Tensions in Practice: Exact noiseless inversion in tension with noise amplification¶
Two-point circular blur · one weakened mode
In a declared two-point blur, each output is 0.55 of its own input plus 0.45 of its neighbor. This preserves the pair’s mean but scales half their difference by 0.1. For input (2, 0), that contrast falls from 1 to 0.1. Recovering it by multiplying by 10 also multiplies any measured contrast error by 10. Limiting that inverse gain to 5 reduces error amplification but under-recovers an error-free contrast.
Recover the error-free signal
Undo the declared blur exactly when its measurements and kernel can be trusted.
Limit amplified measurement error
Reduce how strongly perturbations in a weakly transmitted mode affect the result.
Why these aims pull against each other
The missing contrast and its measurement error arrive in the same weakened mode. A lower inverse gain shrinks both, so noise restraint costs noiseless fidelity.
Choose an arrangement to see what changes and what remains difficult.
Finite illustrative comparisons. Text states carry the meaning; color is not a measured score or universal preference.
What this choice protects
What it costs
When it fits
Compare the arrangements
Use gain 10
Multiply measured contrast by 10 and leave the separately preserved mean unchanged.
| Measured | Recovered | Error | |
|---|---|---|---|
| No error | 0.10 | 1.00 | 0.00 |
| Error .02 | 0.12 | 1.20 | 0.20 |
| Error .10 | 0.20 | 2.00 | 1.00 |
- What it protects
- The error-free measurement 0.1 reconstructs the true contrast 1.
- What it costs
- A contrast error of 0.02 becomes reconstruction error 0.2; error 0.10 becomes 1.
- When it fits
- Fits when the kernel is known and measurement accuracy supports the required inverse gain.
Illustration note: The finite setting and values are editorial assumptions, not measured effects or recommended operating settings. The table uses three stipulated nonnegative contrast errors, not a random noise distribution.
Use gain 5
Multiply measured contrast by 5 rather than 10; preserve the mean as before.
| Measured | Recovered | Error | |
|---|---|---|---|
| No error | 0.10 | 0.50 | 0.50Bias |
| Error .02 | 0.12 | 0.60 | 0.40 |
| Error .10 | 0.20 | 1.00 | 0.00 |
- What it protects
- A change of 0.02 in the measurement changes the estimate by 0.1 rather than 0.2.
- What it costs
- Even the error-free measurement reconstructs contrast 0.5, introducing a bias of −0.5.
- When it fits
- Fits when limiting sensitivity is worth the declared bias and the shrinkage policy is justified for the task.
Illustration note: The finite setting and values are editorial assumptions, not measured effects or recommended operating settings. This selected gain is an illustrative regularized inverse, not an optimized parameter.
What this illustration does—and does not—establish
Convolution: Deconvolution Promise versus Ill-Conditioning (measurement) supplies the ill-conditioned inverse and regularization boundary. The finite kernel and modal calculation make amplification and bias inspectable.
- The circular two-point fixed kernel is linear and shift-invariant; this is not an arbitrary matrix relabeled as convolution.
- Error means absolute difference from the declared true contrast 1. The last regularized value happens to equal truth for that selected error; it is not evidence of universal noise correction.
- Both modes have nonzero gains. A mode erased completely cannot be recovered by increasing an inverse gain.
Source entries
Convolution
Convolution: Deconvolution Promise versus Ill-Conditioning (measurement) supplies the conflict examined here.
Deconvolution Promise versus Ill-Conditioning (measurement)
The failure mode is naive deconvolution that amplifies noise catastrophically: where the kernel's Fourier transform is near zero, dividing by it explodes measurement error into the reconstruction.
Structural Tensions
deconvolution must be regularized, not run as a clean algebraic inverse.