Restoration of images

S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans Pattern Anal Machine Intelligence 6 721-741 (1984). 

The development of specialist software for the analysis of electron micrographs has equipped researchers with a variety of computational tools to analyse different types of sample. These methods are all based on the premise that a micrograph is a simple projection of the object and therefore have much in common. The main steps include (i) pre-processing of images, (ii) restoration of images, (iii) enhancement of images, (iv) determination of orientations and (v) reconstmction of the three-dimensional distribution of density. The result obtained must then be validated and interpreted. 

Sadot, D., Rosenfeld, R., Shuker, G., and Kopeika, N. S. (1995). High resolution restoration of images distorted by the atmosphere, based upon average atmospheric MTF. Opt. Eng. 34, 1799-1807. 

Iterative Identification and Restoration of Images, R. L.Lagendijk, X Biemond ISBN 0-7923-9097-0... 

The classical computer tomography (CT), including the medical one, has already been demonstrated its efficiency in many practical applications. At the same time, the request of the all-round survey of the object, which is usually unattainable, makes it important to find alternative approaches with less rigid restrictions to the number of projections and accessible views for observation. In the last time, it was understood that one effective way to withstand the extreme lack of data is to introduce a priori knowledge based upon classical inverse theory (including Maximum Entropy Method (MEM)) of the solution of ill-posed problems [1-6]. As shown in [6] for objects with binary structure, the necessary number of projections to get the quality of image restoration compared to that of CT using multistep reconstruction (MSR) method did not exceed seven and eould be reduced even further. 

The main point of our elaboration is, that the Gibbs measure (4) of the potential lattice under interest ctin be considered as a nontrivial prior in the Bayes formula for the conditional probability, applied to the problem of image restoration ... 

To investigate the influence of the noise on the quality of image restoration a specific procedure was implemented, which included ... 

In this figure the next definitions are used A - projection operator, B - pseudo-inverse operator for the image parameters a,( ), C - empirical posterior restoration of the FDD function w(a, ), E - optimal estimator. The projection operator A is non-observable due to the Kalman criteria [10] which is the main singularity for this problem. This leads to use the two step estimation procedure. First, the pseudo-inverse operator B has to be found among the regularization techniques in the class of linear filters. In the second step the optimal estimation d (n) for the pseudo-inverse image parameters d,(n) has to be done in the presence of transformed noise j(n). 

The adaptive estimation of the pseudo-inverse parameters a n) consists of the blocks C and E (Fig. 1) if the transformed noise ( ) has unknown properties. Bloek C performes the restoration of the posterior PDD function w a,n) from the data a (n) + (n). It includes methods and algorithms for the PDD function restoration from empirical data [8] which are based on empirical averaging. Beeause the noise is assumed to be a stationary process with zero mean value and the image parameters are constant, the PDD function w(a,n) converges, at least, to the real distribution. The posterior PDD funetion is used to built a back loop to block B and as a direct input for the estimator E. For the given estimation criteria f(a,d) an optimal estimation a (n) can be found from the expression... 

For restoring of three-dimensional SD is used stated above approach. Under restoring of tomographic images for the base undertakes a function of three-dimensional total image, which after double differentiation and inverse projecting describes sought SD 8 (1) ... 

Approach to restoring of stresses SD in the three-dimensional event requires for each pixel determinations of matrix with six independent elements. Type of matrixes depends on chosen coordinate systems. It is arised a question, how to present such result for operator that he shall be able to value stresses and their SD. One of the possible ways is a calculation and a presenting in the form of image of SD of stresses tensor invariants. For three-dimensional SDS relative increase of time of spreading of US waves, polarized in directions of main axises of stresses tensor ... 

Koshovy V. V. Methods of restoring of the acoustical images and their applications to nondestructive testing in civil engineering // Proc. Int. Symp. Non-Destructive Testing in Civil Engineering - Berlin, Sept. 26-28, 1995. - V.2. -P. 1153-1156. 

Wilson D L, Kump K S, Eppell S J and Marchant R E 1995 Morphological restoration of atomic force microscopy images Langmuir 265... 

We have seen how to properly solve for the inverse problem of image de-convolution. But all the problems and solutions discussed in this course are not specific to image restoration and apply for other problems. 

Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1992, Numerical Recipes in C, Cambridge University Press, 2nd edition Richardson, W.H., 1972, Bayesian-based iterative method of image restoration, JOSA 62, 55... 

If the ends of the rows wrap around so that the leftmost element of each row is the rightmost element of the row above, and so on, the matrix is called circulant. The foregoing properties have significance when noniterative solutions to the matrix-posed problem are sought by discrete Fourier transform. Andrews and Hunt (1977) provide numerous details and references, especially for the case in which these linear methods are applied to the restoration of two-dimensional images. 

The restoration for low z suffers a bit more overshoot at the gradient edges. This is consistent with a photon-bunching phenomenon that occurs as the solution to estimator (34). Notice that our estimator (61) includes Eq. (34). If Eq. (34) per se is solved, the answer is that all N photons jam into the resolution cell xm that has the largest degeneracy zm over m = 1,..., M. The other cells are empty, and the estimated spectrum is a spike. The presence of image data, and of noise, will compromise this extreme behavior. However, there is still a tendency to bunch photons in some cells and deplete them from others, as seen in Fig. 6(a-c). 

The benchmark comparison with these results would be the ordinary least-squares restoration of the given image. This was carried through. However, the results cannot be plotted because, owing to the noise level in the data, the output was highly oscillatory, varying in size by 106, and showing no resemblance to the true object. 

Wilson, D.L., K.S. Kump, S.J. Eppell, and R.E. Marchant. 1995. Morphological restoration of atomic-force microscopy images. Langmuir 11 265-272. 

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