# Image Enhancement for Remote Photoplethysmography in a Low-Light Environment

Lin Xi1, Weihai Chen1, Changchen Zhao∗2, Xingming Wu1, and Jianhua Wang1
1 School of Automation Science and Electrical Engineering, Beihang University,
Beijing 100191, China 2 College of Information Engineering, Zhejiang
University of Technology, Hangzhou 310023, China<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Correspnding Author: Changchen Zhao, email<EMAIL_ADDRESS>
###### Abstract

With the improvement of sensor technology and significant algorithmic
advances, the accuracy of remote heart rate monitoring technology has been
significantly improved. Despite of the significant algorithmic advances, the
performance of rPPG algorithm can degrade in the long-term, high-intensity
continuous work occurred in evenings or insufficient light environments. One
of the main challenges is that the lost facial details and low contrast cause
the failure of detection and tracking. Also, insufficient lighting in video
capturing hurts the quality of physiological signal. In this paper, we collect
a large-scale dataset that was designed for remote heart rate estimation
recorded with various illumination variations to evaluate the performance of
the rPPG algorithm (Green, ICA, and POS). We also propose a low-light
enhancement solution (technical solution) for remote heart rate estimation
under the low-light condition. Using collected dataset, we found 1) face
detection algorithm cannot detect faces in video captured in low light
conditions; 2) A decrease in the amplitude of the pulsatile signal will lead
to the noise signal to be in the dominant position; and 3) the chrominance-
based method suffers from the limitation in the assumption about skin-tone
will not hold, and Green and ICA method receive less influence than POS in
dark illuminance environment. The proposed solution for rPPG process is
effective to detect and improve the signal-to-noise ratio and precision of the
pulsatile signal. Our dataset is available at
https://github.com/xilin1991/Large-scale-Multi-illumination-HR-Database.

###### Index Terms:

image enhancement, remote photoplethysmography, low-light environment, remote
heart rate monitoring technology.

##  I INTRODUCTION

The advances in semiconductor technology enables us to measure one’s
physiological parameters in a non-contact way just by means of simple
instrumentations such as consumer-level digital camera [1, 2, 3] and cellphone
[4]. The technology is usually called remote photoplethysmography (rPPG) or
image plethysmography (iPPG). The non-contact rPPG does not need any direct
physical contact with the subject, which introduces great convenience in
numerous applications such as driving monitoring, fitness monitoring[5, 6],
home healthcare [7, 8], face anti-spoofing [9], etc.

However, there exists one question that is crucial but has not gained much
attention: the impact of insufficient illuminance on the rPPG measurement. For
the experimental environment, the performance of the existing approach is
verified under well-lit conditions and there are no datasets acquired under
various illuminance to compare the accuracy of the existing rPPG algorithm. As
a consequence, monitoring heart rate during activities that occur in evenings
or insufficient light environments, such as sleeping, driving train or car,
etc., can remain a new challenge. But in the past, the researcher put less
attention to this issue and done not some work.

The challenges of extracting physiological signal in low light environment are
threefold. First, the region of interest (ROI) cannot be reliably detected.
Most object detection and tracking algorithms determine the target based on
rich features contained in the images, e.g., Scale-Invariant Feature Transform
(SIFT) [10], Histogram of Gradient (HOG) [11], or Convolutional Neural Network
(CNN) based features [12]. However, insufficient illuminance attenuates the
light intensity of the scene, resulting in not enough features inside the
target for the trackers or low color contrast between the target and the
background. Second, the physiological signal captured by the camera is
extremely subtle and this problem is more significant in low-lighting
conditions. A decrease in the amplitude of the pulsatile signal will lead to
the noise signal to be in the dominant position. Third, the pulse extraction
models may fail because their assumptions no longer hold. For example, the
skin reflection model [13, 14] relies heavily on the skin-tone direction and
the pulsatile color variation direction in RBG space. The directions will
change in low light condition because the signals in blue and red channels are
almost zeros.

In this paper, we propose a novel framework to extract rPPG pulse information
in low light condition. Inspired by recent progress in low-light image
enhancement techniques [15, 16, 17], we employ image enhancement algorithm
prior to pulse extraction for accurate heart rate measurement such that the
physiological signals would not be influenced by illumination and existing
rPPG methods can be combined directly with the processing pipeline. The
solution consists of three main steps: 1) video enhancement; 2) ROI detection
and tracking; and 3) heart rate extraction.

The main contributions of this article are summarized as follows:

1) To the best of our knowledge, we provide the first solution for robust rPPG
measurement under low light environment, which consists of an image
enhancement module and a pulse extraction module.

2) We are the first to systematically analyze the light illumination on the
rPPG measurement. The accuracy of several rPPG algorithms are tested on a wide
variety of light illuminance in terms of mean absolute error (MAE), root mean
squared error (RMSE) and signal-to-noise ratio (SNR) [14].

3) We build large-scale dataset with the illuminance covering a wide range
from 1 to 100 lux. To the best of our knowledge, there is no publicly
available datasets for the research of rPPG measurement under low light
environment.

The rest of this article is organized as follow: Section II introduces the
related works of existing publicly available remote heart rate datasets and
remote heart rate estimation. Section III presents a framework of low-light
video enhancement for remote heart rate measurement in details. Experimental
setup and results are reported in Section IV and V, respectively. Section VI
concludes the whole article.

##  II RELATED WORK

###  II-A Publicly Available Datasets for Remote Heart Rate Estimation

Many of the published methods conducted the experiment on private datasets,
which cause difficulties in fair comparisons of different methods. But there
are a lot of publicly available datasets that were designed for rPPG methods.
One of them is the COHFACE dataset [18], which is composed of 160 videos and
physiological signals collected from 40 healthy individuals. The environment
on the scene was changed once as to create two types of lighting conditions,
studio and natural. At the same time, Tulyakov et al. introduced a new dataset
MMSE-HR [19] , which was part of the MMSE database [20] and the subjects‘
facial expressions were more varied. Besides, the MAHNOB-HCI dataset [21] was
introduced by Li et al [22]. for evaluating their method performance. However,
the datasets mentioned above were either designed for other application or
published earlier.

TABLE I: A summary of publicly available datasets for remote heart rate
estimation

Name | Date published | Illumination | Subject | Recording devices | Number of videos | Size | Total duration
---|---|---|---|---|---|---|---
MAHNOB-HCI [21] | 2012 | Normal | 27 | RGB camera (60fps) | 522 | 25.2 GB | 527 min
COHFACE [18] | 2016 | Normal | 40 | RGB camera (20fps) | 160 | 368 MB | 160 min
MMSE-HR [19] | 2016 | Normal | 40 | RBG camera (25fps) | 102 | - | -
OBF [23] | 2018 | Normal | 106 | RGB camera (60fps)/NIR camera (30fps) | 212 | - | 1060 min
PBD-RPPG [24] | 2018 | 3 level illuminance | 3 | RBG camera | 21 | 16.8 GB | 63 min
VIPL-HR [25] | 2018 | Dim/Bright illuminance | 107 | RGB camera (25fps)/NIR camera (30fps) | 3203 | 48 GB (compressed) | 1189 min

In the last two years, there are also a few publicly available datasets
specially designed for the task of remote heart rate estimation. In 2018,
Xiaobai et al. proposed a database designed for HR and heart rate variability
(HRV) measurement [23]. Since this database aims at HRV analysis and all the
situations in this database are well-controlled, making it very easy for
remote HR estimation. Hoffman et al. released a public benchmark dataset for
testing rPPG algorithm performance (PBD-RPPG) [24]. In the datasets, Videos
accompanied by ECG measurements were recorded on three participants with three
different skin tones. Niu et al. introduced a large-scale multi-modal HR
dataset (named as VIPL-HR) [25], which contains 2451 visible light videos and
752 near-infrared videos of 107 subjects. As we can see from Table I, the
existing public-domain datasets for remote heart rate estimation are limited
in the normal light environment.

###  II-B Remote Heart Rate Estimation

Over the last decades, researchers have been investigating the feasibility of
remote heart rate estimation using a non-contact-based device such as a
camera. Pioneering research includes Verkruysse et al. [1] detect PPG signals
captured by regular color digital cameras. This research reveals the green
light (approximately 530 nm wavelength) has been found to give superior
results compared to red or blue light [1, 26] and also compared to infra-red
[27]. Poh et al. applied BSS-based model to remote heart rate estimation [2].
They separate the PPG signal from the red, green, and blue color channel in
the video via Independent Component Analysis (ICA) [2], and then Fourier
transformation was applied. Another kind of rPPG algorithm is based on the
skin reflection model [13] which exploits the interaction between light and
skin. The CHROM [14] uses a linear combination of the chrominance signals by
assuming a fixed skin-tone to white-balance the image. In the later work of
[14], Wang et al. comprehensively investigates the algorithmic principles of
rPPG in a mathematical context with optical and physiological reasoning and
proposes a new method, Plane-Orthogonal-to-Skin (POS) [13] , which project
original RGB signal to extract pulsatile signals for remote heart rate
estimation.

This article investigates the robustness of remote heart rate estimate
algorithms (Green, ICA, POS) of under various illumination conditions,
especially in the low-light environment, further study is carried out to
assess their performance under various light intensity.

