59 233

# Local Gamma Augmentation for Ischemic Stroke Lesion Segmentation on MRI

Jon Middleton† Corresponding Author<EMAIL_ADDRESS>Department of Computer
Science, University of Copenhagen  Cerebriu A/S  Marko Bauer Equal
contribution. Cerebriu A/S  Jacob Johansen  Department of Computer Science,
University of Copenhagen  Cerebriu A/S  Mathias Perslev  Cerebriu A/S  Kaining
Sheng  Department of Computer Science, University of Copenhagen  Cerebriu A/S
Department of Diagnostic Radiology, Rigshospitalet, Copenhagen, Denmark
Silvia Ingala  Cerebriu A/S  Department of Diagnostic Radiology,
Rigshospitalet, Copenhagen, Denmark  Mads Nielsen  Department of Computer
Science, University of Copenhagen  Pioneer Centre for Artificial Intelligence,
University of Copenhagen  Cerebriu A/S  Akshay Pai  Cerebriu A/S  Department
of Diagnostic Radiology, Rigshospitalet, Copenhagen, Denmark

###### Abstract

The identification and localisation of pathological tissues in medical images
continues to command much attention among deep learning practitioners. When
trained on abundant datasets, deep neural networks can match or exceed human
performance. However, the scarcity of annotated data complicates the training
of these models. Data augmentation techniques can compensate for a lack of
training samples. However, many commonly used augmentation methods can fail to
provide meaningful samples during model fitting. We present local gamma
augmentation, a technique for introducing new instances of intensities in
pathological tissues. We leverage local gamma augmentation to compensate for a
bias in intensities corresponding to ischemic stroke lesions in human brain
MRIs. On three datasets, we show how local gamma augmentation can improve the
image-level sensitivity of a deep neural network tasked with ischemic stroke
lesion segmentation on magnetic resonance images.

##  1 Introduction

The potential of deep neural networks to identify and localise pathological
tissues in medical images continues to motivate research into novel training
methods. In theory, deep architectures can achieve arbitrarily high levels of
performance. In practice, however, these algorithms are trained on
insufficient quantities of data. Overfitting arises as a result, and the full
potential of these algorithms remains unrealised.

Traditional approaches to the data scarcity problem use data augmentation to
expand the size of the dataset. A typical setup selects a collection of data-
agnostic operations which come from parameterized families of spatial and
intensity transformations. Examples include affine transformations, elastic
transformations, global gamma transformations, additive and multiplicative
noise transformations, blurring transformations, and brightness
transformations. An unlimited collection of novel samples can then be obtained
by fixing probability distributions on the parameter spaces and sampling
augmentation parameters from them.

Though a dataset can be enlarged in this fashion, the approach may not provide
relevant samples for image models tasked with classifying or segmenting
regions of interest. Moreover, the augmentations themselves are not guaranteed
to have an appreciable impact on model training. For example, the literature
often claims that elastic deformations improve deep neural network performance
[4, 1, 3, 2]. Nonetheless, these transformations do not appear in all well-
known augmentation pipelines (for example, [5]), and some research indicates
that these transformations can impair a network’s performance [7, 6]. The
latter examples fit into the claim in [8] that the effectiveness of a data
augmentation pipeline depends crucially on the content of a dataset.

Thus, for models tasked with identifying and localising pathological tissues
in medical images, practitioners are developing augmentation techniques that
target these tissues. Recent work [4, 10, 9, 6, 11] has focused on local
augmentations that expose deep networks to additional examples by isolating
and altering regions of interest.

We contribute a simple, targeted method of local data augmentation for a
pathology segmentation model and apply this method to the problem of ischemic
stroke lesion segmentation on multi-modal magnetic resonance images. The
method exploits both the single-parameter family of gamma transformations and
ischemic stroke lesion segmentation maps to mitigate an intensity bias in the
training data. We compare local gamma augmentation against a baseline without
it, and we show an improvement in image-level sensitivity as a result.

Figure 1: Depiction of local gamma augmentation. An image $I(x)$ and its gamma
transformation $I_{\gamma}(x)$ are multiplied pointwise with a segmentation
map and combined to produce an image $I_{M,\gamma}(x)$ featuring an
augmentation of a pathology.

##  2 Related Work

Gamma transformations frequently appear in probabilistic data augmentation
pipelines for deep segmentation networks. The training pipeline for nnU-Net
[5] uniformly samples parameters for both gamma compression and gamma
expansion, with a bias toward gamma expansion parameters. A similar gamma
augmentation scheme for a tumour segmentation network appears in [3], with
equal weighting of gamma compression and gamma expansion. In [12], random
channel-wise gamma augmentations are applied to RGB images of retinal vessels,
with a strong bias toward gamma expansion. In [13], random gamma compressions
augment the saturation and value channels of HSV images of retinal vessels. In
all cases, the gamma transformations are applied to the whole image and are
uninformed by any features of the dataset.

Other applications for global gamma augmentations include generative modeling,
out-of-distribution benchmarking, and unsupervised domain adaptation. In [14],
log-normally distributed gamma transformations assist in the construction of
synthetic MRI training data for a segmentation network. In [15], gamma
augmentations help to produce synthetic training data for a contrast-agnostic
MRI-to-CT generation network. In [16], the intensity distribution shifts
induced by gamma transformations are used to assess the robustness of image
segmentation networks. Moreover, the authors observe that gamma compression
impairs a segmentation network’s ability to localise white-matter
hyperintensities. In [17], a GAN framework is trained to simulate global
intensity distribution shifts using instance-specific gamma transformations.

Local image augmentations arise in many training schemes. CarveMix [11]
augments MRI training data by inserting pathologies from one image into a
second image. Moreover, for brain tumours, the authors use deformable image
registration to generate plausible intracranial mass effects. CarveMix is also
used in [4] to augment MRIs featuring multiple sclerosis lesions. Mask-based
augmentations which corrupt image data, such as random erasing [18] and CutOut
[19], have been explored in medical image analysis scenarios and show mixed
results. The authors in [6] show that random erasing, which replaces random
image patches with independent and identically distributed point intensities,
impairs the performance of a melanoma classification model. In [9], a white
matter tract segmentation algorithm is fine-tuned on MRIs in which random
subsets of white matter tracts are cut out. In [10], the authors observe that
masked image modelling can improve the performance of a prostate lesion
classifier.

##  3 Methods

###  3.1 Gamma transformations

Let $I(x)$ be an image with intensity values in the range $[0,1]$. A gamma
transformation is defined by a choice of $\gamma>0$ via $I(x)\rightarrow
I(x)^{\gamma}.$ Thus a gamma transformation yields a pointwise, non-linear
transformation of $I$.

Gamma transformations generalize to images with arbitrary values by
conjugation with min-max normalization:

$I(x)\longrightarrow(m_{2}-m_{1})\left(\frac{I(x)-m_{1}}{m_{2}-m_{1}}\right)^{\gamma}+m_{1},$
(1)

where $m_{1}=\min_{x}I(x)$ and $m_{2}=\max_{x}I(x)$.

A choice of $\gamma$ between 0 and 1 gives a gamma transformation which
increases intensities and is called a gamma compression. For $\gamma$ greater
than 1, the defined transformation decreases intensities and is called gamma
expansion.

Figure 2: Gamma transformations of a diffusion weighted image from the
ISLES-2022 dataset. The center column shows the original image. The top row
contains globally gamma compressed images to the left and globally gamma
expanded images to the right. The bottom row consists of the same gamma
transformations restricted to the location of an acute ischemic stroke.

###  3.2 Local gamma augmentations

To adapt gamma transformations to a local region of interest, we alter the
min-max normalization in equation 1 as follows. Given a binary function
$M(x)$, let $S=\\{x:M(x)=1\\}$ be the set of foreground points of $M$. Set
$\tilde{m}_{1}=\min_{x}S$ and $\tilde{m}_{2}=\max_{x}S$. Then the map

$I_{\gamma}(x)=(\tilde{m}_{2}-\tilde{m}_{1})\left(\frac{I(x)-\tilde{m}_{1}}{\tilde{m}_{2}-\tilde{m}_{1}}\right)^{\gamma}+\tilde{m}_{1}$
(2)

applies a non-linear transformation which is normalized relative to the
foreground $S$.

