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 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)   

whereand  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