Accurate reconstruction of EBSD datasets by a multimodal data approach using an evolutionary algorithm
AAccurate reconstruction of EBSD datasets by a multimodal data approach usingan evolutionary algorithm
Marie-Agathe Charpagne a, ∗ , Florian Strub b , Tresa M. Pollock a a Materials Department, University of California, Santa Barbara, CA 93106, USA b Univ. Lille, CNRS, Centrale Lille, Inria, UMR 9189 - CRIStAL, F-59000 Lille, France
Abstract
A new method has been developed for the correction of the distortions and / or enhanced phase di ff erentiation inElectron Backscatter Di ff raction (EBSD) data. Using a multi-modal data approach, the method uses segmented imagesof the phase of interest (laths, precipitates, voids, inclusions) on images gathered by backscattered or secondaryelectrons of the same area as the EBSD map. The proposed approach then search for the best transformation tocorrect their relative distortions and recombines the data in a new EBSD file. Speckles of the features of interest arefirst segmented in both the EBSD and image data modes. The speckle extracted from the EBSD data is then meshed,and the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is implemented to distort the mesh until thespeckles superimpose. The quality of the matching is quantified via a score that is linked to the number of overlappingpixels in the speckles. The locations of the points of the distorted mesh are compared to those of the initial positions tocreate pairs of matching points that are used to calculate the polynomial function that describes the distortion the best.This function is then applied to un-distort the EBSD data, and the phase information is inferred using the data of thesegmented speckle. Fast and versatile, this method does not require any human annotation and can be applied to largedatasets and wide areas. Besides, this method requires very few assumptions concerning the shape of the distortionfunction. It can be used for the single compensation of the distortions or combined with the phase di ff erentiation.The accuracy of this method is of the order of the pixel size. Some application examples in multiphase materials withfeature sizes down to 1 µ m are presented, including Ti-6Al-4V Titanium alloy, Rene 65 and additive manufacturedInconel 718 Nickel-base superalloys. Keywords:
EBSD, image segmentation, distortions, multi-modal data, CMA-ES
1. Introduction ff erentiation by EBSD Electron Backscattered Di ff raction (EBSD) is a powerful tool for gathering local crystallographic data in multi-phase materials. This technique enables a fast mapping of microstructures with a good angular resolution (typically ∗ corresponding author Email address: [email protected] (Marie-Agathe Charpagne)
Preprint submitted to Materials Characterization March 11, 2019 a r X i v : . [ phy s i c s . d a t a - a n ] M a r .5 ◦ ), based on the collection of di ff raction patterns and the analysis of Kikuchi bands on those patterns. Initially im-plemented in scanning electron microscopes (SEM), it has been adapted to transmission microscopy (TEM) under thename of Transmission Kikuchi Di ff raction (TKD) [1, 2]. In the past years, tridimensional EBSD (3D EBSD) by serialsectioning has also emerged as a new tool to reconstruct and analyze materials in three dimensions [3–8]. To indexthe collected patterns, most commercial software use algorithms based on Hough transformations to detect the char-acteristic bands on the Kikuchi patterns. This process has the advantage of speed but su ff ers from some limitations.Because of the di ff raction patterns they produce, some phases are intrinsically challenging to index with this method.This is the case for the finely distributed β phase in α − β Titanium alloys, for example. Another well-known limita-tion of this method happens in the presence of two phases of similar structures. This is the case of γ and γ (cid:48) phases inNickel and Cobalt-base superalloys. The ordered L γ (cid:48) phase produces extra very low-contrast bands compared tothe Face Centered Cubic (FCC) γ , that are undetectable by Hough transforms. In the case of phases producing similarKikuchi patterns but having significantly di ff erent chemistries, one can employ multi-modal data acquisition. Nowelland Wright [9] have proposed the simultaneous collection of Xray energy dispersive spectroscopy (XEDS) data, alongwith EBSD data. This method has been shown to enable phase di ff erentiation in complex alloys, and reveal uniquecrystallographic relationships between phases [10–12]. However, XEDS-EBSD requires a significant di ff erence in thechemistry of the phases of interest and a minimum number of counts on the XEDS spectra, which can lower the speedof data collection. The acceleration voltage also has to be high enough to enable the detection of the characteristicX-rays emission of the elements of interest. Increasing the acceleration voltage would also result in more counts onthe EDS spectra and increase the indexing speed, but is detrimental to the spatial resolution. Indeed, the XEDS inter-action volume is about one micrometer at 20 kV in Nickel alloys and increases greatly with the acceleration voltage.Using the di ff erent chemical composition of the phases, other methods involving multi-modal data combination havebeen implemented. Phases of di ff erent composition exhibit di ff erent intensities on back-scattered electrons (BSE)images, due to their di ff erent Z number. Payton and Nolze [13] have proposed a method that involves the acquisitionof BSE images on a tilted sample, using specific BSE detectors located above the phosphorous screen of the EBSDdetector. Those authors have shown that this method enables a more accurate phase di ff erentiation than XEDS-EBSD,since the contrast on the back-scattered images is very sensitive to the variations in chemical composition. They alsodemonstrate the capability of that method for segmenting small features precisely, with typical dimensions of the orderof a few micrometers. The limit of this technique is that not all the EBSD detectors are equipped with such built-indiode sensors. The collection of back-scattered data is usually made by relatively large retractable BSE detectors thathave several quadrants, at a 0 ◦ tilt angle. On the other hand, EBSD data is collected at a 70 ◦ tilt angle. This, amongother reasons, leads to complex distortions in the EBSD versus the BSE data, which makes their precise recombinationvery challenging. 2 .2. Distortions in EBSD data EBSD data is known to su ff er from distortions that arise from many instrument and detector artifacts [14]. Thefollowing types of distortions can be distinguished: • First order spatial distortions: those are induced when tilting the sample at 70 ◦ . An area that would be a squareat 0 ◦ tilt turns into a trapezoid if the rotation axis of the stage is not perfectly parallel to the surface of thesample. The borders of the trapeze are also misaligned with those of the non-tilted area. Those distortions canbe modeled by a first order a ffi ne transformation, involving a translation and a rotation. • Second order spatial distortions: the electron beam is deflected by a set of lenses that induce a barrel distortionon the final image. This distortion is mostly visible at low magnification. It can be modeled by a second orderpolynomial function. • Space-time distortions: when a sample is being exposed for some time to the electron beam, charging e ff ectsoccur. The accumulation of charges on the surface of the sample leads to the deflection of the beam duringscanning. This e ff ect occurs in any scanning process. It is usually the most pronounced in the beginning of thescan and becomes more stable as the scan progresses. A good example is shown in [15]. The amount of driftis related to the conductivity of the sample itself, the conductivity of the mounting system that is being used,the voltage and aperture, as well as the cleanliness of the scanned surface. The amount of drift is also moreimportant at high than low magnifications [16]. There is no simple physical model to describe the pattern of thedrift distortions.Given the complexity of the net resulting distortion, it cannot be easily calculated.
