# Adversarial Fine-tuning using Generated Respiratory Sound to Address Class
Imbalance

June-Woo Kim1,4, Chihyeon Yoon1, Miika Toikkanen2, Sangmin Bae3,4, Ho-Young
Jung1
1Department of AI, Kyungpook National University
2ALI Co., Ltd, 3KAIST AI, 4RSC LAB, MODULABS
Republic of Korea
{kaen2891, chichi8969<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>corresponding author

###### Abstract

Deep generative models have emerged as a promising approach in the medical
image domain to address data scarcity. However, their use for sequential data
like respiratory sounds is less explored. In this work, we propose a
straightforward approach to augment imbalanced respiratory sound data using an
audio diffusion model as a conditional neural vocoder. We also demonstrate a
simple yet effective adversarial fine-tuning method to align features between
the synthetic and real respiratory sound samples to improve respiratory sound
classification performance. Our experimental results on the ICBHI dataset
demonstrate that the proposed adversarial fine-tuning is effective, while only
using the conventional augmentation method shows performance degradation.
Moreover, our method outperforms the baseline by 2.24% on the ICBHI Score and
improves the accuracy of the minority classes up to 26.58%. For the
supplementary material, we provide the code at
https://github.com/kaen2891/adversarial_fine-
tuning_using_generated_respiratory_sound.

##  1 Introduction

Deep generative models (DGMs) have become popular due to their potential to
address data scarcity via augmentation. Among recent advancements, methods
such as generative adversarial networks (GANs) [11], variational autoencoders
(VAEs) [18], and diffusion probabilistic models [14] are gaining attraction.
Notably, previous studies [1, 4] have demonstrated that training a model on a
mixture of real samples and synthetic samples generated by DGMs can be an
effective approach to better utilize the limited data. In medical domain, DGMs
have been successfully leveraged to synthesize medical data in a variety of
categories, including retinal images [5, 15], CT and MRI scans [26, 30, 29],
as well as X-rays [21, 22]. However, synthesis is more challenging when it
comes to sequential medical data, including respiratory sound [19, 16, 28],
due to its complex temporal dynamics, high dimensionality and relative lack of
benchmarks.

In this paper, we aim to generate high-fidelity respiratory sound samples
using DGMs and then combine these synthetic samples with real data to improve
the respiratory sound classification task, especially for imbalanced lung
sound disease classes. Figure 1 illustrates the overall process of our
approach split into phases 1 and 2. In phase 1, we introduce a simple method
for generating respiratory sound samples using a conditional neural vocoder
inspired by the recent success of audio diffusion models [14, 20] in obtaining
realistic audio. However, the discrepancy between synthetic and real samples
can introduce problems related to distribution inconsistency, which can
degrade the performance of respiratory sound classification models as the
proportion of synthetic data in the training set increases.

Figure 1: In phase 1, we generate the respiratory sound samples using the
audio diffusion model as conditional neural vocoder. In phase 2, we use the
proposed adversarial fine-tuning method to address the distribution
inconsistency between synthetic and real samples for training the respiratory
sound classification model.

In phase 2, we propose a simple yet effective _adversarial fine-tuning_ method
motivated by [9] to learn the model that is distribution-agnostic between real
and synthetic data. The adversarial fine-tuning method relies on a
discriminator network feedback to move features obtained from real and
synthetic samples closer to each other, while simultaneously training a
classifier to predict the respiratory sound label.

Our experimental results on the ICBHI [27] dataset demonstrate that the
proposed adversarial fine-tuning method effectively aligns the features from
synthetic and real data, leading to improved performance while simply
combining synthetic and real samples for training resulted in performance
degradation. Specifically, our method achieves a 2.24% ICBHI Score improvement
over the baseline and up to 26.58% accuracy improvement of the minority
classes. Our contributions are: $(i)$ We show the successful generation of
high-fidelity respiratory sound samples with audio diffusion model as
conditional neural vocoder $(ii)$ We demonstrate adversarial fine-tuning on
respiratory sound data, which can overcome data distribution inconsistency
between synthetic and real samples $(iii)$ We present that the proposed method
enables the synthetic and real training samples to be used more effectively,
considerably improving performance in the imbalanced abnormal lung disease
class.

##  2 Method

#### Audio Diffusion Probabilistic Model

Diffusion probabilistic models [14] are a type of deep generative model that
use a Markov chain to gradually add Gaussian noise
$\mathcal{N}(x_{t};\sqrt{1-\beta_{t}}x_{t-1},\beta_{t}I)$ into a complex data
distribution. The posterior $q(x_{1},...,x_{T}|x_{0})$ called _diffusion
process_ or _forward process_ is defined by a fixed Markov chain that
transforms the input data $x_{0}$ to a latent variable $x_{1},...,x_{T}$
according to a variance schedule $\beta_{1},...,\beta_{T}$:

$q_{\theta}(x_{1},...,x_{T}|x_{0})\\!:=\prod_{t=1}^{T}q(x_{t}|x_{t-1}),\,\,\,\,\,\,\,\,\,\,\,\,q(x_{t}|x_{t-1}):=\mathcal{N}(x_{t};\sqrt{1-\beta_{t}}x_{t-1},\beta_{t}I)$
(1)

The joint distribution $p_{\theta}(x_{0},...,x_{T-1}|x_{T})$ called _reverse
process_ is defined by a Markov chain with learned Gaussian transitions
starting at $p(x_{T})=\mathcal{N}(x_{T};0,I)$:

$p_{\theta}(x_{0},...,x_{T-1}|x_{T})\\!=\prod_{t=1}^{T}p_{\theta}(x_{t-1}|x_{t})$
(2)

where the transition probability $p_{\theta}(x_{t-1}|x_{t})$ is parameterized
as $\mathcal{N}(x_{t-1};\mu_{\theta}(x_{t},t),\sigma_{\theta}(x_{t},t)^{2}I)$
with shared parameter $\theta$, and the both $\mu_{\theta}$ and
$\sigma_{\theta}$ are calculated with the diffusion-step and $x_{t}$.

In this work, we use the audio diffusion model [20] which is neural vocoding
conditioned on Mel-spectrogram as a conditional neural vocoder to reconstruct
the respiratory sound raw waveform. To this end, we employ the `DiffWave_BASE`
model for our audio diffusion model, which consists of a stack of 30 residual
layers, each with 64 residual channels as well as bidirectional dilated
convolution with kernel size 3, and the dilation is doubled at each layer
within each block. To ensure that the output of the audio diffusion model has
the same length as the Mel-spectrogram, a transposed 2D convolution upsampler
is provided for the conditioned 2D Mel-spectrogram.

In our conditional neural vocoder setting, the outputs of the upsampler are
added to the dilated convolutions in each residual layer for reconstruction.
In other words, our audio diffusion model is conditioned on the Mel-
spectrogram, which means that it uses a lot of prior knowledge to guide the
generation process. This makes it easier to generate realistic samples and
reduces the need for large amounts of training data. This is especially
beneficial in the medical domain, where data is often scarce.

#### Adversarial Fine-tuning

While augmenting real data with synthetic samples can be beneficial, we found
in early experiments that in our case the distribution mismatch between the
two types of data degraded the performance of the classification model. To
overcome this issue, we propose a simple yet effective Adversarial Fine-Tuning
(AFT) method inspired by [9]. The proposed method consists of two losses with
label classifier $\mathcal{L}_{\text{CE}}$ and data discriminator
$\mathcal{L}_{\text{Dis}}$:

$\displaystyle\mathcal{L}_{\text{CE}}\\!=-\frac{1}{N}\sum_{i=1}^{n}\\!\,y_{i}\\!\,\log\,\\!(\hat{y_{i}}),\quad\mathcal{L}_{\text{Dis}}\\!=-\frac{1}{N}\sum_{i=1}^{n}\\!\,d_{i}\\!\,\log\,\\!(\hat{d_{i}}).$
(3)

where $\mathcal{L}_{\text{CE}}$ and $\mathcal{L}_{\text{Dis}}$ are CE loss
with label $y$ and data type label $d$, and the predicted probabilities
$\hat{y}$ and $\hat{d}$ are obtained by passing through the classifier and
data discriminator, respectively. To ensure that the learned features cannot
distinguish between the synthetic and real samples, gradients from
$\mathcal{L}_{\text{Dis}}$ are multiplied by a negative constant during the
backpropagation. The final training objective is
$\mathcal{L}_{\text{Final}}=\mathcal{L}_{\text{CE}}+\lambda\,\\!\mathcal{L}_{\text{Dis}}$
where $\lambda$ is a regularization parameter drawn from [9]. The AFT aims to
reduce classification error while ensuring learned features are consistent
across data types.

##  3 Experimental Setup

#### ICBHI and Mixed Dataset

We used the ICBHI [27] dataset for respiratory sound tasks, following the
official train-test split (60/40%). The training (4,142) and test sets (2,756)
contain four classes: _normal_ (49.8%/57.29%), _crackle_ (29.3%/23.55%),
_wheeze_ (12.1%/13.97%) and _both_ (8.8%/5.19%), hereinafter referred to as
$C_{n}$, $C_{c}$, $C_{w}$ and $C_{b}$, respectively. We generated synthetic
samples for each minority class to balance them with the majority class, and
then mixed these with real data. We denote it _Mixed-ICBHI_ datasets as
follows: _Mixed-500, …, Mixed-N, …, Mixed-5k_ where the number $N$ refers to
the total amount of samples per class. We prioritize real samples so that
synthetic samples are only added if the sample count is less than $N$. We used
the _Specificity_ , _Sensitivity_ and their arithmetic mean, hereinafter
referred to as $S_{p}$, $S_{e}$, and $Score$, respectively [27]. For ICBHI
details and additional statistics on the Mixed-ICBHI dataset, see Appendix B
and C.

#### Audio Diffusion Model

For the data pre-processing, we fixed all of the data length as 4 seconds and
extracted the 4,142 respiratory sound samples from the ICBHI dataset as
80-dimensional Mel-spectrograms. For the audio diffusion model, we trained the
DiffWave [19] on the ICBHI training set from scratch. To this end, we used a
linearly spaced schedule for the diffusion variance schedule parameter
$\beta_{t}$ $\in[1\times 10^{-4},0.02]$, 50 and 6 denoising diffusion steps
for training and evaluation, respectively. We then trained the model for 1M
training steps with Adam [17] optimizer, a learning rate of 1e-4, and a batch
size of 16.

