# FCSN: Global Context Aware Segmentation
by Learning the Fourier Coefficients of Objects in Medical Images

Young Seok Jeon  Hongfei Yang  Mengling Feng Manuscript received xx xx, 2022;
revised xx xx, xxxx , … , and xx xx, xxxx; accepted xx xx, xxxx. Date of
publication xx xx, xxxx; date of current version xx xx, xxxx.This research is
supported by the National Research Foundation Singapore under its AI Singapore
Programme (Award Number: AISG-GC-2019-001 and AISG-GC-2019-002) and the NMRC
Health Service Research Grant (MOH-000030-00).YS. Jeon, H. Yang, M. Feng are
with the Saw Swee Hock School of Public Health and Institute of Data Science,
National University of Singapore, Singapore (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>ephfm@nus.edu.sg).Equal
contribution: Young Seok Jeon and Hongfei Yang.Corresponding author: Mengling
Feng (ephfm@nus.edu.sg).

###### Abstract

The encoder-decoder model is a commonly used Deep Neural Network (DNN) model
for medical image segmentation. Conventional encoder-decoder models make
pixel-wise predictions focusing heavily on local patterns around the pixel.
This makes it challenging to give segmentation that preserves the object’s
shape and topology, which often requires an understanding of the global
context of the object. In this work, we propose a Fourier Coefficient
Segmentation Network (FCSN)—a novel DNN-based model that segments an object by
learning the complex Fourier coefficients of the object’s masks. The Fourier
coefficients are calculated by integrating over the whole contour. Therefore,
for our model to make a precise estimation of the coefficients, the model is
motivated to incorporate the global context of the object, leading to a more
accurate segmentation of the object’s shape. This global context awareness
also makes our model robust to unseen local perturbations during inference,
such as additive noise or motion blur that are prevalent in medical images.
When FCSN is compared with other state-of-the-art models (UNet+, DeepLabV3+,
UNETR) on 3 medical image segmentation tasks (ISIC_2018, RIM_CUP, RIM_DISC),
FCSN attains significantly lower Hausdorff scores of 19.14 (6%), 17.42 (6%),
and 9.16 (14%) on the 3 tasks, respectively. Moreover, FCSN is lightweight by
discarding the decoder module, which incurs significant computational
overhead. FCSN only requires 22.2M parameters, 82M and 10M fewer parameters
than UNETR and DeepLabV3+. FCSN attains inference and training speeds of
1.6ms/img and 6.3ms/img, that is 8$\times$ and 3$\times$ faster than UNet and
UNETR.

Medical Image Segmentation, Global Context Aware Learning, Decoder-Free
Segmentation.

##  1 Introduction

Over recent years, we have witnessed increasing popularity in the applications
of Deep Neural Network (DNN) for various medical image segmentation tasks. The
encoder-decoder model [1, 2] is currently the most widely adopted DNN approach
for the segmentation task. Given enough training data, the encoder-decoder
models can extract local patterns from an image that are associated with
labels at each spatial coordinate. However, due to its heavy reliance on local
patterns, the model often fails to exploit the global contexts that
potentially help to nullify nuisance local variations.

Specifically, in medical imaging tasks where the risk of misclassification is
high, we need a model that is robust to many unpredictable local variations by
incorporating the global contexts. Taking the segmentation of optic cup in
retinopathy as an example which is demonstrated in figure 1, the following
problems are difficult to address unless the model learns the global context:

  * •

anatomically, the shape of an optic cup is always like a single filled oval,
but current DNN often give segmentation with multiple components or with holes

  * •

an optic disc has a smooth contour, but current DNN give contours with sharp
corners or unnecessary zigzags

  * •

retinopathy images from different sources are likely to suffer from different
degradations, which cause generalization problems for current DNNs.

In this paper, we argue that these problems, which are either ignored or
indirectly treated in the conventional encoder-decoder segmentation models,
can be effectively addressed if we train the DNN to directly predict the
shape, size and location of an object.

Figure 1: Comparison of encoder-decoder model (upper) and Fourier Coefficient
Segmentation Network (FCSN) (lower). Unlike the encoder-decoder model, which
makes a coordinate-wise prediction of an object, our FCSN predicts the complex
Fourier coefficients of the object’s masks, which requires the learning of
broader contextual information. Moreover, FCSN is more memory-efficient with
the absence of a decoder.

###  1.1 Encoder-decoder Segmentation Model

As shown in the first row of figure 1, modern segmentation models typically
adopt an encoder-decoder structure which models a conditional probability of
predicting label $y_{hw}$ given an input $\mathbf{x}$ at each spatial
coordinate $h,w$ ($i.e.$ $\text{p}(y_{hw}|\mathbf{x})$). The model is then
optimized to maximize the likelihood of the spatially summed log probability
($i.e.$
$\text{argmax}_{\text{p}}\sum_{hw}y_{hw}\log\text{p}(y_{hw}|\mathbf{x})$),
assuming spatial independence across the coordinates. Based on the structure
of the model and the way in which the model is optimized, the existing
encoder-decoder model will make a prediction mainly relying on local patterns
and often does not utilize the global context of the image at all. This
absence of global context can cause inconsistency in segmentation performance,
especially for the tasks that assume specific global priors.

Most of the existing works on global context learning aim to solve the problem
by proposing a more flexible (general) model structure that offers the model
an opportunity of capturing global patterns [3, 4, 5]. However, offering the
opportunity does not necessarily mean that the model will explore the new
aspect of learning. There is a possibility that the model will still focus on
finding local shortcut evidence and hence fails to focus on the global
evidences [6]. Also, when the network is trained under a data constraint,
higher flexibility could negatively impact the model performance. In this
regard, we argue that increasing the model flexibility alone is an unstable
solution to the global context learning problem.

###  1.2 Contribution

We propose a novel segmentation model—Fourier Coefficient Segmentation Network
(FCSN) that lifts segmentation to a shape prediction task, where the shape is
represented as Fourier coefficients. As shown in figure 1, FCSN perceives the
segmentation mask as a smooth function in a complex domain, which can be
accurately approximated as complex Fourier coefficients. We use Fourier
Transform to extract the Complex Fourier coefficients of the contour of the
mask. Hence, FCSN learns the global shape of an object by predicting its
Fourier coefficients, and during inference, a contour is retrieved with
Inverse Fourier Transform.

To motivate how predicting Fourier Coefficients helps to learn global context,
imagine we want to segment an ellipse-shaped object, which can be precisely
described by three complex Fourier Coefficients $z_{-1},z_{0},z_{1}$. The
$z_{0}$ describes the center of the ellipse, and $z_{-1}$ and $z_{1}$
determine lengths and orientations of the semi-major and semi-minor axes.
Thus, for a DNN to make a precise prediction of the three coefficients, the
model must learn to perceive the whole ellipse as a single object. This is in
contrast to the traditional encoder-decoder model, where the model makes
predictions only by looking at the local structure of the object.

Also, we propose to add a Fourier differentiable spatial to numerical
transform (F-DSNT) module [7] to improve the accuracy of Fourier coefficient
prediction and also to reduce memory consumption. One could view the
coefficient prediction as a typical regression problem and introduce fully-
connected (FC) layers on top of the spatially flattened feature. However, FC
layers have several drawbacks: 1) they are over-parameterized, affecting the
generalizability, 2) it assumes a fixed input shape, and 3) the output range
is not bounded. Instead, DSNT drives the encoder module to produce heatmaps
that represent the probability distributions of Fourier coefficients. DSNT
does not introduce any trainable parameter and works with any input shape.

We evaluate the performance of FCSN on three Medical image segmentation tasks,
including skin lesion, optic disc, and optic cup segmentations. FCSN
outperforms state-of-the-art segmentation models such as DeepLab-v3+ and
U-Net+ when eveludated with Hausdoff Distance. Furthermore, as our model can
attend to global features, its performance does not degrade from local
perturbations such as contrast change, additive noise, or motion blur. Lastly,
our model is lightweight, requiring less computational cost by discarding the
decoder module that has been indispensable in the modern segmentation model
and incurs a considerable memory overhead.

Figure 2: Overview of the FCSN architecture (left) and differentiable spatial
to numerical transform (DSNT) (right)

##  2 Related Work

###  2.1 Encoder-decoder Models

FCN [8] and U-Net [1] were the early few DNN models that proposed encoder-
decoder structure for semantic segmentation. However, the two approaches often
produced noisy predictions that contained holes or non-smooth contours,
implying that the models failed to understand the global context. The issue
had been addressed broadly in two ways while preserving the encoder-decoder
structure: by 1) increasing the receptive field size and 2) introducing a
regularizer that penalizes non-smooth prediction.