##  III METHOD

We propose a solution for remote heart rate estimation under low-light
condition, which consists of three stages: low-light enhancement, ROI
detection and tracking, and remote heart rate estimation. The flowchart is
shown in Fig. 1. The purpose of the first stage is to enhance the input video
frame where visibility of the physiological signal is improved. The output is
an enhanced RGB image for each frame in the video sequence. Then the ROI that
contains the pulsatile signal will be obtained from enhanced video frames in
the second stage. The last stage is heart rate extraction from raw traces in
the ROI.

Figure 1: Overview of the improved low-light video enhancement solution for
rPPG.

###  III-A Low-light Enhancement

Regular remote heart rate estimation methods perform well in normal light
environment, but the regular pipeline may eventually lead to failure if the
videos are recorded under more challenging conditions, e.g., low-light
illumination. Inspired by Retinex model [28, 29], we can enhance the video in
the low-light environment via removing the impact of illumination to
facilitate the process of remote heart rate estimation. Referring to Retinex
model, the frame of low-light video is decomposed into reflectance and
illumination, which can be modeled as:

$\textbf{L}=\textbf{R}\circ\textbf{T}$ (1)

where L is the frame in the video, R and T represent the reflectance and the
illumination map, respectively. The operator $\circ$ denotes the element-wise
multiplication. By transforming (1), we can recover the enhanced result R by
the division is element-wise and T is the key to recover the enhanced image.
We observe that the green channel contains the strongest pulsatile signal
among the RGB color channels under low-light condition. Our initial
illumination map adopts the value of the green channel to prevent mixing of
noise from other channels. The operation is as follows:

$\widehat{\textbf{T}}\leftarrow\textbf{L}^{G}$ (2)

for each individual pixel, where G denotes green channels in the three color
channels.

In this work, we aim to preserve the over-all structure, smooth the textural
details, and amplify the pulsatile signal. To this end, based on our employed
initial illumination map $\widehat{\textbf{T}}$, we solve the following
optimization problem, which is proposed in LIME [17],

$\mathop{\min}_{\textbf{T}}\parallel{\widehat{\textbf{T}}}-\textbf{T}\parallel^{2}_{\textbf{{F}}}+\alpha\parallel\textbf{W}\circ\nabla\textbf{T}\parallel_{\textbf{1}}$
(3)

where $\parallel\cdot\parallel_{\textbf{{F}}}$ and
$\parallel\cdot\parallel_{\textbf{1}}$ denote the Frobenious and $\ell_{1}$
norms, respectively, $\alpha$ is the coefficient to balance the involved two
terms, and W is the weight matrix, and $\nabla\textbf{T}$ is the first order
derivative filter. In Equation (3), initial map $\widehat{\textbf{T}}$ refers
to green channel to preserve physiological signal.

###  III-B ROI Detection and Tracking

The purpose of stage 2 is to obtain the ROI that contains the pulsatile signal
from the enhanced frame of video. We first use the Viola-Jones detection
algorithm [30, 31] to detect the location of a face on the first enhanced
frame of the input video, then adopt Discriminative Response Map Fitting
(DRMF) method [32] to find the coordinates of 66 facial landmarks inside the
face region. We use $n=4$ points out of 66 landmarks to define our ROI and
generate a rectangular mask of the ROI as the yellow bounding box shown in
Fig. 2.

Figure 2:  ROI detection and tracking. The white maker symbol $*$ indicate 66
facial landmarks and we select $n=5,13,29,51$ points to define the initial ROI
boundaries. The yellow line shows the ROI boundary, inside which feature
points are detected and tracked.

The rules of defining the ROI is as follows: the first one is to exclude the
mouth and eye region; the second one is that the left and right boundaries of
the ROI do not exceed the cheek width. Therefore, we select $n=5,13,29,51$
points to define initial ROI boundaries, and then indent the boundaries of ROI
from initial boundaries above.

Then we use Staple tracker [33] to counter the problem of ROI movement for the
subsequent frames. Given the ROI of frame $i$, spatial averaging [2] is
applied to compute the average value of pixels in each color channel within
the ROI. This results in the raw traces:

$\textbf{C(i)}=\left[\vec{C}^{R}(i),\vec{C}^{G}(i),\vec{C}^{B}(i)\right]^{T}$
(4)

for $i=1,...,L$, where L is the number of frames in the video.

###  III-C Heart Rate Estimation

Three published rPPG algorithms (Green, ICA and POS) are used to extract the
blood volume pulse from the enhanced video to compare the performance of these
algorithms under low-light conditions.

Green (Verkruysse, Svaasand & Nelson, 2008 [1]): In this case, we first select
the green color channel vector $\vec{C}^{G}(i)$ from the raw traces. The
signal, $\vec{C}^{G}(i)$, was selected as the channel with the greatest
frequency power in the range 0.7 and 2.5 Hz.

ICA (Poh, McDuff & Picard, 2010 [2]): By referring to priors approach [34], we
detrended the raw traces based on smoothness with $\lambda=100$. Then, ICA
method is applied to the normalized detrended signal and filtered using a
zero-phase, 3rd-order Butterworth bandpass filter with pass band frequencies
of [0.7 2.5] Hz (corresponding [42 150] beat per minute, BPM).

POS (Wang, Brinker, Stuijk & de Haan, 2017 [13]): In this method a projection
plane orthogonal to the skin-tone is used for pulse extraction. A moving
window of length 1.6 seconds was set for the raw traces. For each window of
data, the raw traces data are normalized by their respective mean to give
$\widetilde{\textbf{C}}_{n}(i)\in\mathbb{R}^{3\times L_{W}}$, where $L_{W}$ is
the length of window. Next, the normalized color matrix,
$\widetilde{\textbf{C}}_{n}(i)$, was multiplied by the projection matrix P to
give $\textbf{S}(i)$. The resulting outputs of the window-based analysis was
used to construct the blood volume pulse signal in overlap add fashion.

##  IV EXPERIMENTAL SETUP AND IMPLEMENTATION DETAILS

This section presents the experimental setup for the benchmarking. First, a
self-collected large video dataset recorded under various illumination
conditions is introduced. Next, evaluation metrics are presented. Finally, we
present the parameters of the experiment.

###  IV-A Dataset

We recruited 15 healthy subjects (12 male, 3 female, 18 to 30 years old) in
this experiment and a total number of 165 video sequences were recorded under
various illuminations. The dataset will be available
publicly111https://github.com/xilin1991/Large-scale-Multi-illumination-HR-
Database. The detail of the experimental setup is described as follow:

Apparatus: Logitech HD pro webcam C930E color camera was used to record videos
of $640\times 480$ pixels, 30 fps, 60s, uncompressed bitmap format. PPG signal
was measured using CONTEC CMS50E.

Experimental environment: The video is captured in a darkroom as shown in Fig.
3, which is covered with a blackout cloth in order to isolate from ambient
light. A split type illuminometer, BENETECH GM1030 lux meter, is used as the
instrument for measuring light intensity and brightness in the darkroom.

Figure 3: The appearance of darkroom and experimental setup.

Participants: Fifteen participates were recruited in our experiment. In the
first step of the experiment, we measure the light intensity received in the
region of face via the illuminometer which is placed against the light source
and was positioned in front of the participant‘s face. During the experiment,
each of the 15 individuals was asked to sit 1 meter away from the camera, and
the camera was positioned at the participant‘s nose level as shown in Fig. 3.

###  IV-B Evaluation Metrics

Signal-to-Noise-Ratio (SNR) is adopted from [14] to evaluate the quality of
the extracted pulse signal in comparison with the ground truth signal. It is
derived by the ratio of the first two harmonic ranges to the remaining parts
of the pulse signal in the frequency domain. The valid frequency range
considered in SNR is [42 240] (equivalent to [0.7 4] Hz).

Heart rate accuracy. We extract the instantaneous heart rate estimation from a
set of 51 overlapping 10-second windows for each 1-minute video. The first
10-second windows in the video as initialization is used to calculate the
tenth-second instantaneous heart rate. We choose mean absolute error (MAE) and
root mean squared error (RMSE) to evaluate the accuracy of heart rate
estimation.

###  IV-C Parameters Determination

We chose to vary the illuminance for three compared rPPG methods (Green, ICA,
POS) in order to understand the influence of light intensity on accuracy of
pulse recovery. For completeness and anticipating effects on rPPG signal, we
varied illuminance in the range: {$10^{0},\ 10^{0.2},\ 10^{0.4},\ 10^{0.6},\
10^{0.8},\ 10^{1.0},\ 10^{1.2},\ 10^{1.4},\ 10^{1.6}\\\ ,\ 10^{1.8},\
10^{2.0}$} lux (equivalent to {1.0, 1.6, 2.5, 4.0, 6.3, 10.0, 15.8, 25.1,
39.8, 63.1, 100.0} lux).

TABLE II: Summary of the BVP SNR, MAE and RMSE for the rPPG algorithms under
various light intensity in original video and different enhancement screens.

