Bản chất của hình ảnh y sinh học (Phần 2)

2
Image Quality and Information Content
Several factors a ect the quality and information content of biomedical images
acquired with the modalities described in Chapter 1. A few considerations
in biomedical image acquisition and analysis that could have a bearing on
image quality are described in Section 2.1. 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 and/or 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 di cult proposition. Similarly, the
nature of the information conveyed by an image is di cult to quantify due
to its multifaceted characteristics in terms of statistical, structural, perceptual, semantic, and diagnostic connotations. However, several measures have
been designed to characterize or quantify a few speci c 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 certain phenomena, could then be used to assess variations in image quality and
information content. We shall explore several such measures in this chapter.

2.1 Di culties in Image Acquisition and Analysis
In Chapter 1, we studied several imaging systems and procedures for the
acquisition of many di erent types of biomedical images. The practical application of these techniques may pose certain di culties: 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 di culties that one might encounter in biomedical image acquisition
and analysis.
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Accessibility of the organ of interest: Several organs of interest in
imaging-based investigation are situated well within the body, encased in protective and di cult-to-access regions, for good reason! For example, the brain
is protected by the skull, and the prostate is situated at the base of the bladder near the pelvic outlet. Several limitations are encountered in imaging
such organs special imaging devices and image processing techniques are required to facilitate their visualization. Visualization of the arteries in the
brain requires the injection of an X-ray contrast agent and the subtraction
of a reference image see Section 4.1. Special transrectal probes have been
designed for 3D ultrasonic imaging of the prostate 92]. Despite the use of
such special devices and techniques, images obtained in applications as above
tend to be a ected by severe artifacts.
Variability of information: Biological systems exhibit great ranges of inherent variability within their di erent 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 pattern is often clouded by signi cant 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. Imaging conditions and parameters could cause further ambiguities due to the
e ects 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
pro le in another. Furthermore, the notion of shape roughness is nonspeci c and open-ended. 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 a ect the
appearance of tumors and masses in images. The use of multiple views and
spot magni cation 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 wellbeing) for brief periods of time to permit improved imaging. However, cardiac activity, blood circulation, and peristaltic movement are not under one's
volitional control. The rhythmic contractile activity of the heart poses challenges in imaging of the heart. The pulsatile movement of blood through the
brain causes slight movements of the brain that could cause artifacts in angiographic imaging see Section 4.1. Dark shadows may appear in ultrasound
images next to bony regions due to signi cant attenuation of the investigating
beam, and hence the lack of echoes from tissues beyond the bony regions along
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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 X-ray 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 1.21.
State-of-the-art multislice and helical-scan 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
photon-counting process to a certain speci c phase of the cardiac cycle by
using the electrocardiogram (ECG) as a reference see Figure 1.27 and Section 3.10. Although nuclear medicine imaging procedures take several minutes, the almost-periodic activity of the heart permits the cumulative imaging
of its musculature or chambers at particular positions repeatedly over several

cardiac cycles.

Energy limitations: In X-ray 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, low-energy X-ray photons are absorbed more readily than high-energy photons by the skin and
breast tissues, thereby increasing the radiation dose to the patient. A compromise is required between these two considerations. Similarly, in TEM, a
high-kV electron beam would be desirable in order to minimize damage to
the specimen, but a low-kV beam can provide improved contrast. The practical application of imaging techniques often requires the striking of a trade-o
between con icting 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 organizations 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 bene ts 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 bene ts. Patient safety concerns may preclude
the use of a procedure that may yield better images or results than others,
or may require modi cations to a procedure that may lead to inferior images. Further image processing steps would then become essential in order to
improve image quality or otherwise compensate for the initial compromise.
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2.2 Characterization of Image Quality

