Spectral Deconvolution Sample Clauses

Spectral Deconvolution. Spectral filtering methods exploit structure of the matrix to efficiently compute the singular (or spectral) value decomposition of K, and use this information to con- struct Kr† . The spectral filtering algorithms include many well known techniques for image deblurring such as the Wiener filter [49] and the pseudo inverse filter. But general approaches, such as truncated spectral decompositions and Tikhonov regularization [57] also belong to this group. Whether or not these techniques work well depends on special structure of the PSF (and hence of K) and on the imposed boundary conditions [57]. The computational efficiency of spectral filtering methods for image deblurring with a spatially invariant PSF requires efficient discrete Fourier transform (DFT) and discrete cosine transform (DCT) routines. Although these deconvolution al- gorithms can be very efficient, and they are fairly easy to implement they have many limitations. First, efficient implementation requires the blur to have a very special structure, and this almost always means spatially invariant. In the case of spatially variant blurs, DFT and DCT based methods do not provide the right basis to use in filtering algorithms. It is possible to generalize the filtering ideas, using the singular value decomposition, but generally these approaches are very expensive. One exception is if the space variant blur is separable (i.e., the blurring operation can be separated into components involving a single vertical and a sin- gle horizontal blur). In this case, the matrix K can be represented as a Kronecker product of two smaller matrices. Another limitation of spectral filtering methods is that it is not possible to include additional constraints, such as nonnegativity, in the reconstruction algorithms. Figure 2.1 shows a comparison between the spec- tral and iterative methods; in practice the reconstruction quality is usually much better when an iterative algorithm is used. PSF Blurred image Restored image (spectral) Restored image (iterative) Figure 2.1: Spectral vs. iterative image deblurring. Boundary Conditions If the blur is assumed to be spatially invariant, then the PSF is the same regardless of the position of the point source in the image field of view. In this case, if we also enforce periodic boundary conditions, then K has a circulant matrix structure [57], and the spectral factorization K = F∗ΛF , (2.3) where F is the DFT matrix; a d-dimensional image implies F is a d-dimensional DFT mat...