# A Closer Look at Spatial-Slice Features Learning for COVID-19 Detection

†1Chih-Chung Hsu, ‡1Chia-Ming Lee, 2Yang Fan Chiang, 1Yi-Shiuan Chou,
1Chih-Yu Jiang,1Shen-Chieh Tai, 1Chi-Han Tsai
1Institute of Data Science, National Cheng Kung University, Taiwan
2Department of Electrical Engineering, National Cheng Kung University, Taiwan
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Conventional Computed Tomography (CT) imaging recognition faces two
significant challenges: (1) There is often considerable variability in the
resolution and size of each CT scan, necessitating strict requirements for the
input size and adaptability of models. (2) CT-scan contains large number of
out-of-distribution (OOD) slices. The crucial features may only be present in
specific spatial regions and slices of the entire CT scan. How can we
effectively figure out where these are located? To deal with this, we
introduce an enhanced Spatial-Slice Feature Learning (SSFL++) framework
specifically designed for CT scan. It aims to filter out OOD data within the
entire CT scan, enabling us to select crucial spatial slices for analysis by
reducing 70% redundancy totally. Meanwhile, we proposed Kernel-Density-based
slice Sampling (KDS) method to improve the stability during training and
inference stage, therefore speeding up the rate of convergence and boosting
performance. As a result, the experiments demonstrate the promising
performance of our model using a simple EfficientNet-2D (E2D) model, even with
only 1% of the training data. The efficacy of our approach has been validated
on the COVID-19-CT-DB datasets provided by the DEF-AI-MIA workshop, in
conjunction with CVPR 2024. Our code is available at
https://github.com/ming053l/E2D.

##  1 Introduction

Computed Tomography (CT) [53]has become essential in detecting and managing
diseases. This technology excels at revealing abnormalities within the body,
such as ground-glass opacities and bilateral patchy shadows, which are crucial
for the early detection and monitoring of diseases. In diagnosing COVID-19,
doctors rely on analyzing lung CT scans of patients. However, since a single
patient’s CT scan can include hundreds of images, manual examination becomes a
time-consuming task, especially when doctors have to evaluate CT scans from
dozens or hundreds of patients. This may result in false negatives when
dealing with numerous scans.

With the rapid development of deep learning (DL), DL methods [17, 51, 26, 18,
25, 48, 63] have gained prominence for their ability to quickly and accurately
identify COVID-19 features while efficiently handling large volumes of data.
Furthermore, convolution neural networks (CNNs) have proven to be more
effective than methods based on frequency-domain [68, 49] and low-level
features for CT image analysis [41].

To address the terribly spreading COVID-19, Kolliaz _et al._ proposed the
COVID-19-CT-DB dataset [2, 3, 34, 35, 37, 39, 36, 38], which encompasses a
vast amount of labeled COVID-19 and non-COVID-19 data, advancing the DL
methodology and tackling the challenge faced by the huge requirement of high
quality dataset for DL-based analysis. Many researchers have designed several
methods to deal with COVID-detection task [11, 29, 30, 66].

Despite the effectiveness of CT imaging as a tool for detecting abnormalities,
it suffers from varying resolutions and quality due to different data servers
and scanning machines. The resolution and number of slices in CT images can
differ based on the specific scanning machine used, potentially compelling
researchers to devise more complex network architectures. Additionally,
medical analysis for COVID-19, unlike typical DL-based tasks that focus solely
on performance and applications, necessitates maintaining the explainability
of model predictions for security and safety reasons [12, 47, 11].

Inspired by [57], Tran _et al._ presented that factorizing the 3D convolution
filters (R3D) into separate spatial and temporal components (R(2+1)D) can
yielding significantly gains in accuracy for action recognition. Its
effectiveness have been demonstrated by several works on the fields of Video
Understanding (VU) [42, 20, 44, 6] and Human Action Recognition (HAR) [64,
58]. One video may contains huge redundant information, such as noise from the
audio track or each frame, and meaningless background, these factors make it
difficult to train the model well [7], resulting in a significant increase in
potential costs for data collecting. Likewise, CT scans can be regarded as a
special case of video, it also contains various noise resulted from machine
aging, and non-important spatial-slice pattern due to its imaging process
[53]. Therefore, the different convolution methods on CT-scan is worthy of
discussion.

Figure 1: The brief illustration for SSFL++. It aim to reduce redundancy in
spatial and slice dimension on whole CT-scan to improve model and data
quality. (1)Left: original CT-scan. (2)Middle: after reduction at spatial.
(3)Right: after reduction at slices.

In this work, we introduce a Spatial-Slice Feature Learning (SSFL++) method,
an unsupervised approach designed to reduce computational complexity by
effectively removing out-of-distribution (OOD) slices and redundant spatial
information. Furthermore, previous works [30, 11] have struggled to identify
the most influential slices while considering global sequence information.
Based on this observation, we believe there is room for improvement.
Therefore, we propose the Kernel-Density-based Slice Sampling (KDS) strategy,
which leverages Kernel Density Estimation to simultaneously achieve both
objectives. Experimental results have demonstrated our 2D model’s outstanding
performance, even in the face of data insufficiency.

Our novelties and contributions can be briefly divided into two parts as
following mentions:

  * •

Improved spatial-slice feature learning module: SSFL++ is a morphology-based
approach for CT scans that removes redundant areas in both spatial and slice
dimensions. This significantly reduces computational complexity and
efficiently identifies the Regions of Interest (RoI) without the need for
complicated designs or configurations. Remarkably, we were able to eliminate
70% of the area in the COVID-19-CT-DB datasets without any degradation in
performance.

  * •

The comparison between 2D, (2+1)D, and 3D for CT-scan is discussed: To
facilitate the development of related research, we conducted a thorough
discussion on the use of 2D, 2+1D, and 3D convolutions for CT scan data in
COVID-19 detection. Based on experimental results, we believe that the 2D
convolutional architecture holds more potential for future applications
compared to 3D and 2+1D convolutions.

  * •

Density-aware slice sampling method: Coupled with SSFL++’s ability to
adaptively remove redundant spatial areas and slices, KDS further adaptively
samples the most crucial slices while preserving global sequence information.
This approach enhances data efficiency and strengthens the model’s few-shot
capabilities. Experimental results have shown that our E2D model maintains
strong and robust performance under scenarios with few CT scans and slices.

##  2 Related Work

In this section, we introduce the related works on COVID-19 recognition in
recent years, along with traditional spatial-temporal feature learning for
Video Understanding (VU) and Human Action Recognition (HAR). The philosophy
behind these approaches is important for analyzing CT-Scans.

###  2.1 Region of Interests for Computed Tomography

Background. CT [53] harnesses X-rays, which encircle a specific plane of the
human body, while detectors on the opposite side capture the resultant
signals. This technique exploits the differential attenuation of X-rays by
various tissues, combined with signals obtained from multiple irradiation
angles traversing the body, to compile a sinogram. This sinogram facilitates
the reconstruction of cross-sectional imagery [4, 5]. Nonetheless, the CT
imaging paradigm, necessitating multi-angular signal acquisition for
reconstruction, engenders scans replete with extraneous data, potentially
escalating labor costs.

Although this technology has been around for a long time, designing a robust
and reliable Region of Interest (RoI) selection algorithm for CT scans remains
an open problem. Noise and redundancy harm model performance. In recent years,
most researchers have still focused on enhancing the feature extraction
pipeline [45], or improving the quality of image reconstruction [27], to
address the aforementioned challenges. Cobo _et al_. [14] suggested that
standardizing medical imaging workflows could improve the performance of
radiomics and deep learning systems. Jensen _et al_. [32] proposed enhancing
the stability of CT radiomics with parametric feature maps. Gaidel _et al_.
[23] introduced a greedy forward selection-based method for lung CT images,
but its development was limited due to a lack of robustness against data
shifting and noise.

###  2.2 COVID-19 Recognition

In recent years, substantial progress has been achieved in developing methods
for COVID-19 recognition. Kollias _et al_. [34] have contributed to this field
by analyzing the prediction results of deep learning models based on latent
representations. Chen _et al_. [11] integrated maximum likelihood estimation
with the Wilcoxon test, adopting a statistical learning perspective to
adaptively select slices and design models with explainability.