#### Respiratory Sound Classification

To prepare the data for training, we fixed the duration of all synthetic and
real samples to 5 seconds and extracted 128-dimensional log Mel filterbank
features with a window size of 25 ms and an overlap size of 10 ms. We then
normalized the log Mel filterbank features using the mean and standard
deviation of -4.27 and 4.57, as described in [3]. We trained the
classification model using pretrained Audio Spectrogram Transformer [10] (AST)
model with the Adam optimizer, a learning rate of 5e-5, and a batch size of 32
for 50 epochs. To ensure the stability of our results, we trained our model
using a fixed set of five random seeds for all experiments.

Table 1: Respiratory sound classification performance on ICBHI test set
according to various mixed sample amounts using the AST [10] fine-tuning as
described in [3]. No Aug. denotes only the real ICBHI dataset is used for
training. We only report the ICBHI _Score_ (%). Bold denotes the best result.

| training dataset |
---|---|---
method | No Aug. | Mixed-500 | Mixed-800 | Mixed-1k | Mixed-1.5k | Mixed-2k | Mixed-3k | Mixed-5k
| AST FT
---
59.55 | 59.92±0.82 | 59.99±1.17 | 59.81±0.36 | 59.65±0.30 | 59.18±0.65 | 59.04±0.32 | 58.56±0.84
| AFT
---
- | 61.79±0.47 | 60.89±0.78 | 60.8±1.05 | 60.03±1.14 | 60.64±0.45 | 59.96±0.38 | 59.74±0.6

##  4 Results

###  4.1 Effectiveness of Adversarial Fine-tuning

To validate the proposed AFT, we compared it against AST fine-tuning (AST FT)
with only cross-entropy (CE) loss on several Mixed-ICBHI datasets under the
same conditions. As in Table 1, the AST FT performance decreased as the number
of augmented samples in the ICBHI dataset increased, while the AFT
outperformed it in each case, reaching the best Score on _Mixed-500_. Based on
the result, as $N$ increases, the distribution mismatch between synthetic and
real samples increases, therefore leading to reduced performance. Our method
mitigates this to a degree, but still benefits more in smaller $N$. We further
explore how our method affects the performance of minority classes. We report
their accuracy on the ICBHI test set for AST FT with no augmentation, and AST
FT and AFT on Mixed-500 and Mixed-2k. As in Table 2, directly fine-tuning on
mixed data did not improve the performance of the minority classes overall.
However, our proposed method improved their accuracies by up to 26.58%,
especially in $C_{b}$. These results show that our method can most effectively
enhance the performance of minority classes despite using synthetic samples
that would otherwise degrade them. For additional confusion matrices of Table
2, see Appendix D.

###  4.2 Comparison on ICBHI Dataset Results

Table 2: Accuracy (%) of the abnormal class on the ICBHI test set for AST
fine-tuning and AST adversarial fine-tuning models trained on different
datasets. Bold denotes the best result.

|  | method (dataset)
---|---|---
|  | AST FT | AST FT | AST FT | Adversarial FT | Adversarial FT
class | ratio | (No aug.) | (Mixed-500) | (Mixed-2k) | (Mixed-500) | (Mixed-2k)
crackle ($C_{c}$) | 23.55% | 45.45 | 42.84 | 42.86 | 44.07 | 46.99
wheeze ($C_{w}$) | 13.97% | 36.62 | 36.1 | 22.08 | 37.92 | 30.12
both ($C_{b}$) | 5.19% | 15.38 | 9.09 | 7.69 | 41.96 | 35.66

Table 3: Overall comparison of the ICBHI dataset for the respiratory sound
classification task. We compared previous studies that followed the official
60-40% split for the training/test set. Scores marked with $*$ denote the
previous state-of-the-art performance. Best and second best results.

method | architecture | pretrain |  $S_{p}$ (%) |  $S_{e}$ (%) |  Score (%)
---|---|---|---|---|---
RespireNet [8] (CBA+BRC+FT) | ResNet34 | IN | 72.30 | 40.10 | 56.20
Wang et al. [32] (Splice) | ResNeSt | IN | 70.40 | 40.20 | 55.30
Nguyen et al. [25] (CoTuning) | ResNet50 | IN | 79.34 | 37.24 | 58.29
Moummad et al. [23] (SCL) | CNN6 | AS | 75.95 | 39.15 | 57.55
Bae et al. [3] (Fine-tuning) | AST | IN + AS | 77.14 | 41.97 | 59.55
Bae et al. [3] (Patch-Mix CL) | AST | IN + AS | 81.66 | 43.07 | $\text{\bf 62.37}^{\textbf{*}}$
AFT on Mixed-500 [ours] | AST | IN + AS |<EMAIL_ADDRESS>0.99}$ |<EMAIL_ADDRESS>1.3}$ |<EMAIL_ADDRESS>0.47}$

Table 3 presents an overall comparison of various methods for lung sound
classification on the ICBHI dataset. Our proposed method trained with
Mixed-500 achieved a Score of 61.79%, outperforming the AST FT model by 2.24%,
which is comparable to the state-of-the-art model. This demonstrates the
efficacy and potential of our proposed method, indicating its capability for
addressing the issues with synthetic data.

###  4.3 Qualitative Analysis

Figure 2 provides visual comparison of spectrograms randomly sampled per class
from the test set and the results generated by our diffusion model when
conditioned on these spectrograms. The generated spectrograms per class are
visually similar to the original sample, which demonstrates the capability to
generate high-fidelity audio, yet introduce small realistic variations that
provide some value for augmentation.

Figure 2: Comparison of spectrograms per each class randomly chosen from the
test set and the generated results.

##  5 Conclusion

We presented a simple method for generating realistic respiratory sound
samples using an audio diffusion model. We further introduced adversarial
fine-tuning to address the distribution inconsistency between synthetic and
real samples. Our results show that our method can effectively improve the
performance of imbalanced abnormal classes, demonstrating its ability to
address the challenges of using synthetic data. We believe that our method can
be helpful in various other datasets and could be used to supplement other
augmentation methods.

## Acknowledgments and Disclosure of Funding

This research was supported by the MSIT(Ministry of Science and ICT), Korea,
under the ITRC(Information Technology Research Center) support
program(IITP-2023-2020-0-01808) supervised by the IITP(Institute of
Information & Communications Technology Planning & Evaluation), and by Brian
Impact, a non-profit organization dedicated to the advancement of science and
technology.

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##  Appendix A Related Works

#### Respiratory Sound Classification

The ICBHI [27] dataset is a well-known benchmark for respiratory sound
classification. Various neural network-based approaches have been developed
for this task, including residual blocks [8, 25, 32], CNN [23], pretrained
models on ImageNet [8, 25, 32], AudioSet [23], and Audio Spectrogram
Transformer (AST) [10]. To address the challenge of limited data, previous
studies have proposed various learning protocols, including device-specific
fine-tuning [8], mixup as well as splicing audio augmentation [32], task-
specific co-tuning [25], supervised contrastive learning [23], and patch-mix
contrastive learning [3]. Instead of focusing on previous data augmentation
methods, this paper addressed the challenge of using synthetic samples
generated by deep generative models. To this end, we first trained a pre-
trained AST [3] model on the ICBHI dataset as described in [10]. We also
trained the model on the Mixed-ICBHI dataset, which contains both synthetic
and real samples. We then showed that our proposed adversarial fine-tuning
method can overcome the data distribution inconsistency between synthetic and
real samples.

#### Deep Generative Models

Recent advances in DGMs, such as GAN [11], VAE [18], and diffusion models
[14], have attracted significant attention. This is because DGMs can be used
to generate synthetic samples to mitigate data scarcity issues. They have been
applied to medical images, such as retinal images [5, 15], CT and MRI scans
[26, 30, 4, 29, 12, 2, 33, 6], and X-rays [21, 22] which received additional
interest due their applicability in diagnosing COVID-19 cases. Several
approaches have also been introduced to generate synthetic sequential medical
data, such as respiratory sounds [19, 16, 28], EEG recordings [7, 13, 31], and
ECG signals [24, 34]. Unlike previous studies on respiratory sound, our work
was the first attempt to successfully generate high-fidelity respiratory sound
samples using an audio diffusion model [20] which is neural vocoding
conditioned on Mel-spectrogram as a conditional neural vocoder.

##  Appendix B ICBHI Dataset Details

Table 4: Overall details of the ICBHI [27] respiratory sound dataset. |  number of respiratory samples (ratio)
---|---
label |  train |  test |  sum
Normal |  2,063 (49.8%) |  1,579 (57.29%) |  3,642
Crackle |  1,215 (29.3%) |  649 (23.55%) |  1,864
Wheeze |  501 (12.1%) |  385 (13.97%) |  886
Both |  363 (8.8%) |  143 (5.19%) |  506
Total |  4,142 |  2,756 |  6,898
Table 5: Overall details of Mixed-ICBHI dataset with synthetic and real
samples.

|  mixed dataset (synthetic ratio, %)
---|---
label |  Mixed-500 |  Mixed-800 |  Mixed-1k |  Mixed-1.5k |  Mixed-2k |  Mixed-3k |  Mixed-5k
normal |  0 |  0 |  0 0 |  0 |  0 |  31.23 |  58.74
crackle |  0 |  0 |  0 |  19.00 |  41.11 |  59.50 |  75.70
wheeze |  0 |  37.38 |  49.90 |  66.60 |  75.72 |  83.30 |  89.98
both |  27.40 |  54.63 |  63.70 |  75.80 |  82.40 |  87.90 |  92.74

The ICBHI [27] dataset is a well-known benchmark for respiratory sound
classification. The ICBHI dataset consists of 6,898 respiratory cycles, with a
total duration of approximately 5.5 hours. The dataset is officially split
into a training set (60%) and a test set (40%), with no patient overlap
between the two sets. As shown in Table 4, the training and test sets contain
4,142 and 2,756 samples respectively and are categorized into four classes,
_normal_ (49.8%/57.29%), _crackle_ (29.3%/23.55%), _wheeze_ (12.1%/13.97%) and
_both_ (8.8%/5.19%), respectively. For all our experiments, we resampled all
the samples to 16 kHz. For the metrics, we used Sensitivity ($S_{e}$),
Specificity $(S_{p})$, and their arithmetic mean Score as described in [27].

##  Appendix C Mixed Dataset Details

As described in Table 5, we mixed the synthetic samples with the real data to
create Mixed-ICBHI datasets as follows: _Mixed-500, …, Mixed-N, …, Mixed-2k_
where the number $N$ refers to the total amount of samples per class. We
prioritize real samples so that synthetic samples are only added if the sample
count is less than $N$ (i.e., _Mixed-500_ only contains synthetic samples from
$C_{both}$).

