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Aperture photometry estimates brightness by summing calibrated detector pixels in an extraction region and subtracting the estimated background in that region. For $n$ equally weighted pixels, let $S$ be their total accumulated signal in electrons, $b$ the estimated background electrons per pixel over the same exposure, and $\Delta t$ the exposure duration. Then $\hat{r}=(S-nb)/\Delta t$ is a background-subtracted electron rate. Kepler applied aperture photometry to calibrated pixels with a fitted sky background. ^[ref].

This measurement trades source collection against background: enlarging the extraction region can retain more starlight but includes more sky brightness and neighbouring sources. Photon shot noise, read noise, and uncertainty in background subtraction contribute to its uncertainty. Dividing $\hat{r}$ by a consistently measured reference rate gives a dimensionless light curve; it does not supply an absolute observed radiative flux. If contaminating light remains, the reference includes it and the inferred exoplanet transit depth suffers transit dilution.

In the first toy figure, the source's reference rate is $100,000$ electrons per second, the summed sky contribution is $4,000$ electrons per second, and a known multiplicative response is $g(t)=1+0.0003t$ with $t$ in hours from mid-transit. The artificial exoplanet transit depth is $0.010$, its total duration four hours, and its flat-bottom duration $2.8$ hours; the straight ingress and egress are a toy approximation, not an exact exoplanet transit photometry model. Each synthetic sample averages a $0.5$-hour exposure and adds seeded normal distribution noise with relative standard deviation $0.0006$. Subtracting the known sky rate and dividing by the known response and reference rate recovers the dip. Real observations require estimating those corrections and carrying their uncertainty.

For source electrons $S$, sky electrons $B$, dark-current rate $d$, total exposure $t$, $M$ reads and rms read noise $r$, independent variances add to $S+B+dt+M{r}^2$. calibration can remove reproducible means but cannot remove the science exposure's random photon or dark-current fluctuations. A finite background estimate and imperfect calibration add further uncertainty.

A photodetector bias frame is an exposure of effectively zero integration time used to estimate the electronic offset and read noise. For independent identical bias reads, the variance of their difference is twice the single-read variance; subtracting an estimated mean bias removes the offset but adds the uncertainty of that estimate.

Diffraction-limited resolution is a scale for distinguishing details with a specified ideal diffraction response and a specified criterion. For two equally bright, mutually incoherent optical point sources imaged through a uniformly transmitting, unobscured circular optical pupil, the Rayleigh criterion places the centre of one Airy disk at the first zero of the other. At small angles in air this separation isHere $D$ is pupil diameter, $\lambda$ is wavelength in air, and $j_{1,1}$ is the first positive zero of the order-one Bessel function of the first kind. This is the radius to the first zero, not the diameter of the central spot. Fraunhofer diffraction gives the radial peak-normalized light intensityThe removable value at zero follows from $J_1(q)\sim q/2$. The pupil Fourier transform reduces to a radial integral by circular symmetry, which produces the Bessel function of the first kind; its first zero supplies the factor $1.22$.^[ref]

In the usual scalar circular-pupil approximation for a fluorescence microscope, the corresponding object-plane Rayleigh criterion isHere ${\lambda }_0$ is the vacuum emission wavelength, and the object-side numerical aperture already includes the medium's refractive index. Do not divide ${\lambda }_0$ by $n$ again. For small cone angles in air, $\mathrm{NA}\simeq D/(2f)$ and image-plane radius $f{\theta }_{\mathrm{R}}$ agrees with this form. The assumptions include a single wavelength, ideal optical focus, and mutually incoherent emitters. High numerical aperture can require a vector description with polarization of an electromagnetic wave and emitter orientation; confocal microscopy has its own excitation-and-detection response. The same $0.61$ formula therefore does not describe every microscope or illumination method.^[ref]

In this original model, ${\lambda }_0=550\mathrm{nm}$ and numerical apertures $0.25$ and $0.50$ give first-zero radii $1.342\mu \mathrm{m}$ and $0.671\mu \mathrm{m}$. Each point spread function is divided by its own peak to compare shape. The pupil axes are transverse wave vector components multiplied by ${\lambda }_0/(2\pi )$, so pupil radius equals numerical aperture. A wider pupil accepts a wider range of transverse wave vectors, producing a narrower image. Increasing magnification alone enlarges the same spot and does not reduce its object-plane radius.