Biomedical images are typically complex sources of several items of information. 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 di erent medical imaging systems
comparison of images obtained using di erent imaging parameter settings of a given system
comparison of the results of several image enhancement algorithms
assessment of the e ect of the passage of an image through a transmission channel or medium and
assessment of images compressed by di erent data compression techniques at di erent rates of loss of data, information, or quality.
Specially designed phantoms are often used to test medical imaging systems for routine quality control 104, 105, 106, 107, 108]. Bijkerk et al. 109]
developed a phantom with gold disks of di erent 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. 110] and Barrett 111] for examples. It is clear that not only are FOMs important, but so
is the methodology for their comparison. Kayargadde and Martens 112, 113]
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 de nition
| the clarity with which details are reproduced 114] | which is typically
expressed in terms of image sharpness. This notion was rst mentioned by
Higgins and Jones 115] in the realm of photography, but is valid for image
evaluation in a broader context. Rangayyan and Elkadiki 116] present a
survey of di erent methods to measure sharpness in photographic and digital
images (see Section 2.15). Because quality is a subjective notion, the results

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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 104, 105, 106, 107, 108, 117].
Subjective and objective judgment should agree to some degree under de ned
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.

2.3 Digitization of Images

The representation of natural scenes and objects as digital images for processing using computers requires two steps: sampling and quantization. Both of
these steps could potentially cause loss of quality and introduce artifacts.

2.3.1 Sampling

Sampling is the process of representing a continuous-time or continuous-space
signal on a discrete grid, with samples that are separated by (usually) uniform
intervals. The theory and practice of sampling 1D signals have been well
established 1, 2, 7]. In essence, a band-limited signal with the frequency of
its fastest component being fm Hz may be represented without loss by its
samples obtained at the Nyquist rate of fs = 2 fm Hz .
Sampling may be modeled as the multiplication of the given continuoustime 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, fs Hz ). Therefore, the Fourier
spectrum of the sampled signal is periodic, with a period equal to fs Hz . A
sampled signal has in nite bandwidth however, the sampled signal contains
distinct or unique frequency components only up to fm = fs =2 Hz .
If the signal as above is sampled at a rate lower than fs Hz , an error known
as aliasing occurs, where the frequency components above fs =2 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 fs Hz , the original signal may be recovered
from its sampled version by lowpass ltering and extracting the base-band
component over the band fm Hz from the in nite 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
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is of in nite 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 base-band components without signi cant
reduction in gain over the band fm Hz .
In practice, in order to prevent aliasing errors, it is common to use an
anti-aliasing lter prior to the sampling of 1D signals, with a pass-band that
is close to fs =2 Hz , with the prior knowledge that the signal contains no

signi cant energy or information beyond fm fs =2 Hz . Analog spectrum
analyzers may be used to estimate the bandwidth and spectral content of a
given 1D analog signal prior to sampling.
All of the concepts explained above apply to the sampling of 2D signals or
images. However, in most real-life applications of imaging and image processing, it is not possible to estimate the frequency content of the images, and
also not possible to apply anti-aliasing lters. Adequate sampling frequencies need to be established for each type of image or application based upon
prior experience and knowledge. Regardless, even with the same type of images, di erent sampling frequencies may be suitable or adequate for di erent
applications.
Figure 2.1 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
high-resolution 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 di erent 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.

2.3.2 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 2n possible quantized
levels, spanning the range 0 2n ; 1]. If n = 8 bits are used to represent each
pixel, there can exist 256 values or gray levels to represent the values of the
image at each pixel, in the range 0 255].
It is necessary to map appropriately the range of variation of the given
analog signal, such as the output of a charge-coupled 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
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(a)

(b)

(c)

(d)

FIGURE 2.1

E ect of sampling on the appearance and quality of an image: (a) 225 250
pixels (b) 112 125 pixels (c) 56 62 pixels and (d) 28 31 pixels. All four
images have 256 gray levels at 8 bits per pixel.