####  2.1.1 Broader Receptive Field

For a unit in the prediction of a network, the theoretical receptive field
(TRF) of this unit refers to the region in the input image that contributes to
the prediction of this unit. For convolution neural networks, the TRF is
usually only a fraction of the input image, which depends on the architecture
and filter sizes of the networks. To make more global aware predictions, the
TRF must be large enough to cover the whole region that contains information
related to the prediction.

In the literature several methods have been proposed to increase TRF. In [4]
the authors proposed ParseNet which incorporated a global context feature that
is generated using a global pooling operation in feature embedding. In [9, 10]
the authors proposed non-local U-Nets which included Transformer modules [11]
to extract long-range features. In [2] the authors proposed DeepLab with
Atrous Convolution module that extracts features with varying receptive field
sizes using dilated convolution.

As observed in [12], the effective receptive field (ERF) can be very different
from theoretical receptive field. The ERF is defined as the collection of
pixels inside TRF that have non-negligible impact on the prediction. It is
found in [12] that for neural networks prior training, the ERF is usually
smaller than TRF, and a proper training is needed to enlarge ERF. Therefore,
models with large TRF may not be capable of effectively understanding global
context. In [13], the authors proposed the Lovász metric, which is a convex
function that approximates the Intersection over Union (IoU) metric. Since IoU
is calculated over the whole image, the proposed metric can facilitate global
learning.

####  2.1.2 Regularizing Prediction

Another approach to promote smooth segmentations is to adopt regularization on
the models or the predicted masks. In [14] the authors proposed the ACNN-Seg
for predicting high resolution segmentation masks from low resolution images.
They introduced an extra autoencoder (AE) network to regulate segmentation
outputs, such that the AE would produce similar features for both the
predicted masks and the ground-truths.

More recently, the authors in [15, 16] proposed to add spatial regularization
to softmax activation functions in order to minimize total variation of
predictions, such that the predicted masks are more robust to various local
perturbations in the images.

###  2.2 Segmentation via Shape

For image segmentation most DNNs make per-pixel predictions for segmentation
masks. One way to obtain more regularized prediction is to predict the shape
of the segmentation mask, which effectively reduce the output dimensionality
and complexity.

In [17, 18] the authors proposed DNNs that learn parametrization of boundary
curves via piecewise Bézier curves. However, the Bézier parametrization does
not necessarily converge to the true boundary curve. In [19], the authors
proposed to predict polar coordinates of sampled points on boundary curves for
instance segmentation.

There are not many DNN approaches that utilize Fourier transforms for
segmentation. In [20], the authors used DNN to learn Fourier coefficients of
sampled points on boundary curves for instance segmentation. However, they
regarded the $x$ and $y$ coordinates of boundary points as two sequences of
real numbers and applied Fourier transforms independently. In our approach, we
regard the boundary curve as a sequence in the complex domain, and we apply
Complex Fourier Transform only once to get Fourier coefficients.

##  3 Proposed Method

As shown in figure 2, our DNN model consists of four modules. The first module
$\text{CNN}_{\theta}$ is a feature extraction module that takes an image as
its input. Any standard CNN backbone can be adopted. The second module
$\text{UP}_{\theta}$ generates heatmaps which represent the discrete
probability distribution functions (PDF) of Fourier coefficients. The third
module F-DSNT “softly" picks up the most probable Fourier coefficient from
each of the PDFs. The last module FT recovers segmentation masks from the
predicted Fourier coefficients. To understand our approach, we explain how we
convert masks to Fourier coefficients first.

####  3.0.1 FT : Segmentation Masks to Fourier Coefficients

Let $Y$ be a binary segmentation mask. We regard $Y$ as a function on the
complex domain $D=\\{x+jy:-1\leq x,y\leq 1\\}$, where $Y(x+jy)=1$ for
foreground and $Y(x+jy)=0$ for background. Let $\alpha:[0,1]\to\mathbb{C}$ be
a parametrization of the boundary curve of foreground. We assume $\alpha$ is a
complex valued smooth curve with $\alpha(0)=\alpha(1)$. Given the boundary
curve $\alpha$, the region enclosed by $\alpha$ is the segmentation region.

The Fourier coefficients $\\{z_{n}\ \in\mathbb{C}\\}$ of the boundary curve
$\alpha(t)$ is defined by

$z_{n}=\int_{0}^{1}\alpha(t)e^{-2\pi jnt}\,\mathrm{d}t$ (1)

for $n=\dots,-1,0,1,\dots$, where $j$ is the imaginary unit. The original
boundary curve $\alpha$ can be fully recovered from the Fourier coefficients
$\\{z_{n}\\}$ by taking the Inverse Fourier transform defined by

$\alpha(t)=\sum_{n=-\infty}^{\infty}z_{n}e^{2\pi jnt}.$ (2)

Therefore, instead of making a direct prediction of the segmentation mask $Y$,
it is possible to predict the Fourier coefficients $\\{z_{n}\\}$ and recover
the mask $Y$ with Inverse Fourier transform.

Figure 3: Original mask (second row) and masks generated from boundary curves
with $21$ Fourier coefficients (third row).

Predicting Fourier coefficients forces the training of DNN to utilize global
context better. As suggested by equation (1), the Fourier coefficients, which
we predict, are obtained by integrating global information on the boundary
curve. This forces DNN models to learn the global context of an image better,
facilitating to make more spatially consistent segmentation.

It is usually sufficient to only learn to predict the lower Fourier
coefficients which encodes the location and the general shape of the boundary
curve $\alpha$. This is because the coefficients $\\{z_{n}\\}$ are
concentrated on small absolute values of $n$ when $\alpha$ is smooth: In fact,
if $\alpha$ is $k$-times continuously differentiable, then $z_{n}$ converges
to $0$ faster than $1/|n|^{k}$ for large $n$. Discarding higher Fourier
coefficients can be regarded as a regularization that smooths ground-truth
boundary curves. Figure 3 shows segmentation masks obtained by only taking
$z_{n}$ for $-10\leq n\leq 10$.

####  3.0.2 $\mathbf{UP_{\theta}}$ : Probability Distribution of Coefficients

Given a feature extracted from a raw input using a CNN module,
$\text{UP}_{\theta}$ generates heatmaps which represent the discrete PDFs of
possible Fourier coefficients. ($i.e.$
$\\{\text{p}(z_{n}|\mathbf{x})\\}_{-k}^{+k}=\text{UP}_{\theta}\circ\text{CNN}_{\theta}$).
$\text{UP}_{\theta}$ module consists of a 2D transposed convolution layer with
$2*k+1$ kernels, followed by a softmax activation across spatial axes. 2D
transposed convolution layer projects input feature to a higher spatial
resolution; thus, the generated heatmaps are more granular. We apply softmax
to normalize the heatmaps such that it is non-negative and sum to one.

####  3.0.3 $\mathbf{F-DSNT}$ : Selecting the Most Probable Coefficients

Finding the most probable coefficient from each discrete PDF ($i.e.$
$\hat{z}_{n}=\text{argmax}\;\text{p}(z_{n}|\mathbf{x})$) is not
differentiable. To make it differentiable, we adopt DSNT [7] which can be
viewed as a soft-argmax operation. This is done by calculating the
expectations of the PDFs. As shown in figure 2, the expectations are
calculated by performing a weighted sum of discrete PDF with real and
imaginary coordinate values.

For the original implementation of DSNT in [7], the PDFs are assumed to have
spatial range $[-1,1]\times[-1,1]$. In our model, we multiply the output of
our DSNT module with scaling constants estimated by checking the range of each
Fourier coefficient from the training dataset. This is equivalent to
increasing the resolution of PDFs for higher Fourier coefficients which are
usually close to zero.

