# Fusion of complementary 2D and 3D mesostructural datasets using generative
adversarial networks

Amir Dahari<EMAIL_ADDRESS>Dyson School of Design Engineering,
Imperial College London, London SW7 2DB Steve Kench  Dyson School of Design
Engineering, Imperial College London, London SW7 2DB Isaac Squires  Dyson
School of Design Engineering, Imperial College London, London SW7 2DB Samuel
J. Cooper<EMAIL_ADDRESS>Dyson School of Design Engineering,
Imperial College London, London SW7 2DB

###### Abstract

Modelling the impact of a material’s mesostructure on device level performance
typically requires access to 3D image data containing all the relevant
information to define the geometry of the simulation domain. This image data
must include sufficient contrast between phases to distinguish each material,
be of high enough resolution to capture the key details, but also have a large
enough 3D field-of-view to be representative of the material in general. It is
rarely possible to obtain data with all of these properties from a single
imaging technique. In this paper, we present a method for combining
information from pairs of distinct but complementary imaging techniques in
order to accurately reconstruct the desired multi-phase, high-resolution,
representative, 3D images. Specifically, we use deep convolutional generative
adversarial networks to implement super-resolution, style-transfer and
dimensionality expansion. Three open-access datasets of energy materials are
used to validate the quality of the volumes generated and demonstrate the
widespread applicability of the approach, each illustrating a distinct
material morphology and imaging approach. Having confidence in the accuracy of
our method, based on key mesostructural metrics, we then demonstrate its power
using a real data pair from a lithium-ion battery electrode, where the
resulting 3D high-resolution image data is not available in the literature. We
believe this data-driven approach is superior to previously reported
statistical material reconstruction methods, both in terms of its fidelity and
ease of use. Furthermore, much of the data required to train this algorithm
already exists in the literature, waiting to be combined. As such, our open-
source code could precipitate a step change in the computational materials
sciences by generating the hard to obtain high quality image volumes necessary
to simulate behaviour at the mesoscale.

##  1 Introduction

The geometrical arrangement of porous or composite materials at the mesoscale
can significantly impact the performance of the devices they constitute. In
order to accurately model the physical processes that mediate this
relationship, it is necessary to have geometric information about the
distribution of the various phases. This is typically derived from imaging
data, which must have the following four properties: Firstly, it must be
three-dimensional (3D), since many percolation networks are not reducible to
two dimensions. Secondly, the image data must be of high enough resolution to
capture the key details. Thirdly, it must have a large enough field-of-view
(FoV) to be representative of the material in general. Finally, it must be
possible to confidently differentiate between each of the material phases
present. It is rarely possible to obtain data with all four of these
properties from a single imaging technique 3, meaning that the resulting
simulation domains may be lacking some vital information, which can undermine
the value of any conclusions drawn from these numerical analyses.

Lithium-ion battery cathodes exemplify this kind of challenging imaging
scenario. They are usually porous composites where the solid phase is
comprised of a ceramic “active material” (AM) that is a lithium intercalation
compound (e.g. nickel manganese cobalt oxide or NMC), as well as a polymeric
binder material containing carbon black called the “carbon binder domain”
(CBD) which provides both mechanical integrity and electronic conductivity.
The pores are filled with a liquid electrolyte and must be well percolated to
facilitate transport of the dissolved lithium ions. These electrodes are
typically around 100 $\muup$m thick with particle size distributions varying
along their thickness; whereas the CBD may have relevant features below 100 nm
that critically impact performance.

Imaging these cathode materials in 3D using X-ray computed tomography (XCT)
enables a large, representative field-of-view (FoV) to be observed; however,
current XCT technology does not provide sufficient contrast for the CBD phase
to be confidently differentiated from the pores 27. Furthermore, even if the
contrast could be enhanced, the resolution would be insufficient (c. 100 nm)
to observe the CBD’s nanoscale features 22. It is also common to image cathode
materials with focused ion beam scanning electron microscopy (FIB-SEM), which
can provide much higher resolution data (c. 10 nm), but the imaged regions are
usually quite small as a result. SEM is better than XCT at distinguishing
between the solid phases (i.e. AM vs. CBD); however, since SEM is based on
detecting electrons that have escaped the imaging surface (either
backscattered or secondary), it is difficult to distinguish between material
that is on the desired imaging plane compared to the inner surface of pores.
This “pore-backs” problem has several proposed solutions, including
infiltration 19 and advanced image tracking 24. A recent paper by the authors
describes a strategy to overcome this issue using an in-situ platinum
deposition approach called kintsugi imaging 5. An additional common limitation
of FIB-SEM generated volumes is that uniform spacing in the “slicing”
direction is hard to guarantee 14 and so any observed anisotropy may be
artificial.

The 2D SEM imaging technique is, however, far better at differentiating the
CBD phase through the use of various contrast enhancement techniques 18.
However, as highlighted in Gayon-Lombardo et al. 10, for isotropic materials,
although the statistical information present in a representative two-
dimensional (2D) slice is sufficient to extract basic metrics such as volume
fraction and surface area, certain key performance metrics for physics
simulation are only available in 3D domains, such as percolation and the
tortuosity factor.

To close this gap, our proposed method achieves an accurate reconstruction of
high-resolution (high-res), large FoV and multi-phase 3D volumes by fusing
together information from complementary 2D and 3D imaging techniques. For
this, we use a machine learning framework called generative adversarial
networks (GANs)11. GANs have previously been used to (1) enhance the
resolution of (“super-resolve”) colour photographs20 and (2) transfer style
between images (e.g. converting a photograph to a painting)16. Furthermore,
Kench and Cooper recently published a GAN architecture able to (3) generate 3D
volumes from a single representative 2D slice 17 that can be (4) arbitrarily
large and therefore representative of the material in general. We propose a
GAN architecture that combines these 3 capabilities, designed specifically for
the generation of materials mesostructure phase maps.

Unlike the state-of-the-art in the field of mesostructural generation 25, our
approach does not require the user to select and extract characteristic
metrics in order to generate or modify their images, but instead it learns the
key features and relationships directly from the data. Also, unlike a recent
paper15 that presents a 2D to 2D super-resolution (super-res) mesostructure
technique, our approach not only super-resolves 3D mesostructure, but is also
able to add missing phases, change existing ones and capture anisotropy.
Further recent related studies can be seen in 8, 6, 13.

Figure 1: The SuperRes model inputs and output evaluated on the isotropic NMC
cathode XCT dataset (Case study 1) 29, with a scale factor of 4. The different
phases are pore (black), active material (gray) and binder (white). The top
row shows the inputs and output of the model. For training, the model requires
a high-res multi-phase 2D image and a low-res binary 3D volume ($\times 4$
stretched to make comparison to the super-res version easier), and the output
is a super-res volume of the low-res volume with the same multi-phase
characteristics of the high-res 2D image. The bottom row shows a comparison
between a random cross section in the same position of the low-res and super-
res volumes, and as well as the middle region of the high-res slice to compare
the nature of the fine details.