We then complete the local gamma transformation by using $M(x)$ as a mask to
mix intensity-augmented pathological tissue in $S$ with normal tissue outside
$S$:

$I_{M,\gamma}(x)=(1-M(x))\odot I(x)+M(x)\odot I_{\gamma}(x),$ (3)

where $\odot$ denotes pointwise multiplication. Local gamma augmentation is
depicted in Figure 1, with a segmentation map indicating diseased brain
tissue.

When used to augment image data, gamma transformations are randomly chosen by
sampling $\gamma$ from a probability distribution. Typical choices include the
log-normal distribution for a weakly constrained value of $\gamma$ or a Beta
distribution supported in a fixed interval. Rather than sampling gamma
parameters as in [5], in which $\gamma$ is drawn from the uniform distribution
on the interval $[0.7,1.5]$, we instead sample from a mixture of uniform
distributions:

$\gamma\sim\frac{1}{2}\,U(0.7,1.0)+\frac{1}{2}\,U(1.0,1.5).$ (4)

Thus we effectively partition this interval into a half-open gamma compression
interval $[0.7,1.0)$ and a closed gamma expansion interval $[1.0,1.5]$. During
each choice of $\gamma$, one of the two intervals is selected with equal
probability.

Figure 2 depicts various local and global gamma transformations applied to a
diffusion-weighted image from the ISLES-2022 dataset. In this image, one can
see a clear increase in contrast between healthy brain tissue and diseased
brain tissue for $\gamma<1$, whereas this contrast is decreased for
$\gamma>1$. This differs from global gamma transformations, in which the local
contrast between healthy and pathological tissue remains.

##  4 Data

###  4.1 Training Data

The models were trained on a multi-site in-house brain MRI dataset consisting
of 1308 studies, comprising of 420 ischemic strokes, 265 tumors and 234
hemorrhages, as well as 530 normal studies without abnormalities. Each study
consisted of three MRI sequences: fluid-attenuated inversion recovery (FLAIR),
diffusion-weighted imaging (b0, b1000, and ADC), and either susceptibility-
weighted imaging (SWI) or T2* gradient echo (T2*GRE). These sequences were
annotated by an in-house medical team and verified by a certified
neuroradiologist.

Medical experts classify presentations of ischemic stroke on MRI according to
their age [20]. The various classes include _hyperacute_ ischemic stroke,
which has the shortest time window from stroke onset. When imaged sufficiently
early, hyperacute ischemic stroke lesions appear hyperintense in a DWI
sequence and isointense in a FLAIR sequence [21]. Notably, the MRIs in our
training set underrepresent hyperacute ischemic strokes, as shown in Table 1.
This yields a bias in ischemic stroke signal intensities between the DWI and
FLAIR sequences.

In addition, we include two publicly available datasets for training. These
include the Multimodal Brain Tumor Segmentation (BraTS) [23, 24, 22] Challenge
2019 and Ischemic Stroke Lesion Segmentation (ISLES) Challenge 2022 datasets
[25, 26].

Pathology | Type | Count
---|---|---
Ischemic Stroke | Acute/Subacute | 342
Hyperacute | 15
Hemorrhagic | 63
Hemorrhage | Intra-Axial | 137
Extra-Axial | 97
Tumor | Intra-Axial | 95
Extra-Axial | 170
Infection | Intra-Axial | 27
Extra-Axial | 2
Table 1: Representation of tissue pathologies in training data. Of note is the
relative under-representation of hyperacute infarcts.

All sequences were acquired in different resolutions and acquisition planes.
Accordingly, as part of the pre-processing step, DWI sequences were resampled
to isotropic resolution, while FLAIR and SWI/T2* GRE sequences were resampled
and co-registered to DWI space.

| MVS | RH | WUS
---|---|---|---
| Sensitivity | Specificity | Sensitivity | Specificity | Sensitivity | Specificity
U-Net | 0.781 | 0.928 | 0.757 | 0.899 | 0.566 | N/A
U-Net+LG | 0.844 | 0.910 | 0.814 | 0.895 | 0.736 | N/A
Table 2: Image-level sensitivity and specificity for ischemic stroke detection
across three MRI datasets for a baseline U-Net versus a U-Net trained with
local gamma augmentation. Specificity for the WUS dataset is non-applicable
since all samples in the dataset contain ischemic strokes.

###  4.2 Test Data

Our methods were evaluated on three independent private datasets, yielding a
train-test split that generally arises in a real-world scenario, with a
mixture of public and private data for the training set and private data
entirely sourced from external sites for testing. The test data consist of DWI
(b0 and b1000), FLAIR, and ADC sequences for each patient. At training time,
the test data consisted of image-level labels only, rather than the pointwise
annotations that are customarily supplied to a segmentation model. The three
private datasets are elaborated below:

MVS. A multi-site dataset consisting of 387 studies obtained with Siemens
scanners, comprising of 139 ischemic strokes, 93 tumors, 76 hemorrhages, as
well as 98 studies with no abnormalities, which were annotated on image level
by two neuroradiologists.

RH. A consecutively sampled dataset of 262 acquired brain MRI scans in a
routine setting from Rigshospitalet (RH) in Copenhagen, Denmark. The dataset
has further been consecutively enriched with positive findings of ischemic
strokes (hyperacute, acute, and subacute), brain hemorrhage (intra-axial and
extra-axial) and brain tumors (intra-axial and extra-axial) from three other
hospital sites in the Capital Region of Denmark. This resulted in a multi-site
cohort consisting of 487 scans including 96 ischemic strokes, 80 tumors, 76
hemorrhages, as well as 162 completely normal scans. Furthermore, 73 scans
have significant findings that are not attributed to ischemic stroke,
hemorrhage, or tumors, e.g., inflammatory disease, pineal cysts, aneurysms,
and cavernous malformations. The pathology type of the scans in the dataset
was classified by a medical doctor with 3 years’ experience in brain imaging
using original radiological report impressions as reference.

WUS. A collection of brain MRI scans featuring 51 patients who awoke with
symptoms of ischemic stroke which were not present before falling asleep. This
_wake-up stroke_ dataset contains two distinct consecutively sampled cohorts,
assembled using distinct stroke protocols, The first cohort contains 27 adult
patients, and the second cohort consists of 24 adult patients. Both cohorts
feature visible stroke lesions on the DWI sequence for all patients, as
patients without visible stroke lesions were excluded. Of the 51 patients, 16
were determined to have a mismatch in ischemic stroke signal intensities
between the DWI sequence and the FLAIR sequence. All studies were annotated on
an image level according to patient reports.

##  5 Experiments and Results

###  5.1 Experimental Setup

All experiments used one 40GB Nvidia A100 Tensor Core GPU for model training.
The model architecture largely follows the implementation in [27]. The
architecture has a depth of five and incorporates padded convolution, instance
normalization, and ReLU activations.

Models were trained for 500 epochs with a batch size of 2 and 4-step gradient
accumulation on randomly extracted $128\times 128\times 128$ patches, with
deep supervision, combining cross-entropy and generalized dice loss, and using
an Adam optimizer with an initial learning rate of 0.0001 and a polynomial-
rate schedule.

We trained two U-Net models: a baseline U-Net trained without local gamma
augmentation, and a U-Net trained with local gamma augmentations applied to
DWI sequences, which we call U-Net+LG. Augmentations for _both_ models
include: Rician noise, Gaussian noise, Gaussian blurring, contrast
transformations, brightness transformations, global gamma correction, bias
field correction, histogram equalization, elastic deformation, rotation,
scaling, mirroring, random channel shifts and ghosting.

Of note is that a local gamma augmentation scheme as described in Section 3.2
does not account for images without a pathology. Such images are paired with a
segmentation map $M(x)$ which is identically zero. Such an $M(x)$ reduces
Equation 3 to the identity map. To avoid this reduction for such images, we
set $M(x)=1$, so that a global gamma transformation can be applied to the
patch.

###  5.2 Evaluation

We used image-level sensitivity and image-level specificity to evaluate the
performance of a baseline U-Net and a local gamma augmented U-Net+LG on an
ischemic stroke classification task. As segmentation models, U-Nets are
typically evaluated with pointwise metrics such as the Dice-Sørensen
coefficient or the Jaccard index. Our method of model evaluation, namely
classification instead of segmentation, is required due to the lack of
segmentation maps as annotations for the data set. To treat a segmentation
model as a classification model, we define a U-Net prediction to be positive
if at least one voxel in the model’s output is predicted as corresponding to
ischemic stroke.

