Experimental Veriﬁcation of 2D Digital Image Correlation for Scanning Electron Microscopy

(1)

Experimental Verification of 2D Digital Image Correlation for Scanning Electron Microscopy

Enrico Werkema

s2401479, e.werkema@student.rug.nl University of Groningen

Bachelor Thesis Materials Science Group September - November 2015

(2)

Acknowledgements

I am very much thankful to dr. ir. E.T. Faber for his daily guidance for my bachelor thesis and also for his support during the Scanning Electron Microscope experiments. His talks gave a good insight into which direction my research needed to go.

Secondly I would like to express my gratitude to prof. dr. J. Th. M. de Hosson and dr. V. Ocelik for examining this thesis and having me in their research group. Their support for the Scanning Electron Microscope helped me to perform the experiments and also provided valuable feedback on the methodology.

Finally I would like to thank Leo and Diego for offering me their samples for my research.

Enrico Werkema, October 2015

(3)

Abstract

Reliable 3D surface topography reconstructions can be made from 2D Scanning Electron Microscopy (SEM) images [2]. This method uses a set of images taken at different angles which are compared by a 2D digital image correlation (DIC) algorithm. The 2D DIC algorithm calculates the displacement fields between the images. A literature study and implementation of this algorithm in Matlab was done by J.T. Ouwerling [6] and this DIC implementation was used for the experiments. The key focus is the determination of the error introduced in the 2D SEM-DIC calculations and how these errors can be reduced. 2D SEM-DIC introduces two types of errors: errors associated with imaging methods in the SEM and errors introduced by DIC. Understanding these errors and quantifying them can improve results for the use of 3D surface topography. Results show a good accuracy (small error of 0.11 pixels) for numerically generated images.

SEM induced errors are detected for various samples and is in the order of 0.5 pixels (3nm). The DIC implementation is also shown to be flexible with high noise images and complex structures. Low detail images still remain a problem for the DIC program. A comparison with the commercial DIC program Aramis has shown similar results and verifies the correctness of the custom DIC program.

The root mean square error of the difference in displacement between Aramis and the custom DIC is determined to be of the order of 0.5nm (0.08 pixels).

(4)

Introduction

This chapter shows the structure of this report and states the objectives of the research.

The background information is included to give further meaning to these objectives.

1.1 Background information

The research is based on the article of Faber et al. [2], where a method for calibration-free quantitative 3D topography reconstruction is introduced. The importance of 3D topography reconstructions in scanning electron microscopy (SEM) is that it provides knowledge about contrast variations in the images. The proposed method uses 3 images at various tilt angles. The set of data consist of a reference and (tilted) target images which are compared using 2D digital image correlation (2D DIC), from which the displacement field between reference and target image is obtained. This calibration-free 3D method is different from the traditional methods, since it derives the rotation parameters from the images. This removes the need of complex SEM calibrations, because the precise knowledge about the geographical orientation of the sample is not needed. This 3D topography reconstruction method is summarized in Figure 1.1.

The displacement field between reference and tilted target images is currently obtained by the Aramis 5.3 software, which compares optically acquired images. The resulting displacement field from Aramis is assumed to be correct, but it doesn’t give any error estimation. Also the exact working principles behind the software is unknown, which makes it difficult to optimize the DIC results. Finally Aramis does not take into account SEM specific noise (time varying drift and spatial distortions), which can influence errors significantly [3].

These arguments show that for scientific research it is desirable to produce a more reliable method for displacement field calculations. For this purpose a Matlab software system has been set up by J.T. Ouwerling [6] in which the implementation is well documented.

This program is capable of producing a displacement field from two SEM images. The exact working principles behind this software system are described in Chapter 3.

(7)

Figure 1.1: Scheme of the working principle for obtaining 3D surface topography, without the need of calibrations. The left image summarizes the tilting process of two images and the right image shows the basics of rotation, translation and observations.[2]

1.2 Research goals

The main focus of this research is to determine the error DIC calculations and reducing SEM specific noise. Error estimation in DIC is important to state the relevance of the results for scientific purposes. Also reducing SEM specific noise, like time varying drift and spatial distortions, improves the results and makes it more relevant for scientific research.

To investigate the errors and noise several experiments have been performed. The goals are summarized as follows:

1. DIC accuracy: Understanding the errors of DIC calculations gives insight into the accuracy and gives information about how to improve the results. The implementation of the DIC software could also cause errors and should be tested to ensure a correctly working program.

2. Experimental verification: Different experiments have to show correctness of error estimation in DIC and error reduction of SEM noise.

3. SEM noise detection: SEM images are evaluated to detect their errors in DIC.

Reduction of SEM specific noise from images improves accuracy. Time varying drift and spatial distortions have significant effects on images acquired by the SEM.

1.3 Approach

To obtain the research goals the approach has been divided into 6 parts. The first parts involve changing and optimizing the software to perform the experimental research. After that the experiments and comparison with other DIC software gives more insight into the accuracy of the software and will give information about possible improvements.

(8)

1. Literature study: The first step is to perform a literature study on error analysis of digital image correlation and effects of scanning electron microscope noise. Previous articles are used to determine the best method to apply for the experiments.

2. Optimizing DIC software: In order to perform the experiments the DIC software has to be improved and fixed. The current program has known issues with large images and small subsets. These bugs have to be removed to be able to perform every experiment.

3. Implementation strain analysis: The displacement fields obtained through the DIC software give limited information. An implementation of strain fields will give more information and are helpful for determining the overall error.

4. Experimental verification: Experimental verification of the DIC software is split into two parts. The first focuses on experimenting with numerically simulated images, which have known applied strains. This part will give insight into problems that are caused by the DIC calculations. The second part focuses on real experiments and gives better insight into SEM specific issues.

5. Software comparison: To further dive into problems that are caused by the DIC software, a comparison between Aramis and our custom DIC software is done.

6. Improvement analysis: Discussion and conclusions of the experimental results are translated into possible improvements of the current DIC implementation.

1.4 Structure of the report

The first part of the report gives an introduction into the important principles behind the research. Chapter 2 introduces principles behind scanning electron microscopy (SEM) and discusses the noise that can influence SEM images. Chapter 3 describes the DIC implementation of the software and the mathematics behind it. Chapter 4 combines the experimental setup and results for all experiments. Chapter 5 and 6 summarizes the discussion of experiments and the conclusions respectively.

(9)

Chapter 2

Introduction to Scanning Electron Microscopy

The scanning electron microscope (SEM) was an important breakthrough in the field of materials science. Scanning a sample with the use of a focused electron beam significantly improves resolution and detail, in comparison to traditional optical microscopes. Also the interaction between the electrons and the atoms in the sample provides information about surface topography and composition. This chapter explains the working principles behind the SEM and discusses the noise it will generate during measurements.

