Dot Analyzer for SEM micrographs

an ImageJ/Fiji python script for dot spacing and order analysis of SEM micrographs

License

The Dot_Analyzer script runs under the BSD-3 License

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This document describes the workings of the ImageJ/Fiji python script Dot_Analyzer.py, written by Philippe Girard (email).

This Python script was written to analyse SEM micrograph as shown here:

Fig. 1: A typical SEM micrograph.

1. Installation instructions

• Download the python script file into your computer.
• For the script to appear in the menu, you should saved in the ImageJ2.app/scripts or the ImageJ2.app/plugins/Scripts directory (or a subdirectory thereof).
• Otherwise use File > Open… to open the script Dot_Analyzer.py in the script Editor of ImageJ/Fiji and Click Run on the bottom of the script editor window (you can also go to : Run > Run in the Script Fiji menu).

  
  Fig. 2: Script Dot_Analyzer.py in the script Editor of ImageJ/Fiji.

2. The “Parameters” main window.

When you start the plugin, you have to define different parameters for the analysis and to select the different diagrams/plots that you want to visualize (Fig. 3).

Fig. 3: The “Parameters” main window

You must indicate:
1. `Select the image to analyse`: Choose the micrograph image.
2. `Measure scale bar on the image`: If you do not know the size in nm of a pixel or if your image is not automatically calibrated spatially, you should select this option. In this case before the analysis a dialog box is asking the user to draw a rectangle to fit with the scale bar (it is recommanded to zoom in the region of the scale bar usually at the bottom left of the micrograph) as shown in Fig 4.

Fig. 4: Scale bar measurement

After this dialog, a second dialog box (Fig 5) is asking for the known distance in nm written at the scale bar of the micrograph

Fig. 5: "Known Distance" Dialog box

3. If the distance in pixels and the corresponding distance in nm are known, the previous step can be bypassed (by ticking off the previous checkbox) and these 2 numeric fields: `Distance in pixels` and `Known Distance in nm` can be used for pixel conversion. For example, in a field-emission scanning electron microscope (FE-SEM, LEO-1530, LEO, Oberkochen, Germany), the parameters are 200 nm = 171 pixels at 100.00 KX magnification.
4. `Minimal size of particles in pixels`: Set the minimum size (in pixels^2) to exclude objects that appear in the binary image that are clearly not of interest. This parameter is the same that appears in “Size ( ^2)” of the “Analyze Particles” window.
5. `Automatic Threshold`: if the checkbox is ticked on, the image is automatically thresholded with the "Triangle" method. Otherwise, the Threshold window (Image ▷ Adjust ▷ Threshold…) is displayed as shown in Fig. 6 for allowing the user to manually select the thresholding method or to interactively explore the threshold value that segments every dots (in red).

Fig. 6: Manual threshold

6. `Choose the diagram to display`: `Voronoi diagram` = the Voronoi tesselation [**[1-4]**](#references) or `Voronoi/Delaunay diagram`= the Delaunay triangulation [**[3-4]**](#references) superimposed to the Voronoi tessalation. * The Voronoi diagram is a simple mathematical construct that has proved useful in fields as diverse as environmental studies, cell biology, crystallography, transportation planning, and communications theory. Given a set of points (the center of mass of each gold-dot), the Voronoi diagram defines a series of cells surrounding each point. Each cell contains all points that are closer to its defining point than to any other point in the set. Subsequently, the “borders” of the cells are equidistant between the defining points of adjacent cells.By doing so, the number of borders give you the number of closest neighbors. For more information about the Voronoi diagram, see the [Wikipedia webpage](http://en.wikipedia.org/wiki/Voronoi_diagram).
* Delaunay diagram (or triangulation) . In graph theory, the Delaunay triangulation corresponds to the dual graph of the Voronoi tessellation. By dual, I mean to draw a line segment between two Voronoi vertices if their Voronoi polygons have a common edge, or in more mathematical terminology: there is a natural bijection between the two which reverses the face inclusions. This diagram gives the distance between the closest neighbors of each gold-dot. For more information about the Voronoi diagram, see the [Wikipedia webpage](http://en.wikipedia.org/wiki/Delaunay_triangulation).
7. `Besag’s L Function`: The Besag's L Function is based on the Ripley's K Function. Details of various theoretical aspects of K are in books [**[5-7]**](#references). Ripley’s K function is a popular tool to analyze mapped spatial point pattern. It is defined without edge correction as:

If points are distributed independently from each other, g(ρ)=1 for all values of ρ, so K(r) = πr². This value is used as a benchmark:

• K(r) > πr² indicates that the average value of g(ρ) is greater than 1. The probability to find a neighbor at the distance ρ is then greater than the probability to find a point in the same area anywhere in the domain: points are aggregated.
  