The Rayleigh criterion is a useful visual convention, not an absolute information barrier. For equal incoherent sources with total detected radiant flux $F$ and separation vector $\mathbf{s}$, the ideal image isBelow the Rayleigh criterion this still differs from $Fp(\mathbf{x})$; for small $\mathbf{s}$, symmetry cancels the linear change and the first change is quadratic in separation. Estimating two unknown source positions is then increasingly sensitive to noise, background, relative brightness and uncertainty in the point spread function. Localizing one isolated source and recovering an arbitrary extended object are different inference tasks. Measurements that separate optical modes can also extract separation information differently from direct image light intensity measurement.^[ref]

There remains a real limit for arbitrary images in this model: the optical transfer function is zero above the transmitted spatial frequency band. Its cutoff for scalar incoherent imaging is ${\nu }_c=2\mathrm{NA}/{\lambda }_0$ in object-plane cycles per unit length. The Nyquist–Shannon sampling theorem therefore suggests object-referred detector pixel pitch $p_{\mathrm{pix}}\le {\lambda }_0/(4\mathrm{NA})$, with finer sampling providing margin. Finite detector pixels also integrate over their area, suppressing contrast; coarse sampling can cause aliasing. Finer sampling cannot alone recover spatial frequencies removed by the optics, and Fourier deconvolution amplifies noise where the optical transfer function is small. Photon shot noise, read noise, optical aberrations and defocus can all worsen practical discrimination. A useful resolution claim therefore reports the object model, criterion, wavelength, numerical aperture, sampling and signal-to-noise ratio in photon counting.

George Elwood Smith reports fabricating MOS capacitors, thermally generating the first packet, and clocking it to an output that pushed it into the substrate for electric current detection. A later integrated eight-bit device demonstrated imaging. Dark or reversed-clock control traces are not specified; feasibility alone does not calibrate charge-transfer efficiency.^[ref]

A discriminating repeat would compare identical cycles with and without initial carrier injection, then suppress or reverse the transfer sequence while retaining output switching. An electric current transient synchronized with gate switching can include capacitor feedthrough; only the difference tied to initial electric charge and the expected transfer delay tests packet transport. These are proposed controls, not controls documented in the lecture. For an optical version, a blocked exposure estimates dark current and read noise; a spatially isolated illuminated optical detector pixel exposes any deferred charge in CCD readout trail. Photon shot noise persists even with ideal transfer.

An electron-multiplying CCD adds a high-field multiplication register before the output amplifier. Impact ionization generates a large mean gain, reducing input-referred read noise but adding multiplication excess noise.

A photon of energy $E$ fully absorbed in silicon produces on average $E/w$ electron-hole pairs, with variance $FE/w$ rather than necessarily a Poisson variance. Thus ${\sigma }_E=\sqrt{FwE}$ and Gaussian FWHM is $2\sqrt{2\ln 2}{\sigma }_E$. Representative silicon values $w\simeq 3.65$ eV and $F\simeq 0.12$ give about $156$ eV FWHM at $10$ keV. read noise in split-event pixels, trapping and gain errors broaden that limit; optical pixel size alone does not specify the energy resolution.^[ref]

Let independent exposure sums $A,B$ have mean $N$ ADU at each detector pixel, with pure Poisson distribution noise and fixed gain $g$. The delta method gives $A/B\simeq 1+(A-N)/N-(B-N)/N$, so $\mathrm{Var}(A/B)\simeq 2/(gN)$ and $g\simeq 2/(N{E}^2)$. For nonuniform detector pixel means $N_i$, an unweighted spatial ratio variance is instead $(2/g)⟨1/N_i⟩$. This high-count approximation requires linear unsaturated data, bias subtraction and negligible read noise.

