Denoising Algorithms
Output noise, SNR gain and transfer function of box, Gaussian and median filters on white Gaussian noise, plus the VisuShrink wavelet threshold σ√(2 ln N).
Output noise σ
6.667
0.3333 × the input σ (3 × 3 window)
SNR gain
9.54 dB
20 log₁₀(σ_in / σ_out) on white noise
Half-response frequency f₅₀
0.2098 cy/px
H(0.25 cy/px) = 0.333 along one axis
VisuShrink threshold
99.907
σ√(2 ln N) for N = 512 × 512 coefficients
SNR gain on white Gaussian noise
Transfer function along one axis
Key Formulas
Linear filter: σ_out² = σ²·Σw², Σw = 1 (white noise)
Box k × k: σ_out = σ/k, H(f) = sin(πkf) / (k·sin πf)
Gaussian: g_i ∝ exp(−i²/2σ_s²), |i| ≤ ⌈3σ_s⌉; σ_out → σ/(2√π σ_s), H → exp(−2π²σ_s²f²)
Median of n = k²: σ_out² = σ²·∫x² n!/(m!)² Φ^m (1−Φ)^m φ dx, m = (n−1)/2; → πσ²/(2n)
VisuShrink: T = σ√(2 ln N)
Additive white Gaussian noise, no edge effects; the box and the Gaussian are separable, so H is the same along x and y. A negative H is contrast reversal. The median is about 2/π as efficient as the mean on Gaussian noise but keeps edges and rejects impulse noise. Bilateral, non-local means, BM3D and learned denoisers adapt to the image content and have no closed-form noise gain, so they are not modelled; VisuShrink only gives the soft-threshold level, not the output noise.
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
Additive white Gaussian noise without edge effects. Box and Gaussian filters are linear (σ_out² = σ² Σw²); the median's variance is that of the median of k² independent normals, by quadrature. Bilateral, non-local means, BM3D and learned denoisers depend on the image and are not modelled.
- Gonzalez R. C., Woods R. E. (2018). Digital Image Processing, 4th ed. Pearson. Box, Gaussian and median spatial filters.
- David H. A., Nagaraja H. N. (2003). Order Statistics, 3rd ed. Wiley. Distribution of the sample median; variance 1 − √3/π for n = 3 and π/(2n) for large n. doi:10.1002/0471722162
- Donoho D. L., Johnstone I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81, 425–455. Universal threshold σ√(2 ln N). doi:10.1093/biomet/81.3.425