2. Method
In this context, we propose a method that enables the compensation of the distortions in EBSD data and addsphase di ff erentiation if wanted, by accurately matching it with BSE images collected at a 0 ◦ tilt angle, at individualpixel precision and over broad areas. To do so, the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) isused to calculate the relative distortion function between the EBSD data and the BSE image. With regard to the recombination of multi-modal data, one first intuitive approach is to use sets of matching pointsover the area of interest in complementary images. Some methods aiming at correcting the distortions in EBSDmaps have been implemented using this principle [15]. They require the manual selection of a set of reference andcorresponding points in the BSE image and the EBSD data. The higher the number of pairs of points, the greater theprecision. However, this step can be time-consuming and does or made di ffi cult by the lack of contrasted featuresin both EBSD and BSE data. Furthermore, it does not scale up to large or tridimensional datasets, as it may require3uman annotations for every slice in the dataset. In most 3D EBSD datasets, the process of data collection consists insuccessive movements of the stage to positions that are practical for machining, imaging, and collection of di ff ractionpatterns. The placement of the stage from a slice to another is never exactly the same [17]. This modifies the set ofreference points on every slice.Here, we propose to use speckles of similar features without any human annotations, allowing the recombination ofmulti-modal data to scale up to large EBSD datasets. To do so, a high number of automatically generated referencepoints are combined by using microstructural features as speckles, such as pores, a second-phase, or precipitates.Since the goodness of the superimposition of the speckles can be quantified, it can be used as a parameter to maximize,enabling an automated matching process. Therefore, quantifying the goodness of the superposition of the speckles iscritical. Defining a similarity measure between the speckles
In order to superpose the EBSD and BSE speckles, a quantitative measurement of their similarity is needed. TheDice similarity metric [18] (also known as F1-score) describes the relative overlap between the segmented images.More precisely, this score measures the overlap between segmented pixels. Formally, given the binary EBSD speckle I EBS D ∈ { , } I × J and the binary BSE image I BS E ∈ { , } I × J where 1 encodes the segmented object and 0 thebackground image, and I × J their domain of definition, the Dice similarity for is computed as follows:Similarity( I EBS D , I BS E ) = |I EBS Dseg ∩ I
BS Eseg ||I
EBS Dseg | + |I BS Eseg | (1)where | . | encodes the size of the underlying set and I . seg only corresponds to the segmented pixels in the image. Thus, |I EBS Dseg ∩ I
BS Eseg | corresponds to the number of matching pixels and |I EBS Dseg | and |I BS Eseg | respectively correspond to thetotal number of pixels in the EBSD and BSE images. This similarity measure counts the number of overlappingsegmented pixels normalized by the sum of segmented pixels. It ranges from 0 (no overlapping segmented pixels) to1 (perfect matching). The value of the S imilarity f unction ( I EBS D , I BS E ) is referred to as the ’score’ in the following.As the Dice similarity is both intuitive, robust and easy to compute, it has been extensively used in several imagesegmentation applications such as medicine [19].
The global process of the reconstruction is shown in figure 1. Starting from the initial EBSD data and the BSEimage, two speckles are generated. In the first step (initial alignment), the BSE speckle is rescaled so its pixel sizecorresponds to that of the EBSD data. The rescaling is done using a nearest neighbor interpolation. The BSE speckleimage is also rotated and translated so it is pre-aligned to the EBSD data. To do so, a grid-search over a set of a ffi netransformations is performed, to distort the re-sized speckle over three parameters: two translations along the x andy directions, t x , t y , and a rotation of angle α . The user is free to choose the widths of the translation and the rotationangle, as well as their increment size. The minimum increment for the translations is one pixel. The step size for4he angle can be refined as much as wanted. All the possible translations and rotations within the provided rangesare tested. The transformation leading to the best score is saved and applied to the BSE speckle. The latter is thenused in the second step, as the reference speckle for the compensation of the distortions. In the second step (CMA-ESon fig. 1), the EBSD speckle is meshed on a regular grid, and the grid is distorted using the CMA-ES optimizer,in an iterative process. The initial and new locations of the points of the grid are used as pairs of control points, toestimate the distortion function. At each iteration, the score produced by the distorted EBSD speckle is calculated.The distortion function corresponding to the distorted speckle that exhibits the best score is used to determine the newlocation of the points in the new EBSD file and the segmented BSE image is used to fill in the phase data (if needed).The parametrization and calculation of the distortion function are explained in the following. Figure 1: Schematic representation of the reconstruction algorithm.
Formally, the goal is to find the distortion function f that maximizes the similarity between the EBSD and theBSE speckles. This leads to the optimization procedure described in eq. 3: f ∗ = argmax f (cid:16) Similarity( I EBS D , I BS E ) (cid:17) (2) = argmax f (cid:16) | f ( I EBS D ) seg ∩ I BS Eseg || f ( I EBS D ) seg | + |I BS Eseg | (cid:17) (3)5s described in section 1, many overlapping distortion phenomena occur while collecting the EBSD data, leadingto a non-linear distortion. In the following, the nature of the distortion function is assumed to be polynomial. Itis a very weak assumption as polynomial functions can approximate a large range of complex functions. Besides,the proposed method can be easily extended to other function approximators such as Gaussian processes, trees etc.Formally, a polynomial distortion of degree P is defined as follows: f ( x , y ) = x (cid:48) = (cid:80) Pn = (cid:80) nk c xk , n − k x k y n − k y (cid:48) = (cid:80) Pn = (cid:80) nk = c yk , n − k x k y n − k (4)For instance, given a polynomial fit of degree 2, the coordinates ( x , y ) of a pixel are transformed into ( x (cid:48) , y (cid:48) ): f ( x , y ) = x (cid:48) = c x , + c x , x + c x , y + c x , xy + c x , x + c x , y y (cid:48) = c y , + c y , x + c y , y + c y , xy + c y , x + c y , y (5)Given a polynomial order, the goal is to find the optimal set of weights c ∗ that maximizes the similarity measure.This set of weights c can be computed by two means: by directly looking into the space of parameters c or bygenerating intermediate matching points that are used to perform a polynomial regression.While the first approach may be more direct, the manifold of parameters c is very di ffi cult to explore as a smallchange of parameters can lead to drastically di ff erent distortions. Therefore, in the proposed method, a set of match-ing points is generated, that is used to regress the polynomial coe ffi cients c . As a small change in the location ofthe matching points lead to a close distortion, this process allows a better granularity in exploring the space of thedistortions. More precisely, a regular grid and a distorted one are generated, and then the coe ffi cients c are determinedby a least-squares fitting method that maps the distorted grid to the regular one. This function is described below: • Generate a distorted mesh grid M = (( x , y ) , . . . , ( x N , y N )) • Generate a regular mesh grid M (cid:48) = (( x (cid:48) , y (cid:48) ) , . . . , ( x (cid:48) N , y N , )) • Find the distortion f parametrized by c such as c = argmin ˆ c (cid:80) Ni || (cid:0) x (cid:48) i y (cid:48) i (cid:1) − f ˆ c ( x i , y i ) || where, || . || is the euclidean norm. Note that the dimension of the mesh grid is tuned by the user via the numberof points N and / or the step-size between points. The influence of that parameter is discussed in section 4. As aresult, finding the optimal distortion f ∗ requires to find the optimal set of matching points M ∗ that better describes theunderlying distortion. These matching points are then used to estimate the distortion itself. Counter-intuitively, thedistorted grid may not match physical reality as discussed further in section 4.4.Finally, the polynomial distortion function is called from the python package Sklearn [20] and
Skimage [21]. Thenext section describes how the pairs of matching points are created, using the CMA-ES optimizer.6 .4. Using CMA-ES as a randomized black-box optimization to correct the distortions
The goal of the optimization is to find the function f that maximizes the S imilarity function between the specklesby finding the best distorted mesh grid M (which is non-di ff erentiable). To do so, a black-box optimizer method isused. Black-box optimization algorithms are e ffi cient at solving non-linear, non-convex optimization problems, incontinuous domains. They also overcome the deficiencies of the derivative-based methods in complex multidimen-sional landscapes that are rugged, noisy, have outliers or local optima, or when no error-gradient is available. Allthose features make the black-box optimizer relevant for the present problem, where the shape of the distortion isunknown and the process of superposing speckles intrinsically implies the S imilarity function to reach local maxima.The influence of noise present on the EBSD speckle is discussed in section 4.5.The black-box algorithm that is used in the present case is CMA-ES, which stands for Covariance Matrix Adap-tation Evolutionary Strategy. CMA-ES has become a standard tool for continuous optimization and has been appliedin various fields of research, such as image recognition for biology [22, 23], energy [24], chemistry [25, 26]. Though,to the authors knowledge, it has not been applied in the field of electron microscopy. CMA-ES belongs to the familyof evolutionary strategies (ESs), they are iterative algorithms, based on the principles of natural selection. CMA-ESinvolves a parametrized distribution (a multivariate normal distribution) that evolves throughout the iterations [27].ESs follow four distinct steps: initialization, sampling, evaluation and update. The initialization step consists in ini-tializing the probability distribution parameters and sampling an initial population of individuals accordingly. At eachiteration, a new generation of individuals is created. Those individuals are candidate solutions to the optimizationproblem, which goodness can be evaluated on a fitness function . After evaluation of the fitness of each individual inthat population, the statistics of the distribution are updated according to the algorithm at hand. A new population isre-sampled according the new distribution and the process is repeated until a termination criterion is reached. Thoseoperations are described below:Initialize the distribution parameters θ For each generation g = , , ... number o f iterations :Sample λ individuals x , x , ..., x λ ∈ R n according the probability distribution P θ ( x )Evaluate them on the fitness functionUpdate θ ← F ( θ, x , x , ..., x λ , S imilarity ( x ) , S imilarity ( x ) , ..., S imilarity ( x λ ))where P θ is a probability distribution that describes where the good solutions are believed to be and F ( . ) is someupdate rule. In the CMA-ES, P θ is a multivariate normal distribution [27]. Those steps are detailed in the following. Initialization.
