| Academic Open Internet Journal ISSN 1311-4360 |
Volume 21, 2007 |
SEGMENTATION OF CHROMOSOME SPREAD IMAGES
USING TRANSFORM BASED CURVE EVOLUTION METHODS
- STANDARDIZATION AND QUANTIFICATION
A.Prabhu Britto
Center for Medical Electronics, Department of Electronics and Communication Engineering, Anna University, Chennai 600 025 INDIA
Dr. G.Ravindran
Faculty of Information and Communication Engineering, Anna University, Chennai 600 025 INDIA
Abstract:
In this research, Characterization of Discrete Cosine Transform (DCT) based Gradient Vector Flow (GVF) Active Contours as a suitable boundary mapping technique for Chromosome spread images is done. Suitability of the characterized parameters governing the DCT based GVF Active Contour formulation as standardized values for boundary mapping similar classes of Chromosome spread images is also established. Characterization studies have shown that each parameter has an optimal range of values within which good boundary mapping results can be obtained for various chromosomes in similar class of images. Statistical testing validates the experimental results of characterization. Standardization of characterized parameters is then carried out using a different dataset comprising of similar class of chromosome spread images. Error Quantification is done to validate Standardization.
Keywords: Gradient Vector Flow, Active Contours, Chromosome, Boundary Mapping, Characterization, Standardization
1. Introduction
The classical boundary mapping techniques, namely, region growing, relaxation labeling, edge detection and linking suffer from limitations. Usage of only local information may lead to incorrect assumptions during the boundary integration process leading to errors. Noise and artifacts can possibly cause incorrect segmentation or boundary discontinuities in segmented objects [9]. Therefore, this research work used Discrete Cosine Transform (DCT) based Gradient Vector Flow (GVF) Active Contours to obtain accurate segmentation (boundary mapping) results from a class of chromosome spread images having variability in shape, size and other image properties. The Characterization of the boundary mapping technique is expected to yield a set of parameter values that can be applied to obtain good boundary mapping in similar class of chromosome spread images. Further to characterization, Standardization of the technique is attempted by applying the same set of parameter values yielded by characterization study.
Active Contour or Deformable Curve is a high-level boundary mapping technique, with the main advantage of having the ability to generate closed parametric curves from images. The incorporation of a smoothness constraint provides robustness to noise and spurious edges. The focus is on parametric deformable curves, which provide a compact, analytical description of object shape. A class of parametric Active Contours called Gradient Vector Flow (GVF) field Active Contours is chosen for boundary mapping in chromosome spread images.
2. Active Contour Models
Active Contours also called as Snakes or Deformable Curves, first proposed by Kass [7] are energy minimizing contours that apply information about the boundaries as part of an optimization procedure. They are generally initialized by automatic or manual process around the object of interest. The contour then deforms itself from its initial position iteratively, in conformity with nearest dominant edge feature, by minimizing the energy composed of the Internal and External forces, converging at the boundary of the object of interest. The Internal forces computed from within the Active Contour enforce smoothness of the curve and External forces that are derived from the image help to drive the curve toward the desired features of interest during the course of the iterative process.
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 [11], and the model evolves until equilibrium of all forces is reached, which is equivalent to a minimum of the energy function. The energy function is thus minimized, making the model active.
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]. To define a closed curve, c(0) is set to equal c(1). A discrete model can be expressed as an ordered set of n vertices as vi = (xi,yi)T with v=(v1,…,vn). The large number of vertices required to achieve any predetermined accuracy could lead to high computational complexity and numerical instability11.
Mathematically, an active contour model
can be defined in discrete form as a curve
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 [12]. The first order derivative discourages stretching while the second order derivative discourages bending. The weighting parameters of tension and rigidity govern the effect of the derivatives on the snake.
The external energy function Eext is derived from the image so that it takes on 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, 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. 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 [7]. 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 drawbacks. Boundary concavities leave the contour split across the boundary. Capture range is also limited. Methods suggested to overcome these difficulties, namely multiresolution methods [8], pressure forces [2], distance potentials [3], control points [4], domain adaptivity [5], directional attractions [1] and solenoidal fields [10], however solved one problem but introduced new ones [13]. Hence, a new class of external fields called Gradient Vector Flow fields [13,14] was suggested to overcome the difficulties in traditional active contour models.
