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# SPATIO-TEMPORAL GRAPH-RNN FOR POINT CLOUD PREDICTION

###### Abstract

In this paper, we propose an end-to-end learning network to predict future
frames in a point cloud sequence. As main novelty, an initial layer learns
topological information of point clouds as geometric features, to form
representative spatio-temporal neighborhoods. This module is followed by
multiple Graph-RNN cells. Each cell learns points dynamics (i.e., RNN states)
by processing each point jointly with the spatio-temporal neighbouring points.
We tested the network performance with a MINST dataset of moving digits, a
synthetic human bodies motions and JPEG dynamic bodies datasets. Simulation
results demonstrate that our method outperforms baseline ones that neglect
geometry features information.

Index Terms—  Point Cloud, Graph-based representation learning, Point-based
models.

##  1 Introduction

Point clouds (PCs) sequences provide a flexible and rich geometric
representation of volumetric content, quickly becoming an attractive
representation for applications such as autonomous driving [1], mixed reality
application services [2], cultural heritage [3]. This has motivated intense
research toward PC processing, with strong focus on _static_ PCs, leaving the
_dynamic_ PC processing (DPC) usually overlooked. In this work, we focus on
DPC processing and specifically on the prediction of point cloud dynamics.
Namely, given PC frames $P_{1},P_{2},\ldots,P_{t}$ we are interested in
predicting $\hat{P}_{t+1}$, _with no prior knowledge_ on the ground truth
$P_{t+1}$.

In the current literature, DPCs processing has been approached from two
overlapping directions: (1) motion estimation (ME) and motion compensation
(MC) for PC compression; (2) 3D motion flow prediction (usually deep-learning
based) for high-level tasks (e.g., gesture recognition). Both approaches share
a common challenge: extraction of temporal correlations between sequential PC
frames, challenged by the irregular structure and by the lack of explicit
point-to-point correspondence. At the same time, these two directions have
fundamentally different goals and setups: the former aimed at extracting the
motion vector from two _known_ consecutive frames, the latter focused on a
much more challenging task of prediction of future _unknown_ PC frames. This
requires learning both the short- and long-term PC trajectory. Another key
difference lies in the developed solutions: ME mainly addresses the lack of
correspondence either by projecting the 3D PC into the 2D domain and adopting
mature algorithms from 2D video compression [4] or by developing 3D ME
methodologies, preserving the volumetric information of the PCs [5, 6, 7].

Motion flow prediction involves deep learning processing instead, the
irregular and unordered structure of PC prevents the immediate adoption of
convolution neural networks. Within this framework, PointNet [8] has become a
pillar work for static PC processing, capable of learning directly from raw PC
data with no pre-processing: each point in the PC is processed independently
and all point features are aggregated subsequently. Modeling points
independently achieves permutation invariance, but at the price of losing the
geometric relationship between points, a key piece of information in PCs that
we aim at retaining. To learn the dynamic behavior of sequential data, recent
works [9, 10, 11] has extended PointNET architecture to recurrent neural
networks (RNNs), predicting the 3D motion flow of PCs. In the PointRNN model
[11], for example, each point is processed individually by RNN cells with the
output being the point state (i.e., the motion of the point). Each point state
is extracted by aggregating state information from neighboring points. The
neighborhood of a point of interest is defined as the $k$ nearest neighbor
($k$-nn) points in the previous frame, where the proximity is measured based
on the point coordinates. This methodology inherits the ability to capture the
dynamic behavior of sequential data from RNN models, as well as permutation
invariance from PointNet architecture. However, it suffers from the same
shortcoming of PointNet: _lack of geometric relationship between points_ which
may lead to _i)_ loss of structure during PC reconstruction; _ii)_ poor $k$-nn
neighborhood as grouping points only based on coordinates might connect two
points close in space but not belonging to the same segment, hence not sharing
the same motion.

In this paper, we seek to bridge the gap between graph-based representations
of PC [12, 13, 14] and deep learning motion flow prediction. We propose an
end-to-end architecture, where an initial pre-processing step learns
topological information of PC as geometric features, and leverage on those
learned features to form more representative local neighborhoods of points
than PointRNN models. From the learned features, a Graph-RNN constructs
spatio-temporal $k$-nn graphs. This results in spatio-temporal aggregation of
points that share common local features instead of only points coordinates.
The temporal correlations from the spatio-temporal graph are aggregated to
learn point states. The Graph-RNN learns points states, which retain the model
dynamic information over time and allow to model long-term point trajectory.
The proposed solution has been validated on moving MNIST point cloud dataset
used in the literature [11] as well as on a synthetic human b odies motions
and JPEG dynamic bodies datasets [15, 16]. Simulation results demonstrate that
our method can make correct PC predictions showing its ability to accurately
group points and model long-term relationships while preserving the spatial
structure.

##  2 Proposed Method

Fig. 1: Scheme of the complete hierarchical architecture, composed of four
main components: Sampling and Grouping (SG); GNN for Features Learning ;
Graph-RNN (diagram of it’s operation included in dashed area) ; States
propagation (SP).

We denote a point cloud frame consisting of $n$ points by
$P_{t}=\\{p_{1,t},p_{2,t},\ldots p_{n,t}\\}$ with $p_{i,t}\in\mathbb{R}^{3}$
being the euclidean coordinates of point $i$ in $P_{t}$. Each PC has
additional attributes (i.e., point color) denoted by
$C_{t}=\\{c_{1,t},c_{2,t},\ldots c_{n,t}$, with $c_{i,t}\in\mathbb{R}^{3}$ the
associated color component. Given a point cloud sequence
$\mathcal{P}=(P_{1},P_{2},...,P_{T})$ composed by $T$ frames and additional
attributes $\mathcal{C}=(C_{1},C_{2},...,C_{T})$, our goal is to predict the
coordinates of future point clouds $\hat{P}_{T+1},\ldots,\hat{P}_{T+Q}$, with
$Q$ being the prediction horizon.

To reach this goal, we proposed an interactive framework (Fig. 1), which
allows us to predict future trajectories of points via RNN cells. At each
iteration, the network processes one input frame $P_{t}$ and its color
attribute $C_{t}$ giving as output the prediction of the successor frame
$\hat{P}_{t+1}$. The architecture is composed of two phases: _i)_ a dynamics
extraction (DE) phase where the PC dynamic behaviour is captured in the form
of point states, _ii)_ a PC reconstruction phase where the states are
concatenated and used to output the PC prediction. In the DE phase, as key
novelty, we pre-process the point cloud to extract point features that carries
local geometry information. Specifically, a initial graph neural network (GNN)
module transforms the 3D space into an higher dimensional feature space and
sends the output to a Graph-RNN cell. In each cell, each point is processed
independently to preserve permutation invariance. Specifically, each point
state is extracted by aggregating information from its $k$-nn neighborhood.
After the Graph-RNN cells, the PC reconstruction phase begins. The states are
propagated and processed by a fully connected layer (FC) to estimate motion
vectors, used to predict the next frame $\hat{P}_{t+1}$. Before each Graph-RNN
cell, the point cloud is down-sampled. It is then up-sampled to its original
size before the final FC layer. The down-sampling and up-sampling blocks are
implemented as in [17] and we refer the readers to Section A of the Appendix.
for further information. The intuition for the design hierarchical
architecture is to learn states at multiples scales: the first Graph-RNN cell
handles a dense PC and learns states in local regions (corresponding to local
motions), while the last Graph-RNN cell learns states in a sparser PC with
more distant points included in the neighborhood (corresponding to more global
motions).