The distribution P θ encodes the distribution of the distorted spatial mesh grid. More precisely, the dis-tribution is a multi-variate normal distribution where each dimension encodes either the x-coordinate or y-coordinateof a given point of the grid. For example, if the mesh is a 4x4 grid, the multi-variate normal distribution has 32 di-mensions (16 points of 2 coordinates each). The initial distribution is defined by centering the mesh distribution on aregular grid. The standard deviation then encodes the initial acceptable moving distance of the matching points. Both7he spacing of the points of the grid and the standard deviation are chosen by the user. In practise, CMA-ES slowlyand iteratively distorts the regular grid, in order to retrieve a set of control (matching) points that encode the distortion.The concept of mesh grid here is to understand as an array of points that do not have any topological connectivity. Inother words, the permutation of two points of the grid is allowed and does not a ff ect the regression of the distortionfunction. The properties of the mesh are discussed in more detail in section 4.4. Sampling.
New distorted meshes are generated following the distribution P θ . The coordinates are rounded, andmatching points outside the range of the speckle dimension are kept. Evaluation.
The fitness function is the
S imilarity described in eq. 1. The coe ffi cients c of the distortion function arefirst regressed by matching the distorted mesh grid to the regular one. The similarity measure is then computed andreturned as the fitness score. Update.
The basic CMA equation for sampling the search points at the generation g + x ( g + k ← m ( g ) + σ ( g ) N (0 , C ( g ) ) , f or k = , , ..., λ. (6)where x ( g + k ∈ R n is the k − th o ff spring of the generation g + m g ∈ R n is the mean value of the distribution atthe generation g . σ ( g ) ∈ R + is the step size at the generation g . N (0 , C ( g ) ) is a multivariate normal distribution witha mean of zero and covariance matrix C g . C g ∈ R nxn is the covariance matrix of the distribution at the generation g . It is symmetric definite positive and describes the geometrical shape of the distribution. The initial value of σ , σ , is picked by the user and C = I . They both evolve throughout the iterations, as the population evolves. Theself-adaptation of those parameters is the key point for the rapid convergence of the optimization[28]. The equationsgoverning the update of σ ( g ) , C ( g ) and m g are described in Appendix A. In the present case, this update shifts thespatial distribution of the points of the distorted grid in order to search for the optimal distortion function. Termination.
This process is repeated until the termination criterion is reached. In the present case, a criterion basedon the maximum number of steps is used. More advanced methods can be used to detect when the optimization (andthus the fitness score) starts plateauing. If several speckles are known to have the same distortion, they also can be usedas a validation criterion to avoid over-fitting. Once the CMA-ES algorithm is finished, the means of the distribution P θ are used as the final distorted mesh grid. Once the maximum number of calls has been reached, the polynomial function leading to the best superpositionof the speckles is applied to the EBSD data. In this process, the original grid of the EBSD is kept and the Euler angles,confidence indexes and other data is interpolated using the polynomial function. The points of the grid that end upcontaining no data are filled with zero-values for the confidence indexes and the (0 , ,
0) triplet of Euler angles. Therotated and translated segmented BSE image is used to fill-in the phase data of the new EBSD file. For that reason, and8s already mentioned, the segmentation of the BSE image has to be done as accurately as possible. Some applicationexamples are shown in section 3.
The full distortion pipeline was developed in Python. The code source, the hyperparameters and one data set areavailable at https://github.com/MLmicroscopy .