####  3.0.4 Loss Function

Our loss function is a combination of weighted $L_{1}$ and $L_{2}$ losses plus
the Jensen-Shannon (JS) divergence regularization. Given a batch of $M$ input
images $\\{\mathbf{x}^{(m)}\\}$, our predicted coefficients
$\\{\hat{z}_{n}^{(m)}:-k\leq n\leq k\\}$, and the ground truth Fourier
coefficients $\\{z_{n}^{(m)}:-k\leq n\leq k\\}$, the loss function is

$\displaystyle\text{Loss}(z_{n},\hat{z}_{n})=\frac{1}{M}\sum_{m,n}\Big{\\{}$
$\displaystyle
w_{n}\left(|\hat{z}_{n}^{(m)}-z_{n}^{(m)}|+|\hat{z}_{n}^{(m)}-z_{n}^{(m)}|_{2}^{2}\right)$
(3)
$\displaystyle+\text{JS}(p(\hat{z}_{n}^{(m)}|\mathbf{x}^{(m)})||\mathcal{N}(\hat{z}_{n}^{(m)},\sigma
I_{2}))\Big{\\}},$

where $p(\hat{z}|\mathbf{x})$ is the PDF generated by our $\text{UP}_{\theta}$
module. The $w_{n}$’s are weight constants that we introduce to promote the
learning of higher Fourier coefficients which are much smaller than lower
coefficients, defined as

$w_{n}=\min\left\\{1+\frac{1}{\max_{i}|z_{n}^{(i)}|+\varepsilon},10\right\\}.$

The $\text{JS}(p(\hat{z}|\mathbf{x})||\mathcal{N}(\hat{z},\sigma I_{2}))$ is
the JS divergence between the PDF $p(\hat{z}|\mathbf{x})$ and the bivariate
normal PDF $\mathcal{N}(\hat{z},\sigma I_{2}))$ with the same mean. The
covariance $\sigma$ of the bi-normal PDF is a hyper parameter. The JS
regularzation is minimized when the heatmap matches with the Gaussian
distribution, thus making sure our heatmaps of Fourier coefficients are
unimodal and concentrate nicely around true locations of the Fourier
coefficients.

##  4 Experiments

###  4.1 Evaluation Metrics

(a)

(b)

(c)

(d)

Figure 4: (a) Ground truth, (b)–(d) three predictions with the same dice value
$0.9$ but Hausdorff distances (smaller is better) $11.3$, $26.9$ and $93.3$
respectively. Note that the star shape in (b) is smaller than that in (a).
Table 1: Dice & Hausdorff comparison between FCSN and baseline encoder-decoder
models on 3 tasks. The standard deviation (std) is computed from 5-fold
results. The best result is in bold.

ISIC_2018 RIM_CUP RIM_DISC Models Haus$\pm\text{std}$ Dice$\pm\text{std}$ Haus
Dice Haus Dice # Parameter (M) # Flops (G) UNet 25.00 $\pm$ 1.00 0.89 $\pm$
0.01 22.35 0.78 10.78 0.96 31.39 55.84 UNet+ 24.06 $\pm$ 0.80 0.89 $\pm$ 0.01
22.25 0.77 11.79 0.96 36.63 138.16 DeepLabV3 (ResNet50) 20.79 $\pm$ 1.09
$\mathbf{0.90\pm 0.01}$ 21.69 0.77 10.92 0.96 39.63 40.99 DeepLabV3+
(ResNet50) 20.80 $\pm$ 0.76 $\mathbf{0.90\pm 0.01}$ 22.25 0.77 11.27 0.96
39.76 43.31 DeepLabV3+ (ResNet50 + Lovász) 20.44 $\pm$ 1.34 $\mathbf{0.90\pm
0.01}$ 20.60 0.78 10.63 0.96 39.76 43.31 UNETR (VIT-B-16) 20.50 $\pm$ 0.76
$\mathbf{0.90\pm 0.01}$ 18.98 0.78 10.95 0.96 105.32 32.14 FCSN (ResNet50)
20.21 $\pm$ 0.88 0.88 $\pm$ 0.01 18.15 0.77 9.85 0.96 29.71 $\mathbf{23.54}$
FCSN (DResNet26) 20.14 $\pm$ 1.00 0.88 $\pm$ 0.01 18.07 0.77 9.59 0.96
$\mathbf{22.16}$ 83.01 FCSN (DResNet50) $\mathbf{19.14\pm 0.86}$ 0.88 $\pm$
0.01 $\mathbf{17.42}$ $\mathbf{0.78}$ $\mathbf{9.16}$ 0.96 29.71 98.11

Let $Y$ be a segmentation mask, and let $\hat{Y}$ be a mask predicted by a DNN
model. To measure model performance, we use both the dice metric and the
Hausdorff distance defined by

$\mathrm{H(Y,\hat{Y})}=\max\left\\{\sup_{Y(y)=1}d(y,\hat{Y}),\sup_{\hat{Y}(y)=1}d(y,Y)\right\\},$
(4)

where $d(y,Y)$ is the Euclidean distance from the point $y$ to the target in
$Y$, and $d(y,\hat{Y})$ is defined similarly. The smaller the Hausdorff
distance is, the better the approximation of $\hat{Y}$ is to $Y$, and
$\mathrm{H(Y,\hat{Y})}=0$ means $Y$ and $\hat{Y}$ coincides completely.

The dice metric is widely used in evaluating segmentation models. However, the
dice metric is not sensitive to changes of shape and topology of the masks.
This is demonstrated in figure 4, where (a) is the ground truth, and (b)–(d)
are three predictions with the same dice value $0.9$. However, it is clear
that Figure 4(b) gives the best segmentation, while the shape of the
segmentation in (c) is wrong, and the topology of the segmentation in (d) is
wrong. On the other hand, the Hausdorff distance is more sensitive to changes
in shape and topology, and it can successfully pick up the best segmentation.

###  4.2 Datasets

We use two publicly available medical image datasets: 1) ISIC-2018 [21] and 2)
RIM-ONE-DL [22].

####  4.2.1 ISIC-2018

ISIC-2018 dataset contains 2,594 and 100 dermoscopic images with ground truth
segmentation for training and validation, respectively. The test dataset is
not publicly available. Hence, following conventions of other papers using
ISIC, we report the final evaluation results using 5-fold cross-validation on
the training dataset.

####  4.2.2 RIM-ONE-DL

RIM-ONE-DL dataset contsists of 313 and 172 retinographies from normal and
glaucoma patients respectively. All images include a manual segmentation of
disc and cup that have been assessed by experts. The dataset contains 341 and
149 training and testing samples respectively. As suggested by the dataset
provider, we perform a simple train-test split evaluation.

###  4.3 Implementation Details

During training and inference, images are resized to have a size $256\times
256$. For data augmentations, we used ColorJitter, random crop, and random
flip for the RIM dataset, and we replaced random crop by resizing and random
crop for the ISIC dataset. For all our training, we trained for 500 epochs
with a batch size of $8$, and we used the Adam optimizer [23] with a learning
rate of $3e^{-4}$ without weight decay. To generate Fourier Coefficients, we
sampled $71$ points on boundary curves and used FFT to get the Fourier
Coefficients, where the model only learns $21$ lower coefficients ($i.e.$
$\\{z_{n}\\}_{-10}^{+10}$). All our codes will be made publicly available
later upon acceptance.

Figure 5: Computational efficiency of FCNS and baseline models in 4 different
aspects: Floating point operations per second (FLOPs), inference & train speed
(ms/img), and model size (M).

###  4.4 Results

####  4.4.1 Precise Shape Prediction

We compare the performance of FCSN with different backbone settings against
state-of-the-art segmentation models, including UNet [1], UNet+ [24] and
DeepLab-v3+ [2] with/without the lovász-softmax loss [13], and UNETR [25]
(UNet with Transformer). We perform experiments on 3 segmentation tasks: ISIC
skin lesion, RIM_CUP, and RIM_DISC. The model performance is accessed with
Hausdorff and Dice metrics.

As shown in Table 1, for all instances, FCSN achieves a significantly lower
Hausdorff score while maintaining a competitive Dice score, supporting that
the shape of generated mask closely matches with ground truth. Also, we
observe greater performance gain from RIM_CUP and RIM_DISC tasks that have
much smoother contours than ISIC. We note that the performance of FCSN
improves when we use DResNet [26] backbone that produces higher resolution
output. Also, using a deeper backbone further improves the performance.

Figure 6: Comparison of Effective Receptive Field (ERF)

####  4.4.2 Robustness to Perturbations

We test robustness of models to four types of perturbations at inference:
Gaussian noise, Salt & Pepper noise, contrast changes, and motion blur. We
chose Gaussian and Salt & Pepper noises because they are the most common
additive and impulsive noises respectively. Contrast change and motion blur
are typical degradation in medical images. The results are summarized in
figure 7, where level of perturbation increases along the x-axis. Comparing
with the DeepLab-v3+ (with Lovász loss) and the UNETR models, our method is
more robust, especially for the two noises, where our method can give almost
consistent predictions regardless of noise level; on the other hand, the
predictions of the DeepLab-v3+ model deteriorate heavily as noise level
increases. Metrics of results of the UNETR model are either similar to that of
DeepLab-v3+, or lie between the DeepLab-v3+ and our method.

Figure 8 shows examples of segmentation results for images with perturbations.
For images with noise or contrast change, the DeepLab-v3+ method omitted large
portions of target areas, and the UNETR failed to correctly segment the RIM
cup with Salt & Pepper noise, while our method consistently give reasonable
segmentation for all cases. For the image with motion blur, the DeepLab-v3+
and UNETR methods wrongly included large portion of background area. All the
predictions of the DeepLab-v3+ have either wrong shape or wrong topology. On
the other hand, our method gives satisfactory segmentation results.

