Deconvolution

Deconvolution is a process used in signal and image processing to reverse the effects of convolution with a known or estimated kernel, often a point spread function. In many systems, an original signal x is observed as y = h * x + n, where h is the impulse response (or point spread function in imaging) and n represents noise. Deconvolution seeks to recover x from y given h.

Because convolution with noise is smoothing, direct inversion is unstable; deconvolution is an ill-posed inverse problem. Small errors in y or h can cause large errors in x. Regularization, prior information, or statistical models are often employed. In blind deconvolution, h is also unknown.

Common methods include inverse filtering, which attempts to compute x = h^{-1} * y but requires precise h and noise-free data and is typically impractical. Wiener filtering accounts for signal and noise spectra to minimize mean squared error. Iterative methods such as Richardson–Lucy deconvolution are based on likelihood with Poisson statistics and are widely used in astronomy and microscopy. Tikhonov regularization adds a penalty on the solution norm. Total variation regularization preserves edges. More generally, maximum a posteriori estimation and Bayesian methods can incorporate priors and sparsity constraints.

Applications span astronomical image deblurring, optical and electron microscopy, medical imaging (MRI, CT), photography, and seismology. In practice, accurate knowledge of the point spread function is crucial; when h must be estimated, the problem becomes blind deconvolution, which is more challenging and less stable.

Evaluation metrics include visual quality and quantitative measures such as signal-to-noise ratio and PSF fidelity. Deconvolution techniques are also used in deblurring of audio signals and other time series.