###  5.3 Results

Table 2 summarises our results. We observe an increase in image-level
sensitivity from 0.781 to 0.844 for the MVS dataset and an increase in image-
level sensitivity from 0.757 to 0.814 for the RH dataset. These increases are
accompanied with small reductions in image-level specificity on both datasets.

For the WUS dataset, the improvement is even greater: an increase in image-
level sensitivity from 0.566 to 0.736. We speculate that this larger increase
in sensitivity is due to the influence of local gamma augmentation. In
particular, the local gamma augmentation scheme exposes U-Net+LG to more
examples of MRIs with signal differences between the DWI sequence and the
remaining sequences. We also mention that specificity is not reported for WUS:
a specificity score for this dataset is meaningless since every image contains
an ischemic stroke.

###  5.4 Discussion

The simplicity of local gamma augmentation, namely its use of a one-parameter
family of intensity transformations for data augmentation, means that it is
very straightforward to implement and hence is amenable to further
exploration. One potential research avenue would be to identify better choices
of the gamma augmentation parameter. Our method uses a mixture distribution to
select augmentation parameters, but this distribution is fixed. In this way,
the chosen augmentation parameters are data-independent. Other research
avenues could pursue the use of targeted gamma augmentations for other
applications, such as adversarial training or pseudo-healthy synthesis.

A serious limitation of the method is its dependence on pointwise labels:
without segmentation maps, only image-level gamma augmentations can be
applied. Another limitation is the dependence on pathological tissues which
are characterized by intensity differences relative to normal tissue. Not all
brain diseases are characterized by intensity differences. Hence it is
questionable that local gamma augmentation (or local intensity augmentation of
any kind) would be effective.

##  6 Conclusion

We presented local gamma augmentation, an intensity-based method of data
augmentation which, in a supervised setting, can be used to target
pathological tissues in human brain MRIs. We applied this method to compensate
for an intensity bias in the training set due to an over-representation of
non-hyperacute ischemic strokes. This method was compared against a baseline
data augmentation method without local gamma augmentation on three MRI data
sets. The results suggest that restricting gamma transformations to ischemic
stroke lesions during training of a segmentation model can enhance image-level
sensitivity without impairing image-level specificity.

## References

  * [1] Eduardo Castro, Jaime S. Cardoso and Jose Costa Pereira  “Elastic deformations for data augmentation in breast cancer mass detection”  In _2018 IEEE EMBS Int. Conf. Biomed. Heal. Informatics_ 2018-Janua IEEE, 2018, pp. 230–234  DOI: 10.1109/BHI.2018.8333411
  * [2] Philip Novosad, Vladimir Fonov and D. Collins  “Accurate and robust segmentation of neuroanatomy in T1-weighted MRI by combining spatial priors with deep convolutional neural networks”  In _Hum. Brain Mapp._ 41.2, 2020, pp. 309–327  DOI: 10.1002/hbm.24803
  * [3] Marco Domenico Cirillo, David Abramian and Anders Eklund  “What Is the Best Data Augmentation for 3D Brain Tumor Segmentation?”  In _Proc. - Int. Conf. Image Process. ICIP_ 2021-Septe, 2021, pp. 36–40  DOI: 10.1109/ICIP42928.2021.9506328
  * [4] Berke Doga Basaran, Paul M. Matthews and Wenjia Bai  “New lesion segmentation for multiple sclerosis brain images with imaging and lesion-aware augmentation”  In _Front. Neurosci._ 16, 2022  DOI: 10.3389/fnins.2022.1007453
  * [5] Fabian Isensee, Paul F. Jaeger, Simon A.A. Kohl, Jens Petersen and Klaus H. Maier-Hein  “nnU-Net: a self-configuring method for deep learning-based biomedical image segmentation”  In _Nat. Methods_ 18.2 Springer US, 2021, pp. 203–211  DOI: 10.1038/s41592-020-01008-z
  * [6] Fábio Perez, Cristina Vasconcelos, Sandra Avila and Eduardo Valle  “Data Augmentation for Skin Lesion Analysis”  In _Lect. Notes Comput. Sci. (including Subser. Lect. Notes Artif. Intell. Lect. Notes Bioinformatics)_ 11041 LNCS, 2018, pp. 303–311  DOI: 10.1007/978-3-030-01201-4˙33
  * [7] Davood Karimi and Ali Gholipour  “Improving Calibration and Out-of-Distribution Detection in Deep Models for Medical Image Segmentation”  In _IEEE Trans. Artif. Intell._ 4.2 IEEE, 2023, pp. 383–397  DOI: 10.1109/TAI.2022.3159510
  * [8] Randall Balestriero, Leon Bottou and Yann LeCun  “The Effects of Regularization and Data Augmentation are Class Dependent”, 2022, pp. 1–21  arXiv: http://arxiv.org/abs/2204.03632
  * [9] Wan Liu, Qi Lu, Zhizheng Zhuo, Yaou Liu and Chuyang Ye  “One-Shot Segmentation of Novel White Matter Tracts via Extensive Data Augmentation”  In _Lect. Notes Comput. Sci. (including Subser. Lect. Notes Artif. Intell. Lect. Notes Bioinformatics)_ 13431 LNCS, 2022, pp. 133–142  DOI: 10.1007/978-3-031-16431-6˙13
  * [10] Alvaro Fernandez-Quilez, Christoffer Gabrielsen Andersen, Trygve Eftestøl, Svein Reidar Kjosavik and Ketil Oppedal  “3D Masked Modelling Advances Lesion Classification in Axial T2w Prostate MRI”  In _Proc. North. Light. Deep Learn. Work._ 4, 2023, pp. 1–8  DOI: 10.7557/18.6787
  * [11] Xinru Zhang, Chenghao Liu, Ni Ou, Xiangzhu Zeng, Zhizheng Zhuo, Yunyun Duan, Xiaoliang Xiong, Yizhou Yu, Zhiwen Liu, Yaou Liu and Chuyang Ye  “CarveMix: A simple data augmentation method for brain lesion segmentation”  In _Neuroimage_ 271.March, 2023  DOI: 10.1016/j.neuroimage.2023.120041
  * [12] Xu Sun, Huihui Fang, Yehui Yang, Dongwei Zhu, Lei Wang, Junwei Liu and Yanwu Xu  “Robust Retinal Vessel Segmentation from a Data Augmentation Perspective”  In _Lect. Notes Comput. Sci. (including Subser. Lect. Notes Artif. Intell. Lect. Notes Bioinformatics)_ 12970 LNCS, 2021, pp. 189–198  DOI: 10.1007/978-3-030-87000-3˙20
  * [13] Pawel Liskowski and Krzysztof Krawiec  “Segmenting Retinal Blood Vessels with Deep Neural Networks”  In _IEEE Trans. Med. Imaging_ 35.11, 2016, pp. 2369–2380  DOI: 10.1109/TMI.2016.2546227
  * [14] Benjamin Billot, Douglas N. Greve, Oula Puonti, Axel Thielscher, Koen Van Leemput, Bruce Fischl, Adrian V. Dalca and Juan Eugenio Iglesias  “SynthSeg: Segmentation of brain MRI scans of any contrast and resolution without retraining”  In _Med. Image Anal._ 86, 2023, pp. 102789  DOI: 10.1016/j.media.2023.102789
  * [15] Lotte Nijskens, Cornelis, AT Berg, Joost JC Verhoeff and Matteo Maspero  “Exploring contrast generalisation in deep learning-based brain MRI-to-CT synthesis”  In _Phys. Medica_ 112.May Elsevier Ltd, 2023, pp. 102642  DOI: 10.1016/j.ejmp.2023.102642
  * [16] Lyndon Boone, Mahdi Biparva, Parisa Mojiri Forooshani, Joel Ramirez, Mario Masellis, Robert Bartha, Sean Symons, Stephen Strother, Sandra E. Black, Chris Heyn, Anne L. Martel, Richard H. Swartz and Maged Goubran  “ROOD-MRI: Benchmarking the robustness of deep learning segmentation models to out-of-distribution and corrupted data in MRI”  In _Neuroimage_ 278.April Elsevier Inc., 2023, pp. 120289  DOI: 10.1016/j.neuroimage.2023.120289
  * [17] Pengfei Diao, Akshay Pai, Christian Igel and Christian Hedeager Krag  “Histogram-Based Unsupervised Domain Adaptation for Medical Image Classification”  Springer Nature Switzerland, 2022, pp. 755–764  DOI: 10.1007/978-3-031-16449-1˙72
  * [18] Zhun Zhong, Liang Zheng, Guoliang Kang, Shaozi Li and Yi Yang  “Random Erasing Data Augmentation”  In _Proc. AAAI Conf. Artif. Intell._ 34.07, 2020, pp. 13001–13008  DOI: 10.1609/aaai.v34i07.7000
  * [19] Terrance DeVries and Graham W. Taylor  “Improved Regularization of Convolutional Neural Networks with Cutout”  In _arXiv_ , 2017  arXiv: http://arxiv.org/abs/1708.04552
  * [20] Julie Bernhardt, Kathryn S. Hayward, Gert Kwakkel, Nick S. Ward, Steven L. Wolf, Karen Borschmann, John W. Krakauer, Lara A. Boyd, S. Carmichael, Dale Corbett and Steven C. Cramer  “Agreed definitions and a shared vision for new standards in stroke recovery research: The Stroke Recovery and Rehabilitation Roundtable taskforce”  In _Int. J. Stroke_ 12.5, 2017, pp. 444–450  DOI: 10.1177/1747493017711816
  * [21] Götz Thomalla, Bastian Cheng, Martin Ebinger, Qing Hao, Thomas Tourdias, Ona Wu, Jong S. Kim, Lorenz Breuer, Oliver C. Singer, Steven Warach, Soren Christensen, Andras Treszl, Nils D. Forkert, Ivana Galinovic, Michael Rosenkranz, Tobias Engelhorn, Martin Köhrmann, Matthias Endres, Dong-Wha Kang, Vincent Dousset, A. Sorensen, David S. Liebeskind, Jochen B. Fiebach, Jens Fiehler and Christian Gerloff  “DWI-FLAIR mismatch for the identification of patients with acute ischaemic stroke within 4·5 h of symptom onset (PRE-FLAIR): a multicentre observational study”  In _Lancet Neurol._ 10.11 Elsevier Ltd, 2011, pp. 978–986  DOI: 10.1016/S1474-4422(11)70192-2
  * [22] Bjoern H. Menze, Andras Jakab, Stefan Bauer, Jayashree Kalpathy-Cramer, Keyvan Farahani, Justin Kirby and al.  “The Multimodal Brain Tumor Image Segmentation Benchmark (BRATS)”  In _IEEE Trans. Med. Imaging_ 34.10, 2015, pp. 1993–2024  DOI: 10.1109/TMI.2014.2377694
  * [23] Spyridon Bakas, Hamed Akbari, Aristeidis Sotiras, Michel Bilello, Martin Rozycki, Justin S. Kirby and al.  “Advancing The Cancer Genome Atlas glioma MRI collections with expert segmentation labels and radiomic features”  In _Sci. Data_ 4.July The Author(s), 2017, pp. 1–13  DOI: 10.1038/sdata.2017.117
  * [24] Spyridon Bakas, Mauricio Reyes, Andras Jakab, Stefan Bauer, Markus Rempfler, Alessandro Crimi and al.  “Identifying the Best Machine Learning Algorithms for Brain Tumor Segmentation, Progression Assessment, and Overall Survival Prediction in the BRATS Challenge”, 2018  arXiv: http://arxiv.org/abs/1811.02629
  * [25] Carlo W. Cereda, Søren Christensen, Bruce C.V. Campbell, Nishant K. Mishra, Michael Mlynash, Christopher Levi, Matus Straka, Max Wintermark, Roland Bammer, Gregory W. Albers, Mark W. Parsons and Maarten G. Lansberg  “A benchmarking tool to evaluate computer tomography perfusion infarct core predictions against a DWI standard”  In _J. Cereb. Blood Flow Metab._ 36.10, 2016, pp. 1780–1789  DOI: 10.1177/0271678X15610586
  * [26] Arsany Hakim, Søren Christensen, Stefan Winzeck, Maarten G. Lansberg, Mark W. Parsons, Christian Lucas, David Robben, Roland Wiest, Mauricio Reyes and Greg Zaharchuk  “Predicting Infarct Core From Computed Tomography Perfusion in Acute Ischemia With Machine Learning: Lessons From the ISLES Challenge”  In _Stroke_ 52.7, 2021, pp. 2328–2337  DOI: 10.1161/STROKEAHA.120.030696
  * [27] Olaf Ronneberger, Philipp Fischer and Thomas Brox  “U-Net: Convolutional Networks for Biomedical Image Segmentation”  In _Lect. Notes Comput. Sci._ 9351 Springer International Publishing, 2015, pp. 234–241  DOI: 10.1007/978-3-319-24574-4˙28


