Academic Open Internet Journal

www.acadjournal.com

Volume 16, 2005

 

 

Segmentation of Chromosome Spread Images using

Transform based Curve Evolution Methods

and Their Characterization

 

A.Prabhu Britto1 and Dr.G.Ravindran2

1Center for Medical Electronics, Dept. of ECE, Anna University, Chennai, INDIA  britto_albert@ieee.org

2Chairman, Faculty of Information and Communication Engineering, Anna University, Chennai INDIA

 

 

 

Abstract

Characterization of Discrete Cosine Transform (DCT) based Gradient Vector Flow (GVF) Active Contours as a suitable segmentation technique is done for Chromosome spread images having variability in shape and other image properties.  Evaluation of this technique in terms of robustness is done.  It is found experimentally that a unique set of parameter values of the technique is required for segmentation every chromosome image.  Characterization studies have shown that each parameter has an optimal range of values within which good segmentation results can be obtained for various chromosomes in similar class of images.  Statistical testing validates the experimental results.

 

Keywords: Gradient Vector Flow, Active Contours, Chromosome, Segmentation, Discrete Cosine Transform, Characterization

 

 

 

1. Introduction

 

The classical segmentation 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 leading to errors.  Other difficulties include imaging conditions which introduce further variability in image characteristics.  Noise and artifacts can possibly cause incorrect segmentation or boundary discontinuities in segmented objects [1].  Therefore, this work obtains accurate segmentation (boundary mapping) results using Discrete Cosine Transform (DCT) based Gradient Vector Flow (GVF) field Active Contours from a class of chromosome spread images having variability in shape, size and other image properties.  The characterization of the segmentation technique is expected to yield a set of parameter values that can be applied to obtain good segmentation results in similar class of chromosome spread images.

 

 

Active Contours are a high-level curve evolution technique that can be used for segmentation.  Its main advantage is the ability to generate closed parametric curves from images and incorporation of a smoothness constraint that provides robustness to noise and spurious edges.  The focus is on parametric deformable curves (active contours).  Parametric representations of Active Contour models provide a compact, analytical description of object shape.  A class of parametric Active Contours called Gradient Vector Flow (GVF) [2] field Active Contours is chosen for segmentation in chromosome spread images. 

 

 

 

2. Active Contour Models

 

Active Contours also called as Snakes or Deformable Curves, first proposed by Kass et al. [3] 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 iteratively from its initial position by minimizing the energy composed of the Internal and External forces in conformity with nearest dominant edge feature converging to the boundary of the object of interest.   The energy is composed of the Internal forces computed from within the Active Contour that enforce smoothness of the curve and External forces derived from the image that 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 [4], 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 minimized, thus 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 vi = (xi,yi)T with v=(v1,…,vn).  The large number of vertices required to achieve accuracy could lead to high computational complexity and numerical instability [4].  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 [2].  The first order derivative discourages stretching and the second order derivative discourages bending. The weighting parameters of tension and rigidity, viz., α and β govern the effect of the derivatives on the snake.

 

 

The external energy function Eext is derived from the image so that it takes on its smaller values at the features of interest such as boundaries and guides the active contour towards the boundaries.  The external energy is defined by

 -- (3)

where Gσ(x,y) is a two-dimensional Gaussian function with standard deviation σ, 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 [3].  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 [5], pressure forces [6], distance potentials [7], control points [8], domain adaptivity [9], directional attractions [10] and solenoidal fields[11], however solved one problem but introduced new ones[12].  Hence, a new class of external fields called Gradient Vector Flow fields [12, 13] was suggested to overcome the difficulties in traditional active contour models. 

 

 

 

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 [12, 13].

 

 

The GVF field points towards the object boundary when very near to the boundary, but varies smoothly over homogeneous image regions extending to the image border.  Hence the GVF field can capture an active contour from long range from either side of the object boundary and can force it into the object boundary.  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 [3, 13].

