fft-cnr

FFT-based contrast-to-noise ratio estimation from a single frame.

Measure the contrast-to-noise ratio (CNR) of a 1-D signal profile from a single acquisition. You do not need repeat frames or a separate background region.

Here CNR means the peak signal amplitude above the baseline, divided by the RMS of the noise. The contrast is the peak's height above the baseline, the part that stands out from the background, so dividing it by the noise gives the ratio that governs whether the feature can be detected and measured. This is a single-peak contrast, distinct from the two-region (difference-of-means) CNR measured with two hand-drawn regions in MRI or CT.

fft-cnr uses the Fourier transform to separate the slowly varying signal from the rapid, point-to-point noise. The package automatically finds the frequency boundary between the two (using a model-selection score, the Akaike information criterion or AIC) and returns a CNR estimate with a 95% confidence interval.

Full documentation is at fft-cnr.readthedocs.io.

Installation

Requires Python 3.10 or later (tested on 3.10 through 3.13). If you do not have Python installed, download it from python.org and follow the installer instructions (on Windows, check "Add Python to PATH" when prompted).

Install into a virtual environment so the package and its dependencies stay isolated from your other projects:

python -m venv .venv
source .venv/bin/activate   # Windows: .venv\Scripts\activate
pip install fft-cnr

The only runtime dependencies are numpy (>=1.24) and scipy (>=1.10); pip installs them automatically.

Quick start

import numpy as np
from fft_cnr import fft_cnr

# Simulate a 1-D intensity profile (e.g., from a microscopy line scan)
# with additive detector noise
rng = np.random.default_rng(0)
x = np.arange(256, dtype=float)
signal = 10.0 * np.exp(-0.5 * ((x - 127) / 20) ** 2)
noisy = signal + rng.normal(0, 1.0, 256)

result = fft_cnr(noisy)

print(f"CNR:       {result.cnr:.1f}")
print(f"CNR 95%CI: ({result.cnr_ci95[0]:.1f}, {result.cnr_ci95[1]:.1f})")
print(f"Amplitude: {result.amplitude:.2f}")
print(f"Noise RMS: {result.noise_rms:.3f}")

CNR:       9.8
CNR 95%CI: (6.9, 12.7)
Amplitude: 9.74
Noise RMS: 0.991

The output is deterministic given the seed, so this snippet reproduces the same numbers on each run. The package exposes three names: the fft_cnr function and its result types CNREstimate and NoiseModel.

How it works

A smooth signal spread over many points concentrates its energy in a few low-frequency Fourier components. Uncorrelated, point-to-point noise spreads its energy across all frequencies. So in the power spectrum the signal sits at low frequency and the noise dominates the high-frequency end, which is where the two can be separated.

1. The input profile is mean-subtracted and tapered with a window function (a tapered cosine, or Tukey, window by default) to suppress edge artifacts in the Fourier transform.
2. The power spectrum is estimated by averaging the transforms of overlapping segments, which is smoother than a single transform (the averaged-segment, or Welch, method). More segments give the noise estimate more degrees of freedom, which tightens its confidence interval; this does not by itself improve the CNR point estimate.
3. The power spectrum typically shows high power at low frequencies, where the signal lives, that levels off to a flat noise floor at higher frequencies. The algorithm finds this transition by fitting two line segments in log-log space. It then uses the Akaike information criterion (AIC, a standard score that penalizes overfitting) to pick the boundary that best balances fit quality against model complexity.
4. The spectral bins at and below the boundary are set to zero, the remaining high-frequency bins are transformed back to real space, and the RMS of that reconstructed noise is taken. This RMS is then scaled to undo the window energy and the fraction of bins kept, so it estimates the full-band noise level.
5. Signal amplitude is estimated by one of three methods (see below), and CNR = amplitude / noise RMS.
6. A 95% confidence interval on CNR is computed by first-order error propagation (the delta method), combining the amplitude error and the chi-squared noise interval.

