
Citation 
 Permanent Link:
 http://digital.auraria.edu/AA00002039/00001
Material Information
 Title:
 Wavelet analysis of images of malignant tumors
 Creator:
 Stewart, Samuel Burns
 Place of Publication:
 Denver, Colo.
 Publisher:
 University of Colorado Denver
 Publication Date:
 2002
 Language:
 English
 Physical Description:
 45 leaves : illustrations ; 28 cm
Thesis/Dissertation Information
 Degree:
 Master's ( Master of Science)
 Degree Grantor:
 University of Colorado Denver
 Degree Divisions:
 Department of Electrical Engineering, CU Denver
 Degree Disciplines:
 Electrical Engineering
 Committee Chair:
 Bialasiewicz, Jan T.
 Committee Members:
 Radenkovic, Miloje
Moore, Linda
Subjects
 Subjects / Keywords:
 Cancer  Diagnosis ( lcsh )
Wavelets (Mathematics) ( lcsh ) Cancer  Tomography ( lcsh ) Gaussian processes ( lcsh ) Cancer  Diagnosis ( fast ) Cancer  Tomography ( fast ) Gaussian processes ( fast ) Wavelets (Mathematics) ( fast )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Bibliography:
 Includes bibliographical references (leaf 45).
 General Note:
 Department of Electrical Engineering
 Statement of Responsibility:
 by Samuel Burns Stewart.
Record Information
 Source Institution:
 University of Colorado Denver
 Holding Location:
 Auraria Library
 Rights Management:
 All applicable rights reserved by the source institution and holding location.
 Resource Identifier:
 50710588 ( OCLC )
ocm50710588
 Classification:
 LD1190.E54 2002m .S73 ( lcc )

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Full Text 
WAVELET ANALYSIS OF IMAGES OF MALIGNANT TUMORS
by
Samuel Bums Stewart
B.A., West Virginia University, 1974
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Electrical Engineering
2002
This thesis for the Master of Science
degree by
Samuel Bums Stewart
has been approved
by
Jan T. Bialasiewicz
Miloje Radenkovic
V Date
Stewart, Samuel Bums (M.S., Electrical Engineering)
Wavelet Analysis of Computed Tomography Generated Images of Malignant Tumors
Thesis directed by Professor Jan T. Bialasiewicz
ABSTRACT
Wavelet analysis of veterinary images generated by computed tomography is used to
enhance certain image features. Malignant tumors often are difficult to define in
terms of their extent. Wavelet analysis using wavelets based on Gaussian functions is
capable of differentiating tissues by characterizing the local regularity of the images.
Gaussian functions and their derivatives are used to calculate the modulus maxima of
localized regions using multiscale analysis. Modulus maxima are used to indicate
differences in image characteristics such as regularity and important features such as
edges and transients.
This abstract accurately represents the content of the candidates thesis. I recommend
its publication.
Signe
m
Jan T. Bialasiewicz
ACKNOWLEDGMENT
My thanks to Dr. Susan Kraft of the College of Veterinary Medicine, Colorado State
University for providing the images of tumors used in this research.
CONTENTS
Figure...................................................................vii
Chapter
1. Introduction..........................................................1
1.2 The Nature of Cancerous Tissues.......................................1
1.3 The Imaging System....................................................2
1.4 Problem Statement.....................................................4
1.5 Description of Wavelets...............................................7
1.5.1 Discrete Wavelet Conventions.........................................8
1.5.2 Calculating Wavelet Transforms.......................................8
1.5.3 Wavelet Dilation.....................................................9
1.5.4 Wavelet Reconstruction..............................................11
1.6 Previous Research....................................................11
1.6.1 Wavelets Based on the Gaussian Function.............................11
1.6.2 Calculating Modulus Maxima..........................................13
1.6.3 Function Regularity from Modulus Maxima at Different Scales.........15
2. Methods and Materials.............................................. 17
2.1 Regions of Interest..................................................17
2.2 Algorithm Development................................................23
2.2.1 Computing Wavelet Transforms Using Circular Convolution.............25
v
2.2.2 Computing Dilated Wavelets Using Algorithm a Trous...............27
2.2.3 Creating the Direction, Modulus, and Threshold Modulus Maxima
Images...........................................................28
3. Image Analysis...................................................30
3.1 Image Analysis of Sarcoma on the Jaw of a Dog....................30
3.2 Image Analysis of a Soft Tissue Sarcoma on the Flank of a Dog....35
3.3 Image Analysis of Nasal Tumor in a Dog...........................38
3.4 Tumor on the Leg of A Dog........................................41
4. Conclusion.......................................................44
References.............................................................45
vi
FIGURES
Figure
1.1 Sarcoma Near the Jaw of a Dog...........................................5
1.2 Regions of Interest.....................................................6
1.3 Plot of Pixel Intensity from Detail 1.2.2 in Figure 1.2.................7
1.4 Plot of Pixel Intensity from Detail 1.2.1 in Figure 1.2.................7
1.5 Quadratic Spline and Dilated Quadratic Spline.....................9
1.6 Fourier Transform, Quadratic Spline....................................10
1.7 Fourier Transform of Dilated Quadratic Spline....................10
1.8 Cubic Spline and Quadratic Spline......................................12
2.1 Sarcoma Near the Jaw of a Dog..........................................18
2.2 Regions of Interest 2.2.1 and 2.2.2....................................19
2.3 Soft Tissue Sarcoma of the Flank of a Dog..............................20
2.4 Region of Interest 2.4.................................................20
2.5 Nasal Tumor of a Dog...................................................21
2.6 Region of Interest 2.6.................................................21
2.7 Tumor on the Leg of a Dog..............................................22
2.8 Region of Interest 2.8.................................................22
2.9 Outline of Algorithm for Computing Modulus Maxima Images at Different
Scales................................................................23
vii
2.10 Approximation and Detail Images..................................24
2.11 Direction Images.................................................24
2.12 Modulus Images...................................................25
2.13 Modulus Maxima Images............................................25
2.14 Impulse Response of Filters H(co) and G(co)......................26
2.15 Circular Convolution Algorithm...................................27
2.16 Algorithme a Trous...............................................28
2.17 Subroutine for Creating Image Direction Matrix...................28
2.18 Subroutine for Creating Image Modulus Matrix.....................28
2.19 Subroutine for Calculating Image Modulus Maxima Matrix...........29
3.1 Modulus Maxima Images for ROI 2.2.2, Tumor........................31
3.2 Modulus Maxima Images for ROI 2.2.1, Healthy Tissue...............31
3.3 Approximations and Details Images for ROI 2.2.2, Tumor............32
3.4 Approximations and Details Images for ROI 2.2.1, Healthy Tissue...33
3.5 Modulus Images for ROI 2.2.2, Tumor...............................33
3.6 Modulus Images for ROI 2.2.1, Healthy Tissue......................34
3.7 Direction Images for ROI 2.2.2, Tumor.............................34
3.8 Direction Images for ROI 2.2.1, Healthy Tissue....................34
3.9 Modulus Maxima and Original Image for ROI 2.4....................35
3.10 Approximations and Details Images for ROI 2.4...................36
viii
3.11 Modulus Images for ROI 2.4......................................36
3.12 Direction Images for ROI 2.4....................................37
3.13 Modulus Maxima Images for ROI 2.6...............................38
3.14 Approximations and Details Images for ROI 2.6...................39
3.15 Modulus Images for ROI 2.6......................................39
3.16 Direction Images for ROI 2.6....................................40
3.17 Modulus Maxima Images for ROI 2.8...............................41
3.18 Approximations and Details Images for ROI 2.8...................42
3.19 Modulus Images for ROI 2.8......................................42
3.20 Direction Images for ROI 2.8....................................43
IX
1. Introduction
This dissertation describes the use of wavelet analysis applied to the problem of
image enhancement. Image enhancement is the processing of images to improve
their appearance to the viewer. Although image quality and overall appearance of
the image can be goals of enhancement, the focus is to improve the intelligibility of
the image. The data set studied is medical images of dogs with various types of
cancers. The result of the image enhancement is to better define the shape and
extent of malignant tissue. The decision to use wavelets to enhance medical
images is based on the nature of cancerous tissues and on the process used to image
the diseased and healthy tissue.
1.2 The Nature of Cancerous Tissues
The basic ideas in the following discussion of cancerous tissues is adapted from [1]
and [2]. The terms malignant neoplasm and cancer have the same meaning.
Neoplasms are characterized as an abnormal mass of tissue that is uncoordinated
with normal tissue. The mass has no purpose and competes with normal tissues for
nutrients. Malignant neoplasms have the potential to grow rapidly, destroy
contiguous tissues and disseminate throughout the body leading to death. Tumors
are swellings that may have benign causes and are not necessarily related to cancer.
However the term tumor will be equated with cancer in this discussion. The
images studied in this thesis are of malignant neoplasms. Sarcomas are cancers
1
arising from mesenchymal (fleshy) tissue. Often tumors are composed of poorly
differentiated cells with a close resemblance to each other but there may be more
than one cell type. There are basically two components to a tumor, the proliferating
neoplastic cells, called the parenchyma and supportive tissues including connective
tissue and blood vessels called stroma. In general, the bulk of the tumor is the
parenchyma cells. The nuclei of the cells are disproportionately large. The
undifferentiation of the tumor cells is referred to as anaplasia. The degree of
anaplasia and evidence of invasion of surrounding tissue are the major criteria by
which a diagnosis of cancer is made microscopically. Breast cancer is an example
of a tumor with a large stroma component. Because this type of tumor is quite
visible on mammograms and because the tumors are often benign, there is a
substantial volume of research in breast cancer image enhancement with the goal of
differentiating benign from malignant tumors. Soft tissue tumors such as sarcomas
may be much more difficult to differentiate from healthy tissues using medical
imaging. To summarize, malignant neoplasms have certain physical differences to
healthy tissues, in particular, a higher degree of disorganization and a higher
nuclear content.
1.3 The Imaging System
The basic ideas presented in the following discussion are adapted from [3]. The
process used to create the images used in this study is computed axial tomography
2
or CAT, which is sometimes referred to as computed tomography or CT. The term
CT scan will be used to refer to the process of creating the images using xrays in
the process of computed tomography, or to the actual image created using
computed tomography. The basic process of computed tomography is to measure
the attenuation of xrays through the patient and generate an image from the
attenuation data. The CT machine generates a highly collimated beam of xrays
that only passes through a cross section or volume of the material to be imaged.