We now provide more details on the key modules that are part of our
contributions: GNN-based pre-processing and Graph-RNN cells.

###  2.1 GNN for Feature Learning

Given $P_{t}$ and $C_{t}$ as input, we construct a directed $k$-nn _coordinate
graph_ $\mathcal{G}^{C}_{t}=(P_{t},\mathcal{E}^{C}_{t})$ with vertices $P_{t}$
and edges $\mathcal{E}^{C}_{t}$. Each edge connects a point to its $k$-nearest
neighbors based on euclidean distance. The graph includes self-loop, meaning
each point is also connected to itself. Given the coordinate graph as input,
the GNN module learns the geometric features $F_{t}\in\mathbb{R}^{n\times
d_{f}}$. The GNN is composed of $L$ layers, and in each layer features are
learned by aggregating information along the edges.

Inspired by [12], we learn the features by taking into account the
relationship (similarity) between neighboring points. At the $l$-th layer of
the GNN, the _edge features_ $e^{l}_{i,j}$ are learned for each point $i$ and
for each neighboring node $j$. This is done by concatenating the input point
feature $f_{i,t}^{l-1}$ and the point coordinates $p_{i,t}$, with the geometry
and color displacement/difference between the points $i$ and $j$ ($\Delta
p_{ij}$, $\Delta c_{ij}$, respectively). We then apply a symmetric aggregation
operation on the edge features associated with all the edges emanating from
each point. More formally, the edge features ($e^{l}_{i,j}$) and the output
point features ($f^{l}_{i}$) are obtained as follows:

$\displaystyle e^{l}_{i,j}$
$\displaystyle=h_{F}^{l}(f_{i,t}^{l-1}\;;p_{i,t}\;;\Delta p_{ij}\;;\Delta
c_{ij})$ (1) $\displaystyle f^{l}_{i}$
$\displaystyle=\operatorname*{\textit{MAX}}_{j:(i,j)\in\mathcal{E}^{C}}\big{\\{}e^{l}_{i,j}\big{\\}}$
(2)

where $h_{F}$ is a nonlinear learnable function that can be implemented with a
multi layer perceptron (MLP), ’;’ identifies the concatenation operation and
$\operatorname*{\textit{MAX}}$ represents the element-wise max pooling
function. Note that for the first layer $l=1$, we set $f_{i,t}^{0}$ as a null
entry and the output of the $L$-th layer is the geometric feature
$F_{t}=[f_{1,t}^{L},f_{2,t}^{L},\ldots,f_{n,t}^{L}]$.

###  2.2 Graph-RNN

Each Graph-RNN cell $c$ receives the feature $F_{t}^{c-1}$ and
$P_{t}^{c-1}=[p^{c-1}_{1,t},p^{c-1}_{2,t},\ldots,p^{c-1}_{n,t}]$ as input,
with $F_{t}^{0}$ being the output of the previous GNN module. Given it
iterative nature, the Graph-RNN cell takes into account the input and also its
own output $(P^{c}_{t-1},F^{c}_{t-1},S^{c}_{t-1})$ calculated at the previous
interaction $(t-1)$. The cell extracts the inner state
$S_{t}^{c}=[s^{c}_{1,t},s^{c}_{2,t},\ldots,s^{c}_{n,t}]\in\mathbb{R}^{n\times
d_{s}}$, with $s^{c}_{i,t}$ being the state of point $p^{c}_{i,t}$,
representative of the point dynamic behavior. The new state is added to the
unchanged coordinates $P^{c-1}_{t}$ and features $F^{c-1}_{t}$ and outputted
as $(P^{c}_{t},F^{c}_{t},S^{c}_{t})$. Similarly to [11], we consider three
sequential Graph-RNN cells.

The Graph-RNN operation is the depicted in Fig. 1 (dashed box). As first step,
we compute a _spatio-temporal feature graph_ $\mathcal{G}^{F}_{t}$, in which
each point is connected to $k$ nearest neighbors based on the feature
distance. Specifically, for each input point $p_{i,t}^{c-1}$, we compute the
pairwise distance between $f_{i,t}^{c-1}$ and features of other points
$f_{j,t}^{c-1}$ (features input) and $f_{j,t-1}^{c}$ (features from points in
the past PC). We force our implementation to take the equal number of points
$k$ from $P^{c-1}_{t}$ as from $P^{c}_{t-1}$ to avoid a one-side selection. In
details, this is a spatio-temporal graph since each point is connected to
points in the same PC (_spatial relationship_) and points in the past PC
(_temporal relationship_). Once the features graph is constructed, we learn
edge features similarly to the GNN module. For the edge $(i,j)$, we
concatenate the state of point $i$ ($s_{i}$), the state of point $j$
($s_{j}$), and the coordinate, the feature and the time displacement
($\Delta_{p_{i,j}},\Delta_{f_{i,j}}\Delta_{t_{i,j}}$) between the two points.
The concatenation is then processed by a shared MLP $(h_{S})$. All edge
features are then max pooled to a single representation into the update state
$s_{i,t}$. Formally,

$\displaystyle e^{c}_{i,j}$
$\displaystyle=h_{S}^{c}(s^{c-1}_{i,t};s^{c^{\prime}}_{j,t^{\prime}};\Delta
p_{ij};\Delta f_{ij};\Delta t_{ij})$ (3) $\displaystyle s^{c}_{i,t}$
$\displaystyle=\operatorname*{\textit{MAX}}_{j:(i,j)\in\mathcal{E}^{F}}\big{\\{}e^{c}_{i,j}\big{\\}}$
(4)

When learning output states $S^{c}_{t}$, the Graph-RNN cell considers the
states in the previous frame $S^{c}_{t-1}$. This means that the network learns
point movements taking into consideration the previous movements of points,
allowing the cell to retain temporal information. The states act as a memory
retaining the history of movements and enabling for network to model long-term
relationships over time.

###  2.3 Training

The architecture in Fig. 1 has multiple learnable parameters (in GNN, Graph-
RNN, FC), which are end-to-end trained. We consider a supervised learning
settings in which the loss function relates to the prediction error between
ground truth point cloud $P_{t}$ and the predicted one $\hat{P}_{t}$. To
evaluate the prediction error, we adopt the Chamfer Distance (CD) and Earth
Moving Distance (EMD) between $P_{t}$ and $\hat{P}_{t}$ evaluated as follows
[18]:

$\displaystyle d_{CD}(P,\hat{P})$ $\displaystyle=\sum_{p\in
P}\min_{\hat{p}\in\hat{P}}||p-\hat{p}||^{2}+\sum_{p\in\hat{P}}\min_{p\in
P}||\hat{p}-p||^{2}$ (5) $\displaystyle d_{EMD}(P,\hat{P})$
$\displaystyle=\min_{\theta:P\xrightarrow{}\hat{P}}\sum_{p\in
P}||p-\theta(p)||^{2}$ (6)

where $\theta:P\xrightarrow{}\hat{P}$ is a bijection. The loss function used
to train the network then is given by the sum of CD and EMD distances, namely
$\mathcal{L}(P,\hat{P})=d_{CD}(P,\hat{P})+d_{EMD}(P,\hat{P})$.