• Inversely, K(r) < πr² indicates that the average neighbor density is smaller than the average point density on the studied domain. Points are dispersed. K(r) is estimated by the ratio of the average number of neighbors on the density, estimated itself by the total number of points divided by the domain area λ which is the density (number per unit area = N/area of the picture) of event. Given that only the points in a bounded window of observation can be studied, edge correction is necessary to obtain precise estimates. The weighted edge-corrected function K(r) is defined as:
  

  

  where I(r_ij ≤ r) is an indication function with values either 0 if the condition does not hold or 1 if the condition holds and where the weight w_ij(r) is the proportion of the circumference of a circle centered at the gold-dot i passing through the gold-dot j and that is inside the region of interest (ROI), which is defined as
  

  

  where r_ib is the distance from the gold-dot i to the nearest boundary, r_ib1 and r_ib2 are the distances from gold-dot i to the nearest two boundaries. The first case when the circle is within the ROI, the second case is when the circle intersects with only one border and the last case is when the circle intersects two borders in a corner. Note that w_ij(r) could be unbounded as r increases in practice. Following the recommendation by Ripley, w_ij(r) could be restricted to be less than or equal to 4 for the gold-dot i having distance to j greater than the distance from the gold-dot i to the nearest boundary.
  The Besag's L function is just a normalization of the Ripley's function:
  

  

  because for a homogenous Poisson process, the Ripley function is K(r) = πr². So the Besag's function is a measure of the deviation from a Poisson distribution and it is very useful because L(r) has the advantage of linearizing K(r) and stabilizing its variance and has an expected value of zero for Poisson distribution. So L(r) can also be considered as a measure of the clusterization.
  
  

1. Pair correlation function is measured with the Epanechnikov kernel [8] and an Ohser-Stoyan edge corrector factor [9]. The estimation of the pair correlation function g(r) can be obtained by determining all pairs of gold-dots having inter-gold-dot distance in some small interval and counting their numbers. Since g(r) is a density function, a more elegant method can be employed. Following the recommendation of Penttinen et al. [6], a kernel estimator is used for g(r). The chosen kernel function is the Epanechnikov kernel:
   

   

The kernel δ is very important because it determines the degree of smoothness of the function. Based on Penttinen et al. [8],

in this study. Then, the pair correlation function can be estimated as:

where w(.) is the Epanechnikov kernel function defined above, λ=N/A is the estimated density (gold-dots per unit area), r_ij is the distance between the gold-dots i and j and s(r) is the edge correction factor. For rectangular or square plots, the Ohser–Stoyan edge correction factor [9] can be adopted:

where b₁ and b₂ are the side lengths of the image (that means the height and the width) and s(r) < width x height.

1. Bond-orientational correlation function: The (global) bond-orientationel order parameter 6 was introduced by D. R. Nelson and B. I. Halperin to characterize the structural order in 2D systems [10-11]. It is given:
   

   

   with ψ₆ is the local value for the particle i located at r=(x,y):
   

   

   where I is the imaginary unit (I²=-1), θ_ij is the angle between the particles i and j and an arbitrary but fixed reference axis and n_i the number of nearest neighbors of the dot i. The bond-orientational correlation length ξ₀ was extracted from the “zero-momentum” correlation function of ψ₆ which is called bond-orientational correlation function g₆(r) and defined by:
   

   

   where the denominator is related to the pair correlation function g(r) and:
   

   

For a 2D system (or the quasi long-ranged bond orientational order of the hexatic state), the envelope of this correlation function decay to zero exponentially [12-14]:

Hence, ξ₀ is a measure for the typical size of the single crystalline domain, i.e., larger is ξ₀ and larger is the crystalline domain (and better is the order). The bond-orientational correlation length ξ₀ is determined by fitting log(g₆(r)) with a line Ar+B by using the fitting algorithm described in Numerical Recipes Section 15.2. The plugin shows the function g₆(r) with the exponential fit in red and the length ξ₀ (with the χ² test) in the same plot.

1. Save Spacing and Order in a table this checkbox indicates that you want to save the spacing and the order in a text file. If you select this option, a “Save Spacing & Order” window will appear at the end of the analysis (Fig. 17).
   

3. Analysis

Preprocess step

First, the image is cropped (89% of the original height) to remove the information bar at the bottom of the micrograph (Fig 7).

Fig. 7: Original image (Fig. 1) with the rectangular selection used to crop at 89% the height of the image.

Then the image is converted to a binary image to reveal the spots (by automatic or interactively thresholding depending of the checkbox Automatic Threshold) and a rolling ball of 10 pixels is applied to substract the background to the image (Fig 8).

Fig. 8: Thresholded image.

An overlay of the image with the substracted background (in grey) and the thresholded image (in red) is displayed (Fig. 9) to allow the user choosing to restart the threshold step (Fig. 10).

Fig. 9: The Overlay: Substracted-background image (in grey) / threshold image (in red).

Fig. 10: Restart or not Window.

