This is a continuation of a series of posts about blur management for landscape photography. The series starts here.

In the last post, I looked at blur circle profiles that resulted from a mixture of diffraction, pixel aperture, and defocus, and compared those profiles to the blur circle limits that I had derived from approximations. In this post, I’d light to add precision to the comparison by plotting the *encircled energy* as a function of distance from the center of the blur circles.

Let’s look at one such graph:

This is similar to what you saw yesterday, but I’ve only plotted the right half of the profile. We don’t lose any information by doing that, since the profiles are symmetric about their centers. The modeled aperture is f/8. What you’re looking at above is a cross-section of the blurs from each source. The yellow spike near the y-axis is the defocus blur, and in this case there’s none of that since the subject is perfectly in focus. The magenta box is the blur from the pixel aperture, which is assumed to be a pillbox with a diameter of the pixel pitch times the square root of the fill factor. The red curve is that of the diffraction. I convolved all three to get the blue curve, which is the combined effect of all three modeled blurs. And finally, the vertical line represents the limits of the blur circles that I’ve been showing you, using something called EED70 for the Airy disk diameter. The blur circle limits cross the convolved solution at about a quarter of the peak value.

There’s a new curve in the above graph. It’s green, and it’s the amount of energy that is encircled as you get further away from the center of the blur circle. It crosses the limit line at about 70%.

Here’s the same setup with the camera focused at 1000 meters:

Again, the encircled energy line crosses the limit line at about 75%.

One more, with the lens focused at 8 meters, so that defocus dominates.

Crosses at about 85%.

What’s a good encircled energy to use for the DOF optimizer? It doesn’t make any difference to the optimal f-stops and focused distances that are found. I’m not uncomfortable with the way these graphs look. In the limit as the image becomes progressively more unfocused, the limits I’m using will converge to the normal CoC diameters.

As a check, I did a run with the focal point and the subject at 30 m (which provides no defocus blur), and the fill factor set to 0.00001% (which provides no sampling blur):

Now all we have is diffraction, and this curve is substantially the same as the one in this document.

I can make it even closer by increasing the range of distances, but that’s good enough.

Brandon Dube says

How do you calculate EE? I’ve found the naive method to be garbage unless you’re ridiculously oversampled, and Baliga and Cohn’s way is not all that reliable either, +/- 2% at best, and it “turns around” and heads for zero after a large enough radius.

JimK says

I’m just progressively integrating as the radius increases.

runningSum = 1.0;

result = ones(originRow+1,1); %assigns 1 to center

for r = originRow+1:nRows

runningSum = runningSum + 2 * pi * (r – originRow) * profile(r);

result(r – originRow + 1) = runningSum;

end

I’m sampling at 100 nm spacing.

Note the earlier graphs I posted were wrong. Sorry.

Brandon Dube says

Well, that’s about Q=50, which I would say is (super) ridiculously oversampled.

Are you sure the results are good? For the in-focus case, the airy radius is 5.4 microns, at which point you would expect 83.8% encircled energy from just diffraction. The blur is larger with other factors, so we should expect less EE, maybe something like 60% even. Your plot looks to be around 90% or so, which would be the case for the second zero of the airy pattern (90.9%, anyway).

I can send you the analytic form for the EE of the airy pattern if you want to do some investigation of (technique) vs truth. I suspect you have a domain-not-large-enough problem.

JimK says

I’d be happy to see the analytic form. I see what you mean about the domain size. I’ll look at that in the morning.

Thanks.

Brandon Dube says

Sure — the equation is 1 – J0^2(r) – J1^2(r), where J0 and J1 are bessel functions.

I emailed you a plot I made with prysm – it has a curve for just the airy disk, numerically derived using Baliga and Cohn’s method, and for the AD convolved with a pixel aperture equal to 5.3*.76 um in diameter, calculated the same way. You should expect wiggles in the EE, related to the zeros of the airy disk. It seems the pixel aperture serves to wash those away, which makes intuitive sense since the rings will blur into the ‘valleys’ so to speak.

JimK says

I found an error in the EE calculation. Now it’s:

runningSum = 0.0;

result = zeros(originRow+1,1);

for r = originRow:nRows

runningSum = runningSum + …

pi * ((r + 1 – originRow)^2 – (r – originRow)^2) * profile(r);

result(r – originRow + 1) = runningSum;

end

I revised the graphs and added a diffraction-only graph to the post as a check . It looks pretty much the same as the plot that I found in a set of Cal Tech lecture notes:

http://web.ipac.caltech.edu/staff/fmasci/home/astro_refs/PSFtheory.pdf

Thanks for all your help.

JimK says

Even better: trapezoids instead of rectangles for the integration:

runningSum = 0.0;

result = zeros(originRow+1,1);

for r = originRow:nRows

runningSum = runningSum + …

pi * ((r + 1 – originRow)^2 – (r – originRow)^2) * …

((profile(r) + profile(min(r+1,nRows))) / 2.0);

result(r – originRow + 1) = runningSum;

end

Jack Hogan says

Hi Jim,

Defocus disk diameter = 8NW20. Based on your 1000m plot above, at f/8 that would mean that W20 is 16/64=0.25um about 1/2 lambda. Take a look at the 1/2 lambda PSF in Figure 2 below: not quite a top hat

https://www.strollswithmydog.com/simple-model-for-sharpness-in-digital-cameras-spherical-aberration/

Brandon Dube says

8 fno squared, not 8 fno. The correct notation is W020, not W20.

Jack Hogan says

No, the diameter of the geometrical defocus disk is 8N*W20, do your homework before speaking out of your derriere.

As for W20 vs W020 see H.H.Hopkins – and whatever.