MEDICAL IMAGING THROUGH HISTOGRAM EQUALIZATION AND CANNY
EDGE DETECTION METHOD
MAZLINDA BINTI IBRAHIM
This dissertation is submitted as a partial fulfillment for the award of the Master of
Science (Mathematics)
Faculty of Science
Universiti Teknologi Malaysia
NOVEMBER 2009
To my beloved family ma, ayah, abe kie, akak& abe yahya, lina, juwe, udin, and asri,
not to forget my friend, Ita, love you all so much!
ACKNOWLEDGEMENT
With this opportunity, I would like to express my appreciation to Prof. Dr
Shaharuddin Bin Salleh as my supervisor for his guidance and supportive suggestion
during the whole development of this project. I also would like to express deepest thanks
to my family and my friends.
Last but not least, I am grateful towards all authorities which helped directly or
indirectly for this project. Thanks to everyone.
ABSTRACT
Medical images edge detection is an important work for object recognition of the
human organs. It is an important pre-processing step in surgery planning to detect the
location and size of cancerous tissues or tumor. Today, many edge detection methods
were introduced in order to obtain the better result. One of the most popular methods is
Canny edge detector. But, Laplacian and Sobel method are still taken into account to the
researcher of edge detection. Computer Tomography scan is one of the modalities in
medical imaging. Image reconstruction methods were used to obtain the final images in
CT scan. Test images from CT scan were used to apply the Canny, Laplacian and Sobel
methods using simulation with C++ programming. Comparisons were made and the
result show that the combination of Canny edge detector and histogram equalization
perform well to the tested images.
ABSTRAK
Pengesanan rangka dalam imej perubatan adalah satu langkah yang penting
untuk pengenalpastian organ-organ manusia. Ia juga merupakan langkah awal yang
penting dalam perancangan pembedahan untuk mengenalpasti lokasi dan saiz tisu yang
berkanser ataupun tumor. Hari ini, banyak kaedah untuk pengesanan rangka
diperkenalkan yang bertujuan untuk mendapat hasil yang lebih baik. Salah satu kaedah
yang terkenal adalah pengesanan rangka Canny. Tetapi kaedah Sobel dan Laplacian
masih mendapat perhatian daripada pengkaji pengesanan rangka. Pengesan komputer
tomografi
adalah salah satu kaedah dalam memperoleh imej perubatan. Kaedah
pembangunan semula imej digunakan untuk mendapat imej terakhir dalam pengesan
komputer tomografi. Imej yang diuji adalah daripada pengesan komputer tomografi
untuk mengaplikasikan kaedah Canny, Laplacian dan Sobel menggunakan simulasi
dengan pengaturcaraan C++. Perbandingan dibuat dan keputusan menunjukan
kombinasi pengesanan rangka Canny dan kesamarataan histogram bertindak dengan
baik keatas imej yang diuji.
TABLE OF CONTENTS
CHAPTER
SUBJECT
PAGE
COVER
i
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENT
iv
ABSTRACT
v
ABSTRAK
vi
TABLES OF CONTENTS
vii
LIST OF TABLES
xi
LIST OF FIGURES
xii
LIST OF SYMBOLS
xiv
LIST OF APPENDICES
xv
CHAPTER 1 INTRODUCTION
1.1
Introduction
1
1.2
Problem Statement
3
1.3
Objectives
4
1.4
Scope
5
1.5
Project Outline
5
CHAPTER 2 EDGE DETECTION
2.1
Introduction
7
2.2
Image Processing
8
2.3
Edge Detection
8
2.4
Edge Detection via Image Spatial Gradients
12
2.5
Laplacian Edge Detection
15
2.6
Sobel Edge Detection
16
2.7
Canny Edge Detection
17
2.7.1
Detection Criteria
19
2.7.2
Localization Criteria
20
2.7.3
Only Response to Single Edge
22
2.7.4
Detector for Step Edge
24
2.8
Histogram Equalization
27
2.9
Conclusion
29
CHAPTER 3 MEDICAL IMAGING
3.1
Introduction
30
3.2
Image Reconstruction from Projection
31
3.3
X-ray or Computer Tomography
35
3.3.1
37
A Basic Model for Tomography
3.4
Radon Transform
40
3.5
Conclusion
44
CHAPTER 4 MODEL AND SIMULATION
4.1
Introduction
45
4.2
Algorithms for Laplacian and Sobel Edge Detection 46
4.3
Algorithm for Canny Edge Detection
48
4.4
Histogram Equalization
50
4.5
Simulation
51
4.6
X-ray Tomography Images
54
4.7
Conclusion
56
CHAPTER 5 RESULT AND DISCUSSION
5.1
Introduction
57
5.2
Result of Edge Detection
58
5.2.1 Laplacian
58
5.2.2
Sobel Edge Detection
59
5.2.3
Canny Edge Detection
61
5.3
Histogram Equalization
66
5.5
Discussion
67
5.6
Conclusion
68
CHAPTER 6 CONCLUSION DAN RECOMMENDATIONS
6.1
Introduction
69
6.2
Conclusion
70
6.3
Recommendations
71
REFERENCES
72
APPENDIX
75
LIST OF TABLES
TABLE NO.
3.1
SUBJECT
Attenuation coefficient of human tissues
for 100-keV x-ray.
PAGE
37
LIST OF FIGURES
FIGURE NO.
SUBJECT
PAGE
2.1
Ideal step edge.
10
2.2
Ramp in step edge.
10
2.3
Histogram equalization.
27
3.1
Flat region, showing a simple object
31
in x-ray tomography.
3.2
Result of back projection of simple object.
32
3.3
The beam and detector rotated about 90D
33
and back projection.
3.4
The sum of two projections.
33
3.5
Reconstruction with 1,2,3,4 and 32 back projections.34
3.6
CT scanner configuration with a true
36
fan beam of x-ray.
3.7
Normal representation of a straight line.
40
3.8
Geometry of a parallel x-ray beam.
41
3.9
Shepp-Logan phantom and its sinogram.
42
3.10
Flow of the filter back projection algorithm.
43
4.1
Algorithm for Laplacian edge detection.
46
4.2
Algorithm for Sobel edge detection.
47
4.3
Algorithm for smoothing with Gaussian
48
and differentiation.
4.4
Algorithm for non maximal suppression.
49
4.5
Algorithm for hysterisis thresholding.
50
4.6
Algorithm for histogram equalization.
51
4.7
Layout of the simulation.
53
4.8
Simulation of Sobel Multidirectional with
53
Threshold=500.
4.9
Sample image in x-ray tomography.
55
4.10
Image after cropping.
55
5.1
Laplacian edge detection of the liver.
58
5.2
Edge detection using Sobel of the liver.
59
5.3
The result using Sobel multidirectional
60
edge detection method.
5.4
Image after histogram equalization.
61
5.5
Gaussian smoothing of the liver.
62
5.6
Image differentiation of the liver.
63
5.7
Image after locate the local maxima pixels.
64
5.8
Image after thresholding using hysterisis.
65
5.9
Canny edge detection without performing
66
histogram equalization.
LIST OF SYMBOLS
f ( x, y )
-
Two dimensional light intensity function
-
Convolution
I
-
Convolution kernel
Sx
-
Horizontal kernel
Sy
-
Vertical kernel
I ( x)
-
Image function in one dimension
G ( x)
-
Edge function in one dimension
n( x)
-
Additive white Gaussian noise in one dimension
W
-
Operator width
SNR
-
Signal to noise ratio
xave
-
Average distance between zero crossing
xzc
-
Mean distance between zero crossings
A
-
Amplitude of the step edge
V
-
Standard deviation or sigma in Gaussian function
I
-
The intensity of the x-ray beam
P
-
Linear attenuation coefficient
G
-
Impulse function
ƒ{ f ( x, y )}
-
Radon transform
LIST OF APPENDIX
APPENDIX
A
SUBJECT
Coding of the simulation.
PAGE
75
CHAPTER 1
INTRODUCTION
1.1
Introduction
Image means picture. It can be analogue or digital. An analogue image is an
image which exists on a photographic film, video magnetic tape or signal from a
broadcast transmission (Joshi, 2001). Meanwhile, a digital image is a representation of
two dimensional image using ones and zeros or in binary numbers. Digital image data is
represented in the form of grey scale, corresponding to the intensity. It can be processed
using the digital computer. The term digital image usually refers to raster images also
called bitmap images. Raster images have a finite set of digital values called picture
elements or pixels.
Interest in digital image processing techniques dates back to the early 1920’s
when digitized pictures of world news events were first transmitted by submarine cable
between New York and London. Application of digital image processing concepts,
however did not become widespread until the middle 1960’s, when third generation
digital computers began to offer the speed and storage capabilities required for practical
implementation of image processing algorithms. Since then, this area has experienced
vigorous growth, having been a subject of interdisciplinary study and research in such
fields as engineering, biology, computer science, medicine and chemistry. The result of
these efforts have established the value of image processing techniques in a variety of
problems ranging from restoration and enhancement of space-probe pictures to
processing of fingerprints for commercial transactions.
Image processing is an area of study for manipulating and modifying images.
Image processing involves techniques for image enhancement, restoration and
smoothing by moving, copying, deleting and modifying the contents of the pixels in the
image. Before an object can be extracted from scenery, a way of assessing it shape and
size is needed. This can be done using edge detection or segmentation, both of which are
low-level image processing task, or their combination (Gudmundsson et all, 1998). The
success of the higher level recognition is, therefore highly dependent on this lower level
process.
Edge is the boundary between two regions, with relatively distinct grey level
properties (Joshi, 2001). In a continuous image, a sharp intensity transition between
neighbouring pixels is considered as an edge. Laplacian and Sobel are classical
mathematical methods for the edge detection. The Laplacian edge detection method
uses a two dimensional linear filter to approximate the second order derivative of pixel
values of the image. The Sobel edge detection method uses two dimensional linear
filters to process vertical edges and horizontal edges separately. One of the popular
methods in edge detection is the Canny Edge Detector (Heath et all, 1997). This method
was introduced by John Canny on 1986.
Due to the important of edge detection in image processing many methods were
developing. In real world machine vision problems, numerous issues such as variable
scene illumination make edge and object detection difficult. There exists no universal
edge detection method which works well under all condition (Panetta and Wharton,
2008).
The past 25 years have seen remarkable developments in medical imaging
technology. Universities and industry have made huge investments in inventing and
developing the technology needed to acquire images from multiple imaging modalities,
such as CT, MRI, and Ultrasound. Every modality has its own working principle and
system that enable us to obtain the final images. For example, X-ray computed
tomography (CT) images are sensitive to tissue density and atomic composition, and the
x-ray attenuation coefficient (Hajnal et all, 2001).
Image reconstructions methods were used in order to obtain the final images in
the medical imaging modalities. As an example, in x-ray tomography, image
reconstruction from projection method such as the filtered back projection method was
applied. After the image was reconstruct, we now can apply the image processing
techniques as desired.
1.2
Problem Statement
Given an image of size m u n . In what way can the image be reconstructed so as
to produce binary versions which clearly display the high and low intensity pixels? The
binary version is an image with black and white colour which the white colour shows the
boundary of the object in the image. The white color indicates the edges and the black
color indicate no edges. In medical field, the binary version of the image is a preliminary
stage in any medical decision. The selection of edge detection method is also an
important step because not all method can detect the desire edges.