2.1 Principles underlying the SEM

In the SEM a beam of focused high-kinetic electrons is created. When a specimen surface is irradiated by the electron beam, these electrons are measured and contain various types of information. Figure 2.1 shows the different types of information that can be obtained from the electrons. Figure 2.1(b) indicates the area from which the electrons originate.

Figure 2.1: Figure (a): Overview of different information obtained by electrons. (b) Area from which the type of electrons originate. [4]

SEM can be used for various purposes, depending on the type of information that is measured. Secondary electrons (SE) are important for topographical research of surfaces and is used for this research. The number of secondary electrons depend greatly on the incident angle, due to the Compton scattering effect. Also the energy of these electrons are

(10)

quite low, which causes them only to be emitted from a thin layer on the surface. This makes secondary electrons most suitable for topographical research of surfaces.

The SEM scans the surface by moving the finely focused electron beam in both the X- and Y-direction. The secondary electrons are detected while the beam moves over the surface. High resolution is obtained by generating a fine electron beam and detecting the electron signals efficiently.

2.2 SEM specific noise

The intensity distribution of the obtained SEM images is based on the amount of irradiated electrons. This is depicted in a grayscale image with pixel values ranging from 0 to 255. SEM images are typically under influence of spatial and time-varying drift distortions. These distortions have to be taken account for, because they are quite significant in SEM images [3].

When stress relaxation is apparent, it can also increase the drift distortions. In addition to distortion, SEM images can also be under influence of other random noise. The noise originates from the statistical behaviour of electrons in the electron gun (production of electrons) and the interaction of electrons with the sample. Also the electronic circuit that detects the secondary electrons contains noise and can amplify noise from the electrons.

2.2.1 Spatial distortions

Spatial distortion is similar to the distortion in optical lenses, but in SEM it is hard to take account for due to the complex electromagnetic focusing and scanning processes [3]. This makes well-known distortion correction models for light optics systems not suitable for the SEM. A distortion correction model is developed by Sutton et al [13]. that is capable of removing distortions in SEM measurements. Spatial distortion is a function of displacement and thus can it be quantified by using differences in displacement fields of calibration images.

The method of Sutton et al. to reduce spatial distortions is through a calibration process.

The effect of spatial distortion over the whole image is extracted from translating the sample by known amounts. This calibration process involves performing small translations and consequently obtaining pairs of images before and after translating. The set of images from the translation (calibration) process and the actual experiment is then used to extract spatial distortions.

2.2.2 Drift distortions

Drift distortion is time dependent and can be caused by the heating of components (electron beam), motion of the sample stage or by the sample itself. Charging effects of the sample and SEM stage are also quite significant and interfere with the electron beam. The electron beam takes time to measure different parts of the sample: the beam starts at the top left corner and moves in rows to the right bottom corner. This introduces time dependency on charging effects. Drift distortion is due to this effect not constant over the image. In the

(11)

article of Kammers et al. [3] it is shown that drift at low magnifications commonly appears as vertical gradients in the displacement field and at large magnification it appears as more complex gradients in the displacement field.

Methods for removing drift distortion is similar to that of spatial distortion. It uses image pairs taken from the sample without imposing additional translations. These images contain differences that are caused by the time dependent nature of drift distortions and can be extracted from comparison between images from calibration and actual measurement.

2.2.3 Stress relaxation

When using drift distortion corrections, the sample should not undergo stress relaxation.

Stress relaxation in combination with the drift correction model as described above leads to greater distortions [3].

A technique to reduce stress relaxation effects uses an unloaded pattern on a separate sample for calibration images. When measuring a sample in a tensile test, this unloaded pattern will be placed next to it. The difference between the unloaded and loaded image is used to successfully extract stress relaxation effects.

If changes in the hardware are not possible, the stress relaxation could also be reduced by again using the calibration images. The calibration images contain a load drop due to the stress relaxation. By linking the relaxation-induced displacement to the load drop, this stress relaxation can be modeled. The load drop is measured as function of time and are later translated to the exact location of each pixel.

2.2.4 Random noise

Besides correction for spatial and drift distortions, noise of random effects in SEM images must also be reduced. A general method for reducing random noise is the use of image integration. Image integration uses multiple images where every pixel value is integrated.

Other SEM parameters are also capable of reducing noise, such as longer dwell time and high beam current. Longer dwell time means longer time that the electron beam stays at one location, which averages out random effects. Higher beam current has a similar averaging effect as the beam contains more electrons (i.e. more data).

(12)

Chapter 3

Digital Image Correlation algorithm

Digital Image Correlation (DIC) is the method for quantifying the differences between images. DIC can calculate full-field displacement fields of small deformations in materials with nanometer resolution. This chapter will give an overview of the DIC implementation used in the experiments. This chapter will briefly discuss the algorithms in this DIC program, for a more in depth explanation of why these algorithms are implemented the reader is referred to [6]. This chapter is organized as follows: (1) overview of the basic principles, (2) basic definitions, (3) subset deformation, (4) correlation criteria, (5) optimization strategies, (6) obtaining full field displacements and (7) obtaining the corresponding strains.

3.1 Overview

The basic concept behind DIC is the optimisation of a correlation coefficient between a reference and target image. Optimizing this correlation coefficient results in the optimum of overlap between the images. With the use of this correlation coefficient the displacements are calculated, which is represented by the vector p. The vector p contains the x- and y- displacement (u, v, respectively) and their derivatives in both directions.

p = u v u_x u_y v_x v_y)^T (3.1)

The reference and target image contain grayscale pixel values that represent the intensity at the (x,y) coordinate. These images are divided into subsets of the same size, which are the points where the displacement vector p is calculated. Before starting the DIC algorithm the images are first interpolated to obtain sub pixel information.

The algorithm starts with an initial guess from reference to target image by the user.

A correct initial guess is crucial for a correct working DIC program, as it depends on a reliability guided calculation strategy, which will be explained in more detail in Chapter 3.6. This initial guess will be improved by using two numerical methods, which calculates an

(13)

approximation of the displacements and its gradient in both directions (complete p vector) using an iteration method.

After the displacements of the initial guess are calculated, the program will start the reliability guided DIC calculations to obtain the rest of the displacements. The resulting p vector from the initial guess is used as a guess for the neighbouring subsets. This numerical method for calculating displacements will be discussed in more detail in Chapter 3.5.3.

When all subsets (points) have been calculated, the program will return the full field displacements. Consequently this displacement data is used to calculate the strains. This will involve a smoothing algorithm to obtain accurate strain results.

3.2 Definitions

• Subsets: The images are divided into subsets of size (m x n) pixels. A subset is compared between reference and target image and will result into one displacement vector (see displacement vector).

• Grid spacing: The subsets are spread apart by a grid of (a x b) pixels. This grid spacing gives the possibility to have subsets overlap.

• Region of interest (ROI): The region of interest is defined as the area which is evaluated in the DIC program. The current program is limited to only square ROI’s.