Figure 7: Hausdorff distance (smaller the better) of inferences with
perturbations.  Red: DeepLab-v3+ (Lovász loss),  Green: UNETR,  Blue: FCSN.
FCSN is more robust to perturbations, especially for heavy noises. Figure 8:
Visual comparison of predicted masks with perturbations.

####  4.4.3 Global Context Awareness

Here, we empirically prove that the two major strengths of FCSN, precise shape
prediction and robustness to perturbations, indeed arise from the model’s
global context awareness. We propose to use the Effective Receptive Field
(ERF), initially proposed by Luo et al. [12], as the method to measure the
global context awareness of models. ERF measures how much each input pixel
contributes to the model prediction. Mathematically, this is done by computing
the partial derivative of an arbitrary output unit $y_{i}$ with respect to
input tensor $\mathbf{x}$ $\text{i.e}\;\partial y_{i}/\partial\mathbf{x}$,
measuring how much $y_{i}$ changes as $\mathbf{x}$ changes by a small amount.
ERF is therefore a natural measure of the importance of $\mathbf{x}$ with
respect to $y_{i}$.

Figure 6 shows the comparison of ERF for various models. We observe that FCSN
visually attains significantly bigger ERF size compared to baseline models
across all tasks, strongly supporting our global context awareness argument.

####  4.4.4 Computational Efficiency

We compare the computational efficiency of FCSN against baseline segmentation
models. Specifically, we measure models’ floating-point operations per seconds
(FLOPs), inference time (ms/img), training time (ms/img), and parameter number
(M). We compare FCSN with ResNet50 backbone against UNet, DeeLab with ResNet
50 backbone, and UNETR with VIT-B-16 backbone. During the measure of FLOPs,
inference & training time, we set the input size to 256 $\times$ 256\. The
results in Figure 5 shows the computational efficiency of FCSN in all of the 4
aspects. Comparing with the least performing model for each of the aspect,
FCSN requires 58% less FLOPs, 8$\times$ faster training and inference speed,
and 5$\times$ less parameter number. Note that the computation overheads from
Fourier transform and inverse Fourier transform are small, which are
equivalent to two 1d convolution layers with kernel size of 21 and input size
21. Empirically, these two transforms only take 0.05ms/img.

Our model has high computational efficiency because our model does not contain
a conventional decoder. For most segmentation models employing neural network
approach, they contain decoders which have several layers of 2d convolution
and up-sampling operations. This will introduce a large amount of model
parameters and heavy computations. On the other hand, our model only contains
the encoder, and the prediction of Fourier coefficients is based on the F-DSNT
layer, which incur little computation and does not contain learnable
parameters.

###  4.5 Ablation

####  4.5.1 Impact of DSNT

For comparison, we remove $\mathbf{UP_{\theta}}$ and $\mathbf{DSNT}$ parts of
our model and connect the feature maps from our backbone to FC layers to get
Fourier coefficients. Experiment results in table 2 show that for the Dice
metric, the DSNT approach consistently gives better results, while for the
Hausdorff metric, the DSNT approach gives better results in most of the cases.

|  |  | Epoch Number
---|---|---|---
Tasks | Metric | Heads | 100 | 200 | 300 | 400
ISIC | Dice | DSNT | 0.872 | 0.884 | 0.886 | 0.887
FC | 0.861 | 0.870 | 0.873 | 0.876
Haus | DSNT | 21.6 | 20.07 | 20.11 | 19.8
FC | 21.7 | 19.91 | 21.03 | 20.3
RIM_CUP | Dice | DSNT | 0.737 | 0.762 | 0.768 | 0.772
FC | 0.735 | 0.756 | 0.757 | 0.761
Haus | DSNT | 19.16 | 18.62 | 18.39 | 18.47
FC | 18.46 | 19.04 | 18.59 | 19.09
RIM_DISC | Dice | DSNT | 0.949 | 0.953 | 0.956 | 0.957
FC | 0.949 | 0.951 | 0.951 | 0.953
Haus | DSNT | 10.0 | 9.45 | 9.42 | 9.46
FC | 10.4 | 10.23 | 10.56 | 10.16

Table 2: Dice and Hausdorff metrics of our model with DSNT or FC head.

####  4.5.2 Impact of JS Divergence

We study the effect of the Jensen-Shannon divergence regularization on our
model by removing the regularization or by altering $\sigma$ in the covariance
$\sigma I_{2}$ of the 2d Gaussian PDF. As seen from table 3, the introduction
of the regularization greatly improves model performance, but our model is not
sensitive to the choice of $\sigma$.

##  5 Limitation and Future Works

There are a couple future research directions that can make the proposed FCSN
more robust.

####  5.0.1 3D shape learning

MRI and CT scans are 3D in nature. To apply the current FCSN structure to 3D
segmentation tasks, the 3D scan must be interpreted as independent slices.
However, the independent assumption across the slices could lead to an
inconsistent mask prediction. As a solution to this, one can generalize our
framework by modifying our 2D F-DSNT module to a 3D version of it.

| $\sigma=0.005$ | $\sigma=0.01$ | $\sigma=0.015$ | no JS
---|---|---|---|---
Dice | 0.887 | 0.884 | 0.886 | 0.843

Table 3: Dice on ISIC for our model with various regularisation. Figure 9:
Plots of real parts of Fourier coefficients for a batch of $256$ images. Blue
lines: Predictions by FCSN.  Red lines: Ground truth. Upper: First Fourier
Coefficients. Lower: 10th Fourier Coefficients.

####  5.0.2 High Variance of Higher Frequency Coeficients

Figure 9 shows that FCSN can give accurate predictions for the first Fourier
coefficients, but there are greater mismatch for the higher frequency
coefficients. One way to tackle this problem is to introduce multi-heads with
various resolutions, where the high resolution head can promote learning of
higher Fourier coefficients.

####  5.0.3 Multi-object Segmentation Task

To extend FCSN to multi-instance segmentation cases such as multi-organ
segmentation, one could regard FCSN as a segmentation head in MaskRCNN [27].

####  5.0.4 Learning other transforms

FCSN learns to predict Fourier Coefficients for segmentation, and it works
well for targets with smooth boundary. However, if the target boundaries
contain sharp corners, one may consider modifying FCSN to learn coefficients
from more general transforms, like wavelet or tight frame transform. The idea
is to use a proper family of base functions that are more efficient in coding
boundaries curves.

##  6 Closing Remarks

In this paper we propose FCSN, a novel and lightweight segmentation model that
segments an object by predicting the Fourier coefficient of the object’s
contour. Our model is designed to incorporate the global context of an image,
leading to more accurate segmentation that better preserve the shape and
topology of the object. Moreover, the global context awareness makes our model
robust to unseen local perturbations during inference.

Our approach is the first step towards a systematic study of performing
segmentation by predicting coefficients of mask decomposition. There are many
other approaches besides predicting Fourier coefficients. For instance, one
can use wavelet or tight frame transforms to obtain more efficient
decomposition for boundary curves with sharp corners.

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# LINEAR KINETIC EFFECTS OF CORE PLASMA ON
LOW FREQUENCY ALFVÉN AND ACOUSTIC
EIGENMODES IN TOKAMAKS

I. CHAVDAROVSKI
Korea Institute of Fusion Energy, Daejeon, Republic of Korea
Email<EMAIL_ADDRESS>F. ZONCA
C.R. ENEA Frascati, Rome, Italy  L. CHEN
Dept. of Physics and Astronomy, Univ. of California, Irvine CA, U.S.A.
Institute for Fusion Theory and Simulation, Zhejiang University, Hangzhou, PR
China

###### Abstract

The resonant and non-resonant effects of core plasma on the excitation of low
frequency modes with $\omega<\omega_{BAE}$, such as Beta-induced Alfvén
Acoustic Eigenmodes (BAAEs) and Kinetic Ballooning Modes (KBMs) are examined
in the framework of the generalized fishbone-like dispersion relation. The
formalism of the fishbone-like equation contains all the necessary ingredients
to describe the features of these low frequency fluctuations, and explain
experimental findings. Core plasma properties (diamagnetic frequency and
precession resonance with trapped ions) strongly affect the excitation of the
modes, and in the case of BAAEs more effectively than the energetic particles.
The diamagnetic frequency of the core plasma also contributes to the coupling
of the BAAEs with the KBMs, thus affecting the excitation and polarization of
both modes. Energetic particles can still provide a non-resonant drive to some
of the low frequency modes.

