Sample Area per Pixel

One pixel looks at a square of specimen. The side of that square is the pixel pitch divided by the total magnification, multiplied by the bin factor; the area is that side squared.

Three knobs, then: the sensor's pitch, the optics' magnification, and the bin setting in the software. At 40× a 4.6 µm pixel sees a square 115 nm across, 0.0132 µm². A 6.5 µm pixel sees 162.5 nm, 0.0264 µm² — twice the area. Bin that 4.6 µm camera 2×2 and it sees 230 nm, 0.0529 µm², twice the area of the 6.5 µm camera and four times its own.

Not one of those knobs touches the specimen. The molecules, their spacing, their brightness and the background they sit on are identical in every case. What changes is how finely the field is cut up before anyone counts anything — which is why a number reported per pixel always describes the cutting as well as the thing being cut.

The Sampling Comparator on this page renders one specimen, defined in nanometers, onto each configuration's own grid. The three comparisons below are its first three modes. A fourth, Compare two systems, answers a different question and is described with the two cameras.

Take the 4.6 µm camera and bin it 2×2. Nothing new is collected: the camera reads every pixel as before and then adds each block of four together. The simulation keeps the same noisy readout underneath both panels, which is exactly the physics — change the bin factor and the photons do not move, only the boxes drawn around them.

Per pixel, a centered punctum's SNR rises from 4.62 to 7.85. That is 1.70×, not the 2× the textbook gives, and the difference is worth understanding: the 2× assumes all four pixels carry equal signal. A punctum does not fill the superpixel. The center pixel held 18% of its light; the 2×2 block holds 55% — three times as much signal, against four times the background.

Per resolution element, the SNR is 8.88 before and after, to the second decimal. Summed over the same 297 nm square of specimen, the same photons give the same answer however they were grouped on the way. Push to 4×4, though, and a 460 nm superpixel is wider than the element itself; now two puncta inside one superpixel are counted as one, and no renormalization recovers them.

Turn on the shading and one more thing shows. Background spread falls from 54.3% to 16.0% at 4×4 — but shading's share of what remains climbs from 3% to 29%. Binning cleans the random part and leaves the uneven part, so the more you bin, the more of your background problem is the kind binning cannot fix.

Now put the 4.6 µm and 6.5 µm cameras side by side behind the same microscope, at the same magnification, on the same specimen, neither binned.

Per pixel, the 6.5 µm camera reads 6.32 against 4.62 — 37% higher. It is tempting to read that as a better camera. It is a bigger square: twice the specimen area per pixel, so a larger share of each punctum's light lands in its brightest pixel. The number describes the grid.

Per resolution element the two read 9.09 and 8.88, 2.3% apart. The 6.5 µm camera stays marginally ahead, but for a different reason: its quantum efficiency is 0.95 against 0.90. The 4.6 µm camera's sub-electron read noise, its main selling point, is worth about 1% here, because at 150 photons per punctum read noise is under 2% of the variance. It becomes decisive only when a spot dims to a few photons per pixel. A difference this size is within what a filter change or a fresh alignment would move.

One difference between real instruments is deliberately left out: how much emitted light each microscope delivers to its camera. That is set by the optics — the relays, dichroics and surfaces between objective and sensor — not by the camera, and a camera comparison should not quietly absorb it. Measure it separately and report it as a difference between the systems.

The Comparator's Compare two systems mode puts it back. Each instrument here has one transfer optic passing about 90% of the light; the line scanner also has a five-channel splitter that passes about 90% on average, and less on the channels further down its cascade. On the splitter's average channel the spinning disk leads per element by about 3%, the reverse of the camera order. Channel by channel the gap runs from level to about 5%, so a system comparison has to name the channel. Remove one transfer, as a planned scanner revision does, and the paths reach parity on average: the order returns to the one the cameras set. The per-channel split in the simulation is a model with equal loss at every dichroic, not a measurement of any one splitter.

The per-element figure is the comparison. Everything the per-pixel figure added on top of it was the ratio of the two sensors' pitches, squared.

Comparisons often arrive assembled from two demonstrations, each configured to look its best. Bin the 4.6 µm camera 2×2 and leave the 6.5 µm camera unbinned, and per pixel the 4.6 µm camera looks 1.24× better. Swap which one is binned and the 6.5 µm camera looks 2.02× better.

The ranking reversed and neither sensor changed. Per resolution element the two stay within a few percent throughout. The per-pixel figure was measuring whichever camera had been binned.

One detail the simulation flags rather than hides: binning the 6.5 µm camera 2×2 makes 325 nm superpixels, wider than the 297 nm element, so its figure there is per superpixel and two puncta inside one would merge. A configuration can win a per-pixel comparison by giving up exactly the resolution a counting experiment needs.

The defense is procedural. Ask for both instruments at the same bin setting, or ask for figures per unit specimen area. A comparison that will not provide either is comparing configurations, and should say so.

Binning one camera, setting two cameras side by side, and binning one but not the other look like three different problems. They are one problem seen three times.

In each, a quantity reported per pixel changed while the specimen did not. It changed because the pixel changed — its size set by pitch, magnification and bin together — and a pixel is not a fixed amount of anything physical. Divide the same light into bigger squares and each square holds more of it. That is arithmetic about squares, and it reads like a property of the camera.

The remedy is the same in all three: report against a fixed area of specimen. The resolution element is the natural one, because it is the smallest area the optics can distinguish and so the smallest area worth comparing. Normalized that way, the binned and unbinned camera agree to the decimal, the two cameras agree within 2.3%, and the trap disappears. The pattern is not special to puncta: for a uniform patch of specimen the same two cameras read 44% apart per pixel and under 2% apart per element.

Two things survive the normalization, and they are the ones that matter for counting. Sampling coarser than the element merges neighbors, and nothing afterward separates them. And uneven illumination is not a per-pixel artifact at all; it lives in the specimen's light, no unit removes it, and only flat-field correction does.

Everything above assumes the magnification is known. Frequently it is only assumed.

Objective magnification is a ratio between the objective and the tube lens its maker designed against, and makers do not agree: 200 mm for Nikon and Leica, 180 mm for Olympus, 165 mm for Zeiss. A 60× objective on the wrong stand can deliver 54× with the engraving unchanged, and camera adapters and magnification changers move it further.

Sample area goes as the square of magnification, so a 10% error in magnification is a 19% error in area and in every per-area figure built on it. Nothing in the image looks wrong.

Image a stage micrometer on every objective and camera pairing in use, count pixels between the rulings, and store the result in the acquisition metadata, where the analysis will read it. Repeat after any change to the light path.