Restoration: degradation, denoise, super-resolution

Restoration is the part of imaging that pays for itself in products. But it is very easy to do it by piling up models. The disciplined way is to write down the degradation first, then invert it.

y = D( H( x ) ) + n

# x  clean image
# H  blur / resampling / compression already baked into the file
# D  downsample or quantise   (super-resolution: estimate from this lossy observation)
# n  noise                    (denoise: this is what you undo)
# y  what you actually have

# Every restoration task is: given y, estimate x
# — and the art is in modelling H, D, n and the prior or extra measurements.

That one line separates the three tasks that are constantly confused:

Denoising — `H` and `D` are identity; you are estimating `x` despite `n`.

Deblurring — `n` is small; you are undoing `H` (and it is ill-posed: many `x` produce the same `y`).

Single-image super-resolution — downsampling `D` removes or mixes detail. The inverse is not unique, so an estimate needs a prior (often learned); extra captures can also provide measurements. One low-resolution image alone cannot uniquely determine arbitrary missing detail.

Which brings up the metric problem, and it is the thing most teams get wrong.

· PSNR / MSE measure pixelwise distortion. When several sharp details are plausible from the same degraded input, minimising average pixel error can favor a smooth compromise that scores well but looks soft. This is a perception–distortion tradeoff, not a rule that every high-PSNR result must be blurry.

· SSIM compares local structure and can complement pixel error, but it is not a universal measure of human preference.

· Perceptual metrics such as LPIPS compare learned feature representations. They capture different aspects of similarity; no single metric is a complete quality oracle.