##  3 Experiments

We implemented the end-to-end network described in Sec.2 in the case of $L=3$
layers within the GNN module and $C=3$ RNN cells111The code has been made
available at https://github.com/pedro-dm-gomes/Graph-RNN.. We consider both
short-term and long-term prediction, with the former predicting only one
future frame $Q=1$ (ground truth frame $P_{t}$ is used to predict the next
frame $\hat{P}_{t+1}$) while the latter predicting $Q=T/2$ future frames with
$\hat{P}_{t}$ being used to predict the next frame $\hat{P}_{t+1}$. As
baseline models we consider: (1) Copy Last input model which simply copies the
past PC frame instead of predicting it; (2) PointRNN ($k$-nn) model [11],
which neglects geometry information. In our experiments, we considered the
following datasets: 222We provide detailed information in the supplementary
information in (sec B) of the Appendix. Moving MNIST Point Cloud, created by
converting the MNIST dataset of handwritten digits into moving point clouds,
as in [11], each sequence contains 20 ($T$) frames with either 128 (1 digit)
or 256 points (2 digits) .
Synthetic Human Bodies Activities, synthetically generated by us following
[15] using the online service Mixamo [19]333Due to copyright restriction
imposed by Mixamo, we cannot provide the dataset publicly available.  in
combination with the 3D animation software Blender [20].
JPEG Pleno 8i Voxelized Full Bodies, four benchmark sequences: longdress,
loot, redandblack, and soldier [16].
In the two last datasets, each PC sequence contains 12 ($T$) frames and is
downsampled to $4,000$ points. The network is trained with the Synthetic Human
Bodies Activities dataset, which provides different levels of movements
(walking, jumping, dancing, etc) and tested on both datasets.

To better understand our system, we visualized the learned features for one PC
from the Synthetic Human Bodies dataset. Fig. 2 depicts in sequence: the point
cloud, the learned features $F_{t}$, the output state $S_{t}$, the
reconstructed motion vector $M_{t}$, and the predicted PC. Principal Component
Analysis (PCA) is used for the features visualization. It is worth noting that
features can segment the PC into regions sharing similar topology (leading to
meaningful neighborhood in the features graph) and states are able to capture
the movement of moving parts –e.g., leg and foot taking a step forward. The
states are directly translated into motion vectors, used to make accurate
prediction of the future frame. A more complete comparison can be deduced from
Fig. 3, depicting resultant and ground truth PCs for the MINST dataset.
Interestingly, the predicted digits are sharper and clearer in the Graph-RNN
prediction than the PointRNN. This demonstrates that while both models capture
the correct motion, the Graph-RNN is better at preserving the spatial
structure over time. This is a direct effect from learning geometric features.

We now provide more quantitative results for both the MNIST dataset (Table 1),
and the Synthetic Human Bodies and JPEG datasets (Table 2). For all datasets,
the proposed Graph-RNN outperforms the PointRNN solution as well as the
baseline solutions. From Table 1, it is worth noting that the hierarchical
implementation (in which PC is sampled between RNN-cells) leads to a better
prediction compared to the “Basic” (not down-sampled) counterpart. This is
expected as the hierarchical structure learns states at different resolution.
Finally, the model “Graph-RNN (color)” considers the color attributes when
learning features, resulting in a more meaningful spatio-temporal neighborhood
[21] and therefore in a better prediction.

Fig. 2:  Multiple representation steps of short-term prediction Graph-RNN on
Bodies dataset

Fig. 3: Visualization of moving MNIST point cloud prediction with two moving
digits (Hierarchical architecture)

MINST
---
| Long-term Prediction
| One Digit | Two digit
Method | CD | EMD | CD | EMD
Copy last input | 262.46 | 15.94 | 140.14 | 15.8
| PointRNN | 5.86 | 3.76 | 22.12 | 7.79
Basic | GraphRNN | 2.43 | 2.40 | 13.66 | 6.13
| PointRNN | 2.25 | 2.53 | 14.54 | 6.42
Hierarchical | Graph-RNN | 1.22 | 1.86 | 4.62 | 3.97

Table 1:  Prediction error of PointRNN and Graph-RNN with $k$-nn on the moving
MNIST point cloud dataset.

| Synthetic Human Bodies | JPEG Dynamic Bodies
---|---|---
| Short-Term | Long-Term | Short-Term | Long-Term
Method Hierarchical | CD | EMD | CD | EMD | CD | EMD | CD | EMD
Copy Last Input | 0.161 | 0.153 | 0.247 | 0.408 | 0.0004 | 0.029 | 0.0020 | 0.058
PointRNN | 0.007 | 0.104 | 0.066 | 0.257 | 0.0005 | 0.034 | 0.0024 | 0.082
Graph-RNN | 0.005 | 0.078 | 0.077 | 0.248 | 0.0003 | 0.026 | 0.0018 | 0.074
| Graph-RNN
---
(color)
0.004 | 0.071 | 0.063 | 0.219 | 0.0003 | 0.025 | 0.0014 | 0.053

Table 2:  Prediction error of PointRNN and Graph-RNN on the Synthetic Human
Bodies and JPEG datasets

##  4 Conclusion

This paper proposes end-to-end learning network to process dynamic PCs and
make accurate predictions of future frames. We design a Graph-RNN cell that
can leverage learned features, describing the local topology, to form spatio-
temporal graphs, from where temporal correlations can be extracted.
Experimental results demonstrate the network’s ability to model short and
long-term motions while preserving the spatial structure.

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SPATIO-TEMPORAL GRAPH-RNN FOR POINT CLOUD PREDICTION

SUPPLEMENTARY MATERIAL

This supplementary material provides additional details of the proposed
framework.

In Sec A we provide details on hierarchical structure. Sec B includes
additional information of the datasets used in the experiments. Sec C provides
implementation details of the architecture. Lastly Sec D provides
visualization and analysis of additional experiments.

##  Appendix A Hierarchical structure details

In this paper, we proposed a hierarchical architecture, where before each
Graph-RNN cell the point cloud and the associated components are down-sampled
by a _Sampling and Grouping_ (SG) module. In a second phase, the point cloud
is up-sampled to the original number of points _State Propagation_ (SP)
module. The SG and SP modules were developed in the PointNET++ [17] work. This
section includes a description of the modules operations for the method
proposed in this paper, for a more complete description we refer the reader to
the original [17] work.

###  A.1 Sampling and Grouping

The Sampling and Grouping module takes a point cloud with $n$ points and uses
the farthest point sampling (FPS) algorithm to sample $n^{\prime}$ points. The
sampled points are defined as centroids of local regions. Each region is
composed of the $k$ closest neighborhood points to the centroid point. The
features and states of the points in a region are max pooled into a single
feature and state representation. This representation becomes the feature and
the state of the centroid point. The SG module outputs the $n^{\prime}$
sampled points and their updated feature and state.

