Blind deconvolution
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In applied mathematics, blind deconvolution is a deconvolution technique that permits recovery of the target object from set of "blurred" images in the presence of a poorly determined or unknown point spread function (PSF). Regular linear and non-linear deconvolution techniques require a known PSF. For the "blind" case a set of multiple images (data cube) of the same target object is preferable, each having dissimilar PSFs. The blind deconvolution algorithm is then able to restore not only the target object but also the PSFs. A good estimate of the PSF is helpful for quicker convergence but not necessary.
Iterative methods include Richardson-Lucy deconvolution, and expectation-maximization algorithms.
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[edit] Concept
Suppose we have a signal transmitted through a channel. The channel can usually be modelled as a linear system, so the receptor receives a convolution of the original signal with the impulse response of the channel. If we want to reverse the effect of the channel, to obtain the original signal, we must process the received signal by a second linear system, inverting the response of the channel. This system is called an equaliser.
If we are given the original signal, we can use a supervising technique, such as finding a Wiener filter, but without it, we can still explore what we do know about it to attempt its recovery. For example, we can filter the received signal to obtain the desired spectral power density. This is what happens, for example, when the original signal is known to have no autocorrelation, and we "whiten" the received signal.
Whitening usually leaves some phase distortion in the results. Most blind deconvolution techniques use higher-order statistics of the signals, and permit the correction of such phase distortions. We can optimize the equaliser to obtain a signal with a PDF approximating what we know about the original PDF.
[edit] High-order statistics
Blind deconvolution algorithms often make use of high-order statistics, with moments higher than two. This can be implicit or explicit.
[edit] Gaussianity
The output of a linear system usually has a Gaussian output, in accordance with the central limit theorem. Blind deconvolution algorithms seek equalisers which maximize the "non-Gaussianity" of recovered signals. Such techniques usually don't work with Gaussian signals, since they have higher cumulants equal to zero.
[edit] Algorithms
Important algorithms for blind deconvolution are:
- Richardson-Lucy deconvolution
- Constant modulus algorithm
- Decision-directed estimation
- Shalvi-Weinstein algorithm
- Bussgang blind deconvolution
- Godard algorithm
[edit] See also
- Channel model
- Inverse problem
- Regularization (mathematics)
- Blind equalization