Dot dectection analysis

The function “ParticleAnalyzer” (“Analyze/Analyze Particles…”) is applied to detect the position of the different spots (the parameter “Min size” you have selected at the beginning is use here to remove all the spot below this size in pixel^2). This process permits to measure the centre of mass of each spot. Then a Voronoi analysis is applied to these dot position and the voronoi cells are listed as ROI in Roi Manager. Each voronoi cell (that are not at the border of the image) are color-coded (using the LUT "glasbey inverted") in an 8-bit binary image according to the number of neighbors of each individual dot (Fig. 11). Mouse hovering over the colored Voronoi ROIs enables you to know the respective number of neighbors in the ImageJ/Fiji main window as the number behind “index=”. If Voronoi/Delaunay diagram is selected, a Delaunay triangulation is superimposed to the previous Voronoi image (Fig. 12)

Fig. 11: Voronoi diagram of Fig. 1.

Fig. 12: Voronoi/Delaunay diagram of Fig. 1.

A Neighbor Bar (Fig. 13) is also displayed in addition to the analysis output to indicate the number of neighbors of each voronoi ROI (=individual dot).

Fig. 13: Neighbor Bar.

For in-depth analysis, different plots can be displayed :

• the Besag’s L Function:
  

Fig. 14: The Besag’s L Function of Fig. 1

• the Pair correlation function:
  

Fig. 15: The Pair correlation function of Fig. 1

• the Bond-orientational correlation function:
  

Fig. 16: The Bond-orientational correlation function of Fig. 1.

Save Result: Spacing and Order

If you have ticked on the checkbox Save Spacing and Order in a table in the “Parameters” main window (Fig. 3), the “Save Spacing & Order” main window (Fig. 17) is appearing to let the user indicating the different characteristics listed below and all these parameters are saved in a new or existed CSV file.

1. First row: Polymer
   • First copolymer (in the list: PS, P2VP, PDMS, PMMA) with the number of monomer (Fig. 17a at left)
   • Second copolymer (Fig. 17a at right)
2. Second row:
   • Loading rate of gold (between 0 and 1),
   • Polymer concentration (in mg/ml)
3. Third row:
   • Select if you have use the dipping method (False) or spinning method (True). If you select spinning method the “dipping speed” is disabled and in grey because you do not have to specify it (Fig. 17b).
     
   • Dipping speed (in Volts).
4. Fourth row:
   • Date of the file when it was created (automatically filled with the information of the image),
   • Add in an existed file (Yes = True, No = False). The created file is automatically saved in the folder of the analysed image.

Fig. 17: The “Save Spacing & Order” main window.

The CSV File created contains the following parameters of the analysis:

1. Filename (without extension),
2. Polymer, (example: PS[52400]-P2VP[28100])
3. Loading,
4. Concentration (mg/ml),
5. Speed (V),
6. Date (yy/mm/dd),
7. Number of dots,
8. Spacing in nm,
9. Stdev (Standard deviation) in nm,
10. Sterror (Standard error) in nm,
11. Order Parameter

This file can be opened in ImageJ/Fiji as a Results Table (Fig. 18) or in Excel (with File ▷ Import)

Fig. 18: The parameters of the analyzed image (from Fig. 17) imported in ImageJ/Fiji as a Results Table.

References

[1] G. F. Voronoï. Deuxième mémoire: recherches sur les paralléloèdres primitifs. J. Reine Angew. Math., 136:67–181, 1909.

[2] G. M. Voronoï. Nouvelles applications des paramètres continus à la théorie des formes quadratiques. deuxième Mémoire: Recherches sur les parallélloèdres primitifs. J. Reine Angew. Math., 134:198–287, 1908.

[3] Okabe, A et al., K. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd ed. New York: Wiley, 2000.

[4] F. Aurenhammer. Voronoi diagrams - a survey of a fundamental geometric data structure. ACM Transactions on Mathematical Software, 23:469-483, 1996.

[5] Ripley B. D. Spatial Statistics, Wiley, New York, 1981.

[6] Diggle P.J. Statistical Analysis of Spatial Point Patterns, Academic Press, New York, 1983.

[7] Cressie N.A.C. Statistics for Spatial Data, Wiley, New York, 1991.

[8] Penttinent et al. Marked point processes in forest statistics. For. Sci. 38, 806-824, 1992.

[9] Stoyan et al. Stochastic Geometry and its applications, Wiley, New York, 1987.

[10] B. I. Halperin and D. R. Nelson, Phys. Rev. Lett. 41, 121, 1978.

[11] D. R. Nelson and B. I. Halperin, Phys. Rev. B 19, 2456, 1979.

[12] C.A. Murray, in Bond-orientational order in condensed matter systems, edited by K. Strandburg, Springer-Verlag, New York, pp. 137–215, 1992.

[13] D.R. Nelson, Defects and geometry in condensed matter physics, Cambridge University Press, Cambridge, 2002.

[14] D.R. Nelson, M. Rubinstein, Phil. Mag. A. 46, 105, 1982.