Efficient and highly accurate reconstruction of discrete orientation distribution functions by integral approximation of relative probabilities

8/12/2019 Efficient and highly accurate reconstruction of discrete orientation distribution functions by integral approximation

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Efficient and highly accurate reconstruction of

discrete orientation distribution functions by

integral approximation of relativeprobabilities

Philip Eisenlohr Franz RotersMax-Planck-Institut fur Eisenforschung, Max-Planck-Str. 1, 40237 Dusseldorf,

Germany

Abstract

This paper is concerned with the reconstruction of texturesgiven as discrete, i.e.piecewise constant, orientation distribution functions (ODFs)by a limited num-ber, N, of equally weighted orientations. The N orientations are sampled fromthose Ncentral orientations for which the respective Euler space boxes overlapwith the fundamental zone of the discrete original ODF. Taking strong and inter-mediate experimental and random artificial textures as test-case we compare proba-bilistic and deterministic reconstruction schemes. The quality of ODF reconstructionis quantified by the root mean squared deviation as well as by the correlation factorof a linear regression between the probabilities in the original and reconstructed

ODF. The quality of both reconstruction schemes exhibits a scaling with N

/N.In terms of both quality measures, the deterministic sampling scores higher thanprobabilistic sampling for given N/N. However, if N Ndeterministic sam-pling progressively sharpens the reconstructed texture with decreasing N/N dueto systematic over-weighting of orientations with originally high probability at theexpense of low-probability ones. Therefore, a combined method is proposed, whichfor N < Ndraws a random subset ofN orientations from a population contain-ing Norientations, where this population is itself generated by prior deterministicsampling from the discrete ODF. The reconstruction quality achieved by this hybridmethod is naturally identical to deterministic sampling for N N and asymptot-ically declines with decreasing N to settle at the levels of probabilistic samplingfor N < N/10without systematic sharpening of the reconstructed texture.

Key words: ODF reconstruction, discrete lattice orientations, anisotropyPACS: 02.70.-c, 07.05.Tp, 62.20.Fe, 81.05.Bx

Corresponding author. Tel.: +49 211 6792983; fax: +49 211 6792333.Email address: [email protected] (Philip Eisenlohr).

Preprint submitted to Computational Materials Science 23 August 2007


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from a given ODF have been published. The proposed schemes can be dividedinto two methodologically different groups.

In the first group, the number of required orientations is prescribed and theirindividual, variable weights get selected. This can be achieved by construct-

ing a nearly equi-distant grid in Euler

space as suggested by Kocks et al.[7] and Helming et al. [8]. The weight of orientations on this grid is fittedby comparing recalculated to measured pole figures. Any weights below athreshold may be dropped to reduce the number of orientations considered.Another option, called the Limited Orientation Distance method, was in-troduced for discrete ODFs by Toth and Van Houtte [9]. They suggested aprocedure which iteratively starts at that seed orientation which exhibitsthe maximum ODF intensity. All intensity of neighboring orientations havinga disorientation to the seed below a threshold value, i.e. having limited ori-entation distance, is concentrated into the seeds weight and then subtractedfrom the ODF. Recently, a sophisticated method also employing a threshold

in disorientation was suggested by Melchior and Delannay [10]. Starting froma large ensemble of orientations chosen in a probabilistic manner from a givendiscrete ODF (see next paragraph for details), these orientations are groupedtogether to form grains which then get assigned the mean of their constituentorientations. The grouping respects that the maximum disorientation amongconstituent orientations in a single grain remains below a given threshold (ofa few degrees).

For the second group, each sample out of a prescribed number, N, has equalweight and the orientations required for a statistical representation of theODF are chosen. The straightforward probabilistic sampling would accept arandomly selected orientation gi with probability

p(gi) = f(gi) sin i

max(f(g) sin )gZ. (4)

Toth and Van Houtte [9] (see also [10]) elegantly increased the computa-tional efficiency of such sampling by introducing a cumulative probabilitydensity of orientations. Its inversion is then used as a mapping-function be-tween N uniformly distributed numbers in the range [0; 1] and orientations.Since an integration path connecting boxes in Euler space is essential totheir STAT method, it can only operate on discrete representations of an

ODF. The expected value for the resulting volume fraction, i , of orientationgi in both of these probabilistic sampling methods is

i f(gi)sin i sincep(gi) f(gi)sin i, i.e. they produce unbiased discretizations of the originalODFs [9] (see also Eq. (5)).