Furthermore, Hou _et al_. proposed a method based on contrastive learning to
enhance feature representation. Turnbull _et al_. applied a 3D ResNet [28] for
COVID-19 severity classification. Hsu _et al_. [29] introduced a two-step
model that combines 2D feature extraction with an LSTM [19] and Vision
Transformer [16]. They presented a 2D and (2+1)D approach [30], achieving
outstanding results in the AI-MIA 2023 COVID-19 detection competition.

###  2.3 Spatiotemporal Feature Learning for Video

Video analysis is crucial for computer vision, as videos contain far more
information than single images. This analysis focuses on extracting
spatiotemporal features, with traditional methods relying on optical flow [55,
8] and trajectory analysis [50, 67]. With the advent of deep learning (DL), a
strategy employing 2D Convolution Neural Networks (CNNs) was proposed [22,
65]. This strategy includes temporal feature pooling to aggregate features
from different frames for classification. Subsequently, approaches combining
CNNs with Recurrent Neural Networks (RNNs) and Long Short-Term Memory (LSTM)
networks [1] were introduced, aiming to capture the long-range dependencies
across various frames. 3D convolution kernels (C3D [56], I3D [9]) are used in
video understanding, capturing channel interactions and local interactions
simultaneously. However, they lead to a computational burden and have been
regarded as an inefficient approach.

Subsequently, strategies offering greater efficiency were introduced, such as
the Non-local network [59], S3D [62], CoST [43], SlowFast [21], and CSN [58].
These methods more efficiently learn the spatiotemporal features of videos by
either reducing the number of sampled frames or replacing the use of 3D
convolution with (2+1)D convolution. The prevailing consensus has moved away
from the necessity of utilizing a large number of video frames or 3D
convolution as the optimal approach for learning spatiotemporal features.
Similarly, considering the resemblance between CT scans and videos, it is
plausible to learn the feature representation of CT scans using only a small
number of slices, without relying on 3D CNNs.

##  3 2D, (2+1)D, 3D Convolutions for CT Scan

In this section, we discuss the three types of convolutions within framework
of COVID-19 detection. The detailed architecture is described in Section 5-1.

2D: 2D Convolution over the sampled slices. The use of 2D convolution networks
for extracting spatio-temporal features from 3D-cube data faces certain
limitations, such as the requirement for strong spectral band or temporal
continuity. Without these prerequisites, 2D convolutions may struggle to
perform effectively due to their focus on spatial features and a lack of
comprehensive sequence modeling. In applications involving CT scans, 2D
convolutions are generally considered less effective compared to architectures
like 2+1D convolutions, CNN-LSTM, or CNN-RNN, which are capable of capturing
spatiotemporal features more efficiently. However, previous 2D CNN approaches
often involve pre-processing, where crucial slices are selected and sampled to
serve as inputs for the network. This sampling process tends to be overly
simplistic, for instance, by manually selecting slices with the least
artifacts or best quality, or randomly selecting a few slices to train a 2D
CNN model. This limits the network’s potential for global sequential modeling.

(2+1)D: 1D Convolution over the extracted features on different dimension. The
2+1D model is widely regarded as the greatest solution for CT analysis due to
its exceptional performance and lower computational costs compared to 3D
models. Typically, the 2+1D model performs best as it first extracts features
on the spatial scale before modeling the sequences of these extracted
features, effectively achieving both. However, according to our experiments,
it tends to be less robust in situations with limited samples. This is because
CT scans vary greatly in terms of resolution or the number of slices, making
the 2+1D model more sensitive to the quantity of training data compared to 2D
models. Additionally, we believe a potential concern with the 2+1D model is
its difficulty in augmentation since spatial features are encoded into the
latent space, the implicit learning approach limits its scalability and
interpretability in clinical applications.

3D: 3D Convolution over the contiguous slices. _Compared with 2D and 2+1D, 3D
is a heavy computational resource burden for COVID-19 detetion._ The
differences between CT scans and conventional videos lie in several key
aspects. Firstly, videos typically contain a significantly larger number of
frames compared to the number of slices in a CT scan. Secondly, videos enhance
their spatio-temporal coherence through frame rates (FPS), whereas the spatial
relationships between slices in CT scans are relatively weaker. Lastly, slices
in CT scans often exhibit redundancy at the beginning and end, which does not
substantially contribute to analysis.

In conclusion, the advantages and weaknesses of these three methods can be
itemized as follows:

  * •

2D: Training and testing pipeline are simple. The model is robust no matter
when few-scan or few-slice. Easy to augment. There are multiple methods which
provide an explaniability for 2D model’s prediction, such as GradCAM++ [10],
SHAP [46]. Uneasy to capture sequential information unless dedicated design.

  * •

(2+1)D: The performance is optimal when there is enough training data and the
length of the CT slice sequence is sufficiently large, allowing it to capture
sequential information. However, it becomes unstable with only a few scans or
slices; the pipeline is complicated. It is also difficult to explain and
augment.

  * •

3D: Training and testing pipeline are simple. Can capture sequential
information. Worst performance. Highest computational complexity. Unstable
when few-scan and few-slice. Hard to explain and augment.

We believe 2D-CNNs have the potential to become mainstream for COVID-19
detection tasks. To enhance the ability of 2D-CNNs to learn sequence
information from CT scans, we have designed the KDS method. This approach
helps overcome the limitations of 2D-CNNs in this regard, with details to be
introduced in Section 4.2.

##  4 Methodology

###  4.1 Spatial-Slice Feature Learning

In this section, we introduces our proposed SSFL++, which aim to figure out
the RoIs in spatial and slice dimension, mainly based on the simple but
effective computed morphology method and formulation of optimization problem.

Figure 2: The illustration of spatial steps in proposed SSFL++.

Spatial Steps. The most importance concern is that CT-scan alway exists large
black area between every single CT slice’s background, and it will distort the
RoI area when resizing to fixed shape to neural network, leading to feature
vanish. In order to deal with this, a low-pass filter with a window size of
$k\times k$ is applied to all CT slices $\mathbf{Z}$ to eliminate a noises.
The low-pass filtering operator can be defined as:

$\mathbf{Z}_{\text{filtered}}(i,j)=\frac{\sum_{p=-k}^{k}\sum_{q=-k}^{k}w(p,q)\times\mathbf{Z}(i+p,j+q)}{\sum_{p=-k}^{k}\sum_{q=-k}^{k}w(p,q)}$
(1)

where $w(p,q)$ represents the weight at position $(p,q)$ in the filter kernel.
The above formula can determine the segmentation $\mathbf{Mask}$ of the
filtered slices by a threshold $t$:

$\mathbf{Mask}[i,j]=\begin{cases}0,\,\text{if}\,\mathbf{Z}_{\text{filter}}[i,j]<t\\\
1,\,\text{if}\,\mathbf{Z}_{\text{filter}}[i,j]>=t\end{cases}$ (2)

where i, j denote as an pixel for every single CT slice $\mathbf{Z}^{c}$,
which resolution is $x$ $\times$ $y$. A Cropped region
$\mathbf{Z}_{\text{crop}}^{c}$ can be calculated by:

$\displaystyle\text{min}(\mathbf{Z}_{\text{crop}}^{c}(x))=\min\\{i\mid\mathbf{Mask}[i,j]=1,\forall
i\\}$ (3)
$\displaystyle\text{max}(\mathbf{Z}_{\text{crop}}^{c}(x))=\max\\{i\mid\mathbf{Mask}[i,j]=1,\forall
i\\}$
$\displaystyle\text{min}(\mathbf{Z}_{\text{crop}}^{c}(y))=\min\\{j\mid\mathbf{Mask}[i,j]=1,\forall
j\\}$
$\displaystyle\text{max}(\mathbf{Z}_{\text{crop}}^{c}(y))=\max\\{j\mid\mathbf{Mask}[i,j]=1,\forall
j\\}$

$\mathbf{Z}_{crop}^{c}$ is yielded accordingly, we can further resize the
resolution of $\mathbf{Z}_{\text{crop}}^{c}$ to $H$$\times$$W$ for the slice
steps and as an input of neural network. Spatial Steps in proposed 4SFL
effectively filter out non-lung tissue regions (also known as RoIs in spatial
dimension), and reduce computational complexity, as the Figure 2 illustrated.