###  A.2 State Propagation

In the SG modules, the original point set is down-sampled. However, in our
prediction task, we want to obtain the point states for all the original
points. The chosen solution is to propagate states from subsampled points
$n^{\prime}\times d_{s}$ to the original points $n\times d_{s}$. To this end,
for every down-sampling SG module, there is a corresponding up-sampling SP
module, with a skip link connection between them as shown in Figure 1. The SP
module receives the target points we want to propagate the states into using
skip connections, and interpolates the state’s values $S$ of $n^{\prime}$
points at coordinates of the $n$ points, using inverse distance weighted
average based on k-nearest neighbors. The interpolated states on $n$ points
are then concatenated with states from the SG module. The concatenation of
both states is passed through an MLP to update every point state. The process
is repeated until we have propagated states to the original set of points.

An additional advantage of the hierarchical architecture provided by the SG
and SP modules is a reduction of computational power [11]. This is a result of
the reduced number of points processed in the layer after the down-sampling
operations. Not only does the hierarchical architecture allow us to achieve
better performance (more accurate predictions), informal evaluation during our
experiments also confirmed a reduction of computation required.

##  Appendix B Dataset details

This section provides details on point cloud datasets used in experiments.

###  B.1  Moving MNIST Point Cloud

The Moving MNIST Point Cloud dataset is a small, simple, easily trained
dataset that can provide a basic understanding of the behavior of the network.

The dataset is created by converting the MNIST dataset of handwritten digits
into moving point clouds. The sequences are generated using the process
described in [11]. Each sequence consists of $20$ consecutive point clouds.
Each point cloud contains one or two potentially overlapping handwritten
digits moving inside a $64\times 64$ area. Pixels whose brightness values
(ranged from 0 to 255) are less than 16 are removed, and 128 points are
randomly sampled for one digit and 256 points for two digits. Locations of the
pixels are transformed to $(x,y)$ coordinates with the $z$-coordinate set to 0
for all points.

###  B.2 Synthetic Human Bodies

Open datasets for dynamic point clouds are limited, especially if interested
in complex dynamic movements and not only body translation. Hence, we created
synthetic data set of animated human bodies, similarly to [15]. We use the
online service Mixamo [19] to create multiple models of animated characters.
Next, we used the 3D animation software Blender [20] to render the animations
and to extract one mesh per frame. The mesh is converted to a high-resolution
point cloud by randomly sampling $8000,0000$ points from the faces of the
mesh. The point cloud is further downsampled to $4,000$ points using FPS to
reduce memory and processing cost during experiments

The Human Bodies training dataset consists of $15$ character models each
performing $20$ animations, for a total of $300$ sequences, we were careful to
select a diverse group of activities. Each sequence contains $50$ frames, $12$
consecutive frames are randomly selected at each training step. The dataset is
further augmented by using multiple sampling rates.

The test dataset consists of $5$ models denoted: Andromeda, James, Josh, Pete
and Shae. All performing the same 9 activities: ’Big Jump’, ’Climbing Up
Wall’, ’Entering Code’,’Jazz Dancing’, ’Turn and Kick’, ’Running’, ’Stabbing’,
’Walking Backwards’, ’ Walk with Rifle’. We again use different sampling rates
to expand the dataset to a total of $152$ sequences.

Fig. S1: Test characters: (a) Andromeda (b) Bryce (c) James (d) Pete (f) Shae

###  B.3 JPEG Pleno 8i Voxelized Full Bodies

The dynamic voxelized point cloud sequences in this dataset are known as the
8i Voxelized Full Bodies (8iVFB). There are four sequences in the dataset,
known as longdress, loot, redandblack, and soldier, pictured below. In each
sequence, the full body of a human subject is captured by 42 RGB cameras
configured in 14 clusters. The point clouds are originally high resolution
with over $700,000$ points. The dataset is scaled by a factor of $0.0018$ and
subsequently translated $(-0.37426165;-0.03379993;-0.29201281)$ to match the
Human Bodies training data scale and general position. The data is then
downsampled to 4,000 points using FPS.

Fig. S2: JPEG Test characters: (a) longdress (b) loot (c) redandblack (d)
soldier

FPS was chosen for the last downsample operations because it better coverage
of the entire point cloud, compared with random sampling. This dataset is only
used for evaluation, of the models trained with the Synthetic Human Bodies
dataset.

##  Appendix C Implementation details

This section describes the parameters and specifications of the proposed
framework.

###  C.1 Training details

The models are trained using the Adam optimizer, with a learning rate of
$10^{-5}$ for all datasets. The models trained with the MNIST dataset are
trained for $200,000$ interactions with a batch size of 32. For the Synthetic
Human Bodies dataset, the PointRNN and Graph-RNN models a batch size of 4 is
set and trained for $200,000$ interaction in long-term prediction task and for
$150,000$ interaction in short-term prediction task. The Graph-RNN (color)
model that considers point clouds with color is trained for $200,000$
interaction for both tasks with a batch size of 2. For all models, the
gradients are clipped in range $[5,5]$.

###  C.2 Architecture Specifications

This section provides the specification for each of the main components:
Sampling and Grouping (SG); The Graph neural network (GNN) for features
learning; Graph-RNN cells; States propagation (SP); Final Fully connected
layer (FC);

The Graph-RNN model is implemented the same way for the MNIST dataset and the
Synthetic Human Bodies. For the MNIST dataset, we compare Graph-RNN results
with the original PointRNN results ($k$-nn Model). However, since the original
PointRNN paper [11] did not perform experiments on the Synthetic Human Bodies
dataset, we choose the $k$-values and dimensions to adapt the PointRNN
framework to the dataset. To have a fair comparison between our proposed
Graph-RNN and Point-RNN, we tried to keep the frameworks as similar as
possible while preserving the design choices of each one.

The architecture specifications of both Graph-RNN and PointRNN are displayed
in Table S1.

Specifications
---
| number of
---
output points
Graph-RNN model | Point-RNN model
hierarchical | basic | Components | k | ouput channels | Components | k | ouput channels
n/2 | - | SG | 4 | - | SG | 4 | -
n/2 | n | GNN layer 1 | 16 | 64 | - | - | -
n/2 | n | GNN layer 2 | 16 | 128 | - | - | -
n/2 | n | GNN layer 3 | 8 | 128 | - | - | -
n/2 | n | Graph-RNN cell 1 | 8 | 256 | PointRNN cell 1 | 24 | 256
n/4 | - | SG | 4 | - | SG | 4 | -
n/4 | n | Graph-RNN cell 2 | 8 | 256 | PointRNN cell 2 | 16 | 256
n/8 | - | SG | 4 | - | SG | 4 | -
n/8 | n | Graph-RNN cell 3 | 8 | 256 | PointRNN cell 3 | 8 | 256
n/4 | - | SP 1 | - | 256 | SP 1 | - | 256
n/2 | - | SP 2 | - | 256 | SP 2 | - | 256
n | - | SP 3 | - | 256 | SP 3 | - | 256
n | n | FC 1 | - | 126 | FC1 | - | 128
n | n | FC 2 | - | 3 | FC 2 | - | 3
- | - | Graph-RNN (color) model | - | - | -
n | n | FC color 1 | - | 126 | - | - | -
n | n | FC color 2 | - | 3 | - | - | -

Table S1: Architecture specifications. Each component is described by tree
attributes, i.e number of output points, number of neighborhoods ($k$) and
number of output channel.