A charge-coupled device converts absorbed photons into stored charge in semiconductor potential wells. An exposure accumulates a pixelated image; clocked electrodes transfer rows toward a serial register and output amplifier. Quantum efficiency, pixel sampling and optical throughput connect collected photons to the expected signal. In ordinary optical imaging it primarily counts photoelectrons rather than independently measuring every photon's energy.

The CCD noise budget combines source and sky shot noise, dark-current fluctuations and read noise. For an aperture containing $n$ pixels, known background expectation $N_b$, total dark expectation $N_d$, source expectation $N_s$ and per-pixel read noise $r$,Estimating the background from a finite separate region adds its own uncertainty. Cooling reduces dark current; longer integration reduces the relative importance of one read but can encounter saturation, cosmic rays or changing conditions. Full well capacity, amplifier gain, linearity, blooming and charge-transfer efficiency limit useful detector dynamic range.

Bias frames establish the electronic offset; matched dark frames characterize time- and temperature-dependent dark signal. Flat fields calibrate pixel-to-pixel response and illumination, with appropriate attention to wavelength. Bad-pixel masks, repeated exposures and dithering help reject defects and cosmic rays. A calibrated PSF and astrometric solution are needed for reliable photometry and positions. High quantum efficiency and stable linear integration are advantages, but an uncalibrated count array is not yet an astronomical flux map.

The detector is intended for arcsecond-scale astronomical seeing, rather than the telescope's diffraction limit. A circular aperture of this area has diameter $1.13\mathrm{m}$, giving a Rayleigh criterion angle about $0.11$ arcsec at the observing wavelength. One-arcsecond pixels conventionally sample an astronomical seeing profile of roughly two arcseconds across with two samples; a one-arcsecond profile would be undersampled. The pixel scale alone does not determine an exact astronomical seeing width.

Use the supplied surface brightness as radiative flux integrated over the photometric passband: its units contain no inverse hertz. Treat $h\nu$ as the representative photon energy, and assume unit throughput and quantum efficiency. A genuinely frequency-specific brightness would require a bandwidth. With angles measured consistently in arcseconds, an annulus has area $2\pi rdr$ square arcseconds, soThis is flux per collecting area; collected energy is $AtF$. There is no extra conversion to steradians when both brightness and angular area use square arcseconds.

For Poisson photon counting, a pixel with expected count $N$ has energy $e=Nh\nu$ and standard deviation ${\sigma }_e=h\nu \sqrt{N}$. Thus the local four-sigma rule gives $N=4\sqrt{N}$, or 16 photons at the boundary. With pixel area $\Delta \Omega =1{\mathrm{arcsec}}^2$,Consequently $r_m=(1-16/32.6991)/k=10.2138$ arcsec andIntegrating to the model's outer radius $R=1/k=20$ arcsec instead givesThe pixel cut retains 51.6% of this total. It is an imposed isophotal photometry rule, not a fundamental limit: averaging annuli or fitting a justified profile can recover signal below the per-pixel cut.

In practice, sky background photons add shot noise even after subtraction of the mean sky. read noise, dark current, imperfect flat-field correction, throughput calibration and atmospheric extinction also matter. At large radius the dominant uncertainty can be a tiny error in the sky level, scattered light or the wings of the point spread function, rather than source photon statistics. Mask foreground stars and background objects, compare blank-sky regions and dithered exposures, and check profile recovery by inserting artificial extended sources. astronomical seeing and finite pixels convolve the intrinsic profile; model that response when extrapolating the total.

The luminosity convergence of an outer surface-brightness profile follows directly: $I\propto r^{-p}$ contributes $dF\propto r^{1-p}dr$, hence $F(<R)\propto R^{2-p}$ at large radius for $p<2$. For either shallow index considered, ever larger annuli dominate: extending the law to infinity gives an unphysical divergent luminosity. Such a law must eventually steepen or truncate. Over a finite range, however, substantial light can lie below a surface-brightness selection threshold and be mistaken for sky. This biases galaxy luminosities, the luminosity function and the inferred diffuse light budget; distant diffuse galaxies become especially incomplete because bolometric cosmological surface-brightness dimming scales as $(1+z{)}^{-4}$. An assumed finite-aperture total is therefore a model-dependent quantity.