$q(x\oplus\xi\oplus t)~{}=~{}\xi(x)+t^{2},\hskip 14.22636pt\forall x\in
V,\xi\in V^{*},t\in\mathbb{F}^{1/2}.$

The polar space ${\cal S}_{q}$ defined by $q$ in $\mathrm{PG}(\widetilde{V})$
is isomorphic to ${\cal S}_{f}$, the isomorphism being induced by the linear
mapping from $\overline{V}$ to $\widetilde{V}$ which maps every vector
$x\oplus\xi\in\overline{V}$ onto the $\phi$-singular vector
$x\oplus\xi\oplus\xi(x)^{1/2}\in\widetilde{V}$. This isomorphism yields the
universal embedding $\tilde{\varepsilon}:{\cal
S}_{f}\rightarrow\mathrm{PG}(\widetilde{V})$ of ${\cal S}_{f}$. The subspace
${\cal Q}=\langle[V],[V^{*}]\rangle$ of ${\cal S}_{f}$ is mapped by
$\tilde{\varepsilon}$ onto $\tilde{\varepsilon}({\cal
S}_{q})\cap[\overline{V}\oplus\\{0\\}]$.

###  5.2 Hyperbolic quadrics of infinite rank

Without assuming any hypothesis on $\mathbb{F}$, let
$q:\overline{V}\rightarrow\mathbb{F}$ be the non-degenerate quadratic form
defined on $\overline{V}$ as follows:

$q(a\oplus\alpha)~{}=~{}\alpha(a),\hskip 14.22636pt\forall a\in V,~{}\alpha\in
V^{*}.$ (19)

The bilinearized of $q$ is the symmetric bilinear form $f_{q}$ defined as
follows:

$f_{q}(a\oplus\alpha,b\oplus\beta)~{}=~{}\alpha(b)+\beta(a),\hskip
14.22636pt\forall a,b\in V,~{}\alpha,\beta\in V^{*}.$ (20)

Let ${\cal S}_{q}$ be the polar space defined by $q$. Note that $f_{q}$ non-
degenerate, no matter what $\mathrm{char}(\mathbb{F})$ is. Hence the inclusion
mapping of ${\cal S}_{q}$ in $\mathrm{PG}(\overline{V})$ is the unique
embedding of ${\cal S}_{q}$. As in Section 5.1, the subspaces $[V]$ and
$[V^{*}]$ are opposite generators of ${\cal S}_{q}$ and span
$\mathrm{PG}(\overline{V})$. Hence the embedding of ${\cal S}_{q}$ in
$\mathrm{PG}(\overline{V})$ (which is the unique embedding of ${\cal S}_{q}$)
is tight.

###### Remark 12

When $\mathrm{char}(\mathbb{F})=2$ the form $f_{q}$ is alternating and, with
$f=f_{q}$, the polar space ${\cal S}_{q}$ is the same as the subspace of
${\cal S}_{f}$ called ${\cal Q}$ in Section 5.1.2. As noticed at the end of
Section 5.1.2, the space ${\cal Q}$ admits an embedding in
$\mathrm{PG}(\overline{V}\oplus\\{0\\})$ which however, since ${\cal S}_{q}$
admits a unique embedding, is isomorphic to the natural embedding of ${\cal
Q}={\cal S}_{q}$ in $\mathrm{PG}(\overline{V})$.

###### Lemma 5.5

For every line $\ell$ of $\mathrm{PG}(\overline{V})$, if $|\ell\cap{\cal
S}_{q}|>2$ then $\ell\subseteq{\cal S}_{q}$.

