C18 · Recovering information
Restoration and inverse problems
Before recovery, describe how measurements were degraded. Even a good model cannot restore information it destroyed.
Theory
The observation is not the unknown original
The model separates unknown u, blur operator H and disturbance n. A PSF describes what a point becomes; its DFT is the transfer function. Defocus and motion have different PSFs. Here a known symmetric horizontal filter acts periodically.
Dividing by H amplifies disturbances where transfer is small and fails at zero. Tikhonov with replaces it by bounded . Stability costs bias and detail. The retained synthetic reference makes genuine error measurement possible.
The restoration trade-off
The observation uses a known horizontal blur. Lower λ and see which frequencies get amplified. Add interference: the multiplier cannot distinguish it from useful signal. Compare restoration with the known reference.
Exact values are in the point inspector
Grayscale 0…1: black to white.
Exact values are in the point inspector
Grayscale 0…1: black to white.
Inspect the selected point
What the plot shows
Where blur nearly erased signal, division also amplifies interference. λ bounds amplification; H=0 means absent information for every λ.
- Blur transfer H
- Gain H/(H²+λ)
Weak regularization amplifies unstable frequencies; strong regularization suppresses detail. A zero transfer frequency remains unrecoverable even without interference.
- MSE
- 0.0005
- RMSE
- 0.023
- PSNR (range 0–1)
- 32.757 dB
Parameters and details
Periodic horizontal blur [1,2,1]/4; frequencies use H/(H²+λ). The zero transfer frequency at Nyquist cannot be recovered. This is Tikhonov, not blind deconvolution or Richardson–Lucy.
Select an image point. Arrows move it; Home and End reach row edges. Viewing does not change the calculation.
Changing the example requires JavaScript and Canvas. Computed images remain available above.
An unknown pixel is different from a black one
A separate mask marks missing data; otherwise black is wrongly treated as an observation. Harmonic filling fixes known boundaries and repeatedly averages unknown neighbors. Jacobi uses the previous state simultaneously, not a partly updated row.
This continues smooth fields without inferring hidden objects or complex textures. Biharmonic inpainting uses another operator; generative filling uses a learned prior. A plausible generated result is not known to be recovered truth without a reference.
Fill only missing data
The mask marks a missing rectangle. Add averaging iterations and watch the known neighbors.
Exact values are in the point inspector
Grayscale 0…1: black to white.
Exact values are in the point inspector
Grayscale 0…1: black to white.
Inspect the selected point
What the plot shows
Outside the mask values stay fixed; inside, a smooth continuation spreads. It may be plausible without recovering the unknown object.
Parameters and details
Masked pixels update to their four-neighbor mean (Jacobi); known values stay fixed. The mask is separate data. Smooth continuation cannot infer hidden objects and is not biharmonic inpainting.
Select an image point. Arrows move it; Home and End reach row edges. Viewing does not change the calculation.
Changing the example requires JavaScript and Canvas. Computed images remain available above.
Restoration methods do not assume the same noise
Reference and further methods
Frequency Wiener uses signal/noise power ratios; local Wiener denoising is a different task. Richardson–Lucy relates to a nonnegative Poisson observation model and iterates. Excess iterations can amplify artifacts. Blind deconvolution estimates PSF and source together, increasing ambiguity.
Aligned multiple frames can add actual measurements for HDR, denoising or super-resolution. That differs from one image plus a strong prior. Comparisons need common registration, scales, masks and wrong-PSF tests. A sharp appearance alone proves little.
Practice
Test the idea under another condition. Follow the task above the images; point inspection helps explain the result. This exploration is not scored.
Choose a trade-off for one fixed observation
The interference is stronger than in the worked example. Begin with a very small λ, then increase it without changing the observation. Inspect small oscillations and a real edge separately; use the reference stage to compare.
Exact values are in the point inspector
Grayscale 0…1: black to white.
Exact values are in the point inspector
Grayscale 0…1: black to white.
Inspect the selected point
What the plot shows
Where blur nearly erased signal, division also amplifies interference. λ bounds amplification; H=0 means absent information for every λ.
- Blur transfer H
- Gain H/(H²+λ)
There is no universally best λ: suppressing oscillations costs detail. MSE is available only because the clean synthetic reference is known.
- MSE
- 0.0422
- RMSE
- 0.2055
- PSNR (range 0–1)
- 13.745 dB
Parameters and details
Periodic horizontal blur [1,2,1]/4; frequencies use H/(H²+λ). The zero transfer frequency at Nyquist cannot be recovered. This is Tikhonov, not blind deconvolution or Richardson–Lucy.
Select an image point. Arrows move it; Home and End reach row edges. Viewing does not change the calculation.
Changing the example requires JavaScript and Canvas. Computed images remain available above.
Games
Test your explanation
Question 1 of 5
Transfer is zero at one frequency. What does one observation reveal there?
Exact values are in the point inspector
Grayscale 0…1: black to white.
Exact values are in the point inspector
Grayscale 0…1: black to white.
Exact values are in the point inspector
Grayscale 0…1: black to white.
Exact values are in the point inspector
Grayscale 0…16: black to white.
Exact values are in the point inspector
Grayscale 0…1: black to white.
Exact values are in the point inspector
Grayscale 0…1: black to white.
Choose one answer. It is checked immediately.
Limitations and references
Limits of this example
Implemented: bounded periodic Tikhonov and harmonic fill. Disturbance is deterministic, not a calibrated sensor model. No RAW recovery, blind estimation or neural-model download.