How Tall Was al-Kibar?
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projecting a box of dimensions 47mx47mxh onto a distant plane oriented normal to the line of sight, one can generate a perspective transform that should approximate this image for the right value of h. I've posted an excel spreadsheet that does the calculation.
The projected image of the cube for the best fit height is shown, with just the corners, along with the corner locations from the image. And the variation of the sum of squares of differences, as a function of the height h:
The best-fit height is 25.5 m.
Update: a few other parameters. The uncertainty seems to be in the neighborhood of 2-3 meters. One might more accurately quote it as 53+-5% of the sidewall length. The fit concludes that the cameraholder was about 80 meters from the building. The lower right pixel is harder to locate; I should really have assigned a larger uncertainty in the fit.
posted by Andrew Foland | 7:07 AM
6 Comments:
Another way of estimating the height, if you have a good location for the photographer, is to use the point where the hill behind intersects with the left and right edges of the building.
Google Earth gives heights (in lower info bar for mouse position) and a convenient "ruler tool", so this is fairly easy - I just judged from point hill heights where the skyline would be. I had a go from a visually estimated photographer position, but that came out with implausible heights of 11.8m and 11.53m - but the consistancy of the left and right edge heights suggests this might work well from an accurate photographer position.
Thanks for this calculation.
Now: what is the height of the North Korean building that is ostensibly the model for this one.
And what about that building that supposedly has the supports for the curtain wall on it?
rwendland--thanks for the suggestion, that's a very nice cross-check for sanity. If I get to it at all (which seems unlikely), it won't be before Monday night; feel free to try yourself, here are my coordinates:
The coordinate system is:
The origin is at ground level, on the corner of the building that is nearest to the viewer.
Positive z is upwards.
Positive x is along the wall that goes out from the corner towards the right in the image.
Positive y is along the wall that goes from the corner towards the left in the image.
In this coordinate system, the fit locates the camera at
(-43.9, -58.3, 3.3) meters
The 3.3 m camera height was a very comforting check to me. The very first time, I fixed the camera height at 2 m. This time I let it float. It comes back 3.3 +- 1 m. Given, as CKR has pointed out, that we don't know a lot about the grading of the site, and that the uncertainty is nearly a meter, to me that's consistent with a reasonable height for the camera.
BTW the center axis line of the camera is aimed, according to the fit, at (-40.6, -54.0 , 3.5) in the same coordinate system.
I also increased the uncertainty allowance for the lower right point. Between floating the camera height, and increasing the lower right uncertainty, the height changed to 25.1, well within the expected uncertainty of 2-3 meters.
Do note that the meter-scale has been set here by the Isis estimate of 47mx47m. If you think it's a different size, scale distances accordingly.
CKR--I don't think the pre-curtain-wall building image contains enough info for me to get a height measure by this method.
I know, Andrew. I thought last night, after I had turned off my computer, that my comment was bossy and that, anyway, there wasn't enough information in those photos.
I remain highly skeptical that they can be matched up with the actual builing, though.
Maybe I'll do that for my computer-games time this afternoon.
There is another rough but easy check to see that your height estimate must be pretty close. Whether looking at your scatterplot or the photo, the two leftmost and two rightmost vertices come close to being the vertices of a true rectangle. This means that the corresponding interior plane of the building is nearly parallel to the picture plane (otherwise the images of at least one pair of opposite edges would converge noticeably to a vanishing point).
In this situation, regardless of the lateral offset of this interior rectangle, the perspective image of the rectangle will again be a rectangle of the same proportions. Thus we can estimate the height/width ratio of the interior rectangle by simply measuring that of its dashed image; I get about 0.38.
The height h of the building is then approximately
h = (0.38)*47*sqrt(2) = 25m.
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