One of the available methods in edge detection is Canny edge detection. Canny
edge detector used the first derivative of a Gaussian G ( x) as the optimal filter where;
§ x2 ·
G ( x) exp ¨ 2 ¸
© 2V ¹
where V is the standard deviation of the Gaussian function and x is the value from
origin in one dimension case. Then, the edge point is defined to be a local maximum in
the direction of n of the operator Gn applied to the image I , where Gn
n is normal to the edge direction. At local maximum, we have
w
Gn I
wn
n ˜’G where
0 . Then the
problem becomes to find the location of local maxima in the image. Since an image
contains higher number of pixel, we need to use the simulation to find the edges.
1.3
Objectives
The objectives of this study include:
a)
To apply the Laplacian , Sobel and Canny edge detection method on the
medical images.
b)
To develop a simulation of Laplacian, Sobel and Canny edge detection
method.
c)
To compare the performance of Laplacian, Sobel and Canny edge
detection using visual assessment.
1.4
Scope
This project use Laplacian, Sobel and Canny edge detector for detecting the
edges. Comparisons were made to determine which method is the best using visual
assessment. Laplacian and Sobel edge detection have one parameters meanwhile
Canny edge detection has three parameters. The parameters values are just any values
and we do not study the best parameters values for each method. We apply the method
to the medical images from x-ray tomography. Some discussion about image
reconstruction from projection in x-ray tomography also included. The test image is the
liver image which we aim to locate a tumor.
1.5
Project Outline
This project contains six chapters. The presentation of this project started in
Chapter 1 with discussion of problem statement and objectives. It also includes scope
and project outline. Chapter 2 discusses the edge detection methods. These include the
Laplacian and Sobel method first. The chapter also introduces the Sobel
multidirectional, the Canny edge detection method and histogram equalization.
Chapter 3 begins with a look at medical images. The discussion also includes
image reconstruction from projection in x-ray tomography. This lead to Radon transform
for filter back projections in x-ray tomography.
Then, in Chapter 4, our model will be presented, which apply edge detection on
medical images. The simulation was used using C++ programming. We tested the model
using medical image from x-ray tomography. Result and discussion presented in Chapter
5. Then Chapter 6, conclude the entire project. Supportive recommendations are given
so that interested readers can follow the details.
CHAPTER 2
EDGE DETECTION
2.1
Introduction
This chapter discusses the background of image and edge detection. The concept
of image mathematically and detail on the methods of edge detection were also given.
Laplacian edge detection method uses a two dimension linear filter to approximate the
second order derivative of the pixel value. Meanwhile, Sobel edge detection method
uses two dimension linear filters to process vertical and horizontal edges separately.
Canny edge detector uses the first derivative of a Gaussian as the optimal filter. Canny,
1986 defined three performance criteria in order to come out with Canny edge detector.
Histogram equalization is a technique in image enhancement which aims to equally
distribute the grey level in the image.
2.2
Image Processing
According to Baxes, 1994 digital image processing operations can be broadly
grouped into five fundamental classes. Edge detection is the one in the class of image
enhancement. The five classes are image enhancement, image restoration, image
analysis, image compression, and image synthesis.
Image refers to a two dimensional light intensity function f ( x, y ) , where x and
y denote the spatial coordinates and the value f is proportional to the brightness or grey
level of the image at the point (Rafael, 1977). It is sometimes useful to view the image
function in perspective with the third axis being brightness.
A digital image is an image f ( x, y ) which has been discretized both in spatial
coordinates and in brightness. We may consider a digital image as a matrix whose row
and column indices identify a point in the image and the corresponding matrix element
identifies the grey level at that point. The elements of such a digital array are called
image element, pictures elements, pixels or pels.
2.3
Edge Detection
Edge detection is one of the most commonly used operations in image analysis
and there are probably more algorithms in the literature for enhancing and detecting
edges than any other single subject. The reason for this is that edges form the outline of
an object. Edge is the boundary between two regions, with relatively distinct grey-level
properties (Joshi, 2001). In a continuous image, a sharp intensity transition between
neighbouring pixels is considered as an edge.
Edge is the boundary between an object and the background, and indicates the
boundary between overlapping objects (Parker, 1997). This means that if the edges in an
image can be identified accurately, all of the object can be located and basic properties
such as area, perimeter, and shape can be measured. In medical field, we can study about
the volume and growth of the tumor using edge detection. Since computer vision
involves the identification and classification of objects in an image, edge detection is an
essential tool.
Edgels are edge elements that have significant local variations in image
brightness, characterized by the position x and the orientation T of the brightness
variation. Sequence of edgels forming smooth curve that is edge. Every pairs of pixels
with differing intensities could be considered as potential edge. Edge detection
contributes significantly to algorithms for feature detection, segmentation and motion
analysis. Edge detection can be used to early detect the existence of cancerous tissues in
a patient.
Ideal step edge illustrated in Figure 2.1 adapted from Parker,1997 is the most
common type of edges. In this one dimensional example, the edge is simply a change in
grey level occurring at once specific location that is at pixel 10. The greater the change
in the level, the easier the edge is to detect, but in the ideal case this can be seen quite
easily. Other types of edges are ridge and roof edges (Canny, 1986).
In Figure 2.1 and 2.2 the dotted line shows where the image was sliced to give
the illustrated cross section. Figure 2.2 shows that the change of grey level extent some
number of pixels. The first complication occurs because of digitization. It is unlikely that
the image will be sampled in such a way that all of the edges happen to correspond
exactly with a pixel boundary. The actual position of the edge in Figure 2.2 is considered
to be the centre of the ramp connecting the low grey level to the high one. This is the
ramp in the mathematical word only, since after the image has been jagged appearance
of a staircase.
Figure 2.1:
Ideal step edge.
Figure 2.2:
Ramp in step edge.
The second complication is the problem of noise. Due to a great many factors
such as light intensity, type of camera and lens, motion, temperature, atmospheric
effects, dust and other, it is very unlikely that two pixels that correspond to precisely the
same grey level in the scene will have the same level in the image. Noise is a random
effect and is characterizable only statistically.
The result of noise on the image is to produce a random variation in level from
pixel to pixel, and so the smooth lines and ramps of the ideal edges are never
encountered in real images. Noise is usually expressed as a probability distribution with
a specific mean and standard deviation. As example of noise is the Gaussian white noise.
This noise characterized by the normal distribution with zero mean and specified
standard deviation.
Edge detection is carried out through a linear operation call convolution, or
sometimes called filtering. A pixel value of intensity f ( x, y ) change its value to
f '( x, y ) through a convolution kernel K , as shown in the Equation 2.1. (Salleh, 2008).
f '( x, y )
f ( x, y ) K
(2.1)
where denote convolution, K is the kernel associate to the edge detection method
chosen, x and y are the spatial coordinate in two dimension and f is correspond to the
grey level.
2.4
Edge Detection via Image Spatial Gradients
Since an edge is defined by a change in grey level, an operator that is sensitive to
this change will operate as an edge detector. A derivative operator does this, one
interpretation of a derivative is the rate of change of a function, and the rate of change of
the grey levels in an image is large near an edge and small in constant areas.
Since image is in two dimensions, we need to consider the level changes in two
directions. For this reasons, the partial derivatives of the image function f ( x, y ) are
used with respect to the x and y . The gradient of an image function f ( x, y ) in terms of
directionally oriented spatial derivatives as
’f ( x , y )
ª wf ( x, y ) º
« wx »
«
»
« wf ( x, y ) »
«¬ wy »¼
wf ( x )
wx
(2.2)
To calculate the derivatives in Equation 2.2 we can use the forward difference
operator with respect to x and y as below
wf ( x, y ) f ( x 'x, y ) f ( x, y )
|
wx
'x
wf ( x, y ) f ( x, y 'y ) f ( x, y )
|
wy
'y
(2.3)
(2.4)
Since an image is discrete, we used the differences instead of derivative at a pixe
and thisl is approximated by the difference in grey levels over some local region. Note
that the above operator is correspond to the correlation of the discretized image function
with the operators of the form
> 1 1@
and
ª1º
« 1»
¬ ¼
Respectively where we take
'x 1 and 'y 1
The one dimensional forward difference operator is defined as
D1 ( x)
f ( x 'x ) f ( x )
'x
(2.5)
The central difference approximation is
D2 ( x )
f ( x 'x) f ( x 'x)
2'x
(2.6)
D2 ( y )
f ( y 'y ) f ( y 'y )
2'y
(2.7)
Similarly,
D2 ( x) and D2 ( y ) have centered nature and can be implemented by convolving the
image function with masks
> 1
0 1@
and
ª1º
«0»
« »
«¬ 1»¼
Which operator is used to compute the gradient, the resulting vector contain information
about how strong the edge is at the picture and what direction is. The magnitude of the
gradient vector is the length of the hypotenuse of the right triangle having sides
and
wf ( x, y )
wx
wf ( x, y )
and this reflects the strength of the edge or edge response at any given
wy
pixel. The direction of the edge at the same pixels is the angle of that the hypotenuse
makes with the axis. Therefore, the edge response or magnitude is approximately given
by:
E
(
wf ( x, y ) 2 wf ( x, y ) 2
) (
)
wx
wy
(2.8)
where E is edge and the direction of the edge is approximately given as
Ed
§ § wf ( x, y ) · ·
¨ ¨ wy ¸ ¸
¨ tan ¨
¸¸
¨ ¨ wf ( x, y ) ¸ ¸
¨ ¨ wx ¸ ¸
¹¹
© ©
1
(2.9)
The edge magnitude will be a real number, and is usually converted to an integer
by rounding. Any pixel having a gradient that exceeds a specified threshold value is said
to be an edge pixel, and others are not. Usually, people use the middle value in the range
of grey level as threshold.
2.5
Laplacian Edge Detection
It is a second order derivative operator
D12 ( x)
f ( x 'x) 2 f ( x) f ( x 'x)
( 'x ) 2
(2.10)
D12 ( y )
f ( y 'y ) 2 f ( y ) f ( y 'y )
( 'y ) 2
(2.11)
It is represented as one dimension form
>1
2 1@ for x direction and for the y
ª1º
direction «« 2 »» .
¬« 1 ¼»
A Laplacian kernel is obtain using 3×3 window operator
ª0 0 0 º ª0 1 0 º
«1 2 1 » « 0 2 0 »
«
» «
»
«¬0 0 0 »¼ «¬ 0 1 0 »¼
ª0 1 0 º
«1 4 1 »
«
»
«¬0 1 0 »¼
Another form of the kernel that include the diagonal direction is
ª 0 1 0 º ª 0 0 0 º ª1 0 0 º ª 0 0 1 º
«0 2 0 » «1 2 1 » «0 2 0 » «0 2 0 »
«
» «
» «
» «
»
«¬0 1 0 »¼ «¬0 0 0 »¼ «¬0 0 1 »¼ «¬1 0 0 »¼
ª1 1 1º
«1 8 1»
«
»
«¬1 1 1»¼
2.6
Sobel Edge Detection
A Sobel filter consists of two convolution kernel for detecting the horizontal and
vertical changes in an image. The kernels make up the gradient vector S
ª Sx º
« S » where
¬ y¼
S x the horizontal kernel is and S y is the vertical kernel. The horizontal kernel S x is
used for smoothing along the direction of the x axis, while the vertical kernel S y
applies to smoothing along the y axis (Salleh, 2008).
Sobel use the central difference operator in Equation 2.6 and 2.7 and smoothing
operator in order to obtain the two kernels.