3. Results
All materials have been characterized in a FEI Versa 3D SEM, equipped with a TSL EDAX EBSD Hikari PlusCamera with an indexing speed of 300 to 400 frames per second and a 4x4 binning. The computer for the distortioncorrection was equipped with a Quad Core processor, 4.2 GHz and 64 GB of RAM. Most computations required about10 to 15 minutes. β phase in α − β Titanium alloy
A sample of Ti-6Al-4V with an α − β structure has been embedded in bakelite and electro-chemically polished.It was then mounted on the SEM stage and copper tape was used to enable electrical conductivity from the sampleholder to the polished surface. An area of 14 x 20 µ m has been characterized with a 20 kV acceleration voltage and astep size of 40 nm. One purpose of collecting this dataset is to better resolve the phase map, using the better qualityof the segmentation of the BSE image, with fine details. This mounting system was purposely used in order to inducea large drift of the beam during the EBSD scan.The orientation map for this sample is shown on fig. 2-a. The corresponding BSE image is shown on fig. 2-d,where the β phase exhibits a lighter contrast than the α phase. Fig. 2-b shows the initial phase map with the β phasein white. This phase map is superimposed to the Index Quality map on fig. 2-c, where the β phase is colored in red.The comparison of fig. 2-b and -e shows that only a small fraction of the β phase was indeed identified as such on theinitial EBSD data. Its initial area fraction is 3.4%, versus 9.9% in the segmented BSE image. The CMA-ES optimizerhas been applied on a 25 x 25 points mesh grid, with an initial standard deviation σ =
20 pixels, using a polynomialorder of 3 and 5,000 iterations. Fig. 2-f shows the final phase map, colored as fig. 2-c. It superimposes well withthe Index Quality map. The final area fraction of β phase is 9.9%, as in the BSE speckle. The contribution of theCMA-ES optimizer to compensate higher order distortions and match all the features of the speckles is discussed insection 4.1. γ and γ (cid:48) phases in Rene 65 superalloy Rene 65 is a polycrystalline Nickel-based alloy that has been designed for turbine disk applications. Its chemicalcomposition (wt%) is Ni-16 Cr 13 Co 3.7 Ti 2.1 Al 4 Mo 4 W 1 Fe 0.7 Nb 0.05 Zr 0.016 B [29]. The sample was9xtracted from a ring that was subjected to a sequence of forging operations followed by solution annealing and agingtreatments, 1065 ◦ C for one hour and 760 ◦ C for 8 hours respectively. It was prepared using conventional metallographytechniques, followed by vibratory polishing using a 0.04 µ m Al O suspension. Its microstructure consists of fine γ matrix grains, which average equivalent diameter is 10-12 µ m . Spherical primary γ (cid:48) precipitates are located on thegrain boundaries, with an equivalent diameter in the range of 1-4 µ m . EBSD maps and corresponding BSE imageshave been acquired using an acceleration voltage of 20 kV with a step size of 0.1 µ m over an area of 150 x 200 µ m .Figure 3-a shows the initial EBSD map colored according to the orientation of the crystals projected along the normalto the polished surface. Figure 3-b shows the corresponding BSE image on which the precipitates exhibit a darkercontrast. Figure 3-c and -d show the BSE and EBSD speckles, respectively, created by segmenting the γ (cid:48) precipitates.Fig. 3-c has been created by segmenting the features which equivalent diameter is smaller than 5 µ m , considering theannealing twin boundaries as internal defects. This criterion enables a rough segmentation of the precipitates. Notethat some precipitates may have been omitted from the segmentation and some small grains may have been mistakenlysegmented as precipitates. The influence of the quality of the EBSD speckle on the goodness of the compensation ofthe distortions is discussed in section 4.5. This segmented image was then used as the EBSD speckle for the CMA-ESoptimizer, along with a segmented BSE image of the same area. The CMA-ES optimizer was used on a mesh gridwith a spacing of 24 pixels between consecutive points, an initial standard deviation of 5 pixels and a polynomial orderof 3. The initial grid is presented on figure 3-e, the final grid is presented on figure 3-f. The superimposed specklesafter applying the CMA-ES optimizer are shown on fig. 3-g. The distortion correction leads to a final score of 0.62after 1,800 iterations. Figure 3-h shows the final phase map with the precipitates colored in red, superimposed withthe Image Quality map displayed in grey scale. The location of the precipitates corresponds well with the locationof the grain boundaries on the Image Quality map, over most of the area of interest. The precision of the method isdiscussed in sections 4.6 and 4.4. The alloy Inconel 718 has been manufactured by the Electron Beam Melting (EBM) technique, as described in[30]. A sample was cut by Electrical Discharge Machining (EDM) and polished parallel the build direction, usingconventional metallography processes followed by 0.05 µ m colloidal silica. Image and EBSD data have been acquiredunder a 20 kV acceleration voltage, using a 0.5 µ m step size over an area of 300 x 700 µ m . Fig. 4-a shows theSecondary Electrons (SE) image that was used for the SE speckle. The beam contamination zone correspondingto the area of the EBSD map exhibits a trapezoid shape. A standard cleaning procedure involving slight cleaningof the EBSD data has been applied, consisting in removing isolated groups of less than 3 pixels having the sameorientation, leading to a change of 1% of the data. Since the pores could not be indexed, the TSL software assignedrandom orientations to those pixels. The inverse pole figure map, projected along the build direction, is shown onfig. 4-b. Some of the grains have their < > axis preferentially aligned with the build direction (vertical), acommon feature of additive manufactured face-centered cubic materials. A partial EBSD speckle of the pores has10een created by segmenting the pixels of lowest Confidence Indexes (CI). On the other hand, the pores have beenaccurately segmented from a SE image of the same area, in order to create the SE speckle. The CMA-ES strategy hasbeen applied, using a 65 x 65 points mesh grid, an initial standard deviation σ =
20 pixels, a maximum samplingof 5,000 iterations and a polynomial order of 3 for the distortion function. Fig. 4-c shows the superimposed specklesafter the CMA-ES strategy. The EBSD speckle consists of light blue points on a transparent background, and theSE speckle consists of dark red points on a light red background. Thus, as shown in the insert on the figure: lightred pixels (labeled ”1” on the insert) correspond to the SE background, dark red pixels (labeled ”2”) correspond topores of the SE speckle that did not match any pores of the EBSD speckle, light blue pixels (labeled ”3”) are pores ofthe EBSD speckle that did not match the SE speckle dark blue pixels (labeled ”4”) are pores that were successfullymatched. From this basis, the final EBSD map is shown on fig. 4-d. Only the points with a confidence index higherthan 0.2 are plotted. The color code is the same as fig. 4-b and the pores are colored in black. Most of those poresare located on the grain and sub-grain boundaries. After applying the distortion, some data points ended up out of theinitial grid, which leads to the black pixels on the edges of fig. 4-d. This dataset illustrates the limits of this method tocompensate the distortions over broad areas when the features are unevenly distributed over the area.
4. Discussion