The present paper focuses on cases where the individual volume fractions ofgrains are equal and fixed, i.e. we are concerned with the second group ofmethods. The purpose of the paper is to simplify and scrutinize an idea orig-

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inally mentioned in the work of Leffers and Juul Jensen [11] (see also [12]).Our sampling scheme uses integral approximation of relative probabilities bysuitably scaling the probabilities in the discrete ODF representation. It isfully deterministic for large N and partly probabilistic for small N as willbe explained in Sections 2 and 5. The reconstruction quality of the integral

approximation and of probabilistic sampling will be quantitatively comparedin Section 4. In the limit of small numbers of sampled orientations, i.e. or-der 102, both methods give identically crude results. However, starting at thecomputationally interesting range of around 103104 sampled orientations theproposed method matches a given texture significantly closer than probabilis-tic sampling does.

2 Method

We want to draw a set ofN samples from a discrete ODF representation withthe aim that the texture resulting from distributing those sampled orientationsoverN grains of equal volume,V /N, reproduces the original polycrystallinetexture as closely as possible.

Euler space is discretized into unit boxes of dimension 1 2centered on a regular 1 grid. For a given crystal and/or sample symmetrythe fundamental zone, Z, of Euler space comprises only N out of the total43/(1 2) grid points. Let the discrete ODF, fi = f(gi) with i =1, 2, . . . , N , be defined at the center of each box.

The volume fraction i = dVi/Vof crystallites with orientations falling intotheith box follows from Eqs. (1) and (2) as

i = fi

82Z

dg122 sin(/2) sin i (5)

withi being the box centers value. As for the continuous ODF, the fi (seeEq. (3)) fulfill the normalization requirement such that

N

i=1

i

1. (6)

Now, based on its volume fraction, i, each orientation, gi, of the N onespresent in the fundamental zone of the discrete ODF is selected for a numberof times,

ni = round (C i) , (7)

1 grid regularity is no requirement but chosen for easier illustration

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to yield a set ofN samples. Without the required rounding to an integer value,settingC N would result in an unbiased discretization of the original ODF.However,Cneeds to be iteratively adjusted to fulfill

N

i=1

ni!

=N . (8)

This iterative procedure is easily solved by, for instance, a binary search algo-rithm. However, depending on the input ODF and the choice ofN there arecases where no exact solution is found due to limited numerical precision inC.

3 Analysis

The benchmark probabilistic sampling was implemented according to Eq. (4)as well as by using the more efficient STAT method of Toth and Van Houtte[9].

Three textures are used exemplarily in this study to assess the quality of theintegral approximation (IA) method outlined in Section 2: two real texturesand an artificial random texture withf(g) = 1 104. The real textures weremeasured by EBSD in a rolled duplex steel on a population of around 105

ferritic (-Fe) and austenitic (-Fe) grains, respectively. Both sets of grainorientations were transformed with OIM Analysis 2 into discrete ODFs of5

5

5 cubic degrees box size (Gaussian spread was set to 5 degrees). The

cubic lattice structure of both phases together with the monoclinic symmetrypresent on the samples longitudinal section allows to reduce the required partof Euler space to the region{[0; ], [0; ], [0; /2]}. To check for the influ-ence ofN, the measured crystal orientation data was additionally processedin OIM Analysis assuming orthotropic sample symmetry. This assumptionis not altering the experimental textures to a great extent, as can be seenby visual comparison of the respective ODFs shown in Fig. 1 and in a morequantitative fashion from the distributions ofi depicted in Fig. 2 which ex-hibit only small variations between the respective monoclinic and orthotropiccases. The orthotropic symmetry assumption further reduces the representa-

tive part of Euler

space by a factor of four to {[0; /2], [0; /2], [0; /2]}.The three-fold symmetry of cubic crystals is leveraged to finally restrict bothabove parts to their region III 3 by scaling allf(gi) with the fraction of overlapbetween region III and the volume of the ith box. Thus, both of the two mon-oclinic and two orthotropic textures are characterized by 363618 = 233282 Version 4.6 by EDAX, Inc., Mahwah, New Jersey, USA.3 this fundamental region ofEulerspace for cubic crystal symmetry is defined by|cos | min (1 + cos2 2)0.5 cos 2 , (1 + sin2 2)0.5 sin 2

