ABSTRACT
Texture is one of the important characteristics of images and texture analysis
is encountered in several areas
We nd around us several examples of texture on wooden furniture
cloth brick walls oors and so on We may group texture into two general
categories quasi periodic and random If there is a repetition of a texture
element at almost regular or quasi periodic intervals we may classify the
texture as being quasi periodic or ordered the elements of such a texture are
called textons or textels Brick walls and oors with tiles are examples
of periodic texture On the other hand if no texton can be identied such
as in clouds and cementwall surfaces we can say that the texture is random
Rao gives a more detailed classication including weakly ordered or
oriented texture that takes into account hair wood grain and brush strokes
in paintings Texture may also be related to visual andor tactile sensations
such as neness coarseness smoothness granularity periodicity patchiness
being mottled or having a preferred orientation
A signicant amount of work has been done in texture characterization
and synthesis see Haralick and Haralick
and Shapiro Chapter for detailed reviews According to Haralick
et al texture relates to information about the spatial distribution of
graylevel variation however this is a general observation It is important
to recognize that due to the existence of a wide variety of texture no single
method of analysis would be applicable to several dierent situations Sta
tistical measures such as graylevel cooccurrence matrices and entropy
characterize texture in a stochastic sense however they do not convey a
physical or perceptual sense of the texture Although periodic texture may
be modeled as repetitions of textons not many methods have been developed
for the structural analysis of texture
In this chapter we shall explore the nature of texture found in biomedical
images study methods to characterize and analyze such texture and investi
gate approaches for the classication of biomedical images based upon texture
We shall concentrate on random texture in this chapter due to the extensive
occurrence of oriented patterns and texture in biomedical images we shall
treat this topic on its own in Chapter
Texture in Biomedical Images
A wide variety of texture is encountered in biomedical images Oriented tex
ture is common in medical images due to the brous nature of muscles and
ligaments as well as the extensive presence of networks of blood vessels veins
ducts and nerves A preferred or dominant orientation is associated with the
functional integrity and strength of such structures Although truly periodic
texture is not commonly encountered in biomedical images ordered texture
is often found in images of the skins of reptiles the retina the cornea the
compound eyes of insects and honeycombs
Organs such as the liver are made up of clusters of parenchyma that are
of the order of mm in size The pixels in CT images have a typical
resolution of mm which is comparable to the size of the parenchymal
units With ultrasonic imaging the wavelength of the probing radiation is of
the order of mm which is also comparable to the size of parenchymal
clusters Under these conditions the liver appears to have a speckled random
texture
Several samples of biomedical images with various types of texture are
shown in Figures and see also Figures and
It is evident from these illustrations that no single approach can succeed in
characterizing all types of texture
Several approaches have been proposed for the analysis of texture in med
ical images for various diagnostic applications For example texture mea
sures have been derived from Xray images for automatic identication of
pulmonary diseases for the analysis of MR images and processing
of mammograms In this chapter we shall investigate the na
ture of texture in a few biomedical images and study some of the commonly
used methods for texture analysis
Models for the Generation of Texture
Martins et al in their work on the auditory display of texture in im
ages see Section outlined the following similarities between speech and
texture generation The sounds produced by the human vocal system may be
grouped as voiced unvoiced and plosive sounds The rst two types
of speech signals may be modeled as the convolution of an input excitation
signal with a lter function The excitation signal is quasiperiodic when we
use the vocal cords to create voiced sounds or random in the case of unvoiced
sounds Figure a illustrates the basic model for speech generation
a b
c d
FIGURE
Examples of texture in CT images a Liver b Kidney c Spine d Lung
The true size of each image is mm The images represent widely dif
fering ranges of tissue density and have been enhanced to display the inherent
texture Image data courtesy of Alberta Childrens Hospital
a b
c d
FIGURE
Examples of texture in mammograms from the MIAS database a
c oriented texture true image size mm d random texture true
image size mm For more examples of oriented texture see Figures
and as well as Chapter
a b
c
FIGURE
Examples of ordered texture a Endothelial cells in the cornea Image cour
tesy of J Jaroszewski b Part of a ys eye Reproduced with permission
from D Suzuki Behavior in drosophila melanogaster A geneticists view
Canadian Journal of Genetics and Cytology XVI
c
Ge
netics Society of Canada c Skin on the belly of a cobra snake Image
courtesy of Implora Colonial Heights VA httpwww implora com See
also Figure
FIGURE
a Model for speech signal generation b Model for texture synthesis Re
produced with permission from A C G Martins R M Rangayyan and R A
Ruschioni Audication and sonication of texture in images Journal of
Electronic Imaging
c
SPIE and IST
Texture may also be modeled as the convolution of an input impulse eld
with a spot or a texton that would act as a lter The spot noise model of
van Wijk for synthesizing random texture uses this model in which the
Fourier spectrum of the spot acts as a lter that modies the spectrum of a
D randomnoise eld Ordered texture may be generated by specifying the
basic pattern or texton to be used and a placement rule The placement rule
may be expressed as a eld of impulses Texture is then given by the convolu
tion of the impulse eld with the texton which could also be represented as a
lter A onetoone correspondence may thus be established between speech
signals and texture in images Figure b illustrates the model for tex
ture synthesis the correspondence between the speech and image generation
models in Figure is straightforward
Random texture
According to the model in Figure random texture may be modeled as a
ltered version of a eld of white noise where the lter is represented by a
spot of a certain shape and size usually of small spatial extent compared to
the size of the image The D spectrum of the noise eld which is essentially
a constant is shaped by the D spectrum of the spot Figure illustrates
a randomnoise eld of size pixels and its Fourier spectrum Parts
a d of Figure show two circular spots of diameter and pixels
and their spectra parts e h of the gure show the random texture
generated by convolving the noise eld in Figure a with the circular
spots and their Fourier spectra It is readily seen that the spots have ltered
the noise and that the spectra of the textured images are essentially those of
the corresponding spots
Figures and illustrate a square spot and a hashshaped spot as well
as the corresponding random texture generated by the spotnoise model and
the corresponding spectra the anisotropic nature of the images is clearly seen
in their spectra
Ordered texture
Ordered texture may be modeled as the placement of a basic pattern or texton
which is of a much smaller size than the total image at positions determined
by a D eld of quasi periodic impulses The separations between the
impulses in the x and y directions determine the periodicity or pitch in the
two directions This process may also be modeled as the convolution of the
impulse eld with the texton in this sense the only dierence between ordered
and random texture lies in the structure of the impulse eld the former uses
a quasi periodic eld of impulses whereas the latter uses a randomnoise
eld Once again the spectral characteristics of the texton could be seen as
a lter that modies the spectrum of the impulse eld which is essentially a
D eld of impulses as well
a b
FIGURE
a Image of a randomnoise eld pixels b Spectrum of the image
in a Reproduced with permission from A C G Martins R M Rangayyan
and R A Ruschioni Audication and sonication of texture in images
Journal of Electronic Imaging
c
SPIE and IST
Figure a illustrates a eld of impulses with horizontal peri
odicity p
x
pixels and vertical periodicity p
y
pixels Figure b
shows the corresponding periodic texture with a circle of diameter pixels
as the spot or texton Figure c shows a periodic texture with the texton
being a square of side pixels p
x
pixels and p
y
pixels Figure
d depicts a periodictextured image with an isosceles triangle of sides
and pixels as the spot and periodicity p
x
pixels and p
y
pixels See Section for illustrations of the Fourier spectra of images with
ordered texture
Oriented texture
Images with oriented texture may be generated using the spotnoise model
by providing line segments or oriented motifs as the spot Figure shows
a spot with a line segment oriented at
o
and the result of convolution of
the spot with a randomnoise eld the logmagnitude Fourier spectra of the
spot and the textured image are also shown The preferred orientation of the
texture and the directional concentration of the energy in the Fourier domain
are clearly seen in the gure See Figure for examples of oriented texture
in mammograms See Chapter for detailed discussions on the analysis of
oriented texture and several illustrations of oriented patterns
a b
c d
e f
Figure g h
FIGURE
a Circle of diameter pixels b Circle of diameter pixels c Fourier
spectrum of the image in a d Fourier spectrum of the image in b
e Random texture with the circle of diameter pixels as the spot f Ran
dom texture with the circle of diameter pixels as the spot g Fourier