After passing through the patient the modified xrays are measured by special
detectors that make quantitative measurements of xray intensity. The data
generated is called relative transmission. Relative transmission is the log of the
ratio of the xray intensity at the source with the intensity at the detector. This data
is processed into an image that can display very small differences in tissue density.
The data can be acquired in two ways. The first method is to rotate the xray
source and detector around the patient one slice at a time. The patient is moved to
the next position and another scan is made. This process generates enough data to
build a three dimensional image of the patient. A helical scan process can also
acquire the data where a volume of tissue is scanned. The relative attenuation of
the xrays by the tissues is dependent upon the density of the tissue. Different
tissues have different linear attenuation coefficients /u, which is a measure of how
3
much attenuation has occurred. Under certain operating parameters bone has a //
of 0.528, fat has a // of 0.185, and blood has a fj. of 0.208. These values of // are
relative values because the absolute value of /u depends on a number of factors
besides the density of the tissue, for example, the energy of radiation.
1.4 Problem Statement
Based on the fact that tumors tend to be physically different from healthy tissues
and that CT imaging can differentiate between different tissue types, it is suggested
that CT images will contain enough information to differentiate between healthy
and diseased tissue. The difference may not be apparent upon visual examination
of an image. For example, Figure 1.1 is a slice from a CT scan of a sarcoma near
the jaw of a dog. Detail 1.1 shows the boundary of the tumor as defined by a
radiologist. The image appearance inside the outline is very similar to the tissue
outside the tumor. This indicates very subtle differences between the healthy and
diseased tissues. Figure 1.2 shows two regions of interest in boxes with the left box
enclosing healthy tissue and the right box enclosing the tumor. Detail 1.2.1 labels a
segment of a row in the healthy tissue and detail 1.2.2 labels a segment of a row in
the tumor. Figures 1.3 and 1.4 show a plot of pixels intensities for each segment.
Comparison of figures 1.3 and 1.4 shows the basic range of pixel intensities is
similar which makes identifying malignant tissue using pixel intensities quite
4
difficult. The problem is to design a system that can enhance images features that
are not visually apparent.
5
6
Row 62 Columns 55:119 from Detail 1.2.2 in Figure 1.2
1QO
m
 95
*Q3
a. 90
0 10 20 30 40 50 60
Columns
Plot of Pixel Intensity from Detail 1.2.2 in Figure 1.2
Figure 1.3
Row 62 Columns 20:04 from Detail 1.2.1 in Figure 1.2
o 90
30 40
Columns
Plot of Pixel Intensity from Detail 1.2.1 in Figure 1.2
Figure 1.4
1.5 Description of Wavelets
Since the range of intensity is insignificant, the transient behavior of the pixel
intensities is used for image enhancement. Wavelets are used to locate and
characterize transients in the images. Fourier analysis can be used for
characterizing transient behavior. However, Fourier analysis doesn't describe the
location of singularities, it only provides the overall signal characteristics.
7
Wavelets are defined as functions whose integral is equal to zero. Wavelets are
localized in both space and frequency. This makes wavelets particularly useful for
characterizing the local regularity of a signal.
1.5.1 Discrete Wavelet Conventions
The space domain designated as either x or n is used in this discussion. The letter n
can refer to a single value when discussing onedimensional signals or as an
abbreviation for ni,n2 in the context of two dimensions where ni refers to a row of
pixels in an image and n2 refers to a column of pixels. Discrete signals are
designated as f[n], discrete smoothing functions as 9[n\, and discrete wavelets as
y/\n\. The Fourier transforms of the same signals are designated /(co), 6{co), and
y/{co). All signals analyzed and used for analysis will be discrete.
1.5.2 Calculating Wavelet Transforms
A wavelet transform of a signal, J[n\, is calculated by convolving the wavelet with
the signal and is designated Wf(n). The convolution of a wavelet with a signal to
form a wavelet transform can be written
+00
W(no) = o~] (L1)
n= oo
which is equivalent to the inner product designation
WfM = (f{n),y/[n0]) (1.2)
8
1.5.3 Wavelet Dilation
Wavelets are dilated using a scaling factor s. A dilated wavelet is designated
y/s (x) where x is the time or space variable and s is the scaling factor. The wavelet
is scaled as follows:
Ys[n} = ~V(nl *) (13)
Figure [1.5] compares a quadratic spline with a dilated quadratic spline scaled by a
factor of s = 2. The Fourier transforms of the functions shown in Figure [1.5] are
shown in Figures [1.6] and [1.7]. The effect of the dilation is to lower the
frequency content of the signal.
1D 5 5 10 1D 5 0 5 10
Space Space
Quadratic Spline and Dilated Quadratic Spline
Figure 1.5
The scaling factor often used for the image analysis is based on the algorithme a
trous. Starting with coefficients of the filters, zeros are inserted between the
9
coefficients using a dyadic sequence of 2j 1 for each scale. This form of dilation
lends itself to fast computational algorithms. Each scale of the wavelet gives a
different level of detail of the signal.
Fourier Transform, Quadratic Spline
Frequency in Radians
Fourier Transform, Quadratic Spline
Figure 1.6
Fourier Transform, Dilated Quadratic Spline
Frequency in Radians
Fourier Transform of Dilated Quadratic Spline
Figure 1.7
10
1.5.4 Wavelet Reconstruction
Wavelet transforms calculated at different scales, Wf(s,n), can be analyzed,
modified appropriately, and then used to reconstruct the original image in a
different form. The reconstruction process recovers the original signal as follows
+00 +00
/m= E E]ysw (i4)
s= 00/7 = 00
Wavelets are used to generate structures called modulus maxima. Modulus
maxima provide information about the regularity of a signal and can be used to
enhance transients in a signal.
1.6 Previous Research
The wavelet analysis algorithms described in the materials and methods section and
the discussion of function regularity as related to modulus maxima are based on the
research and ideas presented in [4], [5], [6], and [7].
1.6.1 Wavelets Based on the Gaussian Function
The derivative of a Gaussian function has been shown to be useful in image
enhancement. The Gaussian function is defined as
n2
G[n] = ^=e21 (1.5)
a^ln
and its first derivative, which is used for calculating wavelet transforms, is
11
G\n\ = 
cr3VIk
2 a1
(1.6)
The Fourier transform of the Gaussian function is
G(a>) =
/^sin('3
col 2
(1.7)
and the Fourier transform of its first derivative is
^() =
^sin(W4)^4
a! 4
(1.8)
G[n] and G'[n] will be referred to respectively as cubic spline and quadratic spline
functions. Their graphs are shown in figure 1.
5 0 5 5 0 5
Space Space
Cubic Spline and Quadratic Spline
Figure 1.8
12
1.6.2 Calculating Modulus Maxima
Let a column or row of pixels from an image define a function /[]. Let /[/?0] be
a particular pixel in f[n ]. The quadratic spline function is the derivative of the
cubic spline. Let G[n\ represent the cubic spline function. Therefore
dd[n\
dn
denotes the quadratic spline function or the wavelet y/s[n]. The wavelet transform
of f[n] is
Wf(s,n) = f[n\y/5[n]
(1.9)
where <8> denotes convolution. Substituting ^ for ^s[n] in (1.9) gives
dn
Wf{s,n) = s^(f[n}ds[n]) (1.10)
dn
Therefore the wavelet transform, Wf{s,n), is the first derivative of the signal
smoothed at the scale s. A wavelet transform at 0 that is locally maximum so that
\Wf{s, 0 1) < Wf(s, 0 ) > Wf{s, n0+l) (1.11)
with at least a strict maximum in either the left or right neighborhood of o Is a
modulus maxima at that point. If a modulus maxima is located at 0 then
dWf(s,nq) _Q
dn
(1.12)
13
For a two dimensional system, such as an image, denote the wavelet transforms of
1 2
the rows as W f (5, n) and the wavelet transforms of the columns as IF /(s, n).
Since the wavelet transforms are derivatives of the columns and rows, they describe
the gradient at a particular pixel (ti\, 2)
V/'(j,2) *n = ^cosct + sinor (1.13)
where n =(cosa, sina). For two dimensions the modulus of the gradient vector is
denoted
Mf(n,s)
Wlf(n,s)
+
Wzf(n,s)
(1.14)
The angle of the wavelet transform vector is
a(n) = tan
1
(W1f{n,s)
WXf{n,s)
(1.15)
This angle represents direction of steepest descent of the pixel intensities. The
modulus maxima in two directions is the local maxima of (1.14) in the direction
normal to a.
14
1.6.3 Function Regularity from Modulus Maxima at
Different Scales
For the following discussion a refers to the Lipschitz exponent. Suppose f is m
times differentiable in [v h, v + h\ and pv is the Taylor polynomial in the
neighborhood of v
k=0
k\
= fly) + /'(v)(r v) +  v)2 +  V)" + ... (1.11)
2! n\
The approximation error is ev (t) = f(t)~ pv (t) and the mth order differentiability
of / in the neighborhood of v yields an upper bound on the error ev (t) when t
tends to v. In other words, the precision of the approximation using a Taylor series
is dependent on the order of the polynomial used to approximate the function.
The order of the approximation is dependent on the differentiability of the function.
The values of k in the Taylor series are integer values and represent the order of the
polynomial. The Lipschitz regularity gives the upper bound with noninteger
exponents, in contrast with k, which can only have integer values, which makes it a
more general measure of the regularity of a function. If a function f[n] is uniformly
Lipschitz a in the interval [a,b] then their exists A>0 such that
15
V(/i,j) e [a,6]x91 + \Wf(s,n)\ < Asa+X'2
(1.12)
where 91+ denotes positive real numbers. Since the scale s varies by powers of 2,
(1.12) is equivalent to
The Lipschitz regularity at a point is the slope of log2Wjf(.s,) as a function of
log2 s for modulus maxima that converges to a point as s goes to 1. For isolated
singularities the modulus maxima that converge to a point w0 are located in cone
such that
Because the presence of modulus maxima indicate sharp transitions in pixel
intensity, they are useful for detecting significant structures in images including
edges. Per equation (1.13), modulus maxima are also useful for characterizing the
local regularity of an image. The usefulness of wavelet analysis in characterizing
the regularity of a signal is dependent on the number of vanishing moments for the
wavelet. A wavelet with one vanishing moment, such as the quadratic spline, can
only characterize functions that have Lipschitz regularity of less than one.