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Fig. 1. Per phase (/-Fe) textures of rolled duplex steel which are used in this study.From left to right: -monoclinic, -orthotropic, -orthotropic, and -monoclinic.Range of,2 and step size in 1 is [0; ], [0; /2] and 15 degrees in case of mono-clinic and [0; /2], [0; /2] and 5 degrees for the case of (enforced) orthotropic samplesymmetry, respectively. 1 = 0 in upper left corner. Note the different maximum

intensities in the ODFs of and.

and 18 18 18 = 5832 discrete fi, respectively, of which only N = 6048and 1512 values are non-zero, i.e. lie in region III. These four ODFs are re-ferred to as -monoclinic, -monoclinic, -orthotropic, and -orthotropic inthe following (see Fig. 1). In the case of the artificial random texture threecombinations of angular resolution and Euler(sub)space were set up: 5 de-grees on{[0;2], [0; ], [0;2]} and{[0; /2], [0; /2], [0; /2]}, and 3 degreeson{[0; /2], [0; /2], [0; /2]}, which results in N = 72 36 72 = 186624,181818 = 5832, and 303030 = 27000 discretefi. In these cases, three-fold cubic symmetry is not used to further shrink the considered fundamentalzone as was done for the experimental textures.

In order to quantify the similarity between the original ODF and its recon-structed versions, two means are used in this study. The first is the root meansquared deviation (RMSD), which in the case of discrete ODFs given on N

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0.0

0.2

0.4

0.6

0.8

1.0

0.1 1 10

F

Fig. 2. Cummulative frequency F of normalized volume fractions i for the fourODFs depicted in Fig. 1.

grid points in Eulerspace can be defined as:

RMSD =

Ni=1

(i i )2 (9)

withi and

i (=ni/N) being the volume fractions corresponding to orien-

tationgiin the original and reconstructed ODF, respectively. Secondly, we usethe slope,a, and the correlation factor,fC, of a linear regression performed on

the set of points (i,

i ) which was introduced by Tarasiuk and Wierzbanowski[13]:

a=N

Ni=1

i

iNi=1

iNi=1

i

NNi=1

ii Ni=1

iNi=1

i

(10)

fC=

Ni=1

i

i /N

i

i

i2

i2

i2

i2 (11)

where x=Ni=1 xi/Ndenotes the arithmetic mean ofx.

4 Result

This section compares the IA method to probabilistic sampling with respect tothe reconstruction quality achieved on strong, intermediate and random tex-tures. As expected, both ways of probabilistic sampling, i.e.based on Eq. (4)

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10-4

10-2

100

102

10-1

100

N*/N

STAT

IA

hybrid

IA

Fig. 3. Normalized RMSD (see Eq. (9)) as function of normalized number of sampledorientations for all seven (four experimental plus three random) investigated ODFs.Crosses, open symbols, and filled symbols indicate results of STAT, IA, and hybridIA reconstruction method, respectively. Note the strong overlap of the seven resultsfor each individual method corresponding to respective master curves.

as well as the STAT method, yielded identical results. Therefore, only thoseof the more efficient STAT method are used as benchmark for comparisons.

4.1 Root mean squared deviation

Figure 3 presents in summary the RMSD between the original and recon-structed versions of the experimental and random textures. Despite the largedifferences among all input ODFs, each result of the STAT as well as of theIA method falls, to a very good approximation, onto two individual mastercurves (all crosses and open symbols in Fig. 3) when normalizing the numberof sampled orientations,N, by the number,N, of non zero-valued boxes, i.e.boxes within the fundamental zone, in each of the discrete input ODFs. (De-

tails regarding the hybrid IA method additionally shown in Figs. 36 will begiven in Section 5.) Respecting the normalization of the ordinate by

N, it is

seen that for fully probabilistic sampling the RMSD is inversely proportionalto

N, which is in accordance with a Poisson process. In contrast, the IAmethod exhibits similar characteristic only in the limit of small (normalized)N/N < 102. For larger sampling numbers N/N > 1 the RMSD falls inproportion to N, which is remarkably more rapid than in the case of theSTAT method.