• Correlation coefficient: To compare two subsets in the mathematical sense, the program uses a correlation coefficient. The correlation coefficient that is used in the calculations is the Zero mean Normalized Sum of Squared Differences (ZNSSD).

• Displacement vector: The transformations on subsets are considered to be small, so only first order displacements are taken into account. The displacements and derivatives of the displacements are in total 6 unknowns which are combined into the p vector.

p = u v ux uy vx vy)^T where u_x = ^∂u_∂x, u_y = ^∂u_∂y, v_x = ^∂v_∂x and v_y = ^∂v_∂y.

• Warp function: A warp function is used to quantify transformations between two sets of points (subsets). The inputs of the warp functions are (1) the local coordinates of a set of points and (2) the displacement vector between the two sets of points. This is also known as a displacement mapping function.

(14)

Figure 3.1: Example of a DIC image with the definition of subset size and grid spacing.

The subset size is m x n pixels and the grid spacing is a x b pixels.

3.3 Subset deformation

The image is divided into subsets which contain (m x n) pixels and are spaced over the image on a grid with spacing of (a x b) pixels. The grid spacing can be used to have subsets overlap. These definitions are illustrated in Figure 3.1.

The subset is translated from reference subset to the target subset. The mathematical notation for the grayscale pixel values is denoted f_ij = f (x_i, y_j) and g_ij = g(x_i, y_j) for the reference and target image respectively. If the transformations are small enough they are typically of a first order kind and can be defined by

x_tar= x_ref + u +∂u

∂x(x_ref − x_ref,0) +∂u

∂y(y_ref − y_ref,0) (3.2) and

y_tar= y_ref+ v + ∂v

∂x(x_ref− x_ref,0) + ∂v

∂y(y_ref − y_ref,0) (3.3) where the index (xtar, ytar) and (x_ref, y_ref) represent the coordinates of target and reference image respectively. The index (x_ref,0, y_ref,0) are the center points of each subset. Equation 3.2 and 3.3 can be written in matrix form with the use of a warp function

ζref,0+ W(ξref, p) =



 x_ref,0 y_ref,0

1



+





1 + u_x u_y u vx 1 + vy v

0 0 1









∆x_ref

∆y_ref 1



 (3.4)

where ζ_ref,0 = (x, y, 1) is a vector containing all x and y coordinates, ξ is an augmented vector containing the local x and y coordinates inside a subset (ξ = ∆x, ∆y, 1T

), ∆x and

∆y are the distances between a point inside the subset and its center. The warp function, also known as displacement mapping function, is used to quantify transformations from reference to target image and is defined as

(15)

W(ξ, p) =





1 + u_x u_y u v_x 1 + v_y v

0 0 1









∆x

∆y 1



 (3.5)

Transformations of a first order kind are sufficient to consider, because the strains on a subset are considered to be small. The possible transformations are illustrated in Figure 3.2 and any linear combination of them can be calculated using the warp function W(ξ, p).

Figure 3.2: This figure shows the possible transformations on a subset. Any linear combination of the 6 transformations can be calculated in the warp function.

3.4 Correlation criteria

The correlation coefficient is a mathematical quantification of the similarities between two subsets. The chosen correlation criterion for this DIC implementation is the Zero mean Normalized Sum of Squared Differences, in which a lower correlation coefficient says that the pixels between two subsets are more equal. This inverse relationship is due to the sum squared differences and can be seen from the definition of the correlation coefficient in Equation 3.6.

C_{ZN SSD}=X f¯_ij qPf¯_ij²

− ¯g_ij qP ¯g²_ij

2

(3.6)

where ¯f_ij = f_ij − f_m and ¯g_ij = g_ij − g_m in which the means are defined as f_m = ¹_nP f_ij and g_m = _n¹ P g_ij. The number n is the total number of pixels analyzed in the subset.

The value of the C_{ZN SSD} ranges from [0,4], where 0 means a perfect correlation. This correlation coefficient is able to take into account brightness differences which are apparent in SEM imaging (see Figure 3.3).

In order to determine the deformations between these subsets, the DIC algorithm uses the correlation coefficient to find an optimum by differentiation. To optimize the correlation

(16)

Figure 3.3: Two identical images obtained with the SEM which contain brightness differences due to charging effects. Mean pixel values are ¯f = 95.24 and ¯g = 88.46 for (a) and (b) respectively.

coefficient for the displacement vector, the warp function as discussed in Chapter 3.3 (subset deformation) is put in the correlation coefficient and results in the formula

CZN SSD =X f (x) − ¯f qPf¯_ij²

−g(x + W(ξ, p)) − ¯g qP ¯g_ij²

2

(3.7)

3.5 Optimization strategies

For obtaining the most accurate displacement fields several optimization strategies are considered. This chapter is organized in a chronological order, the order in which they are calculated in the program. The image quality (amount of information) is first improved using a bi-cubic spline interpolation method. The initial guess that is given by the user is improved by a coarse fine search based on the correlation coefficient. The next step is improving the displacement vector using the Inverse Compositional Gauss Newton (IC-GN) method. The rest of the subsets are only calculated based on the IC-GN method and finding the minimum of the correlation coefficient. These methods are described in detail in this chapter.

3.5.1 Bi-cubic spline interpolation

In the process of optimizing the displacement vector it is evident that the pixel values fij

and gij must be provided. There is no pixel information available between pixels. These subpixels can be provided by using an interpolation scheme. It is shown that good accuracy and continuous derivatives can be acquired using a higher order scheme, in specific the bi-cubic splines interpolation scheme [8]. This interpolation scheme is used for this DIC implementation.

(17)

In this algorithm a region of 2 x 2 pixels is interpolated to what is called an interpolation block. This scheme determines the pixel values and first order pixel gradients at sub-pixel locations. These sub pixel values are determined by

f (x, y) =

3

X

i=0 3

X

j=0

a_ijxⁱy^j (3.8)

in which the coefficients a_ij are the 16 unknowns that need to be solved. The 16 unknowns are determined by the 4 corner pixel values and 12 pixel gradients fx, fy and fxy. These gradient values are obtained by looking at the surrounding pixels in a 4 x 4 pixel block.

3.5.2 Coarse fine search

The user gives the initial guess of the u and v parameters of the displacement vector. This initial guess is improved by a brute force algorithm called coarse fine search. This method searches for the best displacement vector p in a region of 3 x 3 pixels around the initial guess. It tries all possible combinations of u and v with an accuracy of 0.25 pixels (sub-pixel level) and compares the correlation coefficient. The sub pixel values are calculated using the data from the interpolation. This method is implemented to give an improved initial guess for the more sophisticated algorithm that calculates all other subsets, namely the Inverse Compositional Gauss Newton method discussed in the next section.

3.5.3 Inverse Compositional Gauss Newton (IC-GN)

The most important algorithm in the DIC program is the Inverse Compositional Gauss Newton (IC-GN) method for finding the displacement vector p. This method is used in numerical mathematics for finding the root of a function. The root finding problem of the DIC program is to find an optimum of the correlation coefficient, which takes p as a parameter. Thus the derivative of C_{ZN SSD} with respect to p needs to be equated to zero.