Figure 3: The illustration of slice steps in proposed SSFL++. The line graph
in the bottom right corner represents the area of each slice in a CT scan. The
blue area denotes OOD data that have been removed, while the red area
represents the CT slices that have been selected.

Slice Steps. To find the lung tissue region in the CT scan, we used the binary
dilation algorithm [61] to obtain the filled result
$\mathbf{Mask}_{\text{filled}}$. The difference between the $\mathbf{Mask}$
and filled mask $\mathbf{Mask}_{\text{filled}}$ represents the lung tissue
region. The above method can be summarized as the following formula:

$Area(\mathbf{Z})=\sum_{i}\sum_{j}\mathbf{Mask}_{\text{filled}}(i,j)-\mathbf{Mask}(i,j).$
(4)

After the above technique, we can finally obtain a range where $s$ and $e$
denote the starting and ending indexes, respectively, and $n_{c}$ is the
constraint of the number of slices for a single CT scan to select most
importance RoIs in slice dimension with proportion $\alpha$. The optimization
problem can be formulated as following:

$\begin{split}&\underset{s,\,e}{\text{maximize}}\quad\sum^{e}_{i=s}Area(\mathbf{Z}_{i}),\\\
&\text{subject to }\quad e-s\leq n_{c},\\\
&\quad\frac{\sum^{e}_{i=s}Area(\mathbf{Z}_{i})}{\sum_{i=1}^{n_{c}}Area(\mathbf{Z}_{i})}\geq\alpha.\end{split}$
(5)

It is worth noting that we sort all CT slices according to their slice numbers
$n_{c}$, as illustrated in the bottom-right corner of Figure 3.

The spatial and slice steps of proposed SSFL++ follow unsupervised learning
manner, which only follow the prior knowledge of lung-CT-scan. It can be
generalize to other organs or body parts CT-scan. However, it may require
parameter adjustments based on their specific characteristics. Additionally,
with the SSFL++, the visual explanation method can also look RoI more
concentrated, as shown in Figure 4.

Figure 4: The GradCAM++ [10] visualization before and after proposed SSFL++.
By reducing redundancy on the spatial scale, we can implicitly enhance the
visual effectiveness of Explainable AI, thereby facilitating clinical
applications.

###  4.2 Density-aware Slice Sampling

Figure 5: The comparison between random sampling, systematic sampling, and the
proposed KDS method is noteworthy. As illustrated, random sampling fails to
uniformly sample CT slices of varying area sizes, tending to select larger
areas while neglecting global information. This results in greater bias and
randomness during training and inference. On the other hand, systematic
sampling divides the area into equally lengthened sub-intervals before
randomly selecting samples from them. Although this approach can capture
global information, it is ineffective at sampling the most crucial CT slices.
Our proposed KDS method combines the advantages of both methods without their
drawbacks, achieving a better balance. KDS can implicitly improve data
efficiency, thereby enhancing the model’s few-shot capability.

Background. The SSFL proposed by Hsu _et al._ [30] employs a random sampling
method to select slices, which were used for the detection of COVID-19 using
2D and 2+1D CNNs. However, random sampling may potentially introduce bias and
instability when training and inference, and it does not efficiently identify
the most representative CT slices, as shown in Figure 5.

In order to address this, we propose a Kernel-Density-based Slice Sampling
(KDS). It performs kernel density estimation (KDE) on the selected slices-set
[$\mathbf{Z}_{e}$,$\mathbf{Z}_{s}$], adaptively and wisely sampling the most
crucial CT-slices. Meanwhile, it also keeps the sequence information globally
and alleviates the instability during training and inference stage.

Definition. KDE is a classic method to estimate the probability density
function (PDF) of a random variable in a non-parametric manner. It can be
defined as:

${\widehat{f}}_{h}(x)={\frac{1}{s}}\sum_{i=1}^{s}K_{h}(x-x_{i})={\frac{1}{sh}}\sum_{i=1}^{s}K\left({\frac{x-x_{i}}{h}}\right)$
(6)

$K(x,x^{\prime})=\exp\left(-\frac{\|x-x^{\prime}\|^{2}}{2\sigma^{2}}\right),$
(7)

where $h$ is the bandwidth constant, calculated by Scott-rule [52], $K$ is a
Gaussian kernel, $s$ is a smooth factor of estimated density function, (the
higher the smoother, we set it to 100). For a given KDE, we can create several
sub-intervals by calculating its Cumulative Distribution Function (CDF), where
the length of each sub-interval adaptively changes with its $p$-percentile.
The CDF of KDE and its $p$-percentile can be calculated as following formulas:

$F(x)=\int_{-\infty}^{x}\hat{f}_{h}(t)dt,F(q_{p})=p$ (8)

Figure 6: In terms of optimizing procedure, our proposed KDS approach,
compared to the random sampling used by Hsu _et al._ [30], is more capable of
learning the global information of CT scans, thereby accelerating the
convergence rate and enhancing the model performance.

In the proposed KDS method, we determine the probability of slices being
selected in each interval based on the density from KDE, while also ensuring
that each sub-interval has at least one sample selected. This method captures
the global sequential information and increases the probability of selecting
the most crucial CT slices.

##  5 Experiment

| Spatial Area (K) | Slice Length | Spatial $\times$ Slice (M) | Total
---|---|---|---|---
| Before | After |  $\Delta$ $(\%)$ | Before | After |  $\Delta$ $(\%)$ | Before | After |  $\Delta$ $(\%)$
Training | 267.25 | 155.53 | 0.4184 | 285.32 | 142.91 | 0.4983 | 76.25 | 22.22 | 0.7085
Positive | 266.42 | 157.69 | 0.4088 | 295.90 | 148.18 | 0.4985 | 78.83 | 23.36 | 0.7036
Negative | 268.21 | 153.03 | 0.4296 | 273.97 | 137.26 | 0.4981 | 73.48 | 21.00 | 0.7141
Validation | 265.62 | 155.23 | 0.4172 | 281.95 | 141.23 | 0.4984 | 74.89 | 21.92 | 0.7072
Positive | 268.94 | 160.48 | 0.4061 | 280.53 | 140.55 | 0.4984 | 75.45 | 22.55 | 0.7010
Negative | 262.12 | 149.69 | 0.4288 | 283.49 | 141.97 | 0.4984 | 74.30 | 21.25 | 0.7139
(T+V) Positive | 267.25 | 155.53 | 0.4184 | 292.96 | 146.72 | 0.4985 | 78.29 | 22.81 | 0.7085
(T+V) Negative | 267.01 | 152.37 | 0.4294 | 275.78 | 138.16 | 0.4982 | 73.64 | 21.05 | 0.7141
Total | 266.94 | 155.47 | 0.4182 | 284.68 | 142.59 | 0.4983 | 75.99 | 22.16 | 0.7082
Testing | 279.55 | 153.41 | 0.4520 | 309.39 | 154.67 | 0.5003 | 86.48 | 23.72 | 0.7256

Table 1: The reduction in redundant data achieved by the SSFL++ module is
evaluated across three dimensions: spatial, slice, and overall. This approach
quantifies the efficiency of the SSFL++ module in reducing unnecessary
information in CT scans, enabling more focused analysis and processing. By
minimizing data redundancy, the module enhances computational efficiency and
potentially improves the accuracy of subsequent analyses or models applied to
the CT data.

Dataset description. In our experiments, we used a total of 1,684 COVID-19-CT-
DB data, provided by Kollias _et al._ [40]. The dataset information have shown
in Table 2. Our loss function is binary cross-entropy. In order to ensure
stability and fairly check performance during the experiments, group-5-fold-
cross-validation is used. Data augmentation and hyperparameters are kept
consistent in all experiments.

Type | Positive Scan | Negative Scan | Total Scan
---|---|---|---
Training | 703 | 655 | 1358
Valid | 170 | 156 | 326
Total | 873 | 811 | 1684
Testing | - | - | 1413
Type | Positive Slice | Negative Slice | Total Slice
Training | 206608 | 178722 | 385330
Valid | 46042 | 43679 | 89721
Total | 252650 | 222401 | 475051
Testing | - | - | 437185
Table 2: The number of data samples at the scan and slice level.

