Set the question before the measurement
A Siemens star can support a visual focus check, directional comparison or quantitative frequency-response analysis. These outputs use different reading rules. Record which result is required before choosing the pattern, illumination and software.
Convert radius into a geometric frequency
For N complete bright/dark pairs around one turn, the tangential period along a circle of radius r is 2πr/N and its frequency is N/(2πr). Use millimetres for r to obtain lp/mm. Count pairs, not the total number of bright and dark sectors.
A chord-based spacing, 2r sin(π/N), measures a straight chord between equivalent boundaries. It is a different geometric convention from distance along the circle, rather than a correction that makes the circular-period definition inaccurate. Keep the chosen convention consistent when comparing specifications.
Keep object-space and image-space units separate
A radius measured on the physical target gives object-space frequency. A radius measured at the image plane gives image-space frequency. For magnification magnitude |M|, f_image = f_object / |M|. If the radius is measured in pixels, the same formula gives cycles per pixel until the pixel scale is applied.
The Nyquist boundary for one-dimensional sampling is two samples per period, not one. Results close to that boundary are sensitive to phase, orientation and processing. A radial star crosses different pixel-grid directions, so reliable testing needs a margin rather than treating Nyquist as a guaranteed readable frequency.
Make a repeatable visual focus check
Mount the target securely, align its plane to the test geometry and illuminate it uniformly. Adjust exposure to avoid clipping. Sweep through focus in small increments and compare continuous radial detail at matching positions. A smaller unresolved region is useful only when the newly visible detail is genuine and not reversed or aliased.
- Keep the aperture, wavelength, magnification and image processing consistent.
- Compare centre, edge and corner field positions rather than inferring the whole field from one central star.
- For zoom-lens back-focus adjustment, follow the lens-specific flange-focus procedure and specified target distance.
Recognise false detail before reporting a limit
Optical contrast reversal under defocus and digital sampling aliasing can both produce apparent detail beyond a reliable resolution boundary. They can occur together. Changes in focus, sampling or orientation help investigate the cause, but no single visual change proves one mechanism exclusively.
| Observation | What to investigate |
|---|---|
| One direction loses detail earlier | Target tilt, focus, directional motion and optical aberrations. Shape alone does not identify a unique cause. |
| Petals, false spokes or reversed contrast | Sampling aliasing, defocus-related contrast reversal and processing. Reappearing patterns do not prove that fine detail is resolved. |
| Results change strongly with exposure | Highlight clipping, leakage through the dark field and nonlinear tone processing. |
| A sharp centre but soft image edges | Repeat the measurement at edge and corner field positions, then compare focus and alignment. |
Match the quantitative method to the waveform
A binary chrome star contains higher harmonics as well as a fundamental. A sinusoidal star is designed around smooth intensity modulation. For quantitative analysis, establish the star centre, sample the signal along selected radii and angular regions, and map radius to frequency using the chosen geometry. Target modulation, illumination and image linearity must be accounted for.
For a sinusoidal intensity signal with mean I₀ and peak modulation amplitude A, modulation is A/I₀. An MTF estimate compares output modulation with the corresponding input modulation under a suitable linear-system measurement. A measured image contrast by itself is not that ratio.
For an ideal 50% duty-cycle square wave, the fundamental peak amplitude is 4/π times the square-wave half-amplitude. Fourier coefficient conventions and normalisation must be explicit. Do not apply a universal 1.27 multiplier to a measured star contrast and call the result MTF.
Document whether analysis describes the full camera pipeline or a minimally processed capture. Sharpening, denoising, clipping and resampling can change the response. An ISO-based test also needs the correct chart design and acquisition procedure; the name Siemens star alone does not establish conformity.
Treat distortion and processing as part of the method
Distortion changes the relationship between physical radius and image radius. Correct the coordinate mapping or use a method that accounts for local magnification. Resampling an image to remove distortion also changes its frequency response, so record the correction pipeline and avoid treating the corrected result as an untouched optical measurement.
Report what was actually measured
Record target waveform, pair count, usable radius, illumination, field position, optical configuration, exposure and processing. State the visual reading criterion or quantitative analysis method and the coordinate plane used for frequency. A slanted-edge comparison can help investigate disagreement, but neither target automatically establishes traceability or an absolute accuracy claim.