The method works best when the signal occupies low frequencies and the noise is broadband (approximately white). The hard minimum length is 16 points (shorter inputs raise ValueError); reliable knee detection needs more, with measurable bias below roughly 128 points (see Accuracy). Signals with strong high-frequency content that overlaps the noise band will also bias the CNR estimate.

Amplitude estimation

By default (fit_model=None or "peak"), fft_cnr removes the high-frequency noise components (by zeroing frequencies above the signal/noise boundary) and reads the peak of the resulting smoothed profile, measured above a baseline estimated from the outer quarter of the profile on each side. This works for any profile shape and assumes nothing about its functional form. If the signal extends into those outer regions, the baseline is contaminated and the amplitude is underestimated, independent of the noise. The profile therefore needs to extend well beyond the feature: for a Gaussian, a profile 2.7 full widths at half maximum (FWHM) long reads the amplitude about 8.5% low, 4.1 FWHM about 1% low, and 5.4 FWHM within 0.1%. When the profile cannot be made longer, the matched filter and the generalized Gaussian fit do not use this baseline margin and are unaffected.

Two alternatives are available:

• Matched filter (template parameter): when a noise-free template of the expected signal shape is available, a matched filter gives the most precise amplitude estimate and a calibrated standard error. This holds only when the template matches the true shape; a mismatched template biases the amplitude.

• Generalized Gaussian fit (fit_model="generalized_gaussian"): fits a 5-parameter model with a shape exponent that covers profiles from heavy-tailed to flat-topped. Useful when fitted parameters (center, width, shape) are needed in addition to CNR. The fit can degenerate, most often on noise-dominated profiles, for example into a flat-topped model as wide as the profile whose amplitude and baseline cancel. A fit whose amplitude exceeds twice the data's peak-to-peak range, or whose width is pinned at the profile length, is refused at any CNR, and the estimate falls back to the peak method; diagnostics["gaussian_fit_rejected"] names the reason, and diagnostics["gaussian_fit_params"] is then absent, as it is when the fit fails to converge. The screen tests only these two conditions. A runaway fit that meets neither, such as one centered outside the profile, is still reported, so treat a generalized-Gaussian CNR far above the peak-method value on a noisy profile with caution.

The result.amplitude_snr property reports the matched-filter signal-to-noise ratio (amplitude over its standard error). A larger value means the amplitude stands out more clearly above the noise. It is defined only on the matched-filter (template) path, where the standard error is calibrated; on the peak and generalized-Gaussian paths it is NaN, because those standard errors are different quantities that are not comparable to it. Those paths still expose amplitude_se directly for callers that want the raw ratio.

# With a known template
result = fft_cnr(noisy, template=signal)

# With a generalized Gaussian fit
result = fft_cnr(noisy, fit_model="generalized_gaussian")
print(result.diagnostics.get("gaussian_fit_params"))  # absent on fallback

Noise model detection

The CNR above treats the noise as one number, which is correct when the noise is the same at every point (for example, detector read noise). Photon-counting measurements break this assumption. Shot noise grows with the local signal, so the noise under the peak is larger than the average noise. Because cnr divides the peak by the average noise, it overestimates the true peak signal-to-noise ratio, and the size of the error depends on the profile shape.

Setting estimate_noise_model=True makes fft_cnr test for this. It fits the photon-transfer relation (variance = gain * signal + read^2) to its own residual, a single-frame surrogate for a photon-transfer curve that is conventionally measured from many exposures, and attaches a NoiseModel to the result:

result = fft_cnr(noisy, estimate_noise_model=True)
model = result.noise_model

if model.signal_dependent:
    # Noise grows with the signal; result.cnr overestimates the peak SNR.
    # The fitted noise model gives the corrected value:
    print(f"Peak SNR: {model.peak_snr(result.amplitude):.1f}")

model.signal_dependent is True when the fitted gain is statistically significant, False when the noise is consistent with a constant level, and None when the profile's signal range is too small to test (the reason appears in diagnostics["noise_model_skipped"]). To judge significance, the package runs the whole estimation pipeline on about 200 simulated data sets, so the test accounts for the estimator's own artifacts. These extra runs cost roughly half a second at N=512.