Hyperparameter settings. The Adam [33] optimizer was used with a learning rate
of $1e-4$ and a weight decay of $5e-4$. The batch-size is set to $16$.

Data Augmentation. In our experiments, we utilized common augmentation
strategy like HorizontalFlip, RandomScaleShifting to prevent overfitting and
enlarge feature space. Additionally, we find that HueSaturationValue,
RandomBrightnessContrast and CoarseDropout [15] are also used.

Evaluation Metric. We mainly used F1-score in the experiments for model
evaluation. F1-score is a metric used to determine the accuracy of a binary
classification model. It combines the harmonic mean of Precision and Recall.

$\text{f1-score}=2\times\frac{\text{precision}\times\text{recall}}{\text{precision}+\text{recall}},$
(9)

where precision and recall are computed for COVID and non-COVID. The macro
f1-score is the average of the f1-scores for all classes:

$\text{macro f1-score}=\frac{1}{N}\sum_{i=1}^{N}\text{f1-score}_{i}$ (10)

where $N$ is the number of classes, and $\text{f1-score}_{i}$ is the f1-score
for the $i$-th class. These metrics provide a balanced evaluation of the
model’s ability to classify each class accurately and its overall performance
across all classes.

###  5.1 Model Details and Performance Comparison

To provide a more comprehensive comparison and improve future research, we
designed simple E2D, E2+1D, E3D in our experiments. The backbones are all
based on EfficientNet-b3 [54, 60]. The baseline method and detailed pipeline
are as follows:

Baseline: The baseline method is presented in [40], Kollias _et al._ adopted
CNN-RNN to extract feature within all CT-slice. First, all CT-slices are
resized to $224$ $\times$ $224$ to extract feature, then RNN (GRU [13] with
$128$ neurons) analyzed the 2D-CNN (ResNet-50 [28]) features. The output of
the RNN element is then forwarded to a fully connected layer. In addition,
this also includes a dropout layer (the dropout rate is set to $0.8$) before
the fully connected layer.

E2D: From the CT-scans processed by SSFL++, subsequently, we use our proposed
KDS. These sampled slices are resized to $384$ $\times$ $384$ and extracted to
high-representation features.

E2+1D: Similar to E2D, firstly, the CT scans processed by SSFL++ are resized
to $384$ $\times$ $384$. And $100$ slices are selected to be encoded.
therefore, we used 2D encoder to get an encoded vectors. By doing so, the CT
scans will be encoded into latent feature queue, which size is $224$ $\times$
$100$. Subsequently, we randomly sampled $50$ features from latent feature
queue, and utilized a simple 1D convolution with kernel size $1$ $\times$ $1$
in $e$ or $l$ dimensions to capture sequential information.

E3D: We first utilized SSFL++ to remove OOD slices and redundant spatial
information, and then sample a certain number of CT slices for modeling.

The experimental results, as presented in Table 3, highlight the E2D model’s
exceptional performance when paired with KDS on the COVID-19 database 2024
validation set. It also showcases remarkable robustness in few-scan scenarios,
delivering results that instill confidence. Comparatively, the E2D model
utilizing KDS achieves a significant improvement in scan-level f1-score
compared to its counterpart that employs random sampling. This underscores the
capability of 2D convolutions to implicitly capture global sequence
information through an appropriate sampling method. In contrast, the E3D model
demands a large sample size, resulting in limited performance and higher
computational requirements.

Model type | Scans | Sampled slice |  | macro f1-score
---
(slice-level)
| f1-score
---
(scan-level)
| baseline [40]
---
100% | - | - | 78.00
E3D | 1% | 33(random) | - | 32.55
| 50% | 33(random) | - | 78.54
| 100% | 33(random) | - | 86.76
| 100% | 50(random) | - | 87.05
| 100% | 80(random) | - | 90.24
| 100% | 120(random) | - | 91.05
E(2+1)D | 1% | 8(random) | 73.46 | -
| 50% | 8(random) | 87.64 | -
| 100% | 8(random) | 91.39 | -
| 100% | 16(random) | 92.31 | 93.69
E2D | 1% | 8(random) | 88.94 | 92.11
| 50% | 8(random) | 91.52 | 92.42
| 100% | 8(random) | 92.44 | 93.18
| 100% | 16(random) | 92.68 | 93.37
| 1% | 4(KDS) | 91.42 | 96.42
| 1% | 8(KDS) | 91.88 | 99.80
| 100% | 8(KDS) | 93.46 | 100.00
| 100% | 16(KDS) | 94.11 | 100.00

Table 3: Performance comparison between baseline provided by Kollias _et al._
[40], and proposed E2D, E2+1D, E3D on COVID-19-CT-DB validation set.

###  5.2 Ablation Study

Spatial step | Slice step | KDS |  | marco f1-score
---
(slice level)
| f1-score
---
(scan level)
|  |  | 80.41 | 81.26
✓ |  |  | 88.01 | 88.04
| ✓ |  | 90.32 | 90.48
✓ | ✓ |  | 92.68 | 93.37
✓ | ✓ | ✓ | 94.11 | 100.00

Table 4: The ablation study of proposed SSFL++ and KDS on COVID-19-CT-DB
validation set.

| macro-F1 | F1(NONCOVID) | F1(COVID)
---|---|---|---
baseline [40] | 85.11 | 87.48 | 82.74
E2D (Ours) [31] | 94.39 | 95.52 | 93.26

Table 5: The results on COVID-19-CT-DB testing set.

To further analyze the impact of SSFL++ and KDS on the COVID-19 detection
task, the ablation study were conducted, with results presented in Table 4.
All experiments are based on the E2D model, with all experimental
hyperparameters kept constant. The results demonstrate that the proposed
SSFL++ significantly enhances performance, implying the importance of spatial
redundancy in CT scans and efficient slice selection. On the other hand, KDS
further improves the model’s prediction ability at the slice-level and makes
significant progress at the scan-level, achieving convincing performance. KDS
effectively addresses the lack of global sequential modeling capability in
2D-CNN when analyzing CT images.

##  6 Generalizability

Our proposed SSFL++ not only excels in performance on the COVID-19-CT-DB [40]
but also demonstrates commendable efficacy on CT scans from various views and
body parts. We showcased the versatility of SSFL++ by selecting four distinct
types of data, with the results depicted in Figure 7. From top to bottom, the
figures represent the different views or body parts before and after SSFL++.
Specifically, (a) (c) (d) are lung CT scans from the COVID-19-CT-DB dataset,
featuring the axial, sagittal, and coronal views. Meanwhile, (b) involves a
dataset provided by [24], aimed at identifying acute appendicitis from CT
scans of acute abdomen cases.

Additionally, it is important that when using SSFL++ on CT slices of different
body parts or from different views, its hyperparameters may need specific
adjustments. For instance, in the case of (b), the original settings might
select OOD slices rather than the RoI slices.

Figure 7: CT slices from different views and body parts, as well as the
results after processing through the spatial step in our proposed SSFL++, are
presented. From left to right, the sequence represents the process of CT
imaging, where OOD data tend to concentrate at the beginning and the end. The
middle section represents the RoI area. As shown in the figure, SSFL++
performs well under various conditions.

##  7 Conclusion

We conducted a comprehensive analysis of the COVID-19 detection task, noting
that CT scans often contain a large amount of redundant information, which
limits the performance of models. To address this issue, we introduced a
simple morphology-based method for CT images, named Spatial-Slice Feature
Learning (SSFL++), designed to efficiently and adaptively locate the Region of
Interest (RoI). This method effectively reduces redundancy across both spatial
and slice dimensions. Furthermore, to inspire future research, we analyzed the
advantages and disadvantages of 2D, 2+1D, and 3D convolutions on CT data.
After extensive experimentation, we believe that 2D-CNNs hold the greatest
potential in the wild.

To overcome the limitations previously encountered by 2D-CNN in research, we
combined SSFL++ with the further designed KDS, thereby addressing the
instability brought about by random sampling during the training and
inference. Moreover, through the global sequence modeling, we activated the
potential of 2D-CNNs. Finally, our method demonstrated promising results on
the validation and testing sets provided by the DEF-AI-MIA workshop.