##  Appendix D Confusion Matrices Results

To show how the proposed method affects _all the classes_ , Figure 3 provides
the confusion matrices between the AST FT with no augmentation, AST FT and AFT
with Mixed-500 and Mixed-2k, respectively. The proposed method did not degrade
considerably on normal classes and achieved the highest performance compared
to other methods on the most imbalanced classes. Our results demonstrate the
effectiveness and potential of our proposed method, showing its ability to
address the data distribution inconsistency problem with synthetic data,
especially in class imbalanced problems.

(a) AST FT (No Aug.)

(b) AST FT (Mixed-500)

(c) AFT (Mixed-500)

Figure 3: Confusion matrix results of AST FT with no augmentation, AST FT and
AFT with Mixed-500 and Mixed-2k, respectively.


# Searching for groups related to pseudo-composition algebras

Vsevolod A. Afanasev

###### Abstract

We study the class of idempotent-generated pseudo-composition algebras, which
is a subclass of the family of axial algebras. More specifically, we utilise
the group-algebra correspondence, natural to the axial framework in order to
study some automorphism subgroups of such pseudo-composition algebras.

##  1 Introduction

††footnotetext: This work was supported by the Russian Science Foundation,
project 23-41-10003, https://rscf.ru/en/project/23-41-10003/

The theory of axial algebras, started in its current generality by Hall,
Rehren and Shpectorov [3], contains an interesting class of examples:
idempotent-generated pseudo-composition algebras, i.e. commutative algebras
equipped with a non-zero symmetric bilinear form $\varphi$ such that

$x^{3}=\varphi(x,x)x$

for any element of an algebra.

Various properties of these objects, relayed in [5], allowed the authors of
[2] to develop a more axial view of their properties. Historically, axial
algebras were very closely connected to groups: for instance, the Matsuo
algebras, which form a large subclass of the so-called axial algebras of
Jordan type, are constructed from $3$-transposition groups. For more
information on axial algebras, we refer interested readers to the survey [6].

In that regard, the newly established view is very useful as the axial
framework provides easy methods for answering various group-theoretic
inquiries about pseudo-composition algebras. Hence, in this article we study
the group-algebra correspondence for this class and provide multiple
statements that may be used to study the Miyamoto group of any finitely-
generated axial pseudo-composition algebra.

More generally, we aim to use the parameter-based classification obtained in
[2] in order to start finding answers to the following

Question. Which groups can be realised as the Miyamoto groups of axial pseudo-
composition algebras?

Naturally, this question is incredibly grand and initially we wish only to
obtain some interesting examples, in order to notice potential patterns and
restrictions that are imposed on such groups. Another goal we have is to find
some small-dimensional pseudo-composition representations of small non-abelian
finite simple groups.

In the first section we shall study the algebras, generated by two
idempotents, namely $a$ and $b$, and determine, which groups can be realised
as their Miyamoto groups.

We will then make use of these results in order to move on to the algebras
generated by three idempotents, for which we take a twofold approach: both
trying to find some groups explicitly as well as attempting to determine
whether some specific groups are Miyamoto groups for some pseudo-composition
algebras.

For most of the paper, the ground field is assumed to be $\mathbb{R}$, i.e. a
field of real numbers, however we will sometimes consider finite fields,
preemptively informing the reader of such a change.

#### Acknowledgements.

The author would like to thank A. Mamontov, A. Staroletov and an anonymous
referee for their insightful comments and incredibly helpful suggestions.

##  2 The case of $2$-generated algebras

Here we shall consider the pseudo-composition algebras, generated by two
idempotents $a$ and $b$. One need not assume that these idempotents are axes,
however it is known that any primitive idempotent in a pseudo-composition
algebra automatically satisfies all the axial properties [5]. More generally,
idempotent-generated pseudo composition algebras are actually axial, thus any
such pseudo-composition algebra $A$ can be decomposed into a direct sum of
eigenspaces $A_{\lambda}(a)$ with the following fusion law

$\begin{array}[]{|c|c|c|c|}\hline\cr*&1&-1&\frac{1}{2}\\\ \hline\cr
1&1&-1&\frac{1}{2}\\\ \hline\cr-1&-1&1&\frac{1}{2}\\\
\hline\cr\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&-1,1\\\ \hline\cr\end{array}$

Fusion law for $\mathcal{PC}(-1)$ – a class of pseudo-composition algebras.

Additionally, this fusion law is naturally $\mathbb{Z}_{2}$-graded with
$A_{\frac{1}{2}}$ being the ”odd” part of the grading. Using these notions we
can, given an axis $a$, construct an order $2$ automorphism $\tau_{a}$ of $A$:
any element $b\in A$ can be presented as follows:

$b=b_{+}+b_{-},$

where $b_{-}\in A_{\frac{1}{2}}(a),$ while $b_{+}\in A_{-1}(a)\oplus
A_{1}(a).$ Now denote

$\tau_{a}(b)=b_{+}-b_{-}.$

This automorphism is called the Miyamoto involution. We then define the
Miyamoto group of $A$ as the group generated by $\\{\tau_{a}|\ a\text{ is an
axis in A }\\}$ . Note that generally this needs not be the whole automorphism
group of $A$.

Per this axis $\to$ automorphism correspondence, it is clear that the Miyamoto
subgroups for the $2$-generated algebras $\langle\langle a,b\rangle\rangle$,
are $D_{n}$ where $n=|\tau_{ab}|=|\tau_{a}\tau_{b}|$. Therefore to classify
them we only need to find the possible orders of $\tau_{ab}$. By [2], this
matrix depends on the parameter $\alpha=(a,b)$, where $(\cdot,\cdot)$ is the
Frobenius form, corresponding to the axial algebra (for pseudo-composition
algebras it coincides with the bilinear form $\varphi$ from the definition).

The matrix may be written explicitly as follows (we assume that linear
transformations act on the left)

$\tau_{ab}=\frac{1}{9}\left(\begin{array}[]{ccc}64\alpha^{2}-16\alpha-3&24\alpha&32\alpha^{2}+4\alpha-6\\\
8-8\alpha&-3&-4\alpha-2\\\ -32\alpha-4&-12&-16\alpha+1\\\ \end{array}\right)$

here the basis is $a,\ b,\ ab$.

######  Proposition 1.

Let $l\in\mathbb{N}$. Then the following correspondence holds for the field of
real numbers $\mathbb{R}$:

$\begin{array}[]{|c|c|}\hline\cr\text{Order of }\tau_{ab}&\alpha\\\ \hline\cr
2&1/4\\\ 3&-1/8,\ 5/8\\\ 4&1/4\pm 3/(4\sqrt{2})\\\ 5&7/16\pm 3\sqrt{5}/16,\
1/16\pm 3\sqrt{5}/16\\\ 6&1/4\pm(3\sqrt{3})/8\\\ 7&\text{real root of
}x^{3}-\frac{3}{8}x^{2}-\frac{9}{32}x+\frac{13}{512}\text{ or
}x^{3}-\frac{9}{8}x^{2}+\frac{3}{32}x+\frac{43}{512}\\\ 8&1/8\left(2\pm
3\sqrt{2\pm\sqrt{2}}\right)\\\ 9&\text{real root of
}x^{3}-\frac{3}{4}x^{2}-\frac{15}{64}x+\frac{19}{512}\text{ or
}x^{3}-\frac{3}{4}x^{2}-\frac{15}{64}x+\frac{73}{512}\\\ 10&1/16\left(4\pm
3\sqrt{2(5\pm\sqrt{5})}\right)\\\ 11&\vtop{\hbox{$\text{real root of
}x^{5}-\frac{7}{8}x^{4}-\frac{5}{16}x^{3}+\frac{127}{512}x^{2}+\frac{119}{4096}x-\frac{263}{32768}$}\vspace{1mm}
\hbox{$\text{ or
}x^{5}-\frac{13}{8}x^{4}+\frac{7}{16}x^{3}+\frac{145}{512}x^{2}-\frac{337}{4096}x-\frac{197}{32768}$}}\\\
12&1/8\left(2\pm 3\sqrt{2\pm\sqrt{3}}\right)\\\ \hline\cr\end{array}$

The proposition can be proved as follows: for a given order $k$ consider the
matrix

$T=\tau_{ab}^{k}-I.$

Then one can find the greatest common divisor of $T_{i,j}$ and check, whether
it has any real roots. This procedure can easily performed via almost any
modern computer algebra system and can be easily done for orders larger than
$12$.

Remark. It can be seen that Proposition 1 holds for any finite extension of
$\mathbb{Q}$, that contains the necessary square roots.

We have shown $|\tau_{ab}|$ for algebras over $\mathbb{R}$ can have many
values and it can be conjectured that in this case it can actually take on any
value. Another way to obtain the desired orders is to pass to the field of
positive characteristic. For instance, in the case of $\mathbb{F}_{7}$ we have
the matrix looking as follows

$\overline{\tau}_{ab}=\frac{1}{\bar{2}}\left(\begin{array}[]{ccc}\alpha^{2}-\bar{2}\alpha-\bar{3}&\bar{3}\alpha&\bar{4}\alpha^{2}+\bar{4}\alpha-\bar{6}\\\
\bar{1}-\alpha&-\bar{3}&-\bar{4}\alpha-\bar{2}\\\
-\bar{4}\alpha-\bar{4}&-\bar{5}&-\bar{2}\alpha+\bar{1}\\\ \end{array}\right).$

Substituting values of $\alpha$ from this field yields the following orders:

$\begin{array}[]{|c|c|c|c|c|c|c|c|}\alpha&\bar{0}&\bar{1}&\bar{2}&\bar{3}&\bar{4}&\bar{5}&\bar{6}\\\
\hline\cr|\overline{\tau}_{ab}|&4&7&2&7&4&3&3\\\ \end{array}.$

Similarly, for $\mathbb{F}_{11}$ we have

$\begin{array}[]{|c|c|c|c|c|c|c|c|c|c|c|c|}\alpha&\bar{0}&\bar{1}&\bar{2}&\bar{3}&\bar{4}&\bar{5}&\bar{6}&\bar{7}&\bar{8}&\bar{9}&\bar{10}\\\
\hline\cr|\overline{\tau}_{ab}|&5&11&3&2&3&11&5&5&6&6&5\\\ \end{array}.$

It can thus be seen that many orders of $\tau_{ab}$ are reachable for the case
of zero characteristic, however working over positive characteristic can also
be useful in that regard.