Several methods have been proposed in the literature, for correcting the distortions in EBSD data, using variousalgorithms to make up for the drift distortions. Zhang et al. [15] have shown that a thin plate spline function enablescompensation of the distortions in EBSD data. Contrary to most methods, the use of the CMA-ES strategy does notassume any specific shape of the distortion function, nor constrain the order of the polynomial function. The use ofspeckles and a score to quantify the goodness of the superposition enables the matching of both images at the resolu-tion of the pixel. Fig. 5 shows a comparison between the superposition of the speckles in the Ti6Al4V dataset, afterthe compensation of a ffi ne distortions only (a) and after the CMA-ES procedure (b). On a blue background, the EBSDspeckle is colored in red and the BSE speckle in white. Compensating the a ffi ne distortions enables a good overlap ofthe speckles. The CMA-ES optimizer then compensates the finer distortions, even very local, and enables to achievea better matching of the speckles despite an important drift of the beam. Three examples are presented in the insertsof Fig. 5, where the laths are accurately superimposed.Figure 6 shows the maps of the vertical and horizontal components of the distortion function in the Inconel 718material. The origin and x and y directions of the scan are indicated with arrows. The electron beam scanning patternstarts from the top left corner and goes line-by-line from top to bottom. A polynomial order of 3 has been used. Both using a method based on the CI to segment the pores can lead to inaccuracies in the segmentation: some pixels may have been segmented aspores even though they are not (typically those pixels correspond to grain boundaries). f x decreases relatively more than f y , which means that the error on the x’ coordinate decreases alongthis direction, whereas the error on the y’ coordinate remains relatively constant. From a line to another (along the ydirection), the magnitude of f x decreases, which means that the error on the location of the x’ coordinate decreasesfrom a line to another. A similar observation can be made about the vertical component f y . Those trends are consistentwith most drift phenomena, where the first scanned pixels that are the most subjected to drift. Drift phenomena tendto decrease as the scan proceeds: along a line, and from a line to another. A quantitative analysis of those trends canbe found in Appendix B. The influence of the CMA-ES parameters on the goodness of the matching has been studied in the Ti-6Al-4Vdataset. Figure 7 shows the final scores obtained for di ff erent numbers of points per width and height of the image,and initial step sizes. The obtained surface is convex, with a global maximum reached for (35,21), pointed by the redarrow on the figure. For a fixed number of grid points, the score admits a maximum for a given initial step size σ .This behavior is visible on the four points on the far right of the figure, on the part of the envelope highlighted by ablue curve. For 55 grid points per length of the image, the score reaches a maximum of 0.5128 for an initial step sizeof 27 pixels. A similar behavior occurs for any given initial step size but varying numbers of points in the mesh grid.Note that a similar behavior is expected for all data sets, but for di ff erent values of the parameters. The accuracy of the reconstruction of the phases can be quantified by comparing the location of the phase bound-aries and grain boundaries on the reconstructed maps. Such measurements have been performed on the Rene 65dataset and displayed on fig. 8. A subset of the corrected EBSD map has been selected where three line profileshave been traced, crossing isolated or clusters of γ (cid:48) precipitates. The point-to-point misorientation is plotted in blackand superimposed to the phase data, which is plotted in red. The first profile crosses a single precipitate. The phaseboundaries and grain boundaries ”a” and ”b” are indicated on the corresponding profile. The transition from a phaseto the other corresponds well with the crossing of high-angle boundaries. The second profile crosses a cluster of twoprecipitates which were separated on the segmentation of the BSE image. The phase boundaries ”c”, ”d” and ”f” arewell superposed with the grain boundaries. Note that the boundary ”d” consists of a single pixel. The profile alsocrosses an annealing twin boundary ( Σ
3, 60 ◦ rotation around a < > axis) present in one of the precipitates atthe marker ”e”, characterized by a 60 ◦ misorientation. The third profile also crosses two precipitates which were notseparated on the segmented BSE image. The grain boundaries of the first precipitate are labeled ”g” and ”h” on theprofile. The second precipitate contains an annealing twin boundary ”i”, that is crossed right before the grain bound-ary ”j”. This grain boundary is well superimposed with the phase boundary. The error on the location of the phase12 able 1: Precision of the location of the phase boundaries versus grain boundaries in the profiles of Fig. 8, in the Rene 65 dataset Boundary ∆ x ( µ m )a 0.0b 0.0c 0.0d 0.0f 0.1g 0.2j 0.0boundaries has been measured on all the profiles, by comparing their location x Ph . B to that of the grain boundaries x GB through the parameter ∆ x of eq. 7. The results are displayed in table 1. The error is less than 0.2 µ m for all theboundaries (i.e. 2 pixels), reaching ≈ µ m for most of them. The error on the location of the two phase boundariesthat were not exactly correlated with the grain boundaries (”f” and ”g”) corresponds to a misplacement of 1 and 2pixels respectively. This gives an indication of the precision achievable with this correction method, assuming a goodsegmentation of the BSE image is achieved. A segmentation in which the boundaries of the objects have been erodedor blurred impacts the quality of the reconstruction. ∆ x = (cid:107) x Ph . B − x GB (cid:107) (7) During the CMA-ES optimization procedure, new distorted mesh grids are generated to regress the polynomialdistortion. Counter-intuitively, these grids are not constrained to encode a realistic distortion; they are only optimizedsuch that the final distortion is meaningful. Thus, one must not use the distorted mesh grids outside the polynomialregression step, as several di ff erent mesh grids can lead to the same distortion function. Indeed, there exists an infinitenumber of sets of points that extrapolate to the same final function. Those points do not have to always be on the curveitself. Even though phenomena of grid points swapping does not a ff ect the calculation of the distortion function, theauthors suggest that the user sets the σ parameter as less than the distance between two consecutive grid points, ifthe user wants to avoid this phenomenon. To a lesser extent, several polynomial coe ffi cients may also lead to a verysimilar resulting distortion function, only di ff ering around the border of the speckle.Despite this apparent limitation, CMA-ES turns out to be highly reproducible in practice. For example, fig. 9shows the typical pattern of convergence of the CMA-ES algorithm: fig. 9-a shows the evolution of the score as afunction of the number of iterations in the CMA optimizer and 9-b shows the evolution of the step size throughout13he iterations. Starting from a score value of about 0.4, the CMA optimizer enables a quick convergence towards asteady-state value of about 0.7. This steady state is reached after less than 5000 iterations. As the score increases, thestep-size σ decreases with the number of iterations, indicating little variation around the mesh grid ground.The CMA-ES strategy has been applied 100 times on the Rene 65 dataset. Each run consisted in 2000 iterations,with a step size of 75 points, an initial standard-deviation σ of 5 pixels. A polynomial order of 3 was used for thepolynomial function. The CMA optimizer produced an average score of 0.6587 with an associated standard deviationof 0.0015. The similarity score between distorted speckles over the 100 runs was 0.9353 in average, indicating a goodprecision and repeatability of the process. Figure 10-a shows the evolution of the score throughout the iterations: thedark blue curve corresponds to the mean score over 100 runs. The light blue colored area around the curve correspondsto the lower and upper bounds of the score (mean - standard deviation, and mean + standard deviation, respectively).Those values show that the optimization is stable and repeatable over several runs. Some steps are clearly visibleon the first 500 iterations and correspond to the new bounds determined by the creation of new individuals at eachgeneration