INTRODUCTION
Gradient Vector Flow (GVF) active contours have been used for successful segmentation of chromosome spread images (Britto and Ravindran, 2005). This research work used a variant of Gradient vector flow active contour, called Discrete Cosine Transform (DCT) based Gradient Vector Flow (GVF) active contour which has been established as a better segmentation tool for chromosome spread images, compared to GVF Active contours and its variants (Britto and Ravindran, 2005). The DCT based GVF active contour has hence been characterized (Britto and Ravindran, 2005) and standardized (Britto and Ravindran, 2005) for segmenting chromosome images from chromosome spread images. The segmentation has been accurately done establishing that DCT based GVF active contours are an efficient tool for chromosome image segmentation. From the segmentation results, further insight has been obtained that substantiates that DCT based GVF Active Contours can also be used for classification of chromosome images. This paper discusses analytically, the suitability of DCT based GVF active contours as a tool for chromosome image classification.
Chromosome images: A chromosome is an extended DNA molecule containing
multiple genes and associated proteins. During mitosis and meiosis, condensed
chromosomes form structures that are visible with high-powered light microscopes
(<http://www.colorado.edu/MCDB/MCDB2150Fall/notes00/L0001.html>).
The chromosome spread images are obtained using the following general procedure.
About 5 mL of blood is removed from the patient. If a fetus is being karyotyped,
amniotic fluid is removed from the amniotic sac which surrounds the fetus during
development. This is done with the aid of a large syringe and ultrasound picturing.
There are cells which have come off the fetus in this fluid. The white blood
cells are removed from the blood or the living cells are removed from the amniotic
fluid. These cells are then cultured in a medium in which they undergo mitosis.
Mitosis is stopped at metaphase using chemicals. The cells are then placed onto
a slide spread out. They are viewed under a microscope which is specially adapted
with a camera to take a picture of the chromosomes from one of the cells (<http://home.earthlink.net/~heinabilene
/karyotypes/karyoty.htm>).
Hence, the chromosomes in a spread image have variations in shape caused due to bending effects, variations due to overlaps, variations due to illumination and device dependencies. This causes difficulties in segmenting the chromosome images as the segmenting algorithm or technique needs to be retrained often. Therefore, gradient vector flow active contour, a special class of deformable curves is chosen to perform good and efficient segmentation of chromosome spread images under such difficulties caused by so much of variation.
Chromosome structure and classification: The human chromosome consists
of 22 pairs of autosomes and one pair of sex chromosomes (X and Y chromosome).
Each pair of the autosomes and each of the sex chromosomes vary from each other
and the absolute length of any chromosome varies depending on the stage of mitosis
in which it was fixed (Fig. 1). However, the relative position
of the centromere is constant, which means that the ratio of the lengths of
the two arms is constant for each chromosome. This ratio is an important parameter
for chromosome classification (<http://arbl.cvmbs.colostate.edu/hbooks/genetics/medgen/chromo/chromo
somes.html>).
Chromosomes are arranged into seven groups based on size and centromere location.
The centromeres can be found in the middle of the chromosome (median), near
one end (acrocentric), or in between these first two (submedian) (<http://home.earthlink.net/~heinabilene
/karyotypes/karyoty.htm>)
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Group A: Chromosomes 1-3 are largest with median centromere. |
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Group B: Chromosomes 4-5 are large with submedian centromere |
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Group C: Chromosomes 6-12 are medium sized with submedian centromere |
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Group D: Chromosomes 13-15 are medium sized with acrocentric centromere |
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Group E: Chromosomes 16-18 are short with median or submedian centromere |
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Group F: Chromosomes 19-20 are short with median centromere |
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Group G: Chromosomes 21-22 are very short with acrocentric centromere. |
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Chromosome X is similar to Group C. |
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Chromosome Y is similar to Group G |
DCT based GVF active contours have been used to successfully segment chromosome images. The segmentation observations indicate that DCT based GVF active contours can also be used for chromosome image classification.
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 (Xu and Prince, 2000; Xu and Prince, 1998).
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 (Xu and Prince, 1998; Kass et al., 1987).
The GVF field is defined as the equilibrium solution to the following vector diffusion equation (Xu and Prince, 2000),
Where, ut denotes the partial derivative of u(x,t) with respect to t, ∇2 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.
1, g(|∇f|)∇2 produces a smoothly varying
vector field and hence called as the smoothing term, while h(|∇f|)
(u-∇f) encourages the vector field u to be close to ∇f computed
from the image data and hence called as the data term. The weighting functions
g(A) and h(A) apply to the smoothing and data terms respectively and they are
chosen (Xu and Prince, 1998) as g(|∇f|) = μ and h(|∇f|)
= |∇f|2. g(.) is constant here and smoothing occurs
everywhere, while h(.) 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
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 ∇f 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 ∇f is nearly zero. μ is a regularization parameter that governs the tradeoff between the first and the second term in the integrand in Eq. 2. The solution of Eq. 2 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 (Xu and Prince, 2000).
Discrete Cosine Transform (DCT) based GVF active contours: 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:
The local contrast of the Image at the given pixel location (k,l) is given by
where,
and
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 (Tang and Acton, 2004), which is then subject to selective segmentation by the energy compact gradient vector flow active contour model using Eq. 2.
MATERIALS AND METHODS
The chromosome metaphase image (at 72 ppi 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. 2 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.
