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
m
Hz may be represented without loss by its
samples obtained at the Nyquist rate of f
s
f
m
Hz
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
s
Hz Therefore the Fourier
spectrum of the sampled signal is periodic with a period equal to f
s
Hz A
sampled signal has innite bandwidth however the sampled signal contains
distinct or unique frequency components only up to f
m
f
s
Hz
If the signal as above is sampled at a rate lower than f
s
Hz an error known
as aliasing occurs where the frequency components above f
s
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
s
Hz the original signal may be recovered
from its sampled version by lowpass ltering and extracting the baseband
component over the band f
m
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
m
Hz
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
s
Hz with the prior knowledge that the signal contains no
signicant energy or information beyond f
m
f
s
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
n
possible quantized
levels spanning the range
n
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
r
max
Let the range r
min
r
max
be divided into L parts demarcated by the decision levels R
R
R
R
L
with R
r
min
and R
L
r
max
see Figure Let the L output levels
of the quantizer represent the values Q
Q
Q
Q
L
as indicated in
Figure
The meansquared error MSE in representing the analog signal by its
quantized values is given by
L
X
l
Z
R
l
R
l
r Q
l
pr dr
Several procedures exist to determine the values of R
l
and Q
l
that minimize
the MSE A classical result indicates that the output level Q
l
should lie at the centroid of the part of the PDF between the decision levels
R
l
and R
l
given by
Q
l
R
R
l
R
l
r pr dr
R
R
l
R
l
pr dr
which reduces to
Q
l
R
l
R
l
if the PDF is uniform It also follows that the decision levels are then given
by
R
l
Q
l
Q
l
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
of
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
l
and the decision levels represented by R
l
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
I
i
I
o
where I
i
is the intensity of the light input and I
o
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
C
OD
f
OD
b
OD
where f
OD
and b
OD
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
C
f b
f b
or as
C
f b
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
C
l
whereas that for the inner square on the right is
C
r
The values of C
l
and C
r
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
C
l
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
P
f
l
M
X
m
N
X
n
d
fmn l l L
where the discrete unit impulse function or delta function is dened as
d
k
if k
otherwise
The histogram value P
f
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
X
l
P
f
l MN
The area under the function P
f
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
f
l of the imagegenerating process
p
f
l
MN
P
f
l
It follows that
L
X
l
p
f
l
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
a
b
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
a
b
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
th
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
hp
hp
if p
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
fg
h
f
h
g
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
H
L
X
l
plhpl
Using log
in place of h we obtain the commonly used denition of entropy
as
H
L
X
l
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
L
that is when the various gray
levels are equally likely
H
max
L
X
l
L
log
L
log
L
If the number of gray levels in an image is
K
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
H
p
with H
p
only for p or p no information is conveyed
by events that do not occur or occur with certainty
The joint information H
p
p
p
n
conveyed by n events with probabil
ities of occurrence p
p
p
n
is governed by H
p
p
p
n
logn
with equality if and only if p
i
n
for i n
Considering two images or sources f and g with PDFs p
f
l
and p
g
l
where l
and l
represent gray levels in the range L the average
joint information or joint entropy is
H
fg
L
X
l
L
X
l
p
fg
l
l
log
p
fg
l
l
If the two sources are statistically independent the joint PDF p
fg
l
l
reduces to p
f
l
p
g
l
Joint entropy is governed by the condition
H
fg
H
f
H
g
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
H
f jg
L
X
l
L
X
l
p
g
l
p
f jg
l
l
log
p
f jg
l
l
L
X
l
L
X
l
p
fg
l
l
log
p
f jg
l
l
where p
f jg
l
l
is the conditional PDF of f given g Then H
f jg
H
fg
H
g
H
f
with equality if and only if f and g are statistically
independent Note The conditional PDF of f given g is dened as
p
f jg
l
l
p
fg
l
l
p
g
l
if p
g
l
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
n
g represent the probability of occurrence of the sequence fl
l
l
l
n
g of gray levels in the image f The n
th
order entropy of f is dened
as
H
n
n
X
fl
n
g
pfl
n
g log
pfl
n
g
where the summation is over all possible sequences fl
n
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
n
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
I
f jg
H
f
H
g
H
fg
H
f
H
f jg
H
g
H
gjf
This measure represents the information received or retrieved with the follow
ing explanation H
f
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
H
f
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
H
f
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
Z
x
Z
y
x y dx dy
Note The D Dirac delta function x is dened in terms of its action within
an integral as
Z
b
a
fx x x
o
dx
fx
o
if a x
o
b
otherwise
where fx is a function that is continuous at x
o
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