This holds under certain conditions for multiple dimensions and results in the following 6 equations (one for every vector element of p)

∂C_{ZN SSD}

∂p = 0 or C_p_i = 0 (3.9)

The basic working principle behind IC-GN is the use of a warp function to update a subset. The IC-GN method reverses the roles of the reference subset and target subset. This reversed strategy puts an incremental warp function W(ξ, ∆p) over the reference subset and compares it with the deformed target subset deformed by W(ξ, p). Although the reference image is not deformed, the use of an incremental warp function for the reference subset has computational benefits that will become clear in a moment. This concept is summarized in Figure 3.4.

(18)

Figure 3.4: Schematic overview of the backward matching strategy of the Inverse Compo- sitional Gauss Newton method. The solid (blue) line is the initial subset, the dotted (red) line is the updated/warped subset. [9]

The warp function W(ξ, p) was already defined in Equation 3.5 and the incremental warp function is defined as

W(ξ, ∆p) =





1 + ∆ux ∆uy ∆u

∆v_x 1 + ∆v_y ∆v

0 0 1









∆x

∆y 1



 (3.10)

where ∆p = (∆u, ∆v, ∆u_x, ∆u_y, ∆v_x, ∆v_y)^T denotes the incremental vector that needs to be determined. The incremental warp function can be solved by equating the derivative of the correlation coefficient to zero (Equation 3.9). The definition of the correlation coefficient C_{ZN SSD} is different for the inverse approach and is defined as

CZN SSD(∆p) =X f (x + W(ξ, ∆p)) − ¯f qPf¯_ij²

−g(x + W(ξ, p)) − ¯g qP ¯g_ij²

2

(3.11)

The incremental warp function can be solved by minimizing this expression. Subse- quently the incremental warp function is inverted and used to update the warp function that is imposed on the target subset. This updating of the warp function is defined by:

W(ξ, p_k+1) = W(ξ, p_k) W(ξ, ∆p)⁻¹ (3.12) where p_k is the displacement vector at iteration k. The next step is to find a way to minimize Equation 3.11 to find ∆p. The derivation of this ∆p can be found in Appendix

(19)

?? and results in the equation

∆p = −H⁻¹×X

∇f∂W

∂p

T

(f (x + ξ) − ¯f ) −∆f

∆g(g(x + W(ξ, p)) − ¯g)

!

(3.13)

in which

∇f =

∂f

∂x

∂f

∂y

(3.14)

∂W

∂p =1 ∆x ∆y 0 0 0

0 0 0 1 ∆x ∆y

(3.15) and H is the 6x6 Hessian matrix:

H =X

∇f∂W

∂p

T

∇f∂W

∂p

!

(3.16) Equation 3.16 has no dependence of p and ∆p. As the method iterates to update the warp function of both the reference image and target image, it recalculates those variables.

Since the Hessian has no dependency of those variables, it stays constant in the iterations and the Hessian can be precomputed. This pre-computation of the Hessian has computational benefits and is due to the reversal of the roles of reference and target image.

The program flow is summarized in Figure 3.5. Every IC-GN starts with the input of an initial guess of p. Then it calculates the pixel values and derivatives of the reference subset, as well as the Hessian, which are all constant per iteration. Every iteration contains the following calculations: (1) a new ∆p will be calculated from optimization of the correlation coefficient (equation 3.13), (2) merge ∆p into a new p (Equation 3.12), (3) the iteration procedure will be repeated while the convergence criterion is not satisfied. Then at the end it will return the final displacement vector p_k. The method returns an error status if (a) the maximum iterations are exceeded or (b) the new p is out of bounds, i.e. the target pixels will be outside the image.

(20)

Figure 3.5: Overview of the IC-GN algorithm which is used to calculate the displacements.

(21)

3.6 Method for obtaining Full Field Displacements

From the minimization of the correlation coefficient C_{ZN SSD} in the IC-GN algorithm, follows an approximation of the displacement vector p between reference and target image of one subset. This results in displacement data of a single material point. To obtain displacement data of all subsets the program uses a Reliability Guided DIC (RG-DIC) strategy [10].

This strategy is commonly used in DIC to obtain a robust algorithm. A brief summary is given here below, for more in depth explanations the reader is redirected to the article of Pan et al. [10].

3.6.1 Reliability Guided DIC (RG-DIC)

The computation starts at the subset of the initial guess of the user. After calculating the correlation coefficient C_{ZN SSD} and the deformation parameters of the initial guess point, its status is updated to ”queued”, which will be done by a status variable set to ”1”. This status variable can take the values: 0 = Not analyzed, 1=Queued and 2=Finished. The results of the initial guess are stored in a priority queue, which is formed as a heap data structure that orders the elements inside based on some property. This ordering of the priority queue is done based on the correlation coefficient, where the lowest value of CZN SSD is put in front.

The program will then enter a while loop and at each iteration it will pop the top entry from the queue. The four surrounding subsets around this subset are then analyzed using the displacement data from this queue point as the initial guess for the IC-GN method. The status of these 4 surrounding points are updated to ”queued” and the status of the center queue point is updated to ”finished”. This process is repeated until the queue is empty and all subsets are finished. These steps are visually summarized in an example in Figure 3.6, where the direction of calculation is indicated by the arrow.

(22)

Figure 3.6: The direction of reliability guided DIC calculations. Every block represents a subset. The calculations proceed in the direction of the lowest CZN N SD (indicated by the arrow).

The advantage of this method is its robustness, the bad correlated points (i.e. data points with high C_{ZN SSD}) will be processed last. The effect is that the first approximation of p for the IC-GN method always comes from the queue point with the lowest correlation coefficient. Another positive effect is that an initial guess provided by an adjacent subset is quite good, given that the subset spacing is not too large, because the displacement fields are in general relatively smooth.

3.7 Method for obtaining Full Field Strains

Although the displacement gradients are directly calculated in the IC-GN method (the last 4 terms of the p vector), the results show large variations indicating that they are not reliable. A commonly used method to reduce this variation in displacement gradient data, is interpolation of the displacement field to obtain a smoother displacement field and consequently calculating the strains from differentiation. This means that the gradient data from the IC-GN calculations are ignored. A negative effect of numerical differentiation is that it amplifies the noise contained in the displacement field, thus it is widely accepted that by first smoothing the data by an interpolation method and then using numerical differentiation will improve strain results [8]. A point wise least squares fitting algorithm is used to smooth the displacement fields and is discussed in the next subsection.