Lux | $10^{0.0}$ | $10^{0.2}$ | $10^{0.4}$ | $10^{0.6}$ | $10^{0.8}$ | $10^{1.0}$ | $10^{1.2}$ | $10^{1.4}$ | $10^{1.6}$ | $10^{1.8}$ | $10^{2.0}$
---|---|---|---|---|---|---|---|---|---|---|---
SNR(dB) | Original | Green | -1.06 | -0.75 | 1.13 | 2.13 | 5.06 | 6.32 | 7.32 | 7.01 | 7.98 | 8.53 | 7.51
ICA | -0.50 | -0.08 | 1.18 | 2.72 | 6.03 | 6.61 | 7.83 | 8.98 | 10.15 | 12.33 | 11.65
POS | -2.9 | -2.54 | -1.01 | -0.87 | 2.59 | 4.72 | 6.00 | 7.17 | 8.23 | 9.49 | 9.20
HE | Green | -1.37 | 0.41 | 0.30 | 1.24 | 3.55 | 4.32 | 6.07 | 5.17 | 5.65 | 5.45 | 4.15
ICA | - | - | - | - | - | - | - | - | - | - | -
POS | -2.63 | -1.93 | -1.26 | -0.93 | 2.62 | 4.56 | 5.59 | 6.81 | 8.10 | 9.18 | 9.33
LIME | Green | -0.19 | 0.03 | 1.00 | 0.11 | 1.45 | 3.72 | 5.38 | 6.61 | 6.63 | 9.46 | 6.70
ICA | - | - | - | - | - | - | - | - | - | - | -
POS | -2.44 | -1.77 | -1.25 | -0.91 | 2.23 | 4.64 | 4.54 | 6.59 | 6.66 | 9.17 | 7.40
Improved | Green | -0.26 | 1.57 | 1.30 | 1.63 | 3.62 | 3.07 | 3.65 | 3.28 | 3.16 | 2.91 | 4.04
ICA | - | - | - | - | - | - | - | - | - | - | -
POS | -2.42 | -1.24 | -0.89 | 0.81 | 2.08 | 1.67 | 1.86 | 1.97 | 2.05 | 2.06 | 3.25
MAE(BPM) | Original | Green | 14.93 | 14.06 | 7.87 | 5.35 | 2.45 | 2.24 | 2.35 | 2.41 | 1.72 | 1.47 | 1.76
ICA | 14.96 | 14.54 | 8.04 | 6.76 | 3.59 | 3.39 | 2.32 | 2.34 | 1.80 | 1.17 | 1.69
POS | 59.89 | 51.38 | 29.37 | 23.98 | 5.11 | 1.70 | 1.87 | 1.27 | 0.97 | 0.86 | 0.94
HE | Green | 18.10 | 12.66 | 8.12 | 8.28 | 4.07 | 3.32 | 2.62 | 2.68 | 3.03 | 2.87 | 3.72
ICA | - | - | - | - | - | - | - | - | - | - | -
POS | 59.52 | 51.3 | 30.52 | 27.26 | 4.34 | 1.77 | 1.86 | 1.34 | 0.93 | 0.91 | 0.96
LIME | Green | 15.55 | 12.27 | 7.18 | 8.86 | 6.51 | 3.23 | 6.56 | 1.99 | 3.81 | 1.20 | 4.36
ICA | - | - | - | - | - | - | - | - | - | - | -
POS | 58.31 | 54.36 | 31.98 | 29.08 | 4.30 | 1.92 | 12.49 | 1.34 | 1.60 | 0.91 | 1.29
Improved | Green | 12.12 | 6.01 | 6.30 | 5.35 | 2.97 | 3.06 | 4.48 | 5.21 | 5.60 | 5.73 | 4.94
ICA | - | - | - | - | - | - | - | - | - | - | -
POS | 56.84 | 43.47 | 26.10 | 8.13 | 4.45 | 2.63 | 4.18 | 6.41 | 5.34 | 6.66 | 2.78
RMSE(BPM) | Original | Green | 18.98 | 18.32 | 11.91 | 8.71 | 4.13 | 3.10 | 4.20 | 3.88 | 3.09 | 2.89 | 2.63
ICA | 20.24 | 19.32 | 12.43 | 8.91 | 5.06 | 4.88 | 3.31 | 3.45 | 2.96 | 1.77 | 2.60
POS | 74.88 | 64.73 | 44.53 | 37.02 | 11.36 | 4.01 | 3.88 | 1.71 | 1.34 | 1.17 | 1.19
HE | Green | 22.57 | 16.65 | 12.11 | 12.65 | 6.30 | 5.78 | 4.40 | 4.92 | 5.73 | 5.47 | 6.99
ICA | - | - | - | - | - | - | - | - | - | - | -
POS | 73.95 | 62.83 | 43.80 | 38.48 | 10.26 | 3.79 | 3.47 | 1.81 | 1.30 | 1.25 | 1.21
LIME | Green | 19.69 | 17.10 | 10.95 | 13.32 | 11.28 | 5.33 | 8.09 | 3.08 | 5.77 | 1.81 | 5.88
ICA | - | - | - | - | - | - | - | - | - | - | -
POS | 74.83 | 66.84 | 45.59 | 41.35 | 9.76 | 4.00 | 15.88 | 1.80 | 3.17 | 1.24 | 1.89
Improved | Green | 17.18 | 10.13 | 9.96 | 9.56 | 4.74 | 5.73 | 7.52 | 7.35 | 8.30 | 9.56 | 8.02
ICA | - | - | - | - | - | - | - | - | - | - | -
POS | 74.31 | 57.33 | 40.31 | 16.17 | 8.49 | 4.88 | 8.46 | 9.96 | 8.03 | 13.04 | 4.42

##  V RESULTS AND DISCUSSION

###  V-A Evaluation of ROI Detection

Figure 4: Snapshots of partial original video in the benchmark dataset and
corresponding enhanced frame video. First column: original; second column: HE;
third column: Improved.

We applied the histogram equalization (HE) and the proposed image enhancement
algorithm to the low-light video on each illuminance level. Fig. 4 shows
comparison between original low-light images and corresponding enhanced images
at different illuminance level. After the above-mentioned operations, we
examine the Intersect over Union (IOU) score between the detection ROI on the
original video and the enhanced video frame, including HE and improved LIME.
The average IOU score at each illuminance level is shown in Fig. 5.

Figure 5: IOU result of original video and low-light enhanced videos.

The results show that there is a significant difference in obtained IOU score
between the original video and enhanced frames. Detection exemplars are
plotted in Fig. 6, where ROI detection results on the original image and
enhanced frames are plotted for comparison. It can be observed that the
estimated ROI given from enhanced frames are usually larger than the original
video in the overlapping area with ground truth, which means that most of the
ground truth region can be detected in the enhanced frames. Under all
illuminance levels, ROI detection on enhanced frames obtains IOU score $>0.7$,
but detection on original video have great fluctuations. Compared with
enhanced frames, including HE and Improved LIME, original frame is
significantly smaller than the former two at 1.0 to 2.5 lux, especially IOU
score is less than 0.5 at 1.0 lux. In this case, 5 out of 15 subjects were not
successfully detected ROI. It implies that the performance of face detection
algorithm was hurt by captured video under insufficient light. The gap between
the enhanced frames and the original frame has become less noticeable from 4.0
lux. There is not a very sensible difference between HE and Improved LIME, and
ROI detection on HE enhanced frame can get high precision in very low light
condition. Overall, ROI detection performance on the enhanced frames is stable
and reliable, the performance on low light video has gradually degraded to IOU
score $<0.5$ where there have no clear details of target scenes in the video
frame.

Figure 6: Detected ROI boundary. Yellow rectangle is represent ground truth;
red is detect ROI boundary on improved LIME enhanced video; green rectangle is
represent detection on HE enhanced video frame; and blue is represent the ROI
detection on original video.

###  V-B Performances of State-of-the-art rPPG Algorithms

We evaluate the performances of state-of-the-art rPPG algorithms under varied
illuminance to answer the following questions: 1) how much pulsatile signal is
preserved in the low-light video? 2) whether state-of-the-art rPPG algorithm
can extract heart rate accurately? To answer the first question, we show
spectrograms to compare the quality of the pulsatile signals from the red,
green and blue channel for all illuminance levels. Signal quality is evaluated
by the highest power spectrum distribution in the dominant heart rate. The
results are reported in Fig. 7.

Figure 7: Spectrograms obtained in red, green and blue channel on video
recorded under various illuminance intensity. The color red indicates the
highest amplitude and plot white $*$ symbol on this. Time is on the horizontal
axis, ranging from 0 to 1 minutes, the vertical axis shows the frequency from
0 to 300 BPM.

For all illuminance categories, the distribution of highest power extracted
from the green channel is continuous and the frequency of the highest power
concentrated on a range of ground truth heart rate. The highest power
distribution of red and blue channel is neither concentrated or continuous.
The red and blue channels contain stable plethysmographic signal starting from
15.8 lux. Simultaneously, we can see in Fig. 4 that the details and color of
the original video were recovered from 15.8 lux. But at less than 15.8 lux,
the pulse amplitude obtained in red and blue is small, the noise plays the
leading role. It suggests that the green channel contains the most stable
plethysmographic signal among the three recorded channels under all
illuminance. We also know from the spectrograms that signal quality extracted
from the green channel is highest no matter what the situation, while signal
quality extracted from the red and blue channel much less. In addition,
increasing the illuminance gradually strengthens pulsatile signal preserved in
the green channel. As the result shows, the green channel can preserve the
best pulsatile signal in all cases.

Figure 8: SNR, MAE and RMSE results as a function of illuminance intensity.
Figure 9: Plot of BVP SNR, MAE and RMSE for Green and POS for different low-
light enhancement algorithm.

