Photon-Transfer Analysis for Noise Separation and Dynamic Range Estimation

Hello. In the last lesson, a CMOS pixel converted collected charge into a voltage and sent it through a selected row to column readout circuitry. That architecture gives us several places where measurements can vary: photon arrivals fluctuate, readout adds uncertainty, and nominally identical pixels respond differently.

This lesson turns those effects into a photon-transfer analysis. You will construct a noise-versus-signal curve from raw image pairs, distinguish temporal noise from fixed spatial patterns, identify saturation, and estimate dynamic range. Allow about 40–45 minutes, including the resources.

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What should “noise versus signal” measure?

Imagine holding illumination and exposure constant while acquiring two images. The same pixel will generally report two different values. Part of that difference comes from the random number of photoelectrons collected; part comes from uncertainty in measuring their charge. Now compare different pixels within one image. Their values also differ, even under uniform illumination, but some differences persist from image to image. These are two distinct statistical questions:

• Temporal variation: How much does a given measurement fluctuate between exposures?
• Spatial nonuniformity: How differently do pixels respond under nominally identical conditions?

The familiar photon-transfer sketch puts both on one log–log graph. Watch Richard Crisp’s explanation before using the sketch, but note one important qualification: subtracting two matched frames removes a stable fixed pattern. A fixed-pattern-dominated branch therefore describes spatial variation in an undifferenced image, not the temporal noise estimated from the difference.

A log–log slope is useful because it tests a scaling law, but the graph alone does not tell you which ensemble produced its noise estimate. We will construct both temporal and spatial estimates explicitly.

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Build the temporal photon-transfer curve

Let $S$ be the mean photo-generated signal in electrons per pixel, excluding the dark signal. For independent photon arrivals, photoelectron counts are Poisson-distributed, so their variance is $S$ and their standard deviation is $\sqrt{S}$. Let $K$ be the system gain in digital numbers per electron (DN/electron), and let $r_{\mathrm{DN}}$ be the standard deviation of signal-independent temporal noise in DN, including any appreciable quantization contribution. In a linear, unsaturated operating range,

$$x\equiv \overline{y}-{\overline{y}}_{\mathrm{dark}}=KS,{\sigma }_{\mathrm{temp}}^2=r_{\mathrm{DN}}^2+K^{2}S=r_{\mathrm{DN}}^2+Kx.$$

Here $x$ is the mean dark-subtracted signal in DN. The equation makes the experiment unusually informative: plot temporal variance against mean signal on linear axes. Its intercept estimates $r_{\mathrm{DN}}^2$; its slope estimates $K$. An independently calibrated photon flux is not needed to recover $K$ this way, although measuring quantum efficiency would require calibrated illumination.

Read Sections 2.4 and 2.5 of the EMVA 1288 Linear standard for the noise model and, especially, the reason a pair of images suffices to estimate temporal variance.

To implement Eq. (18), acquire two raw images $y_i^{(0)}$ and $y_i^{(1)}$ at each illumination level; $i$ indexes pixels in a uniformly illuminated region containing $P$ pixels. At the same exposure time, acquire dark images to estimate ${\overline{y}}_{\mathrm{dark}}$. Write $d_i=y_i^{(0)}-y_i^{(1)}$, and let angle brackets denote an average over those $P$ pixels. Then compute

$$x=\frac{⟨y^{(0)}⟩+⟨y^{(1)}⟩}{2}-{\overline{y}}_{\mathrm{dark}},{\hat{\sigma }}_{\mathrm{temp}}^2=\frac{1}{2}⟨(d_i-⟨d⟩{)}^2⟩.$$

The division by two matters: if the two frames have independent temporal fluctuations of equal variance, their difference has twice the variance of either frame. Centering the difference removes a change common to all pixels between captures, such as a small shift in the illumination level. It cannot remove every form of source instability—for example, spatially varying flicker—so check that the light source and camera settings are stable.

The variance relation also explains the low- and intermediate-signal slopes in the image. Where $Kx\ll r_{\mathrm{DN}}^2$, standard deviation is nearly constant: log–log slope zero. Where $Kx\gg r_{\mathrm{DN}}^2$, but before saturation,

$${\sigma }_{\mathrm{temp}}\simeq \sqrt{Kx},$$

giving slope one-half for standard deviation versus signal. If you plot variance on the vertical axis instead, the shot-noise slope is one. Always label which quantity you plotted.

The dark measurement needs care. At short exposures, its temporal variance often approximates a readout-and-quantization floor. At longer exposures, thermally generated electrons contribute dark-current shot noise as well. A dark frame can measure and remove the mean dark signal; subtracting one noisy dark frame from a light frame does not remove its fluctuations and instead adds that frame’s noise. Characterize dark frames at matching exposure time and temperature, particularly for long integrations.

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Measure the pattern that frame differencing removes

Model the response of pixel $i$, away from saturation, as

$$y_{i,j}=b_i+(1+g_i)x_j+{ϵ}_{i,j}.$$

The index $j$ labels exposures. The offset $b_i$ varies between pixels; $g_i$ is a dimensionless fractional sensitivity deviation; and ${ϵ}_{i,j}$ represents temporally varying shot and read noise. Under the flat-field conditions of this model, the spatial offset variation is dark-signal nonuniformity (DSNU), while the slope variation is photo-response nonuniformity (PRNU). Together, persistent spatial variations are often called fixed-pattern noise—though “nonuniformity” better distinguishes them from random changes between frames.

Read the standard’s distinction before interpreting a bright flat field. Its discussion also cautions that patterns in a photograph need not originate in the sensor: illumination gradients and lens shading can produce persistent structure.