To illustrate the capability and accuracy of the proposed method, we present
four case studies, each based on open-access imaging data. In each study, the
paired datasets used for training consist of high-res 2D images and a low-
resolution (low-res) 3D volume, with the aim of producing a super resolved
(super-res) volume. The first three cases serve to validate the accuracy of
the method in distinct scenarios, while the fourth case demonstrates the
application of the approach on real paired datasets from the literature. Since
the first three case studies require access to the ‘ground truth’ in order to
compare the calculated metrics, the low-res volumes were created by blurring
then down-sampling the original high-res versions and in some cases removing
phases or features to showcase additional functionality. More details about
the training data and how it was prepared for analysis can be found in the
Supplementary Information Table 1. The four case studies are as follows:

  1. Case 1:

Battery cathode (validation on an isotropic material with abundant training
data): A low-res, large FoV volume showing only two of the three phases, is
combined with a high-res three-phase 2D image to produce a large, 3D, three-
phase super-res volume. The material is isotropic and training data is
abundant. Detailed statistical comparisons are made to the original 3D high-
res volume from which the low-res volume and 2D slice were obtained. In
addition, an exploration is made on the impact of the scale factor (the ratio
between the scale of the low-res and the corresponding super-res or high-res
data) on the reconstruction quality.

  2. Case 2:

Battery separator (validation on anisotropic material with limited training
data): A low-res, large FoV, two-phase volume is combined with a set of
orthogonal high-res 2D images containing fine features to produce a large, 3D,
super-res volume. The material is anisotropic and training data is limited.
Detailed statistical comparisons are made to the original volume.

  3. Case 3:

Fuel cell anode (validation on isotropic material with abundant training
data): To highlight an alternative operational mode of the algorithm, the low-
res data in this case is undersampled, rather than underresolved. This also
means that all three material phases are represented at both resolutions.

  4. Case 4:

Battery cathode (demonstration): A low-res, large FoV XCT volume showing only
two phases is combined with a high-res, three-phase, 2D SEM image to produce a
large, 3D, three-phase, super-res volume. The material is isotropic and
training data is limited. The ground truth is not known. To the authors
knowledge, the generated volume (c. $2000^{3}$ voxels, for a 96 µm edge length
cubic volume at 50 nm resolution) is the largest three phase Li-ion cathode
mesostructural dataset currently available in the literature, which can be
downloaded via the link in the code repository.

The mesostructures that are examined in this paper are all battery or fuel
cell related, hence the relevance of the mesoscale transport properties that
are extracted. However, our model is material and lengthscale agnostic and can
be applied to diverse scenarios including catalyst beds, rock formations, or
snow and soil packing.

##  2 Results

###  2.1 Case 1: Validation using an isotropic, three-phase Li-ion battery
cathode dataset

Figure 2: Statistical comparison of key mesostructural metrics between the
original high-res 3D volume (blue), the high-res 2D slice (black) and the
reconstructed super-res 3D volume (orange). The left figure shows a comparison
of the volume fraction of the different phases in the material. The middle
figure shows a comparison of the interphase surface area, the fraction of the
number of neighboring voxels of different phases by the total number of faces
in the volume. The right figure shows a comparison of the transport efficiency
for each phase calculated using a diffusivity simulation4. All metric
calculations were obtained using the TauFactor4 package. All figures show a
violin plot distribution of the 3D high-res and the 3D super-res taken from
256 randomly sampled $64^{3}$ volumes, with normalised width for each
distribution and corresponding horizontal lines for the means. The black
pluses mark the metrics measured on the whole 2D image. Note that since
transport efficiency is a volumetric measurement, only the values for the 2D
high-res slice are not included.

To illustrate the method’s capabilities for isotropic, three-phase materials
with abundant training data, an open-access dataset from the National
Renewable Energy Laboratory (NREL) was used 29. This dataset was derived from
XCT imaging of a lithium-ion battery cathode material. The NREL team performed
a binary segmentation on the XCT dataset to show the distribution of the
active material only and then the binder phase was added stochastically based
on statistical information from an SEM image. Full details on the imaging,
segmentation, and augmentation of this data can be found in the repository
associated with Usseglio-Viretta et al.29.

For this experiment, in order to have a very large “ground truth” volume to
showcase the power of the approach, a 3D high-res three-phase volume of
$1024^{3}$ voxels was generated using a SliceGAN 17 trained on the three-phase
volume from 29. Next, in order to imitate a dataset representing the same
region, but captured using XCT at $4\times$ lower resolution, the following
three operations were performed. Firstly, the pore and binder phases were
merged together, since the binder phase is typically indistinguishable from
the pore phase in XCT due to the very low X-ray attenuation of the binder
compared to the active material. Secondly, a Gaussian blur was applied to the
active material phase followed by a down sampling by a factor of 4. As
described in the methods section, by adjusting the kernel size and standard
deviation of this blur operation, this step could be used to simulate “pixel
binning” all the way through to under sampling. Lastly, the low-res volume was
then thresholded to reconvert it into a phase map. This process of creating
the simulated low-res volume is described in Supplementary Information Figure
12.

Figure 3: The results from exploring the effect of different scale factors.
For each scale factor, the inputs for the model were a high-res 2D slice,
whose center can be seen on the right, and a low-res volume which was blurred
and down-sampled accordingly to the scale factor from the original high-res
volume. Every column shows a different experiment for a different scale
factor. The top row shows a cross section of the original volume, the middle
row shows a cross section of a corresponding position in the low-res volume
and the bottom row shows a cross section of a corresponding position in a
generated super-res volume.

The same SliceGAN model was also used to independently generate a 2D high-res
three-phase slice containing $2048^{2}$ pixels. As explained in the
introduction, the decision to merge the pore and binder phases in the low-res
volume stems from the fact that they are very difficult to distinguish using
XCT due to the very low X-ray attenuation of the binder compared to the active
material.

Figure 1 illustrates the workflow and effectiveness of the model. The
different phases of the cathode shown are pore (black), active material (gray)
and binder (white). In addition to demonstrating the generated super-res of
the low-res volume, this scenario also exhibits the 3D in-painting of a phase
present only in the high-res 2D slice.