Then, we investigate the varying tendency of state-of-the-art rPPG algorithm
accuracy and signal quality along with the illuminance in order to answer the
second question. The pulse signals are extracted on the raw video using Green,
ICA, and POS. The MAE, RMSE and SNR result alongside the illuminance are
listed in Table II. Fig. 8 show plots of pulse signal MAE, RMSE and SNR for
three heart rate estimation methods. Fig. 8 (a) shows clearly the SNR of all
methods increase step by step, in which the result given by ICA (blue line) is
better than the other two. It is notable that Green‘s result is less than POS
after 25.1 lux. For Green, ICA, and POS, the precision curve includes MAE and
RMSE observe a significant reduction until 15.8 lux as shown in Fig. 8 (b) and
(c), suggesting that insufficient lighting, such as low-light environment,
limited performance of remote heart rate estimation, and the state-of-the-art
algorithm is not capable of solving the issue in this situation. Especially,
the MAE and RMSE of POS are dropping from 59.90 to 1.70, 74.88 to 4.01. From
1.0 lux to 10 lux, the precision of Green is the highest, followed by ICA and
POS. Combining the spectrograms obtained from Figure. 8, we know the lack of
pulsatile signal contained in the red and blue channel is harmful to the
method rely on three color channels in case of the illuminance less than 15.8
lux. Furthermore, the noise involved in red and blue channels is mixed into
the process of pulse extraction. From 15.8 lux to 100.0 lux, the precision of
POS is the highest, followed by ICA and POS. Three heart rate estimation
algorithm achieves a slight improvement in precision under the illuminance
15.8 to 100.0 lux. In the case of brighter lighting, POS is the most reliable
method.

###  V-C Evaluation of Low-light Enhancement’s Influence

To assess the impact of low-light enhancement on the performance of the rPPG
measurement (Green, POS), we calculate the SNR in the recovered blood volume
pulse signal and the MAE, RMSE in pulse rate compared with the measurement
from the contact-based PPG signal. We apply HE, LIME and proposed image
enhancement algorithm, improved LIME, to enhance the frame from low-light
videos.

Table II shows the numerical results for SNR, MAE and RMSE alongside the
illuminance for each enhancement case. Fig. 9 show plots of the BVP SNR, MAE
and RMSE for different low-light enhancement algorithm. In each plot we show
the results for the raw video, HE, LIME and improved LIME enhanced frame. For
the Green, the SNR for proposed image enhancement is greater than the raw
video measurements at the illuminance values of 4.0 lux and below and is the
highest in the applied enhancement methods. The differences between raw video
and the proposed enhancement decrease as the illuminance increases. As
expected, increasing the illuminance steadily promote the SNR of resulting raw
video and proposed enhancement, but the differences of them are not
particularly obvious. The SNR for HE and LIME is not much different from the
raw video measurements at 1.0 to 4.0 lux. It’s worth noting that both HE and
proposed enhancement improve the accuracy of ROI detection, but only the SNR
of proposed method was increased. This result demonstrates that the proposed
enhancement method not only enhances the visibility of video but also improves
the quality of physiological signal. Above 4.0 lux, the enhancement effect of
the proposed algorithm is not very significant, and more noise is introduced.
Similar results are observed in POS, The improvement of SNR for proposed image
enhancement is more significant at the illuminance values of 6.3 lux and
below. Above 6.3 lux, the effect of the proposed enhancement algorithm on SNR
is not significant. The MAE and RMSE were also calculated for both Green and
POS extracted rPPG in the raw video, HE, LIME and improved LIME enhanced
frame. For Green and POS, the use of the proposed enhancement algorithm
performs better than using other enhancement methods below 6.3 lux and 10.0
lux, respectively. We can see the enhancement methods are more effective for
POS than Green, suggesting that low-light enhancement methods restore the
details in the picture, especially the skin-tone. The result also reflects the
proposed low-light enhancements method is more effective for the case in very
low illuminance.

##  VI CONCLUSION

Pulsatile signal preserved in the video could be damaged by insufficient
illuminance causing the failure of remote heart rate estimation. We performed
a systematic analysis of the illuminance to evaluate the impact on remote
heart rate estimation. We estimate the SNR, MAE and RMSE of state-of-the-art
rPPG algorithm (Green, ICA and POS) to verify their performance and robustness
on a wide variety of light illuminance. Our results suggest that Green and ICA
is more reliable than POS in low-light conditions. As the illuminance level is
increased, the precision of rPPG algorithm drops stably and the SNR goes up,
especially POS. Then we improved the existing low-light image enhancement for
remote heart rate estimation to render it suitable for remote heart rate
estimation. The results demonstrate that the precision of ROI detection and
remote heart rate estimation is increased in enhanced frame video. It reflects
that low-light enhancement improves the visible detail and gains a pulsatile
signal in the low-light video. We prove that the proposed solution can promote
the process of ROI detection, tracking and remote physiological measurements.
We believe that the improvement in SNR of the proposed solution can continue
to increase, which will be further investigated in our future work. Besides,
we build a large-scale dataset that was designed for remote heart rate
estimation recorded with various illumination variations. It facilitates
follow-up research in this area.

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11institutetext: Astronomical Institute of the Czech Academy of Sciences,
Fričova 298, CZ-25165 Ondřejov, Czech Republic 22institutetext: Charles
University, Astronomical Institute, V Holešovičkách CZ-18000, Praha 8, Czech
Republic 33institutetext: Center of Scientific Excellence - Solar and Stellar
Activity, University of Wrocław, Kopernka 11, 51-622 Wrocław, Poland

# Observations and modeling of spectral line asymmetries in stellar flares

J. Wollmann 11 2 2 P. Heinzel 11 3 3 P. Kabáth 11

###### Abstract

Context. Stellar flares are energetic events occurring in stellar atmospheres.
They have been observed on various stars using photometric light curves and
spectra. On some cool stars, flares tend to release substantially more energy
compared to solar flares. Spectroscopic observations have revealed that some
spectral lines, aside from an enhancement and broadening, exhibit asymmetry in
their profile. Asymmetries with enhanced blue wings are often associated with
the presence of coronal mass ejections while the origin of the red asymmetries
is currently not well understood. A few mechanisms have been suggested but no
modeling has been performed yet.

Aims. We observed the dMe star AD Leo using the 2-meter Perek telescope at
Ondřejov observatory, with simultaneous photometric light curves. In analogy
with solar flares, we model the H$\alpha$ line emergent from an extensive
arcade of cool flare loops and explain the observed asymmetries using the
concept of coronal rain.

Methods. We solve the non-LTE radiative transfer in H$\alpha$ within cool
flare loops taking into account the velocity distribution of individual rain
clouds. For a flare occurring at the center of the stellar disc, we then
integrate radiation emergent from the whole arcade to get the flux from the
loop area.

Results. We observed two flares in the H$\alpha$ line that exhibit red wing
asymmetry corresponding to velocities up to 50 km s-1 during the gradual phase
of the flare. Synthetic profiles generated from the model of coronal rain have
enhanced red wings quite compatible with observations.

###### Key Words.:

Stars: flares – stars: late-type – techniques: spectroscopic – radiative
transfer

##  1 Introduction

Stellar flares are energetic events occurring due to the energy release during
magnetic reconnection in the atmospheres of certain stars. Based on analogy
with the Sun it is commonly assumed that they are stellar counterparts of
solar flares. However, contrary to solar flares, we cannot spatially resolve
stellar flaring structures and thus we are tempted to use our global picture
of solar flares to understand stellar flares. This is particularly true when
we consider different geometrical and physical conditions in chromospheric
ribbons and extended hot or cool flare loops. Stellar flares were observed on
a variety of spectral types of stars (Pettersen 1989) but a majority of them
occur on cool dMe type stars. The energy released during flares on such stars
is typically of the order of 1031 – 1034 ergs, but can reach up to 1036 ergs
or even more in case of the so-called superflares (Shibata 2016). That is
about four orders of magnitude more as compared to the largest solar flares
with the energy of 1032 ergs. The reason is that dMe stars are expected to
have stronger and extended magnetic fields compared to the Sun (Crespo-Chacón
2015).

Most observations of stellar flares are photometric ones. These observations
may have a high temporal resolution even in the case of rather faint dMe stars
(which have low effective temperatures) and therefore are useful for the
determination of the flare occurrence and estimation of some flare properties
like the total energy released, for example (Pietras et al. 2022), (Doyle et
al. 2019) and (Medina et al. 2020). However, well-resolved spectral
observations must be used to study stellar flare dynamics, via the detection
of Doppler shifts or line asymmetries. Spectroscopic studies have revealed
significant changes in the spectra during a flare. Aside from changes in the
continuum and spectral line strengths, various chromospheric lines, namely
hydrogen Balmer lines, exhibit significant broadening and profile asymmetry.
The latter is well known as the so-called blue or red asymmetry with enhanced
wing intensities. These are usually assigned to dynamics of the chromospheric
condensation or evaporative processes, both in solar and analogical stellar
cases. Blue wing enhancements can be associated with the presence of coronal
mass ejections (CME), see e.g. Muheki et al. (2020b), Vida, Krisztián et al.
(2019) and Leitzinger et al. (2022). Recently Muheki et al. (2020a) showed
time evolution of the hydrogen H$\alpha$ line asymmetry where the red wing of
the emission line is enhanced in the case of AD Leo dMe star. A similar
enhancement was also detected by Wu et al. (2022) in various lines including
H$\alpha$. In both papers the authors suggest several possibilities to explain
such asymmetries, but no modeling was performed so far. An important
observation is that the red-wing enhancement appears at wavelength positions
that correspond to rather large Doppler velocities exceeding 100 km s-1 which,
in the case of solar flares, was never detected during the gradual phase. In
this paper, we present similar H$\alpha$ line observations of the star AD Leo
conducted with the Ondřejov Echelle Spectrograph (OES) attached to the 2-meter
Perek telescope. Because of large Doppler shifts during the gradual phase, we
suggest here that such red asymmetries are caused by downward flows of cool
plasma blobs along extended flare loops. Such phenomenon is well known in the
case of solar flares where it was called loop prominences, post-flare loops
(but see (Švestka 2007)) or recently ’a coronal rain’ (Antolin et al. 2010).
In order to quantitatively reproduce our observations, we develop an
approximate non-LTE radiative-transfer model to synthesize H$\alpha$ line
profiles from the spatially unresolved extended arcade of cool flare loops and
we compare the results of our simulations with OES observations.