4. Gradient Vector Flow (GVF) Active Contours
Gradient Vector Flow (GVF) Active Contours use Gradient Vector Flow fields obtained by solving a vector diffusion equation that diffuses the gradient vectors of a gray-level edge map computed from the image. 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 [13,14].
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. The GVF active contour model thus has a large capture range and is insensitive to the initialization of the contour. Hence the contour initialization is flexible.
The gradient vectors are normal to the boundary surface but by combining Laplacian and 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. 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 [7,14].
The GVF field is defined as the equilibrium solution to the following vector diffusion equation[6],
--- (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.(7),
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 chosen14 as
and
.
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(x,y)=[u(x,y),v(x,y)] 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 [14].
5. Discrete Cosine Transform (DCT) based GVF Active Contours
Transform theory plays a fundamental role in image processing. The transform of an Image yields more insight into the properties of the image. The Discrete Cosine Transform has excellent energy compaction. Hence, the Discrete Cosine Transform promises better description of the image properties. The Discrete Cosine Transform is embedded into the GVF Active Contours. When the image property description is significantly low, this helps the contour model to give significantly better performance by utilizing the energy compaction property of the DCT.
The 2D DCT is defined as
--- (11)
The local contrast of the Image at the given pixel location (k,l) is given by
--- (12)
where,
--- (13) and
--- (14)
Here, wt denotes the weights used to select the DCT coefficients. The local contrast P(k,l) is then used to generate a DCT contrast enhanced Image [6], which is then subject to selective segmentation by the energy compact gradient vector flow active contour model using Eq.(8).
6. Results and Discussion
The chromosome metaphase image (at 72 pixels per inch resolution) provided by Prof.Ken Castleman and Prof.Qiang Wu (Advanced Digital Imaging Research, Texas) was taken and preprocessed. Insignificant and unnecessary regions in the image were removed interactively. Interactive selection of the chromosome of interest was done by selecting a few points around the chromosome that formed 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 image was made to undergo minimal preprocessing so as to achieve the goal of boundary mapping in chromosome images with very weak edges. The DCT based 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 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 implying 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 iterations were set suitably.
6.1 Characterization Results
DCT based GVF Active Contours were used to boundary chromosome images from chromosome spread images. A few samples are graphically presented here.
![]() |
Fig.1a Sample 1 Fig.2a Sample 2 Fig.3a Sample 3 Fig. 4a Sample 4 Fig. 5a Sample 5 Fig. 6a Sample 6
![]() |
Fig.1b Vector Field Fig.2b Vector Field Fig.3b Vector Field Fig.4b Vector Field Fig.5b Vector Field Fig.6b Vector Field
![]() |
Fig.1c Output Image Fig.2c Output Image Fig.3c Output Image Fig.4c Output Image Fig.5c Output Image Fig.6c Output Image
The figures show original chromosome image samples, their corresponding DCT based GVF fields and boundary mapped chromosome images as output images. For example, Fig.1a shows an original chromosome image sample, Fig.1b shows its corresponding Vector Field and Fig.1c shows its boundary mapped output image, and henceforth.
The graphical outputs show successful boundary mapping of chromosome images using DCT based GVF Active Contours.
6.2 Validation of characterization experiments
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. 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 points11. The tables represent characterization studies for each parameter.
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. Hence, the best parameter value of that table is the one that gives maximum good quality outputs for all samples or a majority of samples, and exhaustive study on every parameter is done by treating the other parameters as constants.
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”, making that particular parameter value an optimal one and that optimal value is chosen. 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 the investigated discrete value of each parameter that gave best output was chosen. 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.