##  3 $3$-generated algebras

In [2], the authors construct $3$-generated axial pseudo-composition algebras
$A=\langle\langle a,b,c\rangle\rangle$ of dimension $8$ with $4$ parameters
$\alpha=(a,b),\ \beta=(b,c),\ \gamma=(a,c)$ and $\psi=(b,ac)$ defining each
algebra’s multiplication table and the Gram matrix. These algebras are
distinguished by the following property: any other $3$-generated axial pseudo-
composition algebra with the similar Frobenius form and parameter values is a
homomorphic image of $A$.

We will call such algebras universal axial pseudo-composition algebras.

The goal and strategy of this section are similar: we intend to construct the
Miyamoto involutions, corresponding to the generators and manipulate the
parameter values in an attempt to discover interesting Miyamoto groups. Said
matrices will now depend on four aforementioned parameters.

We will be constructing the matrices in the following basis

$a,\ b,\ c,\ ab,\ bc,\ ac,\ a(bc),\ b(ac).$

Given that $\tau_{a}(b)=1/3(8\alpha a-b-4ab)$, and $\tau_{a}(c)=1/3(8\gamma
a-c-4ac)$ we construct the matrices via the automorphism multiplicative
property:

$\tau_{a}=\left(\begin{array}[]{llllllll}1&8/3\alpha&8/3\gamma&4/3\alpha&8/3\psi&4/3\gamma&4/3\psi&8/3\alpha\gamma+\beta-2/3\psi\\\
0&-1/3&0&-2/3&0&0&0&-1/3\gamma\\\ 0&0&-1/3&0&0&-2/3&0&\alpha\\\
0&-4/3&0&1/3&0&0&0&2/3\gamma\\\ 0&0&0&0&-1/3&0&-2/3&1/3\\\
0&0&-4/3&0&0&1/3&0&-2\alpha\\\ 0&0&0&0&-4/3&0&1/3&-2/3\\\ 0&0&0&0&0&0&0&-1\\\
\end{array}\right).$

In similar fashion we derive that

$\tau_{b}=\left(\begin{array}[]{cccccccc}-1/3&0&0&-2/3&0&0&-1/3\beta&0\\\
8/3\alpha&1&8/3\beta&4/3\alpha&4/3\beta&8/3\psi&8/3\alpha\beta+\gamma-2/3\psi&4/3\psi\\\
0&0&-1/3&0&-2/3&0&\alpha&0\\\ -4/3&0&0&1/3&0&0&2/3\beta&0\\\
0&0&-4/3&0&1/3&0&-2\alpha&0\\\ 0&0&0&0&0&-1/3&1/3&-2/3\\\ 0&0&0&0&0&0&-1&0\\\
0&0&0&0&0&-4/3&-2/3&1/3\\\ \end{array}\right),$

$\tau_{c}=\left(\begin{array}[]{llllllll}-1/3&0&0&-4/3\beta&0&-2/3&-\beta&1/3\beta\\\
0&-1/3&0&-4/3\gamma&-2/3&0&1/3\gamma&-\gamma\\\
8/3\gamma&8/3\beta&1&-4/3\alpha+8/3\psi&4/3\beta&4/3\gamma&\xi&\xi\\\
0&0&0&-1/3&0&0&1/3&1/3\\\ 0&-4/3&0&0&1/3&0&-2\gamma&2/3\gamma\\\
-4/3&0&0&0&0&1/3&2/3\beta&-2\beta\\\ 0&0&0&4/3&0&0&-1/3&2/3\\\
0&0&0&4/3&0&0&2/3&-1/3\\\ \end{array}\right),$

where $\xi=8/3\beta\gamma+1/3\alpha-2/3\psi$.

We will now try finding the optimal parameter values so that the products of
obtained matrices are of finite order and investigate the groups generated by
them. In this paper we will be mainly focused on obtaining finite groups,
however infinite Miyamoto groups are interesting to study as well.

It is clear that the results found in the previous section may only be of
partial service: while we know that, for instance $\tau_{ab}^{k}$ acts
trivially on $\langle a,b\rangle$ for a certain value of $\alpha$, we do not
know its behaviour on the rest of $A$. Therefore, the best we can hope for is
that some multiple of the order found for the 2-generated case is the order of
$\tau_{ab}$.

######  Proposition 2.

The characteristic polynomial of $\tau_{ab},\ \tau_{bc},\ \tau_{ac}$ lies in
$\mathbb{R}(\alpha),\mathbb{R}(\beta)$ and $\mathbb{R}(\gamma)$ respectively.

We shall first provide the proof for the universal $3$-generated axial pseudo-
composition algebra $A$, dealing with the quotients later.

It suffices to consider $\tau_{ab}$, since other cases are reduced to this one
by permuting the basis vectors and (for $\tau_{ac}$) replacing $b(ac)$ with
$c(ab)$ in the basis (clearly not changing any invariants and thus preserving
the order). Multiplying $\tau_{a}$ and $\tau_{b}$ we get

$\displaystyle\begin{pmatrix}&&&&R_{1}&&&\\\
-\frac{8}{9}\alpha+\frac{8}{9}&-\frac{1}{3}&-\frac{8}{9}\beta&-\frac{4}{9}\alpha-\frac{2}{9}&-\frac{4}{9}\beta&\frac{4}{9}\gamma-\frac{8}{9}\psi&-\frac{8}{9}\alpha\beta-\frac{4}{9}\beta-\frac{1}{9}\gamma+\frac{2}{9}\psi&-\frac{1}{9}\gamma-\frac{4}{9}\psi\\\
0&0&\frac{1}{9}&0&\frac{2}{9}&-\frac{4}{3}\alpha+\frac{2}{9}&-\alpha-\frac{2}{9}&\frac{1}{3}\alpha+\frac{4}{9}\\\
-\frac{32}{9}\alpha-\frac{4}{9}&-\frac{4}{3}&-\frac{32}{9}\beta&-\frac{16}{9}\alpha+\frac{1}{9}&-\frac{16}{9}\beta&-\frac{8}{9}\gamma-\frac{32}{9}\psi&-\frac{32}{9}\alpha\beta+\frac{2}{9}\beta-\frac{16}{9}\gamma+\frac{8}{9}\psi&\frac{2}{9}\gamma-\frac{16}{9}\psi\\\
0&0&\frac{4}{9}&0&-\frac{1}{9}&-\frac{4}{9}&\frac{2}{3}\alpha+\frac{4}{9}&\frac{1}{9}\\\
0&0&\frac{4}{9}&0&\frac{8}{9}&\frac{8}{3}\alpha-\frac{1}{9}&\frac{1}{9}&-\frac{2}{3}\alpha-\frac{2}{9}\\\
0&0&\frac{16}{9}&0&-\frac{4}{9}&\frac{8}{9}&\frac{8}{3}\alpha+\frac{1}{9}&-\frac{2}{9}\\\
0&0&0&0&0&\frac{4}{3}&\frac{2}{3}&-\frac{1}{3}\par\end{pmatrix}$

Where

$R_{1}=(\frac{64}{9}\alpha^{2}-\frac{16}{9}\alpha-\frac{1}{3},\
\frac{8}{3}\alpha,\
\frac{64}{9}\alpha\beta-\frac{8}{9}\gamma-\frac{32}{9}\psi,\
\frac{32}{9}\alpha^{2}+\frac{4}{9}\alpha-\frac{2}{3},\
\frac{32}{9}\alpha\beta-\frac{16}{9}\gamma+\frac{8}{9}\psi,$
$-\frac{32}{9}\alpha\gamma+\frac{64}{9}\alpha\psi-\frac{4}{3}\beta-\frac{4}{9}\gamma+\frac{8}{9}\psi,\frac{64}{9}\alpha^{2}\beta+\frac{8}{9}\alpha\beta+\frac{32}{9}\alpha\gamma-\frac{64}{9}\alpha\psi-\beta+\frac{4}{9}\gamma-\frac{8}{9}\psi,\
\frac{8}{9}\alpha\gamma+\frac{32}{9}\alpha\psi+\frac{1}{3}\beta-\frac{8}{9}\gamma-\frac{2}{9}\psi)$

is the first row of the matrix, put in such form for compactness reasons. The
characteristic polynomial of this matrix turns out as follows:

$x^{8}-\frac{8}{9}(8\alpha^{2}+2\alpha+1)x^{7}+\frac{4}{27}(256\alpha^{3}-48\alpha^{2}-24\alpha+5)x^{6}-\frac{4}{81}(1024\alpha^{4}+512\alpha^{3}-336\alpha^{2}-100\alpha+34)x^{5}+$
$+\frac{2}{81}(4096\alpha^{4}-1024\alpha^{3}-192\alpha^{2}+32\alpha-77)x^{4}-\frac{4}{81}(1024\alpha^{4}+512\alpha^{3}-336\alpha^{2}-100\alpha+34)x^{3}+$
$+\frac{4}{27}(256\alpha^{3}-48\alpha^{2}-24\alpha+5)x^{2}-\frac{8}{9}(8\alpha^{2}+2\alpha+1)x+1.$

Any $3$-generated axial pseudo-composition algebra is a quotient of the
universal $3$-generated axial pseudo-composition algebra by the latter’s
definiton. So we can consider the algebra $A_{1}=A/I$.

The ideal $I$ is generated as a vector space by the elements $v_{1}\ldots
v_{k}$. We add vectors $e_{1}\ldots e_{p}$ to this basis, so that $k+p=8$ and
so consider the basis $e_{1},\ldots e_{p},\ v_{1},\ldots v_{k}$. Since by [4,
Corollary 3.11], we have that $I$ is invariant under the action of the
Miyamoto group, and, in particular $\tau_{ab}$, we can present the matrix for
$\tau_{ab}$ in this basis, as a following block matrix:

$\tau_{ab}=\left(\begin{array}[]{cc}\left[\tau_{ab}\right]&*\\\
0&I\end{array}\right),$

where $I$, by abuse of notation, denotes the elements $v_{i}$ under
$\tau_{ab}$. It is clear that after taking the quotient only
$\left[\tau_{ab}\right]$ will remain and it is known from linear algebra, that
the characteristic polynomial of $\tau_{ab}$ is the multiple of characteristic
polynomials of the diagonal blocks, thus the characteristic polynomial of the
automorphism of $A_{1}$ corresponding to $\tau_{ab}$ divides
$\chi_{\tau_{ab}}$ and thus also lies in $\mathbb{R}[\alpha]$. The proposition
is proved.