spectrum of the image in e h Fourier spectrum of the image in f The
size of each image is pixels Reproduced with permission from
A C G Martins R M Rangayyan and R A Ruschioni Audication and
sonication of texture in images Journal of Electronic Imaging
c
SPIE and IST
a b
c d
FIGURE
a Square of side pixels b Random texture with the square of side
pixels as the spot c Spectrum of the image in a d Spectrum of
the image in b The size of each image is pixels Reproduced
with permission from A C G Martins R M Rangayyan and R A Ruschioni
Audication and sonication of texture in images Journal of Electronic
Imaging
c
SPIE and IST
a b
c d
FIGURE
a Hash of side pixels b Random texture with the hash of side pixels
as the spot c Spectrum of the image in a d Spectrum of the image in
b The size of each image is pixels Reproduced with permission
from A C G Martins R M Rangayyan and R A Ruschioni Audication
and sonication of texture in images Journal of Electronic Imaging
c
SPIE and IST
a b
c d
FIGURE
a Periodic eld of impulses with p
x
pixels and p
y
pixels b Or
dered texture with a circle of diameter pixels p
x
pixels and p
y
pixels as the spot c Ordered texture with a square of side pixels p
x
pixels and p
y
pixels as the spot d Ordered texture with a triangle
of sides and pixels as the spot p
x
pixels and p
y
pix
els The size of each image is pixels Reproduced with permission
from A C G Martins R M Rangayyan and R A Ruschioni Audication
and sonication of texture in images Journal of Electronic Imaging
c
SPIE and IST
a b
c d
FIGURE
Example of oriented texture generated using the spotnoise model in Fig
ure a Spot with a line segment oriented at
o
b Oriented texture
generated by convolving the spot in a with a randomnoise eld c and
d Logmagnitude Fourier spectra of the spot and the textured image re
spectively The size of each image is pixels
Statistical Analysis of Texture
Simple measures of texture may be derived based upon the moments of the
graylevel PDF or normalized histogram of the given image The k
th
central
moment of the PDF pl is dened as
m
k
L
X
l
l
f
k
pl
where l L are the gray levels in the image f and
f
is the
mean gray level of the image given by
f
L
X
l
l pl
The second central moment which is the variance of the gray levels and is
given by
f
m
L
X
l
l
f
pl
can serve as a measure of inhomogeneity The normalized third and fourth
moments known as the skewness and kurtosis respectively and dened as
skewness
m
m
and
kurtosis
m
m
indicate the asymmetry and uniformity or lack thereof of the PDF High
order moments are aected signicantly by noise or error in the PDF and
may not be reliable features The moments of the PDF can only serve as
basic representatives of graylevel variation
Byng et al computed the skewness of the histograms of
mm sections of mammograms An average skewness measure
was computed for each image by averaging over all the sectionbased skewness
measures of the image Mammograms of breasts with increased broglandu
lar density were observed to have histograms skewed toward higher density
resulting in negative skewness On the other hand mammograms of fatty
breasts tended to have positive skewness The skewness measure was found
to be useful in predicting the risk of development of breast cancer
The graylevel cooccurrence matrix
Given the general description of texture as a pattern of the occurrence of gray
levels in space the most commonly used measures of texture in particular of
random texture are the statistical measures proposed by Haralick et al
Haralicks measures are based upon the moments of a joint PDF that
is estimated as the joint occurrence or cooccurrence of gray levels known as
the graylevel cooccurrence matrix GCM GCMs are also known as spatial
graylevel dependence SGLD matrices and may be computed for various
orientations and distances
The GCM P
d
l
l
represents the probability of occurrence of the pair
of gray levels l
l
separated by a given distance d at angle GCMs are
constructed by mapping the graylevel cooccurrence counts or probabilities
based on the spatial relations of pixels at dierent angular directions specied
by while scanning the image from lefttoright and toptobottom
Table shows the GCM for the image in Figure with eight gray levels
bpixel by considering pairs of pixels with the second pixel immediately
below the rst For example the pair of gray levels
occurs times in the
image Observe that the table of counts of occurrence of pairs of pixels shown
in Table and used to compute the rstorder entropy also represents a
GCM with the second pixel appearing immediately after the rst in the same
row Due to the fact that neighboring pixels in natural images tend to have
nearly the same values GCMs tend to have large values along and around the
main diagonal and low values away from the diagonal
Observe that for an image with B bpixel there will be L
B
gray
levels the GCM is then of size L L Thus for an image quantized to
bpixel there will be gray levels and the GCM will be of size
Fine quantization to large numbers of gray levels such as
levels in highresolution mammograms will increase the size of the GCM to
unmanageable levels and also reduce the values of the entries in the GCM It
may be advantageous to reduce the number of gray levels to a relatively small
number before computing GCMs A reduction in the number of gray levels
with smoothing can also reduce the eect of noise on the statistics computed
from GCMs
GCMs are commonly formed for unit pixel distances and the four angles
of
o
o
o
and
o
Strictly speaking the distances to the diagonally
connected neighboring pixels at
o
and
o
would be
p
times the pixel
size For an M N image the number of pairs of pixels that can be formed
will be less than MN due to the fact that it may not be possible to pair the
pixels in a few rows or columns at the borders of the image with another pixel
according to the chosen parameters d
FIGURE
A part of the image in Figure a quantized to bpixel shown
as an image and as a D array of pixel values
TABLE
Graylevel Cooccurrence Matrix for the Image in Figure with
the Second Pixel Immediately Below the First
Current Pixel Next Pixel Below
Pixels in the last row were not processed The GCM has not been normalized
See also Table
Haralicks measures of texture
Based upon normalized GCMs Haralick et al proposed several
quantities as measures of texture In order to dene these measures let us
normalize the GCM as
pl
l
P l
l
P
L
l
P
L
l
P l
l
A few other entities used in the derivation of Haralicks texture measures are
as follows
p
x
l
L
X
l
pl
l
p
y
l
L
X
l
pl
l
p
xy
k
L
X
l
L
X
l
z
l
l
k
pl
l
where k L and
p
xy
k
L
X
l
L
X
l
z
jl
l
jk
pl
l
where k L
The texture measures are then dened as follows
The energy feature F
which is a measure of homogeneity is dened as
F
L
X
l
L
X
l
p
l
l
A homogeneous image has a small number of entries along the diagonal of the
GCM with large values which will lead to a large value of F
On the other
hand an inhomogeneous image will have small values spread over a larger
number of GCM entries which will result in a low value for F
The contrast feature F
is dened as
F
L
X
k
k
L
X
l
L
X
l
z
jl
l
jk
pl
l
The correlation measure F
which represents linear dependencies of gray
levels is dened as
F
x
y
L
X
l
L
X
l
l
l
pl
l
x
y
where
x
and
y
are the means and
x
and
y
are the standard deviations
of p
x
and p
y
respectively
The sum of squares feature is given by
F
L
X
l
L
X
l
l
f
pl
l
where
f
is the mean gray level of the image
The inverse dierence moment a measure of local homogeneity is dened
as
F
L
X
l
L
X
l
l
l
pl
l
The sum average feature F
is given by
F
L
X
k
k p
xy
k
and the sum variance feature F
is dened as
F
L
X
k
k F
p
xy
k
The sum entropy feature F
is given by
F
L
X
k
p
xy
k log
p
xy
k
Entropy a measure of nonuniformity in the image or the complexity of the
texture is dened as
F
L
X
l
L
X
l
pl
l
log
pl
l
The dierence variance measure F
is dened as the variance of p
xy
in a manner similar to that given by Equations and for its sum
counterpart
The dierence entropy measure is dened as
F
L
X
k
p
xy
k log
p
xy
k
Two informationtheoretic measures of correlation are dened as
F
H
xy
H
xy
maxfH
x
H
y
g
and
F
f exp H
xy
H
xy
g
where H
xy
F
H
x
and H
y
are the entropies of p
x
and p
y
respectively
H
xy
L
X
l
L
X
l
pl
l
log
p
x
l
p
y
l
and
H
xy
L
X
l
L
X
l
p
x
l
p
y
l
log
p
x
l
p
y
l
The maximal correlation coecient feature F
is dened as the square root
of the second largest eigenvalue of Q where
Ql
l
L
X
k
pl
k pl
k
p
x
k p
y
k
The subscripts d and in the representation of the GCM P
d
l
l
have
been removed in the denitions above for the sake of notational simplicity
However it should be noted that each of the measures dened above may be
derived for each value of d and of interest If the dependence of texture
upon angle is not of interest GCMs over all angles may be averaged into a
single GCM The distance d should be chosen taking into account the sampling
interval pixel size and the size of the texture units of interest More details
on the derivation and signicance of the features dened above are provided
by Haralick et al
Some of the features dened above have values much greater than unity
whereas some of the features have values far less than unity Normalization
to a predened range such as over the dataset to be analyzed may be
benecial
Parkkinen et al studied the problem of detecting periodicity in tex
ture using statistical measures of association and agreement computed from
GCMs If the displacement and orientation d of a GCM match the same
parameters of the texture the GCM will have large values for the elements
along the diagonal corresponding to the gray levels present in the texture el
ements A measure of association is the
statistic which may be expressed
using the notation above as
L
X
l
L
X
l
pl
l
p
x
l
p
y
l
p
x
l
p
y
l
The measure may be normalized by dividing by L and expected to possess a
high value for an image with periodic texture under the condition described
above
Parkkinen et al discussed some limitations of the
statistic in the
analysis of periodic texture and proposed a measure of agreement given by
P
o
P
c
P
c
where
P
o
L
X
l
pl l
and
P
c
L
X
l
p
x
l p
y
l