x
o
of the delta function is sifted or selected from all of its values The expression
may be extended to all x as
fx
Z
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
Z
x dx
with the conditions
x
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
Z
Z
hx y f d d
Z
Z
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
f
l
x y
Z
x
x y dx
Z
x
x y dx
y
Z
x
x dx
y
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
l
x y
y that is the LSF which we shall denote here as h
l
x y is given by
h
l
x y
Z
Z
h f
l
x y d d
Z
Z
h y d d
Z
h y d
Z
x
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
h
l
x y is independent of x
Let us now consider the Fourier transform of h
l
x y Given that h
l
x y is
independent of x in the present illustration we may write it as a D function
h
l
y correspondingly its Fourier transform will be a D function which we
shall express as H
l
v Then we have
H
l
v
Z
y
h
l
y expj vy dy
Z
y
dy
Z
x
dx hx y expj ux vyj
u
Hu vj
u
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
f
e
x y
Z
y
f
l
x d
Z
y
d
The resulting function has the property
x f
e
x y
if 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
h
e
y
Z
y
h
l
d
Conversely the LSF is the derivative of the ESF
h
l
y
d
dy
h
e
y
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
m
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
by
F u v
Z
x
Z
y
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
as
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
MN
M
X
m
N
X
n
fmn exp
j
mk
M
nl
N
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
M
X
k
N
X
l
F k l exp
j
mk
M
nl
N
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
N
N
X
m
N
X
n
fmn exp
j
N
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
N
N
X
k
N
X
l
F k l exp
j
N
mk nl
In Equations and the normalization factor has been divided equally
between the forward and inverse transforms to be
N
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
uX
expj uX
sin vY
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
u
sin v
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
o
o
and
o
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
as
circr
if r
if r
where r
p
x
y
The Fourier transform of circr may be shown to be
J
where
p
u
v
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
a
b
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
by
o
d Logmagnitude spectrum of the image in c e Image in c
rotated by
o
using nearestneighbor selection f Logmagnitude spectrum
of the image in e Spectra scaled to map to the display range
See also Figure
a b
FIGURE
a Image of a circular disc The radius of the disc is pixels the size of the
image is pixels b Logmagnitude spectrum of the image in a
See also Figures and
seen in Figure d Compare the examples in Figure with those in
Figure
Figure shows two SEM images of collagen bers as seen in freeze
fractured surfaces of rabbit ligament samples and their Fourier spectra The
highly oriented and piecewise linear rectangular characteristics of the bers
in the normal sample in Figure a are indicated by the concentrations of
energy along radial lines at the corresponding angles in the spectrum in Fig
ure b The scar sample in Figure c lacks directional preference
which is reected in its spectrum in Figure d Compare the examples
in Figure with those in Figure
Important properties of the Fourier transform
The Fourier transform is a linear reversible transform that maps an image
from the space domain to the frequency domain The spectrum of an image
can provide useful information on the frequency content of the image on
the presence of oriented or directional elements on the presence of specic
image patterns and on the presence of noise A study of the spectrum of an
image can assist in the development of ltering algorithms to remove noise
in the design of algorithms to enhance the image and in the extraction of
features for pattern recognition Some of the important properties of the
Fourier transform are described in the following paragraphs with illustrations
as required both the discrete and continuous representations of
functions are used as appropriate or convenient
a
b
FIGURE
a Mesh plot of the circular disc in Figure a The radius of the disc is
pixels the size of the image is pixels b Magnitude spectrum
of the image in a
a
b
FIGURE
a Prole of the logmagnitude spectrum in Figure b along the central
horizontal axis b Prole in a shown only for positive frequencies The
frequency axis is indicated in samples the true frequency values depend upon
the sampling frequency
a b
c d
FIGURE
a TEM image of collagen bers in a normal rabbit ligament sample b Log
magnitude spectrum of the image in a c TEM image of collagen bers in
a scar tissue sample d Logmagnitude spectrum of the image in c See
also Figure and Section
a
b
c
d
FIGURE
a SEM image of collagen bers in a normal rabbit ligament sample b Log
magnitude spectrum of the image in a c SEM image of collagen bers in
a scar tissue sample d Logmagnitude spectrum of the image in c See
also Figure and Section
The kernel function of the Fourier transform is separable and symmetric
This property facilitates the evaluation of the D DFT as a set of D
row transforms followed by a set of D column transforms We have
F k l
N
N
X
m
exp
j
N
mk
N
X
n
fmn exp
j
N
nl
D FFT routines may be used to obtain D and multidimensional Fourier
transforms in the following manner
F m l N
N
N
X
n
fmn exp
j
N
nl
F k l
N
N
X
m
F m l exp
j
N
mk
Care should be taken to check if the factor
N
is included in the forward
or inverse D FFT routine where required
The Fourier transform is an energyconserving transform that is