##  1 INTRODUCTION

Electromagnetic modes with frequencies of the order of thermal ions periodic
motion $\omega_{Ti}=(T_{i}/m_{i})^{1/2}/(qR_{0})$ and
$\omega_{Bi}\sim\epsilon^{1/2}\omega_{Ti}$ and precessional motion of trapped
particles $\overline{\omega}_{Di}$, as well as their diamagnetic frequency
$\omega_{*pi}=(T_{i}c/e_{s}B)(\textbf{k}\times\textbf{b})\cdot\nabla(T_{i})/n_{i}\,$,
have been observed on several machines, such as DIII-D and HL-2A tokamaks. One
of these modes named Beta-induced Alfvén acoustic eigenmode (BAAE) has been
theoretically described by MHD theory [1] and shown to have a mixed Alfvénic
and acoustic polarization and frequency
$\omega_{BAAE}\sim(T_{e}/T_{i})^{1/2}\omega_{Ti}$. Several experiments have
reported resonant energetic particle (EP) excitation of low frequency BAAEs,
and wrongly identified them as a cause of EP losses, on a basis of their
frequency range. However, theoretical analysis shows that energetic particles
are not efficient in resonantly driving the BAAEs [2]. Moreover, the BAAEs are
heavily damped by thermal ions and parallel electric field [3, 4], and not
easy to excite. These modes are however coupled with the Kinetic Ballooning
Mode (KBM) branch of low frequency Alfvénic oscillations [5] with frequency
close to $\omega_{*pi}\,$, but significantly smaller damping rate. Diamagnetic
effects of core plasma contribute to the coupling of the BAAEs and KBMs, thus
affecting the excitation and polarization of both modes, and in the case of
BAAEs significantly reducing the damping rate [4]. Precession resonance with
trapped ions which occurs in this frequency range, is expected to have a more
important role for the excitation of the low frequency modes than energetic
particles.

Additionally, non-resonant effects of the background (or hot) plasma might
provide enough drive to destabilize low frequency modes, in which case the
mode frequency is not determined by particle resonance, but by the thermal
plasma diamagnetic frequency $\omega_{*pi}$ [6]. The modes previously
recognized as BAAEs in experiments are much more likely to be excited in this
way, and hence should be identified as reactive modes. A significant number of
experiments on DIII-D tokamak [6] have shown this type of modes appearing in a
”Christmas lights” pattern due to the time evolution of the $q$-profile which
pases through several rational numbers $q=m/n$, with $m$ and $n$ being the
poloidal and toroidal numbers, respectively. Common feature of these modes,
labeled Low Frequency Modes (LFMs) is that they appear near $q_{min}$ of a
reversed shear profile, and they are driven by high electron temperature
$T_{e}$, but modest $\beta_{i}$. Their polarization is Alfvénic and the
instability is reactive in nature.

The closeness of the modes to the ion resonant frequencies has prompted a
development of a kinetic treatment [3] of both circulating [7] and trapped
particles [8, 4] to properly describe these low frequency specimen. Due to the
large number of activities in this frequency range the identification of the
mode has to include the frequency, damping rate and its polarization. The
latter is however difficult to determine experimentally, which is one of the
reasons why analytical tools such as the Generalized fishbone-like dispersion
relation (GFLDR) is useful for systematic study of these low frequency modes.
In this framework the dispersion relation of the mode is given by [5]

$i\Lambda(\omega)=\delta\bar{W}_{f}+\delta\bar{W}_{k}\;\;,$ (1)

which is a unifying picture for various Alfvénic fluctuations, as well as
Energetic particle continuum modes (EPMs). Here, $\Lambda$ is the general
inertia and represents the physics inside the inertial layer, while
$\delta\bar{W}_{f}$ and $\delta\bar{W}_{k}$ are the ideal region background
MHD and energetic particle kinetic contribution, respectively. The scope of
this work is to show that the GFLDR in its most general form contains all the
ingredients necessary to describe the behavior of the low frequency modes and
their mutual interaction.

##  2 THEORETICAL MODEL

We adopt the ballooning formalism in the space of the extended poloidal angle
$\theta$, for low $\beta=O(\epsilon^{2})$ axisymmetric $(s,\alpha)$ plasma
equilibrium, with $s=rq^{\prime}/q$ being the finite magnetic shear and
$\alpha=-R_{0}q^{2}\beta^{\prime}$ the balloning drive. We describe the plasma
with three fluctuating scalar fields: $\delta\phi$, $\delta\psi$ and $\delta
B_{\parallel}$. Eq. (1) is obtained by asymptotic matching of the ”singular”
inertial layer and ”regular” ideal region solutions of the eigenmode
(vorticity) equation for $\delta\psi$ at the modes rational surface [5]. In
the inertial layer the fields vary on a short scale $\theta_{0}\sim O(1)$ and
on a long scale $\theta_{1}\sim(\beta^{1/2})$, while their governing equations
in the long wavelength limit are the vorticity equation [5, 7]

$\displaystyle B\textbf{b}\cdot\nabla\left[{1\over
B}{{k_{\perp}}^{2}\over{k_{\vartheta}}^{2}}\textbf{b}\cdot\nabla\delta\psi\right]+{\omega^{2}\over
v_{A}^{2}}\left(1-{\omega_{\ast
pi}\over\omega}\right)\frac{k_{\perp}^{2}}{k_{\vartheta}^{2}}\delta\phi+{\alpha\over
q^{2}R^{2}}g(\theta)\delta\psi=\Bigl{\langle}\frac{4\pi
e}{k_{\vartheta}^{2}c^{2}}\omega\omega_{di}\delta K_{i}\Bigr{\rangle}\,,$ (2)

and quasi-neutrality condition

$\left(1+\frac{1}{\tau}\right)(\delta\phi-\delta\psi)=\frac{T_{i}}{ne}\langle\delta
K_{i}-\delta K_{e}\rangle\,.$ (3)

Here $\delta K_{i}$ and $\delta K_{e}$ represent the compressional parts of
the perturbed particle distribution function $\delta f_{s}$, for $s=i,e\,$,
while $\langle(...)\rangle=\int d\textbf{v}(...)$ denotes integration in
velocity space. The perpendicular $k_{\perp}$ and poloidal $k_{\theta}$ wave
numbers satisfy
$k_{\perp}^{2}/k_{\theta}^{2}=1+(s\theta-\alpha\sin\theta)^{2}$, while
$g(\theta)=\cos\theta+[s\theta-\alpha\sin\theta\,]\sin\theta\,$,
$\tau=T_{e}/T_{i}$ and $n_{e}=n_{i}=n$. Further, we will use the more
convenient variables $\delta\Phi=(k_{\perp}/k_{\theta})\,\delta\phi$ and
$\delta\Psi=(k_{\perp}/k_{\theta})\,\delta\psi$, which represented in series
of powers of $\beta^{1/2}$ give (see [7] and [8])
$\delta\Psi^{(0)}=\delta\Psi^{(0)}(\theta_{1})$, $\delta\Psi^{(1)}=0$ and
$\delta\Phi^{(1)}\simeq\delta\Phi^{(0)}+\delta\Phi_{s}(\theta_{1})\sin\theta_{0}$.
Here, $\delta\Phi_{s}(\theta_{1})\sin\theta_{0}$ is the sinusoidal a.c. term
of the perturbed electric potential, while
$\delta\Phi_{c}(\theta_{1})\cos\theta_{0}$ was dropped, since
$\delta\Phi_{c}\ll\delta\Phi_{s}$ [7, 8].

For trapped ions the distribution function is written as [8]

$\displaystyle\delta K_{i}=$ $\displaystyle-$ $\displaystyle
Qf_{0i}\left(\frac{e}{m}\right)_{i}\frac{1}{\omega-\overline{\omega}_{di}}\frac{k_{\theta}}{k_{\perp}}\left[\frac{\overline{\omega}_{di}}{\omega}\delta\Psi^{(0)}+\delta\Phi^{(0)}-\delta\Psi^{(0)}\right]$
(4) $\displaystyle-$ $\displaystyle
Qf_{0i}\left(\frac{e}{m}\right)_{i}\frac{1}{(\omega-\overline{\omega}_{di})^{2}-\omega_{b}^{2}}\frac{k_{\theta}}{k_{\perp}}\left[{\overline{\omega}_{di}}\delta\Phi^{(0)}\xi+(\omega-\overline{\omega}_{di})\delta\Phi_{s}\right]\sin\theta\,\,.$

The second term contains the $\overline{\omega}_{d}\pm\omega_{b}$ resonances
and is an essential contribution to the inertial layer dynamics, while the
first one containing $\overline{\omega}_{di}$ resonance is found to be of
order $\xi=k_{\perp}/k_{\theta}$ smaller than the second one. The ratio
$k_{\perp}/k_{\theta}$ is large in the inertial layer, but of $O(1)$ in the
ideal region, hence the precession resonance will not contribute to the
inertial layer dynamics, aside from the quasineutrality condition. In the
ideal region, however we can expect the thermal ion precession resonance to
appear on the right hand side of Eq.(1), and have effect on modes of the order
of $\overline{\omega}_{di}$, i.e. low frequency modes.