##  2 Observations

Stellar flares can be observed via photometric light curves especially using U
and B filters. The typical shape of these light curves during a flare is a
sharp rise of the flux at the beginning of the flare followed by a gradual
fall to the preflare level. Flares can also be observed using the spectra of
the stars, for example using a light curve of an integrated flux of some
spectral lines and continua.

###  2.1 Dataset

For our study, we observed the dMe star AD Leo in three periods (observing
campaigns) during spring 2019, 2020, and 2021 using the 2-meter Perek
telescope at Ondřejov observatory (Czech Republic). Spectra were obtained
using Ondřejov Echelle Spectrograph (OES), a broad range spectrograph with
range 4250 Å–7500 Å for 2019 and 2020 observations and 3900 Å–9000 Å for 2021.
The resolving power of the spectrograph is $R$ $\sim$ 40000 in the H$\alpha$
region. Exposure times were 15 minutes for 2019 and 2020 observations and 10
minutes for 2021 to get an acceptable signal-to-noise ratio. With these
exposure times we are getting at least 9-10 signal-to-noise ratio in the
H$\alpha$ region in the center of the echelle order. Other properties of the
OES and Perek telescope are described by Kabáth et al. (2020). Spectra were
extracted and bias-flat corrected using standard IRAF procedures.

During the 2019 and 2021 campaigns, astronomers from the SPHE (section of
variable stars and exoplanets) group of the Czech Astronomical Society
simultaneously observed AD Leo using different photometric filters from
various locations in the Czech Republic. Reduced light curves by the observers
were provided to us with typical exposure times ranging from 30–90 s.

###  2.2 Line profiles

In order to quantitatively study changes of spectral line profiles during
flares, we need to calibrate the observed spectra to absolute radiometric
units. For that purpose, our calibration method requires a quiescent
calibrated spectrum of AD Leo. We observed a spectrum of ESO
spectrophotometric standard star HD 93521 (Oke 1990) during the campaign in
2020 and just after that a spectrum of AD Leo. From the ratio of the observed
spectra and known calibrated spectrum of HD 93521 we get calibrated quiescent
AD Leo spectrum $F_{\lambda}^{quiescent}$. We used light curves to ensure that
there were no variations (e.g. flaring activity) during exposure of the
quiescent spectrum.

Since the Earth’s atmospheric extinction can vary on timescales of flares and
we also expect that continuum flux changes during the course of a flare, we
needed to account for these variations during our calibration process. For
calibration of spectra during the flare, we assumed that the spectra observed
before (at time $t_{1}$) and after (at time $t_{2}$) a single flare correspond
to the calibrated quiescent spectrum we observed before. To determine these we
used the photometric light curves. Under this assumption, we apply linear
interpolation to calculate expected quiescent spectra during the flare between
just before and after the flare spectra. From the ratio of the observed
spectrum during the flare and calculated expected quiescent spectrum during
the flare we obtain spectra calibrated to absolute units during the flare. The
expression is following:

$F_{\lambda}^{flare}(t)=\frac{f_{\lambda}(t)}{f_{\lambda}(t_{1})+\frac{f_{\lambda}(t_{2})-f_{\lambda}(t_{1})}{t_{2}-t_{1}}\left(t-t_{1}\right)}F^{quiescent}_{\lambda}$
(1)

where $f_{\lambda}(t)$ is uncalibrated flux from telescope during the flare,
$f_{\lambda}(t_{1})$ is uncalibrated flux prior to the flare,
$f_{\lambda}(t_{2})$ is uncalibrated flux after the flare and
$F_{\lambda}^{quiescent}$ is calibrated quiescent flux.

During the flare, the continuum is varying. In the case of the Sun, this was
demonstrated many times, see Kuhar et al. (2016). In the case of stellar
flares, the visible continuum seems to be enhanced much more compared to the
Sun and in the case of strong flares can be fitted by a Planck function at
temperatures around 10000 K (Kowalski et al. 2013). However, all these
observations typically refer to flare ribbons without any discussion of the
possible contribution of the flare loops to the continuum. The only such study
was performed by Heinzel & Shibata (2018) who showed that the arcade of loops
may contribute significantly to the total stellar continuum enhancement. In
the present study, we do not focus on the analysis of the nature of the
continuum variations which may be caused by both the loops and ribbons. The
continuum is certainly enhanced during our observations which is manifested by
the optical light curves. In the observed flare spectra, the continuum
variations are present due to the flare itself and the atmospheric
transmission effects. To eliminate the latter, we interpolate between the
before-the-flare and the after-the-flare ”quiescent” spectra. Certainly, this
linear interpolation does not contain short-term atmospheric fluctuations
which can introduce uncertainties that affect the observed continuum levels.

Figure 1: Difference between H$\alpha$ profile during the flare and the
quiescent observed on AD Leo. We added a constant (dashed line) to every
spectrum increasing with time. We can see that the H$\alpha$ line flux is
enhanced and the profile is asymmetric with the enhanced red wing of the line.

Figure 2: Photometric light curve of AD Leo flare with marked exposure
intervals for spectra shown in Fig. 2. The light curve is a relative
difference in the magnitude in the B filter.

Figure 3: Difference between H$\alpha$ profile during the flare and the
quiescent observed on AD Leo. We added a constant (dashed line) to every
spectrum increasing with time. We can see that the H$\alpha$ line flux is
enhanced and the profile is asymmetric with the enhanced red wing of the line.

Figure 4: Photometric light curve of AD Leo flare with marked exposure
intervals for spectra shown in Fig. 4. The light curve is a relative
difference in the magnitude in the B filter.

We study the effect of flares on the H$\alpha$ line. To show this effect we
plot $\Delta F_{\lambda}$, the flux difference between calibrated line profile
during the flare and the quiescent calibrated profile:

$\Delta
F_{\lambda}(t)=F_{\lambda}^{\rm{flare}}(t)-F_{\lambda}^{\rm{quiescent}}$ (2)

where $F_{\lambda}^{\rm{flare}}(t)$ is calibrated flux observed during a flare
and $F_{\lambda}^{\rm{quiescent}}$ is quiescent calibrated flux. We observed
two flares with good spectra coverage that were not disturbed by other flares
and used our calibration method. Results are shown in the Fig. 2 and 4.
Additionally to show the correlation between changes of the profile of the
H$\alpha$ line and the photometric light curve in the B filter we plot spectra
exposure times along with the light curve in the Fig. 2 and 4. Color blocks in
Fig. 2 and 4 correspond to the exposure time of the spectra in the Fig. 2 and
4 respectively.

The light curve in Fig. 2 shows that the first spectrum was exposed during the
preflare phase and the next spectrum during the impulsive phase and partially
during the gradual phase. The H$\alpha$ line during the preflare phase shows a
little rise in the line center, see Fig. 2. That could be caused by a smaller-
scale flare (the small bump in the light curve) occurring at about 21:20 UT
just prior to the exposure of the first spectrum. The next spectrum shows a
very small decrease in continuum and the line compared to the quiescent flux.
This could be caused by additional uncertainties introduced by linear
interpolation used for calibration. During later stages, the continuum does
not appear to change much. In the later stages of the gradual phase, we can
see that the H$\alpha$ profile exhibits enhancement in the line center and in
the wings. During this flare, the red wing (positive velocity) is stronger
compared to the blue wing (negative velocity) within the range of 15 – 50 km
s-1 creating an asymmetrical profile.

The light curve of the second flare in Fig. 4 shows that the first calibrated
spectrum was exposed during the preflare phase. The corresponding H$\alpha$
profile shows no change in Fig. 4. Next exposure captured the spectrum during
the impulsive phase of a flare (the sharp rise in magnitude) but the H$\alpha$
profile only exhibits a little rise in flux. The next (green) spectrum shows a
little enhancement in the line center and also in the blue part of the wing
creating a small blue wing asymmetry at velocities 15 – 30 km s-1. Following
spectra were observed during the gradual phase (slow fall of flux to preflare
levels in light curves). During this phase, the flux in the H$\alpha$ line
begins to increase compared to the quiescent state. One can observe that
during this phase the red wing (positive velocity) of the line is slightly
higher than the corresponding velocity in the blue wing (negative velocity)
within the range of 15 – 50 km s-1 creating an asymmetrical profile.
Additionally the continuum exhibits an enhancement during the gradual phase.

