Déconvolution

Déconvolution is the process of reversing the effects of convolution to recover the original signal or image from observations blurred by a known or unknown impulse response. In discrete form, an observed sequence y[n] is the convolution of the true signal x[n] with a kernel h[n], plus noise n[n]: y[n] = sum_k h[k] x[n−k] + n[n]. In the frequency domain, Y(f) = H(f) X(f) + N(f). If the blur kernel H is known and well-conditioned, X can be recovered by dividing by H, but noise makes simple inverse filtering unstable when H has small magnitude or zeros.

Various methods address this instability. Inverse filtering uses the exact inverse of H; Wiener filtering regularizes by incorporating estimates of signal and noise spectra to dampen frequency components affected by noise. Tikhonov regularization adds a penalty on solution roughness to stabilize the inversion. Iterative deconvolution methods include the Richardson-Lucy algorithm, popular for images affected by Poisson noise and widely used in astronomy and microscopy. Nonnegative constraints are often imposed for image data. Blind deconvolution aims to estimate both x and h from y, typically with priors or regularization and iterative optimization.

Applications are found in imaging and signal processing: restoring blurred photographs, deblurring astronomical images, improving resolution in fluorescence and electron microscopy, and more generally in any context where a measured signal is a blurred version of a latent one. Challenges include ill-posedness, non-uniqueness, boundary artifacts, and sensitivity to noise; successful restoration relies on appropriate regularization, accurate or plausible blur models, and robust optimization.