Dekonvolution

Dekonvolution is a set of computational techniques used to recover an original signal or image that has been blurred by convolution with a kernel and corrupted by noise. In a common linear model, the observed data y results from convolving the true signal x with a point spread function h (the blur) and adding noise n: y = h * x + n. The goal of dekonvolution is to estimate x from y, given knowledge of h (non-blind dekonvolution) or to estimate h as well (blind dekonvolution).

The problem is typically ill-posed. Convolution with a finite kernel smooths data and can lose information, so naive inversion amplifies noise and artifacts. Solutions often rely on regularization or prior information about the signal, such as nonnegativity, smoothness, sparsity, or known boundary conditions. Noise models influence the approach: Gaussian noise lead to different formulations than Poisson noise, which motivates specific algorithms.

Common methods include inverse filtering and Wiener filtering in the frequency domain for straightforward cases, as well as regularized inverses such as Tikhonov regularization. Iterative approaches are widely used, notably the Richardson–Lucy deconvolution for Poisson-distributed data. Maximum-a-posteriori and Bayesian formulations incorporate priors; total variation and sparsity-based regularizations help reduce ringing and artifacts. Blind deconvolution seeks to recover both x and the blur kernel from the data with appropriate constraints.

Applications of dekonvolution span multiple fields. In astronomy it sharpens telescope images; in fluorescence and light microscopy it improves resolution; in medical imaging, radar, and acoustics it reverse-blurring effects caused by the imaging system or environment. The technique is also used in photography and remote sensing to enhance detail and contrast in blurred data.