Photon transfer curve

The photon transfer curve (PTC) is the standard method for characterizing a CCD, CMOS or infrared focal-plane camera from its own images. Flat-field frames are recorded at a series of exposure levels, and the variance of the pixel values is plotted against their mean, both in digital numbers (DN). Because photon shot noise has a variance equal to the mean number of electrons, the slope of that plot gives the conversion gain $K$ in electrons per DN; with $K$ known, the read noise, full-well capacity and dark current can all be stated in electrons. Typical scientific sensors have conversion gains of 0.1–5 e⁻/DN, read noise of 1–10 e⁻ and full wells of 10⁴–10⁶ e⁻. The method needs a stable, uniform source, but no calibrated power meter.

The variance-mean relation

For a signal of $N$ electrons, the shot-noise variance is $N$ electrons squared. Converted to DN, a mean $\mu = N/K$ has shot-noise variance $N/K^2 = \mu/K$. Adding read noise $\sigma_r$ (in DN) and pixel-to-pixel gain variation, the total spatial variance of a single flat frame is

$$\sigma^2 = \frac{\mu}{K} + \sigma_r^2 + (P_N\,\mu)^2,$$

where $P_N$ is the photo-response nonuniformity, the fixed-pattern noise that scales with signal. Plotted on log-log axes as noise $\sigma$ against $\mu$, the curve has three regions: a flat floor set by read noise, a region of slope 1/2 set by shot noise, and a region of slope 1 set by fixed-pattern noise, followed by a sharp drop at saturation. With $P_N$ = 1%, fixed-pattern noise equals shot noise at $1/P_N^2$ = 10,000 electrons, so a single-frame PTC of such a sensor becomes dominated by fixed pattern well before full well.

Measurement procedure

The usual method uses pairs of frames at each exposure. The offset (bias level) is subtracted, the mean $\mu$ is taken over a uniformly illuminated region, and the variance is computed from the difference of the two frames divided by 2:

$$\sigma^2_{\text{temporal}} = \tfrac{1}{2}\,\mathrm{Var}(A - B).$$

The difference removes the fixed pattern, which is identical in both frames, and leaves only temporal noise. The conversion gain at each level is then

$$K = \frac{\mu}{\sigma^2_{\text{temporal}} - \sigma_r^2}.$$

As an example, a pair of bias frames whose difference has a standard deviation of 3.0 DN gives a read noise of 3.0/√2 = 2.12 DN. At a mean of 10,000 DN above offset, a half-difference variance of 20,004.5 DN² gives $K$ = 10,000/(20,004.5 − 4.5) = 0.50 e⁻/DN. The signal is then 5000 electrons with a shot noise of 70.7 e⁻, and the read noise is 1.06 e⁻. Fitting a line through many exposure levels gives a more reliable $K$ than any single point.

Exposure is normally varied by changing the integration time at fixed illumination, which keeps the spectrum of the source constant. Dark frames at each exposure time remove the contribution of dark current to the mean.

Full well and linearity

As the brightest pixels in the region begin to saturate, they all report the same value, and the variance stops rising and then collapses. The mean at the variance peak, multiplied by $K$, is the full-well capacity; for a 30,000-electron pixel at 0.5 e⁻/DN it appears near 60,000 DN. The same data, plotted as mean against exposure time, give the linearity curve, and a departure from proportionality before the variance peak marks the linear full well. Full well divided by read noise gives the dynamic range.

Pitfalls

An unsubtracted offset shifts every mean and biases $K$ upward at low signal. Nonlinearity in the conversion, common near saturation and in some CMOS pixels at low signal, makes the slope vary with level, so the fit range should be stated. Electron-multiplying CCDs and other devices with avalanche gain add an excess noise factor: with $F$ = 2 in a high-gain EMCCD, the variance doubles, and a naive fit returns a conversion gain half the true value. In CMOS sensors each pixel has its own amplifier, so the PTC of a region is an average over a distribution, and quantization noise of 1/√12 DN matters when $K$ is large. Illumination flicker between the two frames of a pair adds variance that the difference method does not remove.

Common questions

Why does the photon transfer curve work without a calibrated light source?

The shot-noise variance in electrons equals the mean in electrons, a property of Poisson statistics that holds whatever the absolute flux. The ratio of variance to mean in DN therefore gives the scale factor between DN and electrons directly.

Can the PTC measure quantum efficiency?

Not by itself. It converts DN to electrons; obtaining quantum efficiency also requires the number of incident photons per pixel, measured with a calibrated photodiode at the sensor plane.

Which frames should be used for the read-noise floor?

Bias frames of the shortest available exposure, taken in the dark, processed with the same pair-difference method, so that the floor and the shot-noise region share one procedure.

References: J. R. Janesick, Photon Transfer: DN → λ (SPIE Press, 2007); J. R. Janesick, Scientific Charge-Coupled Devices (SPIE Press, 2001); G. C. Holst and T. S. Lomheim, CMOS/CCD Sensors and Camera Systems, 2nd ed. (SPIE Press, 2011).