As an easy consequence of this we have

######  Corollary 1.

Assume $A_{1}=\langle\langle a_{1},\ b_{1},\ c_{1}\rangle\rangle$ is a pseudo-
composition algebra, generated by $3$ idempotents such that
$\varphi(a)=a_{1},\ \varphi(b)=b_{1},\ \varphi(c)=c_{1}$, where $\varphi:A\to
A_{1}$ – a natural homomorphism by the ideal $I$. Then $|\tau_{a_{1}b_{1}}|$
divides $|\tau_{ab}|$.

We will provide another technical statement.

######  Statement 1.

The elements of the first, second and fourth column of $\tau_{ab}^{n}$ are
either $0$ for column index $j$ such that $j\in\\{3,\ 5,\ 6,\ 7,\ 8\\}=J$ or
non-zero and lie in $\mathbb{R}[\alpha]$ otherwise.

This can be proved by obvious induction on $n$. The base clearly holds, so we
proceed to the induction step by considering $n=k+1$. By induction, the only
non-zero elements of the considered columns are $a_{i,1}\ a_{i,2},\ a_{i,4}$,
thus proving the zeroes claim.

The second claim follows from the fact that, while the first, second and
fourth rows are the only ones containing parameters other than $\alpha$, by
induction on $n$, they contain them only in places $a_{i,j}$ for $j\in J$,
while $i=1,\ 2,\ 4$, so they cancel out upon multiplication. The statement is
thus proved. $\square$

This effect can be easily explained by the fact that any $\tau_{ab}$ leaves
any vector $v\in\langle\langle a,b\rangle\rangle$ in this subalgebra, i.e.
$\tau_{ab}^{n}v\in\langle a,b,ab\rangle$.

Because of the previous statement, we are able to provide a simple yet useful
and somewhat universal method for determining suitable parameters (or proving
their nonexistence) for a given degree of $\tau_{ab}$. Consider the elements
$f_{1}(\alpha)=(\tau_{ab}^{k})_{1,1}-1$ and
$f_{2}(\alpha)=(\tau_{ab}^{k})_{1,2}$. These are polynomial in $\alpha$ and
contain no entries of $\beta,\ \gamma$ or $\psi$ (this can be confirmed by
looking at the matrix). It is clear that if there exists $\alpha_{0}$ such
that $\tau_{ab}^{k}(\alpha_{0})=I$ then

$f_{1}(\alpha_{0})=f_{2}(\alpha_{0})=0.$ $None$

Note that one can use any non-zero element from the first, second or fourth
column of the matrix as additional equations or as replacements for $f_{1}$ or
$f_{2}$ and then employ any suitable real root-finding algorithm for
determining $\alpha_{0}$.

Returning to the Miyamoto groups we see that, for the $3$-generated case, the
Miyamoto involutions $\tau_{a}$ and $\tau_{b}$ commute only in rather specific
situations.

######  Proposition 3.

Assume $a,\ b$ and $c$ are axes, generating a pseudo composition algebra
$A^{\prime}$ and, without the loss of generality, that $|\tau_{ab}|=2$. Then
$dim(A^{\prime})\leq 4$.

We start our proof by again working in the universal algebra $A$.

In this case we have

$f_{1}=\frac{1}{81}(4096\alpha^{4}-3072\alpha^{3}-576\alpha^{2}+464\alpha+33)-1,\
f_{2}=\frac{1}{27}(512\alpha^{3}-256\alpha^{2}-64\alpha+24)$

The only common factor of these polynomials is $(\alpha-\frac{1}{4})$ with
$\frac{1}{4}$ setting $|\tau_{ab}|=4$.

Now we move on to the quotient algebras. Consider the ideal $I\subseteq A$,
generated by elements $\tau_{ab}^{2}x_{i}-x_{i}$ where $x_{i}$ are all the
basis vectors outlined at the begining of the section.

We provide these elements here in the form of a matrix (the entries are
multiplied by $9$ for compactness reasons, so the matrix actually contains
images of $\tau_{ab}^{2}9x_{i}-9x_{i}$)

$\left(\begin{array}[]{cccccccc}0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
-\frac{16}{3}\beta+\frac{32}{3}\gamma-\frac{64}{3}\psi&\frac{32}{3}\beta-\frac{16}{3}\gamma-\frac{64}{3}\psi&-16&-\frac{64}{3}\beta-\frac{64}{3}\gamma+\frac{128}{3}\psi&8&8&16&16\\\
0&0&0&0&0&0&0&0\\\
-\frac{20}{3}\beta-\frac{32}{3}\gamma+\frac{64}{3}\psi&\frac{40}{3}\beta-\frac{20}{3}\gamma-\frac{8}{3}\psi&1&-\frac{8}{3}\beta+\frac{64}{3}\gamma-\frac{128}{3}\psi&-14&4&8&8\\\
-\frac{20}{3}\beta+\frac{40}{3}\gamma-\frac{8}{3}\psi&-\frac{32}{3}\beta-\frac{20}{3}\gamma+\frac{64}{3}\psi&1&\frac{64}{3}\beta-\frac{8}{3}\gamma-\frac{128}{3}\psi&4&-14&8&8\\\
-\frac{16}{3}\beta-\frac{4}{3}\gamma+\frac{44}{3}\psi&-\frac{4}{3}\beta+\frac{38}{3}\gamma-\frac{64}{3}\psi&\frac{1}{2}&\frac{44}{3}\beta-\frac{64}{3}\gamma+\frac{56}{3}\psi&2&2&-14&4\\\
\frac{38}{3}\beta-\frac{4}{3}\gamma-\frac{64}{3}\psi&-\frac{4}{3}\beta-\frac{16}{3}\gamma+\frac{44}{3}\psi&\frac{1}{2}&-\frac{64}{3}\beta+\frac{44}{3}\gamma+\frac{56}{3}\psi&2&2&4&-14\\\
\end{array}\right),$

where the rows correspond to the elements of the basis.

It is clear that any ideal, such that the image of the automorphism has order
$2$ on the quotient algebra, must contain $I$. We then note that the four last
columns can be reduced to the following row echelon form:

$\left(\begin{array}[]{cccc}8&8&16&16\\\ 0&18&36&36\\\ 0&0&36&36\\\
0&0&0&18\\\ 0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ \end{array}\right).$

This implies that the span of the vectors $\tau_{ab}^{2}x_{i}-x_{i}$ is
$4$-dimensional, meaning that $dim\ I\geq 4$.

Setting $\beta=1,\ \gamma=1/4=\psi$ we get that $dim(I)=4$ and $\tau_{ab}$ has
order $2$ on $A/I$; keeping the same parameters and setting $b=c$ yields a
$5$-dimensional ideal, which contains $I$, while setting $ab=0=c$ we get a
$6$-dimensional $I$.

######  Statement 2 (c.f. proposition 1).

For the axial pseudo-composition algebra $A$, finitely generated by at least
three axes, matrix $\tau_{ab}$ has its order divide $3,4,5,6,10$ for values of
$-1/8,1/4,\frac{1}{16}\pm\frac{3\sqrt{5}}{16},5/8,\frac{7}{16}\pm\frac{3\sqrt{5}}{16}$
for $\alpha$. The same values work for $\tau_{ac}$ and $\gamma$ and
$\tau_{bc}$ and $\beta$.

###### Proof.

The result for algebras, generated by $3$ axes can be directly confirmed via
working with the universal algebra and referring to Corollary 1. For the
finitely generated case, consider $\tau_{ab}$ for axes $a,\ b\in A$ such that
$\alpha=(a,b)$ is equal to some value from the result’s statement. Since it is
known that any axial pseudo-composition algebra contains a basis consisting of
primitive axes [2, Proposition 2], we can choose any axis from the basis that
is not equal to $a$ or $b$ as $c$ and consider the subalgebra $\langle\langle
a,b,c\rangle\rangle$. By the definition this is a homomorphic image of the
universal axial pseudo-composition algebra $\langle\langle
a^{\prime},b^{\prime},c^{\prime}\rangle\rangle$ with the same parameter
values. We have that $\tau_{a^{\prime}b^{\prime}}^{k}$ (where $k$ is one of
the orders from the result’s statement) fixes $a^{\prime},\ b^{\prime},\
c^{\prime}$, thus being of order $k$ for the universal algebra, and again by
Corollary 1, we have that $\tau_{ab}^{k_{0}}$, such that $k_{0}$ divides $k$,
fixes $a,\ b,\ c$.

By this argument $k_{0}$ depends on the initial choice of an axis $c$. Note
however, that since $\alpha$ is fixed, $k$ does not depend on that choice so
we can pick $k_{0}$ as the biggest divisor of $k$ that appears this way for
some axis $c$, hence $\tau_{ab}^{k_{0}}$ will fix the whole algebra. ∎

This statement is another demonstration of the following effect: The parameter
of a $2$-generated pseudo-composition algebra $\langle\langle
a_{1},a_{2}\rangle\rangle$ with $a_{i}$ being axes, defines the order of the
corresponding automorphism $\tau_{a_{1}a_{2}}$, but only up to a scalar
multiple.

We will now focus our attention on finding groups, that can be generated by
three automorphisms.

For example, by setting $\alpha=\beta=\gamma=-1/8$ we obtain a $3$ generated
group, that is known to be a group of motions of the euclidean plane,
generated by the reflections in the sides of an equilateral triangle. This
group is known to be infinite, however it is possible to pass to its finite
quotient by adding a relator $(\tau_{a}\tau_{b}^{\tau_{c}})^{m}$. More
specifically:

######  Proposition 4.

Let $G=\langle x,y,z\rangle$ be the group generated by involutions $x,\ y,\
z$, such that $|xy|=|yz|=|xz|=3$. Additionally, let $k=|xy^{z}|$. Then $G$ is
a quotient of $k^{2}\rtimes D_{6}$.

Here $k^{2}$ denotes the direct product of two copies of a cyclic group of
order $k$, while $D_{6}$ is the dihedral group of order $6$.