##  3 Model

According to the standard model of solar flares, see e.g. Priest (2014), the
energy released during the magnetic reconnection creates a flux of high
energetic particles (electrons) that travel along the reconnected magnetic
field lines to lower layers of the solar atmospheres. In the region of higher
density, in the chromosphere, these particles transfer their energy to the
surrounding plasma by heating it. These regions are bright sources of hard
X-ray radiation and are known as flare ribbons. Heated plasma in the
chromosphere is pushed down forming the so-called chromospheric condensation
(with velocities of a few tens of km s-1) and simultaneously the upper
chromosphere and low transition region evaporate into the hot loop. These hot
loops then cool down and form cool loops visible in the H$\alpha$ line and
many other lines of different species e.g. Mikuła et al. (2017). Downflows in
the cool loops have much larger velocities than in the chromospheric
condensation and this looks consistent also with stellar observations. This
coronal rain is prominent in H$\alpha$ and high-resolution images and a movie
can be seen for example in Jing et al. (2016). We use this example of an
extended flare-loop arcade as a prototype of a stellar case.

In our model, we solve the non-LTE radiative transfer in the H$\alpha$ line
within flare loops that form an arcade-like structure and we only model a
situation when flare loops are already formed. The model is inspired by the
presence of a coronal rain that occurs on the Sun during solar flares. Our
model is based on two assumptions about the general properties of the flare.
Flare occurs in the center of the stellar disk with respect to the observer
and the plasma in flare loops is moving along the semicircular magnetic field
lines in a free fall. The arcade structure scheme is shown in Fig. 5.

Figure 5: Flare loop arcade structure. Semicircular field lines are shown in
black. The green and red stripes mark the stellar surface and plasma projected
onto the stellar surface in a single stripe is assumed to have the same
velocity. Coordinates are expressed in the radius of the semicircular field
lines.

Figure 6: Scheme for free fall solution of a cloud of plasma falling along the
semicircular magnetic field line. Plane Z = 0 represents stellar surface.

To solve the radiation transfer through the flare loops we use the cloud model
of plasma (plasma is structured into smaller clouds) with the same physical
properties (Tziotziou 2007). These clouds are moving in free fall along the
magnetic field lines towards the stellar surface with initial velocity v0.
Given this setup, the clouds with the same X coordinate in the Fig. 5 (along
the arcade) have the same velocity. This is indicated in the Fig. 5. The
clouds projected onto the stellar surface in the same stripe have the same
velocity. Velocity for a given X position can be obtained by solving Lagrange
equations for Lagrangian:

$L=\frac{1}{2}mr^{2}\dot{\varphi}^{2}-mgr\cos(\varphi)$ (3)

where $g$ is surface gravity acceleration of the star, $r$ is the radius of
the flare loop along which the cloud is moving and $\varphi$ is angle
describing cloud position in Fig. 6. We used explicit 5th order Runge-Kutta
method to solve the equation with the initial condition $\varphi(0)=0$ and
$\dot{\varphi}(0)=\frac{v_{0}}{r}$.

Since the clouds have the same physical properties except for relative
velocity towards the stellar surface we can calculate the spectrum of the
arcade as a sum of radiation over the flare area:

$I^{\rm{flare}}(\nu,\tau=0)=\frac{\sum^{n}_{k=1}I^{\rm{cloud}}(\nu,\tau=0,v_{k})A_{k}}{A^{\rm{flare}}}$
(4)

where $I^{\rm{cloud}}$ is the specific intensity of a cloud seen by a static
observer with respect to the stellar surface, $\nu$ is the frequency, $\tau$
is the optical thickness of clouds of plasma, $v_{k}$ is the vertical
component of the velocity of all clouds in a k-th stripe, $A_{k}$ is the area
of a k-th stripe (area composed of clouds with the same velocity and same
height) and $A^{\rm{flare}}$ is the total area of a flare. In our case it is
the area of the whole extended arcade as is discussed by Heinzel & Shibata
(2018). The radiation coming from clouds must be correctly Doppler-shifted.

###  3.1 Intensity of a single static cloud

The specific intensity of radiation from a single cloud is given as the
solution of the radiative transfer equation

$\frac{dI}{d\tau}(\nu,\tau)=S(\nu,\tau)-I(\nu,\tau)$ (5)

where $S$ is the source function. The formal solution of the equation is a
combination of two terms: the background disc radiation attenuated by the
plasma and the source term of the cloud. The exact formula is:

$I(\nu,\tau=0)=I(\nu,\tau_{0}(\nu))e^{-\tau_{0}(\nu)}+\int^{\tau_{0}}_{0}S(t(\nu))e^{-t(\nu)}dt(\nu)$
(6)

where $\tau_{0}(\nu)$ is the optical thickness of clouds at a given frequency.
Assuming optical depth at the line center $\tau_{0}$ we can express
$\tau_{0}(\nu)=\tau_{0}\phi(\nu)$ where $\phi(\nu)$ is line profile:

$\phi(\nu)=\frac{1}{\sqrt{\pi\Delta}\nu}e^{-\left(\frac{(\nu-\nu_{0})}{\Delta\nu}\right)^{2}}$
(7)

where $\nu_{0}$ is H$\alpha$ line-center frequency and $\Delta\nu$ is:

$\Delta\nu=\frac{\nu_{0}}{c}\sqrt{\frac{2k_{B}T}{m}+v_{t}^{2}}$ (8)

where $T$ is plasma temperature, $k_{B}$ is Boltzmann constant, $m$ is
hydrogen mass and $v_{t}$ is the turbulent velocity of plasma. We assume that
the background radiation incoming to clouds, $I(\nu,\tau_{0}(\nu))$, is equal
to the stellar quiescent radiation without flare.

We thus need to determine the source function S. For the H$\alpha$ line we use
here a two-level atom approximation which allows us to write the source
function as a sum of scattering term and thermal terms (e.g. Heinzel (2019))

$S=\left(1-\epsilon\right)\bar{J}+\epsilon B_{\nu_{0}}$ (9)

where $\epsilon$ is photon destruction probability, $B_{\nu_{0}}$ is Planck’s
function at the line center frequency with the temperature of the plasma in
clouds $T$ and $\bar{J}$ is the mean integrated intensity of the radiation
field inside the cloud. Following Rybicki (1984) we can split $\bar{J}$ as
$\bar{J}=\bar{J}^{\rm dif}+\bar{J}^{\rm dir}$, where $\bar{J}^{\rm dif}$ is
the diffuse part of the intensity and $\bar{J}^{\rm dir}$ is the part due to
direct external illumination of the cloud. Here we present an approximate
formula for $S$ which was obtained in a heuristic way by Heinzel & Wollmann
(2022, in preparation). Writing formally

$S^{\rm dir}=\left(1-\epsilon\right)\bar{J}^{\rm dir}+\epsilon B_{\nu_{0}}$
(10)

we get

$S(\tau_{0}/2)=\frac{S^{\rm
dir}(\tau_{0}/2)}{1-(1-\epsilon)[1-K_{2}(\tau_{0}/2)]}\,,$ (11)

where $K_{2}$ is the function evaluated numerically according to Hummer
(1982). Its value goes to one for $\tau$ approaching zero and decreases with
increasing $\tau$. We thus see that for very small $\tau$ S approaches $S^{\rm
dir}$ as expected. For optically-thin clouds, $\bar{J}^{\rm dir}$ is evaluated
directly as

$\bar{J}^{\rm dir}=\int^{\infty}_{0}J^{\rm inc}(\nu)\phi(\nu)d\nu$ (12)

where $\phi(\nu)$ is the absorption line profile. We can obtain the mean
incident radiation intensity simply using the dilution factor $W$ (e. g.
(Jejčič & Heinzel 2009))

$J^{\rm inc}(\nu)=WI(\nu)\,,$ (13)

where $I(\nu)$ is the stellar quiescent specific intensity and $W$ is

$W=\frac{1}{2}\left[1-\left(1-\frac{R^{2}}{\left(R+H\right)^{2}}\right)^{\frac{1}{2}}\right]\,.$
(14)

$R$ is the stellar radius and $H$ is the height above the stellar surface.
However, for general values of $\tau$ we multiply this direct mean intensity
by a correction factor as follows

$\bar{J}^{\rm dir}=\int^{\infty}_{0}J^{\rm inc}(\nu)\phi(\nu)d\nu\times(1-{\rm
e}^{-\tau_{0}/2})/(\tau_{0}/2)\,.$ (15)

Note that this factor goes to one for $\tau<<1$ and we get an optically-thin
value as in Eq. 12. The above-described approach gives a good estimate of the
H$\alpha$ line source function in externally illuminated clouds, to within a
factor of two. For details and comparisons with exact solutions see Heinzel &
Wollmann (2022, in preparation).

The photon destruction probability $\epsilon$ is:

$\displaystyle\epsilon$
$\displaystyle=\frac{\epsilon^{\prime}}{1+\epsilon^{\prime}}$ (16)
$\displaystyle\epsilon^{\prime}$
$\displaystyle=\frac{C_{32}}{A_{32}}\left(1-\rm{e}^{-\frac{h\nu_{0}}{k_{B}T}}\right)$
(17)

where $A_{32}$ is Einstein coefficient of spontaneous emission and $C_{32}$ is
the collision coefficient which can be approximated according to Johnson
(1972) as:

$C_{32}=n_{e}f(T)$ (18)

where $n_{e}$ is the electron density in the clouds and $f(T)$ is a weak
function of temperature. Values of $f(T)$ for selected temperature values are
listed in Table 1.