However for image analysis, especially transition and edge detection, a wavelet
with one vanishing moment is sufficient.
r o
\og2\Wf{s,n)\<\o%1A + a + log25
V
(1.13)
\n hq\< Cs
(1.14)
16
2. Methods and Materials
The image enhancement process will consist of calculating the modulus maxima of
five regions of interest over three scales. The data generated by wavelet analysis
will be presented as images to allow direct comparison with the original images.
For each region of interest the approximation, horizontal, vertical, direction,
modulus, and modulus maxima images are displayed to allow comparison with the
original images.
2.1 Regions of Interest
Each image is displayed twice. One image will have the tumor outlined by a
radiologist and the second image will have regions of interest defined. Two
regions of interest are marked on Figure 2.2. The left box encloses healthy tissue
and the right box encloses the tumor and some of the surrounding tissue. The other
three regions of interest in Figures 2.4, 2.6, and 2.8 were chosen to include both
healthy and diseased tissue.
17
Sarcoma Near the Jaw of a Dog
Figure 2.1
18
Regions of Interest 2.2.1 and 2.2.2
Figure 2.2
19
nmor
Soft Tissue Sarcoma of the Flank of a Dog
Figure 2.3
Region of Interest 2.4
Figure 2.4
20
Nasal Tumor of a Dog
Figure 2.5
Region of Interest 2.6
Figure 2.6
21
Tumor on the Leg of a Dog
Figure 2.7
Region of Interest 2.8
Figure 2.8
22
2.2 Algorithm Development
Algorithms were written in Matlab, version 5.3.1. The overall algorithm for
computing the modulus maxima is outlined in Figure 2.9. Figures 2.10, 2.11, 2.12
and 2.13 are examples of the outputs generated by the algorithm.
Algorithm inputs: regions of interest in 24 bit bitmap format. Approximate
dimensions of the images are 128 by 128 pixels. All images are gray scale with
256 levels. The threshold level used is 0.1 in all experiments.
Algorithm outputs: Approximation, vertical horizontal, direction, modulus, and
modulus maxima images at scales 2, 4, and 8
1. Scaled approximation, vertical, and horizontal details images are created
1.1 Compute approximation image
1.1.1 Convolving rows of image with smoothing filter
1.1.2 Convolving columns of image with smoothing filter
1.2. Compute vertical details by convolving columns of approximation image with
wavelet filter
1.3. Compute horizontal details by convolving rows of approximation image with
wavelet filter
1.4. Perform Algorithme a Trous on filters
1.5. Loop back to 1 to create next scaled image
2. Create the direction images, implementing equation 1.15
2.1 Find the ratio of the vertical and horizontal details images, for each pixel
2.2 Compute the arctan of the ratios and save the result in the direction image
3. Calculate the modulus image, implementing equation 1.14
4. Find the threshold modulus maxima for the modulus image by comparing each
pixel in the modulus image with the two pixels closest to normal in the angle
image. If the pixel is a local maxima and is greater than a threshold value then the
pixel is not suppressed______________________________________________________
Outline of Algorithm for Computing Modulus Maxima Images at Different Scales
Figure 2.9
23
Direction Images Horizontal Details Vertical Details Approximation
60 100 150
50 100 150
20
40
60
80
100
120
50 100 150 50 100 150
Scale=2 Scale=4
50 100 150
Scale=S
Approximation and Detail Images
Figure 2.10
50 100 150
Scale=2
50 100 150
Scale=8
Direction Images
Figure 2.11
24
50 100 150
Scale=2
Modulus Images
Figure 2.12
C0
0>
0)
20
40
60
00
100
120
50 100 150
Scale=4
Scale=8
Modulus Maxima Images
Figure 2.13
2.2.1 Computing Wavelet Transforms Using Circular
Convolution
The wavelet transforms are calculated by circular convolution of the rows and
columns of the regions of interest with discrete filters with finite impulse response.
The transfer functions used to derive the filters [7] are
H{(o) = eicoll{cos{colT)y, (2.1)
25
sin(
(2.2)
G{co) = ieico12
The corresponding finite impulse responses of the filters H(a>) andG(ry) are show
in Figure 2.14.
n h[n] g[n]
1 0.125
0 0.375
1 0.375 2.0
2 0.125 2.0
Impulse Response of Filters H(co) and G{co)
Figure 2.14
Let x[n]=I(:,l) denote all rows and column 1 in an image. Let h[n] denote a finite
impulse response filter. Let N denote the period of both x[n] and h[n]. The
circular convolution of x[n] and h[n] is defined as
Nl
x[n\ h[n] = ^.x[m]/2[mi] (23)
/=0
where denotes convolution. If the length of x[n] is / and the length of h[n] is m,
then x[n] and h[n] are padded with m +1 1 zeros. The result of the linear
convolution of the zero padded signals is the same finding the inverse Fourier
26
transform of x[a>]h[o)]. The convolution of the signal and filter is accomplished
using the function cc.m shown in Figure 2.15.
2.2.2 Computing Dilated Wavelets Using Algorithme a
Trous
Dilated wavelets are created by inserting 2J1 zeros between each filter value. The
algorithm used is in Figure 2.16.
function yout=cc(h,x)
%cc.m
%x=signal
%h=filter
%yout=filtered signal
N=length(x);
M=length(h);
xpadded=zeros( 1 ,N+M1);
xpadded(l :N)=x;
hpadded=zeros( 1 ,N+M1);
hpadded(l:M)=h;
%
for n=0:Nl
for i=0:Nl
if 0<(ni+N+l) & (ni+N+l)<=N
m=l;
else
m=0;
end
y(i+1 ,n+1 )=xpadded(i+1 )*hpadded(ni+m*N+1);
end
end
yout=sum(y); % This line sums the rows in the matix y
Circular Convolution Algorithm
Figure 2.15
27
hd=[dyadup(hd,2) 0];
Algorithme a Trous
Figure 2.16
2.2.3 Creating the Direction, Modulus, and Threshold
Modulus Maxima Images
Figures 2.17, 2.18, and 2.19 contain the algorithms for computing the direction,
modulus, and the threshold modulus maxima images. A threshold level of 0.1 was
used in all experiments.
[nr,nc]=size( W aveletT ransformV ertical);
ratio=WaveletTransformVertical(r,c)./WaveletTransformHorizontal(r,c);
for r=l :nr
for c=l:3*nc
if ratio(r,c)>=0
angle(r,c)=atan(ratio(r,c));
else
angle(r,c)=piatan(ratio(r,c));
end
end
end
Subroutine for Creating Image Direction Matrix
Figure 2.17
Modulus=sqrt(WaveletTransformVertical.*WaveletTransformVertical+...
W aveletT ransformHorizontal. W aveletT ransformHorizontal)
Subroutine for Creating Image Modulus Matrix
Figure 2.18
28
inc=pi/8;
for r=2:nrl
for c=2:3*ncl
if (angle(r,c)>=( 15 inc)&angle(r,c)<=( 1 *inc))&modulus(r,c)>=Threshold &.. .
modulus (r,c)>= modulus (rl,c)&modulus (r,c) >= modulus (r+l,c)
tmodulus (r,c)=0;
elseif (angle(r,c)>=(l*inc)&angle(r,c)<=(3*inc))& modulus (r,c)>=Threshold
& modulus g(r,c)>= modulus (rl,cl) & modulus (r,c) >= modulus (r+l,c+l)
tmodulus (r,c)=0;
elseif (angle(r,c)>=(3*inc)& angle(r,c)<=(5*inc))&maxmag(r,c)>=Threshold
& maxmag(r,c)>=maxmag(r,c+l) & maxmag(r,c) >= maxmag(r,cl)
tmaxmag(r,c)=0;
elseif (angle(r,c)>=(5*inc)&angle(r,c)<=(7*inc))& modulus (r,c)>=Threshold
& modulus (r,c)>=modulus(rl,c+l) & modulus(r,c) >= modulus(r+l,cl)
tmodulus(r,c)=0;
elseif (angle(r,c)>=(7*inc)&angle(r,c)<=(9*inc))&modulus(r,c)>:=Threshold
& modulus(r,c)>=modulus(rl,c) & modulus(r,c) >= modulus(r+l,c)
tmodulus(r,c)=0;
elseif (angle(r,c)>=(9*inc)&angle(r,c)<=(l 1 *inc))&modulus(r,c)>=Threshold
& modulus(r,c)>=modulus(rl,cl) & modulus(r,c) >= modulus(r+l,c+l)
tmodulus(r,c)=0;
elseif (angle(r,c)>=(ll*inc)&angle(r,c)<=(13*inc))&modulus(r,c)>=Threshold
& modulus(r,c)>=modulus(r,cl) & modulus(r,c) >= modulus(r,c+l)
tmodulus(r,c)=0;
elseif (angle(r,c)>=( 13 *inc)&angle(r,c)<=(l 5 *inc))&modulus(r,c)>=Threshold
& modulus(r,c)>=modulus(rl,c+l) & modulus(r,c) >= modulus(r+l,cl)
tmodulus(r,c)=0;
end
end
end
Subroutine for Calculating Image Modulus Maxima Matrix
Figure 2.19
29
3. Image Analysis
Modulus maxima, approximation, vertical details, horizontal details, modulus, and
direction images are included for each region of interest. Modulus maxima images
have details labeled to indicate possible features that are revealed by the analysis.
3.1 Image Analysis of Sarcoma on the Jaw of a Dog
The original image is characterized by a visually uniform pixel intensity, which
makes defining the boundaries of the tumor difficult. The only morphological
difference between the left and right side of the jaw is a noticeable bulge in the
region of the tumor. In figure 3.1, the modulus maxima image at scale 2 reveals an
area that generally conforms to the shape of the tumor on the right side of the
image. This area is characterized by few maxima and extends from row 20 to row
80 between columns 90 to 110. This would be consistent with intensity transitions
that are gradual. The modulus maxima image at scale 4, which is obtained with a
wavelet with lower frequency content shows a better correlation and displays more
maximal in the same region. An important observation is the number of maxima is
increasing in parts of the tumor area as the scale is increasing, while the modulus
maxima in Figure 3.2 is decreasing as the scale increases. The approximations,
details, and modulus images do not show remarkable differences between healthy
tissue and the tumor. The direction images for the tumor show some conformance
with the shape of the tumor.