3.7.1 Point wise least squares fitting of displacement fields

The displacement vector field has to be divided into new sets of points called the strain calculation window. This window contains (2m + 1) x (2m + 1) discrete points which are interpolated. If the strain calculation window is small enough, it can be approximated by a linear fit, given by

(23)

u(i, j) = a₀+ a₁∗ x + a₂∗ y = a_u,plane+ (∂u

∂x)x + (∂u

∂y)y (3.17)

and

v(i, j) = b₀+ b₁∗ x + b₂∗ y = b_v,plane+ (∂v

∂x)x + (∂v

∂y)y (3.18)

where i,j = -m:m are the local coordinates within the strain calculation window, u(i,j) and v(i,j) are the original displacement fields obtained with the IC-GN method and a0,1,2 and b0,1,2 are the polynomial coefficients that need to be solved. These coefficients are used to calculate the strains.

Equations 3.17 and 3.18 can be written in matrix form as







1 −m −m

1 −m + 1 −m ... ... ...

1 0 0

... ... ...

1 m − 1 m

1 m m









 a₀ a1

a₂



=







u(−m, −m) u(−m + 1, −m)

... u(0, 0)

... u(m − 1, m)

u(m, m)







(3.19)

and is used to calculate the unknown polynomials. The corresponding Green strains of each strain calculation window is then calculated from the polynomials:

Exx= 1 2

2∂u

∂x+ (∂u

∂x)²+ (∂v

∂x)²

= 1 2

2a1+ a²₁+ b²₁

(3.20)

E_xy = 1 2

∂u

∂y + ∂v

∂x +∂u

∂x

∂u

∂y + ∂v

∂x

∂v

∂y

= 1 2

a₂+ b₁+ a₁a₂+ b₁b₂

(3.21)

Eyy = 1 2

2∂v

∂y + (∂u

∂y)²+ (∂v

∂y)²

= 1 2

2b2+ a²₂+ b²₂

(3.22) This process is repeated for all subsets to obtain a full strain field from the displacements.

The size of the strain calculation window can be independently controlled and should be chosen with care. For homogeneous deformations the displacement field is linear and the strain calculation window should be large. While for inhomogeneous deformations the strain calculation window should be based on the best combination of strain accuracy and smoothness. A small strain calculation window is not able to suppress noise, while a larger strain calculation window will lead to incorrect linearization of deformations. The strain calculation window should be chosen between 11 x 11 to 21 x 21 points, according to Pan et al. [8].

(24)

Chapter 4

Experiments and results

The main goal of the experiments is to obtain knowledge about the working principles behind the DIC program and what accuracy it can achieve. Images obtained with the SEM contain various kinds of noises as discussed in Chapter 2. In general, the validity check of the displacement fields become experimentally very difficult when the transformations are not rigid. To eliminate those difficulties the first part of the experiments focuses on numerically generated images. These images are deformed by known amounts to give insight into the accuracy of the DIC program. This is a well known method to investigate DIC displacement fields [1][9][15]. The second part focuses on real SEM images to check the flexibility of the program and how the errors introduced in the SEM will involve the DIC calculations. The comparison with other DIC software, in specific Aramis, could tell more about the flexibility about the program. Also differences in obtained displacement fields could explain errors in the DIC calculations.

Every experiment contains an ”x” amount of images that are compared using DIC.

To denote this correlation between a set of images in the results, a shorter notation is used: ”experiment[ref erence],[target]”. The reference and target images are represented by a number. For example Solder2,1 denotes the experiment on the Solder sample, where DIC is performed from image 2 (reference) to image 1 (target).

All calculations of the DIC program are done with the Matlab 2015a software (version 8.5.0.197613 (R2015a)). Due to long calculation times (up to 48 hours) multiple computers were used for the experiments.

4.1 DIC configuration

The DIC calculations can be optimized by altering the program parameters. These parameters are summarized in Table 4.1 and all have a different influence on the results.

Optimization of parameters focuses on two key points. The first is improving the convergence of the amount of subsets. Better convergence means that more displacement vectors are calculated. The second is improving the displacement vector itself, to obtain higher accuracy in the results. Improving parameters always come at a cost of higher calculation

(25)

time. Currently the program is slow (up to 100 hours per image) making it difficult to vary the parameters, so it should be chosen with care.

Parameter Standard

value Description

Subset size 11 x 11

This defines the width and height of the subsets (in pixels). The subset size has influence on the convergence. A larger subset size results in a larger area per subset, increasing the chance for a subset to convergence to an answer. It is also better able to average out the noise per subset.

Grid spacing 11 x 11

This defines the spacing between center points of subsets (in pixels). Grid spacing has direct influence on the amount of calculated points. A smaller grid spacing increases the amount of points. When the grid spacing is smaller than the subset size, it causes the subsets to overlap.

Convergence crite-

rion 0.01

The convergence criterion determines when the iterations are stopped, i.e. when the displacement vector is assumed to be correct. A lower convergence criterion results in an improvement of the displacement vector, but largely reduces the total convergence.

Maximum itera-

tions 50

This determines after how many iterations the IC- GN is stopped and the subset is assumed not being able to converge.

Pixel precision 0.1

This defines the precision for the correlation algorithm per pixel. The pixel precision squared is the amount of subpixels. A lower pixel precision is expected to obtain better displacement vectors, but has a big influence on the total calculation time.

Table 4.1: Overview of the DIC parameters that can be varied to optimize convergence of subsets or improve the displacement vector.

Experiments on sample images have been done to determine the ideal DIC configuration.

Some results were as expected, such as that the increase of subset size leads to an increase of convergence. The convergence criterion of 0.01 is assumed to be ideal, as stated in the article of Pan et al. [7]. Convergence criterion has a more significant influence on computational efficiency than accuracy, for this reason this parameter will be kept constant for all experiments.

Two parameters show an unexpected result. The first one was that the increase of iterations would lead to an significant increase of total calculation time. This is not the case for most images, in fact the total calculation time is only increasing by small amounts.

The reason for this is that the average iterations is around 3. Increasing the iterations has

(26)

only influence on subsets that are difficult to calculate. Results have shown an improvement of convergence, while the calculation time doesn’t increase significantly. For this reason the maximum iterations is determined to be 50.

The second unexpected result was that the a better pixel precision doesn’t lead to a significant improvement of accuracy, while the calculation time does increase exponentially.

An improvement of the pixel precision by a factor 10 (from 0.02 to 0.2) leads to an increase of calculation time by a factor 50. The most probable reason is that sub pixels doesn’t necessarily keep improving the information that is contained in one pixel. The amount of sub pixels per pixel with a pixel precision of 0.1 and 0.01 is 100 and 10000 respectively.

This increase of sub pixels has a small influence on the accuracy, while the calculation time increases significantly. Due to this reason the standard value for pixel precision is determined to be 0.1.

4.2 Numerical experiments

Numerically generated images are used to investigate errors introduced in the DIC calculations. The first part uses a real image which is deformed by known amounts and then interpolated to obtain correct sub pixel values. The second part uses a data set from the 2D DIC challenge from the Society of Experimental Mechanics.