For all the models, the final fully connected (FC) layer is implemented in
fact by two fully connected layers FC1 and FC2. The FC1 and FC2 layers process
the states to predict the geometry displacement.

The Graph-RNN (color) model that takes color as input has two additional fully
connected layers (FC1 color and FC2 color). Similar to the FC for points, the
FC (color) will take the states as input and predict the color displacement.
The assumption that the color of the points does not change during the
movement, while mostly correct in the case of synthetically generated data,
can not be applied in real-world data. The point’s color can change due to
lighting conditions, or in extremes cases, scene objects can transform, or be
replaced by new ones. The color prediction was not a priority in this work.
The FC (color) does not affect the loss function and has no impact on the
overall method. Nevertheless, in long-term prediction evaluation, we disregard
the prediction of color displacement and consider instead a null color
displacement, meaning all the points keep the same color from frame to frame.
In the future, we intend to explore color prediction, by including a color
evaluation metric in the loss function.

##  Appendix D Extra Results visualization

This section presents visualization examples of prediction on Synthetic Human
Bodies and JPEG datasets. Long-term prediction examples are depicted on the
right side of the next page and short-term prediction examples on left.

Long-term prediction is a very challenging task. In the MNIST dataset, the
moving digits perform simple translation movements. The Graph-RNN can
effectively model these simple motions over a long period, and make a long-
term prediction (10 frames). On the other hand, Synthetic Human Bodies perform
more complex activities (e.g dancing, climbing, running). Since these
activities are composed of irregular movements, with sudden motion changes,
they are incredibly difficult to predict in the long-term.

Figure S3 and Figure S4, show the difficulty in long-term prediction in human
activities. Both the PointRNN and Graph-RNN have trouble at preserving the
spatial structure over time. This was expected since in long-term prediction
the error in each predicted frame is propagated and amplified for each
subsequent prediction. While from Figures S3 and S4, it would appear the
PointRNN is better at preserving the spatial structure, this is because the
PointRNN failed to capture the correct motion. The Graph-RNN, on the other
hand, correctly captured the general motion, losing however some of the point
cloud shape as a result. We can conclude that a better prediction comes with
the risk of higher deformation. The Point-RNN achieved a smaller prediction
error in the long-term prediction of the Human Bodies dataset, by making more
conservative predictions (low movement prediction). In contrast, the Graph-RNN
makes more accurate predictions, however in the cases the predicted motion is
wrong, the error is amplified, resulting in a worse average performance.

Figure S6 and S7 display short-term prediction examples. All the models show
very similar visual results and can make accurate predictions. The Graph-RNN
superior performance can be observed in the small details like in the knee and
foot of the model.

Figures S5 and S8 show examples on the JPEG Human Bodies dataset. The point
clouds sequence from the JPEG dataset has very little movement, confirmed by
the good performance of the Copy Last Input model in both short and long-term
prediction. While the Graph-RNN achieved a smaller prediction error in the
chosen evaluation metrics, it is difficult to see the performance gains by
looking at the prediction visualizations.

Fig. S3: Short-term prediction on ’Pete Running’ sequence Fig. S4: Short-term
prediction on ’Shae Jazz Dancing’ sequence Fig. S5: Short-term prediction on
’JPEG Soldier’ sequence

Fig. S6: Short-term prediction on ’James Running’ sequence Fig. S7: Short-term
prediction on ’Josh Jazz Dancing’ sequence Fig. S8: Short-term prediction on
’JPEG Redandblackdress” sequence


# Lithium abundances and asteroseismology of red giants: understanding the
evolution of lithium in giants based on asteroseismic parameters

Deepak1,2 and David L. Lambert3
1Indian Institute of Astrophysics, Bangalore - 560034, India
2Pondicherry University, R. V. Nagara, Kalapet, Puducherry - 605014, India
3W.J. McDonald Observatory and Department of Astronomy, The University of
Texas at Austin, Austin, TX 78712, USA E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted 2021 April 23. Received 2021 April 21; in original form 2021
February 08)

###### Abstract

In this study, we explore the evolution of lithium in giant stars based on
data assembled from the literature on asteroseismology and Li abundances for
giants. Our final sample of 187 giants consists of 44 red giant branch (RGB),
140 core He-burning (CHeB) and three giants with an unclassified evolutionary
phase. For all 187 stars, the seismic parameters $\nu\rm_{max}$ (frequency of
maximum oscillation power) and $\Delta\nu$ (large frequency spacing) are
available, while $\Delta\Pi\rm_{1}$ (the asymptotic gravity-mode period
spacing) is available for a subset of 64. For some of the CHeB giants, mass
estimates from the asteroseismic scaling relations are found to be
underestimated when compared with mass estimates from isochrones based on
seismic data. Whilst most of the Li-rich giants in the sample have masses less
than 1.5 $M_{\odot}$, they are also present up to and beyond the maximum mass
expected to have suffered a core He-flash, i.e. $M$ $\leq$ 2.25 $M_{\odot}$:
this suggests contributions from other processes towards Li enrichment. To
understand the evolution of giants in the $\Delta\Pi\rm_{1}$ $-$ $\Delta\nu$
plane, we use the Modules for Experiments in Stellar Astrophysics models which
show the presence of mini He-flashes following the initial strong core He-
flash. From the distribution of A(Li) as a function of $\Delta\nu$, which is
similar to the distribution of A(Li) as a function of luminosity, we find no
indication of Li enrichment near the luminosity bump. Also, A(Li) trends to
$\sim$ -1.5 dex near the RGB tip. The data also suggest a decrease in A(Li)
with an increase in $\Delta\Pi\rm_{1}$ for CHeB giants.

###### keywords:

Asteroseismology – Stars: evolution – stars: interiors – stars: abundances –
stars: low-mass – stars: fundamental parameters

††pubyear: 2020††pagerange: Lithium abundances and asteroseismology of red
giants: understanding the evolution of lithium in giants based on
asteroseismic parameters–Lithium abundances and asteroseismology of red
giants: understanding the evolution of lithium in giants based on
asteroseismic parameters

##  1 Introduction

Across broad swaths of astrophysics, lithium continues to challenge our
understanding even as the observational exploration of the element’s abundance
is expanded and refined. Many would consider the leading (and most exciting)
challenge to be the reconciliation of the predicted Li abundance from the
standard Big Bang with estimates of the Li abundance in the most metal-poor
dwarf stars. Among other, also exciting to many, challenges involving
differences between predicted and observed lithium abundances is the one we
consider here, namely the occurrence of lithium-rich low mass giants.

