Optical Lens Resolution Math

Apparently I ruffled a few feathers asserting that LP/mm is an inappropriate measurement for the evaluation of optical lenses, let me explain. As alluded to in previous posts the industry standard for measuring the resolving capabilities of wide ranging types of mediums is Line Pairs per Millimeter (LP/mm) and only makes sense for certain types of equipment like scanners. For optical lenses this standard unit of measurement simply does not make any sense from the standpoint of giving the photographer any real practical information about the resolving power of their lenses.

Manufacturers who do provide MTF graphs, which are usually scaled in LP/mm, are normally giving this information in a single graph for different positions of the lens, for instance the best resolution normally is in the center of a projected image and the resolution normally falls off towards the edges. This information is useful for the comparisons of differing lenses. For instance the Nikon 400mm lens the graph is nearly perfectly horizontal indicating very high quality lens with very little resolution change across the entire profile of the image. The actual numbers in this case are basically meaningless, the real information is the shapes of the plots.

However when trying to compare absolute resolution of one lens to another or one focal length to another this standard measure is entirely inappropriate and I will show the reasons why. The explanation involves some simple math namely geometry and some very basic trigonometry. Fortunately the conversion of LP/mm to an angle is is the same as for the conversion of focal length to the coverage angle for different focal length lenses and sensor sizes. In other words the method can be reduced to a simple formula such that the numbers can easily be plugged into a calculator.

Here is a simple diagram that shows a top down diagram of a typical subject-lens-sensor and the geometric labels.

Trig_triangle

The angle  \alpha = 2*\theta and in order to use the right triangle relationships the following formula in its simplest form is used:

\alpha = 2*arctan(\frac{\frac{s}{2}}{f})

Where \alpha is normally measured in radians, to convert this angle to degrees it needs to be multiplied by 180/pi. then to convert this angle to arc-seconds it must be multiplied by 3600.

For example a lets say a we are using a 35mm film and a 400mm lens then

5.01degrees = 2*arctan(35/2/400)*180/3.14159

is the angle that is covered with the 400mm lens by the 35mm film.

For another example lets calculate the angle of two pixels of a D3s camera sensor using a 400mm lens, the pixel width is 8.45 micro-meters. (this is the minimum angular resolution using the above configuration)

8.7 arc-seconds = 2*arctan(0.0169/2/400)*180/3.14159*3600

Now lets evaluate the measured resolution of two lenses at two different distances. Lets say the industry standard specification was 30 LP/mm for a 400mm lens using the standard Koren chart photographed at the specified distance of 400mm*52 = 20.8 m and for our other test an 800mm lens is used so the target distance would be twice that of the 400mm lens or 41.6m. Using the same target the chart would be projected at the same size onto the camera sensor. Lets say, for this example, that they both measured 30 LP/mm resolution on the image sensor. One might conclude that these two lenses possessed comparable resolution qualities. That assumption would be exceptionally wrong, lets calculate: (one line pair = 1/30mm – 0.0333mm)

for the 400mm lens     17.2 arc-seconds = 2*arctan(0.0333/2/400)*180/3.14159*3600

But for the 800mm lens (400mm lens with a 2.0x teleconverter) the angular resolving power would be twice that of the 400mm lens for the same 30 LP/mm resolution measurement:

8.59 arc-seconds = 2*arctan(0.0333/2/800)*180/3.14159*3600

So for two different lenses that provide identical resolution as measured in LP/mm the actual angular resolution is NOT THE SAME!!!!! Additionally with the longer distance used to measure the 800mm lens the dispersal of light caused by the additional atmosphere and dust between the target and lens would result in additional resolution losses. In other words:

RESOLUTION MEASURED IN LP/mm IS NOT AN APPROPRIATE MEASUREMENT FOR EVALUATING DIFFERENT OPTICAL LENS QUALITIES. (PERIOD)

The ONLY way to evaluate different lenses is to use charts photographed at IDENTICAL distances and conditions, both with lighting as well as atmospheric conditions.

Happy Independence Day.