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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 Lloyd{Max quantization procedure 8, 9, 118, 119] to optimize a quantizer is derived as follows. Let p(r) 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 rmin rmax ]. Let the range rmin rmax ]
be divided into L parts demarcated by the decision levels R0 R1 R2 : : : RL ,
with R0 = rmin and RL = rmax see Figure 2.2. Let the L output levels
of the quantizer represent the values Q0 Q1 Q2 : : : QL;1 , as indicated in
Figure 2.2.
The mean-squared error (MSE) in representing the analog signal by its
quantized values is given by

"2 =

LX
;1 Z R +1
l

l=0 R

(r ; Ql )2 p(r) dr:

(2.1)

l

Several procedures exist to determine the values of Rl and Ql that minimize
the MSE 8, 9, 118, 119]. A classical result indicates that the output level Ql
should lie at the centroid of the part of the PDF between the decision levels
Rl and Rl+1 , given by
R R +1
r p(r) dr
(2.2)
Ql = RRR +1
p(r) dr
R
which reduces to
Ql = Rl +2Rl+1
(2.3)
if the PDF is uniform. It also follows that the decision levels are then given
by
Rl = Ql;12+ Ql :
(2.4)
It is common to quantize images to 8 bits=pixel. However, CT images
represent a large dynamic range of X-ray attenuation coe cient, normalized
into HU , over the range ;1 000 1 000] for human tissues. Small di erences
of the order of 10 HU could indicate the distinction between normal tissue
and diseased tissue. If the range of 2 000 HU were to be quantized into 256
levels using an 8-bit quantizer, each quantized level would represent a change
000 = 7:8125 HU , which could lead to the loss of the distinction as above
of 2256
in noise. For this reason, CT and several other medical images are quantized
using 12 ; 16 bits=pixel.
The use of an inadequate number of quantized gray levels leads to false
contours and poor representation of image intensities. Figure 2.3 illustrates

the loss of image quality as the number of bits per pixel is reduced from six
to one.
l

l

l

l

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Image Quality and Information Content

Q
0

Q
1

R = r
R
0
min 1

Q
2
R
2

Q
3
R
3

...
R
4

...

69

Q
Q
L-2 L-1
R
R = r
L-1 L
max

Quantizer
output levels
Decision levels

gray level r

FIGURE 2.2

Quantization of an image gray-level signal r with a Gaussian (solid line) or
uniform (dashed line) PDF. The quantizer output levels are indicated by Ql
and the decision levels represented by Rl .
The quantized values in a digital image are commonly referred to as gray
levels, with 0 representing black and 255 standing for white when 8-bit quantization 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 2.1 lists a few variables that bear di erent
relationships with the displayed pixel value.

2.3.3 Array and matrix representation of images

Images are commonly represented as 2D functions of space: f (x y). A digital
image f (m n) may be interpreted as a discretized version of f (x y) in a 2D
array, or as a matrix see Section 3.5 for details on matrix representation of
images and image processing operations. The notational di erences between
the representation of an image as a function of space and as a matrix could
be a source of confusion.

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(a)

(b)

(c)

(d)

FIGURE 2.3

E ect of gray-level quantization on the appearance and quality of an image:
(a) 64 gray levels (6 bits per pixel) (b) 16 gray levels (4 bits per pixel) (c) four
gray levels (2 bits per pixel) and (d) two gray levels (1 bit per pixel) All four
images have 225 250 pixels. Compare with the image in Figure 2.1 (a) with
256 gray levels at 8 bits per pixel.

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Relationships Between Tissue Type, Tissue Density, X-ray Attenuation Coe cient, Houns eld
Units (HU ), Optical Density (OD), and Gray Level 120, 121]. The X-ray Attenuation
Coe cient was Measured at a Photon Energy of 103:2 keV 121].
Tissue Density X-ray
Houns eld
type gm=cm3 attenuation (cm;1 ) units