In main sequence stars, lithium survives in a thin outer skin and is destroyed
below the skin. As a star evolves off the main sequence to become a red giant,
the convective envelope deepens and the surface lithium is diluted with
lithium-free material. After the convective envelope has reached its deepest
extent, the surface lithium abundance is diluted by about a factor of 30 to 60
according to standard models of giants near the base of the red giant branch
(RGB) with masses of 1.0 $M_{\odot}$ to 1.5 $M_{\odot}$ and with about solar
metallicity (Iben, 1967). Observations show that the surface lithium abundance
in main-sequence stars, which are unlikely to have destroyed surface lithium,
is weakly dependent on metallicity. Initial abundance is $A$(Li) =
3.3111Elemental abundances are given on the traditional scale, i.e., A(X) =
$\log$N(X)/N(H)+12 where N(X) is the number density of the element. at [Fe/H]
= 0.0 with a minor increase at [Fe/H] = $+0.3$ and a small decrease at [Fe/H]
= $-0.3$ (Lambert & Reddy, 2004; Randich et al., 2020). Such main sequence
stars as giants at the RGB base are expected to have their surface Li
abundance, $A$(Li), diluted to $\simeq 1.9$ at 1.0$M_{\odot}$ to 1.6 at
1.5$M_{\odot}$. Since many main sequence stars, including the Sun, have
depleted their surface Li below the above initial values, as giants at the
base of the RGB, such stars will have a Li abundance below the above estimates
of 1.6 - 1.9.

For the great majority of giants with near-solar metallicity, the abundance
$A$(Li) indeed is within the range $A$(Li) $\leq 1.7$ or so, suggesting that
the picture outlined in the previous paragraph is broadly correct. But there
is approximately a 1% population of giants with $A$(Li) above the expected
maximum. This population was first revealed by Wallerstein & Sneden (1982)’s
discovery of HD 112127 with the abundance $A$(Li) = 3.2, which struck the
discoverers as having “approximately the maximum seen in unevolved stars".
Subsequently, Brown et al. (1989) surveyed 600 giants and reported that only
nine were Li-rich, i.e., $A$(Li) $\geq 1.8$, for a frequency of just over one
per cent. Observational pursuit of Li-rich giants now has now extended at
least up to $A$(Li) = 4.8 (Yan et al., 2020) and the frequency of Li-rich
giants among solar metallicity giants has been confirmed to be low (see, for
example, Deepak & Reddy, 2019; Casey et al., 2019; Deepak et al., 2020;
Martell et al., 2020).

In discussions of Li-rich giants, it has been assumed that a normal giant
would have an abundance no greater than about $A$(Li) = $1.7\pm 0.2$ where the
uncertainty allows for variations in the dilution resulting from the
convective envelope due to initial mass and metallicity and for observational
uncertainties. It has not always been acknowledged explicitly that a main-
sequence star severely depleted in Li may experience appreciable Li enrichment
and yet not appear as a Li-rich giant by usual standards, i.e., an increase of
$A$(Li) to below about 1.7 will not betray it as a Li-rich giant. In a similar
vein, surface Li abundance changes – increases and decreases – occurring as a
star evolves up the RGB beyond the completion of the first dredge-up may
affect its classification as Li-rich.

Thanks in part to astrometric data provided by the Gaia satellite (Gaia
Collaboration et al., 2016; Gaia Collaboration et al., 2018; Lindegren et al.,
2018) recent surveys of Li abundance among solar metallicity low mass giants
have fully confirmed earlier suspicions that the majority of the Li-rich
giants (with $A$(Li) $\geq 1.7$) are core He-burning giants also known as red
clump (RC) giants (Deepak & Reddy, 2019; Casey et al., 2019; Deepak et al.,
2020; Kumar et al., 2020; Martell et al., 2020). This conclusion about the
preference for Li-rich giants to inhabit the RC has been strengthened by data
from asteroseismology’s ability to distinguish RC stars from their
predecessors on the RGB (Silva Aguirre et al., 2014; Jofré et al., 2015;
Bharat Kumar et al., 2018; Singh et al., 2019a; Singh et al., 2019b; Yan et
al., 2020). Termination of evolution along the RGB in low mass stars occurs
with the He-core flash (Thomas, 1967). A suspicion that the He-core flash in
some low mass stars may result in Li enrichment from the conversion of
internal 3He to 7Be and subsequent decay to 7Li was aired by Brown et al.
(1989) and later by Kumar et al. (2011). This suspicion has now been supported
by theoretical work (Schwab, 2020) showing that an ad hoc prescription of
internal chemical mixing may account for 7Li enrichment from the He core flash
and thus in the resultant RC giant. Schwab suggests that the introduced ad hoc
mixing may result from internal gravity waves excited by turbulent convection
arising from the He core flash.

In addition to a striking reinforcement of the idea that Li-rich giants are
found among RC stars, Kumar et al. (2020) argued that observations showed that
low mass stars having evolved to the tip of the RGB had a Li abundance $A$(Li)
$\simeq-0.9$. But since almost all RC stars have a Li abundance above this
limit, they concluded that Li enrichment from the He core flash must occur in
all stars in order that the Li abundance in RC stars be almost without
exception above the Li abundance limit at the RGB’s tip. The average Li
abundance for RC stars is a factor of 40 over the abundance adopted as the
value for the RGB’s tip. Thus, Kumar et al. propose that Li richness for RC
stars be defined as $A$(Li) $\geq-0.9$. Their limit is, of course, much
different than the limit traditionally adopted to label a star as Li-rich,
e.g., $A$(Li) $\geq 1.7$ (Deepak & Reddy, 2019; Deepak et al., 2020). More
clearly, Kumar et al.’s conclusion is that “all low mass stars undergo a Li
production phase between the tip of the red giant branch and the red clump"
(our italics). This conclusion amounts to “a stark tension between
observations and theory". As observers, we may call for an extension of
Schwab’s exploratory calculations through detailed computer simulations.

In this paper, we combine data from the literature on asteroseismology with Li
abundances for giants. Asteroseismology, as well recognized previously, allows
one to distinguish core He-burning (here, CHeB) giants from those on the RGB,
which are H-shell burning. The CHeB giants can further be separated into lower
mass stars that experienced the He-core flash and higher mass stars, igniting
and burning core He quiescently. A mass of $M\sim 2.2M_{\odot}$ for solar
metallicity stars separates low-mass CHeB giants experiencing a He-core flash
from high-mass giants avoiding the flash. Theoretical modelling of RGB and
CHeB giants shows that the measurable asteroseismic parameters may be used to
distinguish these two classes of giants. Furthermore, the CHeB giants of all
masses have similar interior structures and thus similar asteroseismic
properties, but these properties vary according to mass. In this paper, we
explore the Li abundances among giants with asteroseismic data.

Figure 1: The spectroscopic HR diagram for giants in the KRBS sample along
with Kepler DR25 stars in the background. Here, RGB are inert He core stars
with H-shell burning and CHeB are core He-burning giants while ‘Unknown’ are
the ones with unclassified evolutionary phases.

##  2 Data sample

Much of the asteroseismic data on red giants comes from observations conducted
by the Corot (Auvergne et al., 2009) and Kepler satellites (Borucki et al.,
2010). Bedding et al. (2011) provides a history of ground-based (and
satellite) pursuit of asteroseismic data. Photometry of giants provides a
power spectrum exhibiting many frequencies which correspond to pure acoustic
pressure (p modes) and gravity modes (g modes) as well as mixed modes (Beck et
al., 2011). The mixed modes show a complicated spectrum; however, p- and
g-modes are evenly spaced in frequency and period, respectively. The p modes
are often characterized by the frequency of maximum oscillation power
($\nu\rm_{max}$) in $\mu$Hz and large frequency spacing ($\Delta\nu$) in
$\mu$Hz. Unlike p modes, the g waves propagate through the core. They are
characterized by the period spacing $\Delta\Pi_{l}$ defined as the difference
between the periods of g modes of the same degree and consecutive radial
orders. In this work, we use the asymptotic period spacing $\Delta\Pi\rm_{1}$,
which is the asymptotic g mode period spacing for $l=$ 1 in sec.