Figure 2 shows a quantitative comparison between the original high-res volume
and the super-res volume. In the metrics comparison, the ultimate objective
for the super-res volume is to be in good agreement with the 3D high-res
ground truth volume, but since it only sees the 2D image as the high-res
input, the agreement of the metrics depends on the representativeness of the
2D image to the ground truth volume. As can be seen, except for the
pore/binder interphase surface area, the 2D high-res image is representative
of the properties of the 3D volume, thus the super-res volume is in good
agreement with the ground truth volume on all of these metrics. Interestingly,
since the 2D high-res image is the only information for the high-res binder
characteristics for the reconstruction, the super-res volume has good
agreement with it for the Pore/Binder interphase surface area, slightly more
than the 3D original volume. This good agreement with the 2D high-res image
highlights the accuracy of the proposed method, as well as the importance of
using a representative 2D slice for an accurate 3D reconstruction.

As described in more detail in Section 3, the model is able to support a
variety of different scale-ratios between the resolutions of the two input
images. Figure 3 shows the same NMC sample reconstructed from a range low-res
inputs derived from different scale factors. It is clear that as the scale
factor increases, the fidelity to the low-res volumes reduces, which is to be
expected. However, all of the super-res results look like valid cathode
regions, matching the high-res 2D slice characteristics.

For the extreme case of scale factor 8, 1 voxel in the low-res volume
corresponds to $8^{3}=512$ voxels in the high-res volume. This naturally leads
to a large loss of information when converting from the original volume to the
low-res volume. As a result, there are a huge number of different
reconstruction options for the super-res volume, all of which can be realistic
and statistically similar to the original volume.

It is important to note in addition that Figure 3 only shows one of the many
possible super-res outputs that could have been generated for each scale
factor. Supplementary Information Figure 14 shows the existing variability
from different noise inputs for creating super-res volumes. This highlights
the stochastic nature of the approach, as will be explained in the Methods
section. The figure also shows a comparison to the original volume from which
the low-res volume was created. Also, to quantify the change in resolution
from the low-res to the generated super-res volumes, Supplementary Information
Figure 18 shows a comparison in Fourier Space between the images using FFT
magnitude maps.

###  2.2 Case 2: Validation using an anisotropic, two-phase separator dataset

Figure 4: The model results for an anisotropic material, here producing a
super-res 3D battery separator material 7 with a scale factor of 4. The
different model inputs and outputs are the same as in Figure 1, with the
additional input of high-res 2D slices from all three perpendicular
directions. Since two perpendicular directions have the same properties (upper
slice in the top left corner), for simplicity only two high-res 2D slices are
shown out of the 6 facets used for training data.

To present the flexibility and diverse capabilities of the method, an
anisotropic material was also investigated. For this, an open-access dataset
of a porous polymer battery separator material was chosen 31. In the previous
example, a new phase was introduced in the 2D high-res slice that was not
present in the 3D low-res volume. However, an alternative scenario is that the
same phase displays new features when imaged at higher resolution. The
separator material consists of thick lamellae connected by thin fibrils.

Although the XCT data collected by Finegan et al. 7 was able to capture the
lamellae, it did not have sufficient resolution to observe the fibrils.
However, Xu et al. used a stochastic method to add fibrils 31 based on
statistical information taken from an SEM image. In this experiment, the high-
res 3D separator data is $624\times 300\times 300$ voxels of the lamellae and
stochastically added fibrils. The low-res volume is a down-sampling of only
the lamellae with a coarsening scale factor of 1/4. Together with the low-res
volume, the input also consists of high-res 2D slices, which is the set of all
6 outer facets of the high-res 3D volume (analogous to capturing 2D images of
the outer surfaces of a volume). These slices together sum to very limited
amount of image training data (less than one megapixel) compared to the amount
generally required for deep learning imaging applications 30. The fact that
the model still performs well is likely explained by the random sampling
method used on the high-res input data, so a single pixel can appear in many
training instances. However, using fewer facets did result in an unsuccessful
super-res reconstruction, indicating the existence of a lower threshold for
representativity and successful reconstruction in this scenario.

Figure 5: Statistical comparison of key mesostructural metrics between the
original high-res 3D volume, the high-res 2D slices and the reconstructed
super-res 3D volume for the anisotropic separator material in Figure 4. The
metrics description is the same as in Figure 2. Since the material is
anisotropic, transport efficiency measurements were taken along all different
axis. The figure also shows the transport efficiency measurements of the low-
res volume, and the added information gained by super-resolving.

As can be seen in Figure 4 and the corresponding metrics comparisons in Figure
5, although the limited training high-res 2D data may have slightly harmed the
agreement of the metrics, the facets of the original high-res volume had
enough high-res information for the successful anisotropic super-res
reconstruction. As in case study 1, the surface area of the high-res 2D slice
was offset from the global 3D high-res value. The surface area of the 3D
super-res lies between the high-res 2D value and the original volume data
which the low-res data is derived from. This balance demonstrates the ability
of the generated volume to accurately reproduce the statistics of the HR 2D
slice while also resemble the low-res 3D volume training data.

###  2.3 Case 3: Validation using data from a fuel cell anode with three-phase
low-res data and super-sampling

Figure 6: The SuperRes model inputs and output evaluated on the fuel cell
anode FIB-SEM dataset (Case study 3) 12, with a scale factor of 4. The
different phases are pore (black), nickel (gray) and ceramic (white). In this
example, the low-res is being super-sampled, rather than super-resolved. The
arrangement of images is the same as in Figure 1.

To showcase several additional features of the SuperRes method, an open-access
image dataset of a solid oxide fuel cell anode material was chosen 12. The
images were obtained through FIB-SEM serial sectioning and, as can be seen in
Figure 6, this material has three phases: pore (black), metal (gray) and
ceramic (white). In order to demonstrate super-sampling, instead of super-
resolving, all three phases are present in the low-res image.

An additional dimensionless metric called the normalised Triple Phase Boundary
(TBP) density was calculated for this material, as it is of particular
relevance to the performance of SOFC anodes. Each voxel edge makes contact
with four voxels and an edge where all three phases are present in its
neighbours is considered to be a TPB. As with the other metrics, the
calculation was performed using TauFactor 4 on 256 randomly sampled $64^{3}$
volumes. The mean and standard deviation for cubes from the training set were
0.01087 and 0.00121 respectively; whereas they were 0.01028 and 0.00126
respectively for the super-res generations. The exceptional similarity of
these two distributions is further evidence for the effectiveness of the
SuperRes approach.

It is important to note the distinction between under-sampled data (e.g., for
SEM imaging, raster step size $>$ electron beam spot size) and under-resolved
data (i.e., raster step size $<$ electron beam spot size) as this will impact
how a “low resolution” image should be interpreted by the loss function in the
SuperRes algorithm. If the data is under-resolved, we apply a Gaussian blur of
the appropriate size (before downsampling) in order to achieve the relevant
degree of spatial mixing. However, if the data is under-sampled, the super-res
voxels should instead be sampled at the appropriate spatial interval and
compared directly to the low-res. This option is further explained in the
Methods section.