30
Modulus Maxima Images Modulus Maxima Images
20
40
60
80
120
... *" ^ Detail 3.1
:WY. SVj m WSM& 60 gfaSBBfr 20 40 60 80 100 120 ' S..V" Â£::~e,jrt "J X>Tt Â£ ^ "'*Â£1 y 1 ge^~yhr_y.
20 40 60 80100120
Scale=2
20 40 60 80100120
Scale=4
20 40 60 8010020
Scale=8
Tumor Boundary
Added by Radiologist
20 40 60 80100120
Original Image
Modulus Maxima Images for ROI 2.2.2, Tumor
Figure 3.1
Scale=2
2oHsm
40
60
120
20 40 60 80100120
Original Image
20 40 60 80100120
Scale=8
Modulus Maxima Images for ROI 2.2.1, Healthy Tissue
Figure 3.2
31
Horizontal Details Vertical Details Approximation
20 40 60 80100 20 20 40 60 80100120
20
40
60
80
100
120
7
20 40 60 80100120
20^^BH^H
40
60
80
100
120
20 40 60 80100120
Scale=2
60
80
120
20 40 60 80100120
Scale=4
20
40
60
80
100
120
20 40 60 80100120
Scale=8
1
i
Approximations and Details Images for ROI 2.2.2, Tumor
Figure 3.3
32
Modulus Images Horizontal Details Vertical Details Approximation
40
80
120
20 40 60 80100 20
20
40
60
80
100
120
%r
Â£
20 40 60 80100 20
20
40
60
80
100
120
m
20 40 60 8010020
20MDBMH
40
eo
ioo^^^^HQ
120
20 40 60 8010020
20 40 60 80100 23 20 40 60 80100 20 20 40 60 80100 20
Scale=2 Scale=4 Scale=8
Approximations and Details Images for ROI 2.2.1, Healthy Tissue
Figure 3.4
Modulus Images for ROI 2.2.2, Tumor
Figure 3.5
33
Direction Images Direction Images Modulus Images
20 40 60 80100120
Scale=2
20 40 60 80100120
Scale=4
20 40 60 80100120
Scale=8
Modulus Images for ROI 2.2.1, Healthy Tissue
Figure 3.6
20 40 60 80100 20
Scale=2
20 40 60 80100120
Scale=4
20 40 60 80100120
Scale=8
Direction Images for ROI 2.2.2, Tumor
Figure 3.7
20 m 20 Mm
40 40 RgffitejiE 'VRS^h
60 m 50
80 80
100 100
120 120 rt
20 40 60 8010020
Scale=2
20 40 60 80100 20
Scale=4
20 40 60 80100 20
Scale=8
Direction Images for ROI 2.2.1, Healthy Tissue
Figure 3.8
34
3.2 Image Analysis of a Soft Tissue Sarcoma on the
Flank of a Dog
In this case, the wavelet analysis suggests the tumor extends beyond the boundary
suggested by the radiologist. As in the sarcoma on the jaw, the region of the tumor
is characterized by practically no modulus maxima. However, as the scale
increases, the modulus maxima dont increase. The direction images, especially
scales two and four show a general conformance with the shape of the tumor.
Scale=2
100
120
20 ix
40 r "
60
 c 
V3I
Scale=4
20 40 60 80100120
Scale=8
20 40 60 80100120
Original Image
Boundary Drawn by
Radiologist
Modulus Maxima and Original Image for ROI 2.4
Figure 3.9
35
Modulus Images Horizontal Details Vertical Details Approximation
20 40 60 80100 20 20 40 60 8010020
20 40 60 80100120
20 40 60 80100 20
20 40 60 80100 20
Scale8
Approximations and Details Images for ROI 2.4
Figure 3.10
20 40 60 80100 20
Scale=2
20
40
60
80
100
120
20 40 60 80 1 03120
Scale=4
Modulus Images for ROI 2.4
Figure 3.11
36
Direction Images
20 40 60 8010020 20 40 60 80100120
Scale=2 Scale=4
20 40 60 80100120
Scale=8
Direction Images for ROI 2.4
Figure 3.12
37
3.3 Image Analysis of Nasal Tumor in a Dog
The region defined by the tumor is characterized by no modulus maxima. The
tissue outside the tumor boundary in the lower half of the image also shows sparse
modulus maxima.
20
g 40
a 60
 80
100
1 120
o 20 40 60 80100120
5 Scale=2
Scale=4
w
120 X .
20 40 60 80100120
Scale=8
20
40
60
80
100
120
Tumor Boundary
Drawn by Radiologist
20 40 60 80100120
Original Image
Modulus Maxima Images for ROI 2.6
Figure 3.13
38
Modulus Images Horizontal Details Vertical Details Approximation
20 40 60 80100 20 20 40 60 80100 20
20
40
60
80
100
120
20 40 60 80100 20 20 40 60 80100 20
Scale=2 Scale=4
20 40 60 80100 20
Scale=8
Approximations and Details Images for ROI 2.6
Figure 3.14
20 40 60 80100 20 20 40 60 80100 20 20 40 60 80100 20
Seale=2 Scale=4 Scale=8
Modulus Images for ROI 2.6
Figure 3.15
39
Direction Images
20 40 60 80100120 20 40 60 80100120
Scale=2 Scale=4
Direction Images for ROI 2.6
Figure 3.16
40
3.4 Tumor on the Leg of a Dog
The modulus maxima images show a general agreement with the tumor boundary
indicated by the radiologist with the exception that the tumor seems to extend down
the left side of the image. The area of the tumor is characterized by no modulus
maxima. In this case, the amount of modulus maxima doesnt increase as the scale
increases.
Cft
0)
Detail 3.17
Scale=4
Scale=B
Tumor Boundary
Drawn by
Radiologist
20 40 60 80100120
Original Image
Modulus Maxima Images for ROI 2.8
Figure 3.17
41
Modulus Images Horizontal Details Vertical Details Approximation
20
40
100
120
20 40 60 80100 20
* ~ ...
20 40 60 80100120
20 4;* '*
40
60
80
100
120 .....
20 40 60 80100120
Scale=2
40
80
100
12oB^^^^B
20 40 60 80100120
20 40 60 8010Q20
Scale=4
2oBBB
60
120
20 40 60 80100 20
20 40 60 80100 20
Scale=8
Approximations and Details Images for ROI 2.8
Figure 3.18
20
40
60
80
100
120
20 40 60 80100 20
Scale=2
20
80 ^^^BSB
100
20 40 60 8010020
Scale=4
20
40HP!W
60^BÂ§S
1o^^B^B
120
20 40 60 80100 20
Scale=8
Modulus Images for ROI 2.8
Figure 3.19
42
Direction Images
20 40 60 80100120 20 40 60 80100 20 20 40 60 80100 20
Scale=2 Scale=4 Scale=8
Direction Images for ROI 2.8
Figure 3.20
43
4. Conclusion
A common thread in all the enhanced images is the lack of modulus maxima in
regions that generally conform to tumors identified by a radiologist. Tumors are
unorganized tissues and because of this, it is possible there is more uniformity in
their ability to attenuate xrays. This uniformity would tend to reduce the amount
of sharp transitions in pixel intensity. The more differentiated healthy tissues
would have more connective, vascular, and other specialized components all of
which would effect the attenuation of xrays and create more transients. Because
the boundaries between the tumors and the surrounding tissue are difficult to
determine on unenhanced images, the modulus maxima images may have increased
the amount of knowledge about the tumor boundaries. The next step to prove the
reliability of the technique will be to perform microscopic examinations of tissues
that are low in modulus maxima to confirm the correlation with malignant tissues.
44
References
[1] S.L. Robbins and R.S. Cotran, Pathologic basis of disease, W.B. Saunders
Company, Philadelphia, Pennsylvania, 1979.
[2] J.A. del Regato and H.L. Spjut, Cancer diagnosis, treatment, and prognosis,
C.V. Mosby Company, St. Louis, Missouri, 1977.
[3] E. Seeram, Computed tomography: physical principles, clinical applications &
quality control, W.B. Saunders Company, Philadelphia, Pennsylvania, 1994.
[4] J. Canny, A computational approach to edge detection, IEEE Transactions on
Pattern Anaysis and Machine Intell., vol 8, no. 6, pp. 9611005, September 1986.
[5] S. Mallat and S Zhong, Characterization of signals from multiscale edges,
IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 14, no. 7,
pp. 710732, July 1992.
[6] S. Mallet and W.L. Hwang, Singularity detection and processing with
wavelets, IEEE Transactions on Information Theory, vol. 38, no. 2, pp. 617642,
March 1992.
[7] S. Mallat, A wavelet tour of signal processing, 2nd edition, Academic Press,
London, 1998.
45

Full Text 
PAGE 1
WAVELET ANALYSIS OF IMAGES OF MALIGNANT TUMORS by Samuel Burns Stewart B.A., West Virginia University 1974 A thesis submitted to the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Master of Science Electrical Engineering 2002
PAGE 2
This thesis for the Master of Science degree by Samuel Burns Stewart has been approved by Miloje Radenkovic
PAGE 3
Stewart, Samuel Burns (M.S., Electrical Engineering) Wavelet Analysis of Computed Tomography Generated Images ofMalignant Tumors Thesis directed by Professor Jan T. Bialasiewicz ABSTRACT Wavelet analysis of veterinary images generated by computed tomography is used to enhance certain image features. Malignant tumors often are difficult to define in terms of their extent. Wavelet analysis using wavelets based on Gaussian functions is capable of differentiating tissues by characterizing the local regularity of the images. Gaussian functions and their derivatives are used to calculate the modulus maxima of localized regions using multiscale analysis. Modulus maxima are used to indicate differences in image characteristics such as regularity and important features such as edges and transients. This abstract accurately represents the content of the candidate's thesis. I recommend its publication. Si 111
PAGE 4
ACKNOWLEDGMENT My thanks to Dr. Susan Kraft of the College of Veterinary Medicine, Colorado State University for providing the images of tumors used in this research.
