Kronecker product approximations for image restoration with whole-sample symmetric boundary conditions
Information Sciences j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i n s problems, iterative methods may be more attractive; see [22, 28, 36, 11, 45] for more details. Besides, partial differential-equation-(PDE-) based methods have recently received much attention [4, 6, 13, 18, 34, 38, 40] . In general, a spatially invariant image degradation can be linearly modelled as where K is a matrix associated with the point spread function (PSF) and with the boundary
... onditions (BCs), g is the recorded image, f is the original image to be recovered and g is additive noise. In this work, the PSF is assumed to be known or to be estimated from the data. Many techniques are available for estimating the PSF by a variety of means [26, 27, 43] . The exact structure of matrix K depends on the imposed boundary conditions. Four frequently used BCs and their correspondence with K are list here [1, 2, 7, 10, 12, 16, 21, 33, 37] : Periodic boundary conditions. It assumes that the true infinite scene can be represented as a mosaic of a single finite dimensional image, repeated periodically in all directions. The associated matrix K with this BCs is a block circulant with circulant blocks (BCCB) matrix. Zero boundary conditions. In this case, all pixels outside the image domain is zero. With the BCs, the matrix K has a block Toeplitz with Toeplitz blocks (BTTB) structure. Half-sample symmetric boundary conditions. This is usually used reflexive boundary conditions.It assumes that the scene outside the viewable region is a mirror reflection of the scene inside the viewable region. In the case of the frequentlyused reflexive BCs, the matrix K is a block Toeplitz-plus-Hankel with Toeplitz-plus-Hankel blocks (BTHTHB) matrix. Antireflexive boundary conditions. It extends the pixel values across the boundary in such a way that continuity of the image and of the normal derivative are preserved at the boundary. The matrix arising from the BCs has a block Toeplitz-minus-Hankel-plus-Rank-2 with Toeplitz-minus-Hankel-plus-Rank-2 blocks (BTH2TH2B) structure.