Binning

Charge binning is a CCD operation. Charge from adjacent pixels is shifted together on the sensor and the whole block passes through the output amplifier once, so a 2×2 superpixel carries four pixels' worth of signal and one pixel's worth of read noise. That genuinely defeats the read.

Digital binning is what a CMOS sensor does, and it is a different thing wearing the same name. Every pixel is read independently, and the binning is addition afterwards — on the camera's own electronics or in your software, with identical results. Four reads happen, so four read noises add in quadrature: 2R for a 2×2 block, not R.

Both cameras in any modern comparison are CMOS. If a specification sheet, a vendor slide or a glossary tells you that binning avoids read noise, it is describing a sensor technology you are probably not using.

Sum an n×n block on a CMOS sensor. Signal goes up by n². Shot noise, being the square root of the signal, goes up by n. Read noise, added once per pixel read, also goes up by n. So, for a uniform patch of signal:

SNR = n²S / √(n²S + n²R²) = n × S/√(S + R²)

Exactly n, whether shot-noise-limited or read-noise-limited. The offset scales as n² too, which is why a binned frame's pedestal is four or sixteen times the unbinned one and must be subtracted before any arithmetic.

A punctum is not a uniform patch, and it gains less. At 40× / 1.15 NA a centered punctum puts 18% of its light in one 115 nm pixel and 55% in a 2×2 block — three times the signal against four times the background — so its peak-pixel SNR rises 1.70×, not 2×. The simulation on this page shows exactly that.

Where the CCD and CMOS versions part company is the deep read-noise corner, which is precisely where people reach for binning. At 50 electrons per pixel with 0.7 electrons of read noise, charge binning gives 2.01× and digital gives 2.00×, indistinguishable. At 2 electrons per pixel with 1.6 electrons of read noise, charge binning gives 2.63× and digital still gives 2.00×.

The n× improvement above is per pixel, and a pixel is not a fixed amount of specimen. Fix the area instead — measure over the same patch of sample however the data were binned — and the improvement vanishes entirely.

Measured on synthetic fields where the true answer is known, the detectability index over a fixed area comes out at 7.37 unbinned, 7.37 at 2×2 and 7.37 at 4×4. Identical, because summation is associative: binning before you measure and measuring before you sum are the same arithmetic performed in a different order. Binning moves photons between pixels. It does not collect any.

Both statements are true simultaneously, and which one applies is decided entirely by the scale your measurement works at. A per-pixel threshold sees the n× gain. An analysis that integrates over a structure sees nothing at all.

One boundary worth knowing: the invariance holds while the superpixel still fits inside the feature you are measuring. Bin until one superpixel is larger than that, and the smallest window you can read has outgrown the thing you wanted to measure — at which point detectability really does fall.

Binning is usually argued about as a trade of resolution for sensitivity. There is a third term, and on a real instrument it can dominate both.

Random noise falls when you average. Systematic structure does not. Illumination shading, vignetting and fixed-pattern non-uniformity are the same in every frame, so binning suppresses the random part of the noise budget and leaves the systematic part untouched — which means the systematic part's share of the budget grows with every bin step.

With a 30% shading ramp across the field, detectability falls 10% at 1×1 and 34% at 4×4. A 5% pixel-to-pixel gain variation costs 2% unbinned and 11% at 4×4. The binned image looks cleaner and separates foreground from background worse.

So the honest order of operations is: flat-field first, then decide about binning. Binning an uncorrected field makes the uncorrected part of the problem larger, and it does it while the picture is visibly improving.