PAGE 5
CONTENTS Figure .... .......... ................. ..... ............... ................... .... . ......... ............. . .... . . .... vii Chapter 1. Introduction .... . .... . . ........................................................... .......... ...... . . .... . 1 1.2 The Nature of Cancerous Tissues ..................................... .... ......... . .... . . ...... 1 1.3 The Imaging System . ..... .......... .... .... ............ ... ........................................... 2 1.4 Problem Statement . ....... .... . ..... . .... ............................................................... 4 1.5 Description ofWavelets . .......... . ....... . ....... . ... ... . ......................................... 7 1.5. 1 Discrete Wavelet Conventions ........................................................................ 8 1.5.2 Calculating Wavelet Transforms ....... ................................. .......................... 8 1.5.3 Wavelet Dilation .... ..... . .... .... .... .... . .... ................ . . . ..... .... . ..... .... . ............ 9 1 5.4 Wavelet Reconstruction ....... .... ....... . . . . .... .... .... . . . . . .... ..... .................. 11 1 6 Previous Research . .... . . ....... .... .... .... . . . ......... .... .............. ........................ 11 1. 6.1 Wavelets Based on the Gaussian Function ................................... . ......... . . 11 1.6.2 Calculating Modulus Maxima .................. .... ....... . .... ....... . ...... .................. 13 1.6.3 Function Regularity from Modulus Maxima at Different Scales ............. .... 15 2. Methods and Materials .......... .... ....... . ..... . . .... .... ............... . .......... . ..... . ..... 17 2 1 Regions of Interest ....... ......... .... .... . ................. . . ...................... . . .... ....... 17 2 2 Algorithm Development .... .... ....... . ............ .... .... . . .... .... ...................... .23 2.2 1 Computing Wavelet Transforms Using Circular Convolution ............ .... .25 v
PAGE 6
2.2.2 Computing Dilated Wavelets Using Algorithm a Trous .... .... ........ ........ . .27 2.2 3 Creating the Direction, Modulus and Threshold Modulus Maxima Images .................. . .... ..................................... . . . . . . . . ....... .... . . ..... ...... 28 3. Image Analysis .............................................. . ....... . . . . .... . . .... . . . ..... ..... 30 3.1 Image Analysis of Sarcoma on the Jaw of a Dog .... . . . . . .... ...... . . . . . .... 30 3.2 Image Analysis of a Soft Tissue Sarcoma on the Flank of a Dog ............... 35 3.3 Image Analysis ofNasal Tumor in a Dog .... . .... . ..... .... .............. . . .... ... 38 3.4 Tumor on the Leg of A Dog ..... ............................................................ .... .41 4. Conclusion ................. ..... ........... .................................................................. 44 References ... . .... .... ...... . .... ..... .... ...... ..... ...... ....... ................. . .... . ........ . ......... 45 V l
PAGE 7
FIGURES Figure 1 1 Sarcoma Near the Jaw of a Dog . . .... ............. ........... .... . .... ...... . .... . . . ...... 5 1.2 Regions oflnterest ... . . . . . ....... ................ . . . .... ........ .... . .......................... 6 1.3 Plot ofPixel Intensity from Detail1.2.2 in Figure 1.2 .... . ......... .... .... . ......... 7 1.4 Plot of Pixel Intensity from Detail 1.2.1 in Figure 1.2 .... . ........ ....... . . . . .... 7 1.5 Quadratic Spline and Dilated Quadratic Spline ...................... . .... ..... . . . . .... 9 1.6 Fourier Transform, Quadratic Spline ........ . . . . .... ............ . .... ....... . . . . . .... 10 1.7 Fourier Transform ofDilated Quadratic Spline .......... .............. .......... ..... .... 10 1.8 Cubic Spline and Quadratic Spline ............................................................... 12 2.1 SarcomaNeartheJawofaDog ......................... ..... . . ................................ 18 2.2 Regions of Interest 2.2.1 and 2.2.2 ... .......... ....... . . . . . .... .......... . . . .... . .... . 19 2.3 Soft Tissue Sarcoma of the Flank of a Dog ................. ......... .... ............. ..... 20 2.4 Region of Interest 2.4 . . ..... . ...... ... . . . . . . ..... ..... ..... . . .... .... ........ ........... .... 20 2.5 Nasal Tumor of a Dog . ....... .................................... . . ......... . .... . . . ..... ..... 21 2 6 Region of Interest 2.6 ........ . . ..... . ............ . . . . . . ..... . .... . .... . ............ . .... 21 2. 7 Tumor on the Leg of a Dog . ..... . .... . . ..... . ..... . . ....... ........................ ....... 22 2.8 Region of Interest 2.8 ............................... ....... . . . . ............... . .... ............... 22 2.9 Outline of Algorithm for Computing Modulus Maxima Images at Different Scales .... ...... . ............. . .... ......... . . .... . . . ... .............. .......... .... . ..... . ...... .... 23 Vll
PAGE 8
2.10 Approximation and Detail Images .. .. ... . .... ....... . .... . . .............. ................. 24 2.11 Direction Images ..... ...... . . . . . . . . .... ........ ............................................. .... 24 2.12 Modulus Images . . ..... .... . . . . . . .... ................................... . . . . ... ... . .... .... 25 2 .13 Modulus Maxima Images .... ..... . .... ..... .... . .... . . . . .......... ..... . . .................. .25 2.14 Impulse Response of Filters H (OJ) and G( w) .......................... ....... .......... .26 2 .15 Circular Convolution Algorithm ..... ... .. .. .. ................ . . ..... .... ..... . . .............. .27 2 .16 Algorithme a Trous ..... ...................................... . ..... . ..... . ............ ... . ........ 28 2 .17 Subroutine for Creating Image Direction Matrix ...... . .......... . . . ... ... . . .... . .28 2.18 Subroutine for Creating Image Modulus Matrix ..... . ..... .... . ..... ..... . . .... . .28 2 .19 Subroutine for Calculating Image Modulus Maxima Matrix ....................... 29 3.1 Modulus Maxima Images for ROI 2.2.2, Tumor ....... ..... ........ ................... 31 3.2 Modulus Maxima Images for ROI 2.2.1, Healthy Tissue ............................. 31 3.3 Approximations and Details Images for ROI 2.2.2 Tumor ..... .......... .... . ... 32 3.4 Approximations and Details Images for ROI 2.2.1 Healthy Tissue ......... ... 33 3.5 Modulus Images for ROI 2.2.2 Tumor ..... ........ ...... . ..... ............................ 33 3.6 Modulus Images for ROI 2.2.1 Healthy Tissue ...... . ..... ........ .............. .... 34 3 7 Direction Images for ROI 2 2.2 Tumor .................. . ..... . . .................. ...... 34 3.8 Direction Images for ROI 2.2.1 Healthy Tissue ..... . ....... . . ................ .... . 34 3 9 Modulus Maxima and Original Image for ROI 2.4 ........ . . .......... .............. .35 3.10 Approximations and Details Im a ges for ROI 2.4 .... . . . .... . . . .... ........... . . 36 Vlll
PAGE 9
3 .11 Modulus Images for ROI 2.4 . .... .............. . . . ................. . ........................ 36 3.12 Direction Images for ROI 2.4 ....................................................................... 37 3.13 Modulus Maxima Images for ROI 2.6 . .............. . .... .... . . . .... .... ....... .... . .38 3.14 Approximations and Details Images for ROI 2.6 ......................................... 39 3.15 Modulus Images for ROI 2.6 ....................................................................... .39 3.16 Direction Images for ROI 2.6 ...................................................................... .40 3.17 Modulus Maxima Images for ROI 2.8 ..................................................... .... .41 3.18 Approximations and Details Images for ROI 2.8 ............................... ......... .42 3.19 Modulus Images for ROI 2.8 ....................................................................... .42 3.20 Direction Images for ROI 2.8 ................. . ...... . . ..... ....... ...... . .... ......... .... . .43 IX
PAGE 10
1. Introduction This dissertation describes the use of wavelet analysis applied to the problem of image enhancement. Image enhancement is the processing of images to improve their appearance to the viewer. Although image quality and overall appearance of the image can be goals of enhancement the focus is to improve the intelligibility of the image. The data set studied is medical images of dogs with various types of cancers. The result of the image enhancement is to better define the shape and extent of malignant tissue. The decision to use wavelets to enhance medical images is based on the nature of cancerous tissues and on the process used to image the diseased and healthy tissue. 1.2 The Nature of Cancerous Tissues The basic ideas in the following discussion of cancerous tissues is adapted from [1] and [2]. The terms malignant neoplasm and cancer have the same meaning. Neoplasms are characterized as an abnormal mass of tissue that is uncoordinated with normal tissue. The mass has no purpose and competes with normal tissues for nutrients. Malignant neoplasms have the potential to grow rapidly, destroy contiguous tissues and disseminate throughout the body leading to death. Tumors are swellings that may have benign causes and are not necessarily related to cancer. However the term tumor will be equated with cancer in this discussion. The images studied in this thesis are of malignant neoplasms. Sarcomas are cancers 1
PAGE 11
arising from mesenchymal (fleshy) tissue. Often tumors are composed of poorly differentiated cells with a close resemblance to each other but there may be more than one cell type. There are basically two components to a tumor the proliferating neoplastic cells called the parenchyma and supportive tissues including connective tissue and blood vessels called stroma In general, the bulk of the tumor is the parenchyma cells. The nuclei of the cells are disproportionately large The undifferentiation of the tumor cells is referred to as anaplasia The degree of anaplasia and evidence of invasion of surrounding tissue are the major criteria by which a diagnosis of cancer is made microscopically Breast cancer is an example of a tumor with a large stroma component. Because this type of tumor is quite visible on mammograms and because the tumors are often benign there is a substantial volume of research in breast cancer image enhancement with the goal of differentiating benign from malignant tumors. Soft tissue tumors such as sarcomas may be much more difficult to differentiate from healthy tissues using medical imaging. To summarize, malignant neoplasms have certain physical differences to healthy tissues, in particular, a higher degree of disorganization and a higher nuclear content. 1.3 The Imaging System The basic ideas presented in the following discussion are adapted from [3]. The process used to create the images used in this study is computed axial tomography 2