The quasineutrality condition in zeroth order gives [4]
$\delta\Phi^{(0)}=I_{\Phi}(\omega,\omega_{Di,e},\omega_{*i,e})\delta\Psi^{(0)}$,
where $I_{\Phi}(\omega,\omega_{Di,e},\omega_{*i,e})$ represents a deviation
from the ideal MHD limit $\delta E_{\parallel}=0$, obtained for $I_{\Phi}=1$.
For most of the spectrum of the Alfvénic fluctuations, except frequencies near
$\overline{\omega}_{D,ei}$, we have $I_{\Phi}\simeq 1$ thus recovering the
ideal MHD condition. Around the specific precession frequencies of the ions
and electrons, $I_{\Phi}\neq 1$ which gives a finite parallel electric field
as a result of the slow precessional motion of the electrons and ions in
opposite directions. This electric field might have important implications on
the collisionless micro-tearing effects, but also contribute to the loss of
particles on the trapped/passing border.

Next order ($\beta^{1/2}$) quasineutrality condition gives
$\delta\Phi_{s}(\theta_{1})=S(\omega,\omega_{Bi},\overline{\omega_{Di}},\omega_{Ti})\xi\delta\Phi^{(0)}$,
which recalling $\delta\Psi^{(1)}=0$ makes the function $|S|$ a measure of how
much the polarization deviates from pure Alfvénic due to the parallel a.c.
electric field. The polarization term $S$ was calculated from the
quasineutrality condition [8] to obtain:

$\hskip-20.00003ptS=-\frac{N_{1}(\frac{\omega}{\omega_{Ti}})+\Delta
N_{1}(\frac{\omega}{\omega_{Ti}})+\sqrt{2\epsilon}P_{2}(\frac{\omega}{\overline{\omega}_{Di}},\frac{\omega_{Bi}}{\overline{\omega}_{Di}})}{1+\frac{1}{\tau}+D_{1}(\frac{\omega}{\omega_{Ti}})+\Delta
D_{1}(\frac{\omega}{\omega_{Ti}})+\sqrt{2\epsilon}\left[P_{1}(\frac{\omega}{\overline{\omega}_{Di}},\frac{\omega_{Bi}}{\overline{\omega}_{Di}})-P_{2}(\frac{\omega}{\overline{\omega}_{Di}},\frac{\omega_{Bi}}{\overline{\omega}_{Di}})\right]}\,,$
(5)

where
$P_{1}(\omega/\overline{\omega}_{Di},\omega_{Bi}/\overline{\omega}_{Di})$ and
$P_{2}(\omega/\overline{\omega}_{Di},\omega_{Bi}/\overline{\omega}_{Di})$ come
from the trapped particles dynamics:

$P_{1}(\omega/\overline{\omega}_{Di},\omega_{Bi}/\overline{\omega}_{Di})=-2\frac{\omega^{2}}{\overline{\omega}_{Di}^{2}}\left[(1-\frac{\omega_{*n}}{\omega}+\frac{3}{2}\frac{\omega_{*T}}{\omega})G_{2}-\frac{\omega_{*T}}{\omega}G_{4}\right]\,,$
$P_{2}(\omega/\overline{\omega}_{Di},\omega_{Bi}/\overline{\omega}_{Di})=-2\frac{\omega}{\overline{\omega}_{Di}}\left[(1-\frac{\omega_{*n}}{\omega}+\frac{3}{2}\frac{\omega_{*T}}{\omega})G_{4}-\frac{\omega_{*T}}{\omega}G_{6}\right]\,,$

and we have denoted

$G_{n}=\frac{1}{\pi^{1/2}}\int_{-\infty}^{\infty}\frac{e^{-x^{2}}x^{n}}{(\omega/\overline{\omega}_{Di}-x^{2})^{2}-(\omega_{Bi}/\overline{\omega}_{Di})^{2}x^{2}}\,dx\,,$

for $n=2,4,6,8$. The $G_{n}$ integrals yield [8]

$\displaystyle G_{2}$ $\displaystyle=$
$\displaystyle\frac{\overline{\omega}_{Di}/\omega_{Bi}}{\Omega_{1}+\Omega_{2}}\left[\Omega_{1}Z(\Omega_{1})-\Omega_{2}Z(\Omega_{2})\right]\;\;,$
(6) $\displaystyle G_{4}$ $\displaystyle=$
$\displaystyle\frac{\overline{\omega}_{Di}/\omega_{Bi}}{\Omega_{1}+\Omega_{2}}\left[\Omega_{1}^{2}-\Omega_{2}^{2}+\Omega_{1}^{3}Z(\Omega_{1})-\Omega_{2}^{3}Z(\Omega_{2})\right]\;\;,$
$\displaystyle G_{6}$ $\displaystyle=$
$\displaystyle\frac{\overline{\omega}_{Di}/\omega_{Bi}}{\Omega_{1}+\Omega_{2}}\left[(1/2)(\Omega_{1}^{2}-\Omega_{2}^{2})+\Omega_{1}^{4}-\Omega_{2}^{4}+\Omega_{1}^{5}Z(\Omega_{1})-\Omega_{2}^{5}Z(\Omega_{2})\right]\;\;,$
$\displaystyle G_{8}$ $\displaystyle=$
$\displaystyle\frac{\overline{\omega}_{Di}/\omega_{Bi}}{\Omega_{1}+\Omega_{2}}\left[(3/4)(\Omega_{1}^{2}-\Omega_{2}^{2})+(1/2)(\Omega_{1}^{4}-\Omega_{2}^{4})+\Omega_{1}^{6}-\Omega_{2}^{6}+\Omega_{1}^{7}Z(\Omega_{1})-\Omega_{2}^{7}Z(\Omega_{2})\right],$

where $Z(x)=1/\sqrt{\pi}\int_{-\infty}^{\infty}e^{-y^{2}}/(y-x)\,dy$ is the
plasma dispersion function that contains the
$\omega=\overline{\omega}_{Di}\pm\omega_{Bi}$ resonances in the parameters
$\Omega_{1}$ and $\Omega_{2}$ given by

$\Omega_{1}=\frac{\frac{\omega_{Bi}}{\overline{\omega}_{Di}}+\sqrt{(\frac{\omega_{Bi}}{\overline{\omega}_{Di}})^{2}+4\frac{\omega}{\overline{\omega}_{Di}}}}{2}\,\,{\rm
and}\,\,\Omega_{2}=\frac{-\frac{\omega_{Bi}}{\overline{\omega}_{Di}}+\sqrt{(\frac{\omega_{Bi}}{\overline{\omega}_{Di}})^{2}+4\frac{\omega}{\overline{\omega}_{Di}}}}{2}\,\,.$

In Eq. (5), the functions

$D_{1}(x)=x\left(1-\frac{\omega_{*ni}}{\omega}\right)Z(x)-\frac{\omega_{*Ti}}{\omega}x[x+(x^{2}-1/2)Z(x)]$
(7)

and $N_{1}(x)=2(\overline{\omega}_{Di}/{\omega_{Ti}})N(x)\,,$ with

$\hskip-20.00003ptN(x)=\left(1-\frac{\omega_{*ni}}{\omega}\right)[x+(1/2+x^{2})Z(x)]-\frac{\omega_{*Ti}}{\omega}[x(1/2+x^{2})+(1/4+x^{4})Z(x)]\,\,,$
(8)

come from the well circulating particles dynamics [7]. These functions contain
the $\omega=\omega_{Ti}$ resonance and are strictly valid for well circulating
particles. In order to extend their validity virtually to $\omega\simeq 0$,
correction functions

$\Delta
D_{1}(x)=\frac{x}{\pi^{1/2}}\int_{0}^{\infty}e^{-y}\ln\left(\frac{x+\sqrt{2\epsilon
y}}{x-\sqrt{2\epsilon
y}}\right)\left[1-\frac{\omega_{*ni}}{\omega}-\frac{\omega_{*Ti}}{\omega}\left(y-\frac{3}{2}\right)\right]dy$
(9)

and

$\Delta
N_{1}(x)=\frac{\overline{\omega}_{Di}/\omega_{Ti}}{\pi^{1/2}}\int_{0}^{\infty}ye^{-y}\ln\left(\frac{x+\sqrt{2\epsilon
y}}{x-\sqrt{2\epsilon
y}}\right)\left[1-\frac{\omega_{*ni}}{\omega}-\frac{\omega_{*Ti}}{\omega}\left(y-\frac{3}{2}\right)\right]dy\;\;,$
(10)

were added [8]. The corrections also account for the finite trapped particle
fraction and reduce the circulating population by $\sqrt{2\epsilon}$. This
model is a mixture of deeply trapped and well circulating particles, so some
error is always expected when working in the low frequency regime. However, it
has been shown that this reduced model recovers well the low and high
frequency behavior of hot plasmas [8], and even gives new insights which are
consistent with the experiments.