Sy
ª1º
«0» 1 2 1
@
« »>
¬« 1¼»
ª1 2 1º
« 0 0 0 » and S
x
«
»
«¬ 1 2 1»¼
ª1 º
« 2 » 1 0 1
@
« »>
«¬1 »¼
ª 1 0 1 º
« 2 0 2 »
«
»
«¬ 1 0 1 »¼
S y detect change in the vertical direction that means that it has response on
horizontal edges. Meanwhile, S x detect change in the horizontal direction that means
that it has response on vertical edges.
Panetta and Wharton, (2008) used the Sobel multidirectional kernels which apply
the same standard Sobel operator with diagonal orientation. They constructed the
diagonal filter using 5 u 5 block instead of 3 u 3 . The diagonal kernels are as follows:
ª0 0 1
«0 0 0
«
« 1 0 0
«
« 0 2 0
«¬ 0 0 1
2.7
0 0º
2 0»»
0 1»
»
0 0»
0 0»¼
and
ª 0 0 1
«
« 0 2 0
« 1 0 0
«
«0 0 0
«¬ 0 0 1
0 0º
0 0»»
0 1»
»
2 0»
0 0»¼
Canny Edge Detector
John Canny, proposed method in detecting the edges in 1986 in his paper “A
Computational Approach to Edge Detection”. That paper describes a computational
approach to the edge detection. The success of this method depends on the definition of
a comprehensive set of goals for the computation of edge points. Higher concentration
were given to Canny edge detection because it is likely to be better at detecting weak
edges and less sensitive to noise (Y.Lin et all, 2002).
Canny proposed an approach to edge detection that is optimal for step edges
corrupted by white noise. The optimality of the detector is related to three criteria,
detection, localization and one response. Canny’s derivation of a new edge detector is
based on first smoothing the image with Gaussian, find the derivatives, non maximal
suppression and the novel thresholding method that is hysteresis thresholding.
Canny attempts to find the filter f that maximizes the product of SNR times
localization subject to multiple constraints and while the result is too complex to be
solved analytically, an efficient approximation turns out to the first derivative of a
Gaussian function. The next step in Canny edge detector is a non-maxima suppression
step, where pixels that are not local maxima are removed.
As an extra step Canny suggest thresholding using hysteresis rather than simply
selecting a threshold value to apply everywhere. Hysteresis thresholding uses a high
threshold Th and a low threshold Tl . Any pixel in the image that has a value greater
than Th is presumed to be an edge pixel, and is marked as such immediately. Then, any
pixels that are connected to this edge pixel and that have a value greater than Tl are also
selected as edge pixels and are marked too. The marking of neighbors can be done
recursively or by performing multiple passes through the image.
The optimality of the detector is related to three criteria below (Canny, 1986):
1) Good detection:
The detection criterion expresses the fact that important
edges should not be missed and that there should be no spurious responses.
2) Good Localization:
The localization criterion says that the distance between
the actual and located position of the edge should be minimal.
3) Only response to a single edge:
The one response criterion minimizes
multiples responses to a single edge. This is partly covered by the first criterion,
since when there are two responses to a single edge, one of them should be
considered as false. The third criterion solves the problem of an edge corrupted
by noise and works against non-smooth edge operators.
Assumed that the image consists of the edge and subject to white Gaussian noise.
Let I ( x) G ( x) n( x) where I ( x) is image in one dimension G ( x) is edge and n( x) is
the additive white Gaussian noise. The edge detector was assumed to be a convolution
filter f which would smooth the noise and locate the edge. The problem is to identify
the one filter that optimizes the three edge detection criteria.
2.7.1
Detection Criteria
Let the impulse response of the filter f ( x) and denote the edge itself by G ( x)
0 . Then the response of the filter r ( x) is the
and assume that the edge is centered at x
sum of the response to the signal and the noise. Then we have
r ( x)
f ( x) I ( x)
r ( x)
HG H n
f ( x ) G ( x ) n( x ) f ( x ) G ( x ) f ( x ) n( x )
(2.12)
where H G is given by a convolution integral:
W
HG
³ G ( x) f ( x)
dx
(2.13)
W
assuming the filter has a finite impulse response bounded by [W , W ] .
The root mean squared response to the noise n( x) only, will be
1
Hn
ªW 2
º2
n0 « ³ f ( x) dx »
¬ W
¼
(2.14)
where n02 is the mean squared noise amplitude per unit length. Then, the first criterion
the output single to noise ratio, as the quotient of these two responses.
W
SNR
³ G ( x) f ( x)
dx
W
W
n0
³
W
2
f ( x) dx
(2.15)
2.7.2
Localization Criteria
The second criterion is that the edge point is well localized, means the distance
between the points marked by the detector and the centre of the true edge should be
minimized. For the localization criteria, used the reciprocal of the mean squared distance
of the marked edge from the centre of the true edge.
Let H n ( x) be the response of the filter to noise only, and H G ( x) be its response
to the edge, and suppose there is a local maximum in the total response at the point
x
x0 . Then we have,
H n' ( x0 ) H G' ( x0 ) 0
(2.16)
The taylor expansion of H G' ( x0 ) about the origin gives
H G' ( x0 )
H G' (0) H G'' (0) x0 O ( x02 )
From our assumption that edge centre at x
(2.17)
0 , therefore H G' (0) 0 . Our next
assumption is the displacement x0 of the actual maximum is assumed to be small so we
will ignore quadratic and higher terms. From Equation 2.16 and 2.17 we obtain
H G'' (0) x0 | H n' ( x0 )
(2.18)
Now, H n' ( x0 ) is the Gaussian random quantity whose variance is the mean
squared value of H n' ( x0 ) , and is given by
W
'
n
2
2
0
E[ H ( x0 ) ] n
³
2
f ' ( x) dx
(2.19)
W
From Equation 2.18 we obtain
E ª¬ H G'' (0) x0 º¼
2
E ª¬ H n' ( x0 ) º¼
2
W
E ª¬ x02 º¼ H G'' (0) 2
³
n02
f '2 ( x) dx
W
W
W
n02
E ª¬ x02 º¼
³
f '2 ( x) dx
n02
W
f '2 ( x) dx
G x02
W
''
G
H (0)
2
W
n02
E[ x02 ] |
³
³
ª
º
'
« ³ G ( x) f '( x) dx »
¬ W
¼
W
2
2
f ' ( x) dx
W
ª
º
'
'
« ³ G ( x) f ( x) dx »
¬ W
¼
W
2
G x02
(2.20)
where G x02 is an approximation to the standard deviation of x0 . The localization is
defined as the reciprocal of G x0 .
W
³ G ( x) f
'
Localization
'
( x) dx
W
W
n0
³
W
'2
f ( x)
(2.21)
dx
2.7.3
Only Response to Single Edge
This criterion implicitly defines in first criteria, but we make explicit to eliminate
multiple response. We first note that the mean distance between adjacent maxima in the
output is twice the distance between adjacent zero-crossing in the derivative of the
operator output. Then Canny make use of a result from Rice, (1945) that the average
distance between zero crossings of the response of a function g to Gaussian noise is
xave
§ R(0) ·
S¨
¸
© R ''(0) ¹
1
2
(2.22)
where R (W ) is the autocorrelation of g . In this case we are looking for the mean zerocrossing spacing for the function f ' . Now since,
f
R(0)
³
f
g 2 ( x) dx and R ''(0)
f
³ g '2 ( x) dx
(2.23)
f
The mean distance between zero-crossing of f ' will be
§ f 2
·
¨ ³ f ' ( x) dx ¸
f
¸
xzc ( f ) S ¨ f
¨
¸
2
¨ ³ f '' ( x) dx ¸
© f
¹
1
2
(2.24)
The distance between adjacent maxima in the noise response of f , denoted xmax will be
twice xzc and let this distance to be some fraction k of the operator width
xmax ( f )
2 xzc ( f )
kW
(2.25)
The expected number of noise maxima in region 2W is N n where
2W
xmax
Nn
2
k
(2.26)
If a function f satisfies the multiple response constraint in equation 2.24 for a fixed k ,
then the function f w will also satisfies it, assuming W scales with w . The multiple
response criterion is invariant with respect to spatial scaling of f .
Canny use numerical optimization for finding the optimal detector for the
function f which maximizes
W
³
G ( x) f ( x) dx
W
W
³ G '( x) f '( x)
W
n0
dx
W
³
W
2
f ( x) dx
W
n0
³
f '2 ( x) dx
W
subject to the multiple response constraint. He suggests using penalty method to solve
the above problem. Detail about the solution of this problem is out of the scope of this
project and can be found in Canny, (1986).
2.7.4
Detector for Step Edge
Now, we proceed to the problem for finding a detector for step edge. Here the
input G ( x) is step edge. Specifically we set G ( x)
Au1 ( x) where un ( x) is the n th
derivative of a delta function and A is the amplitude of the step. That is,
­0, for x 0
®
¯1, for x t 0
u 1 ( x )
And substituting for G ( x) in Equation 2.11 and 2.17 gives
0
A
³
f ( x) dx
W
SNR
(2.27)
W
n0
³
f 2 ( x) dx
W
A f '(0)
Localization
W
n0
³
'2
f ( x)
(2.28)
dx
W
From above, we see that these criteria improve directly with the ratio A
term in signal to noise ratio of the image.
0
SNR
A
6( f )
n0
where
6( f )
³
f ( x) dx
W
W
³
W
f 2 ( x) dx
n0
the
Localization
A
/ ( f ')
n0
/ ( f ')
where
f '(0)
W
³
f '2 ( x) dx
W
The optimal filter is implicitly defined by
0
6( f )/ ( f ')
³
f ( x) dx
f '(0)
W
W
³
W
2
f ( x) dx
W
³
(2.29)
2
f ' ( x) dx
W
The filter found by Canny in his paper can be approximated by the first derivative of a
Gaussian G '( x) , where
§ x2 ·
G ( x) exp ¨ 2 ¸
© 2V ¹
(2.30)
Suppose we want to detect edge in two dimension of a particular orientation,
then we create a two dimensional mask for this orientation by convolving a linear edge
detection function aligned normal to the edge direction with a projection function
parallel to the edge direction.
Suppose we wish to convolve the image with an operator Gn which is the first
derivative of a two dimensional Gaussian G in some direction n :
G
§ x2 y 2 ·
exp ¨ ¸ and
2V 2 ¹
©
Gn
wG
wn
n ˜’G
(2.31)
n should be oriented normal to the direction of an edge to be detected and although this
direction is not known, we can estimate from
n
’(G I )
’(G I )
(2.32)
where denote convolution.
An edge point is defined to be a local maximum in the direction of n of the
operator Gn applied to the image I . At local maximum, we have
w
Gn I
wn
0
(2.33)
and substituting for Gn and associating Gaussian convolution, we obtain
w2
GI
wn 2
0
(2.34)
At such point, the edge strength will be the magnitude of
Gn I
’(G I )
(2.35)
We can first convolve with a symmetric Gaussian and then compute directional
second derivatives to locate edges and use the magnitude to estimate the edge strength.
In principle, it could be used to implement an edge detector in arbitrary number of
dimensions, by first convolving the image with a symmetric n dimension Gaussian. The
convolution with n dimension Gaussian is highly efficient because the Gaussian is
separable into n one dimensional filters.
2.8
Histogram Equalization
Histogram equalization is a technique in image enhancement. The aims to create
an image with equally distributed brightness levels over the whole brightness scale as in
Figure 2.3. Histogram equalization enhances contras for brightness value close to
histogram maxima and decreases contrast near minima. (Sonka et all, 2008).
Figure 2.3:
Histogram Equalization
The above figure shows the effect of the histogram equalization to the image
histogram. The left graph is the grey level corresponds to the pixels and the left graph
show that histogram equalization equally distributes the grey level to the all pixels in the
image.