Proof. Let $[a\oplus\alpha],[b\oplus\beta]$ and $[c\oplus\gamma]$ be three
distinct points of ${\cal S}_{q}$ on the same line $\ell$ of
$\mathrm{PG}(\overline{V})$. We can assume that
$c\oplus\gamma=a\oplus\alpha+b\oplus\beta$. By assumption
$\alpha(a)=\beta(b)=\gamma(c)=0$. Hence $\alpha(b)+\beta(a)=0$, namely
$f_{q}(a\oplus\alpha,b\oplus\beta)=0$. Therefore
$[a\oplus\alpha]\perp_{f_{q}}[b\oplus\beta]$. Hence $\ell$ is totally
$q$-singular. $\Box$

###### Proposition 5.6

The hyperbolic lines of ${\cal S}_{q}$ have size 2.

Proof. As $f_{q}$ is non-degenerate, every hyperbolic line of ${\cal S}_{q}$
is contained in a projective line of $\mathrm{PG}(\overline{V})$. Lemma 5.5
yields the conclusion. $\Box$

All we have said in Section 5.1.1 on singular subspaces of ${\cal S}_{f}$ is
word by word valid for singular subspaces of ${\cal S}_{q}$, but for replacing
the last condition of (15) wth the following: $p_{1,2}(x)=0$ for every $x\in
C_{1}$. In particular, the generators of ${\cal S}_{q}$ are still
characterized by the two conditions of (16). We have $[W]^{\perp_{f_{q}}}=[W]$
for every generator $[W]$ of ${\cal S}_{q}$ and the analogue of Proposition
5.3 holds: every generator of ${\cal S}_{q}$ admits a complement. Note also
that, since the inclusion mapping of ${\cal S}_{q}$ in
$\mathrm{PG}(\overline{V})$ is the unique embedding of ${\cal S}_{q}$, two
opposite generators of ${\cal S}_{q}$ are complementary only if their union
generates ${\cal S}_{q}$.

###### Lemma 5.7

If a plane $P$ of $\mathrm{PG}(\overline{V})$ contains three distinct lines of
${\cal S}_{q}$, then $P\subseteq{\cal S}_{q}$.

Proof. Let $P$ be a plane of $\mathrm{PG}(\overline{V})$ containing three
distinct lines of ${\cal S}_{q}$. Every line of $P$ meets each of those three
lines non trivially and, if meets no two of them in the same point, then it is
fully contained in ${\cal S}_{q}$ by Lemma 5.5. Consequently, $P\subseteq{\cal
S}_{q}$. $\Box$

###### Proposition 5.8

All non-deep sub-generators of ${\cal S}_{q}$ are hyperbolic.

Proof. Let $N$ be a sub-generator of ${\cal S}_{q}$ and suppose that
$N^{\perp}$ contains two opposite points of ${\cal S}_{q}$. Since
$\mathrm{Aut}({\cal S}_{q})$ acts transitively on the pairs of opposite points
of ${\cal S}_{q}$, we can safely assume that $N^{\perp}$ contains $[a]$ and
$[\alpha]$ for a vector $a\in V$ and a linear functional $\alpha\in V^{*}$
such that $\alpha(a)=1$. We shall prove that $\langle N,[a]\rangle$ and
$\langle N,[\alpha]\rangle$ are the unique generators of ${\cal S}_{q}$ which
contain $N$. Put $V_{0}:=\ker(\alpha)$. Clearly, $\ker^{*}(a)\cong
V_{0}^{\ast}$. Freely regarding $V^{*}_{0}$ as the same as $\ker^{*}(a)$, put
$\overline{V}_{0}:=V_{0}\oplus V_{0}^{*}=\ker(\alpha)\oplus\ker^{*}(a)$. So,
$N=[W]$ for a maximal totally $q$-singular subspace $W$ of $\overline{V}_{0}$.

For a contradiction, let $\langle N,[b\oplus\beta]\rangle$ be a generator of
${\cal S}_{q}$ containing $N$ and different from both $\langle N,[a]\rangle$
and $\langle N,[\alpha]\rangle$. The points $[a]$, $[\alpha]$ and
$[b\oplus\beta]$ are mutually non collinear in ${\cal S}_{q}$. Hence they span
a plane $P$ of $\mathrm{PG}(\overline{V})$ contained in $N^{\perp_{f_{q}}}$
and $P\cap N=\emptyset$ by Lemma 5.7. However,
$\\{[a],[\alpha]\\}^{\perp_{f_{q}}}=[\overline{V}_{0}]$ has codimension 2 in
$\mathrm{PG}(\overline{V})$ and the line of $\mathrm{PG}(\overline{V})$
spanned by $[a]$ and $[\alpha]$ is a complement of
$\\{[a],[\alpha]\\}^{\perp_{f_{q}}}$. Hence $P$ meets
$\\{[a],[\alpha]\\}^{\perp_{f_{q}}}$ non trivially. Let $[c\oplus\gamma]$ be a
point of $P\cap\\{[a],[\alpha]\\}^{\perp_{f_{q}}}$. Then $c\oplus\gamma\in
W^{\perp_{f_{q}}}\cap\overline{V}_{0}$. As in the final part of the proof of
Lemma 5.2, but with $V,V^{*}$ and $\overline{V}$ replaced by $V_{0},V_{0}^{*}$
and $\overline{V}_{0}$ respecttively, we obtain that $c\oplus\gamma\in W$.
This contradicts the fact that, as previously noticed, $P\cap N=\emptyset$.
$\Box$

Deep sub-generators also exists in ${\cal S}_{q}$. For instance, as in ${\cal
S}_{f}$, every hyperplane $H$ of $V^{*}$ such that $\ker(H)=\\{0\\}$ yields a
deep sub-generator $[H]$ of ${\cal S}_{q}$.

###### Corollary 5.9

The polar space ${\cal S}_{q}$ is regular.

Proof. By Proposition 5.6, the space ${\cal S}_{q}$ is hyperbolic (Definition
4.4). All hyperbolic polar spaces are regular. $\Box$

As noticed at the end of Section 5.1.1, opposite generators $[W]$ and
$[W^{\prime}]$ of ${\cal S}_{f}$ exist which are not complementry. The same
occurs in ${\cal S}_{q}$, as we can see by a slight modification of Example
5.4.

###### Example 5.10

Given a basis $(e_{i})_{I\in\mathfrak{I}}$ of $V$, choose the linear
functionals $\eta_{i}$ in such a way that $\eta_{i}(e_{i})=0$ for every $i$
and $\eta_{i}(e_{j})+\eta_{j}(e_{i})=0$ for any choice of
$i,j\in\mathfrak{I}$. To this goal we can use the following trick. Let $P$ be
a partition of $\mathfrak{I}$ in subsets of size 2 and define two sets $P^{+}$
and $P^{-}$ of ordered pairs as follows: for every pair $\\{i,j\\}\in P$,
choose one of the two ordered pairs $(i,j)$ and $(j,i)$ and put it in $P^{+}$,
putting the other one in $P^{-}$. Then define $\eta_{j}(e_{i})=0$ if
$\\{i,j\\}\not\in P$ (in particular, if $i=j$), $\eta_{j}(e_{i})=1$ if
$(i,j)\in P^{+}$ and $\eta_{j}(e_{i})=-1$ if $(i,j)\in P^{-}$.

As in Example 5.4, the vectors $e_{i}+\eta_{i}$ span a subspace $W$ of
$\overline{V}$. The corresponding subspace $[W]$ of
$\mathrm{PG}(\overline{V})$ is a generator of ${\cal S}_{q}$ opposite to $[V]$
but it is not a complement of $[V]$.

###  5.3 Regular hermitian varieties

Assume that $\mathbb{F}$ admits a separable quadratic extension and let
$\sigma$ be an involutory non-trivial automorphism of $\mathbb{F}$. Assuming
that $V$ is a right $\mathbb{F}$-vector space, its dual $V^{*}$ should be
regarded as a left vector space. However, we can re-define it as a right
vector space by a vector-times-scalar multiplication defined as follows:
$\alpha\cdot t:=t^{\sigma}\alpha$ for every $\alpha\in V^{*}$. Accordingly,
$(a\oplus\alpha)t=at\oplus\alpha t=at\oplus t^{\sigma}\alpha$. Put
$\overline{V}=V\otimes V^{*}$, with $V^{*}$ re-defined as said above. We can
define a non-degenerate $\sigma$-hermitian form
$h:\overline{V}\times\overline{V}\rightarrow\mathbb{F}$ by declaring that

$h(a\oplus\alpha,b\oplus\beta)~{}=~{}\alpha(b)+(\beta(a))^{\sigma},\hskip
14.22636pt\forall a,b\in V,~{}\alpha,\beta\in V^{*}.$ (21)

The form $h$ is non-degenerate. Let ${\cal S}_{h}$ be the polar space
associated to it. The inclusion mapping of ${\cal S}_{h}$ in
$\mathrm{PG}(\overline{V})$ is the unique embedding of ${\cal S}_{h}$. The
subspaces $[V]$ and $[V^{*}]$ are opposite generators of ${\cal S}_{h}$ and
$\langle[V],[V^{*}]\rangle={\cal S}_{h}$. Hence the unique embedding of ${\cal
S}_{h}$ is tight.