In the first two case studies, a Gaussian blur was applied before down-
sampling, to simulate what might have been observed using lower resolution XCT
imaging (i.e. a weighted volume average of regions of voxels). However, in
Case 3 it is assumed that both the low-res and high-res data were collected
using SEM. In an SEM, unlike XCT, the “spot size” of the incident electron
beam can be varied independently of the raster step size used to construct the
image. So, each voxel in the low-res image represents the same amount of
material as in the high-res, rather than an average of a larger region, like
it would in XCT. This requires a simple modification to the way in which the
data is downsampled when calculating the voxel-wise loss, excluding the
Gaussian blur as described in the Methods section. Figures 7 and 10 show a
good match in measured statistics between the training data and original
volume, although perhaps not quite as good as in the first two cases.

Supplementary Information Figure 15 shows the existing variability from
different noise inputs for creating super-res volumes with a high scale factor
of 8.

Figure 7: Statistical comparison of key mesostructural metrics between the
original high-res 3D volume, the high-res 2D slices and the reconstructed
super-res 3D volume for the SOFC anode material12 in Figure 6. The metrics
description is the same as in Figure 2.

###  2.4 Case 4: Demonstration using a pair of complimentary datasets from a
battery cathode

Figure 8: The model results for a demonstration isotropic material, combining
XCT low-res volume and SEM high-res 2D slice, with a scale factor of 8.
Resulting in a super-res volume with nanoscale resolution that is hard to
impossible to achieve with current imaging techniques.

The previous case studies were designed to validate the accuracy of the method
using experiments that had the ground truth high-res 3D volume from which the
complementary training images were derived. This way the generated super-res
volumes could be quantitatively compared to the ground truth. Once the
performance has been validated, a natural next step is to test the method on a
real-world problem, where the desired ground truth high-res 3D volume does not
exist, but all of the necessary information is contained in a pair of
complimentary datasets.

For the cathode dataset in the first study, the binder phase was added
stochastically to the XCT imaged volume based on statistical information from
an SEM image of the same material. In this case study, we will use the XCT
dataset without the added binder phase as the low-res input, leaving only a
map of the active material phase, as can be seen in middle top of Figure 8.

The SEM image (secondary electrons) of the same cathode sample used to extract
the binder statistics in 29 will be used as the high-res 2D input. The high-
res SEM image obtained from the same sample offered sufficient contrast to
distinguish between the CBD phase from the pores, allowing it to be segmented
into three phases. This task was performed using Weka segmentation software 1
accompanied by some manual finishes, to try and account for the “pore-backs”
problem described in the introduction. The entire three-phase segmentation
result can be seen in the upper left corner of Figure 8, while the original
SEM image alongside the segmentation can be seen in the Supplementary
Information Figure 16, along with some further explanation of the segmentation
approach in the figure’s caption.

Here, other than a real representation of the CBD phase, another difference to
the first case study is the resolution and the scale factors used. While in
the first case study the low-res volume had a voxel lengthscale of 1.6
$\muup$m and the high-res 2D slice had a pixel lengthscale of 400 nm (scale
factor of 4), in this case study the low-res volume has a voxel lengthscale of
400 nm and the high-res 2D slice has a pixel lengthscale of 50 nm (scale
factor of 8). To the authors knowledge, this is the highest resolution Li-ion
cathode mesostructure volume dataset that contains a realistic representation
of all three phases currently available in the literature.

We mention that although the SEM image had sufficient information to observe
the CBD’s nanoscale features, it had many unwanted artifacts that damaged the
true distribution of the phases in the image. As well as the “pore-backs”
issue, which may cause the binder to appear over represented in the SEM image,
an additional artefact referred to as ’curtaining’ can occur when FIB is used
to prepare cross-section for SEM imaging. Vertical streaks are formed by the
ion beam milling through phases with different densities, which can make
segmentation more challenging 21. This was not totally accounted for during
the manual correction of the segmentation and can be most clearly seen in the
bottom right of the 2D input image in Figure 8. However, we find that this
anisotropic artifact is hardly found in the output super-res volume. This
positive finding can be linked to the isotropic rotational data augmentation
we performed as a preprocessing step, which will be described in the next
section.

The limited high-res 2D training data in this case is not representative of
the low-res 3D volume, which is noticeable from the high active material
volume fraction of 0.58 compared to 0.40 in the low-res volume. This
discrepancy could have originated either from the small spatial area of the
high-res 2D image or the under-classification of active material in the XCT
segmentation 29. We believe that this unrepresentative slice led to slight
mismatch in the comparison of the pore structure between the generated volume
and high-res slice. As can be seen in the high-res slice, pore tends to be
present in the vicinity of particles, especially between close adjacent
particles. The low active material volume fraction and and the low prevalence
of close adjacent particles in the low-res volume led to lower pore volume
fraction in the generated volume.

However, it is clearly visible that the binder phase in the super-res volume
qualitatively matches the elaborate structure of the binder phase in the SEM
image shown in Figure 6. Furthermore, the contrast to the geometry of the
binder phase that was added statistically (based on metrics from the same SEM
image) in the open-access data as shown in Figure 1 is stark. Although there
might be inaccuracies comparing statistics between different resolution
samples, our alternative method of stochastically adding binder to the XCT
volume have binder volume fraction of 0.506 compared to 0.111 in 29, and the
binder phase percolates with a transport efficiency of 0.242 compared to the
almost non-percolating 0.008 transport efficiency of the binder phase in 29.

To further test the statistical alignment of the generated volumes to the
original volumes, three more functional descriptors 28 were performed:
intraphase and interphase 2-point correlation functions and chord length
distribution. These metrics have all shown good agreement between the original
and generated volumes for the first three case studies, and can be viewed in
Supplementary Information Figure 10.

##  3 Methods

The SuperRes model was built on the development of SliceGAN17, a generative
adversarial network architecture that generates 3D mesostructure from a 2D
slice. In SliceGAN, the input of the Generator ($G$) is random noise and the
output is a one-hot encoded 3D mesostructure which is then sliced orthogonal
to $x$, $y$ and $z$ axes to feed the Discriminator ($D$) with 2D images. Here,
as well as feeding $G$ with random noise, the input of $G$ also contains a
low-res 3D mesostructure which we wish to super-resolve (or possibly super-
sample) and augment. The architecture of $G$ is changed accordingly, whilst
$D$ remains essentially unchanged. Although it is a reasonable assumption that
$G$ would learn to use the low-res input structure to generate a high-res
reconstruction with similar features, this is not guaranteed to be the case.
This is because the standard GAN loss function does not include a measure of
the similarity between the input and output image. It is thus possible for the
generated 3D images to look very different from the low-res image.

