11institutetext:  1 Inception Institute of AI (IIAI) 2 Wuhan University 3
SimulaMet
<EMAIL_ADDRESS>
# Progressively Normalized Self-Attention Network for Video Polyp Segmentation

Ge-Peng Ji 1122 Yu-Cheng Chou 22 Deng-Ping Fan✉
Geng Chen 1111 Huazhu Fu 11 Debesh Jha 33 Ling Shao 11

###### Abstract

Existing video polyp segmentation ††footnotetext: G.-P. Ji and Y.-C. Chou
contributed equally. Code: http://dpfan.net/pranet/(VPS) models typically
employ convolutional neural networks (CNNs) to extract features. However, due
to their limited receptive fields, CNNs cannot fully exploit the global
temporal and spatial information in successive video frames, resulting in
false positive segmentation results. In this paper, we propose the novel PNS-
Net (Progressively Normalized Self-attention Network), which can efficiently
learn representations from polyp videos with real-time speed ($\sim$140fps) on
a single RTX 2080 GPU and no post-processing. Our PNS-Net is based solely on a
basic normalized self-attention block, equipping with recurrence and CNNs
entirely. Experiments on challenging VPS datasets demonstrate that the
proposed PNS-Net achieves state-of-the-art performance. We also conduct
extensive experiments to study the effectiveness of the channel split, soft-
attention, and progressive learning strategy. We find that our PNS-Net works
well under different settings, making it a promising solution to the VPS task.

###### Keywords:

Normalized self-attention Polyp segmentation Colonoscopy

##  1 Introduction

Early diagnosis of colorectal cancer (CRC) plays a vital role in improving the
survival rate of CRC patients. In fact, the survival rate in the first stage
of CRC is over 95%, decreasing to below 35% in the fourth and fifth stages
[4]. Currently, colonoscopy is widely adopted in clinical practice and has
become a standard method for screening CRC. During the colonoscopy, physicians
visually inspect the bowel with an endoscope to identify polyps, which can
develop into CRC if left untreated. In practice, colonoscopy is highly
dependent on the physicians’ level of experience and suffers from a high polyp
miss rate [18]. These limitations can be resolved with automatic polyp
segmentation techniques, which segment polyps from colonoscopy images/videos
without intervention from physicians. However, accurate and real-time polyp
segmentation is a challenging task due to the low boundary contrast between a
polyp and its surroundings and the large shape variation of polyps [8].

Significant efforts have been dedicated to overcoming these challenges. In
early studies, learning-based methods turned to handcrafted features [16, 20],
such as color, shape, texture, appearance, or some combination. These methods
train a classifier to separate the polyps from the background. However, they
usually suffer from low accuracy due to the limited representation capability
of handcrafted features in depicting heterogeneous polyps, as well as the
close resemblance between polyps and hard mimics [24]. In more recent studies,
deep learning methods have been used for polyp segmentation [24, 26]. Although
these methods have made some progress, they only use bounding boxes to detect
polyps, and therefore cannot accurately locate the boundaries. To solve this,
Brandao et al.  [5] adopted a fully convolutional networks (FCN) with a pre-
trained model to recognize and segment polyps. Later, Akbari et al.  [1]
introduced a modified FCN to increase the accuracy of polyp segmentation.
Inspired by the success of UNet [19] in biomedical image segmentation, UNet++
[28] and ResUNet [13] are employed for polyp segmentation and achieved good
results. Some methods also focus on area-boundary constraints. For instance,
Psi-Net [17] makes use of polyp boundary and area information simultaneously.
Fang et al.  [9] introduced a three-step selective feature aggregation
network. ACSNet [25] utilized an adaptive context selection based encoder-
decoder framework. Zhong et al.  [27] propose a context-aware network based on
adaptive scale and global semantic context. Introduced more recently, the
current golden standard for image polyp segmentation, PraNet [8], applies area
and boundary cues in a reverse attention module, achieving the cutting-edge
performance. However, these methods have only been trained and evaluated on
still images and focus on static information, ignoring the temporal
information in endoscopic videos which can be exploited for better results. To
this end, Puyal et al.  [18] propose a hybrid 2D/3D CNN architecture. Their
model aggregates spatial and temporal correlations and achieves better
segmentation results. However, the spatial correlation between frames is
restricted by the size of the kernel, preventing the accurate segmentation of
fast videos.

Recently, the self-attention network [22] has shown superior performance in
computer vision tasks such as video object segmentation [10], image super-
resolution [23], and others. Inspired by this, in this paper, we propose a
novel self-attention framework, called the Progressively Normalized Self-
attention Network (PNS-Net), for the video polyp segmentation (VPS) task. Our
contributions are as follows:

  * $\bullet$

Different from existing CNN-based models, the proposed PNS-Net framework is a
self-attention model for VPS, introducing a new perspective for addressing
this task.

  * $\bullet$

To fully utilize the temporal and spatial cues, we propose a simple normalized
self-attention (NS) block. The NS block is flexible (backbone-free) and
efficient, enabling it to easily be embedded into current CNN-based encoder-
decoder architectures for better performance.

  * $\bullet$

We evaluate the proposed PNS-Net on challenging VPS datasets and compare it
with two classical methods (_i.e._ , UNet [19] and UNet++ [28]) and three
cutting-edge models (_i.e._ , ResUNet [13], ACSNet [25], and PraNet [8]).
Experimental results show that PNS-Net achieves state-of-the-art performance
with real-time speed. All the training data, models, results, and evaluation
tools will be released to advance the development of this field.

##  2 Method

Figure 1:  Pipeline of the proposed PNS-Net, including the normalized self-
attention block (see $\S$ 2.1) with a stacked ($\times$R) learning strategy
(see $\S$ 2.2).

###  2.1 Normalized Self-attention (NS)

Motivation.  Recently, the self-attention mechanism [22] has been widely
exploited in many popular computer vision tasks. However, in our initial
studies, we found that introducing the original self-attention mechanism to
the VPS task does not achieve satisfactory results (_i.e._ , high accuracy and
speed).

Analysis.  For the VPS task, multi-scale polyps move at various speeds. Thus,
dynamically updating the receptive field of the network is important. Further,
the self-attention, such as the non-local network [22], incurs a high
computational and memory cost, which limits the inference speed for our fast
and dense prediction task. Motivated by the recent video salient object
detection model [10], we utilize the channel split, query-dependent, and
normalization rules to reduce the computational cost and improve the accuracy,
respectively.

Channel Split Rule. Specifically, given an input feature (_i.e._ ,
$\mathbf{X}\in\mathbb{R}^{T\times H\times W\times C}$) extracted from $T$
video frames with a size of $H\times W$ and $C$ channels, we first utilize
three linear embedding functions $\theta(\cdot)$, $\phi(\cdot)$, and
$g(\cdot)$ to generate the corresponding attention features, which are
implemented by a $1\times 1\times 1$ convolutional layer [22]. This can be
expressed as:

$Q=\theta(\mathbf{X}),K=\phi(\mathbf{X}),V=g(\mathbf{X}).$ (1)

Then we split each attention feature $\\{Q,K,V\\}\in\mathbb{R}^{T\times
H\times W\times C}$ into $N$ groups along the channel dimension and generate
query, key, and value features, _i.e._ ,
$\\{Q_{i},K_{i},V_{i}\\}\in\mathbb{R}^{T\times H\times W\times\frac{C}{N}}$,
where $i=\\{1,2,\cdots,N\\}$.