PAGE 12
or CAT, which is sometimes referred to as computed tomography or CT. The term CT scan will be used to refer to the process of creating the images using xrays in the process of computed tomography or to the actual image created using computed tomography The basic process of computed tomography is to measure the attenuation ofxrays through the patient and generate an image from the attenuation data. The CT machine generates a highly collimated beam ofxrays that only passes through a cross section or volume of the material to be imaged After passing through the patient the modified xrays are measured by special detectors that make quantitative measurements of xray intensity. The data generated is called relative transmission. Relative transmission is the log of the ratio ofthe xray intensity at the source with the intensity at the detector. This data is processed into an image that can display very small differences in tissue density The data can be acquired in two ways. The first method is to rotate the xray source and detector around the patient one slice at a time The patient is moved to the next position and another scan is made. This process generates enough data to build a three dimensional image of the patient. A helical scan process can also acquire the data where a volume of tissue is scanned. The relative attenuation of the x rays by the tissues is dependent upon the density of the tissue. Different tissues have different linear attenuation coefficients f1 which is a measure of how 3
PAGE 13
much attenuation has occurred. Under certain operating parameters bone has a J1 of0.528, fat has a J1 of0.185, and blood has a J1 of0.208. These values of J1 are relative values because the absolute value of J1 depends on a number of factors besides the density of the tissue, for example, the energy of radiation. 1.4 Problem Statement Based on the fact that tumors tend to be physically different from healthy tissues and that CT imaging can differentiate between different tissue types, it is suggested that CT images will contain enough information to differentiate between healthy and diseased tissue. The difference may not be apparent upon visual examination of an image. For example, Figure 1.1 is a slice from aCT scan of a sarcoma near the jaw of a dog. Detail 1.1 shows the boundary of the tumor as defined by a radiologist. The image appearance inside the outline is very similar to the tissue outside the tumor. This indicates very subtle differences between the healthy and diseased tissues. Figure 1.2 shows two regions of interest in boxes with the left box enclosing healthy tissue and the right box enclosing the tumor Detail 1.2.1 labels a segment of a row in the healthy tissue and detail 1.2.2 labels a segment of a row in the tumor. Figures 1.3 and 1.4 show a plot of pixels intensities for each segment. Comparison of figures 1.3 and 1.4 shows the basic range of pixel intensities is similar which makes identifying malignant tissue using pixel intensities quite 4
PAGE 14
difficult. The problem is to design a system that can enhance images features that are not visually apparent. Sarcoma Near the Jaw of a Dog Figure 1.1 5
PAGE 15
Regions of Interest Figure 1.2 6
PAGE 16
100 ..... Ill c Q) 95 c Q) X 0... 90 0 100 ..... Ill c w 95 c Qi X 0::: 90 0 Row 62 Columns 55:119 from Detail1 2 2 in Figure 1 2 10 20 30 40 50 Columns Plot of Pixel Intensity from Detail1.2 .2 in Figure 1.2 Figure 1.3 Row 62 Columns 20:84 from Detail 1 2 1 in Figure 1 2 10 20 30 40 50 Columns Plot ofPixel Intensity from Detaill.2.1 in Figurel.2 Figure 1.4 1.5 Description of Wavelets 60 60 Since the range of intensity is insignificant, the transient behavior of the pixel intensities is used for image enhancement. Wavelets are used to locate and characterize transients in the images Fourier analysis can be used for characterizing transient behavior. However, Fourier analysis doesn't describe the location of singularities, it only provides the overall signal characteristics 7
PAGE 17
Wavelets are defined as functions whose integral is equal to zero. Wavelets are localized in both space and frequency This makes wavelets particularly useful for characterizing the local regularity of a signal. 1.5.1 Discrete Wavelet Conventions The space domain designated as either x or n is used in this discussion The letter n can refer to a single value when discussing onedimensional signals or as an abbreviation for n1, n2 in the context of two dimensions where n1 refers to a row of pixels in an image and n2 refers to a column of pixels. Discrete signals are designated asj[n], discrete smoothing functions as B[n] and discrete wavelets as '!'[ n]. The Fourier transforms of the same signals are designated j ( (1)) B( OJ), and ifJ(w). All signals analyzed and used for analysis will be discrete. 1.5.2 Calculating Wavelet Transforms A wavelet transform of a signal j[ n ] is calculated by convolving the wavelet with the signal and is designated Wf ( n) The convolution of a wavelet with a signal to form a wavelet transform can be written +OO Wf(no) = Lf[n]'l'[n0 n] (1.1) n =oo which is equivalent to the inner product designation Wf(n0 ) = (f(n),lfl[n0 nJ) (1.2) 8
PAGE 18
1.5.3 Wavelet Dilation Wavelets are dilated using a scaling factor s. A dilated wavelet is designated lf/ s (x) where xis the time or space variable and sis the scaling factor. The wavelet is scaled as follows: 1 lf/Jn] = lj/(n/ s) s (1.3) Figure [1.5] compares a quadratic spline with a dilated quadratic spline scaled by a factor of s = 2 The Fourier transforms of the functions shown in Figure [1.5] are shown in Figures [1.6] and [1.7] The effect of the dilation is to lower the frequency content of the signal. 0 .2 Q) o ::I 0 t:: OJ rn 0. 2 Quadratic Spline Dilated Quadratic Spline 0.2 Q) o ::I 0 t:: Cl rn 0. 2 0. 4 5 0 Space 5 10 10 5 0 Space Quadratic Spline and Dilated Quadratic Spline Figure 1.5 5 10 The scaling factor often used for the image analysis is based on the algorithme a trous Starting with coefficients of the filters ze ros are inserted between the 9
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coefficients using a dyadic sequence of 2j 1 for each scale. This form of dilation lends itself to fast computational algorithms. Each scale of the wavelet gives a different level of detail of the signal. 0.8 Ql g 0.6 ..... c: 0.2 Fourier Transform, Quadratic Spline 0 8 Ql g 0 6 ..... c: 0 2 0.08 0.06 0.04 0.02 0 0.02 0 .04 0 .06 0.08 Frequency in Radians Fourier Transform, Quadratic Spline Figure 1.6 Fourier Transform, Dilated Quadratic Spline 0.08 0.06 0.04 0.02 0 0 .02 0 .04 0 .06 0 .08 Frequency in Radians Fourier Transform of Dilated Quadratic Spline Figure 1.7 10
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1.5.4 Wavelet Reconstruction Wavelet transforms calculated at different scales, Wf(s,n), can be analyzed, modified appropriately, and then used to reconstruct the original image in a different form. The reconstruction process recovers the original signal as follows +OO +OO f[n] = I I(/[ n ],Vf s [n J)V/s [n] (1.4) s=oon=oo Wavelets are used to generate structures called modulus maxima. Modulus maxima provide information about the regularity of a signal and can be used to enhance transients in a signal. 1.6 Previous Research The wavelet analysis algorithms described in the materials and methods section and the discussion of function regularity as related to modulus maxima are based on the research and ideas presented in [4], [5], [6], and [7] 1.6.1 Wavelets Based on the Gaussian Function The derivative of a Gaussian function has been shown to be useful in image enhancement. The Gaussian function is defined as n2 1 2 2 G[n]= e u a.fi; (1. 5) and its first derivative, which is used for calculating wavelet transforms, is 11
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n z G'[n] = n e2oz a3J2; The Fourier transform of the Gaussian function is G(w) = (sin(w/2))3 w I 2 and the Fourier transform of its first derivative is VJ(w) = (sin(w/ 4))4 wl 4 (1.6) (1.7) (1.8) G[ n] and G'[ n] will be referred to respectively as cubic spline and quadratic spline functions. Their graphs are shown in figure 1. Q) "U ::::! 0 2 Cubic Spline Q) "U ::::! Quadratic Spline 0.2 0 f==::::1 0 1=!:..\;l C') Cll 0. 2 C') Cll 0. 2 0. 4 L._ ____ .___ ___ ___. 5 0 Space 5 Cubic Spline and Quadratic Spline Figure 1.8 12 0 Space 5
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1.6.2 Calculating Modulus Maxima Let a column or row of pixels from an image define a function f[ n] Let f[ n0 ] be a particular pixel in f[ n] The quadratic spline function is the derivative of the cubic spline. Let B[ n] represent the cubic spline function. Therefore dB[ n] dn denotes the quadratic spline function or the wavelet If/ s [ n] The wavelet transform of f[n] is Wf(s,n) = f[n] lj/.[n] where 0 denotes convolution. Substituting sdB.[n] for lf/Jn] in (1.9) gives dn d Wf(s,n) = s(f[n]B.[n]) dn Therefore the wavelet transform, Wf(s,n), is the first derivative ofthe signal (1.9) (1.10) smoothed at the scale s. A wavelet transform at n0 that is locally maximum so that 1Wf(s,n0 1)1 IWf(s,no)l I Wf(s,no + 1)1 (1.11) with at least a strict maximum in either the left or right neighborhood of n0 is a modulus maxima at that point. If a modulus maxima is located at n0 then oWf(s,no) = O on 13 (1.12)
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For a two dimensional system such as an image denote the wavelet transforms of the rows as W1 f(s,n) and the wavelet transforms of the columns as W2 f(s,n). Since the wavelet transforms are derivatives of the columns and rows they describe the gradient at a particular pixel ( n1 n 2 ) (1.13) where ii =(cosa, sina). For two dimensions the modulus of the gradient vector is denoted (1.14) The angle of the wavelet transform vector is ( ) l(W2 f(n,s)J an =tan 1 W f(n,s) (1.15) This angle represents direction of steepest descent of the pixel intensities. The modulus maxima in two directions is the local maxima of(1.14) in the direction normal to a. 14
