Deconvolution
Blurring is the most familiar inverse problem of all. A camera, a microscope, a seismometer all
convolve the true signal with a blur kernel h:
d(x) = (h * m)(x) + \text{noise}.
Deconvolution is undoing it — recovering the sharp m
from the blurred, noisy d. Convolution is linear, so this is a linear
inverse problem, and the
Fourier transform
diagonalises it: convolution becomes multiplication,
\hat d(k) = H(k)\,\hat m(k).
Why naïve division explodes
The obvious inverse is to divide each frequency by the blur's response:
\hat m(k) = \hat d(k)/H(k). But a blur is a low-pass
filter — H(k) decays to nearly zero at high frequencies. Dividing by it
amplifies high-frequency noise without limit: the singular values of the convolution
operator are exactly the |H(k)|, and the small ones are the trouble. The
result is a reconstruction drowned in oscillation.
The cure is the now-familiar one —
Tikhonov,
in the frequency domain (the Wiener filter):
\hat m(k) = \frac{\overline{H(k)}}{|H(k)|^2 + \alpha^2}\,\hat d(k).
Where |H| \gg \alpha it divides as usual; where
|H| \ll \alpha it backs off toward zero instead of amplifying noise.
Deblur the signal
Two sharp peaks (faint) are blurred and noised into the dashed data; the bold curve is the
regularized deconvolution. Sweep \alpha: too small and high-frequency
noise erupts into wild ringing; too large and the peaks stay smeared together; in
between, the two peaks re-emerge cleanly. That sweet spot is the whole art of the subject, made
visible.
- Blur is convolution; in Fourier space \hat d = H\,\hat m, so the singular values are |H(k)|.
- Naïve \hat d/H amplifies high-frequency noise (small |H|) — ringing.
- The Wiener filter \overline{H}/(|H|^2+\alpha^2) is Tikhonov regularization in the frequency domain.