|
Academic Open Internet Journal |
Volume 15, 2005 |
Boundary Mapping of Chromosome
Images using
Gradient Vector Flow Active
Contours and Investigations
A.Prabhu Britto1
& Dr.G.Ravindran2
1Research
Scholar, Center for Medical Electronics, Dept. of ECE, Anna University, Chennai
600 025
2Chairman,
Faculty of Information and Communication Engineering,
Abstract
In this paper it is proposed to identify and investigate
optimality of a suitable boundary mapping technique for Chromosome images. This work is expected to yield a robust
technique which can be used to boundary map chromosome images from chromosome
spread images, having variability in shape and size. Weak edges are also manifested here. Gradient Vector Flow field Active Contours
are studied and found to have good convergence properties. Boundary mapping using Gradient Vector Flow
Active Contours is done on chromosome spread images. It is found that a unique set of parameter
values for the technique is required for boundary mapping every chromosome
image. Characterization studies have
shown that an optimal range of values exists for each parameter within which
good boundary mapping results can be obtained for various chromosomes in similar
class of images. Statistical testing
validates that the experimental results are significant.
Keywords: Gradient Vector Flow, Active Contours,
Deformable Curves, Chromosome, Boundary Mapping, Characterization
1. Introduction
The development of robust techniques in
Image Processing and Computer Vision Techniques is one of the primary tasks in
aiding Human Interpretation and Automated Machine Perception. Segmentation Techniques are widely used in
various areas in medicine and industry to extract significant meaningful
information from objects of interest in images.
Segmentation assumes the presence of some image property of interest and
attempts to precisely localize areas that possess that property. Variations in image properties demand highly
efficient techniques for information extraction from images. Hence, the development of a robust
segmentation technique that extracts best information content from images belonging
to similar classes of images having variable image properties is an important
task. The extracted information should
be fit to be used subsequently as input for higher image processing routines or
to form conclusive mathematical opinions.
Manual segmentation by a skilled operator
can be influenced by operator bias and reproducible results may be difficult to
achieve. Automated Segmentation using
mathematical techniques is not influenced by bias and the results can always be
reproduced.
Mathematical description of object
boundaries, otherwise called as “Boundary Mapping”, is a fundamental
segmentation approach that can be easily done in high-contrast noise-free
images using only image information by employing low-level techniques,
traditional edge detectors, region growing or mathematical morphology. These techniques are generally computationally
fast and may sometimes require expert interactive guidance. When these techniques are applied for image
segmentation, noise and other artifacts can possibly cause incorrect
segmentation or boundary discontinuities in segmented objects [1].
The classical boundary mapping techniques,
namely, region growing, relaxation labeling, edge detection and linking suffer
from limitations. Edge or seed points
are identified and are used to construct the boundary considering local
information. This may lead to incorrect
assumptions during the boundary integration process generating boundaries that do
not closely approximate the actual object boundary. It is difficult to automate classical
boundary mapping techniques because complexities in shape, variety and
variability exist in anatomical structures in medical images. Other difficulties include imaging conditions
which introduce further variability in image characteristics. Further the clinical significance factor requires
utmost care in processing medical images.
Hence, high-level segmentation techniques
capable of overcoming these difficulties are explored. However, one formidable task in medical
imaging is the boundary mapping of images with weak edges, highly variable
shapes, variable image properties and variable object locations in similar
images. This requires careful selection
of parameters for any technique that is applied for segmentation and the
parameter values may be image specific which cannot be applied optimally across
a class of images to yield better results.
This work aims to obtain accurate
segmentation results from a class of images that have variable properties as
outlined in the earlier paragraph. The
expected outcome would result in obtaining a universal set of parameter values for
the adopted segmentation scheme that can be applied for similar class of images
that have variability in image properties.
Chromosome spread images have variability
in image properties and hence the task of Boundary Mapping of Chromosome spread
images using high-level segmentation techniques is chosen with the perspective
of obtaining a boundary mapping technique that can be applied universally over
a similar class of images. Prospective high-level segmentation techniques for
boundary mapping were studied and Gradient Vector Flow (GVF) Active Contours
were chosen. Further discussion on GVF
Active Contours and its suitability to the task of boundary mapping in
chromosome spread images and its application can be found in subsequent
paragraphs. GVF Active contour is a parametric
representation providing a compact, analytical description of object shape.