The hyperbolic lines of ${\cal S}_{h}$ are the intersections $\ell\cap{\cal
S}_{h}$ for $\ell$ a projective line of $\mathrm{PG}(\overline{V})$ which
meets ${\cal S}_{h}$ in at least two points but is not fully contained in
${\cal S}_{h}$. For instance, if $a\in V$ and $\alpha\in V^{*}$ are such that
$\alpha(a)=1$ then the hyperbolic line through $[a]$ and $[\alpha]$ consists
of $[a]$ and all points $[at\oplus\alpha]$ for $t+t^{\sigma}=0$. If a plane
$P$ of $\mathrm{PG}(\overline{V})$ contains a point $p\in{\cal S}_{h}$ such
that $p^{\perp_{h}}\supseteq P$, then either $P\subseteq{\cal S}_{h}$ of
$P\cap{\cal S}_{h}$ is the union of a set of lines through $p$ and $\ell\cap
p^{\perp_{h}}$ is a hyperbolic line for every line $\ell$ of $P$ which does
not pass through $p$. By these fact we can prove an analogue of Proposition
5.8: if $N$ is a non-deep sub-generator of ${\cal S}_{h}$ then $N^{\perp}$ is
a hyperbolic line. Consequently, ${\cal S}_{h}$ is regular.

Non-deep subgenerators also exist in ${\cal S}_{h}$, as in ${\cal S}_{f}$ and
${\cal S}_{q}$. The results of Section 5.1.1 remain valid for ${\cal S}_{h}$,
with obvious minor modifications. For instance, every generator of ${\cal
S}_{h}$ admits a complement but not all pairs of opposite generators are
complementary. Non complementary pairs of opposite generators can be obtained
by a minor modifcation of the construction described in Example 5.10.

###  5.4 A family of non-regular quadrics

The construction of Section 5.1 can be generalized by replacing $V^{*}$ with a
subspace $V^{\prime}$ of $V^{*}$ such that $\ker(V^{\prime})=0$, possibly
$V^{\prime}\cong V$. We still obtain a symplectic space. As all symplectic
spaces are regular [3], all polar spaces obtained in this way are still
regular. In contrast, if we do the same in Sections 5.2 and 5.3 then the polar
space we obtain admits a tight embedding, since it lives in
$\mathrm{PG}(V\oplus V^{\prime})$ and admits $[V]$ and $[V^{\prime}]$ as
generators, but in general it is not regular, as in the following case.

Let $\mathrm{char}(\mathbb{F})=2$. Given a basis $(e_{i})_{i\in\mathfrak{I}}$
of $V$, define the linear functionals $\eta_{i}$ as in Example 5.4. So,
$\eta_{i}(e_{j})=\delta_{i,j}$ for every choice of $i,j\in\mathfrak{I}$. Put
$V^{\prime}:=\langle\eta_{i}\rangle_{i\in\mathfrak{I}}$ and
$\overline{V}^{\prime}:=V\oplus V^{\prime}$. Clearly,
$\ker(V^{\prime})=\\{0\\}$. With $q$ and $f_{q}$ as in Section 5.2, let
$q^{\prime}$ and $f^{\prime}$ be their restrictions to $\overline{V}^{\prime}$
and $\overline{V}^{\prime}\times\overline{V}^{\prime}$. Then $f^{\prime}$ is
the sesquilinearized of $q^{\prime}$.

Explicitly, the vectors of $V\oplus V^{\prime}$ are sums $v=\sum_{i\in
I}e_{i}t_{i}\oplus\sum_{j\in J}s_{j}\eta_{j}$ for $I$ and $J$ finite (possibly
empty) subsets of $\mathfrak{I}$, with $t_{i}\neq 0\neq s_{j}$ for every $i\in
I$ and $j\in J$, by convention. With this convention, we call the ordered pair
$(I,J)$ the support of $v$. With this convention, if $v=\sum_{i\in
I}e_{i}t_{i}\oplus\sum_{j\in J}s_{j}\eta_{j}$ and $v^{\prime}=\sum_{i\in
I^{\prime}}e_{i}t^{\prime}_{i}\oplus\sum_{j\in
J^{\prime}}s^{\prime}_{j}\eta_{j}$ with supports $(I,J)$ and
$(I^{\prime},J^{\prime})$ respectively. then

$f^{\prime}(v,v^{\prime})=\sum_{i\in I\cap
J^{\prime}}t_{i}s^{\prime}_{i}+\sum_{j\in J\cap
I^{\prime}}s_{j}t^{\prime}_{j}~{}\mbox{ and }~{}q^{\prime}(v)=\sum_{i\in I\cap
J}t_{i}s_{i}.$

Clearly, $f(v,v^{\prime})=0$ for every $v$ if and only if $v^{\prime}=0$,
namely $I^{\prime}=J^{\prime}=\emptyset$. Therefore $f^{\prime}$ is non-
degenerate. Hence $q^{\prime}$ is non-degenerate as well.

Let ${\cal S}_{q^{\prime}}$ be the polar space associated to $q^{\prime}$. As
$f^{\prime}$ is non-degenerate, the inclusion mapping of ${\cal
S}_{q^{\prime}}$ in $\mathrm{PG}(\overline{V}^{\prime})$ is the unique
embedding of ${\cal S}_{q^{\prime}}$. Clearly, $[V]$ and $[V^{\prime}]$ are
opposite generators of ${\cal S}_{q^{\prime}}$ and their join spans
$\mathrm{PG}(\overline{V}^{\prime})$. So, the natural embedding of ${\cal
S}_{q^{\prime}}$ in $\mathrm{PG}(\overline{V}^{\prime})$ is tight.

####  5.4.1 Deep and hyperbolic sub-generators

We have $\dim(\ker^{*}(H)\cap V^{\prime})\leq 1$ for every hyperplane $H$ of
$V$. Therefore, if $H$ is a hyperplane of $V$ and
$[a\oplus\alpha]\in[H]^{\perp}$ then either $\alpha=0$ (hence
$a\oplus\alpha\in V$) or $\ker(\alpha)=H$. In the latter case, the condition
$\alpha(a)=0$ forces $a\in H$. Hence $\langle H,a\oplus\alpha\rangle=\langle
H,\alpha\rangle=\langle H,\ker^{*}(H)\rangle$. It follows that $[H]$ is
contained in at most two generators of ${\cal S}_{q^{\prime}}$, namely $[V]$
and possibly $[H+(\ker^{*}(H)\cap V^{\prime})]$ (if $\ker^{*}(H)\cap
V^{\prime}\neq\\{0\\}$).

Similarly, since $\dim(\ker(H^{\prime}))\leq 1$ for every hyperplane
$H^{\prime}$ of $V^{\prime}$, we obtain that every hyperplane of
$[V^{\prime}]$ is contained in at most two generators of ${\cal
S}_{q^{\prime}}$. So, both $[V]$ and $[V^{\prime}]$ are hyperbolic generators.
Since every hyperbolic generator contains hyperbolic sub-generators, ${\cal
S}_{q^{\prime}}$ admits hyperbolic sub-generators. Hyperbolic sub-generators
are $\perp$-minimal and, if $N$ is a hyperbolic sub-generator and $a,b$ are
opposite points of $N^{\perp}$, then $\\{a,b\\}^{\perp\perp}=\\{a,b\\}$.

So, ${\cal S}_{q^{\prime}}$ admits hyperbolic lines of size 2. Hence all
hyperbolic lines of ${\cal S}_{q^{\prime}}$ have size 2, since
$\mathrm{Aut}({\cal S}_{q^{\prime}})$ acts transitively on the set of pairs of
opposite points of ${\cal S}_{q^{\prime}}$. However, as we shall prove in the
next subsection (Corollary 5.14), ${\cal S}_{q^{\prime}}$ also admits sub-
generators which are neither deep nor hyperbolic. They cannot be
$\perp$-minimal. Therefore,

###### Proposition 5.11

The polar space ${\cal S}_{q^{\prime}}$ is not regular.