Measure counts in electrons so that photon and detector noise can be combined directly. For source electrons $S$, $n$ aperture pixels, total time $t$, sky rate $b$ per pixel, dark-current rate $d$ per pixel, $M$ reads with rms read noise $r$ per pixel, and an independently estimated mean background per pixel of variance ${\sigma }_{\hat{B}}^2$, an ideal aperture-sum budget isThe last term represents the propagated calibration uncertainty if it can be treated as a random variance; systematic residuals can instead bias the measurement and need an explicit error model. For a background estimated from $n_b$ independent, equally exposed sky pixels, one approximation is ${\sigma }_{\hat{B}}^2=[(b+d)t+M{r}^2]/n_b$. Correlated resampling, detector cross-talk or a spatially structured sky invalidates the independent-pixel simplification.

Photon counting and sky. Source photon shot noise has variance $S$ and cannot be calibrated away. Longer integration or greater throughput improves its fractional error as $S^{-1/2}$. Sky photons, including moonlight, airglow and scattered light, add their own Poisson fluctuations. A dark site, appropriate observing time and filter, good baffling and a smaller or optimally weighted source aperture reduce sky contributions. Better seeing or adaptive optics can concentrate a point source into fewer noise-bearing pixels. Sky subtraction removes the mean background but not the random sky photons already recorded, and the sky estimate adds uncertainty of its own.

Dark and read noise. Thermal dark current contributes a mean and shot noise. Cooling reduces its rate; well-matched dark frames remove a reproducible mean and map hot pixels, but do not undo the science exposure's random dark-current fluctuations. Each read adds electronic noise. At fixed total integration, many short exposures add more read variance than fewer long exposures, so use exposures long enough to become sky limited when saturation, cosmic rays and temporal changes permit. Multiple non-destructive reads can help on detectors that support them. Subtracting a bias frame corrects an electronic offset, not the stochastic read noise.

Calibration and non-Poisson effects. Pixel sensitivity differences, illumination gradients, fringes, nonlinearity, imperfect charge transfer and electronic pattern noise leave calibration errors. High-count flat fields, stable operating conditions, linearity calibration, dithering and appropriate detector models help, while the finite signal-to-noise of calibration frames propagates into the result. Cosmic-ray hits, hot pixels, satellite trails and scattered-light artefacts are not ordinary stationary Gaussian noise: identify or mask them, often using several exposures. Overaggressive clipping can remove real transient or compact-source signal.

Atmospheric scintillation adds correlated intensity fluctuations, especially in short bright-star measurements; seeing changes and guiding errors move flux between pixels or apertures. Differential photometry using appropriate reference stars and stable apertures reduces common variations but does not guarantee removal of chromatic or position-dependent effects. For a constant flux-limited source in independent stationary noise the final signal-to-noise usually grows as $\sqrt{t}$ once background limited; a systematic floor or correlated sky can stop that improvement. Distinguish improvement of calibration, which removes reproducible bias, from elimination of intrinsically random photon arrivals, which is impossible.

In a CCD, absorbed optical photons create photoelectrons collected in electrostatic wells beneath gate electrodes. During readout, clock voltages move the charge packets row by row into a serial register; serial clocks move them one pixel at a time to an output node. The node converts charge to a voltage, approximately $\Delta V=Q/C$, which is amplified and digitized. pixel addresses are recovered from the row and serial-clock sequence. Resetting and sampling the output between packets establishes the baseline. Efficient transfer preserves charge; trapping or incomplete transfer produces loss and trailing. This is CCD clocked charge transfer, not independent wiring of an amplifier to every imaging pixel.^[ref]

dark current is thermally generated charge accumulated even without illumination. If its mean rate is $d$ electrons per pixel per second, a single exposure adds mean $dt$ and Poisson variance $dt$. Cooling the detector and using low-defect material suppresses this generation. A dark calibration removes the reproducible mean pattern, but cannot remove the dark electrons' random shot noise in the science exposure. read noise comes from measuring the output packet and resetting the sense node, amplifiers and electronics; write its rms as $r$ electrons per read. correlated double sampling reduces reset and correlated offsets by subtracting the baseline measurement from the signal measurement. Low-noise electronics and an appropriate slower readout reduce the remaining contribution. The CCD noise budget for $M$ separate exposures combined over total time $t$ contains independent dark and read contributions $dt+M{r}^2$, not $(dt{)}^2+M^2{r}^2$. Long exposures reduce repeated read noise, but saturation, cosmic rays and tracking requirements constrain the exposure length. On-chip binning can reduce read-noise penalties for a fixed binned area at the cost of angular sampling.