Table.1 Characterization of Sigma
|
Sample No. |
GVF (DCT) σ |
|||||||||
|
|
0.05 |
0.1 |
0.15 |
0.2 |
0.25 |
0.5 |
0.6 |
0.8 |
1 |
1.2 |
|
1 |
77 |
77 |
77 |
77 |
77 |
29 |
77 |
29 |
13 |
77 |
|
2 |
77 |
77 |
77 |
29 |
13 |
13 |
13 |
13 |
29 |
77 |
|
3 |
97 |
77 |
34 |
29 |
77 |
29 |
78 |
81 |
75 |
78 |
|
4 |
77 |
77 |
29 |
29 |
31 |
70 |
79 |
79 |
79 |
78 |
|
5 |
97 |
97 |
97 |
97 |
98 |
98 |
98 |
98 |
98 |
98 |
|
6 |
86 |
86 |
46 |
38 |
38 |
14 |
38 |
38 |
46 |
78 |
|
7 |
97 |
97 |
97 |
97 |
98 |
98 |
98 |
98 |
98 |
98 |
|
8 |
86 |
86 |
86 |
54 |
98 |
98 |
98 |
98 |
98 |
98 |
|
9 |
77 |
77 |
77 |
77 |
38 |
46 |
15 |
77 |
13 |
79 |
|
10 |
86 |
77 |
13 |
77 |
46 |
65 |
78 |
13 |
78 |
77 |
|
Median |
86 |
77 |
77 |
66 |
62 |
55 |
78 |
78 |
77 |
78 |
In Table 1, the median indicates that the acceptable optimal range of σ is 0.2 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
|
Sample No. |
GVF (DCT) μ |
|||||
|
|
0.05 |
0.075 |
0.09375 |
0.1125 |
0.15 |
0.3 |
|
1 |
23 |
21 |
21 |
23 |
23 |
97 |
|
2 |
21 |
5 |
23 |
23 |
23 |
97 |
|
3 |
30 |
29 |
29 |
46 |
50 |
97 |
|
4 |
23 |
23 |
23 |
40 |
23 |
97 |
|
5 |
98 |
98 |
98 |
97 |
97 |
97 |
|
6 |
48 |
40 |
48 |
48 |
46 |
97 |
|
7 |
98 |
98 |
50 |
50 |
34 |
97 |
|
8 |
98 |
89 |
62 |
97 |
97 |
97 |
|
9 |
71 |
86 |
30 |
71 |
71 |
97 |
|
10 |
23 |
21 |
29 |
71 |
23 |
97 |
|
Median |
39 |
35 |
29 |
49 |
40 |
97 |
In Table 2, the median indicates that the acceptable optimal range of µ is 0.05 to 0.09375. The best value compared qualitatively amongst those tested is 0.075 and hence it is chosen for performing further characterization.
Table 3. Characterization of Alpha
|
Sample No. |
GVF (DCT) α |
||||
|
|
0 |
0.125 |
0.25 |
0.5 |
1 |
|
1 |
7 |
23 |
77 |
71 |
77 |
|
2 |
7 |
30 |
29 |
77 |
30 |
|
3 |
5 |
67 |
78 |
78 |
67 |
|
4 |
23 |
23 |
79 |
80 |
80 |
|
5 |
98 |
98 |
98 |
98 |
97 |
|
6 |
98 |
48 |
40 |
46 |
87 |
|
7 |
98 |
98 |
98 |
97 |
97 |
|
8 |
90 |
86 |
62 |
97 |
94 |
|
9 |
21 |
23 |
23 |
71 |
27 |
|
10 |
5 |
7 |
23 |
21 |
71 |
|
Median |
22 |
39 |
70 |
78 |
79 |
In Table 3, the median indicates that the acceptable optimal range of α extends from 0 to 0.125. The best value compared qualitatively amongst those tested is 0 and hence it is chosen for performing further characterization.
Table 4. Characterization of Beta
|
Sample No. |
GVF (DCT) β |
||
|
|
0 |
0.5 |
1 |
|
1 |
23 |
30 |
71 |
|
2 |
5 |
21 |
21 |
|
3 |
5 |
21 |
31 |
|
4 |
21 |
23 |
71 |
|
5 |
98 |
98 |
98 |
|
6 |
98 |
46 |
70 |
|
7 |
98 |
98 |
98 |
|
8 |
38 |
94 |
13 |
|
9 |
23 |
71 |
71 |
|
10 |
3 |
21 |
30 |
|
Median |
23 |
38 |
71 |
In Table 4, the median indicates that the acceptable optimal range of β extends from 0 to 0.5. The best value compared qualitatively amongst those tested is 0 and hence it is chosen for performing further characterization.