We will provide the proof of this proposition for the sake of completeness:

Proof. Note that

$[xy^{z},yz^{x}]=y^{z}xxzxyxy^{z}yz^{x}=zyxyxy^{z}yz^{x}=zxyzyzyz^{x}=x^{z}z^{x}=(zx)^{3}=e.$

Therefore the subgroup $K=\langle xy^{z},yz^{x}\rangle$ is abelian (being
isomorphic to $k^{2}$). It is also easy to see that it is a normal subgroup of
$G$. Now consider the quotient $G/K$. In it, $\bar{y}=\overline{z^{x}}$,
therefore the quotient is generated by 2 involutions, making it a Dihedral
group of order $2|xy|$, which is equal to $6$. Since the groups $K$ and
$\langle x,z\rangle$ intersect trivially, we conclude that $G$ is a semidirect
product of $K$ and $D_{6}$, concluding the proof. $\square$

As a result we arrive at another family of groups, that may be realised as
Miyamoto subgroups of pseudo-composition algebras. Now we only need to find
the exact possible orders $\tau_{a}\tau_{b}^{\tau_{c}}$ can take. This can be
accomplished by choosing the correct values of $\psi$, so the situation is
somewhat similar to the previous cases, since we only have a single parameter.

Now, the minimal polynomial of this matrix, computed in GAP [1], looks as
follows:

$x^{5}+\frac{1}{81}(-1024\psi^{2}+32\psi+101)x^{4}+\frac{1}{243}\left(-\frac{32768}{3}\psi^{3}-1024\psi^{2}+1120\psi+\frac{404}{3}\right)x^{3}+$
$+\frac{1}{243}\left(-\frac{32768}{3}\psi^{3}-1024\psi^{2}+1120\psi+\frac{404}{3}\right)x^{2}+\frac{1}{81}(-1024\psi^{2}+32\psi+101)x-1$

with one of its roots being equal to 1 and the other roots being

$r_{1,2}=\frac{1}{9}\left(-(16\psi+2)\pm\sqrt{256\psi^{2}+64\psi-77}\right),$

$r_{3,4}=\frac{1}{162}\left(1024\psi^{2}+256\psi-146\pm\sqrt{-(1024\psi^{2}+256\psi-146)}\right).$

One can then manually use these in order to see the following

######  Statement 3.

For $\alpha=\beta=\gamma=-1/8$ the values
$\frac{5}{32},-\frac{1}{8},\frac{1}{64}-\frac{(9\sqrt{5})}{64},-\frac{13}{32}$
of $\psi$ set the order of $ab^{c}$ to a divisor of $3,4,5,6$ respectively.

Note that after fixing the orders of involution products, $\psi$ is the sole
remaining parameter and as such it may be insufficient to obtain a finite
group. We will illustrate that by the following example:

Consider the following presentation of $A_{5}$

$\langle x,y,z|x^{2},y^{2},z^{2},(xy)^{3},(yz)^{3},(xyz)^{3}\rangle.$

We will attempt to find suitable parameters $\alpha,\ \beta,\ \gamma$ and
$\psi$ to obtain $\langle\tau_{a},\ \tau_{b},\ \tau_{c}\rangle\simeq A_{5}$.
Now set $x=\tau_{a},\ y=\tau_{b},\ z=\tau_{c}$. This automatically fixes
$\alpha=\beta=-1/8$ (due to Proposition 1 and Statement 2 this is the only
possible value).

Additionally, in this presentation the elements $xz$ and $xy^{z}$ must have
order $5$. To set the former we use one of the values of $\gamma$ from
Proposition 1 that sets the order to $5$ in the universal axial pseudo-
composition algebra, with the uniqueness of said parameters also being
guaranteed by Proposition 1 and Statement 2, while for $xy^{z}$ we have to
find suitable values of $\psi$. Therefore if no finite groups are produced for
parameters, guaranteeing that order, this group cannot be obtained as a
Miyamoto group of an $8$-dimensional pseudo-composition algebra, generated by
$3$ axes.

We will use the following construction: consider the axis $b^{\tau_{c}}$,
which we will denote $d$. It can be seen, that $d$ has $\tau_{b}^{\tau_{c}}$
as its Miyamoto involution and that $d$ is an image of an idempotent under the
automorphism and thus $d$ is an idempotent itself. We then consider the
subalgebra $\langle\langle a,d\rangle\rangle$, which has
$\tau_{ad}=\tau_{a}\tau_{b}^{\tau_{c}}$. Therefore we need to only derive the
values of $\psi$ from the equality $(a,d)=r$ with $r$ being one of the
parameter values from Proposition 1, corresponding to $5$. If none of these
parameters yield a finite group, our claim follows.

With these parameters (we pick $\gamma=\frac{1}{16}-\frac{3\sqrt{5}}{16}$,
noting that the reasoning is exactly the same for its conjugate).

With this setup, the angle $(a,d)=-\frac{4}{3}\psi+\frac{3\sqrt{5}+1}{48}$. We
need to find four values of $\psi$ such that
$(a,d)\in\\{\frac{1}{16}\pm\frac{3\sqrt{5}}{16},\
\frac{7}{16}\pm\frac{3\sqrt{5}}{16}\\}$.

Thus we get that

$\psi\in\\{-\frac{1+3\sqrt{5}}{32},\ \frac{-1+6\sqrt{5}}{32},\
-\frac{10+3\sqrt{5}}{32},\ \frac{-5+3\sqrt{5}}{16}\\},$

where the values correspond to $\frac{1}{16}+\frac{3\sqrt{5}}{16},\
\frac{1}{16}-\frac{3\sqrt{5}}{16},\ \frac{7}{16}+\frac{3\sqrt{5}}{16},\
\frac{7}{16}-\frac{3\sqrt{5}}{16}$ respectively.

Two main ways will be used to eliminate values of $\psi$. Either
$|\tau_{a}\tau_{d}|=\left|\tau_{a}\tau_{b}^{\tau_{c}}\right|$ is indeed equal
to $5$ and we use GAP to check, whether the group generated by Miyamoto
involutions is finite or if the order is equal to $5k$ (usually $k=2$), in
which case we check, whether the corresponding algebra is simple by computing
the rank of its Gram matrix, as the algebra being simple implies there is no
way to find a non-trivial quotient with the required order.

The first and second values of $\psi$ are setting the order of
$\tau_{a}\tau_{d}$ to $5$ but the group generated by involutions is infinite,
while the latter two values set the order to $10$, with the corresponding
algebras being simple. Hence our claim follows.

We have considered the presentation of $A_{5}$ that has $(xy)^{3}$, $(yz)^{3}$
and $(xz)^{5}$ as relators of product orders. Since all element orders for
this group lie in the set $\\{1,\ 2,\ 3,\ 5\\}$, by Proposition 3, we only
need to consider orders $3$ and $5$ as possible product orders. However all
other presentations can be dealt with in a similar way as for the case
$(xy)^{3},\ (yz)^{5},\ (xz)^{5}$, one of two elements $xz^{y}$ or $yz^{x}$
always has order $2$ or $3$ either eliminating this case due to Proposition 3
or reducing them to the above case by replacing $z$ with its conjugate.
Likewise it can be manually checked that presentations with all generator
products having order $5$ set the order of $xy^{z}$ to $3$ and can be
rewritten into the previously mentioned presentations.

The technique used here, while simple, is generalisable, allowing one to find
potential parameters for the pseudo-composition algebra for a given group,
provided said group’s presentation consists of relators of the form
$a_{i}a_{j}^{w}$ where $a_{i},\ a_{j}$ are among the generating involutions
and $w$ is a word from the group. In particular, this highlights another
connection to the class of $n$-transposition groups, which have this exact
type of relators by definition.

Nonetheless, we cannot confirm the non-existence of a pseudo-composition
algebra generated by $3$ axes for $A_{5}$ as there may exist quotient algebras
with it as a Miyamoto subgroup. It is to be hoped that finer analysis of the
Miyamoto group of a $4$-dimensional pseudo composition algebra generated by
$3$ axes will shed light on the validity of this non-existence hypothesis.

However for the second smallest non-abelian simple group, $PSL(2,7)$, the
situation is more positive:

It is known that $PSL(2,7)$ satisfies the presentation

$\langle t,s|t^{2}=s^{3}=(st)^{7}=[s,t]^{4}\rangle.$

This can be transformed into a presentation with three generating involutions
$a=t,\ b=t^{s}$ and $c=t^{s^{2}}$. It can be derived that $|ac|=|ab|=|bc|=4$
and that $|ab^{c}|=3$. This allows us, by setting
$\alpha=\beta=\gamma=\frac{1}{4}$ and $\psi=\frac{5}{32}$ to obtain
$\langle\tau_{a},\tau_{b},\tau_{c}\rangle=PSL(2,7)$. The rank of the Gram
matrix for the corresponding algebra is equal to 8, meaning that the algebra
is simple.

Therefore our work with two smallest non-abelian finite simple groups can be
summarised in the following

######  Proposition 5.

Let $A$ be the universal axial pseudo composition algebra generated by $3$
axes $a,\ b,\ c$ over the field $\mathbb{R}$. Then we following statements
hold.

  1. 1.

The alternating group $A_{5}$ cannot be generated by Miyamoto involutions
$\tau_{a},\ \tau_{b},\ \tau_{c}$;

  2. 2.

The group $PSL(2,7)$ can be generated by $\tau_{a},\ \tau_{b},\ \tau_{c}$ with
parameter values $\alpha=\beta=\gamma=\frac{1}{4}$ and $\psi=\frac{5}{32}$.

### Case of positive charactersitic

Akin to the 2-generated case, we can try tweaking the characteristic of the
ground field.

######  Proposition 6.

Assume $A=\langle\langle a,b,c\rangle\rangle$ is the universal axial pseudo-
composition algebra defined over $\mathbb{F}_{5}$ with $a,\ b,\ c$ being
idempotents. Then the group generated by $\tau_{a},\tau_{b},\tau_{c}$ is
either solvable or is isomorphic to $5^{5}:A_{5},\ 5^{5}:\leavevmode\nobreak\
S_{5},$ $5^{2}:\leavevmode\nobreak\ (5^{2}:\leavevmode\nobreak\
(SL(2,5):\leavevmode\nobreak\ 2)),\ PSL(2,7),$ $PSL(3,5),PSU(3,5),\ A_{6},\
A_{7}$.

For the listed groups the Gram matrix of $A$ has rank $5$ for the first two
cases and $8$ for all the other ones.

We note that while said groups appear for multiple combinations of parameters
(with them not just being permutations of each other), the rank of the Gram
matrix remains constant for all options.