The main advantage of Active Contour
models is the ability to generate closed parametric curves from images and
incorporation of a smoothness constraint that provides robustness to noise and
spurious edges. It can always be argued
that low-level boundary mapping techniques could be used to boundary map the
same chromosome spread images. But these
techniques do not yield higher intelligence about the boundary maps of
chromosome spread images. Hence, parametric
representations of Active Contour models are chosen as they are capable of
yielding higher intelligence that could be used as input for further processes
designed to carry out clinical investigations.
Active Contours also called as Snakes or
Deformable Curves, first proposed by Kass et al. [2] are energy-minimizing
contours that apply information about the boundaries as part of an optimization
procedure. They are generally
initialized around the object of interest by automatic or manual process. The contour then deforms itself from its
initial position in conformity with nearest dominant edge feature by minimizing
the energy governing the formulation of the active contour.
The Active Contour Models, which are
connected and continuous consider an object boundary as a whole, and make use
of apriori knowledge of object shape to constrain the segmentation
problem. The inherent continuity and
smoothness of the Active Contour models compensates for noise, gaps and other
irregularities in object boundaries.
In this work the various parameters in the
active contour formulation are investigated for an optimal selection. Hence the same optimal set of parameters is
expected to be used for boundary mapping of all chromosomes in spite of
variability in terms of shape and features in similar classes of chromosome spread
images. The expected outcome would
result in obtaining a universal set of parameter values that can be applied for
similar class of images having variability in image properties.
2. Active Contour Models
Active Contour Models or Deformable Models
can be specified as physically motivated, model based techniques for
delineating region boundaries using closed parametric curves or surfaces that
deform under the influence of internal and external forces. All Active Contour models generally follow
the same methodology to map boundaries of objects. First, a closed curve is initialized either
manually or automatically near the boundary of the object of interest. The initial curve is then allowed to undergo an
iterative relaxation process which is guided by internal and external
forces. Internal forces which enforce
smoothness of the curve are computed from within the Active Contour. External forces are derived from the image
and help to drive the curve toward the desired image features of interest during
the course of the iterative process.
The energy function consisting of the
internal and external forces is minimized during the application of an Active
Contour model to images, thus making the model active. The energy minimization process can be viewed
as a dynamic problem where the active contour model is governed by the laws of
elasticity and lagrangian dynamics [3] and the model evolves according to the
forces acting on it until equilibrium of all forces is reached, which is
equivalent to a minimum of the energy function.
The framework of active contour models comprises mainly of the
representation of the model, the energy function and the optimization of the
energy function.
3. Formulation of Active Contour Models
An Active Contour Model can be represented
by a curve c, as a function of its
arc length τ,
-- (1) with τ = [0…1].
A contour is a closed curve, and hence to
define a closed curve c(0) is set to equal c(1). This representation requires an analytic form
of the curve c. Finite differences are
suggested to obtain a polygonal approximation of the curve. Such a discrete model can be expressed as an
ordered set of n vertices vi = (xi,yi)T with v=(v1,…,vn). The
only difficulty with this discrete representation is that a very large number
of vertices are required to achieve accuracy, which in turn could lead to high
computational complexity and numerical instability [3].
This work focuses on parametric deformable
curves, which synthesize parametric curves within an image domain and allow
them to move toward edges under the influence of internal forces coming from
within the model itself and external forces computed from the image data. Hence the scope of this discussion is
restricted to the relevance to this work.
Mathematically, an active contour model
can be defined in discrete form as a curve x
that moves through the spatial domain of an image to
minimize the energy functional
-- (2)
where α
and β are weighting parameters that control the active contour’s tension
and rigidity respectively [4] , and x’(s) and x”(s)
denote the first and second derivatives of x(s) with respect to s. The first
order derivative discourages stretching and makes the snake behave like an
elastic string. The second order derivative discourages bending and makes the
snake behave like a rigid rod. The weighting parameters of tension and rigidity,
viz., α,β govern
the effect of the derivatives on the snake.
The
external energy function Eext is derived
from the image so that it takes on its smaller values at the features of
interest such as boundaries and guides the active contour towards the
boundaries. The external energy is
defined by
-- (3)
where Gσ(x,y) is a
two-dimensional Gaussian function with standard deviation σ and I(x,y) represents the image, and κ is the external force
weight. This external energy is
specified for a line drawing (black on white) and positive κ is used. For other types of images, the external
energy undergoes a slight modification. A motivation for applying some Gaussian
filtering to the underlying image is to reduce noise.
