Image Deconvolution
Wiener deconvolution of a diffraction-limited image: the restored transfer H²/(H² + NSR), the largest filter gain and the finest period restored, against the passband limit λ/(2NA).
The Wiener filter G = H/(H² + NSR) assumes white noise and a flat object spectrum with a noise-to-signal power ratio NSR = 10^(−dB/10). Tikhonov regularisation with the identity is the same filter with NSR as its parameter. Richardson–Lucy and blind deconvolution are iterative and nonlinear: they have no transfer function to plot, and in practice they cannot restore frequencies above f_c either; positivity-constrained extrapolation beyond it is small and noise-limited (Conchello 1998).
Results
Deconvolution raises the contrast of frequencies the optics passed weakly; it does not pass anything above f_c, so no linear method resolves a period below λ/(2NA) in widefield imaging (SIM and STED change the imaging itself). The filter gain peaks at H = √NSR and rolls off above it, so the noise in the restored image is bounded.
Transfer before and after Wiener restoration
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Linear Wiener filter for white noise and a flat object spectrum, applied to the in-focus diffraction-limited MTF of a circular pupil. Iterative methods (Richardson–Lucy, blind) are not modelled; no linear method restores frequencies above 2NA/λ, and positivity-constrained maximum-likelihood extrapolation beyond it is small and noise-limited.
- Goodman J. W. (2005). Introduction to Fourier Optics, 3rd ed., §6.3. Roberts & Company. Incoherent cutoff 2NA/λ and the diffraction-limited MTF of a circular pupil.
- Gonzalez R. C., Woods R. E. (2018). Digital Image Processing, 4th ed., §5.8. Pearson. Wiener filter H*/(|H|² + K) with a constant noise-to-signal ratio K.
- Conchello J.-A. (1998). Superresolution and convergence properties of the expectation-maximization algorithm for maximum-likelihood deconvolution of incoherent images. J. Opt. Soc. Am. A 15, 2609. doi:10.1364/JOSAA.15.002609