I'll send you an email now.

The original linear fit is probably inaccurate for any set of points with duplicate x values, this is most prominent in those with many similar x values. My understanding of this can be illustrated by considering a vertical perimeter segment. Say we have pixels all along the line x=0, through y values {1,2,3,4,5}; what average value of y does a line through these points take at x=0?, 3. Thus we yield the constant function F[x]=3, but this line is a tangent to the fit we want!

As I say, this occurs for any multiple valued x axis data points, and thus influences the majority of our fits. This can be seen when the gradient by average division (my last post) is compared to the original fits. Most results are similar, but not identical!

If you extract and rewrite the average division gradient algorithm, trying to coax the trigonometry towards the data (at present arctan is one of the last things to be applied), I expect that the division by zero errors can be navigated safely. One solution would indeed be to express the vector for each line as the average of the vectors between the constituent points, and then pass the vectors to the vectorangle function and let it worry about all the zero division and infinities.

Once again, I doubt even a well implemented linear fit is what you ultimately want to use given the curvature of these shapes. I'd encourage you to take the cornerpoints data and try algorithms on a small set until you find something you're happy with. My original work was mostly concerned with extracting the correct data, turning pixel data into perimeter cycles.

In a fit of laziness I've used Fit to fit lines to the point sets in cornerpoints, where we are after the line gradients. The issue here is that Fit finds the y values in terms of the x values, and thus isn't appropriate for us where one x has multiple y images. There are various workarounds, but it would be better to just use a more suitable algorithm.

Take each set of points, and calculate the differences between successive points in each dimension. Then total these differences, and divide them to obtain the average gradient along each line. Plug these gradients into some trigonometry for the angle between the two lines.

ListPlot[#, PlotLabel -> 
    N[Abs[-180 / Pi ArcTan[(x - y)/(1 + x y) /. {
        x -> Divide @@ Total[Differences[#[[1]]]], 
        y -> Divide @@ Total[Differences[#[[2]]]]
}]], AspectRatio -> 1, PlotRange -> All] & /@ cornerpoints

This leads to a lot of division by zero complications (that I have not resolved in the above implementation), and makes the fitted lines harder to plot over the data; but most importantly, is always correct!

Hello Dave G., I just read your comments and I am absorbing them now, but wanted to acknowledge them and thank you for them. I am glad this problem has piqued some interest out there. I am cautious in saying this, but it seems to me that extracting this type of individual measurement at this scale in microstructures is a bit novel and a departure from other work in image analysis that I've seen so far. I'll look at the notebook and let you know what I can do with it. Thanks again. Phill.

Hello David. The approach that you describe is one that I have been considering for the last couple of weeks since I first saw a similar one mentioned in the Mathematica Stack Exchange for a similar problem (I think it was an effort to analyze biological structures which I have seen tons of over the years of studying stereology). I believe there is great promise in using such an approach since it effectively samples what are the relative orientations of the solid-liquid surfaces as the appear further and further away from their point of intersection (and this is of great interest to us). Of course the confidence interval for individual measurements would be large due to the noise at the digital "edges" in the binary representation of the particles, however, it is the mean of the population that we want. This is important because of two things. First, the images I have are of metallographic surfaces, or planes cut through the solid samples. It has been established that the measure of an angle formed in such a plane (also known as the "section angle") from a three-dimensional "dihedral" angle depends BOTH on the measure of the dihedral angle itself AND its orientation with respect to the orientation of the plane. Therefore, one cannot measure three-dimensional dihedral angles by observing two-dimensional metallographic planes. However, it has been shown that the population mean of section angles is equal to the population mean of dihedral angles in a given sample! Therefore, we primarily need a robust sample (and hence the desire for automation) of section angle measurements. Also, we have learned that the population distribution of the section angles gives and indication of the distribution of the dihedral angles, and therefore evaluating the frequency distribution and estimating the standard deviation is very useful as well. I have stored grayscale images of many thousands of these angles that are a rich source for this analysis.

As I mentioned, I was considering the disk approach, but I am still a Mathematica neophyte! I know how to develop a list of coordinates for the triple points, and also how to create a disk primitive with a specified radius and coordinates of the center point, however, I have some work to do. I see future steps like this: Use list of vertex coordinates to ask Mathematica to create disks with a defined radius at each vertex, combine the binary image of the microstructure with the newly created image containing the defined disks and use the microstructure image to mask/remove the portions of the disks that are bounded by the circumference of the disk and the two intersecting particle boundaries, then analyse the difference between areas of the original disks and the new areas with those sections removed. The results should be proportional to the angles at the radius distance from the point of intersention, and the trend with respect to changes in the chosen radius would indicate curvature. This is a lot of fun and I can't wait to try it out, but I just don't have the Wolfram chops yet. Any suggestions on code would be worth a lot to me! Much obliged, Phill.

Hello David, thank you very much for your thoughts about this problem. Allow me to add some additional discussion to make some clarifications.

The images that I posted are a small section of an image that is 5464x4080 pixels in width and height. The images were originally captured in grayscale at 150 dpi, and I have them saved in .tif format at this resolution. Of course the routines that I have used to produce the image showing the "corners" identified have also recognized similar features at the edges of the image, however, my plan is to eliminate any such artifacts by "cropping" the processed images along with the data collected from them once the code is satisfactory.

With regard to any approach for the angle measurement itself, there a few things to keep in mind. First, the angle at the triple-point is analogous to the wetting angle described in literature by "Young's Equation" which is used to determine if liquids will spread over the surface of a solid (whether the surface is hydrophobic or hydrophilic). This angle is defined as measured between the normals of the intersecting surfaces at the point of intersection. At later stages in the liquid-phase sintering process the surface energies will tend to dominate the contact angles and produce a classic shape at these points. The images posted here, however, are from a sample at only 1 minute sintering time, which is not enough time yet to achieve this shape in all contacts. Some of the contacts, as you can see, are still somewhat irregular. It is my belief that this sample was a good choice to start with because it is difficult and requires a robust method. My plan was to first try to fit vectors to the surface of the particles beginning at the triple points and sampling pixels at the surface up to a specified range, perhaps 5 or 10 pixels. Once I have a good method to fit these vectors, I am interested in adjusting the number of neighboring "surface pixels" the codes uses to observe the effect on the measurement outcomes. Of course following the classical definition it would be entirely proper to use one the first closest neighbor to fit the vector, but of course these pixels are placed in their positions by means of a process containing inherent errors and approximations and the resulting measurements would produce an unacceptably high variance.

I have posted a section of the original image at the original resolution of 150 dpi for you to look at.

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