 

The GVF field is defined as the equilibrium solution to the following vector diffusion equation [12],  -- (7a)

 -- (7b)

where, ut denotes the partial derivative of u(x,t) with respect to t, is the Laplacian operator (applied to each spatial component of u separately), and f is an edge map that has a higher value at the desired object boundary. The functions in “g” and “h” control the amount of diffusion in GVF.  In Eq. (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 chosen as and  [7].   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 [13].

 

 

 

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 [14], 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 spread image (size 480 x 512 pixels at 72 pixels per inch resolution) provided by the kind courtesy of Prof. Ken Castleman and Prof. Qiang Wu of Advanced Digital Imaging Research Texas, was taken and preprocessed. Insignificant and unnecessary regions in the image were removed interactively.  The chromosome of interest was selected by user selection of a few points on the chromosome spread image that formed the vertices of a polygon.  On constructing the perimeter of the polygon from the selection points, seed points for the initial contour were determined 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 that the goal of segmentation in chromosome images with very weak edges is maintained.  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 Experimental Results

 

DCT based GVF Active Contours were used to boundary map chromosome images from chromosome spread images.  A few samples are 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 DCT based GVF Vector Field and Fig.1c shows its boundary mapped output image.

 

The graphical outputs show successful segmentation of chromosome images using DCT based GVF Active Contours.

 

 

 

6.2 Experimental Validation

 

In order to quantify the performance of a segmentation method, validation experiments are necessary.  Validation is typically performed using one or two different types of truth models.  In this work, ground truth model is not available and hence validation is performed on ordinal or ranking scale and then quantified.  A set of 10 random samples is taken and characterization of each parameter is done.  The outputs were tabulated in ranking order with “1” describing the best quality output and as the quality decreases the rank increases up to rank “97”.  Rank “98” is a special case, where the output image is rejected based on quality or the output image is not available due to numerical instability possibly caused due to the greater number of contour points [4].  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, the best parameter value of that table is the one that gives maximum good quality outputs for all samples or a majority of samples.  Exhaustive study on every parameter has been done, 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.

DCT GVF Sigma

 

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.

DCT GVF Mu

 

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.

DCT GVF Alpha

 

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.

DCT GVF Beta

 

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.

DCT GVF Kappa

 

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 segmentation 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 as 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

DCT GVF Sigma

0.25

[0.2 , 0.5]

[0.1900 , 0.5250]

DCT GVF Mu

0.075

[0.05 , 0.09375]

[0.0475 , 0.0984]

DCT GVF Alpha

0

[0 , 0.125]

[0.0000 , 0.1313]

DCT GVF Beta

0

[0 , 0.5]

[0.0000 , 0.5250]

DCT GVF Kappa

0.625

[0.5 , 0.875]

[0.4750 , 0.9187]

 

 

 

6.3 Statistical Validation

 

The parameters act independently on the segmentation 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 segmentation 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 the results are actually significant.  Therefore the experimental results are valid.  This justifies that a suitable value of parameter κ can be chosen from Table 5, and that the constant values of parameters σ, µ, α, and β used in Table 5 are also valid as these values also have significant influence on the results tabulated in Table 5.  Therefore, the experimental results and the inferences that are discussed in the previous paragraphs are also significant. 

 

 

 

6.4 Validation of Robustness of the Scheme

 

The banding pattern present in the chromosomes gives rise to higher contrast compared to the outer edges, causing the DCT based GVF external field to have a higher strength at the bands.  Therefore, the DCT based GVF Active Contour feels more attraction towards the bands than the outer boundary.  Hence, the contour tends to cross the boundary into the inner regions seeking the bands.

 

 

The chromosome images in the chromosome spread image have variability in shape and size due to the nature of the spread image.  Also, the spatial distribution of the chromosomes is random accompanied by uneven spacing between adjacent chromosomes.  Hence, each chromosome in a chromosome spread image becomes a unique sample demanding unique values of the parameters governing the DCT based GVF Active Contour.  There is also a need for unique number of iterations to converge.

 

 

The small object size of the chromosomes makes the computed DCT based GVF field also to be small.  Hence suitable choice of parameters is necessary; else the Active Contour crosses the boundary and results in a straight line at the axis of the chromosome sample.

