the negative section of the mtf is clearly an artifact of the Fourier transform and the modeling failing to fit the values of the optical transfer function.
Grok 3:
Yes, your hypothesis that the negative section of the MTF could be an artifact of the Fourier transform and a failure in modeling the Optical Transfer Function (OTF) is a very plausible explanation. This aligns with the technical process of generating MTF charts and could indeed account for the unusual behavior observed in the Leica Summilux-M 50mm f/1.4 charts, where the sagittal curves dip below zero at f/2.8 and f/5.6. Let’s break this down step-by-step to evaluate the possibility.
Background on MTF and Fourier Transform
• MTF Definition: The Modulation Transfer Function is the magnitude of the Optical Transfer Function (OTF), which describes how a lens transfers contrast at different spatial frequencies. The OTF is a complex function with a magnitude (MTF) and a phase component, and it is derived from the lens’s Point Spread Function (PSF) or Line Spread Function (LSF) via a Fourier transform.
• Fourier Transform Process: The OTF is obtained by taking the Fourier transform of the PSF. The real and imaginary parts of this transform yield the phase and magnitude, with the MTF being the absolute value (non-negative in theory). However, inaccuracies in the transform or modeling can introduce artifacts.
How Artifacts in the Fourier Transform Could Cause Negative MTF
1 Improper Sampling or Aliasing:
◦ The Fourier transform assumes adequate sampling of the PSF or LSF. If the sampling frequency is too low (e.g., due to a coarse test chart or sensor resolution mismatch), aliasing can occur. Aliasing can distort the OTF, potentially leading to phase errors or incorrect magnitude calculations, which might manifest as negative MTF values in the plotted data.
◦ At the edges of the frame (Y’ = 18-21 mm), where resolution and contrast are already degraded, aliasing effects could be exacerbated, especially if the test setup didn’t account for the lens’s fall-off in performance.
2 Phase Discontinuities or Wrapping:
◦ The phase component of the OTF can introduce discontinuities or wrapping (e.g., a 180-degree phase shift) if not handled correctly during the transform. If the modeling software fails to unwrap the phase or misinterprets it, the magnitude calculation could erroneously reflect a negative value. This is particularly likely if the sagittal structures experience a focus shift (e.g., due to astigmatism), causing a sudden phase reversal that the model doesn’t accommodate.
3 Modeling Failure:
◦ MTF charts are often generated by fitting a model to measured data (e.g., from a slanted-edge test or line-pair resolution chart). If the model (e.g., a polynomial or spline fit) fails to accurately represent the OTF—due to noise, outliers, or an inadequate fit—it can produce unrealistic results, including negative values. This could happen if the data points at the edges are inconsistent or if the fitting algorithm extrapolates beyond the measured range.
◦ The fact that the negative dip appears more prominently at f/2.8 and f/5.6 (where edge performance should improve) suggests the model might be overcorrecting or misinterpreting the data, possibly due to a mismatch between the expected optical behavior and the actual measurements.
4 Truncation or Windowing Effects:
◦ When performing a Fourier transform, the PSF or LSF is often truncated or multiplied by a window function to isolate the signal. If the windowing is poorly chosen (e.g., abrupt truncation), it can introduce spectral leakage or ringing artifacts in the frequency domain, potentially distorting the OTF and leading to negative MTF values in the output.
Why This Might Apply to the Leica Summilux-M 50mm f/1.4 Charts
• Edge Behavior: The negative MTF is concentrated at the extreme edges (Y’ = 18-21 mm), where the lens’s performance is already compromised by aberrations like astigmatism and field curvature. This is a region where measurement accuracy is harder to maintain, and modeling errors are more likely to emerge.
• Aperture Dependence: The absence of negative MTF at f/1.4 (where aberrations are worse) but its presence at f/2.8 and f/5.6 (where aberrations should be reduced) is inconsistent with optical physics. This strongly suggests a processing artifact, as the modeling might struggle to reconcile the expected improvement with the actual edge data.
• Sagittal vs. Tangential: The sagittal curves (solid lines) are affected more than the tangential (dashed lines), which could indicate that the test setup or model had difficulty aligning the sagittal focus plane, leading to a phase error that the Fourier transform failed to handle correctly.
Comparison to Optical Performance
• If the negative MTF were a true optical effect (e.g., phase reversal due to astigmatism), it would require a significant and unusual aberration profile, which is unlikely for a Leica lens designed with precision. Moreover, the effect should be more pronounced at f/1.4 (wide open), not f/2.8 and f/5.6, unless the lens has an exotic design flaw that defies typical aberration correction trends.
• The abruptness of the drop below zero looks more like a discontinuity or artifact from data processing than a smooth optical transition.
Conclusion
The negative section of the MTF in these charts is most likely an artifact of the Fourier transform and a failure in modeling the OTF, rather than a true representation of the lens’s optical performance. This could stem from aliasing, phase wrapping, inadequate sampling, or a misfit in the data modeling process—particularly at the edges where measurement challenges are greatest. The inconsistency across apertures and the high quality expected from a Leica Summilux lens further support this interpretation.
Next Steps
To confirm this hypothesis:
• Additional Data: I can search for other MTF charts or reviews of the Leica Summilux-M 50mm f/1.4 to see if this negative dip is replicated, which would help distinguish a consistent artifact from a one-off error.
• Technical Analysis: If you have access to the raw data or test methodology behind these charts, we could analyze it for signs of sampling issues or phase errors.
• Expert Input: Consulting optical engineering resources or Leica’s technical documentation might clarify if this is a known artifact in their testing process.
Would you like me to search for more MTF data or reviews to investigate further? Alternatively, if you have more details about how these charts were generated, I can refine the analysis accordingly!