An active contour that minimizes E must satisfy the Euler Equation
-- (4), where
and 
comprise the components of a force balance
equation such that 
-- (5)
The internal force Fint
discourages stretching and bending while the external potential force Fext drives the active contour towards the
desired image boundary. Eq. (4) is
solved by making the active contour dynamic by treating x as a function of time
t as well as s. Then the partial derivative
of x with respect to t is then set equal to the left hand side of Eq. (4) as
follows 
-- (6)
A solution to Eq. (6) can be obtained by
discretizing the equation and solving the discrete system iteratively [2]. When the solution x(s, t) stabilizes, the
term xt(s, t) vanishes and a solution of Eq. (4) is achieved.
Traditional active contour models suffer
from a few severe drawbacks. Boundary
concavities leave the contour split across the boundary. Capture range is also limited. Methods suggested to overcome these
difficulties, namely multiresolution methods [5], pressure forces [6], distance
potentials [7], control points [8], domain adaptivity [9], directional
attractions [10] and solenoidal fields[11], however solved one problem but
introduced new ones[12].
Hence, a new class of external fields
called Gradient Vector Flow fields [12, 13] was suggested to overcome the
difficulties in traditional active contour models. Gradient Vector Flow fields are obtained by
solving a vector diffusion equation that diffuses the gradient vectors of a
gray-level edge map computed from the image.
Hence, the active contour models using Gradient Vector Flow (GVF) fields
as the external force are called as Gradient Vector Flow (GVF) Active Contours. The GVF active contour model cannot be
written as the negative gradient of a potential function. Hence it is directly specified from a dynamic
force equation instead of the standard energy minimization network.
The external forces arising out of GVF
fields are non-conservative forces as they cannot be written as gradients of
scalar potential functions. The usage of
non-conservative forces as external forces show improved performance of
Gradient Vector Flow field Active Contours compared to traditional
energy-minimizing active contours [12, 13].
The GVF field points towards the object
boundary when very near to the boundary, but varies smoothly over homogeneous
image regions extending to the image border.
Hence the GVF field can capture an active contour from long range from
either side of the object boundary and can force it into the object boundary. Thus the GVF Active Contour model has a large
capture range. Hence, the GVF active
contour model is insensitive to initialization of the contour and also flexible
in initialization. The gradient vectors
are normal to the boundary surface but by combining the Laplacian and the Gradient,
the result is not the normal vectors to the boundary surface. As a result of this, the GVF field yields
vectors that point into boundary concavities so that the active contour is
driven through the concavities. It is thus
able to move into boundary concavities.
Information regarding whether the initial contour should expand or
contract need not be given to the GVF active contour model. The GVF is very useful when there are
boundary gaps, because it preserves the perceptual edge property of active
contours [2, 13].
Hence, the GVF active contour model is
judged to be the most suitable Active Contour formulation for boundary mapping
of chromosome spread images.
4. Gradient Vector Flow (GVF) Active Contours
The GVF field is defined as the
equilibrium solution to the following vector diffusion equation [12],
-- (7a)
-- (7b)
where, ut
denotes the partial derivative of u(x,t) with respect
to t, 
is the Laplacian operator (applied to each spatial component
of u separately), and f is an edge map that has a higher value at the desired
object boundary. The functions in “g” and “h” control the amount of diffusion
in GVF. In Eq. (7a), 
produces a smoothly
varying vector field, and hence called as the “smoothing term”, while 
encourages the vector field u to be close to 
computed from the image data and hence called as the data
term. The weighting functions 
and 
apply to the smoothing and data terms respectively and they
are chosen as 
and 
[13]. is
constant here, and smoothing occurs everywhere, while grows larger near strong edges and dominates at boundaries.
Hence, the Gradient Vector Flow field is
defined as the vector field v
that minimizes the energy functional
-- (8)
The effect of this variational
formulation is that the result is made smooth when there is no data. When the gradient of the edge map is large,
it keeps the external field nearly equal to the gradient, but keeps field to be
slowly varying in homogeneous regions where the gradient of the edge map is
small, i.e., the gradient of an edge map has vectors point toward the edges, which are normal to the
edges at the edges, and have magnitudes only in the immediate vicinity of the
edges, and in homogeneous regions is nearly zero.
µ is a regularization parameter that
governs the tradeoff between the first and the second term in the integrand in
Eq. (8). The solution of Eq. (8) can be
done using the Calculus of Variations and further by treating u and v as
functions of time, solving them as generalized diffusion equations [13].
