Poisson Noise
Checks whether the noise in a micrograph behaves like the shot noise of a photon-counting detector. In a real capture the noise variance grows linearly with the local brightness, because detecting more photons means more counting noise, so a plot of per-block variance against per-block mean is a straight line with a positive slope. The indicator fits that line and flags an image whose noise does not scale with intensity, the signature of signal-independent noise added by synthesis or post-processing. It works on the pixels alone, with no model.
Technical description
M5 is a deterministic, generator-agnostic test of the noise model. Light detection is a counting process: the number of photons collected in a pixel follows Poisson statistics, for which the variance equals the mean, so after the detector applies a gain the measured variance follows the photon-transfer relation Var = a mean + b, where a is the system gain and b the read-noise variance. This is the defining property of a real optical capture and the basis of detector characterization. A fabricated or heavily processed image breaks it: uniform noise added independently of intensity gives a variance that does not change with brightness, and denoising or synthesis can remove the intensity dependence entirely. M5 tiles the grayscale image into 16 by 16 blocks, takes each block's mean and variance, fits a line of variance against mean by least squares, and reads two quantities: how well the line fits, and whether its slope is significantly positive. The image must be at least 64 by 64 pixels and contain at least three usable blocks with non-constant intensity, or the indicator returns a zero score and records that it was skipped.
How it works
For each block the mean mu and the variance sigma squared of the pixel values are computed, giving one point per block in the variance-versus-mean plane. A least-squares line
sigma^2 = a mu + b
is fit across all blocks, yielding the slope a, the intercept b, the correlation r, and the standard error of the slope. The coefficient of determination R^2 = r^2 measures how much of the variation in block variance is explained by a linear dependence on the mean.
Two signals drive the score. The base score penalizes a poor linear fit: base = max(0, (0.7 - R^2) / 0.7 times 3.0), which is zero once R^2 reaches 0.7 and rises toward 3.0 as R^2 falls to zero, because shot-noise-limited data lie close to the photon-transfer line. The slope penalty checks that the variance actually grows with intensity. The slope a is the system gain and must be positive and statistically significant, so the indicator forms the t-statistic t = a / standard_error(a). A non-positive slope scores the full penalty of 2.0; a positive but weakly significant slope, t below 3.0, scores 2.0(1 - t/3.0); a clearly significant positive slope scores nothing. The slope is read in a gain-agnostic, unit-invariant way: its magnitude is the gain and need not equal one, so a genuine micrograph whose detector gain differs from unity is not penalized, unlike a fixed slope-equals-one rule. The final score is min(5.0, base + slope_penalty). Findings report a poor linear fit and a slope that is not significantly positive. The metadata records R^2, the slope, the slope t-statistic, the intercept, and the block count.
Score thresholds
| Score | Meaning |
|---|---|
| 0 to 1 | Noise variance scales linearly with intensity with a significant positive slope, consistent with photon-counting capture. |
| 2 to 3 | The photon-transfer relationship is weak: a poor linear fit or a slope that is only marginally positive. |
| 4 to 5 | Noise variance is essentially independent of intensity. Consistent with uniform synthetic noise or heavy denoising. |
Why this matters
The noise in a real micrograph is not a nuisance to be wished away; it is a physical fingerprint of the capture. Quantitative fluorescence microscopy rests on the fact that the dominant noise is photon shot noise, whose variance equals the mean signal in photons, so that the signal-to-noise ratio and the achievable precision are set by counting statistics [3]. Detector engineers exploit the same law in reverse through the photon-transfer technique, plotting variance against signal to recover the gain and read noise of a sensor, with the slope of the variance-mean line giving the gain directly [1]. Single-image noise estimation for natural photographs uses the same Poisson-Gaussian model, Var = a signal + b, and fits it to recover the noise level function, confirming that real captures obey the affine variance law and that the slope is the gain rather than a fixed constant [2]. A synthetic or post-processed image typically lacks this intensity-dependent noise, because added noise is applied uniformly and denoising flattens the relationship. By testing whether the variance grows with the mean at a significant positive rate, M5 asks a question that only a genuine photon-limited detector answers cleanly, and it does so without assuming any particular gain.
Limitations
The screen reads noise from block variance, so it assumes that within a 16-pixel block the intensity is roughly uniform and the variance is dominated by noise; a block crossing a strong edge or fine texture has variance from structure, which inflates its point and can distort the fit. Clipping at black and white saturates the variance and bends the photon-transfer line, so an image with large saturated regions is read less reliably. Strong denoising, gamma correction, and lossy compression all reshape the variance-mean relationship and can make a genuine capture look signal-independent or the reverse. The 16-pixel block bounds the spatial resolution, and the screen reports a single global verdict rather than localizing a tampered region. The thresholds are directional rather than exact. Whether the noise level is the same across the frame, as opposed to whether it scales with brightness, is a different question answered by the noise-consistency indicator I3: I3 tests spatial uniformity of the noise floor for splices, while M5 tests the intensity dependence of the noise for the photon-counting law, so the two are complementary.
Theoretical background
M5 rests on the statistics of light detection. Photon arrivals at a detector are a Poisson process, so the number of photoelectrons collected in a pixel has a variance equal to its mean. The electronics convert electrons to digital numbers with a gain a and add a signal-independent read noise of variance b, giving a measured variance Var = a mean + b that is an affine, increasing function of the signal. This relationship is exact for an ideal photon-counting detector and holds closely for real scientific cameras across their working range, which is why it underpins both detector calibration and noise modeling. A line fit in the variance-mean plane therefore has a physical interpretation: the slope is the gain and the intercept is the read-noise floor, and a high R^2 confirms that the noise obeys counting statistics. Synthesis and post-processing have no obligation to reproduce this law, so an image whose noise variance is flat in intensity, or whose points scatter without a linear trend, fails a test that captured light passes. Reading the slope's significance rather than its exact value keeps the test tied to the physics, which fixes the sign and linearity of the relationship but not the gain.
References
- Janesick JR, Klaasen KP, Elliott T. Charge-coupled-device charge-collection efficiency and the photon-transfer technique. Optical Engineering. 1987;26(10):972-980. DOI: 10.1117/12.7974183
- Foi A, Trimeche M, Katkovnik V, Egiazarian K. Practical Poissonian-Gaussian Noise Modeling and Fitting for Single-Image Raw-Data. IEEE Transactions on Image Processing. 2008;17(10):1737-1754. DOI: 10.1109/TIP.2008.2001399
- Waters JC. Accuracy and precision in quantitative fluorescence microscopy. The Journal of Cell Biology. 2009;185(7):1135-1148. DOI: 10.1083/jcb.200903097