To constrain $G$ to generate images that will have the same features as the
low-res input image, we add a loss term given by the voxel-wise mean squared
error (MSE) between the low-res image and a down-sampled version of the super-
res image. If the imaging technique used to obtain the low-res images is truly
“under resolving” the high-res features, then it is reasonable to apply a
Gaussian blur before the down-sampling, such that the low-res can be compared
against a weighted volume average of the voxels that correspond to each voxel
in the low-res.

The standard deviation and size of the 3D Gaussian kernel used to blur the
super-res generations can be tuned to more closely represent the imaging
conditions used. However, the default mode, as applied in Cases 1 and 2, used
a 3D kernel with odd kernel size $k$ defined by the following equation:

$k=\begin{cases}\lceil sf\rceil&\text{if }\lceil sf\rceil\text{ is odd}\\\
\lceil sf\rceil-1&\text{if }\lceil sf\rceil\text{ is even}\end{cases}$

Where $sf$ is the scale factor and the Gaussian kernel weights are distributed
by the default PyTorch 26 $\sigma=0.3\cdot((k-1)\cdot 0.5-1)+0.8$ for the
appropriate kernel size $k$. The super-sampling approach explored in Case 3
can be thought of as a Gaussian blur with a standard deviation of zero
(although, in reality, it makes more sense to just turn the blur off). For
completeness, it should be noted that a Gaussian blur with very large (or
infinite) $\sigma$ would produce a kernel with uniform weighting, which is
analogous to pixel “binning”. However, all of these case produce very similar
results for scale factors $\leq 4$.

Since the low-res input to $G$ is a segmented volume, after down-sampling the
generated volume a low temperature softmax function was applied instead of a
threshold (step) function to result in a near one-hot encoded phase map, which
keeps the voxel-wise loss differentiable. Other than the addition of the
voxel-wise loss, the Wasserstein loss function 2 remains similar to SliceGAN.
This procedure is outlined in Algorithm 1 and in the Supplementary Information
Figure 11.

1:$G$, the generator function; $D$, the discriminator function; $LR$, the low-
res volume; $HR$, the high-res 2D slice; $sf$, the scale factor between $LR$
and $HR$; $b$, threshold for the voxel-wise loss (default $0.005$); $c$, the
coefficient of the voxel-wise loss multiplication (default $10$); $\sigma$ the
downsampling function: a Gaussian blur followed by a trilinear downsample with
a scale factor of $sf$, followed by a differentiable near one-hot encoding (a
softmax with low temperature); All batch operations and optimization and
gradient penalty parameters are not shown for simplicity, we refer the reader
to the codes at the project’s repository for specific parameter details,
including details about the implementation of the gradient penalty.

2:$HR\leftarrow$ Data augmentation step, expanding $HR$ into all 8 different
possible mirrors and rotations of $HR$ by $None$ multiples.

3:

4:while weights of $G$ did not converge do

5: $lr\leftarrow$ Sample a $(\frac{64}{sf})^{3}$ voxels cube uniformly from
$LR$.

6: $z\leftarrow$ Sample a $(\frac{64}{sf})^{3}$ voxels cube of noise from
normal distribution with mean $0$ and standard deviation $1$.

7: $lr\leftarrow concat(lr,z)$ Concatenate the low-res and noise cubes along
the phase dimension.

8: $sr\leftarrow G(lr)$ Generate a $64^{3}$ super-res volume.

9: $sr_{cropped}\leftarrow$ Crop $4$ facets from every outer plane of $sr$
into a $56^{3}$ volume.

10: $sr_{slices}\leftarrow slice(sr_{cropped})$ Slice the volume
$sr_{cropped}$ in all perpendicular planes into $56\cdot 3$ 2D slices.

11: if Training of $G$ (Only every 5 iterations) then

12: $l_{vw}\leftarrow MSE(lr,\sigma(sr))$ Voxel-wise loss between low-res and
downsample of Gaussian blurred super-res volumes.

13: if $l_{vw}<b$ then

14: $l_{G}\leftarrow-D(sr_{slices})$

15: else

16: $l_{G}\leftarrow-D(sr_{slices})+c\cdot l_{vw}$

17: Backpropagate and update the weights of $G$ from the loss $l_{G}$.

18: else Training of $D$

19: $hr\leftarrow$ Sample a $56^{2}$ pixels square uniformly from $HR$.

20: $l_{gp}\leftarrow$ GP($sr_{slices}$, $hr$, $D$) Calculate the gradient
penalty regularization based on real and fake outputs.

21: $l_{D}\leftarrow D(sr_{slices})-D(hr)+l_{gp}$

22: Backpropagate and update the weights of $D$ from the loss $l_{D}$. return
The super-res volume of the full low-res volume $\mathbb{G}(LR)$.

Algorithm 1 SuperRes algorithm for isotropic materials

We emphasize that both the low-res and high-res input data for the model are
one-hot encoded (i.e. already segmented into channels, rather than the raw
grayscale). However, the model could be modified in future to utilise
grayscale or even colour image data directly if some advantage was identified.

Any overfitting of $D$ on the high-res training data can disrupt the learning
process of $G$. This failure mode will enforce $G$ to generate volumes with
high voxel-wise loss to the input volumes, in an attempt to copy the training
data.

Data augmentation can increase the apparent diversity of the training set and
therefore improves the quality of the trained model by reducing overfitting.
This is particularly important when training data is hard to acquire and
scarce. For example, if the material is assumed to be isotropic then a single
2D slice can be transformed into 8 slices of the same size, constituting of
all the different mirrors and $None$ rotations of the original image. It
should also be noted that the 2D high-res training data does not need to come
from a single large image, but can be from a collection of smaller images,
which may be much easier to obtain.

From our experiments, lack of volumetric low-res data was never a barrier for
successful training thus no data augmentation of the low-res 3D input was
performed but can easily be implemented. This can be explained by the extra
dimension in 3D data that multiplies the number of different samples, or the
importance of diversity in the fine features of the 2D high-res input compared
to the coarse, often repeated, 3D low-res structure input.

The voxel-wise loss between the low-res and super-res volumes is only used to
update the parameters if the voxel-wise loss is above the 0.5% threshold. This
threshold is introduced to provide flexibility for the super-res output to
distance from the blocky corners of the phases present in the low-res volume.
More flexibility comes from the concatenated input noise layer to $G$ and is
necessary for a successful reconstruction, as shown in the Supplementary
Information Figure 17.

Algorithm 1 is for isotropic materials and so a minor modification is required
to generate anisotropic materials. This modification is identical to that used
in SliceGAN 17, where $3$ different discriminators can be used for the x-y,
y-z and x-z directions. Note that the last step of the algorithm output is
possible due to the convolutional design of the generator that is detailed in
Supplementary Information Figure 13, that allows for any size of input volume.