For the source and sky calculation, define the count conversion factor $K=At\Delta \nu /(h\nu )$ and the source flux per pixel $s_{\nu }=F_{\nu }/N_{\mathrm{pix}}$. The ideal source and background electron counts are $S=K{s}_{\nu }$ and $B=K{b}_{\nu }{\Omega }_{\mathrm{pix}}$. These assume unit throughput and quantum efficiency, approximately constant quantities across the band and a well-determined mean sky level from many background pixels. Source and sky photons have independent Poisson fluctuations, so the variance after subtracting that mean sky is $S+B$. Subtracting a background removes its mean, not its photon noise. ThereforeThus the required expression follows directly from adding photon-count variances. Finite sky-estimation noise, dark current and read noise would add further variance. A nonunit common efficiency replaces $A$ by the effective collecting area; a nonflat spectrum requires the corresponding frequency integral. For example, estimating the sky from one equally exposed background pixel would add an extra $B$, whereas a large sky region makes that extra term small.

With $A=\pi (1\mathrm{m}{)}^2$ and the supplied photon frequency, the sky flux per pixel is $b_{\nu }{\Omega }_{\mathrm{pix}}=1.75\times 10^{-32}\mathrm{W}{\mathrm{m}}^{-2}\mathrm{H}{\mathrm{z}}^{-1}$. Multiplying arcsecond-squared pixel area by brightness per arcsecond squared already gives a flux: no additional steradian conversion belongs in this product. Rearranging givesUsing $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$ yields

The first case is mainly source-shot-noise limited, while the last is strongly sky limited. At high source brightness $t/R^2\propto F_{\nu }^{-1}$; at low source brightness it approaches $F_{\nu }^{-2}$. Reducing source flux by ten can therefore require nearly a hundred times the exposure once sky noise dominates. These are per-pixel values for the ideal model, not guaranteed integration times including calibration, cosmic rays or read noise.

Increasing the diameter increases both source and sky photon rates as $D^2$ for fixed angular pixels. In this ideal limit the required exposure decreases as $D^{-2}$, whereas the stated telescope cost increases as $D^{2.5}$: collecting area grows only as cost to the power $0.8$. A larger ground telescope does not remove atmospheric photons or recover arbitrary wavefront information lost in a long seeing-limited exposure. Space observations avoid atmospheric airglow, weather and atmospheric turbulence, and offer stable high-resolution imaging and access to very faint structure. They still have zodiacal light, instrumental backgrounds and their own calibration limits. adaptive optics can provide excellent ground-based resolution in suitable fields, but its guide-star, wavelength and field-dependent limitations do not remove the need for space-based continuum imaging. The need is for complementary low-background, stable and high-resolution measurements; the cost exponent alone is not a proof that every imaging programme is cheaper in space.

For X rays, one fully absorbed photon produces many electron-hole pairs in silicon, rather than one optical photoelectron count. At moderate surface brightness, the probability of two photons landing in the same event island during one CCD frame is small. The charge packet or sum over neighbouring split-event pixels then supplies a pulse height for each photon, while its pixel position supplies the image coordinate. Calibrating pulse height against energy produces an event list from which a spectrum can be accumulated for any spatial region. photon pile-up at larger occupancies merges photon energies and corrupts both spectra and event grades, so fast enough frames and event-pattern screening are essential.^[ref]