Denote the input histogram by H ( p ) and recall that the input grey scale is
[ p0 , pk ] . The aims is to find a monotonic pixels brightness transformation q T ( p ) such
that the desired output histogram G (q ) is uniform over the whole output brightness
scale [q0 , qk ] .
The histogram can be treated as a discrete probability density function. The
monotonic property of the transform T implies
k
k
¦ G (q ) ¦ H ( p )
i
(2.36)
i
i 0
i 0
The sums in Equation 2.36 can be interpreted as discrete distribution functions. Assume
that the image has N rows and columns, then the equalized histogram G (q ) corresponds
to the uniform probability density function f whose function value is a constant
N2
qk q0
f
(2.37)
The value from Equation 2.37 replaces the left side of Equation 2.36. The equalized
histogram can be obtained precisely only for the idealized continuous probability
density, in which case the Equation 2.36 becomes
q
1
ds
q q0
q0 k
N2 ³
N 2 (q q0 )
qk q0
p
³ H (s)
ds
(2.38)
p0
The desired pixel brightness transformation T can be derived as
q T ( p)
qk q0
N2
p
³ H (s)
ds q0
(2.39)
p0
The integral in Equation 2.39 is called the cumulative histogram, which is approximated
by a sum in digital value, so the resulting histogram is not equally ideally. The discrete
approximation of the continuous pixels brightness transformation from Equation 2.39 is
q T ( p)
qk q0
N2
p
¦ H (i) q
0
i p0
(2.36)
2.9
Conclusion
Edge is the boundary between an object and the background, and indicates the
boundary between overlapping objects. Ideal step is the most common type of edges
where, the edge is simply a change in grey level occurring at one specific location.
Laplacian and Sobel edge detection use derivative to locate the edges and are simple to
understand compared to Canny edge detection.
According to Canny, 1986, the optimality of the detector is related to three criteria,
detection, localization and one response. The main step in Canny edge detection is
smoothing with Gaussian, find the derivative, non maxima suppression and thresholding
with hysteresis. The impulse response of the optimal edge operator was shown to
approximate the first derivative of a Gaussian. Canny edge detector was proposed which
used adaptive thresholding with hysteresis to eliminate streaking of edge contour.
Histogram is a graph showing the number of pixels in an image at each different
intensity value found in that image. Histogram equalization is the technique by which
the dynamic range of the histogram of an image is increased. Histogram equalization
assigns the intensity values of pixels in the input image such that the output image
contains a uniform distribution of intensities. It improves contrast and the goal of
histogram equalization is to obtain a uniform histogram. This technique can be used on a
whole image or just on a part of an image.
CHAPTER 3
MEDICAL IMAGING
3.1
Introduction
Medical imaging refers to the techniques and processes used to create images of
the human body for clinical purposes such as for medical procedures seeking to reveal,
diagnose or examine disease or including the study of normal anatomy and physiology.
There are many types of medical imaging now days such as computer tomography (CT)
scan, magnetic resonance imaging (MRI), ultrasonography, nuclear imaging and so on.
In this chapter, concentration was given to the x-ray tomography. To obtain the final
images in x-ray tomography, image reconstruction method from projection are used.
One of the available methods in image reconstruction in x-ray tomography is filtered
back projections (FBP). Then, Radon transform is introduced as the initial step in
filtered back projection. After the image was constructed using image reconstruction
method in x-ray tomography, then we can perform any edge detection method.
3.2
Image Reconstruction from Projection
Reconstruction is focus on tomographic medical imaging. The image reconstruction
method can be classified into analytical methods and iterative (or algebraic) method
(Kudo, 2006). Bear in mind that image reconstruction method here are the
reconstruction from projection not the reconstruction in image analysis. The
reconstruction from projection problem is simple in principle and can be explained
qualitatively in a straightforward and intuitive manner. Let consider Figure 3.1 adapted
from Gonzalez, 2008.
Figure 3.1:
Flat region, showing a simple object in x-ray tomography.
Suppose the image in Figure 3.1 is a cross-section of a three dimension region of a
human body, where the background represent uniform tissue while the round object is a
tumor with higher absorption characteristics. Flat beams of x-ray were passed from left
to right and the absorption profile show that the energy of the beam is absorbed more by
the object than the background. Now, the only information available about the object is a
one dimensional absorption signal.
The reconstruction is begin by creating an image based on the one dimensional
absorption signals. The approach is to project the one dimensional signal back across the
direction from which the beam came as in Figure 3.2. The process of back projecting of
one dimensional signal across two dimensional areas sometimes referred as smearing the
projection back across the area. This means duplicating the same one dimensional signal
across the image perpendicularly to the direction of the beam. For example, Figure 3.2 is
created by duplicating the one dimensional signal in all columns of the reconstructed
images or called back projection.
Figure 3.2:
Result of back projection of simple object.
Now, suppose the source is rotated by 90q as in Figure 3.3 left. Repeating this
procedure yields a back projection image in the vertical direction as Figure 3.3 right.
Continue the reconstruction by adding this result to the previous back projection
resulting in Figure 3.4. Now, we can tell that the object of interest is contained in the
square shown, whose amplitude is twice the amplitude of the individual back
projections.
Figure 3.3:
The beam and detector rotated about 90q and its back projection.
Figure 3.4:
The sum of two back projections.
Figure 3.4 shows the intensity where the back projections intersect is twice the
intensity of the individual back projections. From there we can see that the image is
become more exact to the original shape. What happen if we increase the number of
different projection? As the number of projection increases, the reconstructed image
become clearer and well approximated to the shape of the original image as in Figure
3.5, besides getting the image is blurred by a “halo” effect.
Figure 3.5:
Reconstruction with 1,2,3,4 and 32 back projections.
In Figure 3.5 we can see that the net effect after increases the number of back
projections is the brighter region will dominate the result. The back projections with few
or no intersection will fade into the background as the image is scaled for display. The
“halo” effect is which we can seen appears as a “star” whose intensity is lower that the
object but higher than the background.
Back projection is one of the methods in image reconstruction from projection in xray tomography. Other method such as iterative method, direct algebraic algorithms,
three dimensional algorithm and circular harmonic algorithms can be found detail in
Natterer and Wubbeling, 2001.
3.3
X-ray or Computer Tomography
The CT system was invented by Godfrey Newbold Hounsfield of EMI Central
Research Laboratories in Hayes, England. Hounsfield conceived the idea in 1967 and it
was publicly announced in 1972. The first commercial CT machine using x-rays was
limited to making tomographic sections of the brain but acquired the image data in about
4 minutes and the computation time is about 7 minutes per picture.
Mathematics underlying x-ray tomography was done in 1917 by John Radon.
Literally, tomography means slice imaging. X-ray or Computer Tomography (CT)
provides cross sectional images of body organs. The resulting image is clearer compared
to the conventional x-ray. But, the radiation doses in CT are significant higher, and due
to this reason CT is not used widely unless it is really needed (Suri et all, 2002).
A typical CT slice is formed from 512x512 pixels and each pixel will
corresponds to an element of the cut through the patient of about (0.5x0.5)mm². This
dimension determines the limiting spatial resolution of the image. In CT, an X-ray beam,
which is initially generated as a point source, is collimated to form a single slice of a flat
fan shaped x-ray beam.
The x-ray source is rotated around the patient and the two dimensional or three
dimensional distribution of the x-ray radiation densities within the structure being image
are recorded as projections from many angles of view. The information obtained from
the rotation is used to reconstruct the cross sectional images of the body organs using
mathematical reconstruction techniques (Suri et all, 2002).
Figure 3.6:
CT scanner configuration with a true “fan beam” of x-ray.
Figure 3.6 shows a CT scanner configuration with a true “fan beam” of x-rays.
The dotted lines indicate incremental rotation. Multiple detectors are aligned along the
arc of circle whose centre is the x-ray tube focal spot. The x-ray beam is collimated into
a fan beam. Both the x-ray and the detectors rotated about the patient in concentric
circles whose centre approximately coincide with the centre of the patient.
The conventional manner in which CT images are obtained is to keep the patient
stationary during the scanning time required to generate one image. Scanning is then
halted while the position of the patient is incremented in the direction perpendicular to
the imaging plane using a motorized table. The next image is then obtained and the
procedure is repeated for the number of increments required to cover a specified section
in the body.
3.3.1
A Basic Model for Tomography
In x-ray tomography, we reconstruct a three dimension object from its two
dimension slices. Object of interest in x-ray imaging are described by a real value
function on \3 , called the attenuation coefficient. The attenuation coefficient quantifies
the tendency of an object to absorb x-rays of a given energy. This function varies from
point to point within the object and is usually taken to vanish outside. The average of
attenuation coefficient define a linear transformation called the Radon transform
(Epstein, 2003).
The attenuation coefficient is useful for medical imaging because different
anatomical structure have different attenuation coefficient. For example normal and
cancerous tissues have slightly different attenuation coefficient. Below is the attenuation
coefficient in Hounsfield units adapted from Epstein, 2003.
Attenuation coefficient
Table 3.1:
Material
in Hounsfield units
Water
0
Air
-1000
Bone
1086
Blood
53
Fat
-61
Brain white/gray
-4
Breast tissue
9
Muscle
41
Soft tissue
51
Attenuation coefficients of human tissues for 100-keV x-rays.
Model for the interaction of x-rays with matter is rests on the three basic
assumption below, (Epstein, 2003):
1)
No refraction or diffraction: X-ray beams travel along straight lines that are
not bent by the object they pass through.
2)
The x-rays used are monochoramatic: The waves making up the x-ray beam
are all of the same frequency.
3)
Beer’s law: Each material encountered has a characteristic linear attenuation
coefficient P for x-rays of a given energy.
According to Quinto, 2005, if the x-rays are not monochromatic, then lower energy
x-rays get attenuated more than higher energy x-rays, so the average energy of x-ray
increases as they go through the object. This is called beam hardening and can create
reconstruction artifacts.
The intensity, I of the x-ray beam satisfies Beer’s law in Equation 3.1.
dI
ds
P ( x) I
(3.1)
where s is the arclenght along the straight line trajectory of the x-ray beam and P (x) is
the attenuation coefficient at x . From the first assumption the direction of I at x gives
the direction of the straight line along which the beam is traveling.
Suppose that one of the x-ray beams is traveling along the line given
parametrically by ^x0 sv : s  \` . Here v is a unit vector in \3 . The function
i ( s ) I (x0 sv) gives the intensity of the beam at points along this lines, and
m( s )
P (x0 sv) gives the attenuation coefficient. Beer’s law states that
di
ds
m( s )i ( s ) or
Integrating this equation from s
ª i (b) º
log «
»
¬ i(a) ¼
d (log i )
ds
m( s )
a to s b gives
b
³ m( s ) ds
(3.2)
a
Between the points x 0 av and x 0 bv , the intensity of the beam is attenuated by
ª b
º
exp « ³ m( s) ds »
¬ a
¼
Beer’s law describes how the material attenuates each of the one dimension
beams and this process is isotropic because it does not depend on the direction of the
line along which the x-ray is travel. In real measurement, the x-ray source is turned on
for a known period of time. The total energy I i , incident on the object along a given line
l is known. The total energy I 0 emerging from the object along l is then measured by an
x-ray detector. Integrating Beer’s law we obtain
log
I0
Ii
³ P ds
l
(3.3)
Here ds is the arclength parameter along the straight line path l . A perfect
measurement of the ratio
I0
Ii
would therefore furnish the line integral of the attenuation
coefficient along the line l . Indeed model for measurements made in x-ray tomography
as precisely this line integral.
