Linear Systems 2
Topics in Digital Heritage:
Numerical Methods for Digital Reconstruction
Fall 2026
Deconvolution
Blurry \(\mathbf{x}_0\quad\) Sharp \(\mathbf{x}\)
Given a blurry image \(\mathbf{x}_0\) and a known blur kernel \(\mathbf{h}\), we want to recover the original sharp image \(\mathbf{x}\). One strategy is deconvolution, which can be formulated as a linear system: \[\min_{\mathbf{x}} \|\mathbf{h} * \mathbf{x} - \mathbf{x}_0\|^2_2\] where \(*\) denotes convolution.
In this case, the unknown is \(\mathbf{x}\), where in many cases, multiple solutions may exist due to the ill-posed nature of the problem. This can be solved possibly with Tikhonov regularization.
Deconvolution
Convolution Theorem
Convolution \[ (f * g)(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) d\tau \]
Fourier Transform \[ \mathcal{F}\{ x(t) \} = X(f) = \int_{-\infty}^{\infty} x(t) e^{-j 2 \pi f t} dt \]
\[\begin{aligned} \mathcal{F}\{ f * g \} &= \int_{-\infty}^{\infty} (f * g)(t) e^{-j 2 \pi f t} dt \\ &= \int_{-\infty}^{\infty} \left( \int_{-\infty}^{\infty} f(\tau) g(t - \tau) d\tau \right) e^{-j 2 \pi f t} dt \\ &= \int_{-\infty}^{\infty} f(\tau) \left( \int_{-\infty}^{\infty} g(t - \tau) e^{-j 2 \pi f t} dt \right) d\tau \\ &= \int_{-\infty}^{\infty} f(\tau) \left( \int_{-\infty}^{\infty} g(u) e^{-j 2 \pi f (u + \tau)} du \right) d\tau \\ &\qquad\qquad\qquad\qquad\qquad (u = t - \tau;\ dt=du) \\ &= \int_{-\infty}^{\infty} f(\tau) e^{-j 2 \pi f \tau} \left( \int_{-\infty}^{\infty} g(u) e^{-j 2 \pi f u} du \right) d\tau \\ &= \left( \int_{-\infty}^{\infty} f(\tau) e^{-j 2 \pi f \tau} d\tau \right) \left( \int_{-\infty}^{\infty} g(u) e^{-j 2 \pi f u} du \right) \\ &= \mathcal{F}\{f\} \cdot \mathcal{F}\{g\} \end{aligned}\]