The measure has its maximal value of unity when the GCM is a diagonal
matrix which indicates perfect agreement or periodic texture
Haralicks measures have been applied for the analysis of texture in several
types of images including medical images Chan et al found the three
features of correlation dierence entropy and entropy to perform better than
other combinations of one to eight features selected in a specic sequence
Sahiner et al dened a rubberband straightening transform
RBST to map ribbons around breast masses in mammograms into rect
angular arrays see Figure and then computed Haralicks measures of
texture Mudigonda et al computed Haralicks measures using
adaptive ribbons of pixels extracted around mammographic masses and used
the features to distinguish malignant tumors from benign masses details of
this work are provided in Sections and See Section for a dis
cussion on the application of texture measures for contentbased retrieval and
classication of mammographic masses
Laws Measures of Texture Energy
Laws proposed a method for classifying each pixel in an image based
upon measures of local texture energy The texture energy features rep
resent the amounts of variation within a sliding window applied to several
ltered versions of the given image The lters are specied as separable D
arrays for convolution with the image being processed
The basic operators in Laws method are the following
L
E
S
The operators L E and S perform centerweighted averaging symmetric
rst dierencing edge detection and second dierencing spot detection
respectively Nine masks may be generated by multiplying the
transposes of the three operators represented as vectors with their direct
versions The result of L
T
E gives one of the Sobel masks
Operators of length ve pixels may be generated by convolving the L E
and S operators in various combinations Of the several lters designed by
Laws the following ve were said to provide good performance
L L L
E L E
S E E
R S S
W E S
where represents D convolution
The operators listed above perform the detection of the following types of
features L local average E edges S spots R ripples and W
waves In the analysis of texture in D images the D convolution
operators given above are used in pairs to achieve various D convolution
operators for example LL L
T
L and LE L
T
E each of which
may be represented as a array or matrix Following the application of the
selected lters texture energy measures are derived from each ltered image
by computing the sum of the absolute values in a sliding window
All of the lters listed above except L have zero mean and hence the
texture energy measures derived from the ltered images represent measures
of local deviation or variation The result of the L lter may be used for
normalization with respect to luminance and contrast
The use of a large sliding window to smooth the ltered images could lead
to the loss of boundaries across regions with dierent texture Hsiao and
Sawchuk applied a modied LLMMSE lter so as to derive Laws texture
energy measures while preserving the edges of regions and applied the results
for pattern classication
Example The results of the application of the operators LL EE
and WW to the Lenna image in Figure a are shown in
Figure a c Also shown in parts e f of the gure are the sums
of the absolute values of the ltered images using a moving window It is
evident that the LL lter results in a measure of local brightness Careful
inspection of the results of the EE and WW lters shows that they have
high values for dierent regions of the original image possessing dierent types
of texture edges and waves respectively Feature vectors composed of the
values of various Laws operators for each pixel may be used for classifying
the image into texture categories on a pixelbypixel basis The results may
be used for texture segmentation and recognition
In an example provided by Laws see also Pietkainen et al
the texture energy measures have been shown to be useful in the segmen
tation of an image composed of patches with dierent texture Miller and
Astley used features of mammograms based upon the RR oper
ator and obtained an accuracy of in the segmentation of the nonfat
glandular regions in mammograms See Section for a discussion on the
application of Laws and other methods of texture analysis for the detection
of breast masses in mammograms
Fractal Analysis
Fractals are dened in several dierent ways the most common of which is
that of a pattern composed of repeated occurrences of a basic unit at multiple
scales of detail in a certain order of generation this denition includes the no
tion of selfsimilarity or nested recurrence of the same motif at smaller and
smaller scales see Section for a discussion on selfsimilar spacelling
curves The relationship to texture is evident in the property of repeated
occurrence of a motif Fractal patterns occur abundantly in nature as well
as in biological and physiological systems
the selfreplicating patterns of the complex leaf structures of ferns see Fig
ure the ramications of the bronchial tree in the lung see Figure
and the branching and spreading anastomotic patterns of the arteries in the
heart see Figure to name a few Fractals and the notion of chaos are
related to the area of nonlinear dynamic systems and have found
several applications in biomedical signal and image analysis
a d
b e
c f
FIGURE
Results of convolution of the Lenna test image of size pixels see
Figure a using the following Laws operators a LL b EE
and c WW d f were obtained by summing the absolute values of
the results in a c respectively in a moving window and represent
three measures of texture energy The image in c was obtained by mapping
the range out of the full range of to
FIGURE
The leaf of a fern with a fractal pattern
Fractal dimension
Whereas the selfsimilar aspect of fractals is apparent in the examples men
tioned above it is not so obvious in other patterns such as clouds coastlines
and mammograms which are also said to have fractallike characteristics In
such cases the fractal nature perceived is more easily related to the notion
of complexity in the dimensionality of the object leading to the concept of the
fractal dimension If one were to use a large ruler to measure the length of a
coastline the minor details present in the border having smallscale variations
would be skipped and a certain length would be derived If a smaller ruler
were to be used smaller details would get measured and the total length
that is measured would increase between the same end points as before
This relationship may be expressed as
l l
d
f
where l is the length measured with as the measuring unit the size of
the ruler d
f
is the fractal dimension and l
is a constant Fractal patterns
exhibit a linear relationship between the log of the measured length and the
log of the measuring unit
logl logl
d
f
log
the slope of this relationship is related to the fractal dimension d
f
of the
pattern This method is known as the caliper method to estimate the fractal
dimension of a curve It is obvious that d
f
for a straight line
Fractal dimension is a measure that quanties how the given pattern lls
space The fractal dimension of a straight line is unity that of a circle or a D
perfectly planar sheetlike object is two and that of a sphere is three As the
irregularity or complexity of a pattern increases its fractal dimension increases
up to its own Euclidean dimension d
E
plus one The fractal dimension of a
jagged rugged convoluted kinky or crinkly curve will be greater than unity
and reaches the value of two as its complexity increases The fractal dimension
of a rough D surface will be greater than two and approaches three as the
surface roughness increases In this sense fractal dimension may be used as
a measure of the roughness of texture in images
Several methods have been proposed to estimate the fractal dimension of
patterns Among the methods described by
Schepers et al for the estimation of the fractal dimension of D signals
is that of computing the relative dispersion RD dened as the ratio of the
standard deviation to the mean using varying bin size or number of samples
of the signal For a fractal signal the expected variation of RD is
RD RD
H
where
is a reference value for the bin size and H is the Hurst coecient
that is related to the fractal dimension as
d
f
d
E
H
Note d
E
for D signals for D images etc The value of H and
hence d
f
may be estimated by measuring the slope of the straightline ap
proximation to the relationship between logRD and log
Fractional Brownian motion model
Fractal signals may be modeled in terms of fractional Brownian motion
The expectation of the dierences between the values of such a
signal at a position and another at ! follow the relationship
Ejf ! fj j!j
H
The slope of a plot of the averaged dierence as above versus ! on a log
log scale may be used to estimate H and the fractal dimension
Chen et al applied fractal analysis for the enhancement and classi
cation of ultrasonographic images of the liver Burdett et al derived
the fractal dimension of D ROIs of mammograms with masses by using the
expression in Equation Benign masses due to their smooth and ho
mogeneous texture were found to have low fractal dimensions of about
whereas malignant tumors due to their rough and heterogeneous texture had
higher fractal dimensions of about
The PSD of a fractional Brownian motion signal " is expected to follow
the socalled power law as
"
j j
H
The derivative of a signal generated by a fractional Brownian motion model
is known as a fractional Gaussian noise signal the exponent in the powerlaw
relationship for such a signal is changed to H
Fractal analysis of texture
Based upon a fractional Brownian motion model Wu et al dened an
averaged intensitydierence measure idk for various values of the displace
ment or distance parameter k as
idk
NN k
N
X
m
Nk
X
n
jfmn fmn kj
Nk
X
m
N
X
n
jfmn fm k nj
The slope of a plot of logidk versus logk was used to estimate H and
the fractal dimension Wu et al applied multiresolution fractal analysis as
well GCM features Fourier spectral features graylevel dierence statistics
and Laws texture energy measures for the classication of ultrasonographic