 

 

The chromosomes in the spread image have a minor axis length varying between 14 and 17 pixels approximately and major axis length varying between 30 and 80 pixels approximately at 72 pixels per inch resolution.  This causes the DCT based GVF external field to have a high density at corners.  Accompanied with the banding characteristic, the axis lengths force the DCT based GVF Active Contour to map contours at the inner region of the chromosome instead of the actual boundary at the periphery of the chromosome. 

 

 

The weak edges in chromosomes also contribute to the Active Contour to overwhelm weak edges and move into inner regions.  In addition to these inherent difficulties, more difficulty was introduced to validate the robustness of the segmentation scheme.  The image was further degraded by transforming pixels having gray levels greater than 90% intensity in the range [0, 255].  This resulted in degradation of weak edges, giving rise to distorted edges and uneven boundary in the original image, offering more challenges to the task of segmentation using DCT based GVF Active Contours.

 

 

These difficulties make the task of segmentation of chromosomes in chromosome spread images very difficult.  Validations prove that the segmentation has been very successful.  Hence the robustness of the scheme also stands validated.

 

 

 

7. Conclusion

 

Thus, the DCT based GVF Active Contour is established as a good segmentation technique for chromosome spread images. 

 

 

 

8. Acknowledgement

 

The authors wish to thank Prof. Ken Castleman and Prof. Qiang Wu, from Advanced Digital Imaging Research, Texas for their help in providing chromosome images.

 

 

 

9. References

[1]     T. McInerney and D. Terzopoulos, “Deformable models in medical image analysis”, IEEE Proceedings of the Workshop on Mathematical Methods in Biomedical Image Analysis: 171-180, 1996.

[2]     C. Xu and J.L. Prince, “Gradient Vector Flow: A New External Force for Snakes”, IEEE Proc. Conf. on Comp. Vis. Patt. Recog. (CVPR'97) 66-71

[3]     M. Kass, A. Witkin, D. Terzopoulos, “Snakes: active contour models”, Int. J. Comp. Vision 1: 321–331, 1987.

[4]     D. Rueckert, “Segmentation and tracking in cardiovascular MR images using geometrically deformable models and templates”, PhD thesis, Imperial College of Science, Technology and Medicine, London, 1997.

[5]     B.Leroy, I.Herlin and L.D.Cohen, “Multi-resolution algorithms for active contour models”, In 12th Intl. Conf. on Analysis and Optimization of Systems: 58-65, 1996.

[6]     L.D.Cohen, “On active contours and balloons”, CVGIP: Image Understanding, 53(2):211-218, March 1991.

[7]     L.D.Cohen and I.Cohen, “Finite-element methods for active contour models and balloons for 2-D and 3-D images”, IEEE Trans. On Pattern Anal. Machine Intell., 15(11):1131-1147, November 1993.

[8]     C.Davatzikos and J.L.Prince, “An active contour model for mapping the cortex”, IEEE Trans. on Medical Imaging, 14(1):65-80, March 1995.

[9]     C.Davatzikos and J.L.Prince, “Convexity analysis of active contour models”, In Proc. Conf. on Info. Sci. and Sys.:581-587, 1994.

[10]  A.J.Abrantes and J.S.Marques, “A class of constrained clustering algorithms for object boundary extraction”, IEEE Trans. on Image Processing, 5(11):1507-1521, November 1996.

[11]  J.L. Prince and C.Xu, “A new external force model for snakes”, In 1996 Image and Multidimensional Signal Processing Workshop:30-31, 1996.

[12]  C. Xu and J.L.Prince, “Gradient Vector Flow Deformable Models”, In Handbook of Medical Imaging, Academic Press, Sept. 2000

[13]  C.Xu and J.L. Prince, “Snakes, shapes and gradient vector flow”, IEEE Trans. on Image Processing, 7(3):359-369, March 1998.

[14]  Jinshan Tang, S.T. Acton, “A DCT based gradient vector flow snake for object boundary detection”, Image Analysis and Interpretation, 2004. 6th IEEE Southwest Symposium on: 157 – 161, 28-30 March 2004.

 

 

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