Smoothness constraints

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... 

The constraint of finite extent applies to data that exist only over a finite interval and have zero values elsewhere. Let N1 and N2 denote the nonzero extent of the original undistorted data. The restored function u(k) + v(k), then, should have no deviations from zero outside the known extent of the data. To find the coefficients in v(k) that best satisfy this constraint, we should minimize the sum of the squared points outside the known extent of the object. Actually, recovering only a band of frequencies in v(k) implies the additional constraint of holding all higher frequencies above this band equal to zero. This is necessary for stability and is an example of one of the smoothing constraints discussed earlier. We minimize the expression... 

Another approach to dealing with the ill-conditioned nature of Laplace inversion is regularization, also known as parsimony. Regularization involves the imposition of additional constraints designed to favor some distributions over others, consistent with the measured data. For example, Tikhonov s regularization 49,65 adds a smoothing constraint to the least squares minimization, so that... 

The performance index J consists of two terms. The first, 7expt, is the data-fitting term, which measures the difference between experimentally observed and calculated values of the data. The second,. /re , is a regularization term and is used to impose any a priori knowledge, such as smoothness constraints, on the functional form of permeability distribution obtained as a solution. These terms take the following forms ... 

Constraining the inversion to a smooth solution as given by Phillips -Tikhonov - Twomey Eq. (19) has been proven to be very efficient in numerous applications, e.g. [1, 13, 35-40]. In contrast to Eqs. (20, 23-24) where the solution a is constrained to the actual values of a priori estimates a, Eq. (19) constrains only differences (derivatives) between elements of retrieved vector a and does not restrict their values. Therefore, smoothing constraints may be preferable in applications where a priori magnitudes of unknowns are uncertain. For example, a smooth behavior with no sharp oscillations can naturally be expected for atmospheric characteristic y x) such as the size distributions of aerosol concentrations. Correspondingly, filtering of the solutions with strong oscillations of a, = y(x,) (i=l, appears to be a... 

Studies [11-13] originated Eq. (19) did not imply any statistical meaning to the smoothness constrains. Later studies suggest some statistical interpretation to smoothing constraints. For example, studies [18, 32] considered the smoothness matrix as the inverse matrix to the covariance matrix of a priori... 

In retrievals of the function y(x,) in discrete points X , the expectations of limited derivatives of y(x) can be employed explicitly as smoothness constraints. Namely, if the retrieved function is expected to be close to a constant, straight line, parabola, etc., one can use zero m-th derivatives, as follows from Eq. (35), as a priori estimates g , = 0. Using this knowledge as a second source of information about U = y x,), the multi-source Eq. (25) can be written ... 

The strength of smoothness constraints in Eqs. (43-45) is linked with known values of the derivatives of the retrieved y(x). If an explicit analysis of derivatives y/ x is not feasible, the strength of smoothing can be implied from known least smooth of all a priori known y(x). For example, Eq. (45) can be replaced by the inequality [9] ... 

Problem Type Smooth nonlinear functions subject to smooth constraints Method Sequential quadratic programming Author Peter SpeUucci, Technical University Darmstadt, Germany Contact http //www.mathematik.tu-darmstadt.de/ags/ag8/spellucci/... 

In the context of stochastic parameter evolution (SPE) TARMA models, the parameters are assumed to follow stochastic smoothness constraints in the form of linear integrated autoregressive (lAR) models with integration order q (in this context referred to as smoothness-priors order). A smoothness-priors TARMA (SP-TARMA) model thus has parameters that obey the relations (Kitagawa and Gersch 1996)... 

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