Precision of smoothed Hough-space indexed GND calculations

[Omero Felipe Orlandini]

Hello, all!

With how incredibly easy it is to run the new GND commands, I find myself enthusiastically calculating GND density maps for all sorts of EBSD data sets and getting relative distributions that even seem to make some physical sense. However! My impression has been that the study of GNDs is exclusively the domain of 'HREBSD', and not the thugs who use Hough-based indexing because GNDs are simply too elegant and subtle for such a crude orientation measurement. It seems that thanks to Heilscher et al., 2019 'Denoising of crystal orientation maps', non-HREBSD datasets should be able to generate valid GND density values... but to what degree (if you'll forgive the expression)?

Specifically, how are the limitations of Hough-based indexing manifesting in the GND density calculations? Taking for granted that HREBSD has a better angular resolution than traditional indexing, a first question is - how much better? How precise even are modern indexing methods? Is the price you pay for not using HREBSD and instead using low binning, high Hough space resolution, edges, etc. that the lowest GND density you can resolve is 1E-13 instead of 1E-9?

Based on playing around with a handful of datasets, it seems like if the densities are high enough then high-quality normal indexing plus denoising produces apparently sensible results. The little high-GND/high KAM bands on this example quartz grain make some physical sense because it's right next to a crack, but there is a fainter sort of wormy pattern in the GND maps that might just be a funny kind of noise?

I am also a little concerned by Figure 15 in Hielscher et al 2019, because it seems to my human brain that there are pretty different spatial distributions (even maxima) in the HREBSD GND density map compared to the smoothed traditional GND density map.

What do you all think? It is starting to seem to me that these GND maps are sort of the trace element analysis of EBSD, and as an excitable person I worry that it might be quite easy for me to produce fascinating nonsense.

[Omero Felipe Orlandini]

Hello again, all!

I decided to explore this a little further myself using a single crystal of silicon. This is all very new to me, so please correct me if you see any errors in my logic. I don't really know the history of these single silicon crystals except that they are used to calibrate the EBSD system, but it seemed a safe assumption that this was as close to a strain-free material as I was reasonably going to find. By calculating GNDs for a map area that has a very small to zero number of GNDs in reality, whatever the calculation reports ought to be my 'noise floor'. Bonus points for also being really easy to get a great pattern from, and therefore also high angular precision indexing. I collected three 100x100 maps at a step size of 1 micron, moving around slightly, at 1x1, 2x2, and 4x4 binning for the EBSPs. Each of these were smoothed using the default settings of halfquadratic and were indexed using the 6-band 'refined accuracy' mode of Aztec 3.4. I have attached MAD, 1st-order KAM, GND density maps, and the GND value histograms of every pixel for each run.

Still thinking about this sort of like an EDS/WDS question, it seems reasonable to define the lowest GND density that I can measure as the upper 2-sigma of the GND density distributions from these maps (assuming that they have a Gaussian distribution, which is debatable), each of which are just measuring noise/background.

Based on these data, it seems like at 1x1 binning for my Nordlys Nano the smallest GND density I can measure (in silicon, at least) is 1.3E12, 2x2 binning is 2.0E12, and 4x4 binning is 2.3E12. So, having some respect for the sloppiness of these assumptions, it would seem reasonable to say that if I collect a map, am pretty sure that all of the material properties are correct, and end up with apparent GND densities >1E13, then that is likely some kind of real signal.

What am I missing in these assumptions? One thing is that I have very little mathematical understanding of what the halfquadratic smoothing is really doing, and more practically how to tell when I have misused it (something I will continue to try to improve using Bergman et al 2015 https://arxiv.org/pdf/1505.07029.pdf and Heilscher et al., 2019 https://www.researchgate.net/profile/Ralf_Hielscher/publication/335366394_Denoising_of_crystal_orientation_maps/links/5e254380a6fdcc1015782348/Denoising-of-crystal-orientation-maps.pdf).

Another thing is that it might be inappropriate to compare cubic silicon to trigonal quartz as directly as I am here.

Another thing is that maybe the MAD distribution for a particular map dataset needs to be propagated as further uncertainty in the apparent GND densities; this is attractive as I think it would somewhat also account for the effects of sample preparation.

Thank you all very much for your time and effort developing MTEX, and I apologize if these answers are obvious!

Best,

Phil

[Nicolas Piette-Lauziere]

Hey Phil,

I think it is an interesting approach to test how reliable any EBSD method is (not just GND modelling). I suppose that if you compare axes and angles, the crystal symmetry of your reference material does not matter, but if you model grains necessary dislocations you then compare apples from oranges. Perhaps you could look into the curvature tensor from your reference material as a baseline to single out meaningless data?  

Best,

Nicolas

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[Grafulha Morales Luiz]

Hi Phil

I assume you are familiar with the Jiang, Britton and Wilkinson paper from 2013 at Ultramicroscopy, where they discussed extensively the effects of detector binning and stepsizes

https://www.sciencedirect.com/science/article/pii/S0304399112002823

There is also the Kamaya paper from 2011

https://www.sciencedirect.com/science/article/pii/S0304399111000726

or Britton et al. 2013

https://www.sciencedirect.com/science/article/pii/S030439911300212X

Although I don't think using the Si crystal as a proxy for the noise on the maps is a bad idea, maybe you would get a more reliable "level of noise" if you do the map in a quartz single crystal, which may have some defects, but it is closer to the reality of the deformed quartz crystals you are measuring in your rocks. 

cheers

Luiz

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From: mtex...@googlegroups.com <mtex...@googlegroups.com> on behalf of Nicolas Piette-Lauziere <nicolas....@gmail.com>
Sent: 24 September 2020 20:26:14
To: mtex...@googlegroups.com
Subject: Re: {MTEX} Re: Precision of smoothed Hough-space indexed GND calculations

 

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[orla...@colorado.edu]

Hello, Luiz and Nicolas!