in CMA-ES. Based on those statistics, a heat map has also been generated, displayed on fig. 10-b. Thismap shows the segmented BSE speckle colored according to the number fraction of times that a pixel was present ata given (x’, y’) location. In other words, a pixel that appears in white (value 1.0) was always assigned to the γ (cid:48) phase.On the opposite, a pixel that appears in black was never assigned to this phase. The more consistent and precise theoptimization, the sharper the contrast on this map. The shape of the precipitates appears clearly on fig. 10-b, whichis consistent with the good repeatability suggested by fig. 10-a. This map also gives information about the precisionof the reconstruction over the whole map. The center of the precipitates usually appears with a value of 1.0, howevertheir boundaries are usually less precise. The deviation on the reconstruction of the boundaries of the precipitates isillustrated on two examples, labeled ”A” and ”B”. The precipitate ”A” is close to the border of the image, and theprecipitate ”B” is in the center. The inserts on the side of the figure show that the boundaries of those two precipitatesare not reconstructed with the same consistency over the 100 slices. There is only one layer of pixels having a valuelower than 1.0 on the precipitate B, versus 2 to 3 rows on the boundary of the precipitate A. The boundaries of mostof the precipitates located on the edges of the map have a similar coloring. This indicates that the CMA-ES optimizerleads to consistent results in the center of the map, with a deviation of about 1 pixel as the location of the phaseboundaries are approached. At the borders of the map, the consistency on the location of the phase map is about 2 to 3pixels. This corresponds, in this dataset, to an associated error of 0.2 to 0.3 µ m on the location of the phase boundaries.This error remains much smaller than the actual size of the features of interest. In other words, the reconstruction doesnot artificially create additional features (precipitates) nor assign the wrong phase to any feature.Again, it should also be noted that di ff erent final grids can describe the same polynomial distortion function, sincethe calculation of the function is not a ff ected by the permutation of the points of the mesh grid. Moreover, di ff erentpolynomial functions can describe the same physical distortion. This phenomenon is enhanced if the points of thespeckle are sparsely distributed or mostly located in the center of the region of interest.14 .5. Sensitivity to the accuracy of the EBSD speckle As mentioned previously, the use of speckles eliminates the manual selection of reference points. Another advan-tage of using speckles and a score based on the number of superimposed pixels, is that the EBSD speckle can be onlya small subset of the original image. This is the case in the Rene 65 dataset, where the BSE speckle contains all the γ (cid:48) precipitates but the EBSD speckle contains the smallest features in the dataset. Some precipitates are missing on thisspeckle, and small grains are mistakenly segmented as precipitates. This does deteriorate the theoretical maximumscore achievable, but does not prevent a good superposition of the speckles, as already shown on fig. 3-c. The influ-ence of the quality of the EBSD speckle has been studied on the Inconel 718 dataset: some noise was added (wrongfeatures and random noise) and some pores were also removed from the EBSD speckle on purpose, in comparisonwith the one used on 3.3 (fig. 4). The amount of features added or deleted is quantified by the ∆ f parameter, whichquantifies the percentage of pixels points added or deleted to the speckle containing the correct number of pores: ∆ f = numbero f pixelsinwrongspecklenumbero f pixelsinrightspeckle . Figure 11 shows the best score obtained as a function of the percentage of pixels addedor deleted in the EBSD speckle. Empty and filled markers correspond to the initial and final scores in CMA-ES,respectively. The speckle at ∆ f =
0% corresponds to the EBSD pattern in which only the pores were segmented. Itleads to the best score achievable. The deletion of points leads to a drastic decrease of the score. The addition of”wrong” cavities, however, leads to a slight decrease of the score but stagnates to a roughly constant value, as thenumber of points added increases. This indicates that the calculation of the distortions is fairly tolerant to the presenceof noise or wrong elements in the EBSD speckle. However, missing points are more detrimental. γ − γ (cid:48) nickel-base superalloys: comparison with other mapping techniques The γ and γ (cid:48) phases of nickel-based superalloys are well known to be challenging to di ff erentiate, on the singlebasis of their di ff raction patterns. Methods like conventional EBSD with an indexation of phases based on the Houghtransform are not e ffi cient. Other indexation techniques have been proposed, such as the dictionary approach by DeGraef [31]. In this approach, the collected patterns are compared to a dictionary of synthetically generated patterns.This technique enables the indexation of patterns with a better angular resolution [32] as well as the di ff erentiation ofphases in complex alloys [33] and geological materials [34]. However, it can be computationally expensive and -thisfar- has not been able to di ff erentiate γ and γ (cid:48) phases. Combined EDS-EBSD methods have been used in the last yearsbut su ff er from long indexing times (and so, increased beam drifting), post-processing times and limited resolution[12]. Using a completely di ff erent technique but a similar approach to dictionary indexing, with ion channelingimaging, the iCHORD method of Langlois et al. [35] has been shown to enable the di ff erentiation of phases [36].The channeling contrast is used to obtain information about the orientation. On the other hand, the secondary ionsignals from γ and γ (cid:48) phases are su ffi ciently contrasted to enable the identification of phases. In a recent comparativestudy, Vernier et al. [37] have shown that this method enables a more accurate mapping than combined EDX-EBSDmethods, since it enables to identify precipitates of size down to 150 nm. The scanning speed is also 3 times fasterthan EDS-EBSD methods. The time necessary for the acquisition and indexation of the patterns however, is typically15imilar than that of the combined EDS-EBSD method. With a post-processing time of about 15 to 30 minutes, themultimodal data recombination process using the CMA-ES optimizer appears to be a competitive alternative to theother methods proposed in the literature. The only constraint is the acquisition of a BSE image of su ffi cient quality(in terms of contrast), and its precise segmentation. Indeed, in this method, the size of the smallest objects resolveddepends not only on the step size of the EBSD map but also directly on the quality of the segmentation of the BSEimage. Assuming a good segmentation, the spatial resolution of this technique is about a few pixels, which is betterthan EDS-EBSD methods, and comparable to the iCHORD method. It can also be easily implemented in any SEMequipped with a conventional EBSD detector.
5. Conclusion and perspectives
A new method for the compensation of the distortions and improved phase di ff erentiation in EBSD data has beendeveloped. The principle consists in using an electron image of the same area than the EBSD data, taken at a 0 ◦ tiltangle, and using it as a reference. Similar features are segmented out of the EBSD data and electron image. Then, theCMA Evolutionary Strategy is applied in order to match the speckles. The goodness of the superposition is measuredby a score, which ranges from 0 to 1. Crystallographic and phase data are then recombined in a new EBSD file. Thisnew file has the same grid than the initial EBSD file.This method has been applied successfully to nickel-based superalloys and a titanium alloy. Assuming a precisesegmentation of the phases on the electron image, this method can reach a precision of a couple of pixels over broadareas, despite important drift phenomena. It can be used on any EBSD dataset, as long as two speckles of similarfeatures can be generated. It is relatively tolerant to inaccuracies and noise in the EBSD speckle. Full automation andshort computation times make it a competitive post-processing method, when compared to other options proposed inthe literature. Acknowledgements