4.2.1 Real interpolated images

For this experiment a real reference image from the SEM measurements on nanoporous gold is used. This reference image is deformed by known amounts in the (a) x direction, (b) y direction and (c) both x- and y-direction. This is shown in Figure 4.1 with the corresponding deformations. The deformed images are obtained with bi-cubic interpolation to remain correct sub pixel values. The resulting strains are used to determine the absolute error of the DIC program. 6 different experiments were executed with varying DIC parameters as shown in Table 4.2.

Exp. Subset size Grid spacing Pixel precision

1 11 x 11 11 x 11 0.1

2 21 x 21 11 x 11 0.1

3 21 x 21 21 x 21 0.05

4 31 x 31 21 x 21 0.05

5 41 x 41 41 x 41 0.025

6 11 x 11 11 x 11 0.05

Table 4.2: DIC parameters for the interpolated images with in total 6 different experiments.

Results

(27)

Figure 4.1: Reference image for the numerical experiments. Four different deformations are applied: (a) ^∂u_∂x = 0.08, (b) ^∂v_∂y = 0.08 and (c) ^∂u_∂x = ^∂v_∂y = 0.08. The yellow arrows represent the displacement vectors per subset.

The 6 experiments are performed three times with different strains. The results of the homogeneous deformation in the x-direction (^∂u_∂x = 0.08) is shown in Table 4.3, the y-direction (^∂v_∂y = 0.08) in Table 4.4 and the deformation in both x- and y-direction (^∂u_∂x =

∂v

∂y = 0.08) in Table 4.5. The values shown in these tables are the interpolated deformations (see Chapter 3.7). Results show good accuracy in the order of 1E-03, but it is quite constant for different experiments (varying DIC parameters). Also the standard deviation doesn’t include the theoretical value of 0.08.

Exp. Cmean < ux,intp> σux,intp ux,error

1 5.40E-04 0.080085 2.16E-05 1.06E-03 2 4.83E-04 0.080080 2.16E-05 1.00E-03 3 4.86E-04 0.080081 5.13E-06 1.01E-03 4 4.85E-04 0.080079 2.04E-06 0.99E-03 5 4.81E-04 0.080083 9.06E-06 1.04E-03 6 5.40E-04 0.080085 2.16E-05 1.06E-03

Table 4.3: Results of the experiments on homogeneous deformation in the x-direction (^∂u_∂x = 0.08). Cmeanis the average correlation coefficient, < .. > indicates the average, the subscript

”intp” denotes the interpolated values and σ is the standard deviation.

4.2.2 2D DIC challenge

On the website of the community of Society for Experimental Mechanics (SEM) there are several 2D DIC data sets available [11]. These sets offer a good way to perform software testing and verification of DIC methods and are meant for both commercial codes as well for university codes. Exact solutions of the data sets are provided, from which absolute DIC

(28)

Exp. C_mean < v_y,intp > σ_v_y,intp v_y,error 1 3.90E-04 0.080085 2.39E-05 1.06E-03 2 3.21E-04 0.080079 1.89E-06 0.99E-03 3 3.22E-04 0.080080 4.93E-06 1.00E-03 4 3.18E-04 0.080080 1.24E-06 1.00E-03 5 3.15E-04 0.080076 8.14E-06 0.96E-03 6 3.89E-04 0.080084 2.40E-05 1.06E-03

Table 4.4: Results of the experiments on homogeneous deformation in the y-direction (_∂y^∂v = 0.08). Cmeanis the average correlation coefficient, < .. > indicates the average, the subscript

”intp” denotes the interpolated values and σ is the standard deviation.

Exp. Cmean < ux,intp> σux,intp ux,error < vy,intp> σvy,intp vy,error

1 9.39E-04 0.080084 2.79E-05 1.04E-03 0.080080 3.14E-05 0.99E-03 2 8.21E-04 0.080079 6.10E-06 0.99E-03 0.080079 3.89E-06 0.99E-03 3 8.22E-04 0.080080 6.16E-06 1.00E-03 0.080080 5.22E-06 1.01E-03 4 8.22E-04 0.080079 3.50E-06 0.98E-03 0.080079 2.64E-06 0.99E-03 5 8.25E-04 0.080084 1.05E-05 0.96E-03 0.080076 6.06E-06 0.96E-03 6 9.38E-04 0.080083 2.86E-05 1.03E-03 0.080079 3.14E-05 0.98E-03 Table 4.5: Results of the experiments on homogeneous deformation in the x- and y-direction (^∂u_∂x = ^∂v_∂y = 0.08). C_mean is the average correlation coefficient, < .. > indicates the average, the subscript ”intp” denotes the interpolated values and σ is the standard deviation.

(29)

errors can be estimated. The details of creation of the samples are described in a published paper [12].

Sample data set 15 is used for this experiment, which contains a varying strain as function of the y coordinates. The images are obtained using the TexGen method from which 9 different images are acquired. The reference image is shown in Figure 4.2. An Excel file was added which contains the v displacements per pixel row in the y direction and is constant in the x-direction. The error is determined from the absolute difference in obtained v results.

Figure 4.2: Reference image of the 2D DIC challenge from the Society for Experimental Mechanics. Data set 15 contains 9 images with varying strain [11]. The yellow arrows represent the displacement vectors at the subsets.

Results

The challenge provides an Excel file which contains the exact v displacements per pixel row. The results from the DIC measurements are done per subset and has an height of 11 pixels. The displacement per subset is compared with the averaged displacement from the Excel file, by taking the absolute difference in pixels (absolute error). These errors are plotted as function of the subset row number (1 to 90) in Figure 4.3. The displacements are calculated for 7 different images (k50,k100,k150,k250,k300,k350 and k400). The average absolute error is determined to be 1.11 ∗ 10⁻² pixels and the maximum error is 7.09 ∗ 10⁻² pixels which is smaller than the pixel precision of 0.1 pixels. This indicates that the program has a correct accuracy for all subsets.

It is important to note that the Excel file from the 2D DIC challenge has been corrected for a ”Euler-Lagrange” error, which has been detected by an earlier measurement. For more information on this matter the reader is referred to P. Rue [12].

(30)

Figure 4.3: Absolute error (in pixels) determined from data set 15 of the 2D DIC challenge [11]. The error is logarithmically displayed as function of the subset row number, where each subset row contains 11x1000 pixels. The displacements from the 2D DIC challenge have been corrected by an ”Euler-Lagrange” error [12].

4.3 SEM images: Nanoporous Gold

Images obtained with the SEM are used to check the flexibility of the program and to investigate how errors introduced in the SEM will involve the DIC calculations. SEM images can contain various kind of distortions and cause difficulties in DIC. A correct choice of parameters for the program should minimize the errors and optimize the convergence of subsets.

The SEM measurements are done one a XL30 ESEM for high definition images. The electron beam is set to a voltage of 10kV and a spot size of 3. The working distance and magnification vary per experiment.

The experiments are done with a sample of nanoporous gold, which contains enough detail at the nanometer scale to achieve good DIC results. Three different experiments are performed: (1) static images, (2) translated images and (3) tilted images and are described in the next subsections.