Optical Gray level Appearance
density (brightness) in image

lung

< 0:001 lower

low

;700 ;800]

high

liver

1.2

0.18

medium
50 70]

medium medium

bone

1.9

higher

high
low
+800 +1 000]

low

high

dark

gray
white

Image Quality and Information Content

TABLE 2.1

71

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An M N matrix has M rows and N columns its height is M and width
is N numbering of the elements starts with (1 1) at the top-left corner and
ends with (M N ) at the lower-right corner of the image. A function of space
f (x y) that has been converted into a digital representation f (m n) is typically placed in the rst quadrant in the Cartesian coordinate system. Then, an
M N will have a width of M and height of N indexing of the elements starts
with (0 0) at the origin at the bottom-left corner and ends with (M ; 1 N ; 1)
at the upper-right corner of the image. Figure 2.4 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.
column

row
2

f(0,2) f(1,2) f(2,2) f(3,2)

n = 1

2

3

4

f(1,1) f(1,2) f(1,3) f(1,4)

m = 1

1

y = 0

f(0,1) f(1,1) f(2,1) f(3,1)

2

f(2,1) f(2,2) f(2,3) f(2,4)

f(0,0) f(1,0) f(2,0) f(3,0)

3

f(3,1) f(3,2) f(3,3) f(3,4)

x = 0

1

2

3

f(x, y) as a 4x3 function of
space in the first quadrant

f(m, n) as a 3x4 matrix
(in the fourth quadrant)

FIGURE 2.4

Array and matrix representation of an image.

2.4 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 X-ray attenuation coe cient the
intensity of light re ected 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
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display of images. The OD at a spot on a lm is de ned as

OD = log10 IIi
o

(2.5)

where Ii is the intensity of the light input and Io is the intensity of the light
transmitted through the lm at the spot of interest see Figure 2.5. A perfectly
clear spot will transmit all of the light that is input and will have OD = 0
a dark spot that reduces the intensity of the input light by a factor of 1 000
will have OD = 3. X-ray lms, in particular those used in mammography, are
capable of representing gray levels from OD 0 to OD 3:5.
I

o
film with image
(transparency)

Ii
light

FIGURE 2.5

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

2.5 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 di erence
between the maximum and minimum values present in the image. X-ray
lms for mammography typically possess a dynamic range of 0 ; 3:5 OD.
Modern CRT monitors provide dynamic range of the order of 0 ; 600 cd=m2
in luminance or 1 : 1 000 in sampled gray levels.
Figure 2.6 compares the characteristic curves of two devices. Device A
has a larger slope or \gamma" (see Section 4.4.3) than Device B, and hence
can provide higher contrast (de ned in Section 2.6). 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 (X-ray) exposure are
known as Hurter{Dri eld or H-D curves 3].
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3.0

Saturation
Shoulder

2.0

Device A

-

Device B

Optical
density

1.0

-

Toe
Background level
(base, fog, noise)
log (exposure)

FIGURE 2.6

Characteristic response curves of two hypothetical imaging devices.

The lower levels of response of a lm or electronic display device are affected 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 operation. 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, calci cations in the breast and in tumors, as well
as metallic implants such as screws in bones and surgical clips contribute
to high-density areas in images. Mammograms are expected to possess a
dynamic range of 0 ; 3:5 OD. CT images may have a dynamic range of about
;1 000 to +1 000 HU . Metallic implants could have HU values beyond the
operating range of CT systems, and lead to saturated areas in images: the
X-ray beam is e ectively stopped by heavy-metal implants.
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2.6 Contrast

Contrast is de ned in a few di erent ways 9], but is essentially the di erence
between the parameter imaged in a region of interest (ROI) and that in a
suitably de ned background. If the image parameter is expressed in OD,
contrast is de ned as
COD = fOD ; bOD
(2.6)
where fOD and bOD represent the foreground ROI and background OD, respectively. Figure 2.7 illustrates the notion of contrast using circular ROIs.

b

f

FIGURE 2.7

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 contrast will require normalization. If, for example, f and b represent the average
light intensities emitted or re ected from the foreground ROI and the background, respectively, contrast may be de ned as
or as

b
C = ff ;
+b
C1 = f ;b b :

(2.7)