Table 5. Characterization of Kappa
|
Sample No. |
GVF (DCT) κ |
|||||
|
|
0 |
0.5 |
0.625 |
0.75 |
0.875 |
1 |
|
1 |
97 |
7 |
5 |
5 |
5 |
5 |
|
2 |
97 |
3 |
3 |
3 |
1 |
1 |
|
3 |
97 |
21 |
19 |
21 |
30 |
67 |
|
4 |
97 |
7 |
7 |
7 |
23 |
71 |
|
5 |
97 |
98 |
98 |
98 |
98 |
98 |
|
6 |
97 |
98 |
98 |
98 |
86 |
98 |
|
7 |
97 |
98 |
98 |
98 |
98 |
98 |
|
8 |
97 |
86 |
98 |
97 |
98 |
82 |
|
9 |
97 |
7 |
7 |
23 |
23 |
21 |
|
10 |
97 |
21 |
5 |
19 |
19 |
21 |
|
Median |
97 |
21 |
13 |
22 |
26 |
69 |
In Table 5, the median indicates that the acceptable optimal range of κ extends from 0.5 to 0.875. The best value compared qualitatively amongst those tested is 0.625.
Hence the optimal set of parameter values that give good boundary mapping for the given class of chromosome images is σ = 0.25, µ = 0.075, α = 0, β = 0, and κ = 0.625. 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 (indicated in Table 6).
Table 6. Optimal range of DCT based GVF Active Contour parameter values for tested chromosome spread images
|
Parameter |
Parameter Value used for tested spread image |
Acceptable Range of Parameter values |
Acceptable Range of Values at 5% tolerance |
|
GVF (DCT) σ |
0.25 |
[0.2 , 0.5] |
[0.1900 , 0.5250] |
|
GVF (DCT) μ |
0.075 |
[0.05 , 0.09375] |
[0.0475 , 0.0984] |
|
GVF (DCT) α |
0 |
[0 , 0.125] |
[0.0000 , 0.1313] |
|
GVF (DCT) β |
0 |
[0 , 0.5] |
[0.0000 , 0.5250] |
|
GVF (DCT) κ |
0.625 |
[0.5 , 0.875] |
[0.4750 , 0.9187] |
In Table 6, acceptable range of parameter values are tabulated at 5% tolerance levels.
6.3 Statistical validation of characterization experiments
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 7.17082E-08 on Table 5, which rejects the null hypothesis. The very small p-value of 7.17082E-08 indicates that differences between the column means are highly significant. The probability of this outcome under the null hypothesis is less than 8 in 100,000,000. 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 that 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 are also significant.
7. Standardization
Characterization studies have yielded an acceptable optimal range of values for the parameters σ,μ,α,β and κ. To establish that the parameter values are standardized with reference to similar classes of chromosome spread images, standardization experiments are carried out on images from a different dataset, made available by the kind courtesy of Dr.Michael Difilippantonio, Staff Scientist at the Section of Cancer Genomics, Genetics Branch/CCR/NCI/NIH, Bethesda MD.
The same characterized parameter values of σ = 0.25, µ = 0.075, α = 0, β = 0, and κ = 0.625 have been used. Good boundary mapping results have been obtained and the results are shown in the following pages. Each sample is unique as the chromosomes are imaged in a fluid medium, and random bending effects are manifested. Hence it is shown that the DCT based GVF Active Contour, governed by the characterized values of the parameters of σ = 0.25, µ = 0.075, α = 0, β = 0, and κ = 0.625 are able to overcome the variations in the shape of the chromosomes and give good boundary mapping in each of the samples.
A few samples are graphically illustrated in the following pages. The chromosome image is seen in gray scale, while the DCT based GVF Active Contour mapped boundary is shown in red.
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Fig.7 Sample1 Fig.8 Sample2 Fig.9 Sample3 Fig.10 Sample4
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Fig.11 Sample5 Fig.12 Sample6 Fig.13 Sample7 Fig.14 Sample8
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Fig.15 Sample9 Fig.16 Sample10 Fig.17 Sample11 Fig.18 Sample12
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Fig.19 Sample13 Fig.20 Sample14 Fig.21 Sample15 Fig.22 Sample16
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Fig.23 Sample17 Fig.24 Sample18 Fig.25 Sample19 Fig.26 Sample20
From the above illustrations of boundary mapped chromosomes, it is inferred that the set of parameter values σ = 0.25, µ = 0.075, α = 0, β = 0, and κ = 0.625 governing the formulation of the DCT based GVF Active Contours are hence standardized, and can be applied to obtain successful boundary mapping in similar classes of chromosome spread images. Though standardization has been done exhaustively on a large number of samples, graphical illustration is restricted to 20 samples.
