OPTIMIZATION OF TEXTURE MEASUREMENTS . II . FURTHER APPLICATIONS : OPTIMAL SMOOTHING

In our previous paper (Luzin, 1997. Proc. of Workshop "Neutron Textures and Stress Analysis’) the basic principles of the quantitative approach to optimize the texture measurements were outlined. This paper is the report ofadvances in this direction. The quantitative approach is used to solve the smoothing problem. Smoothing by singular integrals with an integral kernel used by Nikolayev and Ullemeyer (1996). Proc. of Workshop "Math. Methods of Texture Analysis", Textures and Microstructures 25, 149158 is used in this paper. It is shown how the optimal smoothing parameter depends on the grain statistics, i.e. the number of grains in the sample. The algorithm for optimal smoothing of real pole density data (pole figures) is proposed. Also, the application of optimal smoothing for solving the central problem of quantitative texture analysis (QTA), i.e. orientation distribution function (ODF) reproduction, is discussed.


INTRODUCTION
The problem of the optimal texture experiment was formulated by the author in his other article (Luzin, 1997).It was shown that the optimal texture experiment can be conducted if the texture is known.When the texture cannot be initially estimated the standard or an overabundant measurement grid is used to prevent loss of information.
How should the data from this experiment be processed?The one possible answer is to smooth the directional data.Some successful attempts have been made to apply this procedure for processing the 357 probability density functions on the sphere (Traas et al., 1993; Schaeben,  1996; Nikolayev et at., 1996).In this paper the smoothing procedure in the form used by Nikolayev et al. (1996) is applied: wj exp wj arccos(,)Tk), l"-h Yk) = Wj   (1) where w is the smoothing parameter.
Usually raw pole figures (PFs) contain statistical noise.High degree of smoothing however leads to loss of information.Only optimal smoothing provides proper smoothing when statistical noise is elimi- nated and, at the same time, oversmoothing is avoided.In this paper, the main attention is paid to the fact that the optimal smoothing parameter (degree of smoothing) depends on the size of the investigated sample (or the number ofgrains) in the texture experiment and the sharpness of the texture.Both facts are built into the consideration in the same way as it was done in (Luzin, 1997).
As a result the solution of the optimal smoothing problem is directly connected with the solution of the optimal measurement problem when the texture is known (Section 2).Optimal smoothing can also be carded out in a self-contained way even if the texture is not initially known (Section 3).

THE OPTIMAL SMOOTHING PROCEDURE
Let us take a sampling of size N from the sample multitude of orienta- tions described by some true ODF ft(g) and the corresponding PFs P, (y-').The actual distributions can be written as (3) n=l They yield the observed (experimental) distributions determined on a certain measurement grid F {-), j 1,..., J: pe / ps hi (.J" N)  hi (Y, N)K(..]j-)dco(.), )7, )-E S2, (4) where K(fi, fin) is the integral kernel which reflects the conditions of the texture experiment.
The recipe for chosing the grid parameter for the given N in the optimal way was already reported (Luzin, 1997).Next, the following problem is of particular interest.Let us assume the experimental PFs P,(., N) are measured for the given number of grains N and the fixed measurement grid F {)-).Can one improve the obtained data (in the sense of RP-value) by the smoothing procedure?In this article the pro- blem is investigated directly by plotting quantitative dependencies of RP-value on the variables of interest.The simplest texture model of the Gaussian distribution with the center at g {0,0,0) and HWHM 19.7 is used for further calculations.
The most informative is the behavior of the RP-value on the smoothing parameter o and the equiangular grid parameter Ao when the number of grains N is fixed for the given texture.This dependence is presented as the surface and sections ofthis surface in Fig. 1.It should be emphasized that the minimum RP-value achieved by smoothing is pre- cisely the limit achieved in the optimal experiment.So the use ofoptimal smoothing on the fixed grid leads to the same result as the optimal