In the ($\Delta\Pi\rm_{1}$,$\Delta\nu$) plane, low mass CHeB and RGB stars
with $M\leq 1.8M_{\odot}$ are well separated (see below) such that the
evolutionary phase of a star may be assigned (Bedding et al., 2011) and, thus,
the variation of Li abundance with evolutionary phase may be explored.
However, for stars with $M$ >1.8 $M_{\odot}$, also known as secondary red
clump giants (Bedding et al., 2011), there is significant overlap between
tracks for the CHeB and RGB phases which makes it challenging to assign an
evolutionary phase based on the classical definition correctly, i.e., based on
the distribution in the ($\Delta\Pi\rm_{1}$,$\Delta\nu$) plane. Their
evolutionary phase, however, can be assigned in a non-classical way as done by
Kallinger et al. (2012) and the one by Elsworth et al. (2017).

We first searched the literature for giants with a known Li abundance in
Kepler’s field of view. From Martell & Shetrone (2013); Anthony-Twarog et al.
(2013); Silva Aguirre et al. (2014); Molenda-Żakowicz et al. (2014); Jofré et
al. (2015); Takeda & Tajitsu (2017); Bharat Kumar et al. (2018); Singh et al.
(2019a); Singh et al. (2019b); Deliyannis et al. (2019) and Yan et al. (2020),
we found a total of 574 giants with measured Li abundances in the Kepler
field.

There are four open clusters (NGC 6791, NGC 6811, NGC 6819 and NGC 6866) in
the Kepler field, while lone open cluster NGC 6633 is in CoRoT field of view.
NGC 6819 is the only well studied with both Li and seismic data (Deliyannis et
al., 2019). For the other three clusters in the Kepler field, Li abundances
are available only for a few stars; for example, Molenda-Żakowicz et al.
(2014) provides Li abundances of five G-type red clump stars of NGC 6811.
These five giants and giants of NGC 6819 are now part of our sample. Li
abundances for the giants of NGC 6866 and NGC 6791 are apparently unavailable.

Most of the studies in the literature provide the LTE Li abundance from the
neutral Li resonance line at 6707 Å. Given the large range in Li abundance
across the sample, the modest corrections for non-LTE effects for a typical
giant seem unimportant. Abundance increases for the non-LTE effect for the
6707 Å Li line, as provided by Lind et al. (2009) can range from close to zero
for the hottest giants to about 0.4 dex for cool, metal-rich giants. For
giants where the 6707 Å line is strong, the abundance analysis including non-
LTE corrections is more reliably made using the excited Li lines at 6103 Å and
8126 Å.

Many of the observations in our initial list of giants are repeat
observations. To eliminate repeat entries, we adopt the Li abundance (along
with stellar parameters) from the spectroscopic study with the highest
spectral resolution. For stars observed at comparable spectral resolutions in
different studies, we adopt the entry with the highest signal-to-noise (SNR)
ratio. If both spectral resolution and SNR are similar, then the entry from
the latest study is considered. This resulted in a final sample of 514 unique
giant stars in the Kepler field for which Li abundances are also available.

To get the asteroseismic data for these giants, we restrict our search to
Bedding et al. (2011); Mosser et al. (2012); Stello et al. (2013); Mosser et
al. (2014); Vrard et al. (2016); Elsworth et al. (2017); Hon et al. (2017); Yu
et al. (2018) and Vrard et al. (2018), along with the source papers of the
giants from which Li abundances are taken. In these studies, we found a total
of 612 asteroseismic data entries for our 514 unique stars. Data entries
mainly include the evolutionary phase classifications along with some of the
seismic parameters, like $\nu\rm_{max}$, $\Delta\nu$ and $\Delta\Pi\rm_{1}$.
For the majority of the entries, the evolutionary phase classification is
given with $\nu\rm_{max}$ and $\Delta\nu$, while $\Delta\Pi\rm_{1}$ is
available for a small number of stars. We eliminate repeat entries by adopting
data from the latest study. While selecting the data, preference is given to
studies in which uncertainties in the seismic parameters are also provided. If
$\Delta\Pi\rm_{1}$ is missing from the selected study for a star, then we add
its most recent measurement (if available) from other studies (and of course,
by giving preference to the study in which uncertainty in $\Delta\Pi\rm_{1}$
is also provided). This resulted in a final sample of 187 unique stars
(hereafter, the KRBS sample, where ‘K’ stands for the Kepler mission, and
‘RBS’ stands for our ‘red-giant branch sample’). In the sample KRBS, both
$\nu\rm_{max}$ and $\Delta\nu$ are available for 187 stars while
$\Delta\Pi\rm_{1}$ is available for only 64 stars.The KBRS consists of 44
RGBs, 140 core-He-burning (CHeB) and three giants with an unclassified
evolutionary phases.222Spectroscopic and asteroseismic data for giants in our
final sample is publicly available on Zenodo at
https://doi.org/10.5281/zenodo.4519625. 333Comments from a post-publication
email communication with Professor Benoît Mosser regarding giants with
unclassified evolutionary phase in our list. 1) The oscillation spectrum of
KIC 5112751 has a poor signal-to-noise ratio. 2) KIC 5024272 and KIC 5024750
have a status as evolved RGB or evolved AGB; a recent method is tried to
disentangle evolutionary stages of these stars (Mosser et al., 2019), but it
fails here due to a too low SNR. 3) KIC 6221548 burns its core helium in the
secondary clump, with a mass of about 2.3 $M_{\odot}$; its location in the
Mass - $A$(Li) diagram seems consistent.

The spectroscopic Hertzsprung-Russell diagram (sHRD) for the sample KRBS along
with stars from 25th Data Release of the Kepler stellar properties catalog
(hereafter, Kepler DR25; Mathur et al., 2017), in the background are shown in
Figure 1. For stars in the sample KRBS, we used effective surface temperature
($T\rm_{eff}$) from the Kepler DR25 while surface gravity (log$g$) estimates
are taken from the papers where Li abundances are taken.

##  3 Results

###  3.1 Asteroseismic mass for sample stars

The mass of a RGB or a CHeB star with asteroseismic data may be determined
using the following well known scaling relations (see e.g. Stello et al.,
2009; Kallinger et al., 2010; Mosser et al., 2010).