5. Results AND CONCLUSION
The chromosome metaphase image (size 480 x 512 pixels at 72 pixels
per inch resolution) was taken and preprocessed. The image was made to undergo
minimal preprocessing so that the goal of boundary mapping in chromosome images
with very weak edges is maintained. Insignificant
and unnecessary regions in the image were removed interactively. Automatic selection of chromosomes is not
done as this work concentrates on information from a single chromosome only, at
any instant of time. The chromosome of
interest was selected by user selection of a few points on the outer periphery
of the chromosome spread image. These
selected points were made to form the vertices of a polygon. On constructing the perimeter of the polygon,
seed points for the initial contour were determined automatically by
periodically selecting every third pixel along the perimeter of the
polygon. The GVF deformable curve was
then allowed to deform until it converged to the chromosome boundary.
The optimum parameters for the deformable curve with respect to
the Chromosome images were determined by tabulated studies. The GVF Active contour is governed by the
following parameters, namely, σ, µ, α, β and κ.
σ determines the Gaussian filtering that is
applied to the image to generate the external field. Larger value of σ will cause the
boundaries to become blurry and distorted, and can also cause a shift in the
boundary location. However, large values
of σ are necessary to increase the capture range of the active contour.
µ is a regularization parameter in the energy functional given by
Eq. (8), and requires a higher value in the presence of noise in the
image.
α determines the tension of the active
contour and β determines the rigidity of the contour. The tension keeps the active contour
contracted and the rigidity keeps it smooth.
α and β may also take on value
zero. This implies that the influence of
the respective tension and rigidity terms in the diffusion equation is
low.
κ is the external force weight that
determines the strength of the external field that is applied. The GVF field was built up in 80 iterations
and the Contour iterations were set to 40.
5.1 EXPERIMENTAL RESULTS
A few output samples are presented here.
The figures show original chromosome image samples, their
corresponding GVF fields and boundary mapped chromosome images.
5.2 Experimental Validation
In order to quantify the performance of a
segmentation method, validation experiments are necessary. Validation is typically performed using one or
two different types of truth models.
When no ground truth model is available, validation is performed on an
ordinal or ranking scale and quantified.
In this work, ground truth model is not available and hence validation
is performed on ordinal or ranking scale and then quantified.
A set of 10 random samples is taken and
characterization of each parameter is done.
The outputs were tabulated in ranking order with “1” describing the best
quality output and as the quality decreases the rank increases up to rank
“97”. Rank “98” is a special case, where
the output image is rejected based on quality or the output image is not
available due to numerical instability possibly caused due to the greater
number of contour points [3]. The tables
represent characterization studies for each parameter. The statistical median is used to judge the
distribution of values for each parameter value for all samples. When the median leans towards the lower
values, i.e., towards “1”, it indicates that almost 50% of the outputs lean
towards “1”, which is a good indicator of obtaining majority good outputs on
various samples, making that particular parameter value an optimal one. In each characterization study, a value for
the parameter is chosen which gives the maximum good quality outputs for all
samples. Since, each table denotes
variation for only one parameter either between the lower and upper limits of
the parameter or between the lower and upper limits giving significantly
different output, with the other parameters taking a constant value, the best
parameter value of that table is the one that gives maximum good quality
outputs for all samples or a majority of samples. Due to the exhaustive study on every
parameter treating the other parameters as constants, statistical testing using
other statistical techniques like PCA or multivariate analysis need not be
done.
The characterization studies reveal that
each parameter sometimes has an optimal range within which it can assume any
value thereby giving majority good outputs for all samples. But for the sake of experimental purposes,
only that investigated discrete value of each parameter that gave optimal best
output was chosen. Hence, the
characterization has yielded optimal values for the GVF parameters which can be
used for similar class of images to obtain good boundary mapping results.
An important point to be noted is that
characterization studies have been performed for those parameter values which
give either significant output or significant difference in performance between
adjacent parameter values. Those
parameter values where there is no significant difference between adjacent
parameter values have not been tabulated.
Also, those parameter values outside the tabulated range which gave no
proper results have not been tabulated.
The following tables show the
characterization experimental results in the ranking scale, where “1” denotes the
best quality output and the rank increases numerically with decreasing quality
up to “98”.
