Estimating Focal Length
Can we estimate the
camera lens's focal length only by looking at the
photograph?
Image Source:
Digital Camera World
Without talking to the photographer, we don’t know the length of
his/her camera’s focal length or even the type of camera used, but we can
come up with a pretty close approximation for f based on estimates
of the geometry of the original scene and the size of the image sensor.
We first need an equation to describe the focal length so we know what
distances we’ll even need to solve for. Something that will help us is
the field of view angle. The horizontal angle encloses the largest object
that can fit in the width of the image sensor, and the vertical
angle encloses that which fits in its height.
In the above diagram, we see how the dimensions of the largest object
captured by the camera are related to the dimensions of the image sensor.
We have the image sensor one focal length away from the lens so that the
"object distance" is at infinity.
The field of view angle in the horizontal direction can be derived
through straight geometry as a function of the object width and depth from
the camera on one side of the lens and as a function of the width of the
sensor and focal length on the other.
We’re then able to equate the ratios and solve for the focal length:
By examining the photograph, we see that the diagonal of the tennis
court plus the surrounding stands stretches across the whole width of the
image. The depth from that horizontal to the approximate location of the
camera will be half the diagonal. But now comes times for the question,
how do we estimate that diagonal distance??
While we may not know the dimensions of a tennis court off the top of
our heads, we can surely estimate the dimensions of a person. The aim for
these types of estimation problems is to keep the calculations simple so
as to be able to “do them on the back of a napkin,” so we’ll keep the
numbers was to work with and say about one meter per seat. This is
definitely an over estimate, but it accounts for the extra space at the
ends of the stands.
There seem to be groups of about ten seats, three on the short side and
fourfive on the long side. Using Pythagorean theorem, we can estimate
that the center diagonal is 60 meters and therefore the depth is 30
meters. Finally, we can say that the width of the image sensor in the
camera is probably about 2 cm.
From these estimations, we were able to calculate that the focal length
was 10mm, which makes sense because lenses with short focal lengths will
give the wideangle perspective that we see in the photograph. And
according to the original
source, this was in fact the focal length of the lens used!
Further Reading
Optical Insights into Renaissance Art, Falco and
Hockney 2000
Physicist Charles M. Falco and artist David Hockney proposed (and
tested) a theory that Renaissance painters used optical elements to aid
them in their work. Based on the real life dimensions of the various
subjects and the size of the canvas, they were able to identify the focal
lengths and diameters of the mirrors or lenses that were allegedly used by
these artists.