(2.8)
Due to the use of a reference background, the measures de ned 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
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di erence, but a ratio. The human visual system (HVS) has bandpass lter
characteristics, which lead to responses that are proportional to di erences
between illumination levels rather than to absolute illumination levels 122].
Example: The two squares in Figure 2.8 are of the same value (130 in the
scale 0 ; 255), but are placed on two di erent background regions of value

150 on the left and 50 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 e ect could be explained by the measure of simultaneous
contrast: the contrast of the inner square on the left, using the de nition in
Equation 2.8, is
; 150
Cl = 130150
= ;0:1333
(2.9)
whereas that for the inner square on the right is
Cr = 13050; 50 = +1:6:
(2.10)
The values of Cl and Cr using the de nition in Equation 2.7 are, respectively,
;0:0714 and +0:444 the advantage of this formulation is that the values of
contrast are limited to the range ;1 1]. 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 just-noticeable di erence
(JND) is important in analyzing contrast, visibility, and the quality of medical
images. JND is determined as follows 9, 122]: For a given background level
b as in Equation 2.8, 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 (f ; b)=b 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. Experiments have shown that the JND is almost constant, at approximately 0:02
or 2%, over a wide range of background intensity this is known as Weber's
law 122].
Example: The ve bars in Figure 2.9 have intensity values of (from left to

right) 155, 175, 195, 215, and 235. The bars are placed on a background of
150. The contrast of the rst bar (to the left), according to Equation 2.8, is
; 150
= +0:033:
(2.11)
Cl = 155150
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: Calci cations appear as bright spots in mammograms. A calci cation that appears against fat and low-density tissue may possess high
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FIGURE 2.8

Illustration of the e ect of the background on the perception of an object
(simultaneous contrast). The two inner squares have the same gray level of
130, but are placed on di erent background levels of 150 on the left and 50
on the right.

FIGURE 2.9

Illustration of the notion of just-noticeable di erence. The ve bars have
intensity values of (from left to right) 155, 175, 195, 215, and 235, and are
placed on a background of 150. The rst bar is barely noticeable the contrast
of the bars increases from left to right.

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contrast and be easily visible. On the other hand, a similar calci cation that
appears against a background of high-density breast tissue, or a calci cation
that is present within a high-density tumor, could possess low contrast, and
be di cult to detect. Figure 2.10 shows a part of a mammogram with several
calci cations appearing against di erent background tissue patterns and density. The various calci cations in this image present di erent levels of contrast
and visibility.
Small calci cations and masses situated amidst high-density breast tissue
could present low contrast close to the JND in a mammogram. Such features
present signi cant challenges in a breast cancer screening situation. Enhancement of the contrast and visibility of such features could assist in improving
the accuracy of detecting early breast cancer 123, 124, 125] see Sections 4.9.1
and 12.10.

2.7 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 dynamic 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 f (m n) of size M N pixels, with gray levels l =
0 1 2 : : : L ; 1. The histogram of the image may be de ned as

Pf (l) =

MX
;1 NX
;1
m=0 n=0

d f (m n) ; l ]

l = 0 1 2 ::: L ; 1

(2.12)

where the discrete unit impulse function or delta function is de ned as 1, 2]
1 if k = 0
(2.13)
d (k) = 0 otherwise:
The histogram value Pf (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:
LX
;1
l=0

Pf (l) = MN:

(2.14)

The area under the function Pf (l), when multiplied with an appropriate scaling factor, provides the total intensity, density, or brightness of the image,
depending upon the physical parameter represented by the pixel values.

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FIGURE 2.10

79

Part of a mammogram with several calci cations associated with malignant
breast disease. The density of the background a ects the contrast and visibility of the calci cations. The image has 768 512 pixels at a resolution of
62 m the true width of the image is about 32 mm.