####  5.4.2 Sub-generators which are not $\perp$-minimal

For $i\in\mathfrak{I}$ put $u_{i}:=e_{i}\oplus\eta_{i}$ and, for
$i,j\in\mathfrak{I}$, put $u_{i,j}:=u_{i}+u_{j}$. So, $U:=\langle
u_{i,j}\rangle_{i,j\in\mathfrak{I}}$ is a maximal totally
$q^{\prime}$-singular subspace of $\overline{V}^{\prime}$. Note that, however,
$U$ is not maximal among the totally $f^{\prime}$-isotropic subspaces of
$\overline{V}^{\prime}$. Indeed $U$ is a hyperplane of $U^{\prime}:=\langle
u_{i}\rangle_{i\in\mathfrak{I}}$ ($=\langle U,u_{k}\rangle$ for any
$k\in\mathfrak{I}$), which is a maximal totally $f$-isotropic subspace of
$\overline{V}^{\prime}$ (but not a totally $q^{\prime}$-singular subspace,
since none of the vectors $u_{i}$ is $q^{\prime}$-singular).

Pick two elements $i_{0}$ and $i_{1}$ of $\mathfrak{I}$, henceforth called $0$
and $1$ for short, and let $\leq$ be a well ordering of $\mathfrak{I}$ with
$0$ and $1$ as the first and second element. Put $U_{0,1}:=\langle
u_{i,j}\rangle_{1<i<j}$. So, $\mathrm{cod}_{U}(U_{0,1})=2$, namely $[U_{0,1}]$
is a sub-sub-generator of ${\cal S}_{q^{\prime}}$. Indeed $U=\langle
U_{0,1},u_{k,1},u_{h,0}\rangle$ for any choice of $h,k\in\mathfrak{I}$ such
that $k\neq 1$, $h\neq 0$ and $\\{k,h\\}\neq\\{0,1\\}$.

###### Lemma 5.12

We have $U_{0,1}^{\perp_{f^{\prime}}}=\langle
U_{0,1},e_{0},e_{1},\eta_{0},\eta_{1},u_{k}\rangle$, for any choice of $k>1$.

Proof. Pick $k\in\mathfrak{I}\setminus\\{0,1\\}$. Clearly
$U_{0,1}^{\perp_{f^{\prime}}}\supseteq\langle
E_{0,1},e_{0},e_{1},\eta_{0},\eta_{1},u_{k}\rangle$. Conversely, let
$v=\sum_{i\in I}e_{i}t_{i}\oplus\sum_{j\in J}s_{j}\eta_{j}$ with support
$(I,J)$ and suppose that $v$ belongs to $U_{0,1}^{\perp_{f^{\prime}}}$. We
shall prove that $v\in\langle
U_{0,1},e_{0},e_{1},\eta_{0},\eta_{1},e_{k}\oplus\eta_{k}\rangle$.

Up to adding a suitable combination of $e_{0},e_{1},\eta_{0}$ and $\eta_{1}$
we can assume that $I,J\subseteq\mathfrak{I}\setminus\\{0,1\\}$. So, let
$I,J\subseteq\mathfrak{I}\setminus\\{0,1\\}$. If $I\neq J$ then we can always
find $i,j>1$ such that $j\not\in I\cup J$ and $i$ belongs to only one of $I$
or $J$. With $i$ and $j$ chosen in this way, let $i\in I\setminus J$, to fix
ideas. Then $f^{\prime}(u_{i,j},v)=t_{i}$. However $t_{i}\neq 0$ since $i\in
I$ and $(I,J)$ is the support of $v$. It follows that
$v\not\perp_{f^{\prime}}u_{i,j}\in E_{0,1}$. This contradicts the assumption
that $v\in E_{0,1}^{\perp_{f^{\prime}}}$.

Therefore $I=J$. Suppose now that, for at least one $i\in I$, we have
$t_{i}\neq s_{i}$. Then, with $j\in\mathfrak{I}\setminus(\\{0,1\\}\cup I)$, we
get $f^{\prime}(u_{i,j},v)=t_{i}+s_{i}\neq 0$. Again, this contradicts the
assumption $v\in E_{0,1}^{\perp_{f^{\prime}}}$. Therefore $t_{i}=s_{i}$ for
every $i\in I$, namely $v=\sum_{i\in I}u_{i}t_{i}\in U^{\prime}$.

We shall now prove that $v\in\langle U_{0,1},u_{k}\rangle$. We shall argue by
induction on $|I|$. If $I=\emptyset$ there is nothing to prove. Let $|I|=1$,
say $I=\\{i\\}$. So, $v=u_{i}t_{i}$. If $i=k$ there is nothing to prove. Let
$i\neq k$. Then $v+u_{i,k}t_{i}=u_{k}t_{i}$. Hence
$v=u_{k}t_{i}+u_{k,h}t_{i}\in\langle U_{0,1},u_{k}\rangle$, as claimed.

Assuming that $|I|>1$, let $i$ and $j$ be distinct elements of $I$. Then
$v+u_{i,j}t_{i}$ has support $(J,J)$ with $J$ equal to either
$I\setminus\\{i,j\\}$ or $I\setminus\\{i\\}$ according to whether
$t_{i}=t_{j}$ or $t_{i}\neq t_{j}$. By the inductive hypothesis,
$v+u_{i,j}t_{i}\in\langle U_{0,1},u_{k}\rangle$. Hence $v\in\langle
U_{0,1},u_{k}\rangle$. $\Box$

###### Proposition 5.13

The star $\mathrm{St}([U_{0,1}])$ of $[U_{0,1}]$ in ${\cal S}_{q^{\prime}}$ is
isomorphic to the quadric ${\cal Q}_{4}(\mathbb{F})$ of
$\mathrm{PG}(4,\mathbb{F})$.

Proof. The star $\mathrm{St}([U_{0,1}])$ is isomorphic to the polar space
associated with the form induced by $q^{\prime}$ on a complement $X$ of
$U_{0,1}$ in $U_{0,1}^{\perp_{f^{\prime}}}$. By Lemma 5.12, we can choose
$X=\langle e_{0},e_{1},\eta_{0},\eta_{1},u_{k}\rangle$. It is straightforward
to check that, with $X$ chosen in this way, the restriction of $q^{\prime}$ to
$X$ defines a copy of ${\cal Q}_{4}(\mathbb{F})$. Note that $[u_{k}]$ is the
nucleus of this quadric. $\Box$

###### Corollary 5.14

Each of the sub-generators of ${\cal S}_{q^{\prime}}$ containing $[U_{0,1}]$
is contained in exactly $|\mathbb{F}|+1$ (hence at least three) generators of
${\cal S}_{q^{\prime}}$.

For every totally $q^{\prime}$-singular subpace $W$ of $\overline{V}^{\prime}$
containing $U_{0,1}$, the generator $[W]$ of ${\cal S}_{q^{\prime}}$ is a sub-
generator of the polar space ${\cal S}_{f^{\prime}}$ associated to
$f^{\prime}$. Explicitly, if $W^{\prime}:=\langle W,u_{k}\rangle$ then
$[W^{\prime}]$ is the (unique) generator of ${\cal S}_{f^{\prime}}$ which
contains $[W]$. (Recall that $[U_{0,1}+\langle u_{k}\rangle]$ is the nucleus
of the quadric $\mathrm{St}([U_{0,1}])$.)

###### Proposition 5.15

None of the generators of ${\cal S}_{q^{\prime}}$ containing $[U_{0,1}]$
admits a complement in ${\cal S}_{q^{\prime}}$.

