We propose to solve the incorrect inverse problem of image restoration using the estimation of quasi-solution, which is the member of some optimized compact set. For such a compact set we choose the subspace, which is spanned on the several first eigenvectors of symmetrized matrix CK equals (C * K + K * C)/2. Here C is an estimation of covariation of primary images, and K is (pseudo)inverse of noise covariation matrix. This representation provides concentration of noiseless images' energy in main spectral components and effective noise suppression under spectrum truncation. The non-orthogonal basic set for representation of distorted images was obtained from CK set by distortion operator. These two sets are used to compact the matrix representation of the distortion operator and its pseudo-inversion by singular decomposition procedure. We use the novel algorithm of training set decorrelation transformation for calculation of CK eigenvectors. In order to improve the restored image we use the iterative nonlinear procedure, which is based on influence function technique. Intermediate image is similar to primary distorted image, but with rough errors cleaned and omitted spots restored as well. A priori information about non-negative sign and limited variation of restored and intermediate images can easily be accounted. The proposed scheme of image processing enables restoration of large images with rough distortions in extremely high noise (up to 100%).
Alexandr G. Chumakov,
Vitalij N. Kurashov,
"Image restoration by nonlinear mapping onto compact set", Proc. SPIE 2424, Nonlinear Image Processing VI, (28 March 1995); doi: 10.1117/12.205243; https://doi.org/10.1117/12.205243