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1.6.3 Function Regularity from Modulus Maxima at Different Scales For the following discussion a refers to the Lipschitz exponent. Suppose f is m times differentiable in [ vh, v + h] and Pv is the Taylor polynomial in the neighborhood of v m lf(k)( ) P v (t) = L v (t v)k k=O k. (1.11) The approximation error is ev(t) = f(t)Pv(t) and the mth order differentiability of f in the neighborhood of v yields an upper bound on the error ev (t) when t tends to v In other words, the precision of the approximation using a Taylor series is dependent on the order of the polynominal used to approximate the function. The order of the approximation is dependent on the differentiability of the function. The values ofk in the Taylor series are integer values and represent the order of the polynomial. The Lipschitz regularity gives the upper bound with noninteger exponents, in contrast with k, which can only have integer values which makes it a more general measure of the regularity of a function. If a function fin] is uniformly Lipschitz a in the interval [ a,b] then their exists A > O such that 15
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'v'(n,s) E [a,b]x9t+, j Wf(s,n)j Asa+l!Z (1.12) where 9t + denotes positive real numbers. Since the scale s varies by powers of 2, (1.12) is equivalent to (1.13) The Lipschitz regularity at a point is the slope of log2jWf(s,n)j as a function of logz s for modulus maxima that converges to a point as s goes to 1. For isolated singularities the modulus maxima that converge to a point n0 are located in cone such that jn nol Cs (1.14) Because the presence of modulus maxima indicate sharp transitions in pixel intensity, they are useful for detecting significant structures in images including edges. Per equation (1.13), modulus maxima are also useful for characterizing the local regularity of an image. The usefulness of wavelet analysis in characterizing the regularity of a signal is dependent on the number of vanishing moments for the wavelet. A wavelet with one vanishing moment, such as the quadratic spline, can only characterize functions that have Lipschitz regularity ofless than one. However for image analysis, especially transition and edge detection, a wavelet with one vanishing moment is sufficient. 16
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2. Methods and Materials The image enhancement process will consist of calculating the modulus maxima of five regions of interest over three scales. The data generated by wavelet analysis will be presented as images to allow direct comparison with the original images. For each region of interest the approximation, horizontal, vertical, direction, modulus, and modulus maxima images are displayed to allow comparison with the original images. 2.1 Regions of Interest Each image is displayed twice One image will have the tumor outlined by a radiologist and the second image will have regions of interest defmed Two regions of interest are marked on Figure 2.2. The left box encloses healthy tissue and the right box encloses the tumor and some of the surrounding tissue. The other three regions of interest in Figures 2.4, 2.6, and 2.8 were chosen to include both healthy and diseased tissue. 17
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Sarcoma Near the Jaw of a Dog Figure 2.1 18
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Regions of Interest 2.2.1 and 2.2.2 Figure 2.2 19
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Soft Tissue Sarcoma of the Flank of a Dog Figure 2.3 Region of Interest 2.4 Figure 2.4 20
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Nasal Tumor of a Dog Figure 2.5 Region of Interest 2.6 Figure 2.6 21
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Tumor on the Leg of a Dog Figure 2.7 Region of Interest 2.8 Figure 2.8 22
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2.2 Algorithm Development Algorithms were written in Matlab, version 5.3.1. The overall algorithm for computing the modulus maxima is outlined in Figure 2.9. Figures 2.10, 2.11, 2.12 and 2.13 are examples of the outputs generated by the algorithm. Algorithm inputs: regions of interest in 24 bit "bitmap" format. Approximate dimensions of the images are 128 by 128 pixels. All images are gray scale with 256 levels The threshold level used is 0.1 in all experiments. Algorithm outputs: Approximation, vertical horizontal, direction, modulus, and modulus maxima images at scales 2, 4, and 8 1. Scaled approximation, vertical, and horizontal details images are created 1.1 Compute approximation image 1.1.1 Convolving rows of image with smoothing filter 1.1.2 Convolving columns of image with smoothing filter 1.2. Compute vertical details by convolving columns of approximation image with wavelet filter 1.3. Compute horizontal details by convolving rows of approximation image with wavelet filter 1.4. Perform Algorithme a Trous on filters 1.5. Loop back to 1 to create next scaled image 2. Create the direction images, implementing equation 1 .15 2.1 Find the ratio of the vertical and horizontal details images, for each pixel 2.2 Compute the arctan ofthe ratios and save the result in the direction image 3 Calculate the modulus image, implementing equation 1.14 4. Find the threshold modulus maxima for the modulus image by comparing each pixel in the modulus image with the two pixels closest to normal in the angle image If the pixel is a local maxima and is greater than a threshold value then the pixel is not suppressed Outline of Algorithm for Computing Modulus Maxima Images at Different Scales Figure 2 9 23
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g 20 i 40 E 60 80 <( 120 20 a! 'til 40 0 60 80 E 100 > 120 (/) 40 c: 60 E 80 t5 100 Q) 5120 50 100 150 50 100 150 50 100 150 Scale=2 20 40 60 80 100 120 20 40 60 80 100 120 50 100 150 50 100 150 50 100 150 Scale=4 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 80 100 120 Approximation and Detail Images Figure 2 .10 50 100 150 S cale=2 20 40 60 80 100 120 50 100 150 Scale=4 Direction Images Figure 2.11 24 20 40 60 80 100 120 50 100 150 50 100 150 50 100 150 Scale=8 50 100 150 Scale=8
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(/) Q) 20 t 40 60 (/) ;:, 00 :g 100 120 50 100 Scale=2 50 Scale=2 150 20 40 60 00 100 120 50 100 Scale=4 Modulus Images Figure 2.12 Scale=4 150 Modulus Maxima Images Figure 2.13 20 40 60 00 100 120 2.2.1 Computing Wavelet Transforms Using Circular Convolution 50 100 150 Scale=S The wavelet transforms are calculated by circular convolution of the rows and columns of the regions of interest with discrete filters with finite impulse response. The transfer functions used to derive the filters [7] are (2.1) 25
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G( m) = ie im 1 2 sin( m I 2) (2.2) The corresponding finite impulse responses of the filters H ( m) and G( (1)) are show in Figure 2.14. n h[n] g[n] 1 0.125 0 0.375 1 0.375 2.0 2 0.125 2.0 Impulse Response of Filters H ( m) and G( (1)) Figure 2.14 Let x[n] = I(:,1) denote all rows and column 1 in an image. Let h[n] denote a finite impulse response filter. Let N denote the period of both x[n] and h[n] The circular convolution ofx[n] and h[n] is defined as N1 x[n] h[n] = Ix[n]h[ni] i = O (2.3) where denotes convolution. If the length ofx[n] island the length ofh[n] ism, then x[ n] and h[ n] are padded with m + l 1 zeros The result of the linear convolution of the zero padded signals is the same finding the inverse Fourier 26
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transform of .X[ (J) ]h [ (J)] The convolution of the signal and filter is accomplished using the function cc.m shown in Figure 2 15. 2.2.2 Computing Dilated Wavelets Using Algorithme a Trous Dilated wavelets are created by inserting 2 j1 zeros between each filter value. The algorithm used is in Figure 2.16 function yout=cc(h,x) %cc.m %x=signal %h=filter %y out =filtered signal N=length(x); M=length(h); xpadded=zeros( 1 ,N + M 1); xpadded(1 :N)=x; hpadded=zeros(l ,N+M1 ); hpadded(1 :M)=h; % forn=O:N1 for i=O:N1 ifO < (ni+N+ 1) & (ni+N+ 1)<=N m=1; else m=O; end y(i+ 1,n+ 1)=xpadded(i+ 1)*hpadded(ni+m*N+ 1); end end yout=sum(y); % This line sums the rows in the matix y Circular Convolution Algorithm Figure 2.15 27
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hd=[dyadup(hd,2) 0]; Algorithme a Trous Figure 2.16 2.2.3 Creating the Direction, Modulus, and Threshold Modulus Maxima Images Figures 2.17, 2.18, and 2.19 contain the algorithms for computing the direction, modulus, and the threshold modulus maxima images. A threshold level of 0 1 was used in all experiments. [ nr,nc ]=size(WaveletTransform Vertical); ratio= WaveletTransform Vertical(r,c )./W aveletTransformHorizontal(r,c ); for r=1:nr for c=1 :3*nc ifratio(r,c)>=O angle(r,c )=atan(ratio(r,c) ); else angle(r,c )=piatan(ratio(r,c) ) ; end end end Subroutine for Creating Image Direction Matrix Figure 2.17 Modulus=sqrt(WaveletTransform Vertical. *WaveletTransform Vertical+ .. WaveletTransformHorizontal. *WaveletTransformHorizontal) Subroutine for Creating Image Modulus Matrix Figure 2.18 28
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inc=pi / 8; for r=2:nr1 for c = 2:3*nc1 if(angle(r, c) >= (15*inc)&angle(r,c)<=(1 *inc))&modulus(r,c)>=Threshold & ... modulus (r c) >= modulus (r1,c)&modulus (r c) >= modulus (r+ 1 c) tmodulus (r,c )=0; elseif(angle(r,c) >=(1 *inc)&angle(r c) <= (3*inc))& modulus (r c)>=Threshold & modulus g(r,c)>= modulus (r1,c1) & modulus (r c) >= modulus (r+ 1,c+ 1) tmodulus (r,c) =O; elseif(angle(r,c) >= (3*inc)& angle(r,c) <= (5*inc))&maxmag(r,c)>=Threshold & maxmag(r c ) > =maxmag(r,c + 1) & maxmag(r,c) >= maxmag(r c1) tmaxmag(r,c )=0 ; elseif(angle(r c) >= (5*inc)&angle(r c) <= (7*inc))& modulus (r c)>=Threshold & modulus (r,c) >=m odulus(r1 c+ 1) & modulus(r c) >= modulus(r+ 1,c1) tmodulus(r,c ) = 0 ; elseif ( angle(r,c ) >= (?*inc )&angle(r c ) <= (9*inc) )&modulus(r,c )>=Threshold & modulus(r c) > =modulus(r1 c) & modulus(r c) >= modulus(r + 1,c) tmodulus(r,c )=0; elseif(angle(r c) >= (9*inc)&angle(r c) <=(11 *inc))&modulus(r c)>=Threshold & modulus(r c) > =modulus(r1,c1) & modulus(r c) >= modulus(r + 1 c+ 1) tmodulus(r c )=0; elseif ( angle(r,c )>=(11 *inc )&angle(r,c ) <= (13*inc ))&modulus(r c )>=Threshold & modulus(r c ) > =modulus(r c1) & modulus(r c) >= modulus(r c + 1) tmodulus(r,c )=0 ; elseif(angle(r,c)>=(13*inc)&angle(r,c)<=(15*inc))&modulus(r,c)>=Threshold & modulus(r c) > =modulus(r1 ,c+ 1) & modulus(r c) >= modulus(r + 1 c1) tmodulus(r c ) = 0 ; end end end Subroutine for Calculating Image Modulus Maxima Matrix Figure 2.19 29