Query-Dependent Rule. To extract the spatial-temporal relationship between
successive video frames, we need to measure the similarity between query
features $Q_{i}$ and key features $K_{i}$. Inspired by [10], we introduce $N$
relevance measuring (_i.e._ , query-dependent rule) blocks to compute the
spatial-temporal affinity matrix for the constrained neighborhood of the
target pixel. Rather than computing the response between a query position and
the feature at all positions, as done in [22], the relevance measuring block
can capture more relevance regarding the target object within $T$ frames. More
specifically, given a sliding window with fixed kernel size $k$ and dilation
rate $d_{i}=2i-1$, we get the corresponding constrained neighborhood in
$K_{i}$ for query pixel $\mathbf{X}^{q}$ of $Q_{i}$ in position $(x,y,z)$,
which can be obtained by a sampling function $\mathcal{F}^{S}$. This is
computed by:

$\mathcal{F}^{S}\langle\mathbf{X}^{q},K_{i}\rangle\in\mathbb{R}^{T(2k+1)^{2}\times\frac{C}{N}}=\Sigma_{m=x-kd_{i}}^{x+kd_{i}}\Sigma_{n=y-kd_{i}}^{y+kd_{i}}\Sigma_{t=1}^{T}K_{i}(m,n,t),$
(2)

where $1\leq x\leq H$, $1\leq y\leq W$, and $1\leq z\leq T$. Thus, the size of
the constrained neighborhood depends on the various spatial-temporal receptive
fields with different kernel size $k$, dilation rate $d_{i}$, and frame number
$T$, respectively.

Normalization Rule. However, the internal covariate shift problem [11] exists
in the feed-forward of input $Q_{i}$, incurring that the layer parameters
cannot dynamically adapt the next mini-batch. Therefore, we maintain a fixed
distribution for $Q_{i}$ via:

$\hat{Q}_{i}=\texttt{Norm}(Q_{i}),$ (3)

where Norm is implemented by layer normalization [2] along temporal dimension.

Relevance Measuring.  Finally, the affinity matrix is computed as:

$\mathcal{M}^{A}_{i}\in\mathbb{R}^{THW\times
T(2k+1)^{2}}=\texttt{Softmax}(\frac{\hat{Q}_{i}\mathcal{F}^{S}\langle\hat{\mathbf{X}}^{q},K_{i}\rangle^{\mathbf{T}}}{\sqrt{C/N}}),~{}\text{when}~{}\hat{\mathbf{X}}^{q}\in\hat{Q}_{i},$
(4)

where $\sqrt{C/N}$ is a scaling factor to balance the multi-head attention
[21].

Spatial-Temporal Aggregation.  Similar to relevance measuring, we also compute
the spatial-temporally aggregated features $\mathcal{M}^{T}_{i}$ within the
constrained neighborhood during temporal aggregation. This can be formulated
as:

$\mathcal{M}^{T}_{i}\in\mathbb{R}^{THW\times\frac{C}{N}}=\mathcal{M}^{A}_{i}\mathcal{F}^{S}\langle\mathbf{X}^{a},V_{i}\rangle,~{}\text{when}~{}\mathbf{X}^{a}\in\mathcal{M}^{A}_{i},$
(5)

Soft-Attention.  We use a soft-attention block to synthesize features from the
group of affinity matrices $\mathcal{M}^{A}_{i}$ and aggregated features
$\mathcal{M}^{T}_{i}$. During the synthesis process, relevant spatial-temporal
patterns should be enhanced while less relevant ones should be suppressed. We
first concatenate a group of affinity matrices $\mathcal{M}^{A}_{i}$ along the
channel dimension to generate $\mathcal{M}^{A}$. Thus, the soft-attention map
$\mathcal{M}^{S}$ is computed by:

$\mathcal{M}^{S}\in\mathbb{R}^{THW\times
1}\leftarrow\texttt{max}\mathcal{M}^{A},~{}\text{when}~{}\mathcal{M}^{A}\in\mathbb{R}^{THW\times
T(2k+1)^{2}N},$ (6)

where the max function computes the channel-wise maximum value. Then we
concatenate a group of the spatial-temporally aggregated features
$\mathcal{M}^{T}_{i}$ along the channel dimension to generate
$\mathcal{M}^{T}$.

Normalized Self-attention.  Finally, our NS block can be computed as:

$\mathbf{Z}\in\mathbb{R}^{T\times{H}\times{W}\times
C}=\mathbf{X}+\mathbf{Y}=\mathbf{X}+(\mathcal{M}^{T}\mathbf{W}_{T})\circledast\mathcal{M}^{S},$
(7)

where $\mathbf{W}_{T}$ is the learnable weight and $\circledast$ is the
channel-wise Hadamard product.

###  2.2 Progressive Learning Strategy

Encoder.  For fair comparison, we use the same backbone (_i.e._ , Res2Net-50)
as in PraNet [8]. Given a polyp video clip with $T$ frames as input (_i.e._ ,
$\\{\mathcal{I}\\}_{t=1}^{T}\in\mathbb{R}^{H^{\prime}\times W^{\prime}\times
3}$), we first feed it into a spatial encoder to extract two spatial features
from the conv3_4 and conv4_6 layers, respectively. To alleviate the
computational burden, we adopt an RFB-like [15] module to reduce the feature
channel. Thus, we generate two spatial features, including low-level (_i.e._ ,
$\\{\mathcal{X}^{l}_{t}\\}_{t=1}^{T}\in\mathbb{R}^{H^{l}\times W^{l}\times
C^{l}}$) and high-level (_i.e._ ,
$\\{\mathcal{X}^{h}_{t}\\}_{t=1}^{T}\in\mathbb{R}^{H^{h}\times W^{h}\times
C^{h}}$)111We set $H^{l}=\frac{H^{\prime}}{4}$, $W^{l}=\frac{W^{\prime}}{4}$,
$C^{l}=24$, $H^{h}=\frac{H^{\prime}}{8}$, $W^{h}=\frac{W^{\prime}}{8}$, and
$C^{h}=32$..

Progressively Normalized Self-attention (PNS).  Most attention strategies aim
to refine candidate features, such as first-order [8] and second-order [22,
21] functions. As such, the strong semantic information in high-level features
might be diffused gradually during the forward pass of the network. To
alleviate this, we introduce a progressive residual learning strategy in our
NS block. Specifically, we first reshape the corresponding high-level features
$\\{\mathcal{X}^{h}_{t}\\}_{t=1}^{T}$ of consecutive input frames into a
temporal feature, which can be viewed as a four-dimensional tensor (_i.e._ ,
$\widetilde{\mathcal{X}}^{h}\in\mathbb{R}^{T\times H^{h}\times W^{h}\times
C^{h}}$). Then we refine $\widetilde{\mathcal{X}}^{h}$ via stacked normalized
self-attention in a progressive manner:

$\widetilde{\mathcal{X}}^{r}\in\mathbb{R}^{T\times H^{h}\times W^{h}\times
C^{h}}=\texttt{NS}^{\times R}(\widetilde{\mathcal{X}}^{h})=\texttt{NS}^{\times
R}(\mathcal{F}^{R}(\\{\mathcal{X}^{h}_{t}\\}_{t=1}^{T})),$ (8)

where $\texttt{NS}^{\times R}$ means that $R$ normalized self-attention blocks
are stacked in the refinement process. $\mathcal{F}^{R}$ is the reshaping
function for the temporal dimension. To allow this block to easily be plugged
into pre-trained networks, the commonly adopted solution is to add a residual
learning process. Finally, the refined spatial-temporal feature is generated
by:

$\\{\mathcal{X}^{r}_{t}\\}_{t=1}^{T}\in\mathbb{R}^{{H}^{h}\times{W}^{h}\times
C^{h}}=\mathcal{F}^{R}(\widetilde{\mathcal{X}}^{h}+\widetilde{\mathcal{X}}^{r}).$
(9)

Decoder and Learning Strategy.  We combine the low-level feature
$\\{\mathcal{X}^{l}_{t}\\}_{t=1}^{T}$ from the spatial decoder and the
spatial-temporal feature $\\{\mathcal{X}^{r}_{t}\\}_{t=1}^{T}$ from the PNS
block via a two-stage UNet-like decoder $\mathcal{F}^{D}$. Thus, the output of
our method is computed by
$\\{\mathcal{P}_{t}\\}_{t=1}^{T}=\mathcal{F}^{D}(\\{\mathcal{X}^{l}_{t}\\}_{t=1}^{T},\\{\mathcal{X}^{r}_{t}\\}_{t=1}^{T})$.
We adopt the standard cross-entropy loss function in the learning process.