The detection is deterministic by default: the same input always produces the same result. Pass your own random generator (rng=np.random.default_rng()) to draw an independent simulation instead.

Estimating a Poisson-Gaussian noise model from a single image is established prior art, most notably the single-image method of Foi et al. (IEEE Trans. Image Process. 17, 1737 (2008)). This detector does not claim to be first at single-frame noise estimation. What it adds is to fold the photon-transfer fit into the spectral decomposition the CNR already computes, gated by a significance test calibrated through the same pipeline.

Correlated (1/f) noise is not corrected

Signal-dependence is one way the constant-noise assumption fails. Spatially correlated, 1/f-type noise is a separate one. It arises when temporal drift maps onto a spatial direction, as in galvo raster scanning or streak cameras. The drift puts noise power at low spatial frequency, where fft_cnr reads signal, so it biases cnr high.

The NoiseModel has fields for this case (spectral_exponent, white_floor, and correlated), but they are reserved and not filled in: the two floats stay NaN and the flag stays None. Single-frame quantitative correction of correlated noise is not supported, and the reason is an identifiability limit. The signal and the 1/f noise both live in the low frequencies, and a single profile gives one measurement, so there is no way to tell how much of the low-frequency power is signal and how much is drift; any split fits the data equally well. Separating them needs a second, independent view in which the signal repeats but the noise does not: multiple frames, an interleaved scan, or a reference channel. Those are also the way to correct correlated noise once it is detected.

Low-frequency baseline and structured background

A smooth baseline, a slow trend or a single low-frequency fringe, sits entirely below the signal/noise boundary, so fft_cnr reads it as signal. With a strong baseline the peak-method cnr can stay high even for a profile with no peak. Two tools address this:

• diagnostics["lowfreq_dominated"] (with the underlying lowfreq_offpeak_ratio) is true when low-frequency structure away from the peak exceeds 2.5 times the noise RMS. It marks profiles where the reported cnr may reflect baseline power rather than the peak. The flag is set on the peak and generalized-Gaussian methods and is not set on the matched-filter path, where the template defines the signal.
• roi restricts the estimate to a window around the peak, removing off-center baseline structure. Pass explicit (start, stop) bounds, or "auto" to size a window to the peak. "auto" locates the largest feature, so pass explicit bounds when an off-center baseline is larger than the peak of interest.

This guard is calibrated for smooth, low-order baselines. Broadband structured background is a different problem. The residual speckle and fringe field left after interferometric scattering microscopy (iSCAT) background subtraction is the representative case: its power spans many spatial frequencies, including the high-frequency band that sets the noise estimate. It both inflates the noise RMS, which suppresses real peaks, and adds low-frequency amplitude, which raises peakless frames, so the peak-method cnr is unreliable and lowfreq_dominated does not detect it. This is correlated noise (see above) and cannot be characterized from a single frame.

For this regime, use the matched filter: pass the known signal shape as template and read amplitude and amplitude_snr rather than cnr. The projection onto the template rejects background that does not match the template shape, so the amplitude and its SNR stay accurate where cnr does not. For iSCAT the template is the interferometric point-spread function. The scripts/validate_iscat_baseline.py simulation documents this behavior.

Return value

fft_cnr returns a CNREstimate dataclass. The first five fields are the primary outputs; cutoff_index, diagnostics, and noise_model are for advanced inspection.

Parameters

Most users will only need x (and optionally template or fit_model). The remaining parameters control internal details of the spectral estimation and rarely need adjustment.

The minimum accepted length is 16 points. Reliable results need more; see How it works for guidance on length.

Accuracy

Monte Carlo validation (200 trials per condition, scripts/validate_accuracy.py) characterizes bias, precision, and confidence-interval coverage across five signal shapes: Gaussian, heavy-tailed (generalized Gaussian, p=1.5), flat-topped (p=4), Gaussian mixture, and Lorentzian. All conditions use additive white Gaussian noise, the regime the method assumes, so the headline accuracy numbers inherit that assumption.

