Blind deconvolution
Encyclopedia
In image processing and applied mathematics, blind deconvolution is a deconvolution
technique that permits recovery of the target scene from a single or set of "blurred" images in the presence of a poorly determined or unknown point spread function
(PSF).
Regular linear and non-linear deconvolution techniques utilize a known PSF. For blind deconvolution, the PSF is estimated from the image or image set, allowing the deconvolution to be performed. Researchers have been studying blind deconvolution methods for several decades, and have approached the problem from different directions.
Blind deconvolution can be performed iteratively, whereby each iteration improves the estimation of the PSF and the scene, or non-iteratively, where one application of the algorithm, based on exterior information, extracts the PSF.
Iterative methods include maximum a posteriori estimation and expectation-maximization algorithm
s. A good estimate of the PSF is helpful for quicker convergence but not necessary.
Examples of non-iterative techniques include SeDDaRA, the cepstrum transform and APEX. The cepstrum transform and APEX methods assume that the PSF has a specific shape, and one must estimate the width of the shape. For SeDDaRA, the provides information about the scene in the form of a reference image. The algorithm estimates the PSF by comparing the spatial frequency information in the blurred image to that of the target image.
.
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 auto correlation, 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 equalizer to obtain a signal with a PSF approximating what we know about the original PDF.
Deconvolution
In mathematics, deconvolution is an algorithm-based process used to reverse the effects of convolution on recorded data. The concept of deconvolution is widely used in the techniques of signal processing and image processing...
technique that permits recovery of the target scene from a single or set of "blurred" images in the presence of a poorly determined or unknown point spread function
Point spread function
The point spread function describes the response of an imaging system to a point source or point object. A more general term for the PSF is a system's impulse response, the PSF being the impulse response of a focused optical system. The PSF in many contexts can be thought of as the extended blob...
(PSF).
Regular linear and non-linear deconvolution techniques utilize a known PSF. For blind deconvolution, the PSF is estimated from the image or image set, allowing the deconvolution to be performed. Researchers have been studying blind deconvolution methods for several decades, and have approached the problem from different directions.
Blind deconvolution can be performed iteratively, whereby each iteration improves the estimation of the PSF and the scene, or non-iteratively, where one application of the algorithm, based on exterior information, extracts the PSF.
Iterative methods include maximum a posteriori estimation and expectation-maximization algorithm
Expectation-maximization algorithm
In statistics, an expectation–maximization algorithm is an iterative method for finding maximum likelihood or maximum a posteriori estimates of parameters in statistical models, where the model depends on unobserved latent variables...
s. A good estimate of the PSF is helpful for quicker convergence but not necessary.
Examples of non-iterative techniques include SeDDaRA, the cepstrum transform and APEX. The cepstrum transform and APEX methods assume that the PSF has a specific shape, and one must estimate the width of the shape. For SeDDaRA, the provides information about the scene in the form of a reference image. The algorithm estimates the PSF by comparing the spatial frequency information in the blurred image to that of the target image.
Mathematical concept
Suppose we have a signal transmitted through a channel. The channel can usually be modeled 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 equalizerEqualizer
Equalizer or equaliser may refer to:*Equalization, the process of adjusting the strength of certain frequencies within a signal*An equalization filter for used audio and similar signals...
.
If we are given the original signal, we can use a supervising technique, such as finding a Wiener filter
Wiener filter
In signal processing, the Wiener filter is a filter proposed by Norbert Wiener during the 1940s and published in 1949. Its purpose is to reduce the amount of noise present in a signal by comparison with an estimation of the desired noiseless signal. The discrete-time equivalent of Wiener's work was...
, 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 auto correlation, and we "whiten" the received signal.
Whitening usually leaves some phase
Phase (waves)
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...
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 equalizer to obtain a signal with a PSF approximating what we know about the original PDF.
High-order statistics
Blind deconvolution algorithms often make use of high-order statistics, with moments higher than two. This can be implicit or explicit.Algorithms
Important algorithms for blind deconvolution are:- Bussgang blind deconvolution
- Constant modulus algorithm
- Decision-directed estimation
- Godard algorithm
- Shalvi-Weinstein algorithm
- SeDDaRA method
See also
- Channel model
- Inverse problemInverse problemAn inverse problem is a general framework that is used to convert observed measurements into information about a physical object or system that we are interested in...
- Regularization (mathematics)Regularization (mathematics)In mathematics and statistics, particularly in the fields of machine learning and inverse problems, regularization involves introducing additional information in order to solve an ill-posed problem or to prevent overfitting...
- Blind equalizationBlind equalizationBlind equalization is a digital signal processing technique in which the transmitted signal is inferred from the received signal, while making use only of the transmitted signal statistics. Hence, the use of the word blind in the name....
- Maximum a posteriori estimation
- Maximum likelihoodMaximum likelihoodIn statistics, maximum-likelihood estimation is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, maximum-likelihood estimation provides estimates for the model's parameters....