4.3.1 Static

The time dependent nature of SEM imaging introduces distortions. This was seen during the measurement of three identical images, where the mean pixel value is decreasing over time due to charging effects. Figure 4.4 shows the three images where the shaded (red) area is the region of interest. Quantification of SEM induced errors is difficult, but it is expected that the DIC results could visualize these errors.

(31)

Figure 4.4: Three static (identical) images obtained with the SEM, which are influenced by SEM noise. The region of interest is shaded red. The mean pixel values are (a) ¯f = 95.24, (b) ¯g = 93.18 and (c) ¯h = 88.33.

Results

The three images are evaluated using the DIC program and results in two displacement fields: (1) Static2,1 and (2) Static2,3. The resulting u and v displacement fields are shown in Figure 4.5, where the axis values are in µm. The images contain gradients in the displacement fields, while theoretically it should be a constant zero field. This is caused by SEM specific errors such as charging effects. The displacement fields of Static2,1 (Figure 4.5(a) and (b)) show a similar gradient in size and direction. The order of magnitude of the SEM errors is several nanometers.

Figure 4.5: The u and v displacement fields from DIC on identical images for Static_2,1((a) and (b) respectively) and Static2,3 ((c) and (d) respectively)).

4.3.2 Translated

To increase complexity for the DIC program, the second experiment uses a set of images obtained of nanoporous gold that are translated with the SEM. The program should be able to recognize these constant transformations (u and v) and should result in a homogeneous displacement field, when assuming the sample is not influenced by internal displacements.

Differences in obtained displacement fields and nonzero gradients give insight into SEM distortions and DIC errors. Figure 4.6 shows the images where the shaded red area is the region of interest.

(32)

Figure 4.6: Two SEM images that have been translated by ∼ 50nm in u and v. (a) and (b) are the obtained SEM images where the shaded (red) area is the region of interest.

Results

To obtain information about the errors as with the static experiment, the translated component of the images must be removed. The translation is approximately ∼ 50nm in both u and v. The correction factors ¯u and ¯v for the displacement field of u and v respectively are determined from the average displacements. The resulting corrected displacement fields are displayed in Figure 4.7. Results show similar SEM induced gradients as with the static experiment, but also shows horizontal and vertical bands. These bands are typical SEM errors and are approximately 400nm apart [5].

Figure 4.7: u and v displacement fields of the translated image set, which are corrected with a constant factor ¯u and ¯v to remove the translated component. The averages ¯u and ¯v are determined from the displacement fields.

(33)

4.3.3 Tilted

More complex transformations are investigated by tilting the sample in the SEM. As the tilting angle gets larger, the distortions become more apparent (see Figure 4.8). The DIC parameters are optimized to obtain as much detail as possible. In total three images of nanoporous gold are obtained in the SEM. The two resulting p vectors are used in a 3D analysis as described in Chapter 4.6.

Figure 4.8: Three SEM images from which (b) and (c) have been tilted at respectively by 5 and 10 degrees with respect to the reference image (a). The region of interest is shaded red.

Results

As the tilt angle increases, the noise in the images also increases. Rough samples could also cause difficulties for the DIC at local points. The results from Figure 4.9 show that the convergence of Tilted_2,1 (a) is perfect and the convergence of Tilted_2,3 contains many failed subsets (b). The convergence results are summarized in Table 4.6, where experiments (a1) & (b1) are done with a subset size of 11x11 pixels and (a2) & (b2) with a subset size of 21x21 pixels. The increase of subset size results in a increase of convergence of 12.6% for (b). Convergence of (a) decreases with 1.5% due to an increase of out of bound subsets.

Exp. Subset size (px) Subsets Convergence Out of bound Failed

(a1) 11x11 7369 99.1% 0.9% 0.0%

(b1 11x11 7369 77.3% 1.7% 21.0%

(a2) 21x21 7369 97.6% 2.4% 0.0%

(b2) 21x21 7369 89.9% 1.1% 9.0%

Table 4.6: Convergence results of the tilting experiment on nanoporous gold. Two measurements are done for different subset sizes: 11x11 and 21x21 (pixels). The image size is 968x968 pixels. Larger subset size results in better convergence.

(34)

Figure 4.9: DIC result of (a) Tilted_2,1 and (b) Tilted_2,3. The subset size is 21x21 pixels and the yellow arrows represent displacement vectors per subset. Blue dots represent subsets out of bounds and red dots subsets that failed calculation.

4.4 SEM images: JEOL

To test the flexibility of the program several data sets have been provided by F. Timis- chl from the development department of JEOL (Japan Electron Optics Laboratory) [14].

These images are challenging for DIC programs due to low contrast, large variations and complex structures. DIC parameters are optimized to achieve the best results. This part of the experiments will focus on qualitative results, i.e. the amount of calculated points (convergence).

4.4.1 Solder

The first data set is a flat sample of non-uniform composition with fine scratches and holes on its surface. Three images are obtained using the SEM with magnification=2500, working distance=11mm and acceleration voltage=20kV. The width of the scan is 51.2µm. In Figure 4.10 is visible that the images contain enough detail, but the image resolution is quite bad (large noise). A larger subset size should be chosen, such that it contains enough detail and averages out the noise.

(35)

Figure 4.10: Three SEM images of a non-uniform composition with scratches and holes on its surface. The images are rotated with respect to each other by unknown amounts. The images are acquired by JEOL [14]. The region of interest is shaded red.

Results

Two DIC measurements were executed for a subset size of 11x11 pixels (a1 & b1) and 21x21 pixels (a2 & b2). The convergence results are summarized in Table 4.7. The first experiment with a small subset size show that DIC has difficulties with the convergence (30%

of the subsets failed calculation). As expected the low convergence is a result from the bad image quality. Although there is enough local detail available, the difference between images are too high for DIC to find a correlation.

Figure 4.11: DIC result of (a) Solder2,1 and (b) Solder2,3 for a subset size of 21x21 pixels. The yellow arrows represent displacement vectors per subset, blue dots subsets out of bounds and red dots subsets that failed calculation.

The convergence is improved by increasing both the subset size and grid spacing to 21x21 pixels (a decrease of total subsets from 7369 to 2025). The resulting displacement fields are shown in Figure 4.11. Results show a convergence of approximately 90% for both Solder2,1 and Solder2,3. A larger subset contains more local detail and is able to average out the noise.

(36)

(a1) 11x11 7369 66.5% 2.4% 31.1%

(b1) 11x11 7369 64.3% 2.4% 33.3%

(a2) 21x21 2025 89.7% 2.1% 8.2%

(b2) 21x21 2025 87.3% 2.1% 10.6%

Table 4.7: Convergence results of the solder data from JEOL. Two measurements are done for different subset sizes: 11x11 and 21x21 (pixels). The image size is 960x960 pixels.