The vorticity equation expanded to the second order in the inertial layer
reduces to the form
$(\partial^{2}/\partial\theta_{1}^{2})\delta\Psi^{(0)}+\Lambda^{2}\delta\Psi^{(0)}=0$,
where the general expression of $\Lambda^{2}$ can be written as [8]

$\Lambda^{2}/I_{\Phi}=\frac{\omega^{2}}{\omega_{A}^{2}}\left(1-{\omega_{\ast
pi}\over\omega}\right)+\Lambda^{2}_{cir}+\Lambda^{2}_{tra}\;,$ (11)

with $\Lambda^{2}_{cir}$[7] and $\Lambda^{2}_{tra}$[8] being the circulating
and trapped particle contributions, respectively. Here

$\Lambda^{2}_{tra}=\frac{\omega^{2}\omega_{Bi}^{2}}{\omega_{A}^{2}\overline{\omega}_{Di}^{2}}\frac{q^{2}}{\sqrt{2\epsilon}}\left[P_{3}+(P_{2}-P_{3})S(\omega,\overline{\omega}_{Di},\omega_{Bi},\omega_{Ti})\right]$
(12)

and

$P_{3}=-2\left[(1-\frac{\omega_{*n}}{\omega}+\frac{3}{2}\frac{\omega_{*T}}{\omega})G_{6}-\frac{\omega_{*T}}{\omega}G_{8}\right]\,\,.$

The circulating particle term in Eq. (11) contains the well circulating
response [7] and the previously mentioned corrections

$\displaystyle\Lambda^{2}_{cir}=q^{2}\frac{\omega\omega_{Ti}}{\omega_{A}^{2}}\left[\left(1-\frac{\omega_{\ast
ni}}{\omega}\right)\left(F\left(\frac{\omega}{\omega_{Ti}}\right)+\Delta
F\left(\frac{\omega}{\omega_{Ti}}\right)\right)-\frac{\omega_{\ast
Ti}}{\omega}\left(G\left(\frac{\omega}{\omega_{Ti}}\right)\right.\right.$
$\displaystyle\left.\left.+\Delta
G\left(\frac{\omega}{\omega_{Ti}}\right)\right)+\frac{\omega\omega_{Ti}}{4\overline{\omega}_{Di}^{2}}\left(N_{1}\left(\frac{\omega}{\omega_{Ti}}\right)+\Delta
N_{1}\left(\frac{\omega}{\omega_{Ti}}\right)\right)S(\omega,\overline{\omega}_{Di},\omega_{Bi},\omega_{Ti})\right]\;,$

which, in the $\epsilon\rightarrow 0$ limit reduces to the well circulating
particle equations of Ref. [7] with:

$\displaystyle\hskip-40.00006ptF(x)$ $\displaystyle=$ $\displaystyle
x\left(x^{2}+3/2\right)+\left(x^{4}+x^{2}+1/2\right)Z(x)\,,$
$\displaystyle\hskip-40.00006ptG(x)$ $\displaystyle=$ $\displaystyle
x\left(x^{4}+x^{2}+2\right)+\left(x^{6}+x^{4}/2+x^{2}+3/4\right)Z(x)\,.$ (13)

The corrections are given by [8]:

$\Delta
F(x)=\frac{1}{\pi^{1/2}}\int_{0}^{\infty}e^{-y}\ln\left(\frac{x+\sqrt{2\epsilon
y}}{x-\sqrt{2\epsilon y}}\right)\frac{y^{2}}{4}dy\;\;,$ (14)

and

$\Delta
G(x)=\frac{1}{\pi^{1/2}}\int_{0}^{\infty}e^{-y}\ln\left(\frac{x+\sqrt{2\epsilon
y}}{x-\sqrt{2\epsilon y}}\right)\frac{y^{2}}{4}\left(y-\frac{3}{2}\right)dy.$
(15)

The fluid limit ($\omega\gg\omega_{ti}$ and $\omega_{*}\rightarrow 0$) of Eq.
(11) is
$\Lambda^{2}=[\omega^{2}-\omega_{BAE}^{2}(1+\delta(\tau,q,\omega)]/\omega_{A}^{2}\,,$
which contains the known BAE accumulation point
$\omega^{2}_{BAE}=2T_{i}(7/4+\tau)/(m_{i}R_{0}^{2})$, with a higher order
correction $\delta(\tau,q,\omega)$ [9]. The Beta-induced Alfvén eigenmode is
located in the gap below the continuum and its frequency is a function of
$\omega_{*pi}$ [7, 4]. At low frequency the same equation gives the
Rosenbluth-Hinton inertia enhancement [10, 11]

$\Lambda^{2}=\frac{\omega^{2}}{\omega_{A}^{2}}\left(1-{\omega_{\ast
pi}\over\omega}\right)(1+1.6q^{2}\epsilon^{-1/2}+0.5q^{2})\,\,,$ (16)

which contains the known ion polarization term, in addition to the ion
compression in toroidal geometry, i.e. neoclassical enhancement. The term
$I_{\phi}$ in Eq. (11) acts as an additional inertia enhancement due to the
slow precessional motion of ions and electrons around the torus in opposite
directions.

Several different derivations of $\delta\bar{W}_{f}$ are known, one of which
shown here [12]

$\delta\bar{W_{f}}\simeq\frac{\pi}{|s|}\left[\frac{s^{2}}{4}-\frac{3\alpha|s|}{2}+\frac{5\alpha^{2}s^{2}}{32}+\frac{45\alpha^{4}}{128}-(1+\frac{\alpha}{2})e^{-\frac{1}{|s|}}\right]\;,$
(17)

is valid for $|s|\sim\alpha^{2}$ and small
$\alpha=(1+\tau)(1+\eta_{i})q^{2}\beta_{i}/\epsilon_{ni}+q^{2}\beta_{E}/\epsilon_{nE}$,
where $\epsilon_{ns}=L_{ns}/R_{0}$ for $s=i,E$, and $L_{ns}$ are the density
gradient scales of the thermal and energetic ions. The $\delta\bar{W}_{f}$
term accounts for the thermal electron temperature in $\tau$ and non-resonant
EP effects ($\beta_{E}$), as well as the sharp spacial gradient of EPs through
$L_{nE}$. Detailed discussion about the $\delta\bar{W}_{k}$ term can be found
in Ref. [13]. Here, due to the low frequency we focus on the resonant
interaction of the modes with the particle precessional motion, neglecting the
bounce frequency resonance. A fairly exhaustive analysis of the modes can be
conducted by using the EP $\delta\bar{W}_{k}$ term of Ref. [13] for trapped
ions

$\hskip-20.00003pt\delta\bar{W}_{kt}=\frac{2\pi^{2}}{|s|}\frac{e^{2}}{mc^{2}}qR_{0}B_{0}\int
d\varepsilon\int
d\mu\left(\frac{\overline{\omega}_{D}}{k_{\vartheta}}\right)^{2}\left(1-\frac{\Delta_{b0}}{(1+\Delta_{b0}^{2})^{1/2}}\right)\frac{\tau_{b}\,QF_{0}}{\overline{\omega}_{D}-\omega}\,,$
(18)

where $Qf_{0}=(\omega\partial_{\varepsilon}+\hat{w}_{\ast})_{s}f_{0}$,
$\tau_{b}=2\pi/\omega_{b}$, while
$\Delta_{b0}^{2}=k_{\theta}^{2}(\rho_{L}^{2}+\rho_{b}^{2})/2$ accounts for
both finite Larmor radius ($\rho_{L}$) and finite banana orbit width
($\rho_{b}$). This term was originally derived for energetic particles [5],
but it also applies to thermal ions and electrons in the small FLR/FOW limit
($\rho_{L},\rho_{b}\rightarrow 0$). While for energetic particles the spacial
gradient term $\hat{w}_{\ast}$ is dominant, for thermal particles the energy
derivative is of the same order. Here, we remind the reader that for thermal
electrons the frequencies in $\delta\bar{W}_{kt}$ should be replaced with
$\omega_{*i}\rightarrow-\tau\omega_{*i}\,$,
$\overline{\omega}_{Di}\rightarrow-\tau\overline{\omega}_{Di}$ and
$\omega_{bi}\rightarrow\sqrt{\tau}\omega_{bi}$, giving another dependence of
the modes on the electron temperature. Since $\omega_{*p}$ and
$\overline{\omega}_{D}$ for electrons and ions are of the same order, both
$\delta\bar{W}_{kt}$ terms should be kept in the analysis on the righthand
side as $\delta\bar{W}_{ki}+\delta\bar{W}_{ke}$. However, due to the opposite
signs of the resonant frequencies they would likely affect different modes, at
different times. Eq. (18) for thermal particles has strong effects in low
frequency regimes ($\omega\sim\omega_{*p},\overline{\omega}_{D}$), but not
around $\omega_{BAE}$.