images of the liver as normal hepatoma or cirrhosis classication accuracies
of and respectively were obtained In
a related study Lee et al derived features based upon fractal analysis
including the application of multiresolution wavelet transforms Classica
tion accuracies of in distinguishing between normal and abnormal liver
images and in discriminating between cirrhosis and hepatoma were
obtained
Byng et al see also Peleg et al Yae et al and Caldwell
et al describe a surfacearea measure to represent the complexity of
texture in an image by interpreting the gray level as the height of a function
of space see Figure In a perfectly uniform image of size N N pixels
with each pixel being of size units of area the surface area would be
equal to N
When adjacent pixels are of unequal value more surface area
of the blocks representing the pixels will be exposed as shown in Figure
The total surface area for the image may be calculated as
A
N
X
m
N
X
n
f
jf
mn f
mn j jf
mn f
m nj g
where f
mn is the D image expressed as a function of the pixel size The
method is analogous to the popular boxcounting method In
order to estimate the fractal dimension of the image we could derive several
smoothed and downsampled versions of the given image representing various
scales and estimate the slope of the plot of logA versus log the
fractal dimension is given as two minus the slope Smoothing and downsam
pling may be achieved simply by averaging pixels in blocks of
etc and replacing the blocks by a single pixel with the correspond
ing average A perfectly uniform image would demonstrate no change in its
area and have a fractal dimension of two images with rough texture would
have increasing values of the fractal dimension approaching three Yae et
al obtained fractal dimension values in the range of with
mammograms Byng et al demonstrated the usefulness of the fractal
dimension as a measure of increased broglandular density in the breast and
related it to the risk of development of breast cancer Fractal dimension was
found to complement histogram skewness see Section as an indicator of
breast cancer risk
FIGURE
Computation of the exposed surface area for a pixel fmn with respect to
its neighboring pixels fmn and fm n A pixel at mn is viewed
as a box or building with base area and height equal to the gray level
fmn The total exposed surface area for the pixel fmn with respect to
its neighboring pixels at mn and m n is the sum of the areas of
the rectangles ABCD CBEF and DCGH
Applications of fractal analysis
Chaudhuri and Sarkar proposed a modied boxcounting method to es
timate fractal measures of texture from a given image its horizontally and
vertically smoothed versions as well as high and lowgrayvalued versions
derived by thresholding operations The features were applied for the seg
mentation of multitextured images
Zheng and Chan used the fractal dimension of sections of mammo
grams to select areas with rough texture for further processing toward the
detection of tumors Pohlman et al derived the fractal dimension of
D signatures of radial distance versus angle of boundaries of mammographic
masses the measure provided an average accuracy of in discriminating
between benign masses and malignant tumors It should be noted that the
function of radial distance versus angle could be multivalued for spiculated
and irregular contours due to the fact that a radial line may cross the tu
mor contour more than once see Section A measure related to this
characteristic was found to give an accuracy of in discriminating between
benign masses and malignant tumors
Iftekharuddin et al proposed a modied boxcounting method to es
timate the fractal dimension of images and applied the method to brain MR
images Their results indicated the potential of the methods in the detection
of brain tumors
Lundahl et al estimated the values of H from scan lines of Xray im
ages of the calcaneus heel bone and showed that the value was decreased by
injury and osteoporosis indicating reduced complexity of structure increased
gaps as compared to normal bone Saparin et al using symbol dynam
ics and measures of complexity found that the complexity of the trabecular
structure in bone declines more rapidly than bone density during the loss of
bone in osteoporosis Jennane et al applied fractal analysis to Xray
CT images of trabecular bone specimens extracted from the radius It was
found that the H value decreased with trabecular bone loss and osteoporosis
Samarabandhu et al proposed a morphological ltering approach to de
rive the fractal dimension and indicated that the features they derived could
serve as robust measures of the trabecular texture in bone For a discussion
on the application of fractal analysis to bone images see Geraets and van der
Stelt
Sedivy et al showed that the fractal dimension of atypical nuclei in
dysplastic lesions of the cervix uteri increased as the degree of dysplasia in
creased They indicated that fractal dimension could quantify the irregularity
and complexity of the outlines of nuclei and facilitate objective nuclear grad
ing Esgiar et al found the fractal dimension to complement the GCM
texture features of entropy and correlation in the classication of tissue sam
ples from the colon the inclusion of fractal dimension increased the sensitivity
from to and the specicity from to Penn and Loew
discussed the limitations of the boxcounting and PSDbased methods in frac
tal analysis and proposed fractal interpolation function models to estimate
the fractal dimension The method was shown to provide improved results in
the separation of normal and sicklecell red blood cells
Lee et al applied shape analysis for the classication of cutaneous
melanocytic lesions based upon their contours An irregularity index related
to local protrusions and indentations was observed to have a higher correlation
with clinical assessment of the lesions than compactness see Section
and fractal dimension
Fourier domain Analysis of Texture
As is evident from the illustrations in Figure the Fourier spectrum of an
image with random texture contains the spectral characteristics of the spot
involved in its generation according to the spotnoise model shown in Fig
ure The eects of multiplication with the spectrum of the randomnoise
eld which is essentially and on the average a constant may be removed by
smoothing operations Thus the important characteristics of the texture are
readily available in the Fourier spectrum
On the other hand the Fourier spectrum of an image with periodic texture
includes not only the spectral characteristics of the spot but also the eects of
multiplication with the spectrum of the impulse eld involved in its generation
The Fourier spectrum of a train of impulses in D is a discrete spectrum with
a constant value at the fundamental frequency the inverse of the period
and its harmonics Correspondingly in D the Fourier spectrum of a
periodic eld of impulses is a eld of impulses with high values only at the
fundamental frequency of repetition of the impulses in the image domain and
integral multiples thereof Multiplication of the Fourier spectrum of the spot
with the spectrum of the impulse eld will cause signicant modulation of
the intensities in the former leading to bright regions at regular intervals
With reallife images the eects of windowing or nite data as well as of
quasiperiodicity will lead to smearing of the impulses in the spectrum of the
impulse eld involved regardless the spectrum of the image may be expected
to demonstrate a eld of bright regions at regular intervals The information
related to the spectra of the spot and the impulse eld components may be
derived from the spectrum of the textured image by averaging in the polar
coordinate axes as follows
Let F r t be the polarcoordinate representation of the Fourier spectrum
of the given image in terms of the Cartesian frequency coordinates u v we
have r
p
u
v
and t atanvu Derive the projection functions in r
and t by integrating F r t in the other coordinate as
F r
Z
t
F r t dt
and
F t
Z
r
max
r
F r t dr
The averaging eect of the integration above or summation in the discrete
case leads to improved visualization of the spectral characteristics of periodic
texture Quantitative features may be derived from F r and F t or directly
from F r t for pattern classication purposes
Example Figure shows the components involved in the generation of
an image with periodic placement of a circular spot and the related Fourier
spectra The spectra clearly demonstrate the eects of periodicity in the
impulse eld and the texture It is evident that the spectrum of the texture
also includes information related to the spectrum of the spot
The spectrum of the texture in Figure f is shown in polar coordinates
in Figure The projection functions derived by summing the spectrum in
the radial and angular dimensions are shown in Figure The projection
functions demonstrate the eect of periodicity in the texture
The spectrum of the image of a ys eye in Figure b is shown in
Figure in Cartesian and polar coordinates the corresponding projection
functions are shown in Figure A similar set of results is shown in Figures
and for the snakeskin image in Figure c The spectra show
regions of high intensity at quasiperiodic intervals in spite of the fact that
the original images are only approximately periodic and the texture elements
vary considerably in size and orientation over the scope of the images
a d
b e
c f
FIGURE
Fourier spectral characteristics of periodic texture generated using the spot
noise model in Figure a Periodic impulse eld b Circular spot
c Periodic texture generated by convolving the spot in b with the impulse
eld in a d f Logmagnitude Fourier spectra of the images in a
c respectively
FIGURE
The spectrum in Figure f converted to polar coordinates only the upper
half of the spectrum was mapped to polar coordinates
Jernigan and DAstous used the normalized PSD values within selected
frequency bands as PDFs and computed entropy values It was expected that
structured texture would lead to low entropy due to spectral bands with
concentrated energy and random texture would lead to high entropy values