8. Error Quantification
Error Quantification is done to assess the experimental results. The error in Boundary Mapping is measured as a difference between axial radius of the original chromosome image sample and the axial radius of the boundary mapped chromosome image sample.
The error in boundary mapping for the samples used for characterization is shown in Table 7.
Table 7. Error in Boundary Mapping for sample images used for characterization
|
Sample No. |
Original Image Major Axis Radius (pixels) |
Contour Image Major Axis Radius (pixels) |
Major Axis Error (Original - Contour) (pixels) |
Major Axis Absolute Error abs(Original - Contour) (pixels) |
Original Image Minor Axis Radius (pixels) |
Contour Image Minor Axis Radius (pixels) |
Minor Axis Error (Original - Contour) (pixels) |
Minor Axis Absolute Error abs(Original - Contour) (pixels) |
|
1 |
19.836534 |
20.614610 |
-0.778076 |
0.778076 |
7.654679 |
8.154604 |
-0.499926 |
0.499926 |
|
2 |
15.531852 |
15.888311 |
-0.356459 |
0.356459 |
7.348310 |
7.831433 |
-0.483123 |
0.483123 |
|
3 |
19.450454 |
20.106271 |
-0.655817 |
0.655817 |
7.748832 |
8.179759 |
-0.430927 |
0.430927 |
|
4 |
35.059325 |
35.516327 |
-0.457002 |
0.457002 |
8.620817 |
9.395544 |
-0.774727 |
0.774727 |
|
5 |
28.890790 |
29.151164 |
-0.260374 |
0.260374 |
7.735980 |
8.187389 |
-0.451409 |
0.451409 |
|
6 |
18.082140 |
18.692496 |
-0.610356 |
0.610356 |
7.470833 |
7.792613 |
-0.321780 |
0.321780 |
|
7 |
26.226850 |
26.181850 |
0.045000 |
0.045000 |
6.625715 |
7.047331 |
-0.421616 |
0.421616 |
|
8 |
26.033162 |
26.489206 |
-0.456044 |
0.456044 |
7.697243 |
8.131959 |
-0.434716 |
0.434716 |
|
9 |
20.525006 |
21.179205 |
-0.654199 |
0.654199 |
8.725828 |
9.510494 |
-0.784667 |
0.784667 |
|
10 |
27.247951 |
27.158149 |
0.089802 |
0.089802 |
6.839096 |
7.067086 |
-0.227990 |
0.227990 |
|
11 |
15.903326 |
15.845969 |
0.057357 |
0.057357 |
7.782958 |
8.142976 |
-0.360018 |
0.360018 |
|
12 |
27.435787 |
27.275522 |
0.160265 |
0.160265 |
8.315512 |
8.523697 |
-0.208185 |
0.208185 |
|
|
|
|
|
|
|
|
|
|
|
Max Absolute Error |
|
|
|
0.778076 |
|
|
|
0.784667 |
|
Min Absolute Error |
|
|
|
0.045000 |
|
|
|
0.208185 |
The maximum absolute error from Table7 is 0.784667 pixel and the minimum absolute error is 0.045000 pixel. The contour is one pixel thick. The initial contour converges to the boundary in step sizes of one pixel. Considering the above two facts, it is quite logical that there could be an error approximating one pixel. Since the errors given by Table7 are less than one pixel, the boundary mapping obtained can be accepted as accurate. This justifies that the parameter values σ = 0.25, µ = 0.075, α = 0, β = 0, and κ = 0.625 that have been used for boundary mapping using DCT based GVF Active Contours can be accepted as characterized values for similar classes of chromosome spread images.
The error in boundary mapping for the samples used for standardization is shown in Table 8.