For clarification, in each different case study experiment a ‘new’ model is
trained independently without any prior knowledge (i.e. from a random
initialisation). It is the same architecture but with different weights after
training, optimised for the specific training data. A future direction for
improving the method could be to use transfer-learning from a model trained on
some general mesostructural data, shortening the learning and convergence time
of the specific application.

Due to the adversarial nature of GANs, training can be unstable. For example,
overfitting of the discriminator on the high-res 2D training data can in turn
degrade the performance of the generator, making its outputs worse from the
perspective of mesostructural generation. Furthermore, Wasserstein GANs do not
have trivially definable convergence criteria 23. Ultimately, since the
measure of whether the model is performing well depends on what you wish to do
with the outputs, both the instability and arbitrary convergence can be
addressed by extracting cheap metrics of interest during the training process
(e.g. volume fractions and surface areas) and using these to help determine
when to stop training.

Around the edges of the super-res volume the uncertainty high. This is because
the low-res volume does not contain any information about the regions adjacent
to its boundaries. So, for example, in the case of the cathode, a pore just
inside the boundary may in reality be next to a particle of active material
that was just outside the imaged region. This would effect the morphology of
the binder phase nearby. This concept is reflected in the non-uniform
information density near the edges of the volume, as described in 17. As such,
it is advisable for users of the SuperRes method to crop their volume by the
equivalent of 1 voxel of the input volume from the generated volume (e.g. in
4$\times$ resolution increase, remove 4 layers from each face), as advised in
the github repository manual of this project. For this reason, the generated
volumes edges are being cropped before being discriminated.

The model is able to support a variety of different scale-ratios between the
resolutions of the two input images. Specifically, the model supports all
rational scale factors in the range between 1 to 8 that divide 64 without a
remainder, which are all scale factors $\frac{64}{d}$ for any integer $d$ in
the range $8\leq d\leq 64$. This set of scale factors can be expanded and is
only limited by current implementation.

Our proposed model does not implement the classic style-transfer algorithm by
Gatys et al. 9, in part because of the 2D to 3D dimensionality mismatch of the
style representation. However, our approach is not too dissimilar to the
framework in 9, as the loss of $G$ consists of a linear combination of the
content representation (the voxel-wise loss) and the style representation
($D$’s score for fake images).

##  4 Discussion

The results of the four case studies make a strong case for the utility and
potential impact of this GAN based super-resolution method. The first, second
and third case studies demonstrated excellent agreement between the metrics
calculated on the 3D high-res ground truth and super-res outputs. The
excellent agreement of the transport efficiency metric is particularly notable
as it is an inherently 3D property. This highlights the power of this method
to perform both dimensionality expansion as well as data fusion, as this
property could not be accurately calculated on the low-res 3D volume or
calculated at all on the high-res 2D data.

There is a small discrepancy between some of the 3D high-res and 3D super-res
metrics, with the super-res metrics more closely aligned to the 2D high-res
metrics rather than 3D high-res metrics. Moreover, certain metrics are only
contained in the high-res 2D training data, such as the binder phase fraction
in the NMC sample. These results illustrate how the model is learning the
distribution of the 2D high-res training data whilst being constrained through
the super-resolution process to match the gross arrangement enforced by the
low-res input.

Figure 9: Exploring the effects of different noise inputs for the generation
of different super-res volumes, generated by the model of Case study 4. The
top row shows a slice of the low-res volume and a $\times 8$ stretched
magnification for easier comparison. This low-res volume input acts as an
input for the model, together with random noise. The middle row shows a
corresponding slice (#2 out of possible 8) of the low-res volume in 3
different super-res volumes, generated with 3 different random noise inputs.
The bottom row shows the same subsection of all the slices, showing the fine
feature variability that exists as a result of different noise inputs.

###  4.1 Stochasticity

It is interesting to note that it is possible to implement this super-res
method using just a low-res volume as the input to $G$, without concatenating
a layer of noise. In this scenario, the model is deterministic and if it is
well trained, its output should represent a likely underlying mesostructure
associated with each neighbourhood of the input. Yet, the addition of noise
offers three advantages over the deterministic approach:

  * •

Limiting $G$ to a single output for a given low-res input could lead to
unwanted homogeneity of the resulting mesostructure, especially when the low-
res input contains many repeated patterns. This is exemplified for an input
featuring only one phase, as shown in Supplementary Information Figure 17.
Unlike the heterogeneity of features generated with the stochastic input,
without added noise $G$ is only able to output a single pattern across the
pore volume, resulting in a failure mode. The figure also shows the scenario
with large patches of pore between particles and a similar but more complex
failure mode is observed.

  * •

There are two ways in which overfitting of $D$ can occur, either $D$ learns
the true dataset as described in the Architecture section, or $D$ can learn
the output distribution of $G$. In this scenario, after $D$ learns $G$’s
outputs, it can simply classify this dataset as fake, and all other data as
true. By adding noise, $G$ can learn to use it to generate greater diversity
in its generated outputs to overcome this kind of overfitting. This is
especially relevant when only a small amount of low-res volume data is
available. There are many possible super-res reconstructions for a single low-
res volume, enlarging $G$’s output space thus helps avoiding overfitting of
$D$ on $G$’s outputs.

  * •

By varying the noise one can generate different mesostructures and explore the
distribution of potential features and properties, allowing the user to
quantify the uncertainty and richness of the super-resolution process.
Additionally, the fact that the noise is spatially distributed with the input
volume, allows for the creation of periodic boundaries of the properties that
are governed by the noise input 10. Therefore, latent space exploration of the
noise input and its use in mesostructure optimization is promising as a
subject of future research.

It is possible to try and replicate the capability of the proposed method
using only the SliceGAN presented by Kench and Cooper 17. This can be
attempted by searching the latent space of a pretrained generator for a
specific high-res 3D mesostructure whose down-sampled version and the input 3D
volume have a low voxel-wise loss. However, this process is often
unsuccessful, landing in unrealistic local minima far from the desired
mesostructure. This is likely because although the latent space of a typical
SliceGAN generator contains a very large space of possible mesostructures, it
does not contain every possible arrangement of features. Thus, it is highly
likely that a specific 3D image above a certain size is not contained within
$G$. SliceGAN is only able to reconstruct mesostructures with the same
statistical properties, as opposed to the exact high-res volume associated
with the available low-res dataset.

Any super-resolution approach will contain a degree of uncertainty in the
generated result. One advantage of the SuperRes approach is that the expected
variation in the generated volume can be explored by varying the noise input
as can be seen in Figure 9 and subsequent figures in the Supplementary
Information.

