Dekonvolúció

Dekonvolúció, in the context of signal processing and imaging, is the process of recovering an original signal or image from data that have been blurred by convolution with a known or unknown kernel. It is the inverse problem of convolution. The common observation model is y = h * x + n, where x is the original signal, h is the blur kernel or point spread function, and n is noise. In discrete form this is a linear system; in continuous form it is an integral equation.

If the blur kernel h (or H in the frequency domain) is known, one can attempt to invert the process. In the frequency domain, Y(f) = H(f) X(f) + N(f), so a naive inverse filter would be X̂(f) = Y(f)/H(f). This approach is highly sensitive to noise and to zeros or near-zeros of H, making direct inversion unstable. Therefore, deconvolution often relies on regularization or probabilistic models. Common methods include Wiener filtering, which balances fidelity to data with noise suppression, Tikhonov regularization, and iterative algorithms such as Richardson–Lucy for Poisson-distributed noise or compressed-sensing-inspired approaches. Blind deconvolution seeks x and h simultaneously when the blur is unknown.

Applications of dekonvolúció span astronomy, where it helps recover sharp images of celestial objects blurred by atmospheric and instrumental PSFs; microscopy, enhancing resolution by removing blur introduced by the optical system; and general image and signal restoration in photography, medical imaging, and seismic data analysis. Challenges include ill-posedness, noise amplification, model mismatch, and the computational complexity of iterative algorithms. Accurate kernel estimation and appropriate regularization are central to effective deconvolution. See also convolution, regularization, and blind deconvolution.