The authors gratefully acknowledge a Vannevar Bush Fellowship, ONR Grant N00014-18-1-3031.We acknowledge the following agencies for research funding and computing support: CHISTERA IGLU and CPERNord-Pas de Calais / FEDER DATA Advanced data science and technologies 2015-2020 Henry Proudhon from MinesParisTech is acknowledged for prolific conversations and exchange of ideas. Olivier Pietquin, J´er´emie Mary andPhilippe Preux from Inria are acknowledged for insightful conversations. Xuedong Shang from INRIA is acknowl-edged for his various comments on CMA-ES. McLean Echlin from the University of Santa Barbara is acknowledgedfor providing the Ti-6Al-4V sample. Mickael Kirka from the Oak Ridge National Laboratory is acknowledged forproviding the additive manufactured Inconel 718 sample. The Carlton Forge Works company (PCC Corporation) isacknowledged for providing the Rene 65 material. 16 eferencesReferences [1] S. Suzuki, Features of Transmission EBSD and its Application, JOM 65 (9) (2013) 1254–1263, ISSN 1047-4838, doi: \ let \ @tempa \ bibinfo@[email protected] / s11837-013-0700-6.[2] M. P. Echlin, M. Straw, S. Randolph, J. Filevich, T. M. Pollock, The TriBeam system: Femtosecond laser ablation in situ SEM, MaterialsCharacterization 100 (2015) 1–12, ISSN 1044-5803, doi: \ let \ @tempa \ bibinfo@[email protected] / J.MATCHAR.2014.10.023.[3] D. Rowenhorst, A. Gupta, C. Feng, G. Spanos, 3D Crystallographic and morphological analysis of coarse martensite: Combining EBSD andserial sectioning, Scripta Materialia 55 (1) (2006) 11–16, ISSN 13596462, doi: \ let \ @tempa \ bibinfo@[email protected] / j.scriptamat.2005.12.061, URL http://linkinghub.elsevier.com/retrieve/pii/S135964620600039X .[4] G. Spanos, D. Rowenhorst, A. Lewis, A. Geltmacher, Combining Serial Sectioning, EBSD Analysis, and Image-Based Finite ElementModeling, MRS Bulletin 33 (06) (2008) 597–602, ISSN 0883-7694, doi: \ let \ @tempa \ bibinfo@[email protected] / mrs2008.124.[5] F. Lin, A. Godfrey, D. J. Jensen, G. Winther, 3D EBSD characterization of deformation structures in commercial purity aluminum, MaterialsCharacterization 61 (11) (2010) 1203–1210, ISSN 1044-5803, doi: \ let \ @tempa \ bibinfo@[email protected] / J.MATCHAR.2010.07.013.[6] M. Calcagnotto, D. Ponge, E. Demir, D. Raabe, Orientation gradients and geometrically necessary dislocations in ultrafine grained dual-phase steels studied by 2D and 3D EBSD, Materials Science and Engineering: A 527 (10-11) (2010) 2738–2746, ISSN 0921-5093, doi: \ let \ @tempa \ bibinfo@[email protected] / J.MSEA.2010.01.004.[7] L. Holzer, M. Cantoni, Review of FIB tomography, in: Nanofabrication using focused ion and electron beams : principles and applications,Oxford University Press, ISBN 9780199734214, 410–435, 2011.[8] A. T. Polonsky, M. P. Echlin, W. C. Lenthe, R. R. Deho ff , M. M. Kirka, T. M. Pollock, Defects and 3D structural inhomogeneity in elec-tron beam additively manufactured Inconel 718, Materials Characterization ISSN 1044-5803, doi: \ let \ @tempa \ bibinfo@[email protected] / J.MATCHAR.2018.02.020.[9] M. M. Nowell, S. I. Wright, Phase di ff erentiation via combined EBSD and XEDS, Journal of Microscopy 213 (3) (2004) 296–305, ISSN00222720, doi: \ let \ @tempa \ bibinfo@[email protected] / j.0022-2720.2004.01299.x.[10] G. West, R. Thomson, Combined EBSD / EDS tomography in a dual-beam FIB / FEG-SEM, Journal of Microscopy 233 (3) (2009) 442–450,ISSN 00222720, doi: \ let \ @tempa \ bibinfo@[email protected] / j.1365-2818.2009.03138.x.[11] D. Child, G. West, R. Thomson, The use of combined three-dimensional electron backscatter di ff raction and energy dispersive X-ray analysisto assess the characteristics of the gamma / gamma-prime microstructure in alloy 720Li¢, Ultramicroscopy 114 (2012) 1–10, ISSN 03043991,doi: \ let \ @tempa \ bibinfo@[email protected] / j.ultramic.2011.11.003.[12] M.-A. Charpagne, P. Vennegues, T. Billot, J.-M. Franchet, N. Bozzolo, Evidence of multimicrometric coherent γ precipitates in a hot-forged γγ nickel-based superalloy, Journal of Microscopy 263 (1), ISSN 13652818, doi: \ let \ @tempa \ bibinfo@[email protected] / jmi.12380.[13] E. Payton, G. Nolze, The Backscatter Electron Signal as an Additional Tool for Phase Segmentation in Electron Backscatter Di ff raction,Microscopy and Microanalysis 19 (04) (2013) 929–941, ISSN 1431-9276, doi: \ let \ @tempa \ bibinfo@[email protected] / S1431927613000305.[14] G. Nolze, Image distortions in SEM and their influences on EBSD measurements, Ultramicroscopy 107 (2-3) (2007) 172–183, ISSN03043991, doi: \ let \ @tempa \ bibinfo@[email protected] / j.ultramic.2006.07.003, URL http://linkinghub.elsevier.com/retrieve/pii/S0304399106001483 .[15] Y. Zhang, A. Elbrønd, F. Lin, A method to correct coordinate distortion in EBSD maps, Materials Characterization 96 (2014) 158–165, ISSN1044-5803, doi: \ let \ @tempa \ bibinfo@[email protected] / J.MATCHAR.2014.08.003.[16] A. D. Kammers, S. Daly, Digital Image Correlation under Scanning Electron Microscopy: Methodology and Validation, Experimental Me-chanics 53 (9) (2013) 1743–1761, ISSN 0014-4851, doi: \ let \ @tempa \ bibinfo@[email protected] / s11340-013-9782-x.[17] K. Mingard, H. Jones, M. Gee, Metrological challenges for reconstruction of 3-D microstructures by focused ion beam tomography methods,Journal of Microscopy 253 (2) (2014) 93–108, ISSN 00222720, doi: \ let \ @tempa \ bibinfo@[email protected] / jmi.12100.[18] L. R. Dice, Measures of the amount of ecologic association between species, Ecology, Wiley Online Library 26 (3) (1945) 297–302.
19] A. A. Taha, A. Hanbury, Metrics for evaluating 3D medical image segmentation: analysis, selection, and tool, BMC medical imaging 15 (1)(2015) 29.[20] Scikit-learn, http://scikit-learn.org/stable/ , [Online; accessed 23-October-2018], 2018.[21] Scikit-image, https://scikit-image.org/ , [Online; accessed 23-October-2018], 2018.[22] O. Ib´a˜nez, L. Ballerini, O. Cord´on, S. Damas, J. Santamar´ıa, An experimental study on the applicability of evolutionary algorithms tocraniofacial superimposition in forensic identification, Information Sciences 179 (23) (2009) 3998–4028, ISSN 00200255, doi: \ let \ @tempa \ bibinfo@[email protected] / j.ins.2008.12.029, URL http://linkinghub.elsevier.com/retrieve/pii/S0020025509000085 .[23] A. Sisniega, J. W. Stayman, J. Yorkston, J. H. Siewerdsen, W. Zbijewski, Motion compensation in extremity cone-beam CT us-ing a penalized image sharpness criterion, Physics in Medicine and Biology 62 (9) (2017) 3712–3734, ISSN 0031-9155, doi: \ let \ @tempa \ bibinfo@[email protected] / / aa6869, URL http://stacks.iop.org/0031-9155/62/i=9/a=3712?key=crossref.84caaff65ba9b2a3f6da8ee49365aca7 .[24] S. S. Reddy, B. Panigrahi, R. Kundu, R. Mukherjee, S. Debchoudhury, Energy and spinning reserve scheduling for a wind-thermal powersystem using CMA-ES with mean learning technique, International Journal of Electrical Power & Energy Systems 53 (2013) 113–122, ISSN01420615, doi: \ let \ @tempa \ bibinfo@[email protected] / j.ijepes.2013.03.032, URL http://linkinghub.elsevier.com/retrieve/pii/S014206151300149X .[25] S.-E. K. Fateen, A. Bonilla-Petriciolet, G. P. Rangaiah, Evaluation of Covariance Matrix Adaptation Evolution Strategy, Shu ffl ed ComplexEvolution and Firefly Algorithms for phase stability, phase equilibrium and chemical equilibrium problems, Chemical Engineering Researchand Design 90 (12) (2012) 2051–2071, ISSN 02638762, doi: \ let \ @tempa \ bibinfo@[email protected] / j.cherd.2012.04.011, URL http://linkinghub.elsevier.com/retrieve/pii/S0263876212001700 .[26] A. C. J. Weber, E. E. Burnell, W. L. Meerts, C. A. de Lange, R. Y. Dong, L. Muccioli, A. Pizzirusso, C. Zannoni, Communication: Moleculardynamics and H-1 NMR of n-hexane in liquid crystals, The Journal of Chemical Physics 143 (1) (2015) 011103, doi: \ let \ @tempa \ bibinfo@[email protected] / \ let \ @tempa \ bibinfo@[email protected] / ICEC.1996.542381, 1996.