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The vectors of the fan and the fluxes have a symmetry between the second and
fourth entry, and therefore we expect that one of $\epsilon_{i}$ will be zero
at the critical point. Notice also that the physical fluxes depends only on
two linear combinations of the $\mathfrak{t}_{a}$. These are the combinations
invariant under

$\mathfrak{t}_{a}\rightarrow\mathfrak{t}_{a}+\sum_{i=1}^{2}\beta_{i}v_{i}^{a}\,.$
(213)

In the free energy (208) this transformation can be re-absorbed in a
redefinition of $\epsilon_{4}$ using
$\sum_{a}v_{i}^{a}c_{1}(L_{a}){T}=-\epsilon_{i}$ and therefore the central
charge depends only on the physical fluxes. We also solve
$M_{a}=\sum_{b}D_{ab}m_{b}$, for example, by
$m_{a}=(0,0,M_{2},M_{1})$.303030Any other choice would be equivalent. The
equation $M_{a}=\sum_{b}D_{ab}m_{b}$ is invariant under $m_{a}\rightarrow
m_{a}+\sum_{i=1}^{2}\gamma_{i}v_{i}^{a}$. This ambiguity can be re-absorbed in
a shift of $b$ in (206). The constraint (207) and the free energy (208) can be
expanded in a series of integral of Chern classes and expressed in terms of
the intersections $D_{a_{1}\ldots a_{k}}$, which are homogeneous of order $k$
in $\epsilon_{i}$. We see then that $b$ and the free energy are homogeneous of
degree one and two in all the $\epsilon_{I}$, respectively. One can check
explicitly that $F$ is extremized at $\epsilon_{2}=0$. The expressions are too
lengthly to be reported so, for simplicity, we restrict to the case
$M_{2}=M_{1}$. We also fix
$\mathfrak{t}_{a}=(\mathfrak{t}_{1},\mathfrak{t}_{2},\mathfrak{t}_{1},\mathfrak{t}_{2})$
using (213) for convenience. The free energy restricted to $\epsilon_{2}=0$
reads

$\displaystyle
F=-\frac{M_{1}^{2}\mathcal{A}}{8p(p(\mathfrak{t}_{1}+\mathfrak{t}_{2})-2\mathfrak{t}_{1}\mathfrak{t}_{2})}\,,$
(214)

where

$\displaystyle\mathcal{A}=\epsilon_{3}^{2}p^{2}[(k-2)\mathfrak{t}_{1}-2\mathfrak{t}_{2}]^{2}+2\epsilon_{3}\epsilon_{4}p[(k-2)^{2}(p-2\mathfrak{t}_{1})\mathfrak{t}_{1}+4(p+(k-2)\mathfrak{t}_{1})\mathfrak{t}_{2}-8\mathfrak{t}_{2}^{2}]$
(215)
$\displaystyle+\epsilon_{4}^{2}[(k-4)^{2}p^{2}-4(k-3)(k-2)p\mathfrak{t}_{1}+4(k-2)^{2}\mathfrak{t}_{1}^{2}+4(k-6)p\mathfrak{t}_{2}-8(k-2)\mathfrak{t}_{1}\mathfrak{t}_{2}+16\mathfrak{t}_{2}^{2}]$
$\displaystyle+4\epsilon_{1}k[\epsilon_{3}p\mathfrak{t}_{1}((k-1)p\mathfrak{t}_{1}-(k-2)\mathfrak{t}_{1}^{2}-p\mathfrak{t}_{2})+\epsilon_{4}(-3(k-2)p\mathfrak{t}_{1}^{2}+2(k-2)\mathfrak{t}_{1}^{3}+p^{2}((k-3)\mathfrak{t}_{1}+\mathfrak{t}_{2}))]$
$\displaystyle+4\epsilon_{1}^{2}[p^{2}((k^{2}+k-3)\mathfrak{t}_{1}^{2}+(k-4)\mathfrak{t}_{1}\mathfrak{t}_{2}-\mathfrak{t}_{2}^{2})+p\mathfrak{t}_{1}(-(2k^{2}+k-2)\mathfrak{t}_{1}^{2}+(10-3k)\mathfrak{t}_{1}\mathfrak{t}_{2}+4\mathfrak{t}_{2}^{2})]$
$\displaystyle+\epsilon_{1}^{2}\mathfrak{t}_{1}^{2}[k^{2}\mathfrak{t}_{1}^{2}+(2k-4)\mathfrak{t}_{1}\mathfrak{t}_{2}-4\mathfrak{t}_{2}^{2}]\,,$

which should be still extremized with respect to $\epsilon_{1}$ and
$\epsilon_{4}$. One easily sees that the critical point is at a non zero value
of $\epsilon_{1}$. This rectifies a result given in Gauntlett:2019pqg  where
it was assumed that the R-symmetry does not mix with the isometries of
$\mathbb{F}_{k}$. The expression (214) for $k=0$ is extremized at
$\epsilon_{1}=0$ and it correctly reduces to the
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ result (210) setting $\epsilon_{1}=0$ and
$\mathfrak{t}_{1}=\mathfrak{t}_{2}$.

##  5 Conclusions

In this paper we have refined the proposal of Martelli:2023oqk , that the
geometry of an extensive class of supersymmetric solutions is captured by a
universal quantity, depending only on the topology of the internal space and
equivariant parameters associated with the expected symmetries of the
solutions. This quantity is an extension of the equivariant volume, familiar
from symplectic geometry, where we have introduced additional moduli dubbed
higher times, which are necessary to parameterize all the fluxes supported by
a given topology. Although we have assumed from the outset that the spaces of
interest are toric, we have indicated that this assumption may be relaxed by
considering for example “non convex” geometries as well as configurations
including a four-sphere, that are not toric geometries in the strict
mathematical sense. It is also possible to extend our construction to
geometries with a number of expected abelian symmetries which is strictly less
than half of the real dimension313131The main difference in this case the
localization formula involves fixed point sets that are not isolated points.
of the manifold/orbifold (or cone over it). It is well known that in many
situations the metric on the internal space (or the cones over these, in the
odd dimensional case) solving the supersymmetry equations may not be
compatible with a Kähler or even symplectic structure. Nevertheless, the
equivariant volume is a robust topological quantity, insensitive to the
details of the metrics. Indeed, it may be regarded as a gravitational analogue
of anomalies in quantum field theory. In all cases that we have analysed, we
extract an extremal function from the equivariant volume and our prescription
for fixing the parameters on which it depends consists of extremizing over
_all_ the parameters that are left undetermined by the flux quantization
conditions. This is consistent with the logic in the case of GK geometry
Couzens:2018wnk  and indeed it is analogous to the paradigm of a-maximization
in field theory Intriligator:2003jj . Geometrically, the existence of critical
points to the various extremal functions that we proposed may be interpreted
as providing necessary conditions to the existence of the corresponding
supergravity solutions and indeed it would be very interesting to study when
such conditions are also sufficient. In any case, if we assume that a solution
exists, then our method calculates the relevant observables, yielding non-
trivial predictions for the holographically dual field theories. It is worth
emphasising that in the procedure of extremization one should allow all the
equivariant parameters not fixed by symmetries to vary, otherwise it is not
guaranteed that the critical point found will be a true extremum of the
gravitational action. We have demonstrated this point in a number of explicit
examples discussed in section 3.2.3 as well as section 4.2.1.