We will provide a single combination of parameters for every group listed in
the proposition. We will not be using the bar notation, even though the
parameter values lie in $\mathbb{F}_{5}$:

$\begin{array}[]{|c|c|c|c|c|c|}\hline\cr\alpha&\beta&\gamma&\psi&T=\langle\tau_{a},\
\tau_{b},\ \tau_{c}\rangle&rank(gram(A))\\\ \hline\cr
3&4&1&1&5^{2}:\leavevmode\nobreak\ (5^{2}:\leavevmode\nobreak\
(SL(2,5):\leavevmode\nobreak\ 2))&4\\\ 3&1&3&2&5^{5}:A_{5}&5\\\
3&0&4&4&5^{5}:S_{5}&5\\\ 3&3&4&4&PSL(2,7)&8\\\ 3&3&1&0&PSL(3,5)&8\\\
1&3&1&0&PSU(3,5)&8\\\ 3&3&1&1&A_{6}&8\\\ 3&3&1&4&A_{7}&8\\\
\hline\cr\end{array}$

The proposition can be confirmed by directly substituting various parameters
into $\tau_{a},\ \tau_{b}$ and $\tau_{c}$.

We note that for solvable groups the rank of the Gram matrix takes all values
from $1$ to $8$, excluding $6$. This suggests that said solvable groups may
prove to be interesting examples if one wishes to attempt classifying the
ideals inside such pseudo-composition algebras.

It is important to note that $T=\langle\tau_{a},\ \tau_{b},\ \tau_{c}\rangle$
is only a subgroup of the Miyamoto group.

Example. Consider the case when $T\simeq A_{6}$. After using the parameter
values from Proposition 6 to the corresponding algebra $A$, one can find that
$b(ac)$ is an idempotent in that algebra. By [5, Proposition 2.7], it is also
an axis, so we will denote it by $d$, thus having a corresponding Miyamoto
involution, denoted by $\tau_{d}$.

Applying the automorphism construction rules, which can be extracted by
generalising [2, Lemma 2.1], we can construct $\tau_{d}$, which, per our
definitions is also a generator in the Miyamoto group. However, as it turns
out $\langle\tau_{a},\ \tau_{b},\ \tau_{c},\ \tau_{d}\rangle\simeq SL(8,5)$.

In other words, the group $T$ can heavliy differ from the whole Miyamoto group
of an algebra, which however should not discourage interest of studying such
subgroups. We also note that the number of involutions in $SL(8,5)$ is
significantly greater than the number of idempotents in $A$, thus showing that
the idempotent $\leftrightarrow$ involution correspondence is not surjective.

Finally, we state the results of our reconnaissance into bigger values of
positive characteristic.

After experimenting with some parameters over fields of bigger characteristic
we have arrived at a

######  Statement 4.

The group $T=\langle\tau_{a},\ \tau_{b},\ \tau_{c}\rangle$ is isomorphic to
$PSL(3,q)$ for the following combinations of $(q,\alpha,\beta,\gamma,\psi)$:

$(7,\ \bar{6},\ \bar{2},\ \bar{1},\ \bar{5}),\;(11,\ \bar{1},\ \bar{1},\
\bar{3},\ \bar{0}),\;(13,\ \bar{8},\ \bar{10},\ \bar{4},\ \bar{2}).$

Similarly, $T$ is isomorphic to $PSU(3,q)$ for the following combinations of
$(q,\alpha,\beta,\gamma,\psi)$:

$(7,\ \bar{6},\ \bar{2},\ \bar{1},\ \bar{1}),\;(11,\ \bar{1},\ \bar{1},\
\bar{3},\ \bar{1}),\;(13,\ \bar{8},\ \bar{10},\ \bar{4},\ \bar{1}).$

This prompts to present the following:

######  Conjecture 1.

There exist parameters $\alpha,\ \beta,\ \gamma,\ \psi\in F$ such that
$T=\langle\tau_{a},\ \tau_{b},\ \tau_{c}\rangle$ is isomorphic to $PSL(3,F)$
and a different set of parameters such that $T$ is isomorphic to $PSU(3,F)$,
where $F$ is any prime field.

This conjecture may be possible to tackle via constructing reasonably good
presentations for these groups, akin to $PSL(2,7)$.

Concerning other questions, computing the full Miyamoto groups and
automorphism groups for algebras with parameter sets we have already
considered is certainly interesting, as it would allow us to deepen our
understanding of said algebras and of automorphisms of axial algebras as a
whole.

## References

  * [1] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.12.1; 2022. (https://www.gap-system.org).
  * [2] I. Gorshkov, A. Mamontov, A. Staroletov. Axial view on pseudo-composition algebras and train algebras of rank 3, arXiv:2309.05237, (2023).
  * [3] J.I. Hall, F. Rehren, S. Shpectorov. Universal axial algebras and a theorem of Sakuma, J. Algebra, 421 (2015), 394–424.
  * [4] S. M. S. Khasraw, J. McInroy and S. Shpectorov. On the structure of axial algebras, Trans. Amer. Math. Soc., 373 (2020), 2135–2156.
  * [5] K. Meyberg, J.M. Osborn. Pseudo-composition algebras, Math Z., 214:1 (1993), 67–77.
  * [6] J. McInroy, S.Shpectorov. Axial algebras of Jordan and Monster type, arXiv:2209.08043 (2022). Author information Vsevolod A. Afanasev, Sobolev Institute of Mathematics, Novosibirsk,
Russia;
Novosibirsk State University, Novosibirsk, Russia.
E-mail address<EMAIL_ADDRESS>


11institutetext: GE HealthCare
11email: {Amin.Honarmandishandiz, Attila.Radics, Rajesh.Tamada, Arpad.Makk1,
Karolina.Glowacka1, Sandeep.Dutta, Michail.Fanariotis<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Automated Identification of Failure Cases in Organ at Risk Segmentation
Using Distance Metrics: A Study on CT Data

Amin Honarmandi Shandiz  Attila Rádics  Rajesh Tamada  Makk Árpád  Karolina
Glowacka  Lehel Ferenczi  Sandeep Dutta  Michael Fanariotis

###### Abstract

Automated organ at risk (OAR) segmentation is crucial for radiation therapy
planning in CT scans, but the generated contours by automated models can be
inaccurate, potentially leading to treatment planning issues. The reasons for
these inaccuracies could be varied, such as unclear organ boundaries or
inaccurate ground truth due to annotation errors. To improve the model’s
performance, it is necessary to identify these failure cases during the
training process and to correct them with some potential post-processing
techniques. However, this process can be time-consuming, as traditionally it
requires manual inspection of the predicted output. This paper proposes a
method to automatically identify failure cases by setting a threshold for the
combination of Dice and Hausdorff distances. This approach reduces the time-
consuming task of visually inspecting predicted outputs, allowing for faster
identification of failure case candidates. The method was evaluated on 20
cases of six different organs in CT images from clinical expert curated
datasets. By setting the thresholds for the Dice and Hausdorff distances, the
study was able to differentiate between various states of failure cases and
evaluate over 12 cases visually. This thresholding approach could be extended
to other organs, leading to faster identification of failure cases and thereby
improving the quality of radiation therapy planning.

###### Keywords:

Anomaly Detection, Quality Assurance, Organ at risk segmentation, Failure case
identification, Distance metrics, Model performance, Radiation therapy
planning, Thresholding

##  1 Introduction

Organ at risk (OAR) segmentation is an essential step in radiation therapy
planning. Accurate delineation of OARs can help to spare these critical
structures from high doses of radiation and prevent unwanted side effects
[21]. The traditional method for OAR contouring is manual contouring, which is
a time-consuming and labor-intensive process. Manual contouring can also be
subjective and prone to inter-observer variability, which can affect the
accuracy and consistency of the contouring [3, 4, 2]. To address these issues,
automatic contouring methods have been developed in recent years. These
methods use algorithms and machine learning techniques to automatically
segment OARs from medical images. Recently, deep learning methods such as
Convolutional neural networks have played a significant role in various
aspects specially when the data is images [9, 18, 19, 5, 20, 22].

According to a study conducted by Fugan et al [7] an auto-contouring (AC)
system was evaluated and found to be in agreement with manual contours within
the inter-observer uncertainty level. Additionally, Altman et al [1] observed
that there were no significant dosimetric differences between treatment plans
using AC software versus manually contouring. These findings suggest that
despite not always being perfect, auto-contours can still be considered
clinically acceptable. However, the performance of automatic contouring
methods can vary depending on the imaging modality, image quality, and
complexity of the OARs being segmented [7]. As such, it is important to
evaluate the accuracy and robustness of automatic contouring methods before
their implementation in clinical practice.

A conventional approach to enhance the performance of OAR models is to improve
the accuracy of the model’s predictions. To achieve this, it is necessary to
identify cases where the model fails to produce accurate predictions. In the
case of OAR models, this requires a visual inspection of the predicted
contours to determine if they are closely overlapping the ground truth. Cases
where there is unsatisfactory overlap between the predicted and ground truth
contours, or where anomalies exist in the predicted images, are selected as
failure cases. These cases are then subjected to further analysis to identify
the cause of the failure, which involves consultation with experts. However,
this approach can be time-consuming, especially when there are many candidate
cases to test and no predefined set of candidates to prioritize for failure
analysis.

In order to address this issue, our paper proposes a method that utilizes
distance metrics to automatically identify candidate cases for failure
analysis. By analyzing 20 cases for 6 organs, we determined a threshold that
could effectively distinguish these candidates. Instead of manually checking
the predicted contours against the ground truth, this approach could save time
by automatically identifying potential failure cases based on the HD and Dice
metrics [6]. We found that different organs had different values for the
proposed threshold, and this method could help decrease the time needed to
find failure cases.

##  2 Methods and Materials

Loss functions play an important role in various tasks. In image segmentation
tasks, the role of the loss function is distinguished by measuring the
difference between the predicted and ground truth segmentation maps. They
guide the training process to optimize model performance. Some common loss
functions in common 3D segmentation tasks can be seen in figure 1 by Jun Ma et
al [13]. According to the study conducted by Shruti et al. Common loss
functions in image segmentation are different and depend on various
parameters, although the popular used ones can be cross-entropy loss, dice
loss, Hausdorff distance (HD), etc. [11]. In the context of OAR segmentation
tasks, the choice of loss functions depends on task-specific requirements such
as the nature of the data, the desired characteristics of the segmentation
output, and the specific task.

Figure 1: The Figure shows loss functions for typical 3D segmentation tasks in
four different category [8].

###  2.1 Dice and Hausdorff Distance Metrics

To evaluate the accuracy of the OAR segmentation, we used the Dice and
Hausdorff distance metrics. The Dice coefficient measures the similarity
between two sets of binary data, while the Hausdorff distance measures the
maximum distance between two sets of points. We computed the Dice and
Hausdorff distance between the predicted segmentation masks and ground truth
masks for each of the six OARs.

