Finite extent

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 

The standard procedure for finding the minimum of a function of several variables is to take the partial derivative with respect to each variable and set the result equal to zero. Taking the derivative with respect to each unknown coefficient and setting the result equal to zero gives (summation interval suppressed) 

Considering in detail the derivative with respect to Ab, we have 

The other derivatives are calculated in a similar manner, and we find for the complete set of equations 

Because of aliasing, the total number of coefficients obtained should not be greater than N. We have a set of 2c linear equations for the 2c unknown coefficients. A number of standard methods are available for solving a set of linear equations. We used the Gauss-Jordan matrix reduction method. 

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In Fig. 8.18, we illustrate this just sufficient distribution in comparison to a hypothetical flaw distribution for an actual material. In this example, we envision a solid of finite extent, which will have a single critical flaw that activates at a minimum stress tensile stress, the population of flaws which activate should increase rapidly, perhaps as illustrated in Fig. 8.18. In contrast, a flaw distribution just sufficient to satisfy the energy balance criterion increases smoothly as JV [Pg.294]

When the potential barrier in case (ii) above is of finite extent and separates two regions at lower potential the possibility arises that a particle may tunnel through the barrier. 

Emission from shock-heated S02 arises from three electronically excited states 3B1, 1B2, 1B1, populated by collisions from the 1A1 ground state and, for finite extents of dissociation, by the recombination of O and SO360. Equilibrium between the emitting states and the ground state... 

Two further comments about formula 8 and its parameters are noteworthy. The number of terms with coefficients l i in the double sum is formally doubly infinite so that formula 8 can represent spectral terms involving arbitrarily large values of V and 7 with data of finite extent the sums become truncated in a systematic and consistent manner conforming to parameters of a minimum number required according to that intermediate model. Instead of exact equalities in formulae 10 and 11, the approximations arise because each term coefficient constitutes a sum of contributions [5],... 

So far, we considered ensembles of indefinitely long steps. A different situation is encountered for surface perturbations of finite extent along both directions of the sur-... 

A point group is the symmetry group of art object of finite extent, such as an atom or molecule. (Infinite lattices, occruring in the theory of crystalline solids, have translational symmetry in addition.) Specifying the point group to which a molecule belongs defines its symmetry completely. 

Dr. Wollaston was a man of very broad interests, as a list of his publications will show. His papers were on such diverse subjects as force of percussion, fairy rings, gout, diabetes, seasickness, metallic titanium, the identity of columbium (niobium) and tantalum, a reflection goniometer, micrometers, barometers, a scale of chemical equivalents, and the finite extent of the atmosphere. He died in London on December 22,1828 (13). 

NOLR Handbook 1111(Ref 30, p 7-15) defined the detonation wave as an intense shock or compressive wave of forward moving material that is supported by the very rapid exothermic decomposition of the explosive immediately behind the shock front. The pressure profile of a detonation wave occurring in a charge of finite extent has the appearance shown in Fig 1. 

The constraint of finite extent has in practice proved to be of only limited value for improving the quality of restorations, whether the process is one of analytic continuation or otherwise. This is borne out, for example, by the work of Howard (Chapter 9). We have now presented the idea of recovering frequencies beyond the cutoff Q, however. We should not find it too surprising if other types of information bring with them the ability to restore these presumably lost frequencies. 

The solutions illustrated in Schell s original publication were indeed entirely positive and showed some resolution improvement over the inverse-filter estimates. The improvement in these examples was not, however, as great as we have come to expect from the best of the newer methods and may not in fact demonstrate the method s real potential. The method does bring with it in a very explicit way, however, the idea that the Fourier spectrum may be extended, on the basis of a knowledge of positivity. Previous studies had focused on the finite extent constraint to achieve this objective. 

The method employs a gradual increase in frequency beyond the data band limit. High-frequency components are not sought until the best values of low-frequency components are found. Because frequencies are not sought above the lowest needed to satisfy the data, the method is inherently smooth. Furthermore, Biraud s method appears to be the first to have simultaneously utilized both the constraint of positivity and that of finite extent with specific limits, the latter being inherent in the sampling. These facts are probably responsible for the impressiveness of the restoration in the original publication (Biraud, 1969), which is reproduced in Fig. 4. 