4.4.2 Mesh

The second data set consists of a metal mesh on an aluminium specimen holder. The images are obtained using the SEM with magnification=75, working distance=11mm and acceleration voltage=20kV. The width of the scan is 1.7mm, which is much larger than the rest of the experiments. Although the image is quite large it contains enough detail on the metal mesh. The difficult part is the complex structure, which could cause problems for the reliability guided displacement calculations (see Figure 4.12). The DIC program is not likely to be able to calculate the black parts inside the mesh.

Figure 4.12: Three SEM images of a metal mesh on an aluminium specimen holder. The images are tilted by unknown amounts with respect to each other. The region of interest is shaded red.

Results

The complex structure of the Mesh is expected to be difficult for DIC, but Figure 4.13 shows a displacement field of the largest part of the mesh. Black spots within the mesh all failed to converge, because there is not enough detail available. This example illustrates the working principle of the reliability guided strategy, while the black parts all fail calculation, the subsets on the mesh still obtain good convergence.

(37)

Figure 4.13: DIC result of (a) Mesh_2,1 and (b) Mesh_2,3. The yellow arrows represent displacement vectors per subset, blue dots subsets out of bounds and red dots subsets that failed calculation.

The image consists of an area of approximately 63% black holes and 37% mesh. A convergence of 37% would than be considered ideal. The convergence is summarized in Table 4.8 and shows that for both Mesh_2,1 and Mesh_2,3 it is almost ideal. In Figure 4.13 the displacement fields are shown, from which the amount of red points on the mesh structure show that the convergence is not ideal. This difference is probably because subsets are squares and the square area will overlap significantly with the black parts.

In the next part a comparison with Aramis is done and shows that Aramis is not capable of calculating this complex mesh structure.

4.5 Comparison with Aramis

Aramis is a commonly used commercial DIC software package. This software uses a method that is comparable to the DIC implementation that is used in this research, although the details of the working principles behind Aramis are unknown. Aramis has several benefits:

firstly the calculations are faster making it possible to vary the DIC parameters more easily.

Secondly Aramis uses multiple starting points (initial guesses), which increases the ability to calculate every subset. For example an image where the DIC is unable to calculate

(a) 21x21 7369 35.4% 1.1% 63.6%

(b) 21x21 7369 36.2% 1.1% 62.8%

Table 4.8: Convergence results of the mesh data from JEOL. The image size is 960x960 pixels and subset size is 21x21 pixels.

(38)

subsets in a large grain boundary, such that the other side of the grain boundary can’t be calculated.

The resulting displacement fields of the two programs are compared to identify any differences or similarities, which is done with the tilted samples of nanoporous gold (see Figure 4.8). The two programs are also compared for its flexibility to achieve displacement fields of complex images. This is done with the metal mesh from JEOL (see Figure 4.12).

Results

The comparison between Aramis and the custom DIC implementation is done by looking at the differences in the resulting displacement fields. The data of the tilting of nanoporous gold (Chapter 4.3.3) is used for this comparison. The subset size (21x21 pixels) and grid spacing (11x11) is the same for both DIC calculations. The resulting grid points from Aramis differ slightly from the custom DIC, making it unusable for comparison. To obtain displacement fields on the same grid, the data is interpolated using a cubic interpolation.

Consequently the displacement field points are calculated on the same x- and y coordinates.

These interpolated displacement fields are shown in Figure 4.14, where (a), (b), (c) and (d) are the u and v displacement fields of Tilted_2,1 ((b) and (d) are obtained with Aramis).

Figures (e),(f),(g) and (h) are the u and v displacement fields of Tilted2,3, where (f) and (h) are obtained with Aramis.

Figure 4.14: Comparison of displacement fields between Aramis and custom DIC. (a), (b), (c) and (d) are the u and v displacement fields of Tilted_2,1, where (b) and (d) are obtained with Aramis. (e),(f),(g) and (h) are the u and v displacement fields of Tilted2,3, where (f) and (h) are obtained with Aramis.

Visually the displacement fields show similar results for Aramis and the custom DIC.

To identify the differences between the fields, the displacement fields of both programs are subtracted for Tilted2,1 in Figure 4.15(a)(b) and for Tilted2,3 in Figure 4.15(c)(d). The intensity scale of the images is in nanometers and shows that the differences are small.

(39)

Figure 4.15: Differences in displacement fields of u and v between Aramis and custom DIC from Tilted2,1 ((a) and (b)) and Tilted2,3 ((c) and (d)).

DIC < u_custom− u_aramis> RM S_u < v_custom− v_aramis> RM S_v Tilted_2,1 3.47e⁻² nm 4.27e⁻¹ nm 6.80e⁻¹ nm 8.96e⁻¹ nm

Tilted_2,3 −1.23e⁻² nm 6.65e⁻¹ nm −1.07 nm 2.20 nm

Table 4.9: Results from the differences between Aramis and custom DIC, where < .. >

denotes the average and RMS is the Root Mean Square error between Aramis and custom DIC.

The results are summarized in Table 4.9 with the average differences and the Root Mean Square error between Aramis and the custom DIC. The error of Tilted2,3 is larger, because the third image was stronger distorted and causes more variation between the two DIC programs.

4.6 3D topography reconstructions

The DIC implementation is developed for the purpose of calibration-free quantitative 3D topography reconstruction [2]. This 3D reconstruction can be made with a data set of 3 images that are tilted, which are compared with 2D DIC. The two resulting displacement vectors are used as input for the 3D reconstruction. The reconstructions are made for the data sets of tilting of nanoporous gold and tilting of the metal mesh from JEOL. This can verify the correctness of the displacement fields and shows it can be used for correct 3D reconstructions.

Results

Figure 4.16(a) displays the 3D reconstruction for nanoporous gold, calculated with the custom DIC. Figure 4.16(b) shows the scatter plot of u and v data (in nm). Figure 4.17 displays the same results, but with the use of the 2D DIC data obtained with Aramis.

Experimental Veriﬁcation of 2D Digital Image Correlation for Scanning Electron Microscopy

Experimental Verification of 2D Digital Image Correlation for Scanning Electron Microscopy

Enrico Werkema

Acknowledgements

Abstract

Contents

Chapter 1

Introduction

1.1 Background information

1.2 Research goals

1.3 Approach

1.4 Structure of the report

Chapter 2

Introduction to Scanning Electron Microscopy

2.1 Principles underlying the SEM

2.2 SEM specific noise

Chapter 3

Digital Image Correlation algorithm

3.1 Overview

3.2 Definitions

3.3 Subset deformation

3.4 Correlation criteria

3.5 Optimization strategies

3.6 Method for obtaining Full Field Displacements

3.7 Method for obtaining Full Field Strains

Chapter 4

Experiments and results

4.1 DIC configuration

4.2 Numerical experiments

4.3 SEM images: Nanoporous Gold

4.4 SEM images: JEOL

4.5 Comparison with Aramis

4.6 3D topography reconstructions