Hi all!
I can't think straight anymore (it's a little too late), that's why I decided
to post here, and maybe someone knows the answer before I'll dig my way
through my uni maths books tomorrow. Just think of it as a brainteaser if you
feel compelled to answer. ;-)
Anyway, here we go:
I have a photography of an object, which I need to process to calculate
the "relative" width of an object based on the projection on the photographic
2D surface.
I decided to go with the "Zentralprojektion" model (sorry, I don't know the
english name, most probably that's the "vanishing point projection", but I'm
not sure), and arrived at the following sum to get an (increasingly better
with increasing n) upper bound on the (relative) width of the projected range
0 <= xs <= xe (both taken from the left side of the image), when the
vanishing point is projected at xv > xe from the left of the image:
d = ( xe - xs ) / n
relwidth = sum(i=0,n)[ d / ( 1 - ( xs + i * d ) / xv ) ]
Relative width meaning that for xs and xe close to 0, the relative width is
close to xe - xs, whereas moving right in the direction of xv it rapidly
increases (probably exponentially, but I didn't check yet).
Just to make a small (ascii) picture of the variables involved:
xe
+----|-----+
+ \ | +
+ \| +
+ \ +
+ |\ +
+ | \ +
+ |*| /| +
+ |*|/ | +
+ | / | +
+ |/ | +
+ / | +
+ /| | +
+--|----|--+
0 xs xv
* being the object to "measure".
What I'm now looking for is the limit with n -> infinity of that sum, not
because I couldn't live with an upper bound, but rather because I have to
implement this (for the biggest part) in integer math, which is pretty close
to impossible with the sum given above.
Anyway, if anybody can nudge me in the right direction where to look for the
limit of this specific type of sum, I'll be immensely grateful!
Thanks in advance!
--
Heiko Wundram
Product & Application Development
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