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0.0

0.5

1.0

1.5

2.0

0 0.5

0.0

0.5

1.0

1.5

2.0

0 0.5 0 0.5 1

0.0

0.5

1.0

1.5

2.0

a

1/2 2N*/N = 1/8

IA

STAT

i/ %

* i/%

hybrid IA

Fig. 4. Correlation between reconstructed i and i from -orthotropic ODF forthree values ofN (N= 1512). Methods used in top, central and bottom row areIA, hybrid IA and STAT, respectively. Thick lines of slope a follow from linearregression (see Eq. (10)). Dashed lines indicate ideal correlation i =i (a= 1).

4.2 Correlation

The second measure of quality to compare original to reconstructed ODFsevaluates the correlation between the volume fractions i and i through alinear regression. This correlation would be perfect if i = i for all i =1, 2, . . . , N , i.e. agreement for all boxes within the fundamental zone of thediscrete ODF. To illustrate the correlation resulting from both discretizationmethods we first ploti againsti for three different N

/Nin Fig. 4 (takingthe -orthotropic texture as example). The correlation resulting from the IAmethod is presented in the top row and that from the STAT method in the

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bottom row of Fig. 4.

The IA method always produces a step-function i (i) due to the scaling andinteger rounding operation involved in Eq. (7). With respect to the slope, a,of the linear regression this leads to a systematic over-weighting i > i, or

a > 1, the larger i (see steep solid line in top row of Fig. 4). As seen fromthe sequence in Fig. 4 (top, left to right), this over-weighting diminishes withincreasingN/Nand the regression slope asymptotically settles at a= 1. Incontrast, the slope in the case of the STAT method is close to one for all threecases visualized. (Note that the total number of data points shown in each ofthe nine frames of Fig. 4 far exceeds the visual impression, hence apparentand calculated linear regression may differ.)

The scatter of points around the linear regression line decreases with increasingN/N for both methods. However, regarding the correlation factor, fC, theIA method is advantageous compared to the STAT method for all N/N

illustrated in Fig. 4 since a step-function shows only limited scatter. The resultof the IA method atN/N= 2 can be regarded as a perfect match in terms ofboth measures,a and fC, while the cloud of points resulting from the STATmethod then still exhibits significant scatter around the ideal correlation.

The dependence ofaandfConN/Nis illustrated in Figs. 5 and 6 over a large

range of normalized sample numbers and for all seven investigated textures.Each data point represents the average from 10 individual reconstructions.

It can be seen from Fig. 5 (bottom) that, irrespective ofN/N, the correlationresulting from the STAT method ison averageideal for all textures investi-

gated. This is a consequence of the fact, that the STAT method is an unbiasedestimator of the original ODF [9]. However, the standard deviation (indicatedby error bars in Fig. 5) increases markedly with decreasing N/N 1. The maximumvalue thataattains for small sample numbers increases in the order of texturesof random, , and , i.e. correlates with the fraction of large i present inthe respective ODFs (see Fig. 2). With increasing N/N the correlation slopedecreases in a rather similar fashion for all textures and the correlation between

i and i finally turns ideal for N/N >1.

The correlation factor, i.e.the degree of linearity betweeni andi, is plottedin Fig. 6 for the measured textures (left and center) and the random textures(right). Again, the normalization,N/N, of the sample number condenses sim-ilar ODFs onto individual master curves. With increasing normalized samplenumber, the correlation factor increases from values close to zero to its maxi-mum value of one. In all cases, this increase offCinitially follows in proportion

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0

2

4

6

8

0

2

4

10-4

10-2

100

102

0

2

4

random

IA

hybrid IA

a

STAT

N*/N

Fig. 5. Slope a resulting from linear regression (see Eq. (10)) as function of nor-malized sample number. The orthotropic and monoclinic sample symmetry for themeasured textures ( and ) are indicated by circles and squares, respectively.

to

N (compare with dashed line in Fig. 6).