due to a uniform distribution of spectral energy Their results indicated that
the entropy values could provide discrimination between texture categories
that was comparable to that provided by spectral energy and GCMbased
measures In addition to entropy the locations and values of the spectral
peaks may also be used as features Liu and Jernigan dened measures
in the Fourier spectral domain including measures related to the frequency
coordinates and relative orientation of the rst and second spectral peaks the
percentages of energy and the moments of inertia of the normalized spectrum
in the rst and second quadrants the Laplacian of the magnitude and phase at
the rst and second spectral peaks and measures of isotropy and circularity of
the spectrum Their results indicated that the spectral measures were eective
in discriminating between various types of texture and also insensitive to
additive noise
Laine and Fan proposed the use of wavelet packet frames or tree
structured lter banks for the extraction of features from textured images
in the frequency domain The features were used for the segmentation of
multitextured images
a
b
FIGURE
Projection functions in a the radial coordinate r and b the angle coordi
nate t obtained by integrating summing the spectrum in Figure in the
other coordinate
a
b
FIGURE
Fourier spectral characteristics of the quasiperiodic texture of the ys eye
image in Figure b a The Fourier spectrum in Cartesian coordinates
u v b The upper half of the spectrum in a mapped to polar coordinates
a
b
FIGURE
Projection functions in a the radial coordinate r and b the angle coordi
nate t obtained by integrating summing the spectrum in Figure b in
the other coordinate
a
b
FIGURE
Fourier spectral characteristics of the ordered texture of the snakeskin image
in Figure c a The Fourier spectrum in Cartesian coordinates u v
b The upper half of the spectrum in a mapped to polar coordinates
a
b
FIGURE
Projection functions in a the radial coordinate r and b the angle coordi
nate t obtained by integrating summing the spectrum in Figure b in
the other coordinate
McLean applied vector quantization in the transform domain and
treated the method as a generalized template matching scheme for the coding
and classication of texture The method yielded better texture classication
accuracy than GCMbased features
Bovik discussed multichannel narrowband ltering and modeling of
texture Highly granular and oriented texture may be expected to present
spatiospectral regions of concentrated energy Gabor lters may then be
used to lter segment and analyze such patterns See Sections
and for further discussion on related topics
Segmentation and Structural Analysis of Texture
Many methods have been reported in the literature for the analysis of texture
which may be broadly classied as statistical or structural methods
Most of the commonly used methods for texture analysis are based
upon statistical characterization such as GCMs see Section and ACFs
Fourier spectrum analysis described in Section may be considered to be
equivalent to analysis based upon the ACF Statistical methods are suitable
for the analysis of random or ne texture with no largescale motifs for other
types of texture and for multitextured images structural methods could be
more appropriate Structural analysis of textured images requires some type
of segmentation of the given image into its distinct or basic components
Texture elements or textons as called by Julesz and Bergen play
an important role in preattentive vision and texture perception Ordered
texture may be modeled as being composed of repeated placement of a basic
motif or texton over the image eld in accordance with a placement rule
see Section The placement rule may be expressed as a eld of impulses
indicating the locations of the repeated textons consequently the textured
image is given by the convolution of the ordered or quasi periodic impulse
eld with the texton Although this model does not directly permit scale
and orientation dierences between the various occurrences of the texton
such jitter could be introduced separately to synthesize realistic textured
images
Vilnrotter et al proposed a system to describe natural textures in
terms of individual texture elements or primitives and their spatial relation
ships or arrangement The main steps of the system include the generation of
D descriptors of texture elements from edge repetition data the extraction
of elements that correspond to the preceding description the generation of
D descriptors of each texture primitive type and the computation of spatial
arrangements or placement rules when the texture is homogeneous and regu
lar The method was used to classify several types of texture including oor
grating raa brick straw and wool The method does not extract a single
version of a texture element or primitive instead all possible repeated struc
tures are extracted The analysis of a raa pattern for example resulted in
the extraction of three primitives
He and Wang dened texture units in terms of the connected
neighbors of each pixel The values in each unit were reduced to the range
f g with the value of unity indicating that the value of the neighboring
pixel was within a predened range about the central pixel value the values
of and were used to indicate that the pixel value was lower or higher than
the specied range respectively The texture spectrum was dened as the
histogram or spectrum of the frequency of occurrence of all possible texture
units in the image it should be noted that the texture spectrum as above is
not based upon a linear orthogonal transform such as the Fourier transform
Methods were proposed to characterize as well as lter images based upon the
texture spectrum
Wang et al proposed a thresholding scheme for the extraction of
texture primitives It was assumed that the primitives would appear as regions
of connected pixels demonstrating good contrast with their background The
primitives were characterized in terms of the statistics of their GCMs and
shape attributes the textured image could then be described in terms of
its primitives and placement rules Tomita et al proposed a similar
approach based upon the extraction of texture elements assumed to be regions
of homogeneous gray levels via segmentation The centroids of the texture
elements were used to dene detailed placement rules
The problem of segmentation of complex images containing regions of dif
ferent types of texture has been addressed by several researchers Gabor
functions have been used by Turner and Bovik et al for texture
analysis and segmentation Gabor functions may be used to design lters with
tunable orientation radial frequency bandwidth and center frequencies that
can achieve jointly optimal resolution in the space and frequency domains
Gabor lters are ecient in detecting discontinuities in texture phase and
are useful in texture segmentation Porat and Zeevi developed a method
to describe texture primitives in terms of Gabor elementary functions See
Sections and for further discussion on Gabor lters
Reed and Wechsler described approaches to texture analysis and seg
mentation via the use of joint spatial and frequencydomain representations in
the form of spectrograms that is functions of x y u v obtained by the ap
plication of the Fourier transform in a moving window of the image a bank of
Gabor lters DoG functions and Wigner distributions Reed et al de
scribed a texture segmentation method using the pseudoWigner distribution
and a diusion regiongrowing method Jain and Farrokhnia presented
a method for texture analysis and segmentation based upon the application
of multichannel Gabor lters The results of the lter bank were processed
in such a manner as to detect blobs in the given image texture discrim
ination was performed by analyzing the attributes of the blobs detected in
dierent regions of the image Other related methods for texture segmenta
tion include wavelet frames for the characterization of texture proprieties at
multiple scales and circular Mellin features for rotationinvariant and
scaleinvariant texture analysis
Tardif and Zaccarin proposed a multiscale autoregressive AR model
to analyze multifeatured images The prediction error was used to segment
a given image into dierent textured parts Unser and Eden proposed
a multiresolution feature extraction method for texture segmentation The
method includes the use of a local linear transformation that is equivalent to
processing the given image with a bank of FIR lters See Section for
further discussion on related topics
If the texton and the placement rule impulse eld can be obtained from
a given image with ordered texture the most important characteristics of
the image will have been determined In particular if a single texton or
motif is extracted from the image further analysis of its shape morphol
ogy spectral content and internal details becomes possible Martins and
Rangayyan proposed cepstral ltering in the Radon domain of the
image see Section to obtain the texton their methods and results are
described in Section
Homomorphic deconvolution of periodic patterns
We have seen in Section that an image with periodic texture may be mod
eled as the convolution of a texton or motif with an impulse eld Linear
lters may be applied to the complex cepstrum for homomorphic deconvolu
tion of signals that contain convolved components see Section A basic
assumption in homomorphic deconvolution is that the complex cepstra of the
components do not overlap This assumption is usually met in D signal
processing applications such as in the case of voiced speech signals where
the basic wavelet is a relatively smooth signal Whereas it would
be questionable to make the assumption that the D cepstra of an arbitrary
texton and an impulse eld do not overlap it would be acceptable to make
the same assumption in the case of D projections Radon transforms of the
same images Then the homomorphic deconvolution procedures described
in Section may be applied to recover the projections of a single tex
ton The texton may then be obtained via a procedure for image
reconstruction from projections see Chapter