Bias

Median estimated-to-true CNR ratio at N=512, by amplitude method:

The peak method stays within 3% of the true CNR for most shapes from CNR=5 upward. The main exception is flat-topped profiles (up to 7% positive bias at CNR=3, 7% negative at CNR=100). The negative bias at high CNR comes from the noise estimate, not the amplitude, so it affects every amplitude method equally: the sharp shoulders of a flat-topped profile put signal power above the signal/noise boundary, where it is counted as noise. That leaked power is fixed while the noise falls, so the noise level reads progressively high as the CNR rises (for p=4 and sigma=20 samples, 2% high at CNR=50, 8% at CNR=100, and 29% at CNR=200, where the CNR reads about 0.77 of the true value and the lowfreq_dominated flag does not fire; a diagnostic for this case is tracked in issue #15). A wider feature of the same shape leaks less and holds its accuracy to higher CNR. The generalized Gaussian fit is accurate when the signal is in the model family but introduces 6--16% positive bias on shapes outside it (Gaussian mixture, Lorentzian). The matched filter tracks the true CNR to within 2% when the template matches the signal shape.

Profile length

Shorter profiles reduce accuracy slightly. At N=128 (peak method, Gaussian signal with sigma=20 samples), the estimator shows 6--8% negative bias at high CNR. This is the baseline margin described under Amplitude estimation, not the noise estimate: the profile is only 2.7 FWHM long, so the outer-quarter baseline sits on the Gaussian's tails and the amplitude reads about 8.5% low even without noise.

Precision

Trial-to-trial scatter (relative standard deviation) scales as roughly 1/sqrt(N): approximately 11% at N=128, 7% at N=256, and 5% at N=512 (peak method, CNR=5). Providing a matched template reduces scatter at low CNR.

Confidence intervals

The 95% confidence intervals contain the true value in 99 to 100% of trials across all tested conditions, so they are reliable but wider than a true 95% interval. The interval combines two terms, the uncertainty in the amplitude and the uncertainty in the noise level, and which term makes it wide depends on the amplitude method.

For the peak and generalized-Gaussian methods the noise term dominates. The degrees of freedom used for the chi-squared noise interval are set conservatively, so that term, and the combined interval, come out wider than nominal.

For the matched-filter method (a supplied template) the amplitude standard error is the exact closed-form error of the white-noise projection: the windowed data projected onto the windowed template, with error sigma divided by the template norm for a flat window. The noise term still dominates the combined interval at the conservative degrees of freedom above, so the matched-filter interval comes out comparable in width to the others while resting on the most precise amplitude estimate.

Noise model detection

Monte Carlo validation of the signal-dependence test (100 trials per condition, scripts/validate_noise_model.py) covers white noise, where the test should stay quiet, and Poisson counts, where it should fire. Conditions span Gaussian and flat-topped profiles, N in {256, 512}, and CNR 5--20; no trial was skipped for insufficient signal range.

On white noise the false-positive rate is consistent with the 5% test level and the fitted parameters recover the truth. On Poisson data the test fired in all 100 trials at these conditions, the fitted gain is within 5% of the true value, and the derived peak SNR tracks the true peak signal-to-noise ratio (ratio 0.99 to 1.03 across N and CNR).

Citation

If you use fft-cnr in published work, please cite it. The concept DOI below always resolves to the latest release; to cite the exact version you used, take that version's DOI from the Zenodo record and report the pinned version (pip install fft-cnr==0.3.3).

Joseph J. Thiebes. fft-cnr: FFT-based contrast-to-noise ratio estimation from a single frame. Zenodo. https://doi.org/10.5281/zenodo.20691435

@software{thiebes_fft_cnr,
  author    = {Thiebes, Joseph J.},
  title     = {fft-cnr: FFT-based contrast-to-noise ratio estimation from a single frame},
  version   = {0.3.3},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.20691435},
  url       = {https://doi.org/10.5281/zenodo.20691435}
}

Citation metadata is in CITATION.cff; GitHub's "Cite this repository" button generates APA and BibTeX from it.