3.4
Radon Transform
A straight line in Cartesian coordinates can be described either by its slope
intercept form y
mx c or as in Figure 3.7 by its normal representation
x cos T y sin T
U
(3.4)
y
U
T
Figure 3.7:
x
Normal representation of a straight line.
The Radon transform maps the spatial domain ( x, y ) to the domain ( U , T ) . Each
point in the ( U , T ) space corresponds to a line in the spatial domain ( x, y ) . Note that
( U , T ) are not the polar coordinate of ( x, y ) . In x-ray tomography each ray beam can be
represented using Equation 3. 4.
Figure 3.8:
Geometry of a parallel x-ray beam.
The projection of parallel ray beam may be modeled by a set of such lines, as in
Figure 3.8. f ( x, y ) is the object of interest, g ( U j , T k ) is the result on the detector or the
projection signal and L( U j , T k ) is the parallel x-ray beam. An arbitrary point in the
projection signal is given by the ray sum along the line x cos T k y sin T k
U j . Then, the
ray sum is a line integral given by (Gonzalez, 2008)
g ( U j ,Tk )
f f
³³
f ( x, y ) G ( x cos T k y cos Tk U j ) dx dy
(3.5)
- f f
where G is the impulse function. If we consider all value of U and T , the preceding
equation generalizes to
g ( U ,T )
f f
³³
- f f
f ( x, y ) G ( x cos T y cos T U ) dx dy
(3.6)
Equation 3.6 gives the line integral of f ( x, y ) along an arbitrary line in the
xy plane, which is the Radon transform. The radon transform ƒ ^ f ( x, y )` is the
cornerstone of reconstruction from projections in x-ray tomography.
In discrete case, Equation 3.6 becomes
g ( U ,T )
M 1 N 1
¦ ¦ f ( x, y)G ( x cosT y sin T U )
(3.7)
x 0 y 0
where x, y, U , and T are discrete variables. If we fix T and allow U to vary, the above
equation simply the sum of the pixels of f ( x, y ) along the line defined by the specified
values of these two parameters. Changing the value of any parameter in the equation
yield different projection as in previous section.
The function g ( U , T ) is often refered to a sinogram because the Radon transform
of an off-centered point source is a sinusoid as in Figure 3.9.
Figure 3.9:
Shepp-Logan phantom and its sinogram(Radon transform).
Figure 3.9 is a density plot of a function that models the attenuation coefficient
of a slice of the human head. The densities in this plot are scaled so that the highest
attenuation (the skull) is white and the lowest (surrounding air) is black. A model of this
sort is called a mathematical phantom and were introduced by Larry Shepp. The right
figure depicts the line integral of this function.
The purpose of image reconstruction is to obtain f ( x, y ) from the Radon
transform. One of method is filtered back projection or FBP. The details of this method
are described in Natterer, and Wubbeling 2001. Figure 3.10 shows the flow of the
filtered back projection algorithm.
Ramp filter
g ( U ,T )
g '( U , T )
f ( x, y )
Back projection
Figure 3.10: Flow of the filter back projection algorithm.
In filtered back projection algorithm, a ramp filter is used to eliminate the “halo”
effect in the back projection image. There are many choices of filter, but ramp filter has
proved to be the optimal filter in FBP. Then, the process is continued to find the inverse
of Radon transform in order to obtain the original image.
3.5
Conclusion
The x-ray tomography used image reconstruction method to obtain the two
dimensional image of three dimensional object. The aim of image reconstruction is to
obtain the image of original object. Radon transform play important role in image
reconstruction because the attenuation coefficient of object display in this form. An
increase in the number of view or projection has the effect of increasing the level of
clearance of final image. Filter back projection algorithms is one of the method in image
reconstruction. In FBP the Radon transform is filtered using the ramp filter and the
process of back projection produces original image of the object. Then, we can apply the
edge detection to the image obtain from x-ray tomography.
CHAPTER 4
MODEL AND SIMULATION
4.1
Introduction
This chapter presents the model of edge detection and simulation work. The
simulation uses C++ programming and images from x-ray tomography to be tested. We
present the algorithms for Laplacian, Sobel and Canny edge detection. Prior to edge
detection, histogram equalization is applied to produce a clearer image. Histogram
equalization is a method in image enhancement to create an image with equally
distributed brightness levels. We deploy the liver image as the test image using the
Laplacian, Sobel and Canny edge detection for detecting the location of the tumor.
Results for the tested imaged will presented in Chapter 5.
4.2
Algorithms for Laplacian and Sobel Edge Detection
Laplacian Edge Detection
1) Using the Equation 2.10 and 2.11, we decide to use the kernel:
ª1 1 1º
«1 8 1»
«
»
«¬1 1 1»¼
2) Read the image I of size m u n .
3) Read the threshold value enter by the user.
4) Calculate the t value
m
t
n
¦¦ I [i][ j ]
i 0 j 0
threshold 2
5) For i 1 up to m 1 and j 1 up to n 1
L
I [i 1][ j 1] I [i ][ j 1] I [i 1][ j 1] I [i 1][ j ] 8 f [i ][ j ] I [i 1][ j ]
f [i 1][ j 1] I [i ][ j 1] I [i 1][ j 1]
6) Take the absolute value of L .
7) If L d t then L
Figure 4.1
edge (white) else L
nonedge (black).
Algorithm for Laplacian edge detection.
Sobel Edge Detection
1) Use the kernel
Sy
ª1 2 1º
«0 0 0»
«
»
«¬ 1 2 1»¼
Sx
and
ª1 0 1º
« 2 0 2 »
«
»
«¬1 0 1»¼
2) Read the image I of size m u n .
3) Read the threshold value enter by the user.
4) Calculate the t value
m
t
n
¦¦ I [i][ j ]
i 0 j 0
threshold 2
5) For i 1 up to m 1 and j 1 up to n 1
Sx
I [i 1][ j 1] f [i 1][ j 1] 2* I [i 1][ j ] 2* I [i 1][ j ] I [i 1][ j 1] I [i 1][ j 1]
S y I [i 1][ j 1] 2* I [i ][ j 1] I [i 1][ j 1] I [i 1][ j 1] 2* I [i ][ j 1] I [i 1][ j 1]
6) Calculate S
Sx S y
7) If S d t then S
Figure 4.2:
edge (white) else S
nonedge (black).
Algorithm for Sobel edge detection.
Figure 4.1 and 4.2 are the algorithm for Laplacian and Sobel edge detection. We
use these algorithms in developing the simulation in Section 4.5. For Sobel
multidirectional, the algorithm is similar to the Sobel, but we need to add another two
terms in calculating the value of S , where S
S x S y S d 1 S d 2 . The kernels for
S d 1 and S d 2 are in Section 2.6.
4.3
Algorithm for Canny Edge detection
Algorithm for smoothing with Gaussian and differentiation adapted from Parker, 1997.
1) Read in the image to be process I .
2) Create 1D Gaussian mask G to convolve with I . The standard deviation V of
this Gaussian is a parameter to the edge detector.
3) Create 1D mask for the first derivatives of the Gaussian in the x and y
directions, call these Gx and Gy .The same value V is used as in step 2.
4) Convolve the image I with G along the rows to give the x component
image I x , and down the columns to give the y component image I y .
5) Convolve I x with Gx to give I x ' and I y
with Gy to give I y '
6) The result up to here can be view with combined the x and y component. The
magnitude of the result (Equation 2.8) is computed at each pixel ( x, y ) as:
M ( x, y )
Figure 4.3:
I ' x ( x , y ) 2 I ' y ( x, y ) 2
Algorithm for smoothing with Gaussian and differentiation.
After smoothing with Gaussian, we need to find the derivative of the image using
the central differentiation formula as in Equation 2.6 and 2.7. The magnitude of the
gradient image is the edge strength given in Equation 2.35. The next step is to find the
local maxima or the location of edges using the algorithm in Figure 4.4 below.
Algorithm for non maximal suppression of directional edge data
1)
Quantize edge directions eight ways according to the 8 connectivity
using Equation 2.9.
2)
For each pixel with non-zero edge magnitude, inspect the two adjacent
pixels indicated by the direction of its edge.
3)
If the edge magnitude of either of these two exceeds that of the pixel
under inspection, mark it for deletion.
4)
When all pixels have been inspected, rescan the image and erase to zero
all edge data marked for deletion.
Figure 4.4:
Algorithm for non maximal suppression.
After having information about location of non maxima pixels, then we perform
hysterisis thresholding to determine whether the pixels are true edges or not. We need to
read the value of low threshold and high threshold enter by the user, and the final values
of thresholding are calculated using the number of pixels that are local maxima.
Algorithm for thresholding with hysteresis
1)
Define the low threshold tlow and the high threshold thigh
2)
Scan all pixel and ark all edges with magnitude greater than thigh as
correct
3)
Scan all pixels with edge magnitude in the range [tlow , thigh ]
4)
If such a pixel below tlow mark as no edges
5)
If such a pixel above tlow and connected to the edges marked as correct
Figure 4.5:
4.4
Algorithm for hysterisis thresholding
Histogram Equalization
Histogram equalization is a technique for contrast enhancement using grey scale
transformation. The aim is to create an image with equally distributed brightness levels
over the whole brightness scale (Sonka et all, 2008). Figure 4.6 is an algorithm to
perform histogram equalization adapted from Sonka et all, 2008.
Histogram Equalization
1) For N u M image of G gray-levels(often 256), create an array H of length
G initialized with 0 values.
2) Form the image histogram: Scan every pixel and increment the relevant
member of H if pixel p has intensity g p , perform
H[g p ] H[g p ] 1
3) Form the cumulative image histogram H C :
H C [0] 0
H C [ p] H C [ p 1] H [ p], p 1, 2,..., G 1
4) Set
§ G 1
·
T [ p ] round ¨
H C [ p] ¸
© NM
¹
5) Rescan the image and write an output image with grey level g q , setting
gq
Figure 4.6:
4.5
T[g p ]
Algorithm for histogram equalization.
Simulation
The simulation was developed using Microsoft Visual C++ version 6.0 using
Microsoft Foundation Classes. Images in the simulation must use .BMP format. On
Windows, a bitmap is represented by 24 bits of data, representing a total of 16 777 126
colors. An image is form from collection of total number of pixels. Each pixel is
represented by a string consisting of 24 bits of binary digits in the MFC function RGB().
Bitmaps can be drawn from Microsoft Paint program. A rectangular area of a
memory dc where the bitmap is located has to be copied into CClientDC device context.
This is done with the CDC::BitBlt() function. BitBlt means Bitwise Block Transfer.
Then, if we want to set the new value of the pixel, the function is SetPixel().
Edge detection is performed through C++ programming using MFC. Figure 4.7
is the graphic layout of the simulation. The simulation has five buttons and four edit
boxes. The buttons are control for performing Laplacian, Sobel, Sobel multidirectional
and Canny edge detection. The button clear deletes the previous operation and updates
the current window.
The first edit box is for the user to enter any desire value of threshold for the
Laplacian and Sobel method. If a higher value is enter then the resulting image becomes
blurred and less edges are detected compared to a lower value. Some recommended
values to be tested are 500, 1000 and 1500. The user can see the differences between
these values. There are three edit boxes for Canny edge detector that are sigma, low
threshold, and high threshold. Figure 4.8 is the result of the simulation after we perform
the Sobel multidirectional edge detection with threshold at 500.
Figure 4.7:
Layout of the simulation.