Table 8. Error in Boundary Mapping for sample images used for standardization
|
Sample No. |
Original Image Major Axis Radius (pixels) |
Contour Image Major Axis Radius (pixels) |
Major Axis Error (Original - Contour) (pixels) |
Major Axis Absolute Error (Original - Contour) (pixels) |
Original Image Minor Axis Radius (pixels) |
Contour Image Minor Axis Radius (pixels) |
Minor Axis Error (Original - Contour) (pixels) |
Minor Axis Absolute Error (Original - Contour) (pixels) |
|
1 |
15.671966 |
16.664304 |
-0.992338 |
0.992338 |
8.632178 |
9.408042 |
-0.775864 |
0.775864 |
|
2 |
27.532960 |
28.455320 |
-0.922360 |
0.922360 |
9.363175 |
10.831798 |
-1.468623 |
1.468623 |
|
3 |
21.352938 |
22.573477 |
-1.220539 |
1.220539 |
9.245948 |
10.668792 |
-1.422845 |
1.422845 |
|
4 |
22.487728 |
22.988902 |
-0.501174 |
0.501174 |
9.487592 |
11.011601 |
-1.524009 |
1.524009 |
|
5 |
30.832858 |
31.631947 |
-0.799089 |
0.799089 |
9.916949 |
11.222596 |
-1.305647 |
1.305647 |
|
6 |
26.527726 |
27.749187 |
-1.221461 |
1.221461 |
10.348389 |
11.772646 |
-1.424257 |
1.424257 |
|
7 |
20.421685 |
22.022552 |
-1.600867 |
1.600867 |
9.300520 |
10.527103 |
-1.226584 |
1.226584 |
|
8 |
28.932339 |
30.032657 |
-1.100319 |
1.100319 |
9.705534 |
11.185087 |
-1.479554 |
1.479554 |
|
9 |
17.361945 |
18.709701 |
-1.347756 |
1.347756 |
8.734208 |
10.217936 |
-1.483728 |
1.483728 |
|
10 |
14.592386 |
15.831201 |
-1.238815 |
1.238815 |
7.461672 |
8.903638 |
-1.441966 |
1.441966 |
|
11 |
21.530209 |
22.035274 |
-0.505066 |
0.505066 |
8.816231 |
10.256872 |
-1.440641 |
1.440641 |
|
12 |
18.516333 |
19.480915 |
-0.964582 |
0.964582 |
7.394475 |
8.839323 |
-1.444848 |
1.444848 |
|
13 |
21.980379 |
22.722356 |
-0.741977 |
0.741977 |
8.888960 |
10.173608 |
-1.284649 |
1.284649 |
|
14 |
19.394772 |
20.504988 |
-1.110216 |
1.110216 |
8.444747 |
9.892300 |
-1.447553 |
1.447553 |
|
15 |
13.283334 |
14.830840 |
-1.547507 |
1.547507 |
7.783275 |
8.997038 |
-1.213763 |
1.213763 |
|
16 |
15.758397 |
16.832960 |
-1.074563 |
1.074563 |
8.689258 |
9.750896 |
-1.061639 |
1.061639 |
|
17 |
20.383368 |
21.569564 |
-1.186196 |
1.186196 |
8.731019 |
10.135830 |
-1.404811 |
1.404811 |
|
18 |
29.016552 |
29.744011 |
-0.727460 |
0.727460 |
8.824018 |
10.088889 |
-1.264871 |
1.264871 |
|
19 |
20.328609 |
21.203247 |
-0.874638 |
0.874638 |
8.570169 |
9.562535 |
-0.992366 |
0.992366 |
|
20 |
38.396106 |
39.384758 |
-0.988652 |
0.988652 |
9.930610 |
11.533310 |
-1.602701 |
1.602701 |
|
21 |
20.593760 |
21.963540 |
-1.369780 |
1.369780 |
8.777635 |
10.172417 |
-1.394782 |
1.394782 |
|
22 |
28.014880 |
29.189257 |
-1.174377 |
1.174377 |
9.106209 |
10.659509 |
-1.553300 |
1.553300 |
|
23 |
35.057991 |
36.212418 |
-1.154428 |
1.154428 |
9.554718 |
11.220442 |
-1.665724 |
1.665724 |
|
24 |
