The removal of blur from an image is one of the most important operations in
image processing and it may be considered a preprocessing operation in all
applications because image interrogation is significantly easier if the image is
of high quality. The removal of blur is called image deconvolution because the
point spread function (PSF), which represents the blur, is deconvolved from the
blurred image in order to compute the restored image. The greatest research
effort is devoted to the problem of blind image deconvolution, that is, the
restored image must be computed in the absence of prior knowledge of the PSF and
the exact image. Many different methods for its solution have been developed, of
which methods based on Bayes’ theorem and resultant matrices are the most
popular.
These methods will be reviewed, and in particular, their fundamentally different
features will be consid ered. The methods for the solution of the problem of
blind image deconvolution that use Bayes’ theorem require prior information,
which is imposed in the form of probability distributions on the PSF and the
exact image. By contrast, priors are not required when resultant matrices are
used, and the deblurred image is obtained by polynomial computations,
specifically, greatest common divisor computations and polynomial
deconvolutions.
The talk will compare the deblurred images obtained from these two methods, and
it will be shown that the methods based on Bayes’ theorem solve the problem of
semiblind image deconvolution because they require that the size of the PSF be
specified. By contrast, this information is calculated in the method that uses
resultant matrices, and it is not, therefore, specified, It will be shown that
resultant matrices have strong structure, which is imposed on the computations
by using structurepreserving matrix methods, and these methods explain the very
good results obtained by resultant matrices. Also, some theoretical properties
of these matrices will be considered and it will be shown that the QR
decomposition of the Sylvester resultant matrix, which is one type of resultant
matrix, allows the size of the PSF to be computed.