The distinction between the application of homomorphic deconvolution for
the removal of visual echoes as described in Section and for the extraction
of a texton is minor An image with visual echoes may contain only one
copy or a few repetitions of a basic image with possible overlap and with
possibly unequal spacing of the echoes the basic image may be large in spatial
extent On the other hand an image with ordered texture typically
contains several nonoverlapping repetitions of a relatively small texton or
motif at regular or quasiperiodic spacing
Although homomorphic deconvolution has been shown to successfully ex
tract the basic wavelets or motifs in periodic signals the extraction of the
impulse train or eld is made dicult by the presence of noise and artifacts
related to the deconvolution procedure
Example An image of a part of a building with ordered arrangement of
windows is shown in Figure a A single window section of the image
extracted by homomorphic deconvolution is shown in part b of the gure
a b
FIGURE
a An image of a part of a building with a periodic arrangement of windows
b A single window structure extracted by homomorphic deconvolution Re
produced with permission from A C G Martins and R M Rangayyan Tex
ture element extraction via cepstral ltering in the Radon domain IETE
Journal of Research India
c
IETE
An image with a periodic arrangement of a textile motif is shown in Figure
a The result of the homomorphic deconvolution procedure of Martins
and Rangayyan to extract the texton is shown in part b of the same
gure It is evident that a single motif has been extracted albeit with some
blurring and loss of detail The procedure however was not successful with
biomedical images due to the eects of quasiperiodicity as well as signicant
size and scale variations among the repeated versions of the basic pattern
More research is desirable in this area
a b
FIGURE
a An image with a periodic arrangement of a textile motif b A single
motif or texton extracted by homomorphic deconvolution Reproduced with
permission from A C G Martins and R M Rangayyan Texture element ex
traction via cepstral ltering in the Radon domain IETE Journal of Research
India
c
IETE
Audication and Sonication of Texture in Images
The use of sound in scientic data analysis is rather rare and analysis and
presentation of data are done almost exclusively by visual means Even when
the data are the result of vibrations or sounds such as the heart sound signals
or phonocardiograms a Doppler ultrasound exam or sonar they are often
mapped to a graphical display or an image and visual analysis is performed
The auditory system has not been used much for image analysis in spite of
the fact that it has several advantages over the visual system Whereas many
interesting methods have been proposed for the auditory display of scientic
laboratory data and computer graphics representations of multidimensional
data not much work has been reported for deriving sounds from visual images
Chambers et al published a report on auditory data presentation in the
early s The rst international conference on auditory display of scientic
data was held in with specic interest in the use of sound for the
presentation and analysis of information
Meijer proposed a sonication procedure to present image data to
the blind In this method the frequency of an oscillator is associated with the
position of each pixel in the image and the amplitude is made proportional
to the pixel intensity The image is scanned one column at a time and the
outputs of the associated oscillators are all presented as a sum followed by
a click before the presentation of the next column In essence the image
is treated as a spectrogram or a timefrequency distribution The
sound produced by this method with simple images such as a line crossing the
plane of an image can be easily analyzed however the sound patterns related
to complex images could be complicated and confusing
Texture analysis is often confounded by other neighboring or surrounding
features Martins et al explored the potential of auditory display pro
cedures including audication and sonication for aural presentation and
analysis of texture in images An analogy was drawn between random tex
ture and unvoiced speech and between periodic texture and voiced speech in
terms of generation based on the ltering of an excitation function as shown in
Figure An audication procedure that played in sequence the projections
Radon transforms of the given image at several angles was proposed for the
auditory analysis of random texture A linearprediction model
was used to generate the sound signal from the projection data Martins
et al also proposed a sonication procedure to convert periodic texture to
sound with the emphasis on displaying the essential features of the texture
element and periodicity in the horizontal and vertical directions Projec
tions of the texton were used to compose sound signals including pitch like
voiced speech as well as a rhythmic aspect with the pitch period and rhythm
related to the periodicities in the horizontal and vertical directions in the im
age Datamapping functions were designed to relate image characteristics to
sound parameters in such a way that the sounds provided information in mi
crostructure timbre individual pitch and macrostructure rhythm melody
pitch organization that were related to the objective or quantitative measures
of texture
In order to verify the potential of the proposed methods for aural analy
sis of texture a set of pilot experiments was designed and presented to
subjects The results indicated that the methods could facilitate quali
tative and comparative analysis of texture In particular it was observed that
the methods could lead to the possibility of dening a sequence or order in
the case of images with random texture and that soundtoimage association
could be achieved in terms of the size and shape of the spot used to syn
thesize the texture Furthermore the proposed mapping of the attributes of
periodic texture to sound attributes could permit the analysis of features such
as texton size and shape as well as periodicity in qualitative and comparative
manners The methods could lead to the use of auditory display of images as
an adjunctive procedure to visualization
Martins et al conducted preliminary tests on the audication of MR
images using selected areas corresponding to the gray and white matter of
the brain and to normal and infarcted tissues By using the audication
method dierences between the various tissue types were easily perceived
by two radiologists visual discrimination of the same areas while remaining
within their corresponding MRimage contexts was said to be dicult by the
same radiologists The results need to be conrmed with a larger study
Application Analysis of Breast Masses Using Tex
ture and Gradient Measures
In addition to the textural changes caused by microcalcications the presence
of spicules arising from malignant tumors causes disturbances in the homo
geneity of tissues in the surrounding breast parenchyma Based upon this
observation several studies have focused on quantifying the textural content
in the mass ROI and mass margins to achieve the classication of masses
versus normal tissue as well as benign masses versus malignant tumors
Petrosian et al investigated the usefulness of texture features based
upon GCMs for the classication of masses and normal tissue With a dataset
of manually segmented ROIs the methods indicated sensitivity and
specicity in the training step and sensitivity and specicity
in the test step using the leaveoneout method Kinoshita et al used
a combination of shape factors and texture features based on GCMs Using
a threelayer feedforward neural network they reported accuracy in
the classication of benign and malignant breast lesions with a dataset of
malignant and benign lesions
Chan et al Sahiner et al and Wei et al in
vestigated the eectiveness of texture features derived from GCMs for dier
entiating masses from normal breast tissue in digitized mammograms One
hundred and sixtyeight ROIs with masses and normal ROIs were exam
ined and eight features including correlation entropy energy inertia inverse
dierence moment sum average moment sum entropy and dierence entropy
were calculated for each region All the ROIs were manually segmented by a
radiologist Using linear discriminant analysis Chan et al reported an
accuracy of for the training set and for a test set Wei et al
reported improved classication results with the same dataset by applying
multiresolution texture analysis Sahiner et al applied a convolutional neu
ral network and later used a genetic algorithm to classify the
masses and normal tissue in the same dataset
Analysis of the gradient or transition information present in the bound
aries of masses has been attempted by a few researchers in order to arrive
at benignversusmalignant decisions Kok et al used texture features
fractal measures and edgestrength measures computed from suspicious re
gions for lesion detection Huo et al and Giger et al extracted mass
regions using regiongrowing methods and proposed two spiculation measures
obtained from an analysis of radial edgegradient information surrounding
the periphery of the extracted regions Benignversusmalignant classica
tion studies performed using the features yielded an average eciency of
Later on the group reported to have achieved superior results with their
computeraided classication scheme as compared to an expert radiologist by
employing a hybrid classier on a test set of images
Highnam et al investigated the presence of a halo # an area around
a mass region with a positive Laplacian # to indicate whether a circumscribed
mass is benign or malignant They found that the extent of the halo varies be
tween the CC and MLO views for benign masses but is similar for malignant
tumors
Guliato et al proposed fuzzy regiongrowing methods for seg
menting breast masses and further proposed classication of the segmented
masses as benign or malignant based on the transition information present
around the segmented regions see Sections and Rangayyan et