Proof. Let $X$ be a complement of $W$ in $\overline{V}^{\prime}$, with
$W\supseteq U_{0,1}$ and $[W]$ a generator of ${\cal S}_{q^{\prime}}$ and let
$W^{\prime}=\langle W,u_{k}\rangle$. So, $[W^{\prime}]$ is the generator of
${\cal S}_{f^{\prime}}$ which contains $[W]$. None of the vectors of
$W^{\prime}\setminus W$ is singular for $q^{\prime}$. Morever, $u_{k}=w+x$ for
suitable vectors $w\in W$ and $x\in X$. So, $x=u_{k}-w\in W^{\prime}\setminus
W$. Hence $q^{\prime}(x)\neq 0$. Therefore $[X]$ is not a singular subspace of
${\cal S}_{q^{\prime}}$. $\Box$

####  5.4.3 Stars of sub-sub-generators

Let $X$ be a sub-sub-generator of ${\cal S}_{q^{\prime}}$. In principle,
$\mathrm{St}(X)$ could be any of the following:

  * (1)

a copy of the hyperbolic quadric ${\cal Q}^{+}_{4}(\mathbb{F})$ of
$\mathrm{PG}(3,\mathbb{F})$;

  * (2)

a copy of the quadric ${\cal Q}_{4}(\mathbb{F})$ of
$\mathrm{PG}(4,\mathbb{F})$;

  * (3)

a non-degenerate quadric of rank 2 in a projective space of dimension at least
5 over $\mathbb{F}$;

  * (4)

a quadratic cone in a projective space of dimension at least 3, with a non-
degenerate quadric of rank $1$ as a basis;

  * (5)

a pair of lines meeting at a point;

  * (6)

one single line.

In case (1) all sub-generators containing $X$ are hyperbolic while in cases
(2) and (3) none of them is hyperbolic nor deep. In case (4) the sub-generator
corresponding to the vertex of the cone is neither hyperbolic nor deep while
all remaining points of $\mathrm{St}(X)$ are provided by deep sub-generators.
In case (5) the meet-point of the two lines of $\mathrm{St}(X)$ corresponds to
a hyperbolic sub-generator of ${\cal S}_{q^{\prime}}$ and the remaining points
of $\mathrm{St}(X)$ correspond to deep sub-generators. Finally, in case (6)
all sub-generators containing $X$ are deep.

Case (1), (2), (5) and (6) actually occur in ${\cal S}_{q^{\prime}}$. We have
shown in Section 5.4.2 how to define a sub-sub-generator $X$ with
$\mathrm{St}(X)$ as in case (2). In order to obtain (1), choose two
hyperplanes $H$ and $H^{\prime}$ of $V$ (or of $V^{\prime}$) such that
$\ker^{*}(H)\cap V^{\prime}\neq\\{0\\}\neq\ker^{*}(H^{\prime})\cap V^{\prime}$
(respectivley $\ker(H)\neq\\{0\\}\neq\ker(H^{\prime})$) and put $X=[H\cap
H^{\prime}]$. When $\ker^{*}(H)\cap
V^{\prime}=\\{0\\}\neq\ker^{*}(H^{\prime})\cap V^{\prime}$ we get (5) and, if
$\ker^{*}(H\cap H^{\prime})\cap V^{\prime}=\\{0\\}$ then we get (6). We do not
know if cases (3) and (4) actually occur.

In cases (1), (5) and (6) all generators of ${\cal S}_{q^{\prime}}$ containing
$X$ are also generators of ${\cal S}_{f^{\prime}}$ while in cases (2) and (5)
they are deep sub-generators of ${\cal S}_{f^{\prime}}$. Several possibilities
for $\mathrm{St}(X)$ are allowed in cases (3) and (4), depending on the
dimensions of the projective space $X^{\perp_{f^{\prime}}}/X$ which hosts the
quadric $\mathrm{St}(X)$ and the dimension of the radical of the
bilinearization of the quadratic form which defines that quadric. Of course,
the range of hypothetical possibilities depends on $\mathbb{F}$. For instance,
if $\mathbb{F}$ is finite then $\mathrm{St}(X)\cong{\cal
Q}_{5}^{-}(\mathbb{F})$ is the unique possibility in (3) while two subcases
are allowed in (4), according to whether the basis of the cone is a conic of
$\mathrm{PG}(2,\mathbb{F})$ or an ovoid of $\mathrm{PG}(3,\mathbb{F})$. In
contrast, if $\mathbb{F}$ is quadratically closed then (3) is impossible and
just one subcase survives in (4), where the basis of the cone is a conic.

If, as we believe, cases (3) and (4) never occur then the generators of ${\cal
S}_{q^{\prime}}$ are partioned in two families, namely the family ${\cal
M}_{1}$ formed by generators of ${\cal S}_{q^{\prime}}$ which are generators
of ${\cal S}_{f^{\prime}}$ and the family ${\cal M}_{2}$ formed by the
generators of ${\cal S}_{q^{\prime}}$ which are (deep) sub-generators of
${\cal S}_{f^{\prime}}$. The family ${\cal M}_{1}$ contains all hyperbolic
generators of ${\cal S}_{q^{\prime}}$ while ${\cal M}_{2}$ contains the non-
hyperbolic ones. Moreover, a non-hyperbolic generator of ${\cal
S}_{q^{\prime}}$ contains no deep or hyperbolic sub-generator.

####  5.4.4 A conjecture

The quadric ${\cal S}_{q}$ described in Section 5.2 is regular because it is
hyperbolic (Proposition 5.8). The proof of Proposition 5.8 relies on an
analogue of Lemma 5.2. In the proof of Lemma 5.2 we have exploited the fact
that $\ker(\ker^{*}(X))=X$ for every subspace $X$ of $V$. If we replace
$V^{*}$ with a subspace $V^{\prime}$ of $V^{*}$ then, in order to repeat that
proof, we need that $\ker(\ker^{*}(X)\cap V^{\prime})=X$ for every subspace
$X$ of $V$. This property holds only if $V^{\prime}=V^{*}$. This remark
suggests the following conjecture.

###### Conjecture 5.16

With $q$ as in Section 5.2, let $V^{\prime}$ be any proper subspace of $V^{*}$
such that $\ker(V^{\prime})=\\{0\\}$. Then the polar space associated to the
form induced by $q$ on $V\oplus V^{\prime}$ is non-regular.

Proposition 5.11 testifies in favor of this conjecture.

## References

  * [1] F. Buekenhout and A. M. Cohen. Diagram Geometry, Springer, Berlin, 2013.
  * [2] I. Cardinali, L. Giuzzi and A. Pasini. Nearly all subspaces of a classical polar space arise from the universal embedding, Linear Alg. Appl. 627 (2021), 287-307.
  * [3] I. Cardinali, H. Cuypers, L. Giuzzi and A. Pasini. Characterizations od symplectic polar spaces, to appear in Adv. Geom..
  * [4] H. Cuypers, P. M. Johnson and A. Pasini. On the embeddability of polar spaces, Geom. Dedicata 44 (1992), 349-358.
  * [5] H. Cuypers, P. M. Johnson and A. Pasini. On the classification of polar spaces, J. Geometry 48 (1993), 56-62.
  * [6] C. Faure and A. Frölicher. Modern Projectve Geometry, Kluwer, Dordrecht, 2000.
  * [7] P. M. Johnson. Polar spaces of arbitrary rank, Geom. Dedicata, 35 (1990), 229-250.
  * [8] P. M. Johnson. Semiquadratic sets and embedded polar spaces, J. Geometry 64 (1999), 102-127.
  * [9] A. Pasini. On polar spaces of infinite rank, J. Geometry, 91 (2008), 84-118.
  * [10] A. Pasini. Synthetic and projective properties of embeddable polar spaces, to appear in Innovations Inc. Geom.
  * [11] A. Pasini and H. Van Maldeghem. Ovoidal maximal subspaces of polar spaces, to appear in Algebraic Combinatorics and the Monster Group, edited by A. A. Ivanov, Cambridge Univ. Press.
  * [12] A. Pasini and H. Van Maldeghem. An essay on Freudenthal-Tits polar spaces, to appear in J. Algebra.
  * [13] S. E. Payne and J. A. Thas. Finite Generalized Quadrangles, 2nd ed. European Math. Soc. (EMS), Zürich, ”009.
  * [14] M. A. Ronan. Embeddings and hyperplanes of discrete geometries, European J. Combin., 8 (1987), 179-185.
  * [15] E. E. Shult. Embeddings and hyperplanes of Lie incidence geometries, in Geoups of Lie Type and Their Geometries (W. M. Kantor and L. Di Martino eds.), London Math. Soc. Lecture Notes Series, 207, 1995, 215-232.
  * [16] E. E. Shult. Points and Lines, Spriger, Berlin, 2011.
  * [17] J. Tits. Building of Spherical Type and Finite BN-pairs, Springer Lecture Notes in Math. 386, Springer, Berlin, 1974.
  * [18] H. Van Maldeghem. Generalized Polygons, Birkäuser, Basel, 1998.