Background

The spectral decomposition approach used here originated in a study of noise effects on diffusion coefficient estimation in chemical transport imaging (Thiebes & Grumstrup, 2024). This package has evolved from the method described in that paper. The power-spectrum estimation, knee detection, and confidence-interval machinery all differ from the original.

Support for non-Gaussian profiles was motivated by work on excess kurtosis in exciton transport (Arévalo Rodríguez et al., 2026).

Methods and background. The package implements established signal-processing and sensor-characterization methods: averaged-periodogram power-spectrum estimation (Welch, 1967), window tapering for harmonic analysis (Harris, 1978), model selection by the Akaike information criterion (Akaike, 1974), matched filtering (North, 1963; Turin, 1960), the estimation-theory framework for standard errors and the Cramér-Rao bound on estimator variance (Kay, 1993), and the photon-transfer characterization of sensor noise (Janesick, 2007; EMVA Standard 1288). Estimating a Poisson-Gaussian noise model from a single image is established separately (Foi et al., 2008); this package folds a photon-transfer fit into the spectral decomposition it already computes rather than reimplementing that method. Full references with DOIs are listed under References below; machine-readable citation metadata is in CITATION.cff.

References

• Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19(6), 716-723. doi:10.1109/TAC.1974.1100705
• Arévalo Rodríguez, E., Meléndez, M., Cuadra, J., & Prins, F. (2026). Journal of Physical Chemistry Letters, 17, 2479-2484. doi:10.1021/acs.jpclett.5c03961
• European Machine Vision Association (2021). EMVA Standard 1288: Standard for Characterization of Image Sensors and Cameras, version 4.0. emva.org
• Foi, A., Trimeche, M., Katkovnik, V., & Egiazarian, K. (2008). Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data. IEEE Transactions on Image Processing, 17(10), 1737-1754. doi:10.1109/TIP.2008.2001399
• Harris, F. J. (1978). On the use of windows for harmonic analysis with the discrete Fourier transform. Proceedings of the IEEE, 66(1), 51-83. doi:10.1109/PROC.1978.10837
• Janesick, J. R. (2007). Photon Transfer. SPIE Press. doi:10.1117/3.725073
• Kay, S. M. (1993). Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory. Prentice Hall. ISBN 978-0-13-345711-7.
• North, D. O. (1963). An analysis of the factors which determine signal/noise discrimination in pulsed-carrier systems. Proceedings of the IEEE, 51(7), 1016-1027. doi:10.1109/PROC.1963.2383
• Thiebes, J. J., & Grumstrup, E. M. (2024). Journal of Chemical Physics, 160, 124201. doi:10.1063/5.0190347
• Turin, G. L. (1960). An introduction to matched filters. IRE Transactions on Information Theory, 6(3), 311-329. doi:10.1109/TIT.1960.1057571
• Welch, P. D. (1967). The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms. IEEE Transactions on Audio and Electroacoustics, 15(2), 70-73. doi:10.1109/TAU.1967.1161901

Acknowledgments

Thanks to Ferry Prins and Enrique Arévalo Rodríguez for discussions on non-Gaussian transport profiles that informed the design of this package.

Contributing and community

Contributions, bug reports, and feature requests are welcome. See CONTRIBUTING.md for development setup, how to run the tests, and how to submit changes. Participation is governed by the Code of Conduct.

Development disclosure

The method, design, and direction of this package are the author's own. It builds on the author's prior work: the spectral contrast-to-noise estimator introduced in Thiebes and Grumstrup (2024) and developed further in the DICE repository (see Background). The estimator's structure, the choices behind each pipeline step, the validation strategy, and the critique that shaped successive revisions all reflect the author's own thinking.

Generative AI tools (Anthropic Claude) assisted primarily with code: drafting implementations from the author's specifications, writing tests, and editing documentation. The author directed this work, reviewed and corrected every contribution, and characterized the estimator by the Monte Carlo validation reported under Accuracy.

License

Released under the MIT License; see the LICENSE file for the full text.