In this work we have focussed on setups involving internal geometries that are
fibrations over four-dimensional orbifolds $\mathbb{M}_{4}$, that may be
interpreted as arising from branes wrapping completely or partially
$\mathbb{M}_{4}$. For example, the case of M5 branes completely wrapped on
$\mathbb{M}_{4}$, yields a proof of the gravitational block form of the trial
central charge, conjectured in Faedo:2022rqx  (and derived in the field theory
side in Martelli:2023oqk ). The case of M5 branes partially wrapped on a two-
cycle inside $\mathbb{M}_{4}$ is still poorly understood from the field theory
side, the best understood setup being the case of
$\mathbb{M}_{4}=\Sigma_{g}\times S^{2}$, where $\Sigma_{g}$ is Riemann surface
of genus $g$ Bah:2019rgq . The full internal space $M_{6}$ may then be viewed
also as the fibration of the second Hirzebruch surface $\mathbb{F}_{2}\simeq
S^{2}\times S^{2}$ over the Riemann surface $\Sigma_{g}$, and interpreted as
the backreaction of a stack of M5 branes at a (resolved)
$\mathbb{C}^{2}/\mathbb{Z}_{2}$ singularity, further wrapped on $\Sigma_{g}$,
yielding insights about the dual four-dimensional field theories. In section
3.2.3 we have discussed the example of $\mathbb{M}_{4}=\mathbbl{\Sigma}\times
S^{2}$, corresponding to M5 branes probing a $\mathbb{C}^{2}/\mathbb{Z}_{2}$
singularity, further wrapped on a spindle $\mathbbl{\Sigma}$ and it would be
interesting to confirm our predictions with a field-theoretic computation. It
would also be nice to extend the methods of Bah:2019rgq  for computing
anomalies to setups where the M5 branes wrap a two-cycle with non-trival
normal bundle in an $\mathbb{M}_{4}$.

In the context of type IIA supergravity, we have analysed the case of D4
branes completely wrapped on a general toric four-orbifold $\mathbb{M}_{4}$,
proving the gravitational block form of the entropy function conjectured in
Faedo:2022rqx . It would be very interesting to reproduce this from a field
theory calculation of the partition function of five-dimensional SCFTs on
$S^{1}\times\mathbb{M}_{4}$, employing the method of Inglese:2023wky  for
performing localization on orbifolds. Finally, we have also discussed the case
of D3 branes partially wrapped on a two-cycle inside $\mathbb{M}_{4}$,
corresponding to type IIB geometries of the form AdS${}_{3}\times M_{7}$,
making contact with the framework of fibered GK geometries studied in
Gauntlett:2019pqg . In particular, we have improved some of the results
previously obtained in Gauntlett:2019pqg , by revisiting some of the examples
discussed there. In this paper we have not discussed geometries associated to
M2 and D2 branes (already briefly mentioned in Martelli:2023oqk ), which are
not naturally related to four-dimensional orbifolds $\mathbb{M}_{4}$, but we
expect that for these our method will generalize straightforwardly. It would
be very interesting to incorporate new classes of supersymmetric geometries in
our framework, such as for example AdS${}_{2}\times M_{8}$ in type IIB in
order to study entropy functions of AdS5 black holes. It is tantalizing to
speculate that our approach may be eventually extended to include geometries
that do not necessarily contain AdS factors.

## Acknowledgements

We would like to thank Alessandro Tomasiello for useful discussions. EC, FF
and DM are supported in part by the INFN. AZ is partially supported by the
INFN and the MIUR-PRIN contract 2017CC72MK003.

##  Appendix A Fixing the Kähler moduli of $\mathrm{AdS}_{5}\times M_{6}$
solutions with $\mathbb{Z}_{2}$ symmetry

In this appendix we verify that for the Calabi-Yau geometry considered in
section 3.2.2 there is a critical point of
$\mathds{V}^{(3)}(\lambda_{A},\lambda_{AB},\epsilon_{I})$ with
$\lambda_{a}=\lambda_{ab}=0$. Even if the group of gauge transformations (19)
has a sufficient number of parameters to potentially gauge away all
$\lambda_{a}$ and $\lambda_{ab}$, in orbifolds $\mathbb{M}_{4}$ with a small
number of vectors in the fan there are often obstructions that make this
impossible.

In the following we will verify that the values of the Kähler moduli
$\lambda_{A}$, $\lambda_{AB}$ given by

$\begin{cases}\lambda_{a}=\lambda_{ab}=0\,,\\\
\overline{\lambda}_{a,d+1}\text{ such that
}\sum_{a}D_{ab}\overline{\lambda}_{b,d+1}=-\overline{\nu}_{M5}\,M_{a}\,,\\\
\overline{\lambda}_{d+1}\text{ such that
}\partial_{\,\overline{\lambda}_{d+1}}\mathds{V}^{(3)}=0\,,\end{cases}$ (216)

are an extremum of $\mathds{V}^{(3)}$, under the constraints imposed by the
flux equations

$\partial_{\lambda_{A}}\mathds{V}^{(2)}(\lambda_{A},\lambda_{AB},\epsilon_{I})=-\overline{\nu}_{M5}\,M_{A}\>.$
(217)

In practice, we will show that there exist a value for the Lagrange parameters
$\rho_{A}$ such that the function

$\mathcal{E}=\mathds{V}^{(3)}+\sum_{A}\rho_{A}\big{(}\partial_{\lambda_{A}}\mathds{V}^{(2)}+\overline{\nu}_{M5}\,M_{A}\big{)}$
(218)

has null derivatives with respect to $\lambda_{A}$, $\lambda_{AB}$. The
equations that we will solve are then

$\frac{\partial}{\partial\lambda_{A}}\,\mathcal{E}\,(\lambda_{A},\lambda_{AB},\epsilon_{I},\rho_{A})=0\,,\qquad\frac{\partial}{\partial\lambda_{AB}}\,\mathcal{E}\,(\lambda_{A},\lambda_{AB},\epsilon_{I},\rho_{A})=0\,,$
(219)

where the $\lambda$ are given by (216) while $\epsilon_{1},\epsilon_{2}$ can
take general values. We will study the case $\epsilon_{1}=\epsilon_{2}=0$
separately.

##### Case $(\epsilon_{1},\epsilon_{2})\neq(0,0)$.

We claim that values of $\rho_{A}$ that solve (219) exist and they are the
solutions of the following linear system:

$\displaystyle\sum_{A}\rho_{A}\big{(}\mathcal{C}_{A}\big{)}|_{b}=h_{b}\>,\qquad
b=1,\ldots,d$ (220) $\displaystyle
h_{b}\equiv-\frac{1}{2}\,\Big{[}(\epsilon_{3}+\sum_{a}c_{1}^{\mathbb{T}}(L_{a}))^{2}(\bar{\lambda}_{d+1}+\sum_{a}\bar{\lambda}_{a,d+1}c_{1}^{\mathbb{T}}(L_{a}))^{2}\Big{]}\big{|}_{b}\>.$

Indeed, when the above equations are satisfied we have

$\displaystyle\partial_{\lambda_{B}}\sum_{A}\rho_{A}\big{(}\partial_{\lambda_{A}}\mathds{V}^{(2)}+\overline{\nu}_{M5}\,M_{A}\big{)}=\int_{\mathbb{M}_{4}}\frac{\sum_{A}\rho_{A}\,\mathcal{C}_{A}\,\mathcal{C}_{B}}{\epsilon_{3}+\sum_{a}c_{1}^{\mathbb{T}}(L_{a})}=$
(221)
$\displaystyle\qquad=-\frac{1}{2}\int_{\mathbb{M}_{4}}\mathcal{C}_{B}\,(\epsilon_{3}+\sum_{a}c_{1}^{\mathbb{T}}(L_{a}))(\bar{\lambda}_{d+1}+\sum_{b}\bar{\lambda}_{b,d+1}c_{1}^{\mathbb{T}}(L_{b}))^{2}=-\,\partial_{\lambda_{B}}\mathds{V}^{(3)}\,,$

where we have used $\lambda_{a}=\lambda_{ab}=0$. This gives us
$\partial_{\lambda_{A}}\mathcal{E}=0$. The
$\partial_{\lambda_{AB}}\mathcal{E}=0$ equations can also be derived from
(LABEL:Lagrange_param_equations) in a similar manner.