al proposed a regionbased edgeprole acutance measure for evaluating
the sharpness of mass boundaries see Sections and
Many studies have focused on transforming the spacedomain intensities
into other forms for analyzing gradient and texture information Claridge and
Richter developed a Gaussian blur model to characterize the transitional
information in the boundaries of mammographic lesions In order to analyze
the blur in the boundaries and to determine the prevailing direction of linear
patterns a polar coordinate transform was applied to map the lesion into polar
coordinates A measure of spiculation was computed from the transformed
images to discriminate between circumscribed and spiculated lesions as the
ratio of the sum of vertical gradient magnitudes to the sum of horizontal
gradient magnitudes
Sahiner et al introduced the RBST method to transform
a band of pixels surrounding the boundary of a segmented mass onto the
Cartesian plane see Figure The band of pixels was extracted in the
perpendicular direction from every point on the boundary Texture features
based upon GCMs computed from the RBST images resulted in an average
eciency of in the benignversusmalignant classication of cases
Sahiner et al reported that texture analysis of RBST images yielded better
benignversusmalignant discrimination than analysis of the original space
domain images However such a transformation is sensitive to the precise
extraction of the band of pixels surrounding the ROI the method may face
problems with masses having highly spiculated margins
Hadjiiski et al reported on the design of a hybrid classier # adaptive
resonance theory network cascaded with linear discriminant analysis # to
classify masses as benign or malignant They compared the performance of the
hybrid classier that they designed with a backpropagation neural network
and linear discriminant classiers using a dataset of manually segmented
ROIs benign and malignant Benignversusmalignant classication
using the hybrid classier achieved marginal improvement in performance
with an average eciency of The texture features used in the classier
were based upon GCMs and runlength sequences computed from the RBST
images
Giger et al classied manually delineated breast mass lesions in ul
trasonographic images as benign or malignant using texture features margin
sharpness and posterior acoustic attenuation With a dataset of ultra
sound images from patients the posterior acoustic attenuation feature
achieved the best benignversusmalignant classication results with an aver
age eciency of Giger et al reported to have achieved higher sensitivity
and specicity levels by combining the features derived from both mammo
graphic and ultrasonographic images of mass lesions as against using features
computed from only the mammographic mass lesions
Mudigonda et al derived measures of texture and gradient us
ing ribbons of pixels around mass boundaries with the hypothesis that the
transitional information in a mass margin from the inside of the mass to its
surrounding tissues is important in discriminating between benign masses and
malignant tumors The methods and results of this work are described in the
following sections See Sections and for more discussion
on the detection and analysis of breast masses
Adaptive normals and ribbons around mass margins
Mudigonda et al obtained adaptive ribbons around boundaries of
breast masses and tumors that were drawn by an expert radiologist in the
following manner Morphological dilation and erosion operations were
applied to the boundary using a circular operator of a specied diameter
Figures and show the extracted ribbons across the boundaries of a
benign mass and a malignant tumor respectively The width of the ribbon in
each case is mm across the boundary mm or pixels on either side of
the boundary at a resolution of m per pixel The ribbon width of mm
was determined by a radiologist in order to take into account the possible
depth of inltration or diusion of masses into the surrounding tissues
In order to compute gradientbased measures and acutance see Section
Mudigonda et al developed the following procedure to extract pixels
from the inside of a mass boundary to the outside along the perpendicular
direction at every point on the boundary A polygonal model of the mass
boundary computed as described in Section was used to approximate
the mass boundary with a polygon of known parameters With the known
equations of the sides of the polygonal model it is possible to estimate the
normal at every point on the boundary The length of the normal at any
point on the boundary was limited to a maximum of pixels mm on
either side of the boundary or the depth of the mass at that particular point
This is signicant especially in the case of spiculated tumors possessing sharp
spicules or microlobulations such that the extracted normals do not cross over
into adjacent spicules or mass portions The normals obtained as above for a
benign mass and a malignant tumor are shown in Figures and
a
b c
FIGURE
a A section of a mammogram containing a circumscribed benign
mass Pixel size m b Ribbon or band of pixels across the boundary
of the mass extracted by using morphological operations c Pixels along the
normals to the boundary shown for every tenth boundary pixel Maximum
length of the normals on either side of the boundary pixels or mm
Images courtesy of N R Mudigonda See also Figure
a
b c
FIGURE
a A section of a mammogram containing a spiculated malignant
tumor Pixel size m b Ribbon or band of pixels across the boundary
of the tumor extracted by using morphological operations c Pixels along the
normals to the boundary shown for every tenth boundary pixel Maximum
length of the normals on either side of the boundary pixels or mm
Images courtesy of N R Mudigonda See also Figure
With an approach that is dierent from the above but comparable Sahiner
et al formulated the RBST method to map ribbons around breast masses
in mammograms into rectangular arrays see Figure It was expected that
variations in texture due to the spicules that are commonly present around
malignant tumors would be enhanced by the transform and lead to better
discrimination between malignant tumors and benign masses The rectangular
array permitted easier and straightforward computation of texture measures
Gradient and contrast measures
Due to the inltration into the surrounding tissues malignant breast lesions
often permeate larger areas than apparent on mammograms As a result tu
mor margins in mammographic images do not present a clearcut transition
or reliable gradient information Hence it is dicult for an automated detec
tion procedure to realize precisely the boundaries of mammographic masses
as there cannot be any objective measure of such precision Furthermore
when manual segmentation is used there are bound to be large interobserver
variations in the location of mass boundaries due to subjective dierences
in notions of edge sharpness Considering the above it is appropriate for
gradientbased measures to characterize the global gradient phenomenon in
the mass margins without being sensitive to the precise location of the mass
boundary
A modied measure of edge sharpness The subjective impression of
sharpness perceived by the HVS is a function of the averaged variations in
intensities between the relatively light and dark areas of an ROI Based upon
this Higgins and Jones proposed a measure of acutance to compute
sharpness as the meansquared gradient along knifeedge spread functions
of photographic lms Rangayyan and Elkadiki extended this concept
to D ROIs in images see Section for details Rangayyan et al
used the measure to classify mammographic masses as benign or malignant
acutance was computed using directional derivatives along the perpendicular
at every boundary point by considering the insidetooutside dierences of
intensities across the boundary normalized to unit pixel distance The method
has limitations due to the following reasons
Because derivatives were computed based on the insidetooutside dier
ences across the boundary the measure is sensitive to the actual location
of the boundary Furthermore it is sensitive to the number of dier
ences pixel pairs that are available at a particular boundary point
which could be relatively low in the sharply spiculated portions of a
malignant tumor as compared to the wellcircumscribed portions of a
benign mass The measure thus becomes sensitive to shape complexity
as well which is not intended
The nal acutance value for a mass ROI was obtained by normalizing
the meansquared gradient computed at all the points on the boundary
FIGURE
Mapping of a ribbon of pixels around a mass into a rectangular image by
the rubberband straightening transform Figure courtesy of B
Sahiner University of Michigan Ann Arbor MI Reproduced with permission
from B S Sahiner H P Chan N Petrick M A Helvie and M M Goodsitt
Computerized characterization of masses on mammograms The rubber band
straightening transform and texture analysis Medical Physics
c
American Association of Medical Physicists
with a factor dependent upon the maximum graylevel range and the
maximum number of dierences used in the computation of acutance
For a particular mass under consideration this type of normalization
could result in large dierences in acutance values for varying numbers
of pixel pairs considered
Mudigonda et al addressed the abovementioned drawbacks by de
veloping a consolidated measure of directional gradient strength as follows
Given the boundary of a mass formed by N points the rst step is to com
pute the RMS gradient in the perpendicular direction at every point on the
boundary with a set of successive pixel pairs as made available by the ribbon
extraction method explained in Section The RMS gradient d
m
at the
m
th
boundary point is obtained as
d
m
s
P
p
m
n
f
m
n f
m
n
p
m
where f
m
n n p
m
are the p
m
pixels available along the
perpendicular at the m
th
boundary point including the boundary point The
normal p
m
is limited to a maximum of pixels pixels on either side of
the boundary with the pixel size being m
A modied measure of acutance based on the directional gradient strength
A
g
of the ROI is computed as