Nicolas, I am sadly not sure I understand what you mean by looking into the curvature tensor. I suspect your knowledge of the overall GND calculation exceeds my own, and if you have time to show me this I would very much appreciate it.

Luiz, thank you for these references! I had not read them all, and they were very interesting. For the most part though, these papers discuss cross-correlated HR-EBSD measurements and not Hough-space indexing. I am not an expert in HREBSD but my impression is that the way the two determine orientations is different enough that they cannot be compared in the way that I mean. HREBSD is clearly more precise and well-suited for GND density observations and there are many interesting questions about its accuracy and precision, but I do not have access to this expensive software. My question here is specific to Hough-space indexing combined with MTEX de-noising.

Kamaya 2010 infers an orientation noise between ~0.08 and 0.4 degrees using EBSP resolutions of 640x480 and lower, with Hough space resolutions of 120 and lower. Unfortunately, from Figures 8 and 9 it is not clear to me which case yielded which background noise levels but it is probably the obvious choices. Britton et al 2013, parts I and II are very cool but seems to be exclusively discussing HREBSD when it comes to precision and accuracy.

Jiang et al 2013 has many interesting observations relevant to the precision of HREBSD, and the exploration of step size vs. dislocation cell size was extremely relevant. The introduction of this paper does show that what I'm really asking about is the angular resolution measured in rads. The GND calculations are an interesting proxy for this metric and it might be the most practically useful for EBSD analysists to gauge this themselves, but it would be better to determine this value directly if possible. Jiang et al 2013 claims that Hough transform based indexing have an angular resolution of 4.3E-3 rads (0.2464 degrees?) and cites Humphreys 2004 'Characterization of fine-scale microstructures by electron backscatter diffraction (EBSD)' for that number.

In the Humphreys 2004 paper, Table 1 lists 'typical' angular resolutions of 1 degree without providing any evidence or discussion of how that number was arrived at. The discussion of angular resolution of Hough-space EBSD is limited to one paragraph, in section 3.4 where 'orientation noise' is described as typically 1-2 degrees citing Randle and Engler 2000 'Introduction to Texture Analysis', and several approaches such as averaging pixels together can improve the angular resolution that are not really relevant to my question.

Randle and Engler 2000 states that ‘The average spatial resolution and accuracy of EBSD are ~200-500 nm and ~1 degree respectively,…’ and goes on to describe various influencing factors like specific materials and geometries. No source is cited for this number and no evidence provided in the text. Figure 7.14 is interesting because it shows misorientation distributions for various probe currents as measured on a single crystal with presumably no defects, showing that a 400 nA current produced a measured misorientation of <0.1 degrees and cited Humphreys et al 1999.

Although Figure 7.14 represents an actual measurement by proxy of the best angular resolution possible (at least, two decades ago) instead of a chain of citations, looking at Humphrey et al 1999 a figure of ~1 degree is asserted, with general considerations and limitations on angular resolution referred back to Krieger Lassen et al 1992 and Wright and Adams 1992. Humphrey et al 1999 goes on to state that as EBSD cameras gain higher pixel resolutions, the best possible angular resolution should increase.

Without going into Dr. Lassen’s work as well, I think it’s safe to say that the commonly cited ~1 degree angular resolution for Hough-space indexing has persisted through major improvements in EBSD cameras and indexing processes and is likely no longer accurate. So what is the angular resolution of a modern system?

From this list, the most recent attempt to actually measure the angular resolution seems to be Kamaya 2011, nine years ago. This study had smaller EBSP pixel resolution than is common today, did not have the improvement to accuracy offered by the MTEX de-noising routines, and was nine years behind the ever-improving (however opaquely and slightly) commercial indexing algorithms - all of which suggest that a modern EBSD study should be capable of angular resolutions better than ~1 degree, and potentially much better? The question I am posing out loud now is how can we all figure out where our own noise floor is if we start pursuing high-sensitivity data like GND distributions.

Sort of following in the Humphreys model, the distribution of m2m.angle./degree values in my 1x1 map of nominally undeformed silicon single crystal is attached below. Does this suggest that the angular resolution of my system+processing for this map is 0.043 +/- 0.042 degrees? Again, there are sensible caveats to extrapolating this to a deformed totally different material, and I 100% agree that a quartz standard (or maybe blank is the better term) would be a much better place to start if I intend this for a quartz analysis.

Thank you all again for your time!

Best,

Phil

[orla...@colorado.edu]

Hi all - sorry, was not clear with the last histogram. While commuting home I realized that it might be useful to have versions with and without the halfquadratic de-noising. I accidentally posted the raw version and statistics earlier, the de-noised version has a mean value of 0.0321 degrees with a standard deviation of 0.0149, although I don't know that the assumption of Guassian distribution is valid for the de-noised values.