##  3 Experiments

###  3.1 Implementation Details

Datasets.  We adopt four widely used polyp datasets in our experiments,
including image-based (_i.e._ , Kvasir [12]) and video-based (_i.e._ , CVC-300
[4], CVC-612 [3], and ASU-Mayo [20]) ones. Kvasir is a large-scale and
challenging dataset, which consists of 1,000 polyp images with fully annotated
pixel-level ground truths (GTs). The whole Kvasir is used for training. ASU-
Mayo contains 10 negative video samples from normal subjects and 10 positive
samples from patients. We only adopt the positive part for training. Following
the same protocol as [4, 3], we split the videos from CVC-300 (12 clips) and
CVC-612 (29 clips) into $60\%$ for training, $20\%$ for validation, and $20\%$
for testing.

Training.  Due to the limited video training data, we try to fully utilize
large-scale image data to capture more appearances of the polyp and scene.
Thus, we train our model in two steps: i) Pre-training phase. We remove the
normalized self-attention (NS) block from PNS-Net and pre-train the static
backbone using an image-based polyp dataset (_i.e._ , Kvasir [12]) and the
training set of video-based polyp datasets (_i.e._ , CVC-300 [4], CVC-612 [3],
and ASU-Mayo [20]). The initial learning rate of the Adam algorithm and the
weight decay are both 1e-4. The static part of our PNS-Net convergences after
100 epochs. ii) Fine-tuning phase. We plug the NS block into our PNS-Net and
fine-tune the whole network using the video polyp datasets, including the ASU-
Mayo and the training sets of CVC-300 and CVC-612. We set the number of
attention groups $N=4$ and the number of stacked normalized self-attention
blocks $R=2$, along with a kernel size of $k=3$. The initial learning is set
to 1e-4, and the whole model is fine-tuned over one epoch. In this way,
although the densely labeled VPS data is scarce, our PNS-Net still achieves
good generalization performance.

Testing and Runtime.  To test the performance of our PNS-Net, we validate it
on challenging datasets, including the test set of CVC-612 (_i.e._ ,
CVC-612-T), the validation set of CVC-612 (_i.e._ , CVC-612-V), and the
test/validation set of CVC-300 (_i.e._ , CVC-300-TV). During inference, we
sample $T$=5 frames from a polyp clip and resize them to 256$\times$448 as the
input. For final prediction, we use the output $\mathcal{P}_{t}$ of the
network followed by a sigmoid function. Our PNS-Net achieves a speed of
$\sim$140fps on a single RTX 2080 GPU without any post-processing (_e.g._ ,
CRF [14]). The speeds of the compared methods are listed in Tab. 1.

###  3.2 Evaluation on Video Polyp Segmentation

Baselines.  We re-train five cutting-edge polyp segmentation baselines (_i.e._
, UNet [19], UNet++ [28], ResUNet [13], ACSNet [25], and PraNet [8]) with the
same data used by our PNS-Net, under their default settings, for fair
comparison.

Metrics.  The metrics used included: (1) maximum Dice (maxDice), which
measures the similarity between two sets of data; (2) maximum specificity
(maxSpe), which refers to the percentage of the samples that are negative and
are judged as such; (3) maximum IoU (maxIoU), which measures the overlap
between two masks; (4) S-measure [6] ($S_{\alpha}$), which evaluates region-
and object-aware structural similarity; (5) enhanced-alignment measure [7]
($E_{\phi}$), which measures pixel-level matching and image-level statistics;
and (6) mean absolute error ($M$), which measures the pixel-level error
between the prediction and GT.

Qualitative Comparison.  In Fig. 2, We provide the polyp segmentation results
of our PNS-Net on CVC-612-T. Our model can accurately locate and segment
polyps in many difficult situations, such as different sizes, homogeneous
areas, different textures, _etc_.

Table 1: Quantitative results on different datasets.  |  |  |  2018$\sim$2019
---
| 2020
---
2021
|  | UNet | UNet++ | ResUNet | ACSNet | PraNet | PNS-Net
| Metrics |  MICCAI [19] |  TMI [28] |  ISM [13] |  MICCAI [25] |  MICCAI [8] | (OUR)
| Speed | 108fps | 45fps | 20fps | 35fps | 97fps | 140fps
CVC-300-TV |  maxDice$\uparrow$ | 0.639 | 0.649 | 0.535 | 0.738 | 0.739 | 0.840
maxSpe$\uparrow$ | 0.963 | 0.944 | 0.852 | 0.987 | 0.993 | 0.996
maxIoU$\uparrow$ | 0.525 | 0.539 | 0.412 | 0.632 | 0.645 | 0.745
$S_{\alpha}\uparrow$ | 0.793 | 0.796 | 0.703 | 0.837 | 0.833 | 0.909
$E_{\phi}\uparrow$ | 0.826 | 0.831 | 0.718 | 0.871 | 0.852 | 0.921
$M\downarrow$ | 0.027 | 0.024 | 0.052 | 0.016 | 0.016 | 0.013
CVC-612-V |  maxDice$\uparrow$ | 0.725 | 0.684 | 0.752 | 0.804 | 0.869 | 0.873
maxSpe$\uparrow$ | 0.971 | 0.952 | 0.939 | 0.929 | 0.983 | 0.991
maxIoU$\uparrow$ | 0.610 | 0.570 | 0.648 | 0.712 | 0.799 | 0.800
$S_{\alpha}\uparrow$ | 0.826 | 0.805 | 0.829 | 0.847 | 0.915 | 0.923
$E_{\phi}\uparrow$ | 0.855 | 0.830 | 0.877 | 0.887 | 0.936 | 0.944
$M\downarrow$ | 0.023 | 0.025 | 0.023 | 0.054 | 0.013 | 0.012
CVC-612-T |  maxDice$\uparrow$ | 0.729 | 0.740 | 0.617 | 0.782 | 0.852 | 0.860
maxSpe$\uparrow$ | 0.971 | 0.975 | 0.950 | 0.975 | 0.986 | 0.992
maxIoU$\uparrow$ | 0.635 | 0.635 | 0.514 | 0.700 | 0.786 | 0.795
$S_{\alpha}\uparrow$ | 0.810 | 0.800 | 0.727 | 0.838 | 0.886 | 0.903
$E_{\phi}\uparrow$ | 0.836 | 0.817 | 0.758 | 0.864 | 0.904 | 0.903
$M\downarrow$ | 0.058 | 0.059 | 0.084 | 0.053 | 0.038 | 0.038
Figure 2:  Qualitative results on CVC-612-T [3]. For more visualization
results please refer to the supplementary material (_i.e._ , PDF file and
videos).

Quantitative Comparison.  Quantitative comparison results are summarized in
Tab. 1. We conduct three experiments on test datasets to verify the model’s
performance. CVC-300-TV consists of both validation set and test set, which
include six videos in total. CVC-612-V and CVC-612-T each contain five videos.
On CVC-300, where all the baseline methods perform poorly, our PNS-Net
achieves remarkable performance in all metrics and outperforms all SOTA
methods by a large margin (max Dice: $\sim$10%). On CVC-612-V and CVC-612-T,
our PNS-Net consistently outperforms other SOTAs.

