Thursday, May 25, 2006

An interesting difference between the human eye and a still camera

Warning: Slipshod handwaving arguments ahead. Rigorous people proceed with care.


There are many comparisons that draw attention to the similarities between the human eye and a still camera. Today, I will point out one interesting but rarely, if ever, mentioned difference.


In the human eye, the signal strength can be assumed to be approximately proportional to the quantity of light falling on the retinal cells.

Simplistically,



Where the signal strength coming out of the eye is a function of light intensity falling on the retina, aperture of the pupil, retina sensitivity, efficiency and other eye-related parameters.


In a still camera, there is one more parameter that dramatically changes the form of the equation- the exposure time.



Here, the fact that light intensity that falls on the sensor can vary with time is highlighted, hence it is not a constant but denoted by a time-dependent function I(t). Aperture, sensor sensitivity (ISO), light transmission efficiency and other camera-related parameters are assumed to be fixed for each exposure.

And, the signal strength is the time integral of the function. Which sort of makes sense, since doubling the exposure time in the presence of a constant light source results in photographs twice as bright.

It is exactly this time integral that allows double exposures to be made, where the film is exposed to two different scenes to create an overlapping image.

If the exposures were made 24 hours apart, then the integral can be taken to be starting from 0 and ending 24 hours later, with I(t) being zero for a large proportion of the time (since the film was not exposed to light between exposures).


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4 Comments:

Blogger 小李飞刀 said...

I was taught that the eye has a log response to intensity.

I guess its so that brightness wont vary so much with distance.

10:44 pm, May 25, 2006  
Blogger Lao Chen said...

Ah, ic. At high intensities, one would expect the response to be nearly saturated, any brighter and the eye would not be able to tell the difference.


I doubt it's got anything to do with distance though. Afterall, the surface area in view per unit solid angle increases as a square of distance, but intensity of a point source decays to the inverse square of distance. So number of visible point sourcess increases as a square of distance from the eye, but each point's brightness decays as an inverse square of the distance. The net result is no change.

So the total quantity of light that reaches the eyes is the same for a forest far away (that spans 600 arc seconds) and a nearby tree (that also spans 600 arc seconds). This, assuming that they are bathed in the same ambient light.


Just to be safe, some definitions:

angle : defined as S/r, where S is the perimeter of the circle that is bounded by the angle, and r is the radius of the circle.

solid angle: defined as A/r, where A is the area of the sphere that is bounded by the solid angle, and r is the radius of the sphere.

A mere extension from 2 to 3 dimensions.

1:41 am, May 26, 2006  
Blogger 小李飞刀 said...

Aha.. yeah, I see your point... a clear example of muddled thinking on my part.

Actually there is a famous cosmology paradox which uses the argument you have illustrated.

Something to do with if the universe is infinite in extent, our skies should be bright as day, as we would receive infinite light energy.

Cant be bothered to type it out.

3:52 am, May 26, 2006  
Blogger Lao Chen said...

Ah yes. That ancient argument that if the universe is infinitely old and infinitely large, then every line of sight away from the earth would terminate on the surface of a star, and the entire sky would not only be bright as daylight, but the whole celestial sphere would be as bright as the sun. Everywhere.

There was really no need to type this out... we already know the result.

4:15 am, May 26, 2006  

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