A
g
N f
max
f
min
N
X
m
d
m
where f
max
and f
min
are the local maximum and the local minimum pixel
values in the ribbon of pixels extracted and N is the number of pixels along
the boundary of the ROI Because RMS gradients computed over several pixel
pairs at each boundary point are used in the computation of A
g
the measure is
expected to be stable in the presence of noise and furthermore expected to be
not sensitive to the actual location of the boundary The factor f
max
f
min
in the denominator in Equation serves as an additional normalization
factor in order to account for the changes in the graylevel contrast of images
from various databases it also normalizes the A
g
measure to the range
Coe cient of variation of gradient strength In the presence of ob
jects with fuzzy backgrounds as is the case in mammographic images the
meansquared gradient as a measure of sharpness may not result in adequate
condence intervals for the purposes of pattern classication Hence statis
tical measures need to be adopted to characterize the feeble gradient varia
tions across mass margins Considering this notion Mudigonda et al
proposed a feature based on the coecient of variation of the edgestrength
values computed at all points on a mass boundary The stated purpose of this
feature was to investigate the variability in the sharpness of a mass around
its boundary in addition to the evaluation of its average sharpness with the
measure A
g
Variance is a statistical measure of signal strength and can be
used as an edge detector because it responds to boundaries between regions
of dierent brightness In the procedure proposed by Mudigonda et al
the variance
w
localized in a moving window of an odd number of pixels
M in the perpendicular direction at a boundary pixel is computed as
w
M
bMc
X
nbMc
f
m
n
w
where M f
m
n n p
m
are the pixels considered at the m
th
boundary point in the perpendicular direction and
w
is the running mean
intensity in the selected window
w
M
bMc
X
nbMc
f
m
n
The window is moved over the entire range of pixels made available at
a particular boundary point by the ribbonextraction method described in
Section The maximum of the variance values thus computed is used
to represent the edge strength at the boundary point being processed The
coecient of variation G
cv
of the edgestrength values for all the points on
the boundary is then computed The measure is not sensitive to the actual
location of the boundary within the selected ribbon and is normalized so as
to be applicable to a mixture of images from dierent databases
Results of pattern classi cation
In the work of Mudigonda et al four GCMs were constructed by
scanning each mass ROI or ribbon in the
o
o
o
and
o
directions
with unitpixel distance d Five of Haralicks texture features dened
as F
F
F
F
and F
in Section were computed for the four GCMs
thus resulting in a total of texture features for each ROI or ribbon
A pixel distance of d is preferred to ensure large numbers of co
occurrences derived from the ribbons of pixels extracted from mass mar
gins Texture features computed from GCMs constructed for larger distances
d and pixels with the resolution of the images being or m
were found to possess a high degree of correlation and higher with the cor
responding features computed for unitpixel distance d Hence pattern
classication experiments were not carried out with the GCMs constructed
using larger distances
In addition to the texture features described above the two gradientbased
features A
g
and G
cv
were computed from adaptive ribbons extracted around
the boundaries of mammographic ROIs including benign masses and
malignant tumors Three leading features with canonical coecients greater
than including two texture measures of correlation d at
o
and
o
and a measure of inverse dierence moment d at
o
were selected
from the texture features computed from the ribbons The classication
accuracy was found to be the maximum with the three features listed above
The two mosteective features selected for analyzing the mass ROIs included
two measures of correlation d at
o
and
o
Pattern classication experiments with masses from the MIAS database
benign and malignant indicated average accuracies of and
using the texture features computed with the entire mass ROIs and the
adaptive ribbons around the boundaries respectively This result supports the
hypothesis that discriminant information is contained around the margins of
breast masses rather than within the masses With the extended database of
masses benign and malignant and with features computed using the
ribbons around the boundaries the classication accuracies with the gradient
and texture features as well as their combination were and
respectively The area under the receiver operating characteristics curves
were respectively and see Section for details on this
method The gradient features were observed to increase the sensitivity but
reduce the specicity when combined with the texture features
In a dierent study Alto et al obtained benignversusmalignant
classication accuracies of up to with acutance as in Equation
with Haralicks texture measures and with shape factors applied
to a dierent database of breast masses and tumors Although combina
tions of the features did not result in higher pattern classication accuracy
advantages were observed in experiments on contentbased retrieval see Sec
tion
In experiments conducted by Sahiner et al with automatically ex
tracted boundaries of mammographic masses Haralicks texture measures
individually provided classication accuracies of up to only whereas the
Fourierdescriptorbased shape factor dened in Equation gave an accu
racy of the highest among shape features texture features and ve
runlength statistics Each texture feature was computed using the RBST
method see Figure in four directions and for distances How
ever the full set of the shape factors provided an average accuracy of the
texture feature set provided the same accuracy and the combination of shape
and texture feature sets provided an improved accuracy of These results
indicate the importance of including features from a variety of perspectives
and image characteristics in pattern classication
See Sections and for discussions on the detection of masses in
mammograms Sections and for details on shape analysis of masses
and Section for a discussion on the application of texture measures for
contentbased retrieval and classication of mammographic masses
Remarks
In this chapter we have examined the nature of texture in biomedical images
and studied several methods to characterize texture We have also noted
numerous applications of texture analysis in the classication of biomedical
images Depending upon the nature of the images on hand and the antici
pated textural dierences between the various categories of interest one may
have to use combinations of several measures of texture and contour rough
ness see Chapter in order to obtain acceptable results Relating statistical
and computational representations of texture to visually perceived patterns
or expert opinion could be a signicant challenge in medical applications See
Ojala et al for a comparative analysis of several methods for the analysis
of texture See Chapter for examples of pattern classication via texture
analysis
Texture features may also be used to partition or segment multitextured
images into their constituent parts and to derive information regarding the
shape orientation and perspective of objects Haralick and Shapiro
Chapter describe methods for the derivation of the shape and orientation
of D objects or terrains via the analysis of variations in texture
Examples of oriented texture were presented in this chapter Given the
importance of oriented texture and patterns with directional characteristics in
biomedical images Chapter is devoted completely to the analysis of oriented
patterns
Study Questions and Problems
Selected data les related to some of the problems and exercises are available at the
site
wwwenelucalgarycaPeopleRangaenel
Explain the manner in which
a the variance
b the entropy and
c the skewness
of the histogram of an image can represent texture
Discuss the limitations of measures derived from the histogram of an image
in the representation of texture
What are the main similarities and dierences between the histogram and a
graylevel cooccurrence matrix of an image
What are the orders of these two measures in terms of PDFs
Explain why graylevel cooccurrence matrices need to be estimated for several
values of displacement distance and angle
Explain how shape complexity and texture graylevel complexity comple
ment each other You may use a tumor as an example
Sketch two examples of fractals in the sense of selfsimilar nested patterns
in biomedical images
Laboratory Exercises and Projects
Visit a medical imaging facility and a pathology laboratory Collect examples
of images with
a random texture
b oriented texture and
c ordered texture
Respect the priority privacy and condentiality of patients
Request a radiologist a technologist or a pathologist to explain how he or she
interprets the images Obtain information on the dierences between normal
and abnormal disease patterns in dierent types of samples and tests
Collect a few sample images for use in image processing experiments after
obtaining the necessary permissions and ensuring that you carry no patient
identication out of the laboratory
Compute the logmagnitude Fourier spectra of the images you obtained in
Exercise Study the nature of the spectra and relate their characteristics to
the nature of the texture observed in the images
Derive the histograms of the images you obtained in Exercise Compute the
a the variance
b the entropy
c the skewness and
d kurtosis
of the histograms Relate the characteristics of the histograms and the values
of the parameters listed above to the nature of the texture observed in the
images
Write a program to estimate the fractal dimension of an image using the
method given by Equation Compute the fractal dimension of the images
you obtained in Exercise Interpret the results and relate them to the nature
of the texture observed in the images