Table 1. Characterization of Sigma
|
gvfsigma |
0.125 |
0.25 |
0.5 |
1 |
1.5 |
|
sample 1 |
45 |
45 |
37 |
45 |
81 |
|
sample 2 |
50 |
34 |
87 |
11 |
12 |
|
sample 3 |
49 |
38 |
37 |
33 |
98 |
|
sample 4 |
48 |
45 |
31 |
31 |
16 |
|
sample 5 |
50 |
98 |
98 |
98 |
98 |
|
sample 6 |
48 |
46 |
46 |
58 |
46 |
|
sample 7 |
98 |
97 |
98 |
98 |
98 |
|
sample 8 |
90 |
50 |
98 |
98 |
98 |
|
sample 9 |
86 |
45 |
52 |
98 |
47 |
|
sample 10 |
77 |
35 |
52 |
82 |
82 |
|
median |
50 |
45 |
52 |
70 |
82 |
In Table 1, the median indicates that the
acceptable optimal range of σ extends from 0.125 to 0.5. The best value compared qualitatively amongst
those tested is 0.25 and hence it is chosen for performing further
characterization.
Table 2. Characterization of Mu
|
gvfmu |
0.005 |
0.01 |
0.1 |
0.15 |
0.2 |
|
sample 1 |
29 |
29 |
45 |
39 |
62 |
|
sample 2 |
29 |
29 |
34 |
33 |
29 |
|
sample 3 |
34 |
34 |
38 |
45 |
50 |
|
sample 4 |
31 |
31 |
45 |
32 |
47 |
|
sample 5 |
98 |
98 |
98 |
90 |
90 |
|
sample 6 |
37 |
36 |
46 |
62 |
61 |
|
sample 7 |
98 |
98 |
97 |
98 |
98 |
|
sample 8 |
91 |
98 |
50 |
97 |
97 |
|
sample 9 |
29 |
29 |
45 |
47 |
45 |
|
sample 10 |
45 |
45 |
35 |
35 |
45 |
|
median |
36 |
35 |
45 |
46 |
56 |
In Table 2, the median indicates that the
acceptable optimal range of µ extends from 0.005 to 0.15. The best value compared qualitatively amongst
those tested is 0.01 and hence it is chosen for performing further
characterization.
Table 3. Characterization of Alpha
|
gvfalpha |
0.2 |
0.25 |
0.3 |
0.35 |
0.4 |
0.5 |
1 |
|
sample 1 |
29 |
87 |
13 |
70 |
13 |
31 |
29 |
|
sample 2 |
29 |
13 |
13 |
29 |
29 |
29 |
29 |
|
sample 3 |
29 |
29 |
29 |
75 |
29 |
29 |
98 |
|
sample 4 |
31 |
31 |
31 |
32 |
31 |
31 |
77 |
|
sample 5 |
98 |
98 |
98 |
98 |
98 |
98 |
98 |
|
sample 6 |
98 |
36 |
38 |
36 |
38 |
36 |
55 |
|
sample 7 |
98 |
98 |
98 |
98 |
98 |
98 |
98 |
|
sample 8 |
98 |
98 |
98 |
98 |
98 |
98 |
36 |
|
sample 9 |
31 |
13 |
31 |
13 |
13 |
13 |
29 |
|
sample 10 |
15 |
15 |
45 |
32 |
29 |
31 |
45 |
|
median |
31 |
34 |
35 |
53 |
30 |
31 |
50 |
In Table 3, the median indicates that the
acceptable optimal range of α extends from 0.2 to 0.5. The best value compared qualitatively amongst
those tested is 0.3 and hence it is chosen for performing further
characterization.
Table 4. Characterization of Beta
|
Gvfbeta |
0 |
0.2 |
0.4 |
0.6 |
0.8 |
1 |
|
sample 1 |
13 |
29 |
29 |
31 |
45 |
46 |
|
sample 2 |
13 |
13 |
77 |
34 |
29 |
46 |
|
sample 3 |
29 |
45 |
77 |
97 |
29 |
97 |
|
sample 4 |
31 |
31 |
31 |
31 |
77 |
31 |
|
sample 5 |
98 |
98 |
98 |
98 |
98 |
98 |
|
sample 6 |
38 |
36 |
36 |
38 |
42 |
37 |
|
sample 7 |
98 |
98 |
98 |
98 |
98 |
98 |
|
sample 8 |
98 |
98 |
98 |
98 |
98 |
98 |
|
sample 9 |
31 |
29 |
31 |
15 |
31 |
31 |
|
sample 10 |
45 |
39 |
47 |
47 |
47 |
45 |
|
Median |
35 |
38 |
62 |
43 |
46 |
46 |
In Table 4, the median indicates that the
acceptable optimal range of β extends from 0 to 0.2. The best value compared qualitatively amongst
those tested is 0 and hence it is chosen for performing further
characterization.