###  3.3 Ablation Study

Table 2: Ablation studies. See $\S$ 3.3 for more details.  |  | Variants
---
| CVC-300-TV
---
| CVC-612-T
---
No. | Base | N | Soft | Norm | R |  maxDice$\uparrow$ |  maxIoU$\uparrow$ | $S_{\alpha}\uparrow$ | $E_{\phi}\uparrow$ |  maxDice$\uparrow$ |  maxIoU$\uparrow$ | $S_{\alpha}\uparrow$ | $E_{\phi}\uparrow$
#1 | ✓ |  |  |  |  | 0.778 | 0.665 | 0.850 | 0.858 | 0.850 | 0.778 | 0.896 | 0.885
#2 | ✓ | 1 |  |  | 1 | 0.755 | 0.650 | 0.865 | 0.844 | 0.850 | 0.779 | 0.896 | 0.891
#3 | ✓ | 2 |  |  | 1 | 0.790 | 0.679 | 0.876 | 0.872 | 0.825 | 0.746 | 0.870 | 0.856
#4 | ✓ | 4 |  |  | 1 | 0.809 | 0.709 | 0.893 | 0.884 | 0.834 | 0.760 | 0.881 | 0.867
#5 | ✓ | 8 |  |  | 1 | 0.763 | 0.663 | 0.867 | 0.842 | 0.787 | 0.702 | 0.841 | 0.829
#6 | ✓ | 4 | ✓ |  | 1 | 0.829 | 0.729 | 0.896 | 0.903 | 0.852 | 0.784 | 0.895 | 0.897
#7 | ✓ | 4 | ✓ | ✓ | 1 | 0.827 | 0.732 | 0.897 | 0.898 | 0.856 | 0.792 | 0.898 | 0.896
#8 | ✓ | 4 | ✓ | ✓ | 2 | 0.840 | 0.745 | 0.909 | 0.921 | 0.860 | 0.795 | 0.903 | 0.903
#9 | ✓ | 4 | ✓ | ✓ | 3 | 0.737 | 0.609 | 0.793 | 0.751 | 0.732 | 0.613 | 0.776 | 0.728

Effectiveness of Channel Split.  We investigate the contribution of channel
split rule under different scales. The results are listed in rows #2 to #5 in
Tab. 2. We observe that #4 (N=4) outperforms other settings (_i.e._ , #2, #3,
and #5) on CVC-300-TV, in all metrics. This improvement shows that an improper
receptive field (RF) harms the ability to excavate temporal information, since
a large RF will pay more attention to the global environment rather than local
motion information. On the other hand, when the split number is too small, the
model fails to capture multi-scale polyps moving at various speeds.

Effectiveness of Soft-attention.  We further investigate the contribution of
the soft-attention mechanism. As shown in Tab. 2, #6 is generally better than
#4 with the soft-attention block on CVC-612-T. This improvement suggests that
introducing the soft-attention block to synthesize the aggregation feature and
affinity matrix is necessary for increasing performance.

Effectiveness of the Number of NS Blocks.  To access the number of normalized
self-attention blocks under different settings, we derive three variants as
#7, #8, and #9. We observe that #8 (PNS-Net setting) is significantly better
than #7 and #9, with $R=2$, in all metrics on CVC-300-TV and CVC-612-T. This
improvement illustrates that too many iterations of NS blocks may cause
overfitting on small datasets (#9). In contrast, the model fails to alleviate
the diffusion issue of high-level features with a single residual block.
Empirically, we recommend increasing the number of NS blocks when training on
larger datasets.

##  4 Conclusion

We have proposed a self-attention based framework, PNS-Net, to accurately
segment polyps from colonoscopy videos with super high speed ($\sim$140fps).
Our basic normalized self-attention blocks can be easily plugged into existing
CNN-based architectures. We experimentally show that our PNS-Net achieves the
best performance on all existing publicly available datasets under six
metrics. Further, extensive ablation studies demonstrate that the core
components in our PNS-Net are all effective. We hope that the proposed PNS-Net
can serve as a catalyst for progressing both in VPS as well as other closely
related video-based medical segmentation tasks. Exploring the performance of
PNS-Net on a larger VPS dataset will be left to our future work.

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# Inversion Ring in Chromonic Twisted Hedgehogs: Theory and Experiment

Federica Ciuchi<EMAIL_ADDRESS>CNR-Nanotec c/o Physics Department,
University of Calabria, Ponte Bucci, Cubo 31C, 87036 Arcavacata di Rende,
Italy  Maria Penelope De Santo<EMAIL_ADDRESS>CNR-Nanotec c/o
Physics Department, University of Calabria, Ponte Bucci, Cubo 31C, 87036
Arcavacata di Rende, Italy  Silvia Paparini<EMAIL_ADDRESS>Department of Mathematics, University of Pavia, Via Ferrata 5, 27100 Pavia,
Italy  Lorenza Spina<EMAIL_ADDRESS>CNR-Nanotec c/o Physics Department,
University of Calabria, Ponte Bucci, Cubo 31C, 87036 Arcavacata di Rende,
Italy  Physics Department, University of Calabria, Ponte Bucci cubo 33B,
Arcavacata di Rende 87036, CS, Italy  Epifanio G. Virga<EMAIL_ADDRESS>Department of Mathematics, University of Pavia, Via Ferrata 5, 27100 Pavia,
Italy

(August 28, 2024)

###### Abstract

_Twisted hedgehogs_ are defects in spherical cavities with homeotropic
anchoring for the nematic director that arise when twist distortions are
sufficiently less energetic than splay (and bend) distortions. They bear a
characteristic _inversion ring_ , where the director texture changes the sense
it spirals about the center of the cavity. This paper applies a quartic twist
theory recently proposed to describe the elasticity of chromonics to explain a
series of inversion rings observed in aqueous solutions of SSY at two
different concentrations. The theory features a phenomenological length $a$,
whose measure is extracted from the data and shown to be fairly independent of
the cavity radius, as expected for a material constant.

##  I Introduction

Among the many lyotropic phases, chromonic liquid crystals (CLCs) have
attracted attention for their potential applications in life sciences [1, 2,
3, 4]. These materials are constituted by plank-shaped molecules that arrange
themselves in stacks when dissolved in (usually aqueous) solution. For
sufficiently large concentrations or small temperatures, the constituting
stacks give rise to an ordered phase, either nematic or columnar [5, 6, 7, 8,
9]. Here, we shall only be concerned with the nematic phase. Numerous
substances have a CLC phase; these include organic dyes (especially those
common in food industry), drugs, and oligonucleotides.

The classical elastic Oseen-Frank theory has been applied to CLCs, but the
twist constant $K_{22}$ has to be taken anomalously small to explain the
spontaneous _double_ twist exhibited by these materials when confined to
cylindrical cavities with degenerate planar anchoring on the lateral boundary
[10, 11]. In particular, $K_{22}$ must be smaller than the saddle-splay
constant $K_{24}$, thus violating one of the inequalities Ericksen [12] had
put forward to guarantee that the Oseen-Frank stored energy be bounded below.
However, as shown in [13], free-boundary problems may reveal paradoxical
consequences stemming from violating Ericksen’s inequalities. If
$K_{22}<K_{24}$, a CLC droplet in an isotropic fluid enforcing degenerate
planar anchoring, is predicted to be unstable against _shape_ perturbations:
it would split indefinitely in smaller droplets, while the total free energy
plummets to negative infinity [13]. This prediction is in sharp contrast with
the wealth of experimental observations of CLC tactoids in the biphasic
region, where nematic and isotropic phases coexist in equilibrium [14, 15, 16,
17, 18].

To resolve this contradiction, a minimalist quartic theory for CLCs was
proposed in [19], which adds to the Oseen-Frank energy density a single
quartic term in the twist measure; hence the name _quartic twist_ theory. It
was shown in [19] that indeed within this theory the total free energy of
chromonic droplets subject to degenerate planar interfacial anchoring remains
bounded below, even if $K_{22}<K_{24}$.

In [20], the quartic twist theory was applied to explain the formation of
_inversion rings_ within spherical cavities enclosing water solutions of SSY
(Sunset Yellow, a dye used in industrially processed food) subject to
_homeotropic_ boundary conditions for the director field $\bm{n}$. The
predictions of the theory were contrasted with some experimental evidence
found in recent literature [21].

Although the comparison was encouraging, it was neither systematic nor self-
consistent: it was based on a single example and used an estimate for a
phenomenological length $a$ featuring in the theory that came from a different
experiment. Thus, the motivation for the present study is to provide a
systematic and self-consistent validation of the quartic twist theory.

We observed a large number of microcavities enclosing a SSY solution at two
distinct concentrations and three different times since the samples were
prepared. We extracted the length $a$ in different physical conditions from
direct measurements of the inversion ring.

The paper is organized as follows. In Section II, we summarize the quartic
twist theory to make our development self-contained. In Section III, we
describe the materials and the experimental methods. Section IV is devoted to
the comparison of the experimental data with the proposed theory. Here we
extract $a$ and study its dependence on concentration and other factors that
may change the physical conditions of the samples as time elapses since their
preparation. Finally, in Section V, we collect the conclusions reached in this
study. The paper is closed by two Appendices: in one, we disaggregate the data
that in the main text were presented in a single universal graph; in the
other, we delve in some statistical details that could have hampered our
presentation in the body of the paper.

