Spatial Frequency Transfer Tuning in Optical Phase Retrieval Systems
Phase retrieval accuracy hinges on balancing low-frequency noise amplification against high-frequency contrast decay through tuned transfer filters.
Contrast Transfer Function
Meaning
Optical imaging systems modify the amplitude of transmitted spatial frequencies depending on the aperture size and aberrations. The contrast transfer function mathematically describes this modulation for sinusoidal intensity patterns as a function of spatial frequency. It quantifies how faithfully the system reproduces the contrast of the object in the final image.
This description applies primarily to incoherent imaging conditions where the intensity distribution is linear.
Frequency Response
Resolution limits of an instrument are determined by the cutoff frequency where the modulation falls to zero. For a perfect system, the contrast transfer function decreases monotonically with increasing spatial frequency. Aberrations like spherical aberration alter this response, often causing the contrast to drop more rapidly or even reverse.
This reversal manifests as a phase shift.
Defocus Influence
Out of focus conditions modify the transfer behavior by introducing phase errors that vary with the square of the spatial frequency. This modulation causes the contrast transfer function to oscillate between positive and negative values, meaning that some spatial frequencies are imaged with inverted contrast. Such oscillations create spurious resolution where fine details appear in the image but do not correspond to real features in the object.
Correcting for this effect is necessary when interpreting high resolution electron micrographs or optical scans. In practice, researchers must acquire images at multiple focus levels to fill the zero crossings where no information is transmitted.
Imaging Filter
Digital reconstruction algorithms utilize the known transmission profile to restore the original object details. By dividing the measured image spectrum by the contrast transfer function, the algorithm boosts weakened frequencies. However, this division also amplifies high frequency noise.
Thus, a regularization term is added to prevent noise amplification from dominating the reconstructed image.