Even though some of you may have already seen the results of my preliminary analysis on this subject posted at
Nemo's site, I've decided to do it all over again a bit differently, with much greater precision and followed with detailed, step-by-step explanations in order for everyone to be able to check my reasoning and verify my final findings. The analysis is based on the following image out of Chad's series, here with marked laser measured distances done by the detectives:
As you can see, there is an interesting coincidence that made me use this certain image for analysis - the markings for 36 and 50 meter measurements are practically in line with the imaginary axis passing along the smaller side fins and through the center of the drone ring. Also, it seems as if the drone is almost touching the canopy of the trees below, so it is probably very close to this imaginary line. This information by itself enables some provisional assessments of the drone size, but by courtesy of the DRT I was provided with yet another image, taken from the exact same spot as the indicated laser measurements, totally unmodified and containing full EXIF information which, among other things, enables precise calculation of field of view angle(s). As this is the basic information for all calculations here, I'm going to refer to this image as the "calibration image" in further text. Here's the basic concept for further analysis:
This represents the "top" view of the situation present in both Chad's and calibration image with the aforementioned measured distances of 36 and 50 meters marked as points A and B, respectively. What we want to find out is the length between these two points, and for this we'll use some basic
trigonometry to crunch the numbers extracted from our calibration image. But first, some clarifications: although they are represented by two simple horizontal blue lines in the above image, what we are actually looking at are planes in 3D space, and those two I've drawn are each containing one of our reference points. Note how their intersections with those red and yellow circles (again, actually spheres in 3D space marking camera equidistant points) are represented in this clip from the calibration image (ignore the length, we didn't calculate it yet
:
The outer edge of thus created circles contains all the points with the same distance that are also closing the same angle with the axis through the center of the image. Our previous basic trigonometry lesson reminded us that tan() of this angle is related to the tan() of half of horizontal field of view angle of the calibration image in the same way as the intersection circle radius is to the half of image width... but let's try with another image containing all formulas:
This is the situation around our point B, marked as X in the above image. For clarity, I've translated it around the previously mentioned circle into X' in order to have it sit on the horizontal image axis. This doesn't affect neither its distance from image center, nor from the camera, so also all the angles in thus defined right triangle (a1-b-c1) still remains the same. Note that we have yet another right triangle - the one that shares the same camera distance leg with the first one (a2-b-c2) with the known angle alpha2 (FoV/2) - now, by formulating our equations in just the right way so that only the ratio is required instead of using real measurements for a2/a1, we can use pixel measurements from the image to deduce the angle alpha1, and then the rest is easy. The value of a2 was calculated in two different ways just to check if everything is all right, and as the final result we get spatial coordinates of point X (also double checked by calculating the
euclidean distance to the camera). After we do the same math for point A, by calculating the euclidean distance between the spatial coordinates of points A and B we get the value already shown in the calibration image:
23.5 meters.
Now, for the fun part - with all the dimensions known as a result of previous calculations, it's not hard to construct the ruler below our line. Here's how it looks in the calibration image:
But what we need is this ruler transferred into the perspective of Chad's image. In order to do that, first we find the same reference points in Chad's image:
There's difference in perspective, resolution, vegetation growth and so on, but I believe this should be the right spots. However, as nothing about Chad's image is known, I'm also going to need the following (IMO, safe) assumption to be able to transfer the ruler: both small side fins are of equal length. With this in mind, here's the final result of this analysis:
There. Feel free to comment on any aspect and inquire about anything that is presented here, especially if you've found an error or think my reasoning about something wasn't correct. If you're interested in trying to analyze this or other images by using similar methods, I've also created a spreadsheet containing formulas for precise FOV calculation from EXIF data, all of the above trigonometry and euclidean distance formulas, and also formulas for creation of the 2D ruler in any 3D perspective that is needed - just PM me.