##  II Quartic Twist Theory

In this study we adopt an elastic theory for CLCs that has already been
developed and applied in a number of papers [19, 20, 22]. In Section IV, we
shall employ it to interpret the experiments described in Section III.

The elastic free energy density that was posited in [19] has the distinguished
feature of envisioning a _double twist_ (with two equivalent chiral variants)
as ground state of CLCs in three-dimensional space. Here we briefly elaborate
on this concept.

First, we introduce the independent scalar _measures of distortion_
$(S,T,B,q)$ for a director field $\bm{n}$; they are defined as follows:
$S:=\operatorname{div}\bm{n}$ is the _splay_ ,
$T:=\bm{n}\cdot\operatorname{curl}\bm{n}$ is the _twist_ , $B:=|\bm{b}|$ is
the modulus of the _bend_ vector
$\bm{b}:=\bm{n}\times\operatorname{curl}\bm{n}$, and $q>0$ is the _octupolar
splay_ [23], derived from

$2q^{2}=\operatorname{tr}(\nabla\bm{n})^{2}+\frac{1}{2}T^{2}-\frac{1}{2}S^{2}.$
(1)

Then, following a terminology proposed by Selinger [24] (see also [25]), we
call _double_ twist a distortion characterized by having _all_ measures of
distortion equal to zero, _but_ a single one, $T$. Somehow ironically, a
_single_ twist is characterized by having _two_ non-zero measures of
distortion related to one another:

$S=0,\quad B=0,\quad T=\pm 2q.$ (2)

A single twist is nothing but what others would call a _cholesteric_ twist.
While a single twist can fill _uniformly_ the whole space [26], a double twist
cannot. The latter can only be realized locally, not globally: it is thus a
_frustrated_ ground state. It was shown in [27] that a double twist can, for
example, be attained exactly on the symmetry axis of cylinders enforcing
degenerate planar anchoring on their lateral boundary.

The simplest way to induce locally a double twist is to add a _quartic twist_
term to the classical Oseen-Frank stored energy density [28, 29] and define
[19]

$W_{\mathrm{QT}}(\bm{n},\nabla\bm{n}):=\frac{1}{2}(K_{11}-K_{24})S^{2}+\frac{1}{2}(K_{22}-K_{24})T^{2}+\frac{1}{2}K_{33}B^{2}+\frac{1}{2}K_{24}(2q)^{2}+\frac{1}{4}K_{22}a^{2}T^{4},$
(3)

where $K_{11}$ is the _splay_ constant, $K_{22}$ is the _twist_ constant,
$K_{33}$ is the _bend_ constant, $K_{24}$ the _saddle-splay_ constant, and $a$
is a _characteristic length_. $W_{\mathrm{QT}}$ is bounded below whenever

$\displaystyle K_{11}\geqq K_{24}\geqq 0,$ (4a) $\displaystyle K_{24}\geqq
K_{22}\geqq 0,$ (4b) $\displaystyle K_{33}\geqq 0,$ (4c)

which in this theory replace the celebrated _Ericksen inequalities_ [12].
When, as we shall assume here, inequalities (4) hold strictly,
$W_{\mathrm{QT}}$ is minimum at the _degenerate_ double twist characterized by

$S=0,\quad T=\pm T_{0},\quad B=0,\quad q=0,$ (5)

where

$T_{0}:=\frac{1}{a}\sqrt{\frac{K_{24}-K_{22}}{K_{22}}}.$ (6)

Here, we shall treat $a$ as a phenomenological parameter to be determined
experimentally. We designate the elastic theory based on $W_{\mathrm{QT}}$ in
(3) as quartic _twist_ theory, as among all possible quartic invariant terms
in $\nabla\bm{n}$ we only consider the one proportional to $T^{4}$ (see also
[26] for other possible quartic elastic theories).

###  II.1 Twisted Hedgehog

The quartic twist theory was applied in [20] to describe the _twisted
hedgehog_ that forms within a spherical cavity enforcing homeotropic alignment
on its boundary. It is known since the seminal work of Lavrentovich and
Terentiev [30] that for a splay constant $K_{11}$ sufficiently larger than the
twist constant $K_{22}$, the _radial_ hedgehog becomes unstable and acquires a
twisted texture (with either sign of chirality equally likely to emerge), see
Fig. 1.

Figure 1: Field lines of $\bm{n}_{\mathrm{T}}$ in (8) in a spherical cavity
enforcing homeotropic boundary conditions. An inversion ring, depicted in
blue, is present on the equatorial plane (orthogonal to the symmetry axis
$\bm{e}$). Black lines are field lines lying on the equatorial plane; red
lines are field lines coming out of the equatorial plane. The whole 3D picture
is obtained by rotating this drawing about $\bm{e}$. This sketch is adapted
from Fig. 2 of [20]; it represent only one chiral variant of twisted hedgehog,
the other is obtained by reversing the sign of $\alpha$ in (8).

More precisely, as proved in [31, 32, 33], the radial hedgehog loses local
stability whenever

$K_{11}>K_{22}+\frac{1}{8}K_{33}.$ (7)

The original proof of this inequality was given for the Oseen-Frank elastic
theory; however, it also applies to the theory associated with the energy
density in (3), as both theories share the same second variation of the energy
functional [27]. For CLCs, inequality (7) holds because $K_{11}$ and $K_{33}$
are customarily comparable, whereas $K_{22}$ is much smaller (nearly by an
order of magnitude).

The field lines shown in Fig. 1 were obtained in [20] by minimizing the total
elastic free energy on the trial family of fields defined as

$\bm{n}_{\mathrm{T}}(\bm{x}):=\mathbf{R}(\alpha(r))\frac{\bm{x}}{r},$ (8)

where $\bm{x}$ is the position vector, $r:=|\bm{x}|$, and $\mathbf{R}(\alpha)$
denotes the rotation of angle $\alpha$ about a symmetry axis $\bm{e}$. Letting
$\alpha$ depend only on $r$, we reduced the total elastic free energy to an
admittedly complicated functional $\mathcal{F}[\alpha]$ of a single scalar
function $\alpha(\rho)$, where $\rho:=r/R$ is the only space variable scaled
to the radius $R$ of the spherical cavity. $\mathcal{F}$ is even, and so the
same energy is assigned to a function $\alpha$ and its opposite. This
degeneracy means that minimizers come in pairs of twisted hedgehogs with
opposite chirality; Fig. 1 represents only one chiral variant. Three
dimensionless parameters feature in $\mathcal{F}$: the reduced elastic
constants $k_{1}:=K_{11}/K_{22}$ and $k_{3}:=K_{33}/K_{22}$, and the ratio
$\lambda:=a/R$.

In the special case where $\lambda=0$, the quartic twist theory reduces to the
classical Oseen-Frank theory. Figure 2

Figure 2: The elastic free energy (positive) minimizer $\alpha_{\lambda}$ for
reduced elastic constants $k_{1}=7.5$, $k_{3}=9.0$ (which apply to a SSY
solution at concentration $c=30\,\mathrm{wt}\%$), and $\lambda=0.94$ is
compared with the minimizer $\alpha_{0}$ according to the Oseen-Frank theory.
The dotted line is drawn at $\alpha=\frac{\pi}{2}$; its intercepts with the
graphs of $\alpha_{\lambda}$ and $\alpha_{0}$ designate the (scaled) radii of
the inversion ring, $\rho^{\ast}:=r^{\ast}/R\doteq 0.52$ and
$\rho^{\ast}_{0}:=r^{\ast}_{0}/R\doteq 0.03$, respectively. As shown in [20],
$\widehat{\alpha}_{0}=\arccos(-1/4)$ is the value of $\alpha_{0}$ at $\rho=0$,
whereas $\alpha_{\lambda}(0)=\pi$.

shows a typical positive minimizer $\alpha_{\lambda}$ of $\mathcal{F}[\alpha]$
compared with the minimizer $\alpha_{0}$ of the classical theory.