The Dice distance, also known as the Dice coefficient, is a widely used metric
for evaluating the similarity or overlap between two sets. In the context of
medical image analysis, it is often used to assess the accuracy of
segmentation algorithms that delineate the boundaries of anatomical structures
or regions of interest.

The Dice distance is calculated as the ratio of the intersection of two sets
to their combined size, and ranges from 0 to 1, where a value of 1 indicates
perfect overlap and a value of 0 indicates no overlap. The formula for Dice
distance is:

$Dice(A,B)=\frac{2\left|A\cap B\right|}{\left|A\right|+\left|B\right|}$ (1)

where $\cap$ represents the intersection of the predicted and actual volumes.
The Hausdorff distance is another metric used to evaluate the accuracy of
segmentation algorithms. It measures the maximum distance between any point on
the surface of one volume and the closest point on the surface of the other
volume. In other words, it quantifies the "worst-case scenario" of
disagreement between two sets. The formula for Hausdorff distance is:

$Hausdorff(A,B)=\max\\{h(a,B),h(b,A)\\}$ (2)

where a and b are points on the surfaces of A and B, respectively, and h(a,B)
and h(b,A) are the distances from a to the closest point on B, and from b to
the closest point on A, respectively.

The Hausdorff distance is often used in conjunction with the Dice distance to
provide a more complete picture of the accuracy of a segmentation algorithm.
While the Dice distance measures the degree of overlap between two sets, the
Hausdorff distance provides information about the extent and location of any
discrepancies between the two sets.

###  2.2 Dataset

We have used two datasets, -i): one containing five organs-at-risk (OARs) that
are commonly segmented in radiation therapy planning from an internal dataset
and -ii) another one contains one organ from a public dataset for lymph node
[16, 15, 17]. The internal dataset (acquired by our clinical partners)
consists of 20 CT images depicting different areas, scanned using different CT
protocols, with slice thickness between 0.5 and 3 mm. These scans were
acquired on volunteers, using various multi-slice CT scanners such as
(Discovery 750HD, Optima540, etc). These scans, unlike the ones in the public
dataset, are not uniform, and differ in CT image parameter settings (e.g.
resolution, pixel spacing).

###  2.3 Model Architecture

Figure 2: The figure shows the 3D auto-encoder model for organ at risk
segmentation [8].

The figure 2 shows a schematic diagram of an auto-encoder model, which is a
type of neural network used for tasks such as dimensional reduction and
feature extraction. The model consists of two parts: the encoder and the
decoder. The encoder takes the input data and compresses it into a lower-
dimensional representation or latent space, while the decoder takes this
representation and reconstructs the original data. This model can be used for
various applications, including image and signal processing.

In our paper, organ contours were created using a novel DL auto segmentation
model (DLAS v1.0, GE HealthCare), that the auto encoder structure is like
figure 2 used in [8]. We employed the encoder part of the model as a feature
extractor to obtain a compressed representation of the input images that
captures the salient features related to the organs of interest. This
compressed representation was then fed into a decoder to generate the organ
segmentation contours. This approach allowed us to improve the accuracy and
efficiency of the segmentation task compared to manual contouring or other
traditional segmentation methods.

###  2.4 Experimental Setup

Firstly, the contours were produced and reviewed by experts radiologist for
five different organs including breasts, Chiasma, Brainstem, Trachea and
Femurs. for each organ there are 20 CTs with their ground truth. Then, we
converted all of the data to nrrd format and fed them to the 3D auto-encoder
model like figure 2 with train, evaluation and test partitions respectively
10,5,5 which were chosen for the model and batch size of 5.

Figure 3: The figure summarizes four situations expressed in Table 1 for
Breasts organ segmentation. The four situations include: A) Shows over
segmentation of the model, extra predictions exists in the output beside the
region of interest, B)Extra predictions with less overlap with ground truth
for the left breast and under segmentation for the right breast organ in some
slices, C) good predictions, D) model has good predictions but there are extra
predictions(over segmentation) in the left breast. Ground Truth contours are
distinguished by red line and predictions are distinguished by green.

For training the model we used Dice as loss function. After contours were
predicted by the model then two metrics including Hausdorff and Dice [14, 12]
used to evaluate the model prediction. After visualizing the predicted
contours, we compared the results with ground truth to find if there are
failure prediction exist in the predicted values. Afterward, we visualize the
predicted failed cases versus good cases as presented in figure 3.

As it is showed in figure 3 for breast organ there are situations where the
predicted organs does not overlap with the ground truth or there might be
extra lesion predicted outside of the expected organ region. To find these
cases we use the characteristics of image distance measures such as Dice and
Hausdorff distances. As the Dice’s characteristic [6] is suitable to find the
overlapped contours and base on characteristics of Hausdorff distance [10]
that is considered for distance calculation, we use this metric to find the
regions which are predicted far from the ground truth as it is sensitive to
the outliers.

Figure 4: The figure shows the results of lymph node contouring in CT images
using a deep learning model.

We also analyised lymph node data(see fig 4) containing abdominal and
mediastinal lymph nodes which are publicly available total of 388 mediastinal
lymph nodes in CT images of 90 patients and a total of 595 abdominal lymph
nodes in 86 patients (for more info see [16, 17, 15]).

##  3 Results and Discussion

After calculating Dice and Hausdorff distances for predicted versus ground
truth, the purpose was to find the less predicted contours in some slices by
low Dice score values. Also, analyzing the HD values could determine if the
output is over segmented or not. As you can see in table 1 with different
combination of Dice and Hausdorff values we are able to decide if output
contours need to consider for post-processing.

We found that setting a threshold for Dice and Hausdorff distance produced for
these six organs could help to distinguish the failure cases, identifying a
significant number of candidate cases while keeping the false positive rate
low. We also found that in most cases, the identified failures were due to
under segmentation between the predicted and ground truth contours, or due to
over segmentation by the model.

We compared our approach to a baseline method that relied on visual inspection
alone and found that our method could identify candidate cases with higher
speed as it is not needed to visually check them. Our approach is also could
be useful in other organs by finding their threshold values.

Table 1: The table summarizes the different situations for anomaly detection in medical imaging based on the Hausdorff distance and Dice similarity coefficient for Breast. The table provides an explanation for each situation based on the predicted results. Situation |  Explanation
---|---
HD $<$ 6 and Dice $\sim$ 0.9 |  Good prediction
6 $<$ HD $<$ 100 and Dice $\sim$ 0.9 |  Good overlap prediction but there are extra predictions connected and close to GT
Hd $>$ 100 and Dice $\sim$ 0.9 |  Good prediction but there are extra predictions which are far from respective organ
Hd $>$ 6 and Dice $<$ 0.8 |  There are extra predictions in slices or far from organ and there is led overlap for some ground truth contours
Hd $<$ 6 and Dice $<$ 0.8 |  There is no extra pridicted contours or lesion but no predicted contours for some GT in some slices

In summary, our research highlights the usefulness of employing a range of
distance metrics for detecting inferior or unsatisfactory segmentation
predictions in organ-at-risk segmentation, which can enhance the performance
of models and facilitate model failure analysis in medical imaging. It should
be noted that the appropriate threshold would vary for different organs, and
in respect of the size of the organs and contrast and other factors it could
change.

In order to evaluate the performance of our proposed approach, First we used
the Table 1 values to identify good cases with good overlap and less out of
box prediction. Then, we select randomly 12 cases to visually inspect the
generated contours. In this step we could see that out of 12 cases which were
randomly selected by this approach, the selected ones have less anomaly inside
the predicted contours.

As you can see in Table 1 to evaluate the model’s performance to detect the
anomalies in the predicted contours, we have considered different situations
based on the combination values of Hausdorff Distance and Dice coefficient.
When Hausdorff distance is less than 6mm and the Dice coefficient is around
0.9, we consider the prediction to be a good one. In the case of Hausdorff
between 6 and 100 and the Dice coefficient around 0.9, we have identified the
prediction as having a good overlap with the ground truth contours but also
having over segmentation because of high value in HD. For Hausdorff distance
greater than 6 and the Dice coefficient around 0.9, we acknowledge the
prediction as a good one, but with extra predictions located far from the
respective organ. Moreover, when Hausdorff is greater than 6 and the Dice
coefficient is less than 0.8, we determine that there are over segmentation in
slices or far from the organ and there is lack of overlap between ground truth
and prediction in some of the slices. Finally, when HD is less than 6 and Dice
also is less than 0.8 percent then there are less predicted values for GT and
less over segmentation because the HD value is small.

Table 2: Thresholds for Hausdorff distance and Dice score for organ segmentation. The table lists the recommended thresholds for breast and lymph node segmentation based on Hausdorff distance and Dice score metrics. The values are based on expert review of a sample dataset. Organ | Dice score | Hausdorff Distance
---|---|---
Breast | 0.9 | 6 mm
Lymph Node | 0.95 | 0.5 mm
Femur | 0.9 | 11 mm
Trachea | 0.85 | 19 mm
Chiasma | 0.66 | 3.4 mm
Brainstem | 0.8 | 10 mm
Figure 5:  The figure shows six contoured organs in different boxes, including
brainstem, chiasma, breast, trachea, lymph node and femur. These organs are
important in radiation therapy planning as they are at risk of damage during
treatment. Accurate segmentation of these organs is essential for successful
treatment, and auto-contouring models are increasingly being used for this
purpose.

As it is showed in fig 5 we visualized different organs and showing their
inaccurate prediction that needs to find automatically by our method. We
analyzed 20 CTs of 6 organs and we could find threshold values for Dice and HD
for these organs as you can see in Table 2 for respective organs. Then we
compared 12 cases of each organs which chosen randomly by setting this
threshold and compare them visually to find out if this method could
distinguish the candidate failure cases.

##  4 Conclusion

In conclusion, our paper proposed a method for identifying potential failure
cases in organ at risk segmentation using distance metrics, which is a
combination of Dice and Hausdorff distances. By analyzing 20 cases of six
different organs and visually comparing the ground truth and predicted
contours, we were able to determine a threshold for each organ that could
potentially decrease the time required for identifying failure cases. Our
findings suggest that this approach could be useful for improving the
performance of auto contouring models in radiation therapy planning in order
to speed up selecting failure cases for post-processing. In future research,
we plan to explore the use of this method for anomaly detection after model
training and for improving the model performance during clinical decision-
making.

##  5 Acknowledgments

We would like to thank all the contributors from the Deep Learning Auto
Segmentation (DLAS) project for facilitating this study.

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