Figure 4.8:
Simulation using Sobel multidirectional with threshold=500.
4.6
X-ray Tomography Images
Final images in x-ray tomography after image reconstruction were .tiff files and
were 512 pixels by 512 pixels. A sample data with cancerous tumor can be seen in
Figure 4.9. The image is being viewed from a top-down perspective with the back
located at the top of the image.
Figure 4.9 is the image with liver tumor. The liver is located on the left hand side
of the image. The tumor can be seen just left of the centre of the liver as a darkish blur.
This image shows through its lack of detail that there is much room for improvement.
The image was cropped before any processing was performed. The main reason
for doing this was to decrease computational time, besides only the liver was of interest.
The image was cropped at rows 30 and 360 and column 90 and 420. Figure 4.10 is the
image after cropping and it can be seen how cropping the image resulted in placing the
liver as the focus of the CT scans.
Figure 4.9:
Sample final image in x-ray tomography.
Figure 4.10: Image after cropping.
4.7
Conclusion
Here we presented the algorithms for Laplacian, Sobel and Canny edge
detection. We also present algorithm of the histogram equalization, one method in image
enhancement. Histogram equalization chosen because the effect on equalizing the grey
level in the image makes the image clearer compared to the original one. An edge
detection method can be implemented effectively using Microsoft Visual C++. The
simulation was develop so that the user can adjust the parameter and chooses the desired
method of edge detection without write the other code. Laplacian and Sobel only have
one parameter meanwhile Canny edge detector has three parameters.
.
CHAPTER 5
RESULTS AND DISCUSSION
5.1
Introduction
Chapter 5 discusses the results from simulations and some discussion. We test
and compare the Sobel, Sobel multidirectional, Laplacian and Canny edge detection
methods using an image on liver. Since Canny contain four main steps in the algorithm,
we also present output for each step. After some comparison on the edge detection
methods, we observe that Canny is the most appropriate method for medical imaging.
Lastly, we presented the result of other test image.
5.2
Results of Edge Detection Methods
The results of different edge detection methods are presented and we compare
these methods using visual assessment. Panette and Wharton (2008) and Zhaou et all,
2005, also used visual assessment to demonstrate the performance of method of edge
detection. The user can try to run the simulation using the various value of threshold for
Sobel Multidirectional, Sobel and Laplacian method.
5.2.1
Laplacian
Figure 5.1:
Laplacian edge detection of the liver.
Figure 5.1 shows the result of applying the Laplacian edge detection to the liver
image with threshold value at 800. The Laplacian edge detection method has been easily
applies here but the method fails to detect the location of the tumor.
5.2.2
Sobel Edge Detection
Figure 5.2:
Edge detection using Sobel of the liver.
Figure 5.2 shows the result when the Sobel method is applied the vertical and
horizontal kernels. We run the method using threshold value of 800. The method also
fails to detect the location of the tumor but is successful in detecting several main edges
in the image. Sobel result well than Laplacian edge detection. The Sobel edge detector
provides good performance and is relatively insensitive to noise (Pitas, 1993), but in this
case Sobel edge detector show the bad performance, maybe due to the source of image.
Figure 5.3:
The result using Sobel multidirectional edge detection method.
The above figure shows the result using Sobel multidirectional method with a
threshold value at 500. Sobel multidirectional performs better than Sobel because, it
manage to detect more edges. We can see that Sobel multidirectional fail to detect the
existence of tumor in the image but the most successful method up to here. Besides,
more edges are detected compared to Sobel two directions in Figure 5.2. Therefore,
Sobel multidirectional should be given more attention in the field of edge detection.
5.2.3
Canny Edge Detection
Before perform Canny edge detection, we apply the histogram equalization
technique to the image. The image that we have after running the histogram equalization
on the liver tumor can be seen in Figure 5.4. It can be seen that the image detail is more
apparent and the tumor and blood vessel can be seen much more easily than the original
image.
Figure 5.4:
Image after histogram equalization.
Figure 5.5:
Gaussian smoothing of the liver.
The image is smoothed by a two dimensional Gaussian function of width
specified by a user parameter to eliminate noises. In this example we use V
2 . Once
the user enters the value of sigma, a mask is computed and the Gaussian smoothing is
performing using standard convolution methods.
Two dimensional convolution with Gaussian takes time but in practice it is
common to approximate this by two one dimensional Gaussian, one aligned with the x axis and the other with y -axis. In Figure 5.5 we can see that the edges tend to be smooth
and less noisy. We also can see that the location of the tumor as a darkish area at the left
side of the liver. The higher value of sigma, the lower is the Gaussian’s sensitivity to
noise.
Figure 5.6:
Image differentiation of the liver.
After smoothing the image, the next step is to find the edge strength by taking
the gradient of the image. The magnitude or the edge strength is approximated as the
square root of the sum of the squared derivatives values. We are using the mask
> 1
0 1@ to perform the derivatives. The above figure is the result of smoothing with a
Gaussian and differentiation in x and y direction. From Figure 5.6 the user can
differentiate the edges and non-edges. The differentiation detects the edges in the
smoothed image by putting the thick white line to those edges and the rest become dark.
Figure 5.7:
Image after locate the local maxima pixels.
Non-maxima suppression is applied to the magnitude of the gradient image to
detect the location of local maxima or the edges. First this process initialized to zero of
the resulting image, then for every pixel (except pixels at the borders) it will check the
magnitude of eight neighbor pixels according to the direction of the gradient. The idea is
the magnitude of the edge pixels should be greater than the magnitude of the gradient of
the pixels on each side of the edge. This process suppresses the non maxima pixel as
the black pixel and the local maxima pixel as the white pixel. In the simulation the
function non_max_supp performs this task. In above figure, we can see there are many
pixels with white color that indicated the pixels as the potential edges.
Figure 5.8:
Image after thresholding using hysterisis.
The thresholding process is applied to the non maxima suppression image. Here,
the potential edges in Figure 5.7 is determine whether the pixels is edges or non edges
The effect of using two value of thresholds prove to be more effective compared to a
single value of threshold. The value of low threshold is 0.6 and the high threshold is 1.5.
The recommends value of low threshold is between 0.20 and 0.60 and for the high
threshold is between 0.60 and 1.0. Actually, the input values are not original values used
in the thresholding with hysterisis. From the input, the new values of low and high
threshold are computed according to the number of pixels that are local maxima. Canny
gave a big contribution in the image processing with introducing the hysteresis
thresholding. Figure 5.8 is the final image in Canny edge detection. We can see the there
exist a tumor with specific location in the image.
5.3
Histogram Equalization
Figures 5.8 and 5.9 are the resulting images with and without performing
histogram equalization before applying the Canny edge detection. Result from the
simulation show that Figure 5.8 is much clearer and smoother than Figure 5.9.
Therefore, histogram equalization improves the Canny edge detection applied to the
medical image. This suggests the histogram equalization should be applied before Canny
edge detection on the medical images.
Figure 5.9:
Canny edge detection without performing histogram equalization.
5.5
Discussion
Laplacian edge detector use the second order derivative in the kernel and the
resulting figure shows that this method fail to detect the main edges in the CT scan
image. Sobel operator is only applied in the horizontal and vertical orientations.
However, it is possible to include the diagonal kernel.
By including a diagonal kernel, edges which are commonly missed by the horizontal
and vertical kernel are detected. However Sobel multidirectional still do not manage to
detect the existence of tumor in the image. The same result obtains from Zhao et all
2005, that Sobel and Laplacian fail to filter the noise in specific image they tested.
Histogram equalization is the process to uniformly distribute the grey level in the
image. It has been shown give an effect when combined with Canny edge detector. We
also try to compare histogram equalization with Laplacian and Sobel but the resulting
images are not much improved with the original edge detection images.
Canny edge detector has been shown to be successfully detecting the location of
the tumor in the liver image, but there is no guarantee that this method wills also success
to other images. It is because even the source is the same, but there are many other
conditions to be considered. Therefore the radiologist should study edge detection
deeply.
Objective measures on how the edge detection performs can be done on the
synthetic images using Pratt’s Figure of Merit (PFOM). Synthetic image is an image
with the known number of edges. Since the CT images are not synthetic, we can only
use visual assessment to compare the performance of edge detection method, as in
Paneta and Wharton, 2008 and Zhao et all, 2005.
Heath et all, 1996 introduces a method based on the experimental psychology
and statistics, in which human rate the output of low level vision algorithms. According
to Shirai, 1994, objective evaluation of vision algorithms is difficult without specifying
the purpose of a total system which includes the algorithm. One possible way is to
compare the performance of an algorithm with that of human vision.
The suffer in understanding Canny edge detection is worth when it success to detect
the tumor in the image. The result obtain agreed with Y. Lin et all, 2002, that the Canny
edge detector better at detecting weak edges and less sensitive to noise.
5.6
Conclusion
Sobel and Laplacian edge detector have one parameter to be adjusted by the user
that is the threshold value. With a smaller value of threshold, less edge are detected.
Canny edge detector has three parameters that are sigma, low threshold and high
threreshold. The higher value of sigma will make the image more blurred and the user
must choose the value of parameters carefully. Canny edge detector successfully detects
the location of the tumor and main edges with the help of histogram equalization. But in
real world problem, this method never promises to always be successful. Therefore, the
radiologist should consider as many as the edge detection method.
CHAPTER 6
CONCLUSION AND RECOMMENDATIONS
6.1
Introduction
In this chapter, we discuss the conclusion and recommendations. We conclude
what we have discussed in previous chapter. Then, some recommendations are provided
for future works. These will benefit those who have interest in edge detection, image
reconstruction and medical imaging.
6.2
Conclusion
Edge is the boundary between two region with huge different in grey level or
intensity. By performing edge detection, we can know information such as location, and
size of object in the image. Edge detection can be perform using derivative and is carried
out through a liner operation call convolution. The convolution kernel is up to the edge
detection method chosen. Laplacian has one kernel meanwhile Sobel has two kernel that
are vertical and horizontal kernel. In Sobel multidirectional there are four kernels to
avoid missing edge in diagonal orientation.
The advantage of Laplacian edge detection is it is easy to understand and to
implement but this edge detector is responds doubly to some edges in the image.
Therefore in practice, the Laplacian edge detection is applied to an image that has first
been smoothed with something such as Gaussian smoothing filter in order to reduce its
sensitivity to noise. Meanwhile Sobel edge detection is slightly more complex than
Laplacian but it has been shown that Sobel performs better than Laplacian edge
detection.
Canny edge detection is more complex compared to Sobel and Laplacian edge
detection, but this method is better at detecting weak edges and smoothing process
reduces the effect of noise. Parameters in Canny are allowed to be adjusted so as to
meet the requirements.
Canny edge detector used smoothing with Gaussian, derivative, non maxima
suppression and hysterisis thresholding. Experiment result show that Canny edge
detector is feasible to medical images and deserves further research. It could be used to
segment the tumor tissues in CT images. We also introduce to the histogram
equalization method to make the Canny edge detector obtain better result.
Medical images edge detection is an important work for object recognition of the
human organs and it is an important pre-pocessing step in medical image segmentation
and three dimension reconstruction (Zhao,et all, 2005). In CT scan, filtered back
projection method was used in image reconstruction from projection method to obtain
final images. We can obtain these images, because different object have difference
attenuation coefficient.
6.3
Recommendation
Today, Canny edge detection receives much attention from researcher. As a
suggestion, studies on the best values for the parameters in Canny edge detection can be
done. Besides, we also can combine Laplacian and Sobel edge detection with hysterisis
thresholding in Canny edge detection. Interested reader can also extend this project with
using the numerical analysis to compare the performance of edge detection methods.
