Simulating microlenses: kicking it up a notch

Preamble

My first stab at simulating microlenses made some strong assumptions regarding the effective shape of the photosite aperture. Reader IlliasG subsequently pointed me to an illustration depicting a more realistic photosite aperture shape --- which happens to be a concave polygon.

At first, it might seem trivial to use this photosite aperture shape in the usual importance sampling algorithm employed by mtf_generate_rectangle. It turns out to be a bit more involved than that ....

The mtf_generate_rectangle tool relied on an implementation of the Sutherland-Hodgeman polygon clipping routine to compute the area of the intersection of the photosite aperture and the target polygon (which is typically a rectangle). The Sutherland-Hodgeman algorithm is simple to implement, and reasonably efficient, but it requires the clipping polygon to be convex, so I required a new polygon clipping routine to allow concave/concave polygon intersections (astute readers may spot that I could simply exchange the clipping/clippee polygons, but I wanted concave/concave intersections anyway). After some reading, it seemed that the Greiner-Hormann algorithm had a fairly simple implementation ...

... but it did not handle the degenerate cases (vertices of clipping/clippee polygons coinciding, or a vertex falling on the edge of the other polygon). Kim's extension solves that problem, but it took me a while to implement.

Effective photosite aperture (with microlenses)

The Suede (on dpreview forums) posted a diagram of the effective aperture shape after taking the microlenses into account. I thumb-sucked an analytical form for this shape, which looks like this (my shape in cyan overlaid on The Suede's image):

The fit of my thumb-sucked approximation is not perfect, but I declare it to be good enough for government work. I decided to call this the rounded-square photosite aperture (that is the identifier used by mtf_generate_rectangle).

I am not sure how to scale this shape relative to the 100% fill-factor square. Intuitively, it seems that the shape should remain inscribed within the square photosite, or otherwise the microlens would be collecting light from the neighbouring photosites too. This type of scaling (as illustrated above) still leaves the corners of the photosite somewhat darkened, which is what we were aiming for. Incidentally, this scaling only gives me a fill-factor of ~89.5%. I guess the "100% fill-factor" claim sometimes seen in connection with microlenses applies to equivalent light-gathering ability, rather than geometric area.

Results

MTF curves for 0-degree step edge

MTF curves for 45-degree step edge

The two plots above illustrate the MTF curves of three possible photosite aperture shapes, combined with an Airy PSF (aperture=f/5.6, photosite pitch=4.73 micron, lambda=550 nm). The first plot is obtained by orienting the step edge at 0 degrees, i.e., our MTF cross-section is along the x-axis of the photosite. In the second plot, the step edge was oriented at 45 degrees relative the photosite, i.e., it represents the diagonal across the photosite.
Both plots include the MTF curves for an inscribed circle aperture, for comparison. Note that the fill-factors have not been normalized, that is, each aperture appears at its native size, which maximizes aperture area without going outside the square photosite's bounds.

Purely based on its fill factor of ~90%, we would expect the first zero of the rounded-square aperture's MTF curve to land between the 100% fill-factor square and the 78% fill factor circle, which is clearly visible in the first plot. In fact, the rounded-square aperture's MTF curve appears to be a blend of the square and circle curves, which makes sense.

The second plot above shows that the rounded-square aperture still exhibits some anisotropic behaviour, but that the effect is less pronounced than that observed with a square photosite (see this article for more details on anisotropic behaviour); this also seems logical given the shape.

In the real world (well, simulated real world, at least)

The MTF curves show some small but measurable differences between the 100% fill-factor square photosite aperture and the ~90% rounded-square photosite aperture response to a step edge. But can you see these differences in an image?

USAF1951-style chart, f/1.4, 100% fill-factor square photosite aperture

USAF1951-style chart, f/1.4, ~90% fill-factor rounded square photosite aperture

Well ... not really (click on the image to see full-size version). I even opened up the aperture to f/1.4 to accentuate the differences in the photosite apertures. Just to show you something, here is a rendering using a highly astigmatic photosite aperture (a rectangle that is 0.01 times the photosite pitch in height, but one times the pitch wide):

USAF1951-style chart, f/1.4, 2% fill-factor thin rectangular photosite aperture

Note that this is basically point-sampling in the vertical direction, but box-sampling in the horizontal direction. This shows up as rather severe aliasing (jaggies) in the vertical direction.

In the real real world

So how do these simulated MTF curves compare to actual measured MTF curves? In a previous post I described the method I used to capture the MTF of a Nikon D40 camera with a sharp lens set to an aperture of f/4. Here is a comparison of the simulated MTF curves to the empirically measured MTF curve.

Nikon D40 at f/4

At first glance it might seem that the 100% fill-factor square photosite aperture simulation is marginally closer to the measured curve, but keep in mind that these simulations were both performed with an OLPF split factor of 0.375. This value of 0.375 was determined by trial and error using the 100% fill-factor photosite simulation --- it is likely that the optimal OLPF split factor for the rounded-square photosite aperture model is different. In fact, I would expect a slightly larger value, say around 0.38 or 0.385 to perform better, purely on the difference in fill factor (100% vs ~90%) between the two simulations.

So yes, you could say I am lazy for not optimizing the OLPF split factor for the rounded-square photosite aperture model right now, but I do not feel comfortable doing any sort of quantitative comparison between the models with only one empirical sample at hand (one measured D40 MTF curve). Until such time as I have sufficient data to perform a proper optimization and evaluation of the models, I will leave it at the following statement: it certainly appears that the rounded-square model is a viable approximation of the photosite aperture of the D40.