typically functions of space hence the units of measurement

in the image domain are m cm mm m etc In the D Fourier domain

the unit of frequency is cyclesmm cyclesm mm

etc Frequency is also

expressed as lpmm If the distance to the viewer is taken into account

frequency could be expressed in terms of cyclesdegree of the visual angle

subtended at the viewers eye The unit Hertz is not used in D Fourier

analysis

In computing the Fourier transform it is common to use the discrete Fourier

transform DFT via the fast Fourier transform FFT algorithm The D

DFT of a digital image fmn of size M N pixels is dened as

F k l

fmn exp

For complete recovery of fmn from F k l the latter should be computed

for k M and l N at the minimum

Then the inverse transform gives back the original image with no error or

loss of information as

fmn

F k l exp

for m M and n N This expression may be

interpreted as resolving the given image into a weighted sum of mutually or

thogonal exponential or sinusoidal basis functions The eight sine functions

for k that form the imaginary part of the basis functions of

the D DFT for M are shown in Figure Figures and show

the rst cosine and sine basis functions for k l that are

the components of the D exponential function in Equation

FIGURE

The rst eight sine basis functions of the D DFT k from top

to bottom Each function was computed using samples

In order to use the FFT algorithm it is common to pad the given image

with zeros or some other appropriate background value and convert the image

to a square of size N N where N is an integral power of Then all indices

in Equation may be made to run from to N as

F k l

fmn exp

mk nl

FIGURE

The rst cosine basis functions of the D DFT Each function was computed

using a matrix

FIGURE

The rst sine basis functions of the D DFT Each function was computed

using a matrix

with k N and l N The inverse transform is

given as

fmn

F k l exp

mk nl

In Equations and the normalization factor has been divided equally

between the forward and inverse transforms to be

for the sake of symme

try

Example the rectangle function and its Fourier transform A

D function with a rectangular base of size X Y and height A is dened as

fx y A if x X y Y

otherwise

The D version of the rectangle function is also known as the gate function

The D Fourier transform of the rectangle function above is given by

F u v AXY

sin uX

expj uX

sin vY

expj vY

Observe that the Fourier transform of a real image is in general a complex

function However an image with even symmetry about the origin will have

a real Fourier transform The exp functions in Equation indicate the

phase components of the spectrum

A related function that is commonly used is the rect function dened as

rectx y

if jxj

jyj

if jxj

jyj

The Fourier transform of the rect function is the sinc function

rectx y sincu v

where

sincu v sincu sincv

sin u

sin v

and indicates that the two functions form a forward and inverse Fourier

transform pair

Figure shows three images with rectangular square objects and their

Fourier logmagnitude spectra Observe that the smaller the box the greater

the energy content in the higherfrequency areas of the spectrum At the lim

its we have the Fourier transform of an image of an innitely large rectangle

that is the transform of an image with a constant value of unity for all space

equal to and the Fourier transform of an image with an innitesimally

small rectangle that is an impulse equal to a constant of unity represent

ing a white spectrum The frequency axes have been shifted such that

u v is at the center of the spectrum displayed The frequency coor

dinates in this mode of display of image spectra are shown in Figure b

Figure shows the logmagnitude spectrum in Figure f with and

without shifting the shifted or centered or folded mode of display as in

Figure b is the preferred mode of display of D spectra

The rectangle image in Figure e as well as its magnitude spectrum

are also shown as mesh plots in Figure The mesh plot demonstrates

more clearly the sinc nature of the spectrum

Figure shows three images with rectangular boxes oriented at

and

and their logmagnitude spectra The sinc functions in the Fourier

domain in Figure are not symmetric in the u and v coordinates as was

the case in the spectra of the square boxes in Figure The narrowing of

the rectangle along a spatial axis results in the widening of the lobes of the

sinc function and the presence of increased highfrequency energy along the

corresponding frequency axis The rotation of an image in the spatial domain

results in a corresponding rotation in the Fourier domain

Example the circle function and its Fourier transform Circular

apertures and functions are encountered often in imaging and image process

ing The circ function which represents a circular disc or aperture is dened

circr

if r

where r

The Fourier transform of circr may be shown to be

where

represents radial frequency in the D u v

plane and J

is the rstorder Bessel function of the rst kind

Figure shows an image of a circular disc and its logmagnitude spec

trum The disc image as well as its magnitude spectrum are also shown as

mesh plots in Figure Ignoring the eects due to the representation of

the circular shape on a discrete grid both the image and its spectrum are

isotropic Figure shows two proles of the logmagnitude spectrum in

Figure b taken along the central horizontal axis The nature of the

Bessel function is clearly seen in the D plots the conjugate symmetry of

the spectrum is also readily seen in the plot in Figure a In displaying

proles of D system transfer functions it is common to show only one half

of the prole for positive frequencies as in Figure b If such a prole

is shown it is to be assumed that the system possesses axial or rotational

symmetry that is the system is isotropic

Examples of Fourier spectra of biomedical images Figure

shows two TEM images of collagen bers in rabbit ligament samples in

crosssection and their Fourier spectra The Bessel characteristics of the

spectrum due to the circular shape of the objects in the image are clearly

a b

c d

e f

FIGURE

a Rectangle image with total size pixels and a rectangle square of

size pixels b Logmagnitude spectrum of the image in a c Rect

angle size pixels d Logmagnitude spectrum of the image in c

e Rectangle size pixels f Logmagnitude spectrum of the image

in e The spectra have been scaled to map the range to the display

range See also Figures and

FIGURE

Frequency coordinates in a the unshifted mode and b the shifted mode of

display of image spectra U and V represent the sampling frequencies along

the two axes Spectra of images with real values possess conjugate symmetry

about U and V Spectra of sampled images are periodic with the periods

equal to U and V along the two axes It is common practice to display one

complete period of the shifted spectrum including the conjugate symmetric

parts as in b See also Figure

a b

FIGURE

a Logmagnitude spectrum of the rectangle image in Figure e without

shifting Most FFT routines provide spectral data in this format b The

spectrum in a shifted or folded such that u v is at the center It

is common practice to display one complete period of the shifted spectrum

including the conjugate symmetric parts as in b See also Figure

FIGURE

a Mesh plot of the rectangle image in Figure e with total size

pixels and a rectangle square of size pixels b Magnitude spectrum

of the image in a

a b

c d

e f

FIGURE

a Rectangle image with total size pixels and a rectangle of size

pixels b Logmagnitude spectrum of the image in a c Rectangle

size pixels this image may be considered to be that in a rotated

d Logmagnitude spectrum of the image in c e Image in c

rotated by

using nearestneighbor selection f Logmagnitude spectrum

of the image in e Spectra scaled to map to the display range