Restored image

By imposing the prior information on the solution one implies, that this solution is suited to the only class of problems which corresponds to the information that is involved. But it should be noticed that these classes can be so wide that real constraints to the restored image are reduced to minimum. 

Fig. 1. Two spatial views of the restored image of the object using 4 projections and CSA...

Fig. 3. One simulated projection of the simulated object with different level of noise (left column) a) 0.0, b) 0.05, c) 0.1, d) 0.15, and the corresponding restored images (right column) using 5 projections.

Figure 3a. Restored image using a simple cutoff frequency (Wcutofi = 80) in the deconvolution.

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 ... 

In the Maximum Entropy Method (MEM) which proceeds the maximization of the conditional probability P(fl p ) (6) yielding the most probable solution, the probability P(p) introducing the a priory knowledge is issued from so called ergodic situations in many applications for image restoration [1]. That means, that the a priori probabilities of all microscopic configurations p are all the same. It yields to the well known form of the functional 5(/2 ) [9] ... 

The importance of distinct a priori knowledge account becomes more perceptible if noisy data are under restoration. The noise / ( shifts the solution of (1) from the Maximum Likelihood (ML) to the so called Default Model for which the function of the image constraint becomes more significant. 

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

To estimate the quality of restoration having both, simulated and reconstructed images, using some kind of quality criterion For this last purpose the following mean square measure was applied ... 

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... 

In sections, where exists high probability of presence of defects, on the base formed in the binary type of projections acoustical tomographic images of only defective structure elements of sections is restored. IT of restoring stipulates such operations ... 

Note some particularities of new USCT method. At first, data collection and search of areas with anomalous (inhomogeneous)SD of acoustic parameters (velocities of spreading of US waves) is joined. As a sought image, on which anomalies is revealed, it is offered total image B (r), which practically is the low frequency copy of restored fimction g(f). As PMF SD of... 

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. 

Fig. 1. Experimental restored acoustical tomographic images of defects in product sections (a-dififerent density 5p 10%, b- porosity x,y- in mm).

The image without treatment (Fig lb) should be compared with the deconvolved image of the rod (Fig. 2b). The dimension of the rod is well restored and the contour is reinforced by the signal processing. 

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

Materials are being developed to exhibit both photochromic and photographic behaviors one such system is based on a substituted indolinospiro-hen opyrene embedded in a polystyrene matrix (26). This system acts as a photochromic system at low exposure in the uv range and at high exposure it functions as a photographic system. The image can be devisualized by heat and can be restored many times with uv irradiation. 

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