Table 1: Value of $f(T)$ for a few selected temperature values. $T$ [K] |  $f(T)$ [cm3 s-1]
---|---
8000 | 2.501 $\times 10^{-7}$
10000 | 2.529 $\times 10^{-7}$
15000 | 2.649 $\times 10^{-7}$
20000 | 2.777 $\times 10^{-7}$

Since the source function in our approximation is constant inside the cloud,
Eq. 6 simplifies to:

$I(\nu,\tau=0)=I(\nu,\tau_{0}(\nu))\rm{e}^{-\tau_{0}(\nu)}+S\left[1-\rm{e}^{-\tau_{0}(\nu)}\right]$
(19)

###  3.2 Intensity of a single moving cloud

When the cloud is moving with respect to the stellar surface we need to
account for Doppler shifts by modifying both terms in the equation 19. We
calculate the intensity coming from the cloud with the vertical component of
velocity $v$ towards the stellar surface in the reference frame connected to
the stellar surface.

Background radiation in the first term is already in the correct frame so it
does not shift but it is attenuated by the plasma in the cloud which ”sees”
the radiation at shifted (higher) frequency.

$I^{\rm{attenuated}}(\nu,\tau=0,v)=I(\nu,\tau_{0}(\nu))\rm{e}^{-\tau_{0}(\nu^{+})}$
(20)

where $\nu^{+}$ is:

$\nu^{+}=\nu\frac{c+v}{c}$ (21)

The second term represents the radiation coming from the cloud which must be
Doppler-shifted to the reference frame, the radiation cloud emits is seen by
the observer at lower frequencies. However, the scattering term in the source
function contains background radiation which for the calculation of
$\bar{J}^{\rm{dir}}$ must be shifted as well.

$I^{\rm{source}}(\nu,\tau=0,v)=S(v)\left(1-\rm{e}^{-\tau_{0}(\nu^{+})}\right)$
(22)

where $S(v)$ is computed using Eq. 10 but the $\bar{J}^{\rm{dir}}$ is due to
the Doppler shifts computed as:

$\bar{J}^{\rm{dir}}(v)=\int^{\infty}_{0}J_{\nu^{-}}^{\rm{inc}}\phi(\nu)d\nu\times(1-{\rm
e}^{-\tau_{0}/2})/(\tau_{0}/2)$ (23)

where $\nu^{-}$ is:

$\nu^{-}=\nu\frac{c}{c+v}$ (24)

The total intensity observed by the observer is a sum of these two terms:

$I^{\rm{cloud}}(\nu,\tau=0,v)=I^{\rm{attenuated}}(\nu,\tau=0,v)+I^{\rm{source}}(\nu,\tau=0,v)$
(25)

Using the formula 4 we can calculate the intensity of the whole arcade.

For the purpose of understanding how the background and source terms affect
the final synthetic spectrum we sum them over the whole arcade, like in Eq. 4:

$I^{\rm{background}}(\nu,\tau=0)=\frac{\sum^{n}_{k=1}I^{\rm{attenuated}}(\nu,\tau=0,v_{k})A_{k}}{A^{\rm{flare}}}$
(26)

and:

$I^{\rm{source}}(\nu,\tau=0)=\frac{\sum^{n}_{k=1}I^{\rm{source}}(\nu,\tau=0,v_{k})A_{k}}{A^{\rm{flare}}}$
(27)

###  3.3 Parameter summary

The model described above has a few parameters and they are summarized below.

We need to know the basic properties of the star on which the loop arcade
forms. Parameters that describe the star in our model are stellar radius $R$
and the quiescent specific intensity $I_{\nu}$.

The macroscopic description of the loop arcade requires the radius of the
semicircular loops $r$, the area of the arcade $A^{\rm{flare}}$ and the
initial velocity of clouds of plasma at the top of the arcade $v_{0}$.

For the description of the plasma in clouds, we use the following parameters:
plasma temperature $T$, the optical thickness of the clouds at the line center
$\tau_{0}$ and turbulent velocity $v_{t}$ in plasma which affects the profile
width $\phi(\nu)$.

##  4 Results

Results of the modeling described above are presented in this section as the
spectra computed for various values of the input parameters. For the quiescent
spectrum illuminating the clouds, we used a Gaussian profile which we obtained
by fitting the calibrated AD Leo observation in the H$\alpha$ region plus the
continuum fit. In reality, AD Leo H$\alpha$ does not have a strictly Gaussian
shape but exhibits a reversal in the line core, shown in the Fig. 7. However,
to demonstrate the effect of flows on our models we assumed that a schematic
input spectrum will provide basic results - note that the wing asymmetry is
mainly sensitive to the incident radiation outside the central reversal. The
fitted Gaussian spectrum is shown in Fig. 7. The stellar radius $R$ for AD Leo
we used is 0.39 R⊙ (Reiners et al. 2009).

Figure 7: The observed calibrated quiescent spectrum of the AD Leo and the
fitted Gaussian profile in the H$\alpha$ region. This spectrum is used as the
background radiation for our model. Compared to the observed AD Leo H$\alpha$
line this profile does not feature a small reversal in the line center.

###  4.1 Modeled profiles

We modeled H$\alpha$ line profiles for various input parameters and the
results are presented in the following grid of synthetic profiles. One
parameter is varied and the rest are fixed. We study the effect of parameters
that describe the plasma in our clouds. In the figures below we plot synthetic
specific intensities calculated using Eq. 4 with total source term and total
attenuated background term as described above.

Temperature variations for selected parameter values are plotted in Fig. 8.
For lower electron density and optical thickness (a – d panels), one can see
that the source term radiation does not change much with increasing
temperature. That is expected as at lower electron densities the photon
destruction probability $\epsilon$ is very low so the scattering term
dominates the source function. For higher electron density (e – h panels) the
higher photon destruction probability significantly amplifies the contribution
of the thermal term in the source function. Higher optical thickness
attenuates the background radiation and also amplifies the source term due to
diffusion.

|
---|---
(a) | (b)
|
(c) | (d)
|
(e) | (f)
|
(g) | (h)
Figure 8: Synthetic H$\alpha$ line profiles. Modeled specific intensity is a
sum of attenuated background radiation and source radiation. Both components
are plotted as well as the sum. This figure shows variations of the modeled
profiles with temperature for two values of electron density and optical
thickness.

Electron density variations for selected parameter values are plotted in Fig.
9. For lower temperature and higher optical thickness panels (a – d) show that
the source term is significantly more amplified with increasing electron
density. For higher temperature and lower optical thickness (e – h panels), we
can see that the source term radiation is increasing again with increasing
electron density. We use the electron densities up to the relatively high
$10^{13}$ $\rm{cm^{-3}}$. These electron densities have been recently found in
cool flare loops on Sun, see Jejčič et al. (2018).

|
---|---
(a) | (b)
|
(c) | (d)
|
(e) | (f)
|
(g) | (h)
Figure 9: Synthetic H$\alpha$ line profiles. Modeled specific intensity is a
sum of attenuated background radiation and source radiation. Both components
are plotted as well as the sum. This figure shows variations of the modeled
profiles with electron density for two values of temperature and optical
thickness.

Optical thickness variations for selected parameter values are plotted in Fig.
10. For low temperature (a – c panels) we can see that the source term
increases a lot and the attenuated background term decreases with increasing
optical thickness.

|
---|---
(a) | (b)
|
(c) | (d)
|
(e) |
Figure 10: Synthetic H$\alpha$ line profiles. Modeled specific intensity is a
sum of attenuated background radiation and source radiation. Both components
are plotted as well as the sum. This figure shows variations of the modeled
profiles with optical thickness.

Turbulent velocity variations for selected parameter values are plotted in
Fig. 11. We used relatively large values of $v_{\rm t}$, up to 50 km s-1,
which is consistent with observations of Mikuła et al. (2017). The results of
our modeling indicate that increasing turbulent velocity significantly widens
red wing enhancement for both lower temperature and higher optical thickness,
as well as for higher temperature and lower optical thickness.

|
---|---
(a) | (b)
|
(c) | (d)
Figure 11: Synthetic H$\alpha$ line profiles. Modeled specific intensity is a
sum of attenuated background radiation and source radiation. Both components
are plotted as well as the sum. This figure shows variations of the modeled
profiles with turbulent velocity for two values of temperature, electron
density and optical thickness.

###  4.2 Comparison with OES observations

To test the model we attempt to fit the synthetic profile differences to our
OES observations. The data presented in Fig. 2 and 4 is the difference between
stellar flux during the flare and quiescent flux of AD Leo. To calculate the
same effect using our model we compute flux seen by the observer on Earth and
we subtract the quiescent radiation we used as the input for our model
expressed as flux seen on Earth. Flux seen by an observer on Earth under all
the assumptions of our model can be approximated as:

$F_{\lambda}=\frac{I^{\rm{flare}}_{\lambda}A^{\rm{flare}}+I^{\rm{quiescent}}_{\lambda}\left(A^{\rm{disc}}-A^{\rm{flare}}\right)}{d^{2}}$
(28)

where $I^{\rm flare}_{\lambda}$ is the synthetic specific intensity calculated
using Eq. 4, $I^{\rm quiescent}_{\lambda}$ is the quiescent specific intensity
of the star, $A^{\rm flare}$ is the area of the flare, $A^{\rm disc}$ is the
area of the stellar disc and $d$ is the distance from the star to the Earth.
Subtracting the quiescent flux seen by the observer we get the formula

$\Delta
F_{\lambda}=\frac{\left(I_{\lambda}^{\rm{flare}}-I_{\lambda}^{\rm{quiescent}}\right)A^{\rm{flare}}}{d^{2}}$
(29)

simulating the formula in Eq. 2. During fitting, we varied model parameters
and tried to get the best match between the results of a model and the
observed H$\alpha$ line. However our model does not include any effect that
enhances the continuum flux from loops (Heinzel & Shibata 2018), so an
additional constant was included to account for the offset.