In Chapter 3, we only discuss image reconstruction from projection briefly and a
little introduction about Radon transform. It is because of the time constraint. Actually,
this topic can be extended to discuss or introduce new or existence image reconstruction
method. This project applied the edge detection methods to the medical images. The
interested reader can actually expanded this topic to other images such as images from
satellite or aircraft in the defense field.
REFERENCES
Epstein, C.L. (2003). Introduction to the Mathematics of Medical Imaging. USA.
Prentice Hall.
Canny, J. (1986). A Computational Approach to Edge Detection. IEEE on Pattern
Analysis and Machine Intelligence. Vol. 8 , Nov. 1986
Chung, S.H, Ho, S.B.(2008). Classification of Liver Diseases MRI Images Using First
Order Statistics-Grid Computing Approach. IEEE.
Gonzalez, R.C. Woods, R.E. (2008). Digital Image Processing. USA. Preason.
Gudmundsson, M. Essam A.El-Kwae, M.R Kabuka. (1998). Edge Detection in Medical
Images Using a Genetic Algorithm. IEEE Transaction on Medical Imaging,
Vol.17, No.3 June 1998.
Hajnal, J.V, Hill, D.L.G, Hawkes, D.J. (2001). Medical Image Registration. USA. CRC
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Heath, M. Sarkar, S. Sanocki, T. Browyer. K. (1998). Comparison of Edge Detection.
Computer Vision and Image Understanding. Vol.69. No.1 Jan 1998.
Hiroyuki Kudo, (2006). Analtical Image Reconstruction Methods for Medical
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Joshi, M.A. (2001). Digital Image Processing An Algorithmic Approach. India. Prentice
Hall.
Lee, M.K Leung, S.W. Pun, T.L Cheung, H.L. Edge Detection by Genetic Algorithm.
IEEE.
Nattere, F. Wubbeling, F. (2001). Mathematical Methods in Image Reconstruction.
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Panetta K.A, Wharton E.J. “ Logarithmic Edge Detection with Application”. Journal of
Computer, Vol. 3. No.9 Sept 2008.
Parker, J.R. (1997). Algorithms for Image Processing and Computer Vision. USA. John
Wiley & Sons.
Pitas,I. (1993). Digital Image Processing Algorithms. UK.Prentice Hall. (1993).
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Rice, S.O. (1945). “Mathematical Analysis of Random Noise”. Journal of Bell Syst
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Shirai, Y. (1994). Reply to Performance Characterization in computer Vision. CVGIP.
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Sonka, M. Hlavac, V. Boyle, R. (2008). Image Processing, Analysis, and Machine
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Suri, J.S, Setarehdan, S.K. Singh, S. (2002). Advanced Algorithmic Approaches to
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Zhao Yu-qian,G.Wei-hua,C.Zhen-cheng,T.Jing-tian,L.Ling-yun. (2005). Medical
Images Edge Detection Based on Mathematical Morphology. IEEE. Engineering
in Medicine and Biology 27th Annual Conference Shanghai, China. 1-4 Sept.
2005.
APPENDIX A:
Coding of the simulation
#include"simulation.h"
BEGIN_MESSAGE_MAP(Csimulation,CFrameWnd)
ON_WM_PAINT()
ON_BN_CLICKED(IDC_MYBUTTON1, OnMyButton1)
ON_BN_CLICKED(IDC_MYBUTTON2, OnMyButton2)
ON_BN_CLICKED(IDC_MYBUTTON3, OnMyButton3)
ON_BN_CLICKED(IDC_MYBUTTON4, OnMyButton4)
ON_BN_CLICKED(IDC_MYBUTTON5, OnMyButton5)
END_MESSAGE_MAP()
Csimulation::Csimulation()
{
Home=CPoint(10,10);
sImage=CSize(M,M);
threshold=0;
Create(NULL,"Medical Imaging Through Histogram Equalization and Canny
Edge Detection",WS_OVERLAPPEDWINDOW,CRect(0,0,1050,655));
ibmp.LoadBitmap(IDB_BITMAP1);
MyButton1.Create("Clear",WS_CHILD|WS_VISIBLE|BS_DEFPUSHBUTTON
,CRect(700,550,775,600),this,IDC_MYBUTTON1);
MyButton2.Create("Laplacian",WS_CHILD|WS_VISIBLE|BS_DEFPUSHBUT
TON,CRect(500,550,575,600),this,IDC_MYBUTTON2);
MyButton3.Create("Sobel",WS_CHILD|WS_VISIBLE|BS_DEFPUSHBUTTO
N,CRect(400,550,475,600),this,IDC_MYBUTTON3);
MyButton4.Create("Canny",WS_CHILD|WS_VISIBLE|BS_DEFPUSHBUTTO
N,CRect(600,550,675,600),this,IDC_MYBUTTON4);
MyButton5.Create("Sobel
Multidirectional",WS_CHILD|WS_VISIBLE|BS_DEFPUSHBUTTON,CRect(225,550,
375,600),this,IDC_MYBUTTON5);
ex.Create(WS_CHILD|WS_VISIBLE|WS_BORDER,CRect(950,520,1000,545),
this,IDC_x);
ey.Create(WS_CHILD|WS_VISIBLE|WS_BORDER,CRect(950,555,1000,580),
this,IDC_y);
ez.Create(WS_CHILD|WS_VISIBLE|WS_BORDER,CRect(950,590,1000,615),t
his,IDC_z);
ew.Create(WS_CHILD|WS_VISIBLE|WS_BORDER,CRect(30,550,80,575),this
,IDC_w);
ew.SetFocus();
}
void Csimulation::OnPaint()
{
CPaintDC dc (this);
CDC memDC;
memDC.CreateCompatibleDC(&dc);
memDC.SelectObject(&ibmp);
dc.BitBlt(Home.x,Home.y,433,433,&memDC,0,0,SRCCOPY);
for (int rows=0;rows<M;rows++)
for (int cols=0;cols<M;cols++)
{
origBuf[M*rows+cols]=dc.GetPixel(cols,rows);
threshold += (double)origBuf[M*rows+cols];
}
dc.TextOut(850,485,"Parameters for Canny");
dc.TextOut(850,520,"Sigma");
dc.TextOut (850,555,"Low Threshold");
dc.TextOut(850,590,"High Threshold");
dc.TextOut(10,485,"Parameter for Sobel and Laplacian");
dc.TextOut(10,525,"Threshold");
}
void Csimulation::OnMyButton1()
{
CClientDC dc(this);
CBrush bkBrush(RGB(255,255,255));
CRect rc;
rc=CRect(450,0,950,450);
dc.FillRect(&rc,&bkBrush);
Invalidate();
}
void Csimulation::OnMyButton2()
{
CClientDC dc(this);
int L;
CString str1;
ew.GetWindowText(str1);
ew.GetWindowText(str1);
NN=atof(str1);
threshold /= (double)(NN*NN);
for (int rows=16-1;rows<M-1;rows++)
for (int cols=16-1;cols<M-1;cols++)
{
L=origBuf[M*rows+cols-1]+origBuf[M*(rows-1)+cols]8*origBuf[M*rows+cols]+origBuf[M*(rows+1)+cols]+origBuf[M*rows+cols+1]+origB
uf[M*(rows-1)+cols-1]+origBuf[M*(rows-1)+cols+1]+origBuf[M*(rows+1)+cols1]+origBuf[M*(rows+1)+cols+1];
L=abs(L);
if (L<=(int)threshold)
L=RGB(0,0,0);
else
L=RGB(255,255,255);
dc.SetPixel(500+cols,2+rows,L);
}
}
void Csimulation::OnMyButton3()
{
CClientDC dc(this);
int Sx,Sy,S;
CString str1;
ew.GetWindowText(str1);
ew.GetWindowText(str1);
NN=atof(str1);
threshold /= (double)(NN*NN);
for (int rows=16-1;rows<M-1;rows++)
for (int cols=16-1;cols<M-1;cols++)
{
Sy=origBuf[M*(rows-1)+cols-1]-origBuf[M*(rows+1)+cols1]+2*origBuf[M*(rows-1)+cols]-2*origBuf[M*(rows+1)+cols]+origBuf[M*(rows1)+cols+1]-origBuf[M*(rows+1)+cols+1];
Sx=origBuf[M*(rows-1)+cols-1]-origBuf[M*(rows1)+cols+1]+2*origBuf[M*rows+cols-1]2*origBuf[M*rows+cols+1]+origBuf[M*(rows+1)+cols-1]origBuf[M*(rows+1)+cols+1];
S=abs(Sx)+abs(Sy);
if (S<=(int) threshold)
S=RGB(0,0,0);
else
S=RGB(255,255,255);
dc.SetPixel(500+cols,2+rows,S);
}
}
void Csimulation::OnMyButton4()
{
CClientDC dc(this);
int i,j;
CString str2,str3,str4;
ex.GetWindowText(str2);
sigma1=atoi(str2);
histogram(origBuf,M,M);
Gaussian_smoothing(HE,sigma1,M,M);
derrivative_x_y(smoothedim,M,M);
non_max_supp(magnitude,delta_x,delta_y,M,M);
ey.GetWindowText(str3);
ez.GetWindowText(str4);
tlow=atof(str3);
thigh=atof(str4);
hysteresis(magnitude,result,M,M,thigh,tlow,aftBuf);
int color;
for (int rows=15;rows<M-1;rows++)
for (int cols=15;cols<M-1;cols++)
{
color=aftBuf[M*rows+cols];
dc.SetPixel(500+cols,10+rows,COLORREF
RGB(color,color,color));
}
}
unsigned char Csimulation::histogram(unsigned char *origBuf, int rows,int cols)
{
int i,j;
double sum,temp1,temp[256];
double H[256],HC[256];
int T[256];
for(i=0;i<256;i++)
{
H[i]=0;
temp[i]=0;
T[i]=0;
HC[i]=0;
}
HC[0]=0;
for (i=0;i<rows;i++)
for (j=0;j<cols;j++)
{
H[origBuf[rows*i+j]]+=1.0;
}
for(i=0;i<256;i++)
H[i]/=(rows*cols);
sum=0.0;
for(i=0;i<256;i++)
{
sum+=H[i];
HC[i]=sum;
}
for(i=0;i<256;i++)
{
