Regularity constraint

When started with a smooth image, iterative maximum likelihood algorithms can achieve some level of regularization by early stopping of the iterations before convergence (see e.g. Lanteri et al., 1999). In this case, the regularized solution is not the maximum fikelihood one and it also depends on the initial solution and the number of performed iterations. A better solution is to explicitly account for additional regularization constraints in the penalty criterion. This is explained in the next section. 

Because the noise usually contaminates the high frequencies, smoothness is a very common regularization constraint. Smoothness can be enforced if 0prior(x) is some measure of the roughness of the sought distribution x, for instance (in 1-D) ... 

There are additionally two other independent constraints on the solution that can be used to improve the stability of the process. One is that each fj be non-negative. The second regularization constraint is to require that for any real sample, the pore size distribution must be smooth. As a measure of smoothness we use the size of the second derivative of f(H) ... 

In addition to the traditional and nontraditional goal constraints described in the previous section, the tactical submodel requires several regular constraints that must be met for the overall solution to remain feasible. These constraints are formulated as follows. 

Secondly, the linearized inverse problem is, as well as known, ill-posed because it involves the solution of a Fredholm integral equation of the first kind. The solution must be regularized to yield a stable and physically plausible solution. In this apphcation, the classical smoothness constraint on the solution [8], does not allow to recover the discontinuities of the original object function. In our case, we have considered notches at the smface of the half-space conductive media. So, notche shapes involve abrupt contours. This strong local correlation between pixels in each layer of the half conductive media suggests to represent the contrast function (the object function) by a piecewise continuous function. According to previous works that we have aheady presented [14], we 2584... 

An implicit edge process is involved in the regularization process where A acts as a scale parameter which gives a constraint on the size of the homogeneous patches and p. comes from ho = -y/ p/A where ho is the threshold above which a discontinuity is introduced. We propose, then to combine these two functionals to obtain a satisfactory solution ... 

In addition to existing as helices in crystals, there is evidence that certain vinyl polymers also show some degree of regular alternation between trans and gauche conformations in solution. In solution, the chain is free from the sort of environmental constraints that operate in a crystal, so the length of the helical sequence in a dissolved isotactic vinyl polymer may be relatively short. 

These local stmctural rules make it impossible to constmct a regular, periodic, polyhedral foam from a single polyhedron. No known polyhedral shape that can be packed to fiU space simultaneously satisfies the intersection rules required of both the films and the borders. There is thus no ideal stmcture that can serve as a convenient mathematical idealization of polyhedral foam stmcture. Lord Kelvin considered this problem, and his minimal tetrakaidecahedron is considered the periodic stmcture of polyhedra that most nearly satisfies the mechanical constraints. 

To illustrate this theory, we consider a one-component fluid with the interaction between the same species given by Eq. (36). Obviously, the model differs from that described in Sec. (II Bl). In particular, the geometrical constraints, which determine the type of association products in the case of a two-component model, are no longer valid. If we restrict ourselves to the case L < cr/2, only dimers and -mers built up of rigid, regular polygons are possible. 

There are cases where non-regular lattices may be of advantage [36,37]. The computational effort, however, is substantially larger, which makes the models less flexible concerning changes of boundary conditions or topological constraints. Another direction, which may be promising in the future, is the use of hybrid models, where for example local attachment kinetics are treated on a microscopic atomistic scale, while the transport properties are treated by macroscopic partial differential equations [5,6]. 

The a priori penalty prior(x) oc — log Pr x allows us to account for additional constraints not carried out by the data alone (i.e. by the likelihood term). For instance, the prior can enforce agreement with some preferred (e.g. smoothness) and/or exact (e.g. non-negativity) properties of the solution. At least, the prior penalty is responsible of regularizing the inverse problem. This implies that the prior must provide information where the data alone fail to do so (in particular in regions where the noise dominates the signal or where data are missing). Not all prior constraints have such properties and the enforced a priori must be chosen with care. Taking into account additional a priori constraints has also some drawbacks it must be realized that the solution will be biased toward the prior. 

There exist many different kind of regularization which enforce different constraints or similar constraints but in a different way. 

Solving such a myopic deconvolution problem is much more difficult because its solution is highly non-linear with respect to the data. In effect, whatever are the expressions of the regularization terms, the criterion to minimize is no longer quadratic with respect to the parameters (due to the first likelihood term). Nevertheless, a much more important point to care of is that unless enough constraints are set by the regularization terms, the problem may not have a unique solution. 

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