Table 5. Characterization of Kappa
|
gvfkappa |
0.1 |
0.2 |
0.25 |
0.3 |
0.35 |
0.4 |
0.45 |
0.5 |
0.6 |
0.8 |
1 |
1.2 |
|
sample 1 |
97 |
97 |
97 |
29 |
29 |
29 |
46 |
29 |
13 |
29 |
29 |
77 |
|
sample 2 |
97 |
77 |
77 |
77 |
29 |
29 |
29 |
38 |
13 |
13 |
38 |
13 |
|
sample 3 |
97 |
13 |
13 |
75 |
34 |
50 |
29 |
11 |
29 |
45 |
74 |
77 |
|
sample 4 |
97 |
78 |
79 |
80 |
29 |
29 |
29 |
29 |
31 |
98 |
98 |
77 |
|
sample 5 |
97 |
97 |
97 |
97 |
98 |
98 |
98 |
98 |
98 |
98 |
98 |
98 |
|
sample 6 |
97 |
97 |
40 |
38 |
37 |
36 |
52 |
46 |
38 |
98 |
98 |
98 |
|
sample 7 |
97 |
97 |
97 |
98 |
97 |
98 |
98 |
98 |
98 |
98 |
98 |
98 |
|
sample 8 |
97 |
97 |
97 |
60 |
97 |
54 |
98 |
98 |
98 |
98 |
98 |
98 |
|
sample 9 |
97 |
78 |
77 |
77 |
77 |
29 |
87 |
46 |
31 |
29 |
29 |
29 |
|
sample 10 |
97 |
49 |
45 |
45 |
45 |
45 |
46 |
46 |
45 |
13 |
13 |
37 |
|
median |
97 |
88 |
78 |
76 |
41 |
41 |
49 |
46 |
35 |
72 |
86 |
77 |
In Table 5, the median indicates that the
acceptable optimal range of κ extends from 0.35 to 0.6. The best value compared qualitatively amongst
those tested is 0.6.
Hence the optimal set of parameter values
that give good boundary mapping for the given class of chromosome images,
subjected to the preprocessing treatment outlined in previous paragraphs is σ
= 0.25, µ = 0.01, α = 0.3, β = 0, and κ = 0.6
A safe limit of 5% tolerance can be
introduced to the optimal range of parameter values to make them suitable for
use in similar classes of chromosome spread images.
Table 6. Optimal range of GVF Active Contour
parameter values for chromosome spread images
|
Parameter |
Chosen Parameter Value for tested spread
image |
Acceptable |
|
|
GVF Sigma |
0.25 |
[0.125 , 0.5] |
[0.1187 , 0.5250] |
|
GVF Mu |
0.01 |
[0.005 , 0.15] |
[0.0047 , 0.1575] |
|
GVF Alpha |
0.3 |
[0.2 , 0.5] |
[0.1900 , 0.5250] |
|
GVF Beta |
0 |
[0 , 0.2] |
[0 , 0.2100] |
|
GVF Kappa |
0.6 |
[0.35 , 0.6] |
[0.3325 , 0.6300] |
This can be extended to other similar
classes of images by performing a Characterization study to find optimal
parameters, and the same parameters can be used for successful boundary mapping
of images in the same class of images.
5.3 STATISTICAL VALIDATION
The parameters act independently on the
boundary mapping scheme. In each
characterization, the effect of other parameters will also be felt as they
assume a definite constant value. In the
course of the characterization study from Table 1 to Table 5, optimum values
for the respective parameters are chosen and applied as constant in the
characterization study of the next parameter in the successive table. In the last characterization study shown in
Table 5, the values of σ, µ, α and β take on the chosen optimal
values and only κ is investigated, thereby yielding a one way
variation. Hence, one way analysis of
variance on Table 5 is sufficient to test the significance of the entire
boundary mapping process. A significant
outcome from Table 5 will justify that the experimental results of Table 5 are
valid, implying that the selected parameter values from Table 1 to Table 4 used
as constants in Table 5 are also valid.