###  4.2 Representativity

During training, $G$ generates $64^{3}$ voxel cubes in such a way that, after
convergence, every 2D perpendicular $64^{2}$ slice of the cube is
indistinguishable by the discriminator from the real training data $64^{2}$
samples. After training, the final output is then generated using an
arbitrarily larger input to the generator. For example, a low-res input volume
of $128^{3}$ with a scale factor of $4$ will output a super-res volume size of
$512^{3}$, while during training generating $64^{3}$ volumes required only
$16^{3}$ low-res inputs. A possible limitation of the SuperRes method is that
the large volume will fail to capture very long-range interactions, since it
was only trained and measured by generating small volumes.

The impacts of limited high-res training data are highlighted in the second
and fourth case studies. We have not encountered low-res training data limits,
nor do we expect this to be encountered by users of the model. This is because
the low-res data typically captures a large FoV in 3D, which should provide
plenty of data, especially after data augmentation.

At first glance, the fact that the low-res volume is used for both training
and evaluating the model (by being used in generating the result), might seem
problematic. However, in the context of super-resolution, generalization is
not an issue because the model’s mission is to generate a super-res image of
the low-res input data given. This also simplifies the requirements on the
user as they are only required to submit two data-sets: The high-res 2D slice
and the low-res 3D volume to be super-resolved; there is no need for a
train/validation/test data division.

We have not rigorously defined what representative is in respect to the high-
res training data. Representativeness is an illusive concept in the
literature, partly because its definition is context dependent. For example, a
sufficiently large volume for confidently establishing volume fractions may
not be large enough to act as a representative domain for simulating
electrochemical processes. Broadly, a representative volume of a
macrohomogeneous material can be seen as one where the statistics of the
volume are acceptably close to the global statistics of the material. However,
this can be hard to establish given limited data. An investigation of the
definition of representativity in energy materials will be presented in a
future study.

In our high-res data, the isotropy is assumed based on our knowledge of the
material, but not quantified. It could be that some degree of anisotropy was
present. An implication for our work is that, for example, the binder phase in
case study 1 might appear to be isotropic based on the three orthogonal
slicing directions, but sampling a slice of a generated volume in an arbitrary
direction may yield different statistics. However, we suspect this will only
have a fairly modest impact on the resulting structures of the cases we
explore, related to the level of anisotropy of the high-res input data.

The transport efficiency 4 (the ratio of the phase fraction to its tortuosity
factor) can be used to explore the longer range interactions in a large
volume, because it is calculated by simulating steady-state diffusion through
the entire volume between two parallel faces. A comparison was made between
the transport efficiencies across the different phases of the super-res large
volume and the mean of the small $64^{3}$ cubes in both the validation case
studies, and no apparent difference was found, with $<0.01$ mean absolute
difference. This positive finding can be linked to the nature of the super-
resolution process, as the bulk structure of the material is already present
in the low-res volume, retaining the long-range structure of the material. For
example, the active material arrangement in the first case study, and the
lamellae arrangement in the second case study.

##  5 Conclusion

The results presented in this paper show the ability of the SuperRes algorithm
to faithfully synthesize a high-res, multi-phase and large FoV volume by
fusing complementary 2D and 3D image data. This was shown across a range of
lengthscales and phase sensitivities. Although this study focused on energy
storage materials, the application of SuperRes is not limited to this domain,
and could be extended to other materials and imaging techniques. For example,
catalyst beds, rock formations, or snow and soil packing, as well as medical
imaging.

By the complementary fusion of 2D and 3D image data, this method can overcome
the physical and experimental constraints of contemporary 3D imaging such as
contrast, resolution and FoV. With realistic, super-resolved, style-
transferred 3D data, more accurate simulations and characterisations of
materials can be performed. Finally, even when possible, imaging high-res 3D
volumes can be expensive and time-consuming. SuperRes offers an alternative,
exchanging expensive machinery and complex techniques with open-source
software.

Future developments of this method might include adjusting the architecture
such that it can more easily capture longer range relationships as the network
can be deeper and the information density could be shared among a larger
region. Other future work might include a combination with other common GAN
approaches such as conditional GANs and transfer learning. The former will
enable interpolation between mesostructures with differing properties, while
the latter aims to speed up the training process. In this study, only imaging
techniques that capture the geometry and phase have been used. An extension of
this method could include a data fusion of techniques that capture a wider
range of crystallographic, chemical and mechanical information, such as
electron backscatter diffraction and energy-dispersive X-ray spectroscopy.
Additionally, this model could be used in an algorithm that applies to an in-
painting problem, whereby missing or erroneous data can be in-painted,
removing the need to repeat long imaging processes containing small errors and
avoiding throwing away potentially useful datasets. Thus, there is broad
potential for expansion of this approach as a tool for material
characterization and optimization.

## Data Availability

The study used open-access training data available from the following sources
as described in the Supplementary Information Table 1. All the generated
volumes and their corresponding input data are available at
https://zenodo.org/record/7104555#.YyxPedJByGY.

## Code Availability

The codes used in this manuscript are available at https://github.com/tldr-
group/SuperRes with an MIT license agreement.

## Acknowledgements

This work was supported by funding from the EPSRC Faraday Institution Multi-
Scale Modelling project (https://faraday.ac.uk/; EP/S003053/1, grant number
FIRG003 received by AD, SK and SJC), and funding from the President’s PhD
Scholarships received by AD. Funding from the Henry Royce Institute’s
Materials 4.0 initiative was also received by AD and SJC to support the
completion and extension of this study (EP/W032279/1). The Titan Xp GPU used
for this research was kindly donated by the NVIDIA Corporation through their
GPU Grant program to SJC. The authors would also like to thank Liam Yasin,
Dominic Williamson, Owen O’Connor, Katie Zeng and Boyd Siripornpitak for their
useful comments, as well as the rest of the tldr group.

## Authorship

AD designed and developed the code for SuperRes, trained the models, performed
the statistical analysis and drafted the manuscript. SK, IS and SJC
contributed to the development of the concepts presented in all sections of
this work, helped with data interpretation and made substantial revisions and
edits to all sections of the draft manuscript.

## Competing interests

The authors declare no competing interests.