Let us now discuss the existence of solutions to the equations
(LABEL:Lagrange_param_equations). The restrictions of
$\sum_{A}\rho_{A}\,\mathcal{C}_{A}$ to the fixed points are not independent,
they satisfy the following linear relation:

$0=\int_{\mathbb{M}_{4}}\sum_{A}\rho_{A}\,\mathcal{C}_{A}=\sum_{b}\frac{\sum_{A}\rho_{A}\big{(}\mathcal{C}_{A}\big{)}|_{b}}{d_{b,b+1}\,\epsilon_{1}^{b}\,\epsilon_{2}^{b}}\>.$
(222)

However, the $h_{b}$ also satisfy the same linear relation, given that the
value of $\overline{\lambda}_{d+1}$ is set by the condition
$\partial_{\,\overline{\lambda}_{d+1}}\mathds{V}^{(3)}=0$, which reads

$0=\frac{1}{2}\int_{\mathbb{M}_{4}}(\epsilon_{3}+\sum_{a}c_{1}^{\mathbb{T}}(L_{a}))^{2}(\bar{\lambda}_{d+1}+\sum_{a}\bar{\lambda}_{a,d+1}c_{1}^{\mathbb{T}}(L_{a}))^{2}=-\sum_{b}\frac{h_{b}}{d_{b,b+1}\,\epsilon_{1}^{b}\,\epsilon_{2}^{b}}\>.$
(223)

We can thus always eliminate one of the equations
(LABEL:Lagrange_param_equations). Considering that shifting
$\rho_{A}\to\rho_{A}+\sum_{I}\alpha^{I}V_{I}^{A}$ with
$\sum_{I}\alpha^{I}\epsilon_{I}=0$ leaves the left hand side of
(LABEL:Lagrange_param_equations) invariant, we can always gauge away
$\rho_{d+1}$323232We note that the $(\epsilon_{1},\epsilon_{2})\neq(0,0)$
hypotheses is needed to set $\rho_{d+1}=0$. and one of the $\rho_{a}$. We are
left with a system of $d-1$ equations in $d-1$ variables that generally is not
singular and thus has a solution.

There is an edge case in which the system of equations must be further
reduced: when there is $\overline{a}\in\\{1,\ldots,d\\}$ such that
$\epsilon_{2}^{\overline{a}}=0$ (and consequently
$\epsilon_{1}^{\overline{a}-1}=0$). Since
$(\epsilon_{1},\epsilon_{2})\neq(0,0)$ and $\mathbb{M}_{4}$ is compact we must
have $\epsilon_{1}^{\overline{a}}\neq 0$, $\epsilon_{1}^{\overline{a}-1}\neq
0$. The $b=\overline{a}-1$ and $b=\overline{a}$ equations are ($\rho_{d+1}$
has been gauged away)

$-\rho_{\overline{a}}\>\epsilon_{2}^{\overline{a}-1}=h_{\overline{a}-1}\,,\qquad-\,\rho_{\overline{a}}\>\epsilon_{1}^{\overline{a}}=h_{\overline{a}}\>.$
(224)

In principle depending on the value of $h_{\overline{a}-1}$ and
$h_{\overline{a}}$ the above equations might not have a solution. However if
we consider that $h_{\overline{a}-1}$ and $h_{\overline{a}}$ can only depend
on $\epsilon_{2}^{\overline{a}-1}$ and $\epsilon_{1}^{\overline{a}}$
respectively,333333By definition $h_{\overline{a}-1}$ and $h_{\overline{a}}$
are the restrictions of an equivariant form on the fixed points
$\overline{a}-1$ and $\overline{a}$. and that in general
$d_{\overline{a}-1,\overline{a}}\,\epsilon_{1}^{\overline{a}-1}=-d_{\overline{a},\overline{a}+1}\,\epsilon_{2}^{\overline{a}}$,
then the only way for the right hand side of (223) to be finite is for

$\frac{h_{\overline{a}-1}}{\epsilon_{2}^{\overline{a}-1}}=\frac{h_{\overline{a}}}{\epsilon_{1}^{\overline{a}}}\,,$
(225)

which means that equations (LABEL:Lagrange_param_equations) are solvable
without issue.

##### Case $(\epsilon_{1},\epsilon_{2})=(0,0)$.

It is not immediately clear whether the solutions to equations
(LABEL:Lagrange_param_equations) are well-behaving in the limit
$\epsilon_{1},\epsilon_{2}\to 0$. However when $\epsilon_{1}$ and
$\epsilon_{2}$ are zero the equations (219) are quite simple and we can solve
them directly.

For $\epsilon_{1}=\epsilon_{2}=0$ we have $c_{1}{T}(L_{a})=c_{1}(L_{a})$ and
thus

$\int_{\mathbb{M}_{4}}c_{1}{T}(L_{a_{1}})\ldots
c_{1}{T}(L_{a_{k}})=\begin{cases}D_{a_{1}a_{2}}\quad&k=2\\\
0&\text{otherwise}\end{cases}\>.$ (226)

From this relation it easily follows that

$\int_{\mathbb{M}_{4}}\frac{c_{1}{T}(L_{a_{1}})\ldots
c_{1}{T}(L_{a_{k}})}{\epsilon_{3}+\sum_{b}c_{1}^{\mathbb{T}}(L_{b})}=\begin{cases}(\epsilon_{3})^{-1}D_{a_{1}a_{2}}\quad&k=2\\\
0&k>2\end{cases}\>.$ (227)

Using the above relations the extremization equations (219) become

$\begin{cases}\partial_{\lambda_{a}}\text{
equation:}&\frac{1}{2}\sum_{b}D_{ab}(\overline{\lambda}_{d+1}^{2}+2\,\epsilon_{3}\,\overline{\lambda}_{d+1}\overline{\lambda}_{b,d+1})+(\epsilon_{3})^{-1}\sum_{b}D_{ab}\rho_{b}=0\\\
\partial_{\lambda_{ab}}\text{
equation:}&\frac{1}{2}\,\epsilon_{3}\,\overline{\lambda}_{d+1}^{2}D_{ab}-\rho_{d+1}D_{ab}=0\\\
\partial_{\lambda_{a,d+1}}\text{
equation:}&-\sum_{b}D_{ab}(\epsilon_{3}^{2}\,\overline{\lambda}_{d+1}\overline{\lambda}_{b,d+1}+\epsilon_{3}\,\overline{\lambda}_{d+1}^{2})-\sum_{b}D_{ab}(\rho_{b}-\rho_{d+1})=0\\\
\partial_{\lambda_{d+1,d+1}}\text{ equation:}&\begin{aligned}
&\tfrac{1}{2}\textstyle\sum_{a,b}D_{ab}(\epsilon_{3}^{3}\,\overline{\lambda}_{a,d+1}\overline{\lambda}_{b,d+1}+6\,\epsilon_{3}^{2}\,\overline{\lambda}_{d+1}\overline{\lambda}_{b,d+1}+3\epsilon_{3}\,\overline{\lambda}_{d+1}^{2})+\\\
&+\textstyle\sum_{a,b}D_{ab}(\rho_{b}-\rho_{d+1})=0\end{aligned}\end{cases}$
(228)

The $\partial_{\lambda_{d+1}}$ equation was omitted because it is trivial:
$\partial_{\lambda_{d+1}}\sum_{A}\rho_{A}\big{(}\partial_{\lambda_{A}}\mathds{V}^{(2)}+\nu_{M5}M_{A}\big{)}=0$
and
$\partial_{\lambda_{d+1}}\mathds{V}^{(3)}=\partial_{\,\overline{\lambda}_{d+1}}\mathds{V}^{(3)}=0$
because of (216).

The solution to (228) is343434When we plug this solution into the left-hand
side of the $\partial_{\lambda_{d+1,d+1}}$ equation we do not get zero
straight away but rather we get the same expression as
$\partial_{\,\overline{\lambda}_{d+1}}\mathds{V}^{(3)}$ which is zero by
(216).

$\rho_{a}=-\epsilon_{3}^{2}\,\overline{\lambda}_{d+1}\overline{\lambda}_{b,d+1}-\frac{1}{2}\,\epsilon_{3}\,\overline{\lambda}_{d+1}^{2}\>,\qquad\rho_{d+1}=\frac{1}{2}\,\epsilon_{3}\,\overline{\lambda}_{d+1}^{2}\>,$
(229)

and thus (216) is the proper extremum of $\mathds{V}^{(3)}$ under the flux
constraints.