[28] N. Hansen, A. Ostermeier, Completely Derandomized Self-Adaptation in Evolution Strategies, Evolutionary Computation 9 (2) (2001) 159–195, ISSN 1063-6560, doi: \ let \ @tempa \ bibinfo@[email protected] / ff , Strategy for Texture Management in Metals AdditiveManufacturing, JOM 69 (3) (2017) 523–531, ISSN 1047-4838, doi: \ let \ @tempa \ bibinfo@[email protected] / s11837-017-2264-3.[31] Y. H. Chen, S. U. Park, D. Wei, G. Newstadt, M. A. Jackson, J. P. Simmons, M. De Graef, A. O. Hero, A Dictionary Approach to ElectronBackscatter Di ff raction Indexing, Microscopy and Microanalysis 21 (03) (2015) 739–752, ISSN 1431-9276, doi: \ let \ @tempa \ bibinfo@[email protected] / S1431927615000756.[32] F. Ram, S. Wright, S. Singh, M. De Graef, Error analysis of the crystal orientations obtained by the dictionary approach to EBSD indexing,Ultramicroscopy 181 (2017) 17–26, ISSN 03043991, doi: \ let \ @tempa \ bibinfo@[email protected] / j.ultramic.2017.04.016.[33] F. Ram, M. De Graef, Phase di ff erentiation by electron backscatter di ff raction using the dictionary indexing approach, Acta Materialia 144(2018) 352–364, ISSN 1359-6454, doi: \ let \ @tempa \ bibinfo@[email protected] / J.ACTAMAT.2017.10.069.[34] K. Marquardt, M. De Graef, S. Singh, H. Marquardt, A. Rosenthal, S. Koizuimi, Quantitative electron backscatter di ff raction (EBSD) dataanalyses using the dictionary indexing (DI) approach: Overcoming indexing di ffi culties on geological materials, American Mineralogist102 (9) (2017) 1843–1855, ISSN 0003-004X, doi: \ let \ @tempa \ bibinfo@[email protected] / am-2017-6062.[35] C. Langlois, T. Douillard, H. Yuan, N. Blanchard, A. Descamps-Mandine, B. Van de Moort`ele, C. Rigotti, T. Epicier, Crystal orientationmapping via ion channeling: An alternative to EBSD, Ultramicroscopy 157 (2015) 65–72, ISSN 0304-3991, doi: \ let \ @tempa \ bibinfo@X@ oi10.1016 / J.ULTRAMIC.2015.05.023.[36] C. Langlois, M.-A. Charpagne, S. Dubail, T. Douillard, N. Bozzolo, Ni-based superalloy: crystalline orientation mapping and gamma-gammaphases discrimination with the iCHORD method, in: European Microscopy Congress 2016: Proceedings, Wiley-VCH Verlag GmbH & Co.KGaA, Weinheim, Germany, 930–931, doi: \ let \ @tempa \ bibinfo@[email protected] / γ - γ nickel-based superalloy microstructures, Materials Characterization 142 (2018) 492–503,ISSN 1044-5803, doi: \ let \ @tempa \ bibinfo@[email protected] / J.MATCHAR.2018.06.015. ppendix A. Update of the parameters in CMA-ES Set m ∈ R n , σ ∈ R + , λ Input C = I , p c = , p σ = , Initialize parameters c c ≈ / n , c σ ≈ / n , c ≈ / n , c µ ≈ µ w / n , c + c µ ≤ , d σ ≈ + (cid:113) µ w n , w i = ...λ such that µ w = (cid:80) µ i = w i ≈ . λ While not terminate, x i = m + σ y i , where y i = N i (0 , C ) , for i = , ..., λ Sampling and variation m ← (cid:80) µ i = w i x i : λ = m + σ y w , where y w = (cid:80) µ i = w i x i : λ Update mean p c ← (1 − c c ) p c + (cid:107) p σ (cid:107) < . √ n (cid:112) − (1 − c c ) √ µ w y w Cumulation for the calculation of Cp σ ← (1 − c σ ) p σ ) + (cid:112) − (1 − c σ ) √ µ w C − y w Cumulation for the calculation of σ C ← (1 − c − c µ ) C + c p c p Tc + c µ (cid:80) µ i = w i y i : λ y Ti : λ , Update of the covariance matrix C σ ← σ exp ( c σ d σ ( (cid:107) p σ (cid:107) E (cid:107)N (0 , I ) (cid:107) − σ . Appendix B. Derivative of the vertical and horizontal components of the distortion function
Figure B.12 shows 2 sets of maps of the partial derivatives of the vertical and horizontal components of thedistortion function reported in fig. 6. As discussed in section 4.1, the variation in color on all four components showsthat the error on the location (x’,y’) varies as the scan progresses. Fig. B.12 also shows that this variation is notlinear. The magnitude of ∂ f x ∂ x (B.12-a) increases from the left to the right of the map, which means that the error on thex’ coordinate increases along each line. This gradient of color is the same along the y component of the map whichmeans that this error remains within the same magnitude from a line to another. The magnitude of this overall gradientremains always much higher than that of ∂ f x ∂ y (fig. B.12-b), which means that most of the error on the x’ coordinatesare accumulated along a line. However, the decreasing magnitude of ∂ f x ∂ y (fig. B.12-c) along the y component of themap, indicates that the error on the x’ location decreases from a line to another. this is consistent with a decreasingdrift e ff ect as the scan proceeds. As concerns the derivatives of the vertical component f y , one can see that most ofthe error on the y’ coordinate occurs along the vertical direction, from a line to another (fig. B.12-d). On the otherhand, ∂ f y ∂ x (fig. B.12-c) is fairly constant and low along a line and from a line to another. Those trends are consistentwith drift phenomena, where the drift is high in the beginning of the scan, and decreases along a line, and from a lineto another. 20 igure 2: Microstructure of the Ti-64 sample: a) Inverse Pole Figure (projected along the normal to the sample surface), b) EBSD speckle segmentedout of the initial phase map, with α phase in black and β phase in white, c) Initial phase map superimposed to the Image Quality map, d) BSEimage used for phase segmentation, e) Segmented BSE image used as a phase map in the un-distorted EBSD file. f) Final phase map colored as c). igure 3: Microstructure of the solution annealed Rene 65 sample. a) EBSD map colored according to the Inverse pole figure projected along thenormal direction to the surface of the sample, b) BSE image from which the precipitates have been segmented, c) BSE speckle, d) EBSD speckle:smallest features of the dataset segmented from the initial EBSD file, considering annealing twins as internal defects, e) initial mesh over the EBSDspeckle, f) final mesh over the EBSD speckle, g) Superimposed EBSD (blue) and BSE (red) speckles, h) Phase map with the precipitates coloredin red superimposed to the pattern quality map of the new EBSD file. igure 4: Microstructure of Additive Manufactured Inconel 718, a) SE image of used for the segmentation of the pores, the trapezoid contaminationzone corresponding to the EBSD area is visible, b) Initial EBSD map colored according to the Inverse Pole Figure projected along the build directionsuperimposed to the Image Quality map in greyscale, c) Superimposed SE (red) and EBSD (blue) speckles after CMA-ES, d) Final EBSD mapsuperimposed to the Image quality map in greyscale, with pores colored in black. igure 5: Speckle superposition after correction of a) A ffi ne distortions only, b) All distortions. On a blue background, the EBSD speckle is coloredin red and the BSE speckle in white.Figure 6: X and Y components of the distortion function in the Additive Manufactured Inconel 718 sample. igure 7: Influence of the initial CMA-ES parameters on the final score, in the Ti-6Al-4V dataset. igure 8: Comparison of the location of phase and grain boundaries in the reconstructed EBSD data of the Rene 65 alloy, on three misorientationprofiles across γ (cid:48) precipitates. igure 9: Typical optimization path with CMA-ES, as a function of the iterations, here on the AM Inconel 718 dataset: a) Value of the similarityfunction at the generation g , b) σ ( g ) : step size at the generation g . igure 10: Convergence and stability of the CMA optimization over 100 runs in the Rene 65 dataset. a) Mean score (dark blue), minimum andmaximum values (light blue), as a function of the number of iterations, b) Corresponding heatmap : the pixels are colored according to the numberfraction of times they were identical among the 100 runs.Figure 11: Influence of the addition or deletion of points on the EBSD speckle. Initial scores and final scores in CMA-ES as a function of thepercent of points added or deleted. igure B.12: Partial derivatives of the vertical and horizontal components of the distortion function along the X and Y coordinates, for the AdditiveManufactured Inconel 718 material. a) ∂ f x ∂ x , b) ∂ f x ∂ y , c) ∂ f y ∂ x , d) ∂ f y ∂ y ..