How Sharp is That Lens?

As I read reviews of photographic equipment, I find the occasionally come across a review in which the reviewer notes that the reviewed lens was a dud and didn't focus properly. How could I tell if I had a dud lens without a way to compare it to others?  Some kind of quantitative method for lens comparison is needed. There are all kinds of sharpness test patterns, but none of them seemed to be easy-to-use. Then I saw an example of a pattern of black and white bars that got progressively closer together. You can tell how sharp the lens is by looking at where the bars mush together to form gray. I looked at this and realized that this is the spacial equivalent of the swept-sine frequency response test used for audio equipment. What if I approached this like a signal processing problem? A basic test of signal processing equipment is the step-response. On a first-order system you can use this to determine its time constant, which is a fundamental metric. This could be the metric I was looking for? I used a laser printer to make a sheet of paper that was half black and white. I took three pictures with an old point and shoot camera: wide, tele, and purposely blurred. The photos were taken at maximum resolution.

WIDE

 TELE

BLUR

Sampling a line pixels from left to right in the middle each image should result in a step response from black to white. To extract the pixel values I used a Ruby script with RMagick:

require 'RMagick'
include Magick
puts ARGV[0]

image = ImageList.new(ARGV[0])
midPointY = image.rows / 2

(0..image.columns).each do |x|
      pixel = image.pixel_color(x,midPointY )
    print x
    print ", "
    print (pixel.red + pixel.green + pixel.blue) / 3
    print "\n"
end

RMagick is so cool! I sent the output of the file to a spreadsheet to compare the three images. 

Look at this! The wide angle has the fastest rise time. You can even see a little 2nd order ringing that's probably due to the compression algorythm. Interestingly there's pre-ringing too because spacial systems are non-causal.

Note that the wide angle is not sharper because it had to be closer to the paper to fill the frame. I've noticed that this 12 year-old camera is just not as sharp on the telephoto setting as it used to be. This graph quantified my observation. How much of a difference is there between wide and tele? We'll zoom in on the data.

In a first-order system, the time constant is measured at 63% of the final value of the step function. There are a couple of sample points in that area, so I'll use those as an approximation rather than interpolating an exact point. Now we can say that the wide-angle setting is more than twice as sharp as the telephoto setting.

There are still some questions I'd like to answer. How can I compare cameras with different pixel resolution? Can I harmonize my results somehow with the lensmakers' specs? Could I perform an average of successive images to get a better accuracy? How could I compare the center of the lens to the edges? Would deconvolution or FFT be useful analysis tools? These are all questions for a later blog post!