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3. Image Analysis Modulus maxima, approximation, vertical details, horizontal details, modulus, and direction images are included for each region of interest. Modulus maxima images have details labeled to indicate possible features that are revealed by the analysis. 3.1 Image Analysis of Sarcoma on the Jaw of a Dog The original image is characterized by a visually uniform pixel intensity, which makes defining the boundaries of the tumor difficult. The only morphological difference between the left and right side of the jaw is a noticeable bulge in the region of the tumor. In figure 3.1, the modulus maxima image at scale 2 reveals an area that generally conforms to the shape of the tumor on the right side of the image. This area is characterized by few maxima and extends from row 20 to row 80 between columns 90 to 110. This would be consistent with intensity transitions that are gradual. The modulus maxima image at scale 4, which is obtained with a wavelet with lower frequency content shows a better correlation and displays more maximal in the same region. An important observation is the number of maxima is increasing in parts of the tumor area as the scale is increasing, while the modulus maxima in Figure 3.2 is decreasing as the scale increases. The approximations, details, and modulus images do not show remarkable differences between healthy tissue and the tumor. The direction images for the tumor show some conformance with the shape of the tumor. 30
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(I) Q) C) (1) 20 {'0 40 E x 60 (1) 80 (I) 100 ::;, 120 :; "0 0 20 40 60 80 10()20 Scale=2 Oetail3 1 20 40 60 80 100 120 20 40 60 8010020 Scale=4 20 40 60 8010020 Original Image Modulus Maxima Images for ROI 2 2.2, Tumor Figure 3.1 20 40 60 80 10()20 Scale=2 20 40 60 80 100 120 20 40 60 80 100 120 20 40 60 0010020 Scale=4 20 40 60 8010020 Original Image 20 40 60 8010020 Scale=B Modulus Maxima Images for ROI 2.2.1, Healthy Tissue Figure 3.2 31
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20 20 40 40 40 E6() 60 60 00 00 00 a 100 100 100 120 120 20 40 60 0010020 20 40 60 0010020 20 40 60 0010020 20 40 60 0010020 20 40 60 0010020 20 40 60 0010020 Scale=2 Scale=4 Scale=8 Approximations and Details Images for ROI 2.2.2, Tumor Figure 3.3 32
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g20 20 20 40 40 40 E60 60 60 80 80 80 100 100 120 120 120 20 40 60 8010020 20 40 60 8010020 20 40 60 8010020 20 40 60 8010020 20 40 60 8010020 20 40 60 8010020 Scale=2 Scale=4 Scale=8 Approximations and Details Images for ROI 2.2 .1, Healthy Tissue Figure 3.4 Modulus Images for ROI 2.2 2, Tumor Figure 3.5 33
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(I) &20 20 20 40 40 40 (1)60 60 60 .280 80 80 100 100 120 120 (I) 20 40 60 8010020 20 40 60 8010020 20 40 60 8010020 Scale=2 Scale=4 Scale=8 Modulus Images for ROI 2.2.1, Healthy Tissue Figure 3.6 &20 20 20 40 40 c; 60 60 60 .!2 80 80 80 100 100 100 i5 120 120 120 """"'"'_,.lll' 20 40 60 8010020 20 40 60 8010020 20 40 60 8010020 Scale=2 Scale=4 Scale=8 Direction Images for ROI 2.2.2, Tumor Figure 3.7 Direction Images for ROI 2.2.1, Healthy Tissue Figure 3.8 34
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3.2 Image Analysis of a Soft Tissue Sarcoma on the Flank of a Dog In this case, the wavelet analysis suggests the tumor extends beyond the boundary suggested by the radiologist. As in the sarcoma on the jaw, the region of the tumor is characterized by practically no modulus maxima. However, as the scale increases, the modulus maxima don't increase. The direction images, especially scales two and four show a general conformance with the shape of the tumor. 20 40 60 0010020 Scale=2 Detai13. 9 20 40 60 80 100 120 20 40 60 00 100 20 40 60 0010020 Scale=4 120 ..__. ______ __, 20 40 60 8010020 Original Image 20 /. ...:_..40 ... : = . .... :'i 60 1 ..... 80 ' / .....:. .:._ ...... ,... ... ) .,.. ... :" 100 .. I :::.).. 120 If J 20 40 60 0010020 Scale=S Boundary Drawn by Radiologist Modulus Maxima and Original Image for ROI 2.4 Figure 3.9 35
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520 20 20 40 40 40 E60 60 60 80 80 80 100 100 <120 120 120 (I) 2040 60801(020 2040 60801(020 2040 60001(020 20 40 60 80 100 120 L.....1 20 40 60 80 100 120 L...____ ____. 20 40 60 80 1C020 20 40 60 80 1(020 20 40 60 80 1C020 20 40 60 8010020 20 40 60 80 1(020 20 40 60 0010020 Scale=2 Scale=4 Scale=8 Approximations and Details Images for ROI 2.4 Figure 3.10 &20 20 20 40 40 40 (1)60 60 60 = 80 80 80 100 100 120 120 20 40 60 0010020 20 40 60 8010020 20 40 60 8010020 Scale=2 Scale=4 Scale=B Modulus Images for ROI 2.4 Figure 3.11 36
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(J) 20 20 40 40 40 c60 60 60 000 00 00 100 100 5120 120 120 20 40 60 0010020 20 40 60 0010020 20 40 60 0010020 Scale=2 Scale=4 Scale=S Direction Images for ROI 2.4 Figure 3.12 37
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3.3 Image Analysis of Nasal Tumor in a Dog The region defined by the tumor is characterized by no modulus maxima The tissue outside the tumor boundary in the lower half of the image also shows sparse modulus maxima 20 40 60 80 100 120 LL..lo..l'""'..:.:..U Scale=2 20 40 60 8010020 Scale=4 20 40 tTumor Boundary 60 Drawn by Radiologist 80 100 120 20 40 60 80100120 Original Image Modulus Maxima Images for ROI 2.6 Figure 3.13 38
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c: 20 20 20 E (1j 40 40 40 E 60 60 60 x 0 80 80 80 .... 0.100 100 100 120 120 20 40 60 8010020 2040 608010020 20 40 60 8010020 20 20 20 a! 40 40 40 Q) a 60 60 60 (ij 80 80 80 (.) ;:: 100 100 100 120 120 120 20 40 60 8010020 20 40 60 8010020 20 40 60 8010020 .!!!. c; 20 20 20 Q) 40 40 40 a 60 60 60 c: 80 80 80 0 100 100 .!::! 100 0 120 ::r: 120 120 20 40 60 8010020 2040 608010020 20 40 60 8010020 Scale=2 Scale=4 Scale=8 Approximations and Details Images for ROI 2.6 Figure 3.14 (/) &20 20 20 40 40 40 (1)60 60 60 .2 80 80 80 100 100 120 120 20 40 60 80100120 20 40 60 8010020 20 40 60 80100120 Scale=2 Scale=4 Scale=8 Modulus Images for ROI 2.6 Figure 3.15 39
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(/) 20 20 e 40 4o 4o c: 60 60 60 080 80 80 1 00 1 00 1 00 Nl'::!.ll!lt 6120 120 120 20 40 60 8010020 20 40 60 8010020 20 40 60 8010020 Scale=2 Scale=4 Scale=8 Direction Images for ROI 2.6 Figure 3.16 40
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3.4 Tumor on the Leg of a Dog The modulus maxima images show a general agreement with the tumor boundary indicated by the radiologist with the exception that the tumor seems to extend down the left side of the image. The area of the tumor is characterized by no modulus maxima. In this case the amount of modulus maxima doesn t increase as the scale mcreases. Scale=2 Detail3.17 20 40 60 80 100 120 20 40 80 80 100 120 20 40 so 80100120 Scale=4 20 40 so 80100120 Original Image 20 40 ( 100 .. 120. . = 20 40 60 80 100120 Scale=8 Tumor Boundary Drawn by Radiologist Modulus Maxima Images for ROI 2.8 Figure 3.17 41
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. 2 20 20 20 40 40 40 E60 60 60 00 00 00 100 100 100 120 120 2040600010020 2040600010020 2040608010020 (J) a; 20 20 20 40 40 (ij 60 60 60 c:oo oo oo 0 t! 100 100 100 120 120 (J) 20 40 60 8010020 20 40 60 0010020 20 40 60 0010020 Scale=2 Scale=4 Scale=8 Approximations and Details Images for ROI 2.8 Figure 3.18 20 20 _g 40 40 40 (1)60 60 60 .2 00 80 00 100 100 120 120 20 40 60 0010020 20 40 60 0010020 20 40 60 8010020 Scale=2 S c ale=4 Scale=8 Modulus Images for ROI 2.8 Figure 3.19 42
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(/) 20 20 40 40 40 c60 60 60 80 80 100 100 100 c5 120 120 120 .:::::... ::::..._.......o 20 40 60 8010020 20 40 60 0010020 20 40 60 8010020 Scale=2 Scale=4 Scale=8 Direction Images for ROI 2.8 Figure 3.20 43
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4. Conclusion A common thread in all the enhanced images is the lack of modulus maxima in regions that generally conform to tumors identified by a radiologist. Tumors are unorganized tissues and because of this, it is possible there is more uniformity in their ability to attenuate xrays. This uniformity would tend to reduce the amount of sharp transitions in pixel intensity. The more differentiated healthy tissues would have more connective, vascular, and other specialized components all of which would effect the attenuation ofxrays and create more transients. Because the boundaries between the tumors and the surrounding tissue are difficult to determine on unenhanced images, the modulus maxima images may have increased the amount of knowledge about the tumor boundaries. The next step to prove the reliability of the technique will be to perform microscopic examinations of tissues that are low in modulus maxima to confirm the correlation with malignant tissues. 44
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References [1] S.L. Robbins and R.S. Cotran, Pathologic basis of disease W.B. Saunders Company Philadelphia Pennsylvania 1979. [2] J .A. del Regato and H.L. Spjut Cancer diagnosis treatment and prognosis, C.V. Mosby Company, St. Louis Missouri, 1977. [3] E. Seeram Computed tomograph y : physical principles clinical applications & quality control W.B. Saunders Company Philadelphia Pennsylvania 1994. [4] J. Canny, "A computational approach to edge detection" IEEE Transactions on Pattern Anaysis and Machine Intell., vol 8, no. 6 pp. 9611005 September 1986. [5] S Mallat and S Zhong, Characterization of signals from multiscale edges", IEEE Transactions on Pattern Anal y sis and Machine Intelligence, vol. 14 no. 7 pp. 710732 July 1992 [6] S. Mallet and W.L. Hwang Singularity detection and processing with wavelets" IEEE Transactions on Information Theo ry vol. 38 no. 2 pp 617642 March 1992 [7] S. Mallat A wavelet tour of signal processing 2nd edition Academic Press London 1998. 45