Hence one way anova
test is performed on the last characterization (Table 5) to judge the
experimental results. At the customary
.05 significance level, one way anova test yields a p
value of 1.00027E-009 on Table 5, which rejects the null hypothesis. The very small p-value of 1.00027E-009
indicates that differences between the column means are highly significant. The
probability of this outcome under the null hypothesis is less than 1 in
1,000,000,000. The test therefore strongly supports the alternate hypothesis
that one or more of the samples are drawn from populations with different means. This implies that the results in Table 5 do
not arise out of mere fluctuations and the results are actually significant. Therefore the experimental results are valid. This justifies that a suitable value of
parameter κ can be chosen from Table 5, and that the constant values of
parameters σ, µ, α, and β used in Table 5 are also valid as
these values also have significant influence on the results tabulated in Table
5. Therefore, the experimental results
and the inferences that are discussed in the previous paragraphs are also
significant.
5.4
VALIDATION OF ROBUSTNESS OF THE SCHEME
The following difficulties were observed during the implementation
of the boundary mapping scheme.
The banding pattern present in the chromosomes gives rise to
higher contrast compared to the outer edges. This characteristic causes the GVF
external field to have a higher strength at the bands. Therefore, the GVF Active Contour feels more
attraction towards the bands than the outer boundary. Hence, the contour tends to cross the
boundary into the inner regions seeking the bands.
The chromosome images in the chromosome spread image have
variability in shape and size due to the nature of the spread image. Also, the spatial distribution of the
chromosomes is random accompanied by uneven spacing between adjacent
chromosomes. Hence, each chromosome in a
chromosome spread image becomes a unique sample demanding unique values of the
parameters governing the GVF Active Contour.
There is also a need for unique number of iterations to converge.
The small object size of the chromosomes makes the computed GVF
field also to be small. Hence suitable
choice of parameters is necessary; else the Active Contour crosses the boundary
and results in a straight line at the axis of the chromosome sample.
The chromosomes in the spread image (at 72 pixels per inch
resolution) have a minor axis length varying between 14 and 17 pixels
approximately and major axis length varying between 30 and 80 pixels approximately. This causes the GVF external field to have a
high density at corners. Accompanied
with the banding characteristic, the axis lengths force the GVF Active Contour
to map contours at the inner region of the chromosome instead of the actual
boundary at the periphery of the chromosome.
The weak edges in chromosomes also contribute to the Active
Contour to overwhelm weak edges and move into inner regions.
In addition to these inherent difficulties, more difficulty was
introduced to validate the robustness of the boundary mapping scheme. The image was further degraded by
transforming pixels having gray levels greater than 90% intensity in the range
[0, 255]. This resulted in degradation
of weak edges, giving rise to distorted edges and uneven boundary in the
original image, offering more challenges to the task of segmentation using GVF Active
Contours.
These difficulties make the task of boundary mapping of
chromosomes in chromosome spread images very difficult. In spite of these difficulties, the GVF
Active Contours were employed to map the boundaries of chromosomes in
chromosome spread images by careful selection of the GVF Active Contour
parameters. The successful results
support the fact that the employed scheme is very robust even in the midst of
so many difficulties and can be used to boundary map chromosomes in similar
classes of chromosome spread images. In
other classes of chromosome spread images, a characterization need to be done,
and then boundary mapping can be successfully done with the optimal parameter
values obtained from the characterization process.
5.5
CONCLUSION
Chromosome
spread images assume characteristics similar to each other, except for the
exact shape of chromosomes that are variable under imaging conditions. The external force that guides the contour to
convergence is calculated from the image, and hence the external force assumes
a value that is characteristic of the image.
Other parameters of the GVF Active Contour are influenced only by the
size of the chromosomes, which will be similar in a class of images that are
imaged under similar conditions.
The GVF Active Contours are well suited to
the task of boundary mapping in chromosome images with the same optimal value
of parameters for a class of images.
This proves that the GVF Active Contours which were earlier used for
boundary mapping based on unique parameter values for every image can also be
used for boundary mapping a given class of images based on optimal parameter
values for all the images in the given class.
This can be extended to other classes of chromosome spread images. From the validations, it is inferred that the
boundary mapping scheme will be successful in other classes of chromosome
spread images also.
6. Acknowledgement
The authors wish to thank Prof. Ken
Castleman and Prof. Qiang Wu, both from Advanced
Digital Imaging Research, Texas for their help in providing chromosome
images.
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