##  3 DISPERSION RELATION OF THE LOW FREQUENCY MODES

The GFLDR contains all the elements of the kinetic thermal ion gap necessary
to identify and study low frequency modes and their polarization, including
the background plasma MHD potential, resonant and non-resonant energetic
particle drive, as well as thermal ion and electron dynamics. Hence, the
equation as roots has both the reactive low frequency instability [6], as well
as several different modes in this frequency range like BAEs, EPMs, KBMs,
BAAEs and MHD modes [14]. For modes located at the vanishing shear, i.e.
$q_{min}$ of a reversed shear profile, the LHS of Eq.(1) should be expanded
around $s=0$, as done in Ref.[13]. The equation $\Lambda=0$ was solved in Ref.
[4] and the accumulation points were obtained for the three modes shown in
FIG. 1., without the presence of EPs. Since the functions in $\Lambda$ are
transcendental, additional roots in the negative complex plane appear, in
which case for each branch only the most unstable mode is physically relevant.
The roots numbered $1-3$ are mixed Alfvén-acoustic, and they do not show
excitation for the parameters considered in this work. For low $\eta_{i}$ and
$\Omega_{*}$ the KBMs is marginally stable and has a frequency close to
$\omega_{*pi}$, i.e, proportional to the $n$\- mode number up to values of
$\omega/\omega_{ti}\sim 1.5$. For increasing $\omega_{*pi}$ this behavior is
modified due to the coupling with the BAAE [4] and BAE [7], and the KBM is
becoming more damped. In FIG. 1. the KBM is shown for $\eta_{i}=1$ and
$\Omega_{*}=0.8$ marked with an ’x’ in the negative complex plane. The
coupling to the KBM is of great importance for the excitation of BAAEs since
they are shown to have a large damping rate [3], that only reduces for high
$\eta_{i}$ values [4].

FIG. 1.:  Accumulation points of BAE, KBM and BAAE in complex plane for
$\eta_{i}=0$, $\tau=1$ and $q=4/3$. The vertical scale is $\omega_{r}$,
horizontal $i\gamma$, thick red line $\gamma=0$. ’$\Delta$’ marks the BAAE for
$\eta_{i}=3.5$ and $\Omega_{*}=1$, while ’x’ marks the KBM for $\eta_{i}=1$
and $\Omega_{*}=0.8$.

The BAAEs are modes with mixed Alfvén-acoustic polarization. In our model, the
polarization of the modes is determined by the $S$ function of Eq.(5). Modes
with Alfvénic polarization have $\delta\Phi^{(0)}=\delta\Psi^{(0)}$ and
$|S|\sim\beta^{1/2}$ or lower, while modes with significant acoustic
component, such as BAAEs are identified by having $|S|\gg\beta^{1/2}$. In the
fluid limit of the theory, around the BAE accumulation point
$|S|\sim\beta^{1/2}$, which is also the case in the opposite limit
$\omega\ll\omega_{bi}$. This explains the vanishing of the $\tau$ term in the
Rosenbluth-Hinton Eq.(16). At low $\eta_{i}$ the KBM is purely Alfvénic, but
for increasing $\eta_{i}$ and $\Omega_{*}$ it becomes of mixed polarization,
which creates an issue with the identification of the mode as a BAAE or KBM.
Detailed discussion of this problem can be found in Ref.[4]. The increase of
the diamagnetic frequency can also change the polarization of the BAAE, while
decreasing its damping rate. In FIG.1. the BAAE mode for $\eta_{i}=3.5$ and
$\Omega_{*}=1$ is marked with ’$\Delta$’. Here, the BAAE is already unstable,
but its frequency is below the EP resonant frequencies and $\omega_{*pi}$. The
mode could be driven by a non-resonant EP drive coming from
$\delta\bar{W}_{f}+\delta\bar{W}_{k}$, as shown by the numerical simulations
with GTC code, but this might result with a change of the polarization to
Alfvénic [15]. This excitation is non-resonant, i.e. BAAEs would weakly affect
EP transport. Additionally, the BAAEs may be excited by the precessional
motion of thermal particles through $\delta\bar{W}_{k}$ in Eq.(18), confirmed
by GTC simulations that show power exchange between one of these low frequency
modes with the thermal ions at the precession frequency.

When discussing the accumulation points ($\Lambda$=0), the electron
temperature only appears through $\tau=T_{e}/T_{i}$ in the polarization term,
so when $S$ is negligible the mode is not affected by $T_{e}$. The KBM
frequency $\omega\simeq\omega_{*pi}$ at low $\eta_{i}$ is not correlated with
the electron temperature. Similarly the accumulation point of the BAAE at
$\eta_{i}=0$ is not strongly affected by $\tau$, even though the usual
assessment is $\omega_{BAAE}\propto(T_{e}/T_{i})^{1/2}$. BAE frequency is
strongly dependent of $\tau$ since the functions in the denominator of Eq.(5)
add up to $-1$, which brings forth the $1/\tau$ term. This is not the case for
BAAEs, so any significant effects of $T_{e}$ on this branch is not coming from
the accumulation point.

It is important to mention that both the terms $\delta\bar{W}_{f}$ and
$I_{\phi}$ also contain $\tau$, so the origin of the effects of $T_{e}$ and
$\nabla T_{e}$ on the modes should be investigated carefully. In the case of
the LFM in Ref.[6], increasing $T_{e}$ drives the mode, but increasing $T_{i}$
and $\nabla T_{i}$ has stabilizing effects, while the mode frequency is
closely following the $n$ mode number. The later is also valid for KBMs at low
$\omega_{*pi}$. The effects of electron temperature on these low frequency
modes can be fully explained when the RHS of Eq. (1) is taken into account, in
which case the mode is localized slightly off the rational surface. The
location of the accumulation point of the mode (rational surface) is
determined by the mode numbers and $q$-profile, while the mode frequency
itself by matching the real part of $i\Lambda$ with
$Re\,(\,\delta\bar{W}_{f}+\delta\bar{W}_{k})$. The imaginary part of
$\delta\bar{W}_{k}$ compensates for the damping on the LHS of GFLDR, i.e.
$Im(i\Lambda)=Im(\,\delta\bar{W}_{k})$.

##  4 CONCLUSIONS

The thermal particles play an important role in the dynamics of the low
frequency modes located in the kinetic thermal ion gap, even more than
energetic particles themselves. The low frequency Alfvénic mode closely
connected with the ion diamagnetic frequency known as KBM, is easier to excite
than the mixed polarization BAAE. The latter is however coupled with the KBM
through the ion diamagnetic frequency, and becomes marginally unstable only
for high $\Omega_{*}$ and $\eta_{i}$, in which case it could be non-resonantly
excited by energetic particles. This however can change its polarization to
Alfvénic. The precession resonance with thermal ions and electrons, as well as
the effects from $\delta\bar{W}_{f}$ due to energetic particle $\beta$ and
$\omega_{*}$ are worthwhile examining as a drive for some of these modes. Out
of the modes mentioned in this work, only the BAEs strongly interact with EPs
in tokamak plasma, and can be deleterious to their confinement. The LFMs are
known to be driven by high electron temperature, which can be traced to the
thermal part of the RHS terms in Eq.(1). The GFLDR is the proper description
to explain the linear excitation of the low frequency branches regardless
weather they are reactive of resonant, since it contains all the relevant
physics of fluctuations in the thermal ion kinetic gap. It is a general and
comprehensive tool for understanding mode structures at low frequencies and
explaining experimental observations.

ACKNOWLEDGMENT

This work is supported by R&D Program through the Korean Institute of Fusion
Energy (KFE) funded by the Ministry of Science and ICT of the Republic of
Korea (KFE-EN2141-7).

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