17.289731 |
18.370185 |
-1.080455 |
1.080455 |
8.024122 |
9.004566 |
-0.980445 |
0.980445 |
|
25 |
22.021634 |
22.461027 |
-0.439393 |
0.439393 |
8.142623 |
9.489822 |
-1.347199 |
1.347199 |
|
26 |
29.784084 |
30.682099 |
-0.898015 |
0.898015 |
9.548830 |
10.755476 |
-1.206646 |
1.206646 |
|
27 |
33.294617 |
34.134478 |
-0.839861 |
0.839861 |
9.542091 |
11.054903 |
-1.512812 |
1.512812 |
|
28 |
24.728291 |
25.920548 |
-1.192257 |
1.192257 |
9.318735 |
10.594356 |
-1.275622 |
1.275622 |
|
29 |
15.355989 |
16.272398 |
-0.916409 |
0.916409 |
7.927053 |
9.153714 |
-1.226661 |
1.226661 |
|
30 |
15.797688 |
16.992960 |
-1.195272 |
1.195272 |
8.164426 |
9.377414 |
-1.212988 |
1.212988 |
|
31 |
19.119752 |
20.198432 |
-1.078680 |
1.078680 |
8.006951 |
9.386859 |
-1.379909 |
1.379909 |
|
32 |
50.529747 |
51.836646 |
-1.306899 |
1.306899 |
9.569420 |
11.155712 |
-1.586292 |
1.586292 |
|
33 |
28.677206 |
29.719430 |
-1.042224 |
1.042224 |
9.347141 |
10.847365 |
-1.500224 |
1.500224 |
|
34 |
18.489779 |
19.566706 |
-1.076928 |
1.076928 |
9.020151 |
10.322819 |
-1.302668 |
1.302668 |
|
35 |
15.369255 |
16.562642 |
-1.193387 |
1.193387 |
8.009787 |
9.352062 |
-1.342275 |
1.342275 |
|
36 |
22.002596 |
22.199599 |
-0.197003 |
0.197003 |
8.269518 |
9.260998 |
-0.991480 |
0.991480 |
|
37 |
22.856044 |
23.861794 |
-1.005750 |
1.005750 |
8.909892 |
10.293682 |
-1.383790 |
1.383790 |
|
38 |
18.296507 |
19.566748 |
-1.270241 |
1.270241 |
9.445701 |
10.707874 |
-1.262173 |
1.262173 |
|
39 |
27.960635 |
29.151157 |
-1.190522 |
1.190522 |
8.545565 |
10.136476 |
-1.590911 |
1.590911 |
|
40 |
18.852903 |
20.423583 |
-1.570681 |
1.570681 |
8.455191 |
9.796682 |
-1.341491 |
1.341491 |
|
41 |
22.698254 |
22.763508 |
-0.065254 |
0.065254 |
8.464508 |
9.752790 |
-1.288282 |
1.288282 |
|
42 |
12.293282 |
13.703855 |
-1.410574 |
1.410574 |
7.320788 |
8.823351 |
-1.502564 |
1.502564 |
|
43 |
11.489264 |
12.447829 |
-0.958566 |
0.958566 |
7.235781 |
8.566689 |
-1.330908 |
1.330908 |
|
44 |
18.432328 |
19.577044 |
-1.144716 |
1.144716 |
7.436791 |
9.010851 |
-1.574060 |
1.574060 |
|
45 |
12.968119 |
14.117667 |
-1.149548 |
1.149548 |
7.803107 |
8.611738 |
-0.808631 |
0.808631 |
|
46 |
14.282209 |
15.730363 |
-1.448154 |
1.448154 |
7.647730 |
8.570545 |
-0.922816 |
0.922816 |
|
47 |
19.454318 |
20.187653 |
-0.733335 |
0.733335 |
7.421526 |
8.633051 |
-1.211525 |
1.211525 |
|
48 |
20.625064 |
21.377546 |
-0.752482 |
0.752482 |
8.360867 |
9.469176 |
-1.108310 |
1.108310 |
|
49 |
24.934309 |
25.765914 |
-0.831605 |
0.831605 |
9.539810 |
10.676653 |
-1.136843 |
1.136843 |
|
50 |
20.374887 |
21.820383 |
-1.445496 |
1.445496 |
8.790606 |
9.990901 |
-1.200295 |
1.200295 |
|
51 |
19.383527 |
19.820843 |
-0.437316 |
0.437316 |
8.478031 |
9.419203 |
-0.941172 |
0.941172 |
|
