Image Restoration Using Inverse Filtering, Wiener Filtering, and Constrained Least Squares Filtering

In signal processing, inverse filtering, Wiener filtering, and constrained least squares filtering represent three widely-used signal restoration techniques. Inverse filtering is primarily employed to recover blurred images or signals by implementing the inverse of the degradation function, which can be expressed in frequency domain as G(u,v) = F(u,v)/H(u,v) where H(u,v) represents the blur transfer function. Wiener filtering addresses noise-induced distortions by incorporating statistical characteristics of both the signal and noise, implementing a minimum mean-square error estimator through the transfer function W(u,v) = [H*(u,v)]/[|H(u,v)|² + S_η(u,v)/S_f(u,v)], where S_η and S_f represent noise and signal power spectra respectively. Constrained least squares filtering performs restoration by simultaneously considering constraint conditions and least squares criteria, typically solving optimization problems through regularization techniques that balance data fidelity and smoothness constraints. These techniques find extensive applications in engineering fields and serve as fundamental knowledge for beginners, with practical implementations often involving Fourier transforms, matrix operations, and regularization parameter selection algorithms.