FIGURE
The dependence of RP-value on the smoothing parameter and the grid parameter plotted as a surface (right) and as its sections (left).The dependencies of the RP-value before pole figures smoothing (dashed circles) and the minimum achived RP-value after optimal smoothing (clear circles) on the grid parameter of the net used.As an illustration, the pairs unsmoothed- smoothed pole figures (100) are plotted.experiment with optimal grid.This is actual only for grid parameters less than the optimal one.For grid parameters greater than the optimal one the smoothing procedure cannot decrease the resultant RP-value.
Different values of N produce surfaces analogous to the surface in Fig. 1 with the following features.The greater the number N the lesser is the optimal smoothing parameter tMop and the minimal achieved RP-value RPmin RP(wopt).
From the multitude ofthe above-mentioned surfaces (scanning by N) information about the dependence of )opt and RPmin Re(Jopt) on N can be extracted.It turns out that these quantities have a very expressed behavior shown in Fig. 3 for the grid parameters A 5 , 15 , 30 and 0" ', U.P ted -0.9 2.0 2.6 3.0 3.5 4.0 4.5 5.0 LoglO(Number of grains) FIGURE 3 The dependencies of the optimal smoothing parameter (left) and mini- mum RP-value (right) on the number of grains in the sample.(Dashed points are unsmoothed data.) the grain numbers in the range N 100-50 000.Due to the fact that the optimal smoothing parameter ")opt and the minimal achieved RP-value, RPmin RP(")opt), are directly connected with the optimal parameters of the optimal grid problem, in the range of the smallest values the curves ")opt--")opt(N) and RPmin RPmin(N) coincide and are independent of the grid parameter Ao.This branch appears as line in double logarithmic where the coefficients A and A2 depend only on the sharpness oftexture and numerically determined constants p and q are p 0.3, q 0.17.
The N-range where the linear law holds is determined by the grid parameter.As N is getting larger and larger and statistical errors decrease, the dependence on N deviates from the linear behavior and tends to some limit.This limit corresponds to approximation errors and is specific for the chosen grid.For the given grid parameter Ao the dependence on N begins to deflect when N achieves the value Nop for which the given grid parameter Ago is close to the optimal one.Then in the limit N > Nopt, the optimal smoothing parameter ,)opt and the minimal RP-value RPrnin achieve their lowest level so and quality of PFs is scarcely affected by the optimal smoothing procedure.

ARE THE OPTIMAL SMOOTHED POLE FIGURES
OPTIMAL FOR THE ODF REPRODUCTION AS WELL?
The question placed in the title of this section is of prime interest for anybody working in the field of QTA.In the frame of the outlined approach the influence of smoothing on the ODF reproduction can be elucidated.
Three sets of PFs are at hand after the ODF reproduction procedure: the exact (true), experimental and the reconstructed PFs.Comparison of these sets for various smoothing degrees gives us information about the goodness of the smoothing procedure for the purposes of ODF reproduction.The quantitative dependencies of all possible RP-values (RP(exp, true), RP(exp, calc) and RP(calc, true)) are shown in Fig. 4 for the above mentioned texture.The Bunge (series expansion up to L 22) method and the component method were used.The optimal smooth- ing parameter opt is described as the position of the RPmin(exp, true).From Fig. 4 the meaning of the optimal smoothing parameter follows.
In the component method has Oop has its minimum value when RP(exp, calc)= RP(calc, true).This means that the set of calculated PFs is at equal distances from the experimental PFs and true PFs sets and the RP-value is the measure of the distance.In the Bunge method, 03op coincides with the actual position of the RPmin(calc, true).These results show the advantages and validity of optimal smoothing.30 aO " "" --,P(EXP,CALC),Bungeme,od]" 4. SMOOTHING OF THE REAL EXPERIMENTAL DATA Relatively simple dependencies of the optimal smoothing parameter 03op and the minimum RP-value give us a hint as to how to apply the smoothing procedure to the real texture data.Since the optimal values cannot be evaluated without knowledge of the texture, i.e. before the experiment, the following alternative can be proposed.
Let the number of grains be known and the grid be fixed.The first approximation of the texture can be done by the component method of ODF reproduction.Then, the optimal values can be evaluated and the procedure of optimal smoothing can be performed.After that the smoothed PFs can be used for the second approximation of the texture and the next estimate of the optimal smoothing parameters.This itera- tive procedure provides the lowest level of statistical errors.

CONCLUSION
In this paper, smoothing of the pole density data defined on the sphere was investigated with respect to grain statistics (the number of grains in the sample).It turns out that for the minimum RP-value between experimental and true PFs exists when the smoothing parameter varies.This optimal smoothing parameter provides the optimal smoothing procedure and minimizes the statistical errors connected with grain statistics.The validity of optimal smoothing is confirmed for two ODF reproduction methods (Bung and component methods).
FIGURE 2The dependencies of the RP-value before pole figures smoothing (dashed circles) and the minimum achived RP-value after optimal smoothing (clear circles) on the grid parameter of the net used.As an illustration, the pairs unsmoothed- smoothed pole figures (100) are plotted.

FIGURE 4
FIGURE 4 The comparison of reconstructed PFs and experimental PFs with respect to the true ones.(N 500 grains, 5 x 5 measurement grid.)