Despite the normlization, for a fixed N/N the resulting fC differ slightlyamong the , , and random textures and are always larger for those ODFswith a larger fraction of largei,i.e.better for sharper textures. The sharpness

of texture also determines the factor in N

/Nby which the results from theIA method surpass those of the STAT method. In the present case, this factorranges from about 10 for the random texture to 100 for the rather sharp texture, see right and left of Fig. 6, respectively.

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10-4

10-2

100

102

0.1

0.2

0.4

0.6

0.8

1

10-4

10-2

100

102

10-4

10-2

100

102

IA

fC

hybrid IA

STAT STAT

hybrid IA

IA

N*/N

STAT

IA

rand

om

Fig. 6. Correlation factor fC resulting from linear regression (see Eq. (11)) as func-tion of normalized sample number. Results on both (left), both (center), andthe three random (right) textures are shifted horizontally against each other by aconstant factor, i.e. the vertical dotted line always corresponds to N/N= 1. Theorthotropic and monoclinic sample symmetry for the measured textures ( and )are indicated by circles and squares, respectively.

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5 Discussion

We demonstrated in the previous section that the fully deterministic dis-cretization of ODFs by means of the IA method shows advantages over prob-

abilistic sampling but also has shortcomings, the most significant being thesystematic over-weighting of large i for small sample numbers, which resultsin a pronounced sharpening of the reconstructed texture. A discriminationline regarding the reconstruction quality can be drawn at normalized samplenumbers N/N 1. IfN/N >1 the reconstructed ODF resulting from theIA method is always superior to the respective result of probabilistic sampling.In the caseN/N


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Fig. 7. ODF of -monoclinic texture (framed at top center, N = 6048, see alsoFig. 1 right) compared to its reconstructed versions from the STAT and hybrid IAmethod (left and right) at N/N= 1 and 1/4 (top and bottom).

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ally being identical to the latter forN N. This notion is readily confirmedin Fig. 3 where the transient between RMSD 1/N and RMSD 1/Nturns slightly sharper for the hybrid IA method. Also, in Fig. 6 the smoothsaturation of fC is virtually absent but fC = 1 is rapidly achieved by thehybrid IA method. Looking at these findings from a reverse angle, it becomes

clear that some quality improvements (i.e. higher fC and lower RMSD) indeterministic over probabilistic sampling are readily lost if more samples aregenerated than actually aimed for,

Ni=1 ni > N

, since that would require tochose a (random) subset from them. Figures 3 and 6 suggest that the (hybrid)IA method becomes equal to the STAT method if only less than 10% of thedeterministically generated orientations enter into the ODF reconstruction.Therefore, adjustment of the scaling factor Cin Eq. (7) should be done to ahigh accuracy to avoid significant deviations from the intended number, N,of orientations.

6 Conclusion

This work compared the quality of reconstructing a given texture by a setof discrete orientations. The two methods tested were probabilistic samplingemploying the efficient STAT method proposed by Toth and Van Houtte [9] aswell as the deterministic integral approximation (IA) method originally men-tioned by Leffers and Juul Jensen [11]. In terms of the root mean squared de-viation (RMSD, see Eq. (9)) the IA method yields equal or better reconstruc-

tions of the ODF than the STAT method. When employing the correlationfactor introduced by Tarasiuk and Wierzbanowski [13] as quality measure theIA method even performs far better compared to the STAT method. However,the IA method has an inherent shortcoming since at small sample numbersit systematically over-weights those orientations which already have a highprobability in the original texture on the expense of those of low probability.This leads to a significant sharpening of the reconstructed ODF compared tothe original one. As this problem is absent in probabilistic sampling owingto its reconstructions being unbiased estimators of the original texture [9],a combination of both methods was proposed (hybrid IA) by altering theIA method: at small sample numbers, a certain degree of probabilistic sam-

pling is reintroduced by selecting a subset of discrete orientations from a fixedpopulation which is itself determined by invoking the IA method. This newhybrid IA method blends the advantages of both former methods. Its recon-struction quality of all strong, intermediate, and random textures investigatedin this paper is equal to deterministic sampling, i.e.far better than probabilis-tic sampling, for N N. At small N/Nthe method does not exhibit thedetrimental texture sharpening inherent to the IA method, thus settles at thequality of probabilistic sampling forN < N/10.

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Efficient and highly accurate reconstruction of discrete orientation distribution functions by integral approximation of relative probabilities

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