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## Supplementary Information

Case | Name | Isotropy | Data | Size (voxels number) | Shown in figure | Phases visible | Resolution | Field of view | Source
---|---|---|---|---|---|---|---|---|---
1 |  Battery cathode | Isotropic | High-res input | $2048^{2}$ |  $256^{2}$ (Both Fig. 1 and 3) | 3 | Medium | Large |  Both inputs were generated using 17 from dataset 29.
Low-res input 1.6 | $320^{3}$ |  $128^{2}$ (Fig. 3) | 2 | low | Large
Low-res input 2 | $256^{3}$ |  $128^{2}$ (Fig. 3) | 2 | Low | Large
Low-res input 4 | $128^{3}$ |  $60^{3}$ (Fig. 1) $128^{2}$ (Fig. 3) | 2 | V. low | Large
Low-res input 8 | $64^{3}$ |  $128^{2}$ (Figure 3) | 2 | V. low | Large
Super-res output | $512^{3}$ |  $240^{3}$ (Fig. 1) $128^{2}$ (Fig. 3) | 3 | Medium | Large
2 |  Battery separator | Anisotropic | High-res input | $624\times 300\ (\times 4)+300^{2}\ (\times 2)$ | $624\times 300\ (\times 2)$ | 2 | High | Small |  High-res input 31 Low-res input 7.
Low-res input | $156\times 75\times 75$ | $156\times 75\times 75$ | 2 | Low | Large
Super-res output | $624\times 300\times 300$ | $624\times 300\times 300$ | 2 | High | Large
3 |  Fuel cell anode | Isotropic | High-res input | $1024^{2}$ | $256^{2}$ | 3 | High | Medium |  Both inputs were obtained from dataset 12.
Low-res input 4 | $64^{3}$ | $64^{3}$ | 3 | Low | Large
All super-res outputs | $256^{3}$ | $256^{3}$ | 3 | High | Large
4 |  Battery cathode | Isotropic | High-res input | $758\times 628$ | $758\times 628$ | 3 | High | Small |  High-res input (SEM) and low-res input (XCT) from dataset 29.
Low-res input | $240^{3}$ | $64^{3}$ | 2 | Medium | Large
Super-res output | $1920^{3}$ | $512^{3}$ | 3 | High | Large

Table 1: Summary of the open-access input data and generate output volumes for
the four scenarios investigated. Further information about each dataset can be
found in the Supporting information  Figure 10: Further statistics comparing
the generated volumes to the original volumes for the first three case
studies. In all plots, the continuous line corresponds to the original volume
and the dashed line corresponds to the generated super-res volume. The three
columns refer to the first three case studies accordingly. The first row shows
the calculation of the intraphase 2-point correlation function, which
calculates the probability of reaching the same phase if moving $d$ voxels in
a certain direction. The second row shows the calculation of the interphase
2-point correlation function, which is similar to the intraphase function, but
calculates the probability of reaching a different phase if moving $d$ voxels
in a certain direction, for $d\in\\{0,\dots,80\\}$. Notice that for the
2-phase separator material, the interphase function is just the complementary
probability of the intraphase function. The last row shows the chord-length
distribution, which is the probability of a voxel to be part of a $d$-length
chord of the same phase in a certain direction. The distance 80 limit was
chosen since the functions converged from this point onwards. For isotropic
materials case studies 1 and 3, only one direction is shown since similar
results were obtained for the other directions. For the 2-phase anisotropic
case study 2, both the ion-flow direction and the perpendicular ‘through‘ the
lamellae sheets direction are shown. Figure 11: Model training diagram.
Gaussian blurring is part of the down sampling as outlined in the Methods
section. Figure 12: The process of creating the low-res input data volume from
the high-res volume for case study 1. For better visualisation, a random
$128^{2}$ sample (later $32^{2}$ after down-sampling) from a random slice was
chosen from the volume. First, (a) shows a sample from the original high-res
3-phase volume. Then, (b) only the active material phase is chosen for
blurring and down-sampling since the binder phase cannot be detected in low
resolution using XCT. (b) is converted to gray-scale to (c) apply a gaussian
blurring kernel and down-sampling (d) to approximately describe the physics
limitation of the imaging technique. The gray-scale volume is then converted
to (e) a 2-phase volume created by using a 0.5 threshold, mimicking a
procedure made by researchers to obtain a segmented volume of the material.
Figure 13: The structure of the generator ($G$) and the discriminator ($D$) in
the GAN architecture. $G$’s structure is dependent on the scale factor used,
and can have 3 different structures. This design enables the same output of
$64^{3}$ cubes from different inputs, that are dependent on the scale factor.
For example, low-res input with a scale factor of $2$ is a $32^{3}$ cube and a
low-res input with a scale factor of $4$ is a $16^{3}$ cube and both will
output a $64^{3}$ super-res cube.  Figure 14: Presentation of different super-
res volumes generated by the model of case study 1 using 3 different random
noise inputs. The structure of this figure is similar to Figure 9 with the
addition of a slice from the original 3D high-res volume that the low-res
volume that was derived from. Figure 15: Presentation of different super-res
volumes generated by the model of case study 3 with scale factor of 8 using 3
different random noise inputs. The structure of this figure is similar to
Figure 14.
In this case of high scale factor, one voxel of the low-res input should
represent $8^{3}=512$ voxels of the original volume. This difference in orders
of magnitude naturally leads to a large loss of information when interpolating
from the original volume to the low-res volume. As a result, there are a huge
number of different reconstruction options for the super-res volume, all of
which will be statistically similar to the original volume. Figure 16:
Segmentation of the 2D SEM image used in case study 4. The segmentation from
greyscale to three-phase image was performed using the Trainable Weka
Segmentation software package 1 , accompanied by some manual finishes to
account for imaging artefacts. The “curtaining” effect is visible as vertical
streaks in the lower right-hand corner of the image. This caused Weka to
mislabel some of the active material phase as binder, which we corrected
manually. The “pore-backs” effect caused some regions that perhaps should have
been labelled as pore, to be labelled as binder instead, which we also
adjusted manually. The total number of manually adjusted pixels represented
less than 1%of the image. The different segmented phases are pore (black),
active material (grey) and binder (white). The original image is taken from an
open source dataset 29. Figure 17: The importance of concatenating noise to
the low-res input of $G$. A comparison is made between models that are trained
with and without the concatenated input noise. Left column shows one slice of
the low-res volume input to $G$, top left is from an all pore volume input and
bottom left is from a volume that contains active material. The middle and
right columns show slices of the outputs from models trained without (middle)
and with (right) concatenated noise inputs. Respectably, the top row are
slices of outputs from the all-pore volume (top left) and the bottom row are
slices of outputs from the volume that contains active material (bottom left).
Figure 18: FFT magnitude plots comparison between the blurred original volumes
and the generated super-res images made on the first two case studies. (a) The
left column shows a slice of the respective volumes in the same location;
column (b) shows the (centrally shifted) logarithmic FFT magnitude plots,
while column (c) shows the plots magnitude profiles from the center towards
higher frequencies by calculating the mean magnitude over receding rings
around the center. Column (d) shows a comparison for each case study of the
different magnitude plots. The profile comparison clearly shows a shifted
elbow presence of higher-frequency information in the generated volumes, which
indicates a higher resolution. For the n-phase image in the first case study,
which is not grey-scale, the map is a mean of the FFT magnitude maps for the
different phase maps (In this case the active material map and binder phase
map). The blurred original volumes were chosen over the low-res input volumes
since these images are in the same voxel-size as the generated volumes, for a
simpler comparison. The low-res input, which is a result of down-sampling and
segmentation of the blurred volume, contain less high fidelity information
than the blurred volume.