Editing Deconvolution (section)

== Description ==
In general, the objective of deconvolution is to find the solution ''f'' of a convolution equation of the form:

: <math>f * g^{-1} = h \, </math>

Usually, ''h'' is some recorded signal, and ''f'' is some signal that we wish to recover, but has been convolved with a filter or distortion function ''g'', before we recorded it. 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.  If we know ''g'', or at least know the form of ''g'', then we can perform deterministic deconvolution. However, if we do not know ''g'' in advance, then we need to estimate it.  This can be done using methods of [[statistics|statistical]] [[estimation theory|estimation]] or building the physical principles of the underlying system, such as the electrical circuit equations or diffusion equations.

There are several deconvolution techniques, depending on the choice of the measurement error and deconvolution parameters:

===Raw deconvolution===

When the measurement error is very low (ideal case), deconvolution collapses into a filter reversing. This kind of deconvolution can be performed in the Laplace domain. By computing the [[Fourier transform]] of the recorded signal ''h'' and the system response function ''g'', you get ''H'' and ''G'', with ''G'' as the [[transfer function]]. Using the [[Convolution theorem]],

: <math>F = H / G \, </math>

where ''F'' is the estimated Fourier transform of ''f''. Finally, the [[Fourier inversion theorem|inverse Fourier transform]] of the function ''F'' is taken to find the estimated deconvolved signal ''f''. Note that ''G'' is at the denominator and could amplify elements of the error model if present.

===Deconvolution with noise===

In physical measurements, the situation is usually closer to

: <math>(f * g^{-1})  + \varepsilon  = h \, </math>

In this case ''&epsilon;'' is [[noise (physics)|noise]] that has entered our recorded signal.  If a noisy signal or image is assumed to be noiseless, the statistical estimate of ''g'' will be incorrect.  In turn, the estimate of ''&fnof;'' will also be incorrect.  The lower the [[signal-to-noise ratio]], the worse the estimate of the deconvolved signal will be. That is the reason why [[inverse filter]]ing the signal (as in the "raw deconvolution" above) is usually not a good solution. However, if at least some knowledge exists of the type of noise in the data (for example, [[white noise]]), the estimate of ''&fnof;'' can be improved through techniques such as [[Wiener deconvolution]].