A measured silicon pair-creation energy $w\simeq 3.65\mathrm{eV}$ gives $N_e=E/w\simeq 2.74\times 10^3$ electrons for a $10\mathrm{keV}$ photon. Let $F$ be the Fano factor of pair production, $n_r$ the number of independently read pixels in the event sum, and $r$ the rms read noise per pixel. ThenThe Fano-limited X-ray CCD energy resolution using the representative silicon value $F\simeq 0.12$ gives an ideal Fano-limited $\Delta E_{\mathrm{FWHM}}\simeq 0.16\mathrm{keV}$ at $10\mathrm{keV}$, or resolving power $E/\Delta E\simeq 64$. Finite read noise, charge loss and event summing broaden this; an order-of-magnitude practical estimate is a few tenths of a keV, but the particular detector's value is not determined by the given optical-pixel parameters. These $w$ and $F$ values are external detector properties, not supplied question data. Treating every pair as independent Poisson production instead would give $0.19\mathrm{keV}$ rms, or $0.45\mathrm{keV}$ FWHM; stating which width is meant avoids conflating those estimates. Radiation-induced charge-transfer inefficiency requires additional gain and spectral-response calibration.^[ref]

The fundamental angular floor is the diffraction limit of a telescope, of order $\lambda /D$. The ideal diffraction grating has $R_g=mW/d$, whereas its slit-limited resolving power has $R\theta D=mW\lambda /d$. Requiring $R\le R_g$ givesThe coefficient depends on the adopted line-resolution and aperture conventions. Below this scale, a narrower spaxel oversamples the same spatial mode rather than creating a new independently resolved element. Equivalently a spatial optical mode occupies etendue of order ${\lambda }^2$.

There is also an engineering floor before arbitrary shrinkage: the optical camera focal length, photodetector size and support clearances do not shrink, while the number of photodetectors increases as ${\theta }^{-2}$. Thus the shrinking-beam model's constant combined optical camera volume is eventually overtaken by fixed per-camera overheads. For fixed brightness, smaller spaxels receive fewer photons, so read noise and photon shot noise can constrain useful sampling before the formal optical floor. Diffraction limits independent spatial information; fixed photodetector and mechanical dimensions limit practical volume reduction.

With constant detected rates, $Q=R_{Q}T$, $B=R_{B}T$. Squaring the previous signal-to-noise ratio in photon counting givesFor $R_Q>0$, nonnegative background and target $Z>0$, choose the positive root:The other root is nonpositive. With no read noise, this reduces to $T=Z^2(R_Q+2R_B)/R_Q^2$; in a read-dominated short exposure it approaches $Z\sqrt{2n}r/R_Q$. If $R_Q=0$, no positive target stellar signal-to-noise is attainable. A zero target can be assigned zero exposure, with its limiting interpretation when all noise terms also vanish. Real quantum efficiency or multiple reads modifies the rates or read-variance term before solving the quadratic.

Use the ideal detection convention of one collected electron per detected photon; otherwise replace photon totals by detected electron totals using the quantum efficiency. Let $A$ be the summed measurement on the star patch and $B^{\prime }$ the independent background-only patch. After additive offsets are removed, their means are $Q+B$ and $B$. Independent Poisson distribution counts have variance equal to their mean, and a single read of $n$ detector pixels contributes $n{r}^2$ to each patch's variance. ThereforeThe two factors of two arise from the independently measured background and the second patch's read noise, not from doubling the stellar signal. The formula assumes equal background means, independent detector pixels and patch measurements, one read per patch, and negligible dark current or other noise. Several exposure reads or correlated detector pixels require their own variance and covariance terms.

For $k=m/2$ independent reads in each sum, read noise contributes $k{r}^2/g^2$ in squared ADU units. HenceUsing the shot-noise-only formula therefore givesThe condition for negligible contamination is $gN\gg k{r}^2$, equivalently a large number of collected electrons per individual frame compared with $r^2$. Summing more identical-level frames does not change that ratio because both $N$ and the read variance scale with $k$. Measure read noise from paired bias images, or fit mean versus noise variance across several illumination levels: the shot-noise slope determines detector gain, while the read term gives an intercept. If $r$ is known, the corrected positive solution satisfies $E^2{N}^2{g}^2-2Ng-2k{r}^2=0$. Offset subtraction removes the mean bias but not the read variance. Correlated read errors need a different covariance calculation.

