Academic Open Internet Journal ISSN 1311-4360

www.acadjournal.com

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 v_i = (x_i,y_i)^T with v=(v₁,…,v_n).  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 E_ext 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, G_σ(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)   

whereand  comprise the components of a force balance equation such that

  --- (5)

The internal force F_int discourages stretching and bending while the external potential force F_ext 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 x_t(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, u_t 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, w_t 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.

Fig.7 Sample1                       Fig.8 Sample2                      Fig.9 Sample3                          Fig.10 Sample4

 Fig.11 Sample5                     Fig.12 Sample6                    Fig.13 Sample7                        Fig.14 Sample8

  Fig.15 Sample9                     Fig.16 Sample10                  Fig.17 Sample11                    Fig.18 Sample12

Fig.19 Sample13                   Fig.20 Sample14                      Fig.21 Sample15                       Fig.22 Sample16

  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