A remarkable feature of the twisted hedgehog is the _inversion ring_ that the
nematic director field $\bm{n}$ exhibits on the plane orthogonal to the
symmetry axis of the distortion texture. There, the direction of winding of
the spiralling field lines of $\bm{n}$ changes; it has the optical appearance
of a disclination (see also Fig. 6 below), but it is _not_ a defect, as
$\bm{n}$ is continuous there.

How the radius $r^{\ast}$ of the inversion ring depends on $\lambda$ was
studied in [20]. For given $\lambda$, $r^{\ast}$ can be read off from Fig. 2:
it falls where the graph of $\alpha_{\lambda}$ intersects the line
$\alpha=\frac{\pi}{2}$. The comparison between $\alpha_{\lambda}$ and
$\alpha_{0}$ suggests that the quartic twist theory may predict inversion
rings appreciably larger than those compatible with the Oseen-Frank theory.
Figure 3 represents the graph of $\rho^{\ast}:=r^{\ast}/R$ against $\lambda$
for the same choice of reduced elastic constants made in Fig. 2.

Figure 3: The scaled radius $\rho^{\ast}:=r^{\ast}/R$ of the inversion ring
for the elastic free energy minimizer $\alpha_{\lambda}$ is plotted against
$\lambda:=a/R$ for the same reduced elastic constants chosen in Fig. 2. The
graph saturates at $\rho^{*}_{\infty}\doteq 0.82$, while $\rho^{*}_{0}\doteq
0.03$ is the limiting value as $\lambda\to 0$.

It is the aim of this paper to provide an experimental validation of the
theory recalled in this section. This will be achieved by extracting measures
of the radius of the inversion ring produced in a number of spherical cavities
trapping a solution of Sunset yellow (SSY) in water inside a polymeric matrix
enforcing homeotropic anchoring on the nematic director. How these cavities
are obtained and with what materials is explained in the following section.

##  III Experimental

The material SSY, whose molecular structure is shown in Fig. 4(a), was
purchased from Sigma-Aldrich and used without further purification. The brute
formula is
$\mathrm{C}_{16}\mathrm{H}_{10}\mathrm{N}_{2}\mathrm{Na}_{2}\mathrm{O}_{7}\mathrm{S}_{2}$
and the molecular weight is $452.37$.

When dissolved in water it stacks in cylinder formations, the length of which
depends on temperature and concentration; the cylinders then give rise to a
nematic phase. At room temperature, the nematic phase exists between ca.
$28$-$34\,\mathrm{wt}\%$ (Fig. 4(b)), [34].

(a) Molecular structure of SSY.

(b) SSY phase diagram.

Figure 4:

SSY emulsions were prepared using polydimethylsiloxane (PDMS) as an immiscible
matrix (Sylgard, kit $184$). PDMS is a liquid polymer that appears oily and
transparent and has a high viscosity [21]. To obtain a homeotropic anchoring
at the interface, PDMS was used pure, as an oil in which to disperse the
liquid crystal solution. PDMS brute formula is
($\mathrm{C}_{2}\mathrm{H}_{6}\mathrm{OSi})_{n}$, the structure of the
repeating unit is shown in Fig. 5.

Figure 5: Molecular structure of PDMS.

For the preparation of the chromonic solution, the SSY powder and water were
mixed in appropriate proportions to obtain solutions containing SSY
$30\,\mathrm{wt}\%$ and $31.5\,\mathrm{wt}\%$. The confinement of chromonic
solutions in spherical geometries is achieved by preparing emulsions. The
emulsions were prepared by adding a small amount of water-based chromonics
solution to the oil matrix (PDMS). By mechanically stirring, a large number of
spherical microcavities were obtained.

For microcavity texture investigations, a small amount of the prepared
emulsion was sandwiched between two laboratory glass slides separated by mylar
stripes of $190\mu\mathrm{m}$. Due to the high viscosity of PDMS, it was not
necessary to seal the cell with tape or glue. Microsphere textures were
studied using a polarized light microscope (Zeiss Axiolab $5$, Axiocam $208$
colour). Images were acquired with a monochromatic red filter whose
transmission spectrum has peak at $639\,\mathrm{nm}$ (with full width at half
maximum equal to $10\,\mathrm{nm}$).

##  IV Comparison with Theory

A typical example of microcavities produced via the method described in
Section III is shown in Figure 6.

Figure 6: Microcavities enclosing a solution of SSY in water at concentration
$c=30\,\mathrm{wt}\%$ and temperature $\mathrm{T}=25\,^{\circ}\mathrm{C}$.
This particular sample was observed two days after preparation. All cavities
allegedly host the same twisted hedgehog, but its symmetry axis is differently
oriented relative to the observer. For the cavities viewed along the
hedgehog’s symmetry axis, the inversion ring is easily identified as a circle;
it does not look circular for the others, which are viewed askew (see also
Fig. 10(b)).

These enclose a SSY solution in water at concentration $c=30\,\mathrm{wt}\%$
and temperature $\mathrm{T}=25\,^{\circ}\mathrm{C}$. A similar experiment was
also performed with a solution at $c=31.5\,\mathrm{wt}\%$ and same
temperature. We conventionally refer to these as Experiment I and II,
respectively. In both experiments, observations of the samples at the optical
microscope were made immediately after having prepared the cell, as well as
one and two days after. These are referred to as Observation $0$, $1$, and
$2$, respectively; cavities seen in one observation are not the same as
cavities seen in another.

###  IV.1 Inversion Ring Statistics

For each cavity that could be clearly discerned in images like the one in Fig.
6, we measured both $r^{\ast}$ and $R$ and then we extracted $a$ from the
value of $\lambda=a/R$ read off from the graph like that in Fig. 3
corresponding to the elastic constants of the material at the appropriate
concentration and temperature. For the physical conditions of Experiment I,
the reduced elastic constants are $k_{1}\approx 7.5$ and $k_{3}\approx 9.0$,
while for those of Experiment II they are $k_{1}\approx 11.0$ and
$k_{3}\approx 10.4$. (These values were derived from the measurements reported
in [35, 36, 37], which we read as follows. Experiment I: $K_{11}\approx
7.6\,\mathrm{pN}$, $K_{22}\approx 1.02\,\mathrm{pN}$, $K_{33}\approx
9.2\,\mathrm{pN}$; Experiment II: $K_{11}\approx 13.8\,\mathrm{pN}$,
$K_{22}\approx 1.25\,\mathrm{pN}$, $K_{33}\approx 13.0\,\mathrm{pN}$.)

The data thus collected for $r^{\ast}$, $R$, and $a$ were more conveniently
organized on a universal graph (depending only on $k_{1}$ and $k_{3}$) that
plots the ratio $r^{\ast}/a$ against $R/a=1/\lambda$. This graph is different
for each experiment; it is obtained from the graph in Fig. 3 via the
transformation

$\frac{r^{\ast}}{a}=\frac{\rho^{\ast}}{\lambda}.$ (9)

This changes the function $\rho^{\ast}$ plotted in Fig. 3 against $\lambda$
into a monotonic function that approaches $0$ as $R/a\to 0$ and grows
nonlineraly until becoming asymptotically linear for large $R/a$,
$r^{\ast}/R\approx\rho^{\ast}_{0}(R/a)$, where $\rho^{\ast}_{0}$ is the
(scaled) radius of the inversion ring predicted by the Oseen-Frank theory. The
rearranged data are shown in Fig. 7 for both experiments.

(a) Experiment I: SSY solution at concentration $c=30\,\mathrm{wt}\%$ and
temperature $\mathrm{T}=25\,^{\circ}\mathrm{C}$. The total number of data
points is $116$. (b) Experiment II: SSY solution at concentration
$c=31.5\,\mathrm{wt}\%$ and temperature $\mathrm{T}=25\,^{\circ}\mathrm{C}$.
The total number of data points is $114$.

Figure 7: Data collected in all observations. They are shown against the
universal graph of the function $r^{\ast}/a$ against $R/a=1/\lambda$ obtained
from the function $\rho^{\ast}$ plotted in Fig. 3 against $\lambda$ via the
transformation (9). Diamonds refer to samples just prepared (Observation $0$),
boxes to samples observed after one day (Observation $1$), dots to samples
observed after two days (Observation $2$). Straight lines apply to the
classical Oseen-Frank theory; they are recovered asymptotically by the quartic
twist theory in the limit as $R/a\to\infty$.