For the flare in the Fig. 2 the fitted synthetic profile difference is shown
in Fig. 13. We fit spectrum observed on 18 April 2019 at 22:07 UT. The fitted
parameters are listed in the Table 2. The flare area from the fit covers
approximately 15 % of the stellar disc surface of AD Leo. The observed profile
difference has a stronger red wing for velocities within 15 – 50 km s-1. The
synthetic profile difference has stronger red wing up to velocities of 100 km
s-1. Compared to the observed profile difference the synthetic difference has
a similar shape but appears to be shifted by a few km s-1 towards red
velocities.

For the flare in the Fig. 4 the result of our fitting is shown in Fig. 12. We
fit the spectrum observed on 19 April 2019 at 23:42 UT. The fitted parameters
are listed in the Table 2. The flare area from the fit covers approximately 16
% of the stellar disc of AD Leo. The observed profile difference has a
stronger red wing for velocities within 15 – 50 km s-1. The synthetic profile
difference has a stronger red wing up to velocities of 100 km s-1. Compared to
the observed profile difference the synthetic difference has a similar shape
but appears to be shifted by a few km s-1 towards red velocities as in the
previous comparison.

Table 2: Fitted model parameter values for the two selected profiles. Flare |  $T$ [K] |  $n_{e}$ [cm-3] | $\tau_{0}$ |  $v_{\rm t}$ [km s-1] |  $r$ [Mm] |  $A$ [Mm2] |  $v_{0}$ [m s-1] | constant [erg Å-1 cm-2 s-1]
---|---|---|---|---|---|---|---|---
18\. 04 . 2019 | 10000 | 1012 | 8.2 | 2 | 75 | 16875 | 50 | 7 $\times 10^{-14}$
19\. 04 . 2019 | 10000 | 1012 | 5.5 | 2 | 80 | 19200 | 50 | 14 $\times 10^{-14}$

Figure 12: Fit of the modeled spectrum difference $\Delta F_{\lambda}$ and
spectrum difference $\Delta F_{\lambda}$ observed on 19 April 2019 during
gradual phase at 23:42 UT in Fig. 4. To account for the observed continuum
enhancements, a constant is added to the synthetic profile difference as a fit
parameter. The synthetic profile has a similar shape compared to the observed
within the range of 15 – 50 km s-1.

Figure 13: Fit of the modeled spectrum difference $\Delta F_{\lambda}$ and
spectrum difference $\Delta F_{\lambda}$ observed on 18 April 2019 during
gradual phase at 22:07 UT in Fig. 2. To account for the observed continuum
enhancements, a constant is added to the synthetic profile difference as a fit
parameter. The synthetic profile has a similar shape compared to the observed
within the range of 15 – 50 km s-1.

##  5 Discussion

Spectral line asymmetries during stellar flares have been observed for several
stars in recent years, e.g. Fuhrmeister et al. (2018), Vida, Krisztián et al.
(2019), Leitzinger et al. (2022), Muheki et al. (2020a), Muheki et al.
(2020b), Koller et al. (2021) or Wu et al. (2022). Enhancements of the blue
wing of spectral lines are often linked to the possible presence of CMEs.
Enhancements in the red wing are associated with downward flows of the plasma,
mostly chromospheric condensation or backward-falling material from an
eruption, or CMEs occurring close to the limb. Velocities in the red asymmetry
can reach several hundred km s-1. Our OES observations of the H$\alpha$ line
indicate velocities up to 50 km s-1 while the observed profile changes are
similar to asymmetries observed by Muheki et al. (2020a).

On the Sun the observed asymmetries in some spectral lines during impulsive
phases of flares are caused by the chromospheric condensation (Kuridze et al.
2015) with velocities reaching a few tens of km s-1. This mechanism could
explain some of the lower-velocity red asymmetries observed on M-type stars
but the presence of high-velocity red asymmetries (hundreds of km s-1)
suggests other mechanisms to be present. A good candidate seems to be coronal
rain, a downflow of plasma usually formed during the gradual phase of flares.
It can reach much higher velocities compared to the chromospheric
condensation, Antolin et al. (2010) reports velocities up to 120 km s-1. On
cool M-dwarf stars, where flare loops are expected to be much larger compared
to the stellar radius than on the Sun, e.g. (Hawley et al. 1995), the
velocities of the coronal rain could reach even higher values.

In our model, we solve the H$\alpha$ radiative transfer inside the cool loop
clouds. Synthetic profiles resulting from the whole loop arcade are
asymmetrical with an enhanced red wing of the line. This enhancement can be
followed up to 200 km s-1 for the parameters we used. Comparing the synthetic
profiles of our model with OES observations we are able to reproduce a similar
shape of the profile with enhanced red wing suggesting that the presence of a
coronal rain could create asymmetries consistent with observations. However,
there are three discrepancies that our model doesn’t explain.

First, the whole synthetic profile difference appears to be somewhat shifted
towards the red by a few km s-1. That could be caused by a flare occurring not
at the center of the stellar disc as our model assumes but rather at a
different position. In our model the flare occurs at the center of the stellar
disc with respect to the observer. In this case, all of the clouds in the
filled arcade of flare loops move away from the observer. If the flare
occurred a little further from the center of the disc, one half of the clouds
close to the top of the arcade would move towards the observer with lower
velocities and the other half of the clouds would move away. The radiation
coming from the clouds close to the top moving toward the observer would
therefore be blue-shifted rather than red-shifted. Additionally, some of the
clouds close to the farther anchor of the flare loops would be covered by the
rest of the arcade and the line-of-sight component of terminal velocity of the
clouds at the other anchor would be lower. This would result in a slightly
stronger blue wing and a slightly weaker red wing virtually shifting the whole
profile difference towards blue velocities. If the flare occurred at the limb
most of the clouds would contribute radiation in the blue wing rather than the
red one possibly creating a blue asymmetry. To account for a general location
of the flare will require a more complex approach and this will be a subject
of our future studies.

Second, our model does not provide any continuum enhancements that are
observed by OES. As discussed in the section 2, the observed continua
enhancements are caused by both the flare and by uncertainties introduced by
using linear interpolation during our flux calibration process. To account for
this we just add a constant to the synthetic profile during fitting. The
continuum enhancements are often linked to flare ribbons which are bright long
and narrow areas. However, Heinzel & Shibata (2018) showed that on cool stars
the flare loops can significantly contribute to the total white-light flux.
Unlike on the Sun, we are unable to resolve these features on stars. The
observed spectra likely contain contributions of both loop and ribbon
components but here we assume that during the gradual phase the loop arcade
dominates. It is well known that in later phases of solar flares the ribbons
fade out while the loops grow and fill a larger and larger area, for example
Jing et al. (2016).

Third, the profile differences of the H$\alpha$ line during the second flare
in the Fig. 4 have a reversal in their center forming a double peak. In our
model, we assume a constant H$\alpha$ source function in the cloud which
leads, by definition, to a non-reversed profile. The reverted profile can be
modeled using the full non-LTE radiative transfer but that is beyond the scope
of this paper and could be a subject of future studies.

##  6 Conclusions

We have observed dMe star AD Leo during the spring periods in 2019, 2020, and
2021, using the Ondřejov Échelle Spectrograph (OES) attached to the 2-meter
Perek telescope at Ondřejov observatory. Simultaneously, in 2019 and 2021 AD
Leo photometric observations were carried out. In this paper, we have studied
the effect of stellar flares on the H$\alpha$ line. During flares, we observed
that the H$\alpha$ line exhibited enhancement in the line center, broadening,
and an asymmetry with enhanced flux in the red wing at velocities within up to
50 km s-1.

In order to explain these H$\alpha$ profile asymmetries, we developed a simple
model based on a direct analogy with solar flares. The model synthesizes
H$\alpha$ profiles emergent from an arcade of flare loops, assuming that the
flare occurs at the center of the stellar disc. The resulting H$\alpha$
profiles are asymmetrical with the enhanced red wing at velocities reaching up
to 200 km s-1.

We attempted to fit the model results to match the observed asymmetries. Our
model yields profiles with a similar shape but the whole profile differences
appear to be slightly shifted towards the red by a few km s-1, moreover, our
model does not produce any continuum enhancements that OES observed. The whole
profile shifting is probably caused by a flare that does not occur at the
center of the stellar disc and thus some contribution from the loop arcade is
not red-shifted but can be blue-shifted thus effectively shifting the whole
profile difference towards the blue wing. More complex geometrical models
(projections) must be used to solve this.

###### Acknowledgements.

This study was supported by the Czech Funding Agency grants GACR 19-17102S and
22-30516K, LTT-20015 for data collection with 2-m telescope in 2020 and by
RVO:67985815 project. This work was also partially supported by the program
”Excellence Initiative \- Research University” for years 2020-2026 at
University of Wrocław, project no. BPIDUB.4610.96.2021.KG. This paper is based
on the results of the diploma thesis of J. Wollmann. We would like to thank
the observers from the SPHE section of the Czech Astronomical Society for
providing us with their photometric observations of AD Leo and namely to H.
Kučáková for her photometric observations using the Mayer’s telescope at
Ondřejov observatory. We are grateful to M. Špoková, R. Karjalainen, J.
Šubjak, and M. Skarka for the reduction of spectra from 2-meter Perek
telescope and their advice during further processing of spectra and to E.
Guenther for his advice during the spectra calibration process. We are also
grateful to the referee for his/her useful comments.

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