The principal thrust of the present chapter is to describe methods that facilitate superior restoration through the use of bounds. Sometimes bounds are implemented in the object domain, for example, positivity or finite extent, and sometimes in the Fourier domain, for example, band limitedness. The effectiveness of this type of prior knowledge was apparent from the early work with constraints. A number of researchers therefore focused their efforts on making use of all possible combinations of constraints and partial data. What could be more suggestive than direct application of bounds to trial solutions by truncation, first in the object domain and then in the Fourier domain ... 

Solutions for objects of finite extent are obtained through the familiar minimum mean-square-error criterion by selecting the Fourier components that minimize... 

Although restorations by this method are superior to those obtained by inverse filtering alone, it should be no surprise that they do not show the dramatic improvements obtainable through the positivity constraint. When, in fact, the integration limits of Eq. (44) are taken well outside the true extent of the object, very little improvement is noted. The applied finite-extent bound must literally butt against the true object for greatest effect. How, then, can the more powerful and useful constraint of positivity be incorporated ... 

Note that other constraints may be added. Finite extent is particularly easy. Combining the two ideas in this section is done merely by using the finite-extent limits with the integral in Eq. (45). 

Zhou and Rushforth (1982) and Rushforth et al. (1982) have explored Howard s finite-extent-bounded method via matrix techniques. Zhou and Rushforth demonstrated the effectiveness of utilizing the maximum amount of prior knowledge. 

Many of the most effective constraints set well-defined limits to the data function (or its spectrum) beyond which the correct function is not allowed to go. An important example of this type of constraint is nonnegativity, whereby the correctly restored function is not allowed to extend below the zero baseline and thereby take on nonphysical negative values. This is an appropriate constraint for spectroscopy and optical images. A further example of the constraints of fixed limits is that of an upper bound to the values of the restoration. Another important constraint of this type is that of finite extent, for which no deviations from zero are allowed for the spatial function over those intervals on the spatial axis that lie outside the known... 

Fig. 4 Restoration by inverse filtering of the low-frequency band followed by spectral restoration of the high-frequency band with the constraint of finite extent, (a) Function produced by inverse filtering of the peak in Fig. 2(c) with six (complex) coefficients retained in the Fourier spectrum, (b) Improved function resulting from the restoration of 16 (complex) coefficients to the spectrum with the constraint of finite extent applied to the region indicated by the tick marks.

Fig. 5 Restoration of the inverse-filtered result shown in Fig. 4(a) with the required summation for the constraint of finite extent taken over the interval indicated by the tick marks, (a) Restoration of 16 (complex) coefficients to the inverse-filtered estimate, (b) Restored function produced by restoring only three (complex) coefficients to the inverse-filtered estimate.

Fig. 6 Restoration of a Fourier spectrum of inverse-filtered noisy infrared peaks with the constraint of finite extent, (a) Two merged infrared peaks, (b) Inverse-filtered infrared peaks with the spectrum truncated after the 10th (complex) coefficient, (c) Spectrum restored by applying the constraint outside the marked region. Five (complex) coefficients were restored, (d) Spectrum restored with the constrained region including the first negative sidelobes and the dip between the peaks as well as all other regions outside the peaks. Sixteen (complex) coefficients were recovered.

Although this procedure was developed from the constraint of minimum negativity, it will easily accommodate other constraints also, with only slight modification. Note that if the summation is not over a different set of data points for each iteration, but over a fixed set of points, the summation need be computed only once, because u(k) and the sinusoids are constant for each value of the variable k. This is true for the finite-extent constraint, in which... 

Fig. 1(b). This is slightly better than the result obtained with the constraint of finite extent in which the summation interval included the first negative sidelobes around the peak, which is shown in Fig. 4(b) of Chapter 9. It is vastly better, however, than the result obtained with the summation interval taken farther away from the peak as shown in Fig. 5 of Chapter 9. 

We have found it to be generally true that the constraint of minimum negativity produces results much superior to those obtained with the constraint of finite extent. The minimum-negativity procedure has also been found to be extraordinarily insensitive to noise and other error. This is contrasted with the equations resulting from the constraint of finite extent, for which usually only a narrow band of coefficients may be permitted restoration to achieve a stable solution. The best overall results, however, are obtained with a combination of the two constraints. 

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