As also recalled in [20], the classical Oseen-Frank theory predicts an
inversion ring whose radius $r^{\ast}_{0}$ is such that the ratio
$r^{\ast}_{0}/R$ depends only on the reduced elastic constants; thus, the
graphs representing the prediction of the classical theory in both panels of
Fig. 7 are straight lines. More precisely, for Experiment I,
$r^{*}_{0}/R=\rho^{*}_{0}\approx 0.03$, while $r^{*}_{0}/R=\rho^{*}_{0}\approx
0.06$ for Experiment II. It is apparent that for chromonics the Oseen-Frank
theory fails to represent our data, whereas the proposed quartic twist theory
seems in good agreement with both experiments, more so for smaller cavities
than for larger ones, for which the inversion ring tends to be larger than
predicted by theory.

All collected data are shown in Fig. 7: they include all observations. The
reader will find in Appendix A separate illustrations for each observation in
both experiments.

Since the measures of both $r^{\ast}$ and $R$ are affected by errors, these
propagate to $a$, in a way made possible to estimate by the predicted
dependence of $\rho^{\ast}$ on $\lambda$ (shown in Fig. 3). For each
observation, we also estimated how the inferred values of $a$ are distributed
compared to the measured values of the radius $R$. Our goal is to show that
the former are less dispersed than the latter, as appropriate for a
phenomenological length that should only be related to the material and its
physical conditions. The comparison is made easier by the _dispersion_
histogram shown in Fig. 8 for a single observation (at a given SSY
concentration).

Figure 8: Dispersion histogram of the data in Fig. 7(a) for the observation
two days after preparation in Experiment I. The measured values $R_{i}$ of the
radius $R$ are black crosses and the estimated values $a_{i}$ of the
characteristic length $a$ are black dots (with corresponding blue error bars).
The red vertical line marks the mean value $\bar{a}\approx 33\,\mu\mathrm{m}$
of $a$.

For completeness, similar dispersion histograms for all other observations are
collected in Appendix B. Despite the noise present in the data, we can be
reasonably confident in the material nature of $a$. Quantitative details on
how we determined the error bars in Fig. 8 are also given in Appendix B.

###  IV.2 Estimates of $a$

The method illustrated in Section IV.1 led us to the estimates of $a$
collected in Table 1.

SSY concentration | Observation $0$ | Observation $1$ | Observation $2$
---|---|---|---
$30\,\mathrm{wt}\%$ | $a\approx 47\,\mu\mathrm{m}$ | $a\approx 41\,\mu\mathrm{m}$ | $a\approx 33\,\mu\mathrm{m}$
$31.5\,\mathrm{wt}\%$ | $a\approx 54\,\mu\mathrm{m}$ | $a\approx 46\,\mu\mathrm{m}$ | $a\approx 35\,\mu\mathrm{m}$
Table 1: Estimates of $a$ for SSY solutions in the physical conditions of
different observations.

The corresponding deviations from the mean values (which do not exceed $9\%$)
are recorded in Appendix B, which also documents the much larger dispersion in
the data set for $R$.

It is worth remarking that for SSY in the same physical conditions as in
Experiment I, in [19] we had estimated $a\approx 6.4\,\mu\mathrm{m}$ from the
director field observations of [10] in cylinders with planar degenerate
anchoring conditions on their lateral boundary. The difference (by nearly one
order of magnitude) between this estimate and those summarized in Table 1
might result from the different experimental settings. By the disparity in the
number of data collected here and in [10] and the fact that in [19] we needed
to determine two fitting parameters ($a$ and $K_{24}$) instead of one, we are
inclined to think that the estimates summarized in Table 1 are more reliable.

Two trends emerge clearly from these findings: $a$ increases with
concentration and decreases as time elapses since cell preparation (see Fig.
9).

Figure 9: Estimated values of $a$ plotted against the time elapsed since cell
preparation for both SSY concentrations, namely, $c=30\,\mathrm{wt}\%$ (black
line) and $c=31.5\,\mathrm{wt}\%$ (blue line). Dots mark the values in Table
1; straight lines are guides for the eye.

###  IV.3 Spiralling Cores

As proved in [20] and illustrated in Fig. 1, the quartic twist theory recalled
in Section II predicts a distortion _spiralling core_ within the inversion
ring of a twisted hedgehog. Here we compare with experiment this qualitative
feature of the theory.

Among the many observed cavities, we focus on two exemplary cases; they are
shown in Fig. 10.

(a) Spherical cavity of radius $R\approx 44\,\mu\mathrm{m}$ observed after one
day from the sample preparation. Taking $a\approx 41\,\mu\mathrm{m}$ from
Table 1, we estimate $r^{*}\approx 23\,\mu\mathrm{m}$. (b) Spherical cavity of
radius $R\approx 105\mu\mathrm{m}$ observed in a just prepared sample. Taking
$a\approx 47\,\mu\mathrm{m}$ from Table 1, we estimate $r^{*}\approx
46\,\mu\mathrm{m}$.

Figure 10: Spherical cavities enclosing a SSY solution at concentration
$c=30\,\mathrm{wt}\%$ and temperature $\mathrm{T}=25\,^{\circ}\mathrm{C}$. The
spiralling core predicted by the quartic twist theory (depicted in green) is
superimposed to the experimental images. The radius of the inversion ring is
estimated by theory (as explained in Section II); its predicted values are in
good agreement with direct measures.

In both cases, SSY is at concentration $c=30\,\mathrm{wt}\%$ and temperature
$\mathrm{T}=25\,^{\circ}\mathrm{C}$: the cavity in Fig. 10(a) was observed one
day after preparation, while the cavity in Fig. 10(b) was observed just at the
time of preparation. In the former instance, the spiralling core is observed
along the symmetry axis of the twisted hedgehog (at right angles with the
plane containing the inversion ring), whereas in the latter it is viewed
askew, along a direction that we estimated at the angle $\pi/8$ from the
symmetry axis.

Based on this evidence, we may say that the quartic twist theory seems also to
be in good qualitative agreement with the experimental observations.

##  V Conclusions

A quartic twist theory for the elasticity of chromomic liquid crystals is
subjected to experimental scrutiny. The spiralling texture of twisted
hedgehogs in spherical cavities enforcing homeotropic anchoring is
characterized by an inversion ring, which can be observed optically.
Measurements of the inversion ring are contrasted with both the classical
Oseen-Frank theory and the quartic twist theory, and shown to be in better
accord with the latter than with the former.

A phenomenological length $a$ features in the proposed theory; it is a
material parameter, supposedly depending on both temperature and concentration
of the chromonic solution. The data collected for the inversion ring agree
with theory if a specific value of $a$ is chosen for each cavity’s radius $R$.
The small dispersion in the values of $a$ compared to the dispersion of radii
is a testament to the material nature of $a$, which depends only on the
physical conditions in which the observation takes place.

The average of $a$ for each observation is found to increase with
concentration and decrease with the time elapsed after sample preparation. The
former trend can be understood by recalling that a similar behaviour is shown
by the ratio of elastic constants $K_{11}/K_{33}$ in homogeneous chromonic
phases (see either Fig. 3 of [35] or Fig. 2c of [37]). According to the
microscopic theory developed in [38] (see also [39]) for the elastic
properties of liquid crystal phases generated by _flexible_ molecules,

$\frac{K_{11}}{K_{33}}=\frac{\ell}{\delta},$ (10)

where $\ell$ is the _contour_ length of the constituting molecules and
$\delta$ is their _persistence_ length. In the present context, since $\delta$
is far less sensitive to concentration than $\ell$, (10) would suggest that
$\ell$ (here the length of CLC aggregates) increases with $c$. Since $a$
increases with $c$ too, one is led to conjecture that $a$ is an increasing
function of $\ell$.

