Sub-Pixel Hot-Spot Detection in IR Imaging

The standard objection — “you need at least 2-3 pixels to detect a feature (Nyquist, characterization, robustness against hot/dead pixels)” — is sound for visible-light imaging. In IR thermography, the physics is different enough that a sub-pixel hot spot can dominate a pixel’s signal by orders of magnitude over the sensor noise floor.

This note works through why, with a worked example and the physical limits where the claim breaks.

The mixing model

A pixel does not measure “a temperature.” It integrates in-band spectral radiance over its instantaneous field of view (IFOV). If a fraction

\[f = \frac{A_{\text{spot}}}{A_{\text{pixel}}}\]

of that IFOV is at temperature \(T_{\text{hot}}\) and the remainder \((1-f)\) is at \(T_{\text{bg}}\), the radiance reaching the detector for that pixel is the area-weighted average:

\[L_{\text{pixel}} = f\, B(T_{\text{hot}}) \;+\; (1-f)\, B(T_{\text{bg}}).\]

For an emissivity-corrected blackbody, the in-band radiance \(B(T)\) rises very steeply with temperature:

  • LWIR (8-14 µm): \(B(T) \propto T^{4}\) to a good approximation (Stefan-Boltzmann).

  • MWIR (3-5 µm): \(B(T) \propto T^{8}\!\dots T^{12}\), since the band sits up the Wien side of the Planck curve. Hot-spot contrast is even sharper.

A 100 × 100 µm hot spot inside a 1 × 1 mm pixel IFOV

A 100 × 100 µm hot spot occupies just 1 % of a 1 × 1 mm pixel IFOV. The pixel’s signal is the area-weighted average of the hot region and the cool background.

A worked numerical example

Quantity

Value

Pixel IFOV \(A_{\text{pixel}}\)

1 mm × 1 mm = 1 mm²

Hot spot \(A_{\text{spot}}\)

100 µm × 100 µm = 0.01 mm²

Area fraction \(f\)

1 %

Background \(T_{\text{bg}}\)

25 °C = 298 K

Hot spot \(T_{\text{hot}}\)

200 °C = 473 K

Band

LWIR (\(B \propto T^{4}\))

Ratio of emissive powers:

\[\frac{B(T_{\text{hot}})}{B(T_{\text{bg}})} = \left(\frac{473}{298}\right)^{4} \approx 6.34.\]

Pixel-averaged radiance:

\[\frac{L_{\text{pixel}}}{B(T_{\text{bg}})} = 0.01 \cdot 6.34 + 0.99 = 1.053.\]

Inverting through \(T^{4}\) to recover an apparent pixel temperature:

\[T_{\text{app}} = T_{\text{bg}} \cdot (1.053)^{1/4} \approx 301.9~\text{K}, \qquad \Delta T_{\text{app}} \approx 3.9~\text{K}.\]

Compare to typical sensor noise floors (NETD, noise-equivalent temperature difference):

Detector

NETD

Cooled MWIR / LWIR (HgCdTe, InSb)

~ 20 mK

Uncooled microbolometer LWIR

~ 30-50 mK

Note

\(\mathrm{SNR} \approx \dfrac{3.9~\text{K}}{0.030~\text{K}} \approx \mathbf{130}.\) A 1 % sub-pixel hot spot sits two orders of magnitude above the noise floor.

In MWIR, with its higher effective exponent, the same scenario pushes the apparent \(\Delta T\) to roughly 8-10 K — easier still.

Why visible-light imaging cannot pull this trick

Reflectance imaging mixes the same way,

\[I_{\text{pixel}} = f\,\rho_{\text{spot}}\,E + (1-f)\,\rho_{\text{bg}}\,E,\]

but the gain is linear in the albedo difference \(\rho_{\text{spot}}-\rho_{\text{bg}}\), not in any power of it. A 1 % sub-pixel speck with 50 % reflectance contrast lifts the pixel by 0.5 % — right at the camera noise floor for an 8-bit sensor (~0.3-1 %).

Planck radiance B(T) proportional to T^4

LWIR radiance scales as \(T^{4}\). The 175 K rise from background to defect more than sextuples the emitted power, so even a 1 % area share contributes a measurable pixel-level signal.

The PSF assist

Even if the defect is physically sub-pixel, the optical point-spread function (Airy disk plus lens aberrations) spreads its photons over typically 2-4 pixels with FWHM on the same order as the pixel pitch. So the sensor response is not a single hot pixel but a small Gaussian-ish blob — easy to distinguish from a stuck-pixel artefact, which is a delta function and is also present in the dark-reference frame after non-uniformity correction (NUC).

Sub-pixel point source spreading into a multi-pixel response blob

Optical blur turns even a sub-pixel point source into a small Gaussian-shaped response across several pixels. Stuck or hot pixels, in contrast, are delta-shaped and stable across frames — they are removed by NUC.

What can break the claim

Two assumptions hide in the math; raise them when a vendor over-promises 1-pixel detection.

  1. Emissivity uniformity. The mixing model assumed \(\varepsilon_{\text{spot}} \approx \varepsilon_{\text{bg}}\). A polished metal flake on oxidized steel can have \(\varepsilon \sim 0.1\) versus \(\varepsilon \sim 0.8\) — the contrast can flip sign or vanish entirely at some viewing angles. The radiance at a hot, low-\(\varepsilon\) surface mostly reflects the room behind the camera.

  2. Atmospheric and window transmission. CO₂, H₂O, and ozone bands attenuate selectively across LWIR and especially MWIR. A long stand-off distance or a viewport (germanium, ZnSe) cuts effective \(\Delta T\). Always quote the band, path length, and window.

Bottom line: detection vs. characterization

Both views — “you need 2-3 pixels” and “1 pixel is enough” — are correct. They are answering different questions, and the disagreement evaporates once you separate the two.

“2-3 pixels” — Characterization view

“1 pixel” — Detection view

Question it answers: “What does the defect look like?”

Question it answers: “Is there a defect here, yes or no?”

Nyquist sampling. To pin down the spatial frequency of a feature without aliasing, you need at least 2 samples per cycle — so the feature must span \(\ge 2\) pixels.

Optics spread the light. A real lens’s PSF blurs even a sub-pixel hot spot over 2-4 pixels (FWHM \(\ge\) 1-2 px). So the defect’s footprint can be 1 pixel even when the sensor’s response covers several.

Morphology needs room. To measure size, shape, or orientation, you need enough pixels to fit a meaningful pattern — typically a \(3 \times 3\) neighbourhood as a minimum.

IR contrast is exponential, not linear. Thermal radiance scales as \(T^{4}\) (LWIR) or steeper (MWIR), so a tiny hot region with a large \(\Delta T\) dominates the pixel’s signal far above NETD.

Single-pixel anomalies are ambiguous. Without context, one bright pixel could be a real defect, a hot/stuck pixel, or shot noise. You can’t tell from just the pixel itself.

Sensor artefacts are removed first. Non-uniformity correction (NUC) and reference-frame subtraction strip out hot/dead pixels, so a remaining 1-pixel anomaly is a real alarm, not a sensor quirk.

Important

Verdict. If the speaker is making a detection claim — “we alarm on hot spots smaller than one pixel” — and it is backed by NUC plus PSF reasoning on a near-blackbody surface, 1 pixel is honest.

If they are claiming measurement at the 1-pixel scale — defect size, shape, orientation, classification — the “2-3 pixels theoretically” objection is right, and Nyquist and morphology are the reasons.

A point-source siren you can hear is not the same as a face you can recognise.