Let $k=m/2$. After offset subtraction, each summed image has mean $N$ analogue-to-digital units per detector pixel. For a constant detector conversion gain $g$ and independent Poisson distribution detected counts, $\mathrm{Var}(A)=\mathrm{Var}(B)=N/g$: summing $k$ independent frames adds both their means and variances. Here $N$ is the mean of the sum, not one frame.

Write $A=N+a$, $B=N+b$. At high counts, the delta method gives $A/B\simeq 1+a/N-b/N$. Independence therefore givesThus the formula is a high-signal gain estimation from flat-field ratios, not an exact identity for a ratio of random variables. A Poisson distribution denominator can even be zero at low counts. The exposures must be linear and unsaturated, offsets removed, and read noise negligible. Being close to saturation increases the signal but does not establish linearity; clipping or charge-induced correlations biases a variance estimate. A large ensemble of suitable detector pixels supplies the measured standard deviation. In normal photodetector terminology $g$ is collected electrons per ADU, consistent with detected photons here only under the stated unit-yield convention; see the instrument gain definition.

If spatial illumination is stable between the two sums, it acts just like the sensitivity factor: the mean ratio remains one but each detector pixel has a different shot-noise variance. A whole-image standard deviation is disproportionately influenced by faint detector pixels, so substituting a global arithmetic mean signal generally underestimates detector conversion gain. If the illumination pattern differs between the sums, their noise-free ratio is not constant; its structure adds apparent variance and causes a further systematic error.

Use locally uniform signal bins or subregions, remove offsets, mask defective or saturated detector pixels, and propagate the local shot and read variances. In the stable-pattern case a useful alternative to pooling raw ratios is the normalized differenceAt high signal and negligible read noise, treating the denominator at its mean gives $\mathrm{Var}(U_i)\simeq (2N_i/g)/(2N_i)=1/g$, independent of the illumination. Thus a variance fit or $g\simeq 1/\mathrm{Var}(U)$ can use a spatially varying flat without the arithmetic/harmonic mismatch. Measured noisy denominators cause higher-order corrections, so independent or smoothed signal estimates can be preferable. Fit and subtract a genuine changing illumination pattern before noise analysis, using sufficient smoothing not to fit away the random noise. Arbitrary flat-field normalization without propagating its noise does not by itself recover the original gain.

If temporal variation is spatially uniform, summing all frames in each group still gives independent Poisson distribution totals, but their means may now be $N_A,N_B$ with ratio $c=N_A/N_B\ne 1$. At high counts and negligible read noise, direct propagation givesAfter dividing the ratio image by its measured mean $c$, the appropriate estimator isThe original formula is recovered only for $N_A=N_B=N$. Uniform brightness drift does not automatically create a spatial pattern or extra ratio scatter beyond these changed noise levels; its leading problem is unequal group normalization. Interleave the groups in time or pair exposures of comparable flux, measure their separate means, and propagate shot and read noise when rescaling. If the spatial pattern or photodetector sensitivity also evolves, the ratio develops systematic structure that must be modeled or those frames rejected. A normalization inferred from finitely many detector pixels introduces a small shared correlation, which vanishes in the large-pixel limit and can be included in a precise uncertainty analysis.

Let $N$ be the input photoelectron count and $\mu =\eta Q$ its mean, with quantum efficiency $\eta$ and mean incident photon count $Q$. Independent arrivals give a Poisson distribution, so $\mathrm{Var}N=\mathbb{E}N=\mu$. Conditional on $N$, the preceding gamma distribution gives $\mathbb{E}[X\mid N]=gN$ and $\mathrm{Var}(X\mid N)=g^{2}N$. The law of total variance therefore yieldsOne contribution is the ordinary photon shot noise; the other is multiplication excess noise. Thus, neglecting read noise and backgrounds,This is an excess-noise factor $\sqrt{2}$ in analog operation: actual photon conversion efficiency has not been halved. At low occupancy, thresholding each pixel as zero or one event can avoid most multiplication noise, but high arrival rates produce coincident events that cannot be counted separately. That is why the high-rate result concerns charge measurement rather than ideal binary photon counting.