Short answers, pulled from the story.
Norbert Wiener of the Massachusetts Institute of Technology laid the foundations for deconvolution and time-series analysis in his 1949 book "Extrapolation, Interpolation, and Smoothing of Stationary Time Series". The work was based on research Wiener had conducted during World War II that had been classified at the time.
In reflection seismology, deconvolution is used to strip a seismic wavelet from a recorded seismogram and recover the Earth-reflectivity function, which reveals the geological structure underground. Enders Robinson and colleagues including Wiener, Norman Levinson, and Paul Samuelson developed the convolutional model for this purpose starting in 1950 at MIT.
Early Hubble Space Telescope images were distorted by a flawed mirror and were sharpened using deconvolution before physical corrections were made. The technique works by determining the point spread function that describes the distortion, then convolving the acquired image with its inverse to recover the original, undistorted image.
Blind deconvolution is a technique used when the distortion function is unknown; it works by systematically testing different possible point spread functions and assessing whether the image improves. It is well established in astronomy and is also used in fluorescence microscopy and fluorescence spectral imaging. The Richardson-Lucy algorithm is the most common iterative method for blind deconvolution.
Deconvolution is applied in tracer kinetics to estimate hormone secretion rates from blood concentration measurements. It is also used to recover blood glucose concentration from interstitial glucose readings, which are a time- and amplitude-distorted version of the actual blood glucose level.
When a noisy signal is treated as noiseless, the statistical estimate of the distortion function g is incorrect, which in turn produces an incorrect estimate of the original signal. The lower the signal-to-noise ratio, the worse the result. Techniques such as Wiener deconvolution improve the estimate by incorporating knowledge about the character of the noise, for example whether it behaves as white noise.