##  Appendix B $\mathrm{AdS}_{3}\times M_{7}$ solutions with the addition of
higher times

In this appendix we revisit the computation of section 4.2, now with the
inclusion of second and triple times in the equivariant volume. For the
$\mathrm{AdS}_{3}\times M_{7}$ solutions we considered in section 4.2 there
was no need to add any higher times. We will now show that it is still
possible to perform the computation even when the equivariant volume is over-
parametrized. The extremization procedure for the parameters in excess plays a
crucial role this time: relations that where automatically verified when
$\mathds{V}^{(2)}$ only included single times are now derived as extremization
conditions. This provides further evidence that extremization is the correct
way to deal with any parameter $\lambda$ that is not fixed by the flux
constraints.

The second degree homogeneous component of the equivariant volume with triple
times is given by

$\displaystyle\mathbb{V}^{\,(2)}(\lambda_{A}\,,\lambda_{AB}\,,\lambda_{ABC}\,,\epsilon_{I})=\int_{\mathbb{M}_{4}}\,\frac{(\tau{T})^{2}}{2\,p\>\mathcal{C}_{d+1}\>\mathcal{C}_{d+2}}\>,$
(230)
$\displaystyle\tau{T}=\sum_{A}\lambda_{A}\>\mathcal{C}_{A}+\sum_{A,B}\lambda_{AB}\>\mathcal{C}_{A}\>\mathcal{C}_{B}+\sum_{A,B,C}\lambda_{ABC}\>\mathcal{C}_{A}\>\mathcal{C}_{B}\>\mathcal{C}_{C}\>.$

We need to impose the following flux constraints:

$-\nu_{D3}\,M_{A}=\partial_{\lambda_{A}}\mathbb{V}^{\,(2)}(\lambda_{A}\,,\lambda_{AB}\,,\lambda_{ABC}\,,\epsilon_{I})=\int_{\mathbb{M}_{4}}\,\frac{\mathcal{C}_{A}\>\tau{T}}{p\>\mathcal{C}_{d+1}\>\mathcal{C}_{d+2}}\>.$
(231)

Proceeding in a similar way as we did in section 4.2, we will set all the
$\lambda$ to zero except for $\lambda_{d+1,\,d+2}$ and
$\lambda_{d+1,\,d+2,\,A}$. This assumption is justified by the fact that in
principle the group of gauge transformation for the single, double and triple
times has enough parameters to gauge away all the $\lambda$ except
$\lambda_{d+1,\,d+2}$ and $\lambda_{d+1,\,d+2,\,A}$.353535Note that
$\lambda_{a,b}$, $\lambda_{a,b,d+1}$, $\lambda_{a,b,d+2}$ and
$\lambda_{a,b,c}$ do not appear inside $\mathds{V}^{(2)}$ unless
$a,b,c\in\\{\overline{a},\overline{a}+1\\}$ for some
$\overline{a}\in\\{1,\ldots,d\\}$. At the end of this appendix we will quickly
check that $\mathds{V}^{(2)}$ does indeed have a critical point for
$\lambda_{a,b}=\lambda_{a,b,d+1}=\lambda_{a,b,d+2}=\lambda_{a,b,c}=0$, thus
verifying the correctness of this choice of $\lambda$. The flux constraints
(231) now read

$-\nu_{D3}\,M_{a}=\frac{1}{p}\int_{\mathbb{M}_{4}}c_{1}{T}(L_{a})\Big{(}\lambda_{d+1,\,d+2}+\sum_{A}\lambda_{d+1,\,d+2,\,A}\>\mathcal{C}_{A}\Big{)}=\sum_{b}D_{ab}\overline{\lambda}_{b}\>,$
(232)

where

$\overline{\lambda}_{a}=\frac{\lambda_{d+1,\,d+2,\,a}-\lambda_{d+1,\,d+1,\,d+2}}{p}+\frac{\mathfrak{t}_{a}(\lambda_{d+1,\,d+1,\,d+2}-\lambda_{d+1,\,d+2,\,d+2})}{p^{2}}\>.$
(233)

Up to gauge transformations, the $\overline{\lambda}_{a}$ are then fixed to be
$\overline{\lambda}_{a}=-\nu_{D3}\,m_{a}$, where the $m_{a}$ are such that
$\sum_{b}D_{ab}m_{b}=M_{a}$.

We notice that the flux constraints did not fix all the $\lambda$, but rather
there is one such parameter left:

$\displaystyle\mathbb{V}^{\,(2)}=$
$\displaystyle\,\frac{1}{2p}\int_{\mathbb{M}_{4}}\mathcal{C}_{d+1}\>\mathcal{C}_{d+2}\,\Big{(}\lambda_{d+1,\,d+2}+\sum_{A}\lambda_{d+1,\,d+2,\,A}\>\mathcal{C}_{A}\Big{)}^{2}$
(234) $\displaystyle=$
$\displaystyle\,\frac{p}{2}\,\nu_{D3}^{2}\int_{\mathbb{M}_{4}}\mathcal{C}_{d+1}\>\mathcal{C}_{d+2}\,\Big{(}\overline{\lambda}-\sum_{a}m_{a}\>c_{1}{T}(L_{a})\Big{)}^{2}\>,$

where

$\overline{\lambda}=\frac{\lambda_{d+1,\,d+2}-\epsilon_{3}\,\lambda_{d+1,\,d+1,\,d+2}}{p\,\nu_{D3}}+\frac{\epsilon_{4}\,(\lambda_{d+1,\,d+1,\,d+2}-\lambda_{d+1,\,d+2,\,d+2})}{p^{2}\,\nu_{D3}}\>.$
(235)

Our procedure prescribes to fix the value of $\overline{\lambda}$ by
extremizing $\mathbb{V}^{\,(2)}$ with respect to it. If we call
$b(\epsilon_{I})$ the extremal value of $\overline{\lambda}$, we find that

$0=\frac{\partial}{\partial\overline{\lambda}}\mathds{V}^{(2)}=p\,\nu_{D3}^{2}\int_{\mathbb{M}_{4}}\mathcal{C}_{d+1}\>\mathcal{C}_{d+2}\,\Big{(}b(\epsilon_{I})-\sum_{a}m_{a}\>c_{1}{T}(L_{a})\Big{)}\>.$
(236)

Notably, the equation we obtain is the exact same as (207). In the context of
the computation without higher times, equation (207) was a trivial relation, a
predictable consequence of the fact that there are only $d-1$ single times but
$d$ fixed points. In the computation of this appendix the same relation is now
derived as an extremization condition.

If we substitute (236) into $\mathds{V}^{(2)}$ we get

$\mathbb{V}^{\,(2)}=-\frac{p}{2}\,\nu_{D3}^{2}\int_{\mathbb{M}_{4}}\mathcal{C}_{d+1}\>\mathcal{C}_{d+2}\,\Big{(}b(\epsilon_{I})-\sum_{a}m_{a}\>c_{1}{T}(L_{a})\Big{)}\sum_{a}m_{a}\>c_{1}{T}(L_{a})\>,$
(237)

which is the same as the main result of section 4.2.

We can quickly verify that the values of $\lambda$ that we have fixed are an
extremum of $\mathds{V}^{(2)}$ by employing the same strategy as appendix A.
We can find the values of the Lagrange parameters $\rho_{A}$ such that the
function

$\mathcal{E}(\lambda_{A}\,,\lambda_{AB}\,,\lambda_{ABC}\,,\epsilon_{I},\rho_{A})=\mathds{V}^{(2)}+\sum_{A}\rho_{A}\big{(}\partial_{\lambda_{A}}\mathds{V}^{(2)}+\nu_{D3}\,M_{A}\big{)}$
(238)

has null derivatives with respect to
$\lambda_{A}\,,\lambda_{AB}\,,\lambda_{ABC}$ by solving the following linear
system:

$\sum_{A}\rho_{A}\big{(}\mathcal{C}_{A}\big{)}|_{b}=-p\,\nu_{D3}\Big{[}\mathcal{C}_{d+1}\>\mathcal{C}_{d+2}\,\Big{(}b(\epsilon_{I})-\sum_{a}m_{a}\>c_{1}{T}(L_{a})\Big{)}\Big{]}\big{|}_{b}\>.$
(239)

Using the same line of reasoning as in appendix A, solutions to this system
exist and thus (237) is the proper extremal value of $\mathds{V}^{(2)}$ (with
respect to the extremization in $\lambda$).

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