Fixed-pattern noise

Fixed-pattern noise (FPN) is the spatial nonuniformity of an image sensor: under identical illumination, different pixels report different values, and the pattern repeats in every frame. It has two parts. Dark signal non-uniformity (DSNU) is the spread in offset, the signal each pixel gives with no light, arising from differences in dark current and in amplifier and readout offsets. Photo-response non-uniformity (PRNU) is the spread in gain, the fraction by which each pixel's response to light differs from the mean, caused by variations in pixel area, quantum efficiency, microlenses and conversion gain. PRNU of a few tenths of a percent to about 1 % rms is common in silicon sensors; InGaAs arrays for the SWIR have larger spreads and more defective pixels. Because the pattern is fixed, it can be measured once and subtracted or divided out, which separates it from the temporal noise sources, read noise and shot noise, that change from frame to frame.

How it adds to the noise budget

Expressed in electrons, for a signal $S$ the variance seen across a uniformly illuminated frame is

$$\sigma^2 = \sigma_r^2 + \sigma_{\mathrm{DSNU}}^2 + S + (P\,S)^2,$$

where $\sigma_r$ is the read noise, $S$ is the shot-noise variance and $P$ the fractional PRNU. The PRNU term grows linearly with signal while shot noise grows as $\sqrt{S}$, so they are equal at $S = 1/P^2$. For $P$ = 1 % that is 10,000 e⁻, where each contributes 100 e⁻ and the total is 141 e⁻. At 40,000 e⁻, within the full-well capacity of many scientific pixels, shot noise is 200 e⁻ and PRNU 400 e⁻; the signal-to-noise ratio there is 89, and it cannot exceed $1/P$ = 100 however much light is collected. A sensor with $P$ = 0.5 % reaches the crossover at 40,000 e⁻.

DSNU matters at the other end, in dark or faint images and long exposures. Its dark-current part scales with integration time and roughly doubles every several kelvin, and hot pixels with dark current far above the median make up its tail.

Measurement

FPN is measured by separating spatial from temporal variation. Averaging $N$ frames taken under the same conditions reduces temporal noise by $\sqrt{N}$ and leaves the fixed pattern untouched; the spatial standard deviation of the averaged frame, after a small correction for the remaining temporal noise, gives FPN. Averaged dark frames at the working exposure and temperature give DSNU. Averaged frames under uniform illumination, for example from an integrating sphere, with the dark average subtracted and divided by the mean, give PRNU. The difference of two individual frames, used to measure read noise, cancels the fixed pattern; this is the standard way to keep the two apart. Column-to-column and row-to-row components are usually evaluated separately, since stripes are visible in an image at amplitudes well below the rms per-pixel noise.

Correction

The standard correction uses a master dark frame $D$ and a master flat frame $F$:

$$I_{\mathrm{corr}} = \frac{I - D}{F - D}\,\langle F - D\rangle.$$

The masters must be averages of many frames, or their own temporal noise is written into every corrected image. A single flat at 20,000 e⁻ carries 0.71 % shot noise per pixel, comparable to the PRNU it is meant to remove; an average of 16 such frames brings this to 0.18 %, and 64 frames to 0.09 %. The dark master must match the exposure time and temperature of the data (and the flat needs its own dark at the flat's exposure), and the flat must match the optical path, since dust shadows and vignetting are corrected along with pixel gain. Infrared cameras apply the same correction internally as a non-uniformity correction table, often a two-point (offset and gain) calibration valid only near the temperature and exposure where it was taken.

Where it matters

In CMOS sensors each pixel and column has its own amplifier, so column FPN and pixel offsets are larger than in CCDs, where all pixels share one or a few output amplifiers; on-chip correlated double sampling removes much of the offset part. Beam profiling, flat-field microscopy, photometry and any measurement comparing intensities across the frame depend on PRNU correction, and the pattern should be removed before computing beam widths or uniformity.

Common questions

Is fixed-pattern noise really noise?

It is a deterministic error, the same in every frame, and is called noise because it appears as random scatter across pixels in a single image. Once measured it can be corrected, unlike read noise and shot noise.

Why does my image show stripes after dark subtraction?

The dark frame was probably taken at a different exposure or temperature, or with too few frames averaged; column offsets in CMOS sensors can also drift between the dark frames and the data.

Does averaging frames reduce fixed-pattern noise?

No. Averaging reduces temporal noise only; the fixed pattern is identical in each frame and survives the average.

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