temp1=(256*H[i]);
temp1=temp1-1;
if (temp1>0.0)
T[i]=(int)temp1;
else
T[i]=0;
}
double gmin,gmax;
gmin=gmax=0;
for(i=0;i<256;i++)
{
if (H[i]!=0)
{
gmin=i;
break;
}
}
gmin=gmin/255;
for(i=255;i>=0;i--)
{
if (H[i]!=0)
{
gmax=i;
break;
}
}
gmax=gmax/255;
for(i=0;i<256;i++)
{
temp[i]=((gmax-gmin)*HC[i])+gmin;
T[i]=(int)(temp[i]*255);
}
for (i=0;i<rows;i++)
for (j=0;j<cols;j++)
{
HE[rows*i+j]=T[origBuf[rows*i+j]];
}
return *HE;
}
int Csimulation::Gaussian_smoothing(unsigned char *origBuf,int sigma,int rows,int
cols)
{
int r,c,rr,cc,windowsize,center;
float dot,sum;
int i;
{
float x,fx,sum=0.0;
windowsize=1+2*ceil(2.5*sigma);
center=windowsize/2;
for(i=0;i<windowsize;i++)
{
x=(float)(i-center);
fx=pow(2.71828,0.5*x*x/(sigma*sigma))/(sigma*sqrt(6.2831853));
kernel[i]=fx;
sum+=fx;
}
for(i=0;i<windowsize;i++)
kernel[i]/=sum;
}
center=windowsize/2;
for(r=0;r<rows;r++)
{
for(c=0;c<cols;c++)
{
dot=0.0;
sum=0.0;
for(cc=(-center);cc<=center;cc++)
{
if(((c+cc)>=0&&((c+cc)<cols)))
{
dot+=(float)origBuf[r*cols+(c+cc)]*kernel[center+cc];
sum+=kernel[center+cc];
}
}
tempim[M*r+c]=dot/sum;
}
}
for(c=0;c<cols;c++)
{
for(r=0;r<rows;r++)
{
dot=0.0;
sum=0.0;
for(rr=(-center);rr<=center;rr++)
{
if(((r+rr)>=0&&((r+rr)<rows)))
{
dot+=tempim[(r+rr)*cols+c]*kernel[center+rr];
sum+=kernel[center+rr];
}
}
smoothedim[r*M+c]=(int)(dot*BOOSTBLURFACTOR/sum+0.5);
if(smoothedim[r*M+c]>M-1)
smoothedim[r*M+c]=M-1;
}
}
return *smoothedim;
}
int Csimulation::derrivative_x_y(int *smoothedim,int rows,int cols)
{
int r,c,pos,sq1,sq2;
for(r=0;r<M;r++)
{
pos=r*M;
delta_x[pos]=smoothedim[pos+1]-smoothedim[pos];
pos++;
for(c=1;c<M-1;c++,pos++)
{
delta_x[pos]=smoothedim[pos+1]-smoothedim[pos-1];
}
}
for(c=0;c<M;c++)
{
pos=c;
delta_y[pos]=smoothedim[pos+M]-smoothedim[pos];
pos+=M;
for(r=1;r<M-1;r++,pos+=M)
{
delta_y[pos]=smoothedim[pos+M]-smoothedim[pos-M];
}
}
for(r=0,pos=0;r<rows;r++)
{
for(c=0;c<cols;c++,pos++)
{
sq1=(int)delta_x[pos]*(int)delta_x[pos];
sq2=(int)delta_y[pos]*(int)delta_y[pos];
magnitude[pos]=(short)(50.0+sqrt((float)sq1+(float)sq2));
}
}
return *magnitude;
}
unsigned char Csimulation::non_max_supp(short
*magnitude,short*delta_x,short*delta_y,int rows,int cols)
{
short z1,z2;
short m00,gx,gy;
float mag1,mag2,xperp,yperp;
int rowcount,colcount,count;
short *magrowptr,*magptr;
short *gxrowptr,*gxptr;
short *gyrowptr,*gyptr;
unsigned char *resultrowptr,*resultptr;
for(count=0,resultrowptr=result,resultptr=result+M*M-1;
count<M;resultptr++,resultrowptr++,count++)
{
*resultrowptr=*resultptr=(unsigned char)0;
}
for(count=0,resultptr=result,resultrowptr=result+M-1;
count<M;count++,resultptr+=M,resultrowptr+=M)
{
*resultptr=*resultrowptr=(unsigned char) 0;
}
for(rowcount=1,magrowptr=magnitude+M+1,gxrowptr=delta_x+M+1,
gyrowptr=delta_y+M+1,resultrowptr=result+M+1;rowcount<M2;rowcount++,magrowptr+=M,gyrowptr+=M,gxrowptr+=M,resultrowptr+=M)
{
for(colcount=1,magptr=magrowptr,gxptr=gxrowptr,gyptr=gyrowptr,resultptr=res
ultrowptr;colcount<M-2;colcount++,magptr++,gxptr++,gyptr++,resultptr++)
{
m00=*magptr;
if(m00==0)
{
*resultptr=(unsigned char)NOEDGE;
}
else
{
xperp=-(gx=*gxptr)/((float)m00);
yperp=(gy=*gyptr)/((float)m00);
}
if(gx>=0)
{
if(gy>=0)
{
if(gx>=gy)
{
z1=*(magptr-1);
z2=*(magptr-M-1);
mag1=(m00-z1)*xperp+(z2-z1)*yperp;
z1=*(magptr+1);
z2=*(magptr+M+1);
mag2=(m00-z1)*xperp+(z2-z1)*yperp;
}
else
{
z1=*(magptr-M);
z2=*(magptr-M-1);
mag1=(z1-z2)*xperp+(z1-m00)*yperp;
z1=*(magptr+M);
z2=*(magptr+M+1);
mag2=(z1-z2)*xperp+(z1-m00)*yperp;
}
}
else
{
if(gx>=-gy)
{
z1=*(magptr-1);
z2=*(magptr+M-1);
mag1=(m00-z1)*xperp+(z1-z2)*yperp;
z1=*(magptr+1);
z2=*(magptr-M+1);
mag2=(m00-z1)*xperp+(z1-z2)*yperp;
}
else
{
z1=*(magptr+M);
z2=*(magptr+M-1);
mag1=(z1-z2)*xperp+(m00-z1)*yperp;
z1=*(magptr-M);
z2=*(magptr-M+1);
mag2=(z1-z2)*xperp+(m00-z1)*yperp;
}
}
}
else
{
if((gy=*gyptr)>=0)
{
if(-gx>=gy)
{
z1=*(magptr+1);
z2=*(magptr-M+1);
mag1=(z1-m00)*xperp+(z2-z1)*yperp;
z1=*(magptr-1);
z2=*(magptr+M-1);
mag2=(z1-m00)*xperp+(z2-z1)*yperp;
}
else
{
z1=*(magptr-M);
z2=*(magptr-M+1);
mag1=(z1-z2)*xperp+(z2-m00)*yperp;
z1=*(magptr+M);
z2=*(magptr+M-1);
mag2=(z2-z1)*xperp+(z2-m00)*yperp;
}
}
else
{
if(-gx>-gy)
{
z1=*(magptr+1);
z2=*(magptr+M+1);
mag1=(z1-m00)*xperp+(z1-z2)*yperp;
z1=*(magptr-1);
z2=*(magptr-M-1);
mag2=(z1-m00)*xperp+(z1-z2)*yperp;
}
else
{
z1=*(magptr+M);
z2=*(magptr+M+1);
mag1=(z2-z1)*xperp+(m00-z1)*yperp;
z1=*(magptr-M);
z2=*(magptr-M-1);
mag2=(z2-z1)*xperp+(m00-z1)*yperp;
}
}
}
if((mag1>0.0)||(mag2>0.0))
{
*resultptr=(unsigned char)NOEDGE;
}
else
{
if (mag2==0.0)
*resultptr=(unsigned char)NOEDGE;
else
*resultptr=(unsigned char)POSSIBLE_EDGE;
}
}
}
return *resultptr;
}
unsigned char Csimulation::hysteresis(short *magnitude,unsigned char*result,int
rows,int cols,float tlow,float thigh,unsigned char*aftBuf)
{
int r,c,pos,numedges,highcount;
int lowthreshold,highthreshold,hist[32768];
short int maximum_mag;
for(r=0,pos=0;r<M;r++)
{
for(c=0;c<M;c++,pos++)
{
if(result[pos]==POSSIBLE_EDGE)
aftBuf[pos]=POSSIBLE_EDGE;
else
aftBuf[pos]=NOEDGE;
}
}
for(r=0,pos=0;r<M;r++,pos+=M)
{
aftBuf[pos]=NOEDGE;
aftBuf[pos+M-1]=NOEDGE;
}
pos=(M-1)*M;
for(c=0;c<cols;c++,pos++)
{
aftBuf[c]=NOEDGE;
aftBuf[pos]=NOEDGE;
}
for(r=0;r<32768;r++)
hist[r]=0;
for(r=0,pos=0;r<M;r++)
{
for(c=0;c<M;c++,pos++)
{
if(aftBuf[pos]==POSSIBLE_EDGE)
hist[magnitude[pos]]++;
}
}
for(r=1,numedges=0;r<32768;r++)
{
if(hist[r]!=0)
maximum_mag=r;
numedges+=hist[r];
}
highcount=(int)(numedges*thigh+0.5);
r=1;
numedges=hist[1];
while((r<(maximum_mag-1))&&(numedges<highcount))
{
r++;
numedges+=hist[r];
}
highthreshold=r;
lowthreshold=(int)(highthreshold*tlow+0.5);
for(r=0,pos=0;r<rows;r++)
{
for(c=0;c<cols;c++,pos++)
{
if((aftBuf[pos]==POSSIBLE_EDGE)&&(magnitude[pos]>=highthreshold))
{
aftBuf[pos]=EDGE;
follow_edges((aftBuf+pos),(magnitude+pos),lowthreshold,M);
}
}
}
for(r=0,pos=0;r<rows;r++)
{
for(c=0;c<cols;c++,pos++)
if (aftBuf[pos]!=EDGE)
aftBuf[pos]=NOEDGE;
}
float percent;
CString str;
percent=float((r*100)/M);
return *aftBuf;
}
void Csimulation::follow_edges(unsigned char *aftBuf,short*magnitude,short lowval,int
cols)
{
short *tempmag;
unsigned char *tempaft;
int i;
int x[8]={1,1,0,-1,-1,-1,0,1},
y[8]={0,1,1,1,0,-1,-1,-1};
for(i=0;i<8;i++)
{
tempaft=aftBuf-(y[i]*M)+x[i];
tempmag=magnitude-y[i]*cols+x[i];
if ((*tempaft==POSSIBLE_EDGE)&&(*tempmag>lowval))
{
*tempaft=(unsigned char)EDGE;
follow_edges(tempaft,tempmag,lowval,M);
}
}
}
void Csimulation::OnMyButton5()
{
CClientDC dc(this);
int Sx,Sy,Sd1,Sd2,S;
CString str1;
ew.GetWindowText(str1);
ew.GetWindowText(str1);
NN=atof(str1);
threshold /= (double)(NN*NN);
for (int rows=17-1;rows<M-2;rows++)
for (int cols=17-1;cols<M-2;cols++)
{
Sy=origBuf[M*(rows-1)+cols-1]-origBuf[M*(rows+1)+cols1]+2*origBuf[M*(rows-1)+cols]-2*origBuf[M*(rows+1)+cols]+origBuf[M*(rows1)+cols+1]-origBuf[M*(rows+1)+cols+1];
Sx=origBuf[M*(rows-1)+cols-1]-origBuf[M*(rows1)+cols+1]+2*origBuf[M*rows+cols-1]2*origBuf[M*rows+cols+1]+origBuf[M*(rows+1)+cols-1]origBuf[M*(rows+1)+cols+1];
Sd1=origBuf[M*rows+cols-2]+2*origBuf[M*(rows+2)+cols1]+origBuf[M*(rows+2)+cols]-origBuf[M*(rows-2)+cols]-2*origBuf[M*(rows1)+cols+1]-origBuf[M*rows+cols+2];
Sd2=-origBuf[M*rows+cols-2]-2*origBuf[M*(rows-1)+cols-1]origBuf[M*(rows2)+cols]+origBuf[M*(rows+2)+cols]+2*origBuf[M*(rows+1)+cols+1]+origBuf[M*row
s+cols+2];
S=abs(Sx)+abs(Sy)+abs(Sd1)+abs(Sd2);
if (S<=(int) threshold)
S=RGB(0,0,0);
else
S=RGB(255,255,255);
dc.SetPixel(500+cols,2+rows,S);
}
}