RESULTS AND DISCUSSION
Characterization of parameters of the DCT based GVF active contour segmentation scheme has yielded the values of σ = 0.25, μ = 0.075, α = 0, β = 0 and κ = 0.625 as characterized parameter values (Britto and Ravindran, 2005). Standardization experiments have established that the characterized parameter values are indeed standardized and can be used for segmenting chromosome spread images from any dataset (Britto and Ravindran, 2005). Therefore, chromosome images from three independent datasets.
Three hundred and eighteen images from three datasets were segmented using
DCT based GVF active contours. A sample image, its DCT based GVF field and its
corresponding segmented image are shown below (Fig. 2a-c).
Chromosome images were drawn from three independent datasets and subjected
to segmentation. A few segmented chromosome image samples are shown below (Fig.
3-42).
Studies on the 318 chromosome images used for segmentation and their corresponding segmented output images revealed the following observations (Table 1).
Though visual inspection by the naked eye does not find errors, from the table
above, it is found that there is a very small segmentation error. The ratio
of the lengths (major axis diameter) of the original chromosome image to the
boundary mapped chromosome image reveals that the ratio is almost constant with
a standard deviation of only 0.02 or 2%. Similarly, the ratio of the areas has
a standard deviation of only 0.04 or 4%.
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| Fig. 2a: |
Sample chromosome image |
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| Fig. 2b: |
DCT based GVF field |
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| Fig. 2c: |
Segmented output image of Fig.1a |
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| Fig. 3: |
Segmented sample 1 |
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| Fig. 4: |
Segmented sample 2 |
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| Fig. 5: |
Segmented sample 3 |
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| Fig. 6: |
Segmented sample 4 |
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| Fig. 7: |
Segmented sample 5 |
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| Fig. 8: |
Segmented sample 6 |
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| Fig. 9: |
Segmented sample 7 |
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| Fig. 10: |
Segmented sample 8 |
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| Fig. 11: |
Segmented sample 9 |
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| Fig. 12: |
Segmented sample 10 |
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| Fig. 13: |
Segmented sample 11 |
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| Fig. 14: |
Segmented sample 12 |
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| Fig. 15: |
Segmented sample 13 |
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| Fig. 16: |
Segmented sample 14 |
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| Fig. 17: |
Segmented sample 15 |
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| Fig. 18: |
Segmented sample 16 |
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| Fig. 19: |
Segmented sample 17 |
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| Fig. 20: |
Segmented sample 18 |
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| Fig. 21: |
Segmented sample 19 |
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| Fig. 22: |
Segmented sample 20 |
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| Fig. 23: |
Segmented sample 21 |
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| Fig. 24: |
Segmented sample 22 |
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| Fig. 25: |
Segmented sample 23 |
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| Fig. 26: |
Segmented sample 24 |
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| Fig. 27: |
Segmented sample 25 |
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| Fig. 28: |
Segmented sample 26 |
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| Fig. 29: |
Segmented sample 27 |
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| Fig. 30: |
Segmented sample 28 |
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| Fig. 31: |
Segmented sample 29 |
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| Fig. 32: |
Segmented sample 30 |
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| Fig. 33: |
Segmented sample 31 |
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| Fig. 34: |
Segmented sample 32 |
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| Fig. 35: |
Segmented sample 33 |
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| Fig. 36: |
Segmented sample 34 |
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| Fig. 37: |
Segmented sample 35 |
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| Fig. 38: |
Segmented sample 36 |
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| Fig. 39: |
Segmented sample 37 |
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| Fig. 40 |
Segmented sample 38 |
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| Fig. 41: |
Segmented sample 39 |
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| Fig. 42: |
Segmented sample 40 |
| Table 1: |
Segmentation observations |
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Hence, the length or area of the boundary mapped image, when multiplied with
its respective mean (from table) will surely approximate the length or area
of its corresponding original image within 5% tolerance. If suitable training
algorithms like neural networks or fuzzy algorithms are utilized (that are already
in use, description of which is beyond the scope of this work), the length and
area of the original images can be obtained from the observations on boundary
mapped images.
The Centroid of the boundary mapped image serves to identify the centre point of the boundary mapped chromosome image. Accompanied with suitable centromere segmentations algorithms (that are already in use, description of which is beyond the scope of this work), information about the exact location of the Centroid in the boundary mapped chromosome image can be extracted.
Hence, from this information, the chromosomes can be classified into Groups A to G (for autosomes) or into 22 pairs of chromosomes and the sex chromosomes (X and Y) or into X and Y chromosomes as per prevalent cytogenetic guidelines for chromosome identification.
Thus, we find that the DCT based GVF active contours which was used for chromosome image segmentation has got good potential for application as a tool for chromosome classification using the same observations generated during the segmentation process. Therefore, the DCT based GVF active contours can also serve as a tool for chromosome classification.
CONCLUSIONS
The discussion yields valuable insight into the capabilities of DCT based GVF
active contours, hitherto employed as an efficient chromosome segmentation tool
to extend as a suitable chromosome classification tool. Hence, the DCT based
GVF active contours can be used as a suitable chromosome classification tool.
ACKNOWLEDGMENTS
The authors extend their heartfelt thanks to Dr. Michael Difilippantonio, Staff
Scientist at the Section of Cancer Genomics, Genetics Branch/CCR/NCI/NIH, Bethesda
MD; Prof. Ekaterina Detcheva at the Artificial Intelligence Department, Institute
of Mathematics and Informatics, Sofia, Bulgaria; Prof.Ken Castleman and Prof.Qiang
Wu, from Advanced Digital Imaging Research, Texas and Wisconsin State Laboratory
of Hygiene-- <http://worms.zoology.wisc.edu/zooweb/Phelps/
karyotype.html for their help in providing chromosome spread images.