For two independent captures at the same exposure, center each image by its own spatial mean and compute their cross-frame spatial covariance:

$${\hat{s}}_{\mathrm{spatial}}^2=⟨(y_i^{(0)}-⟨y^{(0)}⟩)(y_i^{(1)}-⟨y^{(1)}⟩)⟩.$$

A persistent high-response pixel contributes similarly to both factors; independent temporal fluctuations tend to cancel in the average. If offsets and fractional gains are approximately uncorrelated across pixels, the model predicts

$$s_{\mathrm{spatial}}^2(x)\simeq s_b^2+g_{\mathrm{rms}}^2{x}^2.$$

Thus, dark frames estimate the offset-pattern term $s_b$, while sufficiently bright, unclipped flats reveal a PRNU term whose spatial standard deviation grows in proportion to signal. That is the schematic’s unit-slope region. If offsets and gains are correlated, a term proportional to $x$ also appears; do not force every measured spatial curve into the two-term approximation.

This distinction matters when interpreting an apparent improvement. A dark reference and a flat-field calibration can reduce stable offsets and gain patterns, subject to calibration accuracy. They cannot predict the particular photon arrivals in the next exposure. Conversely, increasing exposure improves shot-noise-limited relative precision but does not by itself remove a fractional PRNU pattern. For a sensor-only measurement, use a carefully uniform flat field and examine whether a gradient belongs to the source, optics, or sensor before calling it PRNU.

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Locate the usable ceiling and estimate dynamic range

At the bright end, the mean response eventually bends away from its linear relation with exposure. The limiting mechanism might be charge capacity, a sensing-node or amplifier limit, or digital clipping. Usable saturation capacity is not automatically physical full-well capacity. Once values clip, their measured variance may fall: identical maximum codes look artificially quiet, not unusually precise.

The EMVA standard defines an absolute sensitivity threshold using signal-to-noise ratio (SNR) equal to one and specifies how to choose a saturation point without substantially biasing the mean and variance. Read its definitions and practical fitting procedure; the latter keeps the gain fit below the saturated region.

In electron units, define $r_e=r_{\mathrm{DN}}/K$, including whichever signal-independent temporal terms were retained in the measured dark variance. For the simple linear model, the SNR of photo-signal $S$ is

$$\mathrm{SNR}(S)=\frac{S}{\sqrt{S+r_e^2}}.$$

Setting this SNR to one gives the positive solution

$$S_{\min }=\frac{1+\sqrt{1+4r_e^2}}{2}.$$

If $S_{\mathrm{sat}}$ is the photoelectron signal at the selected usable saturation point, the corresponding linear-model dynamic range is

$$\mathrm{DR}=\frac{S_{\mathrm{sat}}}{S_{\min }},{\mathrm{DR}}_{\mathrm{dB}}=20{\log }_{10}(\mathrm{DR}),{\mathrm{DR}}_{\mathrm{stops}}={\log }_2(\mathrm{DR}).$$

For constant quantum efficiency, taking both endpoints in incident photons rather than generated electrons gives the same ratio. When $r_e$ is large compared with one electron, $S_{\min }$ is close to $r_e$; this yields the familiar saturation-signal divided by read-noise approximation. The exact SNR-one threshold is preferable when read noise is very small. State the threshold convention when reporting a dynamic-range number: requiring SNR greater than one, or including residual spatial nonuniformity, changes the usable lower limit.

For scale, suppose a fitted slope gives $K=0.25$ DN/electron, the measured dark temporal standard deviation is $1.0$ DN, and the usable dark-subtracted saturation signal is $3000$ DN. Then $r_e=4.0$ electrons, $S_{\mathrm{sat}}=12,000$ electrons, and $S_{\min }\simeq 4.53$ electrons. The estimated dynamic range is approximately $2650:1$, or 68.5 dB (about 11.4 stops). This describes the stipulated temporal-noise, linear-response criterion; it does not claim that spatial patterns or scene-dependent processing are absent.

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Putting the analysis together

A defensible measurement uses raw, linear data rather than a tone-mapped image. Keep gain and other camera settings fixed, capture paired flat fields from dark to near clipping, and capture paired dark frames at the exposure times used. For each level, calculate the dark-subtracted mean $x$, the half-variance of the centered frame difference, and the cross-frame spatial covariance. Inspect mean response against exposure as well as noise against signal: the former helps identify nonlinearity, while the latter identifies the noise regimes.

Fit temporal variance versus mean signal only over an adequately linear, unclipped range to obtain $K$ and the temporal floor. Plot temporal standard deviation versus signal on log–log axes if you want to inspect the zero- and one-half-slope regimes. Plot spatial standard deviation separately to look for the PRNU-related unit slope. Finally, select a saturation point with negligible clipping, convert its dark-subtracted signal using $K$, and apply an explicitly stated SNR threshold to report dynamic range.

These results belong to the tested operating conditions. Changing analogue gain, exposure time, temperature, or processing can change the floor or usable ceiling; a curved variance–mean relation may invalidate a single-gain fit. Keeping those conditions and the temporal-versus-spatial distinction visible is what turns a suggestive graph into a reproducible sensor characterization.

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Conclusion

Photon transfer extracts several engineering limits from one controlled sequence of measurements. Paired-frame differences isolate temporal variation: a low-signal floor measures readout-related uncertainty, while the linear variance–mean relation reveals shot noise and system gain. Cross-frame spatial measurements expose persistent offsets and sensitivity differences. The response and clipping behavior establish a usable upper limit; combined with a stated low-signal SNR criterion, that limit gives dynamic range.

Next, the course turns to color-filter-array measurements. The raw pixel values characterized here will become samples of different spectral bands, introducing reconstruction and sampling artifacts that a noise curve alone cannot describe.