Abstract
This paper describes a segmentation method combining a texture based technique with a contour based method. The technique is designed to enable the study of cell behaviour over time by segmenting brightfield microscope image sequences. The technique was tested on artificial images, based on images of living cells and on real sequences acquired from microscope observations of neutrophils and lymphocytes as well as on a sequence of MRI images. The results of the segmentation are compared with the results of the watershed and snake segmentation methods. The results show that the method is both effective and practical.
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Acknowledgement
We are grateful for the support received from the British Council in the form of the travel grant no. WAR 341/235, which allowed us to exchange ideas and jointly develop the combined method.
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Appendix
Appendix
To determine the intersection of the cell’s contour with the line of the profile, the point of the highest curvature of the graph of the Cumulated normalized Prewitt gradient operator Function (called CPF) is calculated. The point of the highest curvature is called the bending point. This is done in three steps, first, the CPF is smoothened F applying a polynomial approximation of discretized Gaussian convolution:
where w(j/RmW) are the weight coefficients of the smoothening operator, and RmW is its width.
Let r be parametrization of the curve:
or of the broken line (Fig. 10):
then
In the second step, the curvature of the smoothened CPF graph is calculated. For a continuous and differentiable curve, the value of the curvature is defined to be the inverse of the radius of the second order tangent circle at a given point (Fig. 10). Unfortunately, this definition fails in our case, since the cumulated gradient function is a non-differentiable broken line. A convenient approximate expression for curvature has, therefore, to be derived:
where
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x p = r x (t−H) x s = r x (t) x n = r x (t + H)
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y p = r y (t−H) y s = r y (t) y n = r y (t + H)
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H width of curvature operator (in t parameter space)
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r the parameterisation of curve in plane (function)
In our case, (9) is applied to the graph. This amounts to substituting r x (t) = t, and r y (t) = F (t) in (9). The fact that a discrete set of the values is given in equally spaced points also has to be taken into account. Therefore, H is regarded as the distance between two consecutive points in the curvature operator.
It should be noted that operator K * may also be used for numerical calculation of the curvature of a smooth curve on a plane: it can be shown that if r(t) is a curve of the class C 2 then
Finally, in the last step, the point of a certain curvature (e.g. maximum curvature point), hence the intersection of the cell’s contour with the profile line is obtained based on the bending operator analysis. Because of the width of the operator arm, the H parameter, the maximum of the bending operator is slightly moved to the last almost vertical path of the s-shaped typical profile line (Fig. 11a), so the strategy for the finding that point is to determine the t for which the bending operator value reaches the value of the standard deviation of the bending operator calculated for all points along the part of the line profile. The part is chosen from the point distanced of H from 0 to the point distanced of H from the end of the profile line, it means for discrete points set H, H + 1, …, L−H. If there are several such points, the strategy of one point determination is following: t is the first nearest point of the t max point, while t max is the point value of K * attains its maximum. The first is determined according to the order of analysis which starts from t max and goes along profile lines in the cell’s central point direction.
In Fig. 11a, there are the cumulated normalized Prewitt gradient function of pixels along 36 profile lines placed radially from cell’s center point (see some of them in Fig. 2f) in 10° angle distance on the cell 1 shown in Figs. 1 and 2. Most of these functions are s-shaped but there are a few in the form of the raising wavy line. Both types of functions are shown in Fig. 11 respectively in path b and c (dashed line). The s-shaped function is typical of a well defined cell border part (b) while the raising wavy function is typical of barely visible or not visible cell contour (c). Both types of functions are shown with the bending operator function of the pixels along the profile line (continuous line) and with straight vertical lines which show the value of the bending operator standard deviation (dotted line). The boundary points are marked as stars in both Fig. 11b and c.
The values of the RmW and H parameters according to the length profile lines L = 160 pixels have been examined. To do this a special tool has been developed in MATLAB. The interface of the tool allows adjusting the values of both parameters with a bar under observation of the position of the cell’s edge for a chosen line or for all profile lines. In the case of moving neutrophil image sequences, the 30 pixels distance of the H arm and the 10 pixels distance of the width of the smoothing operator RmW are chosen experimentally for the length of the profile lines L = 160 pixels. According to the analysis in Sect. 15 for small rounded cells, all the parameters should be shorten proportionally, e.g. in the case of the smallest artificial object (Fig. 8) the following values were used: L = 100, RmW = 5 and H = 16.
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Korzynska, A., Strojny, W., Hoppe, A. et al. Segmentation of microscope images of living cells. Pattern Anal Applic 10, 301–319 (2007). https://doi.org/10.1007/s10044-007-0069-7
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DOI: https://doi.org/10.1007/s10044-007-0069-7