Z
x
Z
y
jfx yj
dx dy
Z
u
Z
v
jF u vj
du dv
This relationship is known as Parsevals theorem
The inverse Fourier transform operation may be performed using the
same FFT routine by taking the forward Fourier transform of the com
plex conjugate of the given function and then taking the complex con
jugate of the result
The Fourier transform is a linear transform The Fourier transform
of the sum of two images is the sum of the Fourier transforms of the
individual images
Images are often corrupted by additive noise such as
gx y fx y x y
Upon Fourier transformation we have
Gu v F u v u v
Most reallife images have a large portion of their energy concentrated
around u v in a lowfrequency region however the presence
of edges sharp features and smallscale or ne details leads to increased
strength of highfrequency components see Figure On the other
hand random noise has a spectrum that is equally spread all over the
frequency space that is a at uniform or white spectrum Indis
criminate removal of highfrequency components could cause blurring of
edges and the loss of the ne details in the image
The DFT and its inverse are periodic signals
F k l F k N l F k l N F k N l N
where and are integers
The Fourier transform is conjugatesymmetric for images with real val
ues
F kl F
k l
It follows that jF klj jF k lj and
F kl
F k l that
is the magnitude spectrum is even symmetric and the phase spectrum is
odd symmetric The symmetry of the magnitude spectrum is illustrated
by the examples in Figures and
A spatial shift or translation applied to an image leads to an additional
linear phase component in its Fourier transform the magnitude spec
trum is unaected If fmn F k l are a Fouriertransform pair
we have
fmm
o
n n
o
F k l exp
j
N
km
o
ln
o
where m
o
n
o
is the shift applied in the space domain
Conversely we also have
fmn exp
j
N
k
o
m l
o
n
F k k
o
l l
o
This property has important implications in the modulation of D sig
nals for transmission and communication however it does not have
a similar application with D images
F gives the average value of the image a scale factor may be re
quired depending upon the denition of the DFT used
For display purposes log
jF k lj
is often used the addition of
unity to avoid taking the log of zero and the squaring may some
times be dropped It is also common to fold or shift the spectrum to
bring the frequency point the DC point to the center and the
folding frequency half of the sampling frequency components to the
edges Figures and illustrate shifted spectra and the
corresponding frequency coordinates
Folding of the spectrum could be achieved by multiplying the image
fmn with
mn
before the FFT is computed Because the
indices m and n are integers this amounts to merely changing the signs
of alternate pixels This outcome is related to the property in Equa
tion with k
o
l
o
N which leads to
exp
j
N
k
o
m l
o
n
expj m n
mn
and
fmn
mn
F k N l N
Rotation of an image leads to a corresponding rotation of the Fourier
spectrum
fm
n
F k
l
where
m
m cos n sin n
m sin n cos
k
k cos l sin l
k sin l cos
This property is illustrated by the images and spectra in Figure
and is useful in the detection of directional or oriented patterns see
Chapter
Scaling an image leads to an inverse scaling of its Fourier transform
fam bn
jabj
F
k
a
l
b
where a and b are scalar scaling factors The shrinking of an image leads
to an expansion of its spectrum with increased highfrequency content
On the contrary if an image is enlarged its spectrum is shrunk with
reduced highfrequency energy The images and spectra in Figure
illustrate this property
Linear shift invariant systems and convolution Most imaging
systems may be modeled as linear and shiftinvariant or positioninvariant
systems that are completely characterized by their PSFs The output
of such a system is given as the convolution of the input image with the
PSF
gmn hmn fmn
N
X
N
X
h fm n
Upon Fourier transformation the convolution maps to the multiplica
tion of the two spectra
Gk l Hk lF k l
Thus we have the important property
hx y fx y Hu vF u v
expressed now in the continuous coordinates x y and u v The char
acterization of imaging systems in the transform domain is discussed in
Section
It should be noted that the convolution multiplication property with
the DFT implies periodic or circular convolution however this type
of convolution may be made to be equivalent to linear convolution by
zeropadding Details on this topic are presented in Section
Multiplication of images in the space domain is equivalent to the con
volution of their Fourier transforms
f
x y f
x y F
u v F
u v
In medical imaging some types of noise get multiplied with the image
When a transparency such as an Xray image on lm is viewed using
a light box the resulting image gx y may be modeled as the product
of the transparency or transmittance function fx y with the light
source intensity eld sx y giving gx y fx y sx y If sx y
is absolutely uniform with a value A its Fourier transform will be an
impulse Su v Au v The convolution of F u v with Au v
will have no eect on the spectrum except scaling by the constant A If
the source is not uniform the viewed image will be a distorted version
of the original the corresponding convolution Gu v F u v Su v
will distort the spectrum F u v of the original image
The correlation of two images fmn and gmn is given by the op
eration
fg
N
X
m
N
X
n
fmn gm n
Correlation is useful in the comparison of images where features that
are common to the images may be present with a spatial shift
Upon Fourier transformation we get the conjugate product of the spec
tra of the two images
!