This is, however, a slippery terrain, as observations of twisted tactoids in
the biphasic regime suggested in [17] that $\ell$ might instead be a
decreasing function of $c$, at least for temperatures just slightly below the
biphasic-to-isotropic transition.111The droplets observed in [17] are in the
biphasic region of phase space, where nematic and isotropic phases of the
solution coexist in equilibrium, although at different concentrations, higher
in the nematic ($c_{\mathrm{N}}$), lower in the isotropic ($c_{\mathrm{I}}$);
see Fig. 4(b). If in [17] $c>c_{\mathrm{I}}$ is the concentration at which the
solution is prepared, then the concentration in the twisted tactoids would be
$c_{\mathrm{N}}>c$.

In our experiments, temperature was not varied, but (different) cavities were
observed at different times. The length $a$ was found to decrease as such a
_curing_ time elapsed. If the conjectured link between $a$ and $\ell$ is to be
trusted, one may interpret this result as suggesting that molecular aggregates
become shorter on average (hence larger in number) as the cure progresses. A
theoretical validation of this conjecture could be found by building a kinetic
model for chromonic molecular aggregation, possibly along lines similar to
those that started to be traced in [40, 41].

## Acknowledgments

The UNICAL Group acknowledges financial support through the Italian PRIN 2022
- PNRR Project No. P2022HM5E4 on _Chirality induction in water based self-
assembling materials_.

##  Appendix A Data Disaggregation

Two distinct experiments were performed and, for each, three observations were
made at different times. Experiments, conventionally denoted I and II, used
different SSY solutions at room temperature. The data that were aggregated per
experiment in Fig. 7 are now disaggregated in Fig. 11.

(a) Experiment I; Observation 0.

(b) Experiment I; Observation 1.

(c) Experiment I; Observation 2.

(d) Experiment II; Observation 0.

(e) Experiment II; Observation 1.

(f) Experiment II; Observation 2.

Figure 11: Disaggregation of the data shown in Fig. 7. Experiment I uses a SSY
solution at concentration $c=30\,\mathrm{wt}\%$ and temperature
$\mathrm{T}=25\,^{\circ}\mathrm{C}$; Experiments II uses a SSY solution at
concentration $c=31.5\,\mathrm{wt}\%$ and temperature
$\mathrm{T}=25\,^{\circ}\mathrm{C}$. Observations are denoted $0$, $1$, or
$2$, according to whether they took place with the cell just prepared, one, or
two days later, respectively.

##  Appendix B Error Analysis and Statistical Dispersion

The radii $R_{i}$ of microcavities and $r^{\ast}_{i}$ of inversion rings were
measured on image reproductions; they were all affected by the same absolute
error $\Delta R=\Delta r^{\ast}=1\,\mu\mathrm{m}$.

(a) Experiment I; Observation 0: $\bar{a}\approx 47\,\mu\mathrm{m}$.

(b) Experiment I; Observation 1: $\bar{a}\approx 41\,\mu\mathrm{m}$.

(c) Experiment I; Observation 2: $\bar{a}\approx 33\,\mu\mathrm{m}$.

(d) Experiment II; Observation 0: $\bar{a}\approx 54\,\mu\mathrm{m}$.

(e) Experiment II; Observation 1: $\bar{a}\approx 46\,\mu\mathrm{m}$.

(f) Experiment II; Observation 2: $\bar{a}\approx 35\,\mu\mathrm{m}$.

Figure 12: Dispersion histograms comparing the different measures inferred for
$a$ to the cavity radii. They are organized according to experiments and
observations, precisely as the panels of Fig. 11; in addition, here $\bar{a}$
denotes the average value of $a$ for each set of conditions. All average
values are collected in Table 1 in the main text.

As a consequence, the corresponding values of $\rho^{\ast}=r^{\ast}/R$ were
listed as $\rho^{\ast}_{i}\pm\Delta\rho^{\ast}_{i}$, where

$\Delta\rho^{*}_{i}:=\frac{r^{*}_{i}}{R_{i}}\left(\frac{1}{r^{*}_{i}}-\frac{1}{R_{i}}\right)\Delta
r^{*}.$ (11)

Since, as shown in Fig. 3, the theoretical link between $\rho^{\ast}$ and
$\lambda=a/R$ is not linear, the symmetric absolute errors
$\pm\Delta\rho^{\ast}_{i}$ propagate asymmetrically to the corresponding
$a_{i}$’s; we write them as $\pm\Delta a_{i}^{\pm}$, where

$\Delta a_{i}^{\pm}=R_{i}\Delta\lambda_{i}^{\pm}+\lambda_{i}\Delta R$ (12)

and $\Delta\lambda_{i}^{\pm}$ are defined by

$\rho^{\ast}(\lambda_{i}\pm\Delta\lambda_{i}^{\pm})=\rho^{\ast}_{i}\pm\Delta\rho^{\ast}_{i}.$
(13)

Thus, we can attribute to the $a_{i}$’s the following relative error,

$\frac{\Delta a_{i}}{a_{i}}:=\frac{1}{2}\left(\frac{\Delta
a_{i}^{+}}{a_{i}}+\frac{\Delta
a_{i}^{-}}{a_{i}}\right)=\frac{\Delta\lambda_{i}}{\lambda_{i}}+\frac{\Delta
R}{R_{i}},$ (14)

where $\Delta\lambda_{i}:=(\lambda^{+}_{i}-\lambda^{-}_{i})/2$.

The dispersion histograms for the $a_{i}$’s and $R_{i}$’s are shown in Fig. 12
for every experiment and observation. Figure 12(c) is the same as Fig. 8 in
the main text; it is reproduced here for completeness.

To estimate the degree of statistical dispersion for both data sets $a_{i}$’s
and $R_{i}$’s, we computed the relative _mean absolute deviation_
$\mathcal{M}$, defined for a set of $n$ data $x_{i}$ by

$\mathcal{M}(x):=\frac{1}{\bar{x}}\frac{1}{n}\sum_{i=1}^{n}|x_{i}-\bar{x}|,$
(15)

where $\bar{x}$ denotes the (arithmetic) average of the $x_{i}$’s. A summary
of the averages $\bar{a}$ and $\bar{R}$, and the mean absolute deviations
about them for all experiments and observations is presented in Table 2.

Experiment | Observation 0 | Observation 1 | Observation 2
---|---|---|---
$\mathrm{I}$ | $\bar{a}\approx 47\,\mu\mathrm{m},\quad\bar{R}\approx 38\,\mu\mathrm{m}$ | $\bar{a}\approx 41\,\mu\mathrm{m},\quad\bar{R}\approx 40\,\mu\mathrm{m}$ | $\bar{a}\approx 33\,\mu\mathrm{m},\quad\bar{R}\approx 32\,\mu\mathrm{m}$
| $\mathcal{M}(a)\approx 8\%,\quad\mathcal{M}(R)\approx 20\%$ | $\mathcal{M}(a)\approx 8\%,\quad\mathcal{M}(R)\approx 24\%$ | $\mathcal{M}(a)\approx 9\%,\quad\mathcal{M}(R)\approx 39\%$
$\mathrm{II}$ | $\bar{a}\approx 54\,\mu\mathrm{m},\quad\bar{R}\approx 68\,\mu\mathrm{m}$ | $\bar{a}\approx 46\,\mu\mathrm{m},\quad\bar{R}\approx 64\,\mu\mathrm{m}$ | $\bar{a}\approx 35\,\mu\mathrm{m},\quad\bar{R}\approx 60\,\mu\mathrm{m}$
| $\mathcal{M}(a)\approx 7\%,\quad\mathcal{M}(R)\approx 17\%$ | $\mathcal{M}(a)\approx 9\%,\quad\mathcal{M}(R)\approx 32\%$ | $\mathcal{M}(a)\approx 9\%,\quad\mathcal{M}(R)\approx 36\%$
Table 2: Averages and corresponding relative mean absolute deviations for data
sets $a_{i}$ and $R_{i}$ in every experiment and observation. The averages
$\bar{a}$’s are the estimates for $a$ listed in Table 1 in the main text.

It is apparent from these data how the microcavity radii $R_{i}$ are much more
scattered about their averages compared to the $a_{i}$. This justifies
considering the averages $\bar{a}$’s as estimates of a phenomenological length
characteristic only of the material and its physical conditions.

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