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80

Biomedical Image Analysis

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 pf (l) of the image-generating process:
(2.15)
p (l) = 1 P (l):

MN f

f

It follows that

LX
;1
l=0

pf (l) = 1:

(2.16)

Example: The histogram of the image in Figure 1.3 is shown in Figure 2.11.
It is seen that most of the pixels in the image lie in the narrow range of 70 ; 150
out of the available range of 0 ; 255. The e ective dynamic range of the image
may be taken to be 70 ; 150, rather than 0 ; 255. This agrees with the dull
and low-contrast 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 1.3 is
in the range 80 ; 90: 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 100 ; 130. 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 2.12 (a) shows the histogram of the image in Figure 1.5
(b). The discrete spikes are due to noise in the image. The histogram of the
image after smoothing, using the 3 3 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 100 ; 180 and
180 ; 255, representing the collagen bers and background, respectively. Most
of the pixels corresponding to the collagen bers in cross-section have gray
levels below about 170 most of the brighter background pixels have values
greater than 200.
Example: Figure 2.13 shows a part of a mammogram with a tumor. The
normalized histogram of the image is shown in Figure 2.14. It is seen that the
histogram has two large peaks in the range 0 ; 20 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 200 ; 255 is
insigni cant.
Example: Figure 2.15 shows a CT image of a two-year-old male patient
with neuroblastoma (see Section 9.9 for details). The histogram of the image
is shown in Figure 2.16 (a). The histogram of the entire CT study of the
patient, including 75 sectional images, is shown in Figure 2.16 (b). Observe
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Image Quality and Information Content

81

4

x 10

2

Number of pixels

1.5

1

0.5

0

0

50

100

150

200

250

Gray level

FIGURE 2.11

Histogram of the image of the ventricular myocyte in Figure 1.3. The size of
the image is 480 480 = 230 400 pixels. Entropy H = 4:96 bits.

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82

Biomedical Image Analysis
3000

2500

Number of pixels

2000

1500

1000

500

0

0

50

100

150

200

250

150

200

250

Gray level

(a)
1800

1600

1400

Number of pixels

1200

1000

800

600

400

200

0

0

50

100
Gray level

FIGURE 2.12

(b)

(a) Histogram of the image of the collagen bers in Figure 1.5 (b) H =
7:0 bits. (b) Histogram of the image after the application of the 3 3 mean
lter and rounding the results to integers H = 7:1 bits.
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Image Quality and Information Content

FIGURE 2.13

83

Part of a mammogram with a malignant tumor (the relatively bright region
along the upper-left edge of the image). The size of the image is 700 700 =
490 000 pixels. The pixel resolution of 62 m the width of the image is about
44 mm. Image courtesy of Foothills Hospital, Calgary.

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84

Biomedical Image Analysis
0.025

Probability of occurrence

0.02

0.015

0.01

0.005

0

0

50

100

150

200

250

Gray level

FIGURE 2.14

Normalized histogram of the mammogram in Figure 2.13. Entropy H =
6:92 bits.
that the unit of the pixel variable in the histograms is HU however, the graylevel values in the image have been scaled for display in Figure 2.15, 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 2.16 (a) in the range 50;150 HU correspond to liver
and other abdominal organs and tissues. The small peak in the range 200 ;
300 HU in the same histogram corresponds to calci ed parts of the tumor.
The histogram of the full volume includes a small peak in the range 700 ;
800 HU corresponding to bone not shown in Figure 2.16 (b)]. Histograms of
this nature provide information useful in diagnosis as well as in the follow up
of the e ect of therapy. Methods for the analysis of histograms for application
in neuroblastoma are described in Section 9.9.

2.8 Entropy
The distribution of gray levels over the full available range is represented
by the histogram. The histogram provides quantitative information on the
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Image Quality and Information Content

FIGURE 2.15

85

CT image of a patient with neuroblastoma. Only one sectional image out of a
total of 75 images in the study is shown. The size of the image is 512 512 =
262 144 pixels. The tumor, which appears as a large circular region on the lefthand side of the image, includes calci ed tissues that appear as bright regions.
The HU range of ;200 400] has been linearly mapped to the display range
of 0 255] see also Figures 2.16 and 4.4. Image courtesy of Alberta Children's
Hospital, Calgary.

© 2005 by CRC Press LLC