Deconvolution

The foundations for deconvolution and time-series analysis were largely laid by Norbert Wiener of the Massachusetts Institute of Technology in his book Extrapolation, Interpolation, and Smoothing of Stationary Time Series (1949).

Usually, h is a distorted version of f and the shape of f can't be easily recognized by the eye or simpler time-domain operations.

The function g represents the impulse response of an instrument or a driving force that was applied to a physical system.

In physical measurements, the situation is usually closer to In this case ε is noise that has entered our recorded signal.

That is the reason why inverse filtering the signal (as in the "raw deconvolution" above) is usually not a good solution.

He worked with others at MIT, such as Norbert Wiener, Norman Levinson, and economist Paul Samuelson, to develop the "convolutional model" of a reflection seismogram.

The reflectivity may be recovered by designing and applying a Wiener filter that shapes the estimated wavelet to a Dirac delta function (i.e., a spike).

The result may be seen as a series of scaled, shifted delta functions (although this is not mathematically rigorous): where N is the number of reflection events,

However, by formulating the problem as the solution of a Toeplitz matrix and using Levinson recursion, we can relatively quickly estimate a filter with the smallest mean squared error possible.

It is usually done in the digital domain by a software algorithm, as part of a suite of microscope image processing techniques.

Deconvolution is also practical to sharpen images that suffer from fast motion or jiggles during capturing.

Early Hubble Space Telescope images were distorted by a flawed mirror and were sharpened by deconvolution.

The usual method is to assume that the optical path through the instrument is optically perfect, convolved with a point spread function (PSF), that is, a mathematical function that describes the distortion in terms of the pathway a theoretical point source of light (or other waves) takes through the instrument.

[3] Usually, such a point source contributes a small area of fuzziness to the final image.

In practice, finding the true PSF is impossible, and usually an approximation of it is used, theoretically calculated[4] or based on some experimental estimation by using known probes.

[3] Blind deconvolution is a well-established image restoration technique in astronomy, where the point nature of the objects photographed exposes the PSF thus making it more feasible.

Division of the time-domain data by an exponential function has the effect of reducing the width of Lorentzian lines in the frequency domain.