52 |
19.327516 |
20.508350 |
-1.180835 |
1.180835 |
8.350576 |
9.233636 |
-0.883060 |
0.883060 |
|
53 |
26.433113 |
27.533623 |
-1.100511 |
1.100511 |
8.610975 |
9.593304 |
-0.982329 |
0.982329 |
|
54 |
12.101642 |
12.949808 |
-0.848166 |
0.848166 |
7.246340 |
8.059449 |
-0.813109 |
0.813109 |
|
55 |
16.634324 |
17.486275 |
-0.851951 |
0.851951 |
8.394216 |
9.252824 |
-0.858608 |
0.858608 |
|
56 |
14.749573 |
16.175581 |
-1.426008 |
1.426008 |
6.833781 |
8.201446 |
-1.367665 |
1.367665 |
|
57 |
26.644550 |
27.732467 |
-1.087917 |
1.087917 |
9.192708 |
10.450890 |
-1.258182 |
1.258182 |
|
58 |
46.708148 |
47.716019 |
-1.007871 |
1.007871 |
9.854564 |
11.425238 |
-1.570674 |
1.570674 |
|
59 |
11.804992 |
12.946824 |
-1.141832 |
1.141832 |
6.838070 |
8.085564 |
-1.247494 |
1.247494 |
|
60 |
14.789925 |
15.845047 |
-1.055122 |
1.055122 |
8.045290 |
9.278818 |
-1.233528 |
1.233528 |
|
61 |
16.053096 |
17.098621 |
-1.045525 |
1.045525 |
8.571831 |
10.044725 |
-1.472894 |
1.472894 |
|
62 |
16.381754 |
17.627738 |
-1.245984 |
1.245984 |
8.109770 |
9.287800 |
-1.178031 |
1.178031 |
|
63 |
24.880002 |
26.125909 |
-1.245907 |
1.245907 |
8.112620 |
9.060594 |
-0.947974 |
0.947974 |
|
64 |
23.946641 |
25.039662 |
-1.093021 |
1.093021 |
12.670524 |
13.276545 |
-0.606021 |
0.606021 |
|
65 |
12.162489 |
13.637709 |
-1.475220 |
1.475220 |
8.477049 |
9.129125 |
-0.652076 |
0.652076 |
|
66 |
29.266172 |
30.914282 |
-1.648110 |
1.648110 |
10.406857 |
11.619620 |
-1.212763 |
1.212763 |
|
67 |
14.633115 |
15.986863 |
-1.353749 |
1.353749 |
7.746963 |
8.767422 |
-1.020460 |
1.020460 |
|
68 |
41.377986 |
42.913452 |
-1.535466 |
1.535466 |
8.388075 |
9.731649 |
-1.343574 |
1.343574 |
|
69 |
30.902650 |
31.722539 |
-0.819889 |
0.819889 |
7.790671 |
9.171754 |
-1.381083 |
1.381083 |
|
70 |
30.486127 |
32.010595 |
-1.524468 |
1.524468 |
7.433136 |
8.958509 |
-1.525374 |
1.525374 |
|
71 |
53.003739 |
54.121765 |
-1.118025 |
1.118025 |
9.009000 |
10.065934 |
-1.056934 |
1.056934 |
|
72 |
23.914936 |
25.087823 |
-1.172888 |
1.172888 |
8.439396 |
9.522967 |
-1.083572 |
1.083572 |
|
73 |
25.988345 |
27.436165 |
-1.447820 |
1.447820 |
7.605870 |
9.074429 |
-1.468560 |
1.468560 |
|
74 |
15.201740 |
16.600042 |
-1.398303 |
1.398303 |
10.073521 |
10.899285 |
-0.825764 |
0.825764 |
|
75 |
27.651519 |
28.806415 |
-1.154896 |
1.154896 |
9.832643 |
11.599894 |
-1.767251 |
1.767251 |
|
76 |
18.248415 |
19.923112 |
-1.674697 |
1.674697 |
8.730340 |
10.555582 |
-1.825242 |
1.825242 |
|
77 |
27.280692 |
28.092240 |
-0.811548 |
0.811548 |
10.746050 |
12.236822 |
-1.490772 |
1.490772 |
|
78 |
31.649805 |
32.705289 |
-1.055484 |
1.055484 |
12.474942 |
13.596004 |
-1.121062 |
1.121062 |
|
79 |
33.039166 |
34.504250 |
-1.465084 |
1.465084 |
10.738993 |
12.331706 |
-1.592714 |