${M_{SR}\over M_{\odot}}=m\
\left({\nu{\rm{}_{max}}\over\nu{\rm{}_{max,\odot}}}\right)^{3}\left({\Delta\nu\over\Delta\nu_{\odot}}\right)^{-4}\left({T{\rm{}_{eff}}\over
T{\rm{}_{eff,\odot}}}\right)^{3/2}$ (1)

here, $T\rm_{eff,\odot}$ = 5777 K, $\Delta\nu_{\odot}$ = 135.1 $\pm$ 0.1
$\mu$Hz and $\nu{\rm{}_{max,\odot}}$ = 3090 $\pm$ 30 $\mu$Hz are reference
solar values (Huber et al., 2011) while $m$ is a scaling factor. The scaling
relation is known to slightly vary for stars depending on their evolutionary
phase. For example, Mosser et al. (2010) found that for giant stars, the value
of $m$ is 0.89 $\pm$ 0.07 for $\Delta\nu_{\odot}$ = 135.5 $\mu$Hz,
$\nu{\rm{}_{max,\odot}}$ = 3050 $\mu$Hz, and $T\rm_{eff,\odot}$ = 5777 K. In
the current study, we use the original scaling relation (i.e., with $m$ = 1).
For more recent updates see Hekker (2020). We also estimated uncertainties in
mass through error propagation. Mass is determined for all the 184 giants for
which asteroseismic data is available; however, estimation of the
corresponding uncertainty is not possible for seven giants due to the
unavailability of uncertainties in seismic data. Mean uncertainty in estimated
mass is 0.15 $\pm$ 0.11 $M_{\odot}$. For 181 of our sample stars, seismic mass
estimates are also available from the original studies reporting the
asteroseismic data. Our mass estimates from equation 1 are lower by only 0.002
$\pm$ 0.018 $M_{\odot}$ than estimates in these studies. This difference and
the slight scatter in the mass differences is likely due to small systematic
and random differences in the adopted effective temperatures and reference
solar seismic parameters.

Instead of just relying on the above scaling relations, we also estimated
seismic masses for our sample stars using PARAM (da Silva et al., 2006; Miglio
et al., 2013).444http://stev.oapd.inaf.it/cgi-bin/param_1.3 PARAM provides
Bayesian estimation of stellar parameters (including mass) using PARSEC
isochrones from Bressan et al. (2012). In the current study, we use PARAM in
its asteroseismology mode with other conditions as default, except Bayesian
priors, which are estimated for the age interval from 0.1 to 14 Gyr. For each
star, the input parameters are $T\rm_{eff}$, [Fe/H], $\nu\rm_{max}$ and
$\Delta\nu$ along with their uncertainties and a star’s evolutionary phase
classification. Hereafter, we refer to estimated masses from PARAM by $M_{\tt
PARAM}$. For seven of our sample stars for which uncertainties in
$\nu\rm_{max}$ and $\Delta\nu$ are not available, we assumed uncertainties of
0.1 and 0.05 $\mu$Hz, respectively. Distribution of masses for our sample
stars estimated from scaling relation and from PARAM are shown in Figure 2.
The scaling relation suggests the mass range of CHeB stars extends down to
$M\simeq 0.5M_{\odot}$ which seems implausible on account of very long main
sequence lifetimes.

Figure 2: $A$(Li) as a function of stellar mass estimated from the scaling
relation (top panel) and from PARAM (bottom panel). Errors in mass estimates
are shown as black error bars. Filled red and blue circles are core He-burning
giants with Li detections and upper limits, respectively. Filled black and
green squares are RGB giants with H-burning in shell with Li detections and
upper limits, respectively.

###  3.2 Li as a function of stellar mass

Variations of Li abundance with stellar mass are potentially valuable clues to
the origins of Li enrichment in RGB and CHeB giants. $A$(Li) as a function of
a star’s mass as estimated from scaling relation ($M_{SR}$) and from PARAM
($M_{\tt PARAM}$) is shown in Figure 2. Errors in the estimated masses are
shown as black error bars. The majority of giants in our sample have $A$(Li)
higher than 1.5. This is a selection bias. However, the lack of low-Li giants
will not affect our analysis as we are more concerned about the presence of
giants with Li higher than the expected value. According to standard models of
stellar evolution, post first dredge-up, A(Li) in low-mass giants is expected
to be lower than $\sim$ 1.5 dex. Hence, all the giants with A(Li) higher than
1.5 can be safely assumed as Li-enriched, and their presence across the range
of stellar parameters, like mass and asteroseismic parameters, is enough to
draw conclusions about processes that may be responsible for Li-enrichment in
these giants. The necessity of having giants with a low A(Li) becomes absolute
only if one wants to estimate the Li-rich giants’ occurrence rate as a
function of these stellar parameters, which is not a goal of this study.
Hence, the lack of low A(Li) giants in our sample will not affect our
conclusions.

Li in our sample of RGB giants has a maximum value of $A$(Li) at an average of
$\simeq 2.3$ (black squares) in stars with masses from about 1-3 $M_{\odot}$.
Other RGB giants (green squares) provide only an upper limit to $A$(Li) with
$A$(Li) extending to $-1.5$ over apparently a narrow mass range centred on
about 1.6-1.8$M_{\odot}$, but this may be an illusion resulting from the small
sample.

Mass estimates for the CHeB giants extend beyond the mass of 2.25$M_{\odot}$,
which is considered to be the maximum mass for a RGB star to experience the
He-core flash, a leading suspect for Li-enrichment via conversion of 3He
(Iben, 1967). The vast majority of the CHeB giants in our sample are Li-rich
and even very Li-rich with $A$(Li) $\geq 4$. Li-rich stars with $A$(Li) $\geq
2$ are not restricted to giants expected to have suffered a He-core flash,
i.e., $M\leq 2.25M_{\odot}$. For a few CHeB giants, the Li 6707Å feature is
undetectable, and an upper limit is set for the Li abundance (blue circles).

Figure 3: Distribution for our sample stars in the $\Delta\Pi\rm_{1}$ $-$
$\Delta\nu$ plane. Used labels are same as in Figure 2.  Figure 4:
Evolutionary tracks for solar-metallicity model stars with masses 0.8 (blue
line), 1.0 (red line) and 1.2 $M_{\odot}$ (black line) in $\Delta\Pi\rm_{1}$
$-$ $\Delta\nu$ plane. The RGB phase is shown in the top and middle panel,
while the CHeB phase is shown in the bottom panel. Arrows indicates the global
direction of evolution of model stars. Open circles in the top panel indicate
the position of the luminosity bump for each of the three masses. In the
middle panel, the luminosity bump occurs at the obvious ‘S-type’ loop in the
tracks. The initial He core flash at $\Delta\nu$ $<$ 0.1 $\mu$Hz) is very
strong. However, the actual peak value may be even higher and missed due to
the finite step size. This and other early episodes of He core flash occur in
a brief phase at a luminosity well above the RC. Figure 5: Variation of
$\Delta\nu$, $\Delta\Pi\rm_{1}$, luminosity and $T\rm_{eff}$ as a function of
time (in million years) during the CHeB phase for solar-metallicity model
stars of masses 0.8, 1.0 and 1.2 $M_{\odot}$. A base-10 log scale is used for
the time axis to highlight the fact that giants in the CHeB phase spend almost
their entire life in a relatively stable state except for the brief He-
flashing region at the start.

###  3.3 Predicted evolution in the $\Delta\Pi\rm_{1}$ $-$ $\Delta\nu$ plane

The $\Delta\Pi\rm_{1}$ versus $\Delta\nu$ distribution for 64 sample stars is
shown in Figure 3. These are the stars in our sample for which
$\Delta\Pi\rm_{1}$ is available. As expected, both RGB (filled squares) and
CHeB (filled circles) are well separated. Indeed, the differences between
$\Delta\Pi\rm_{1}$ and $\Delta\nu$ were the primary criteria for assigning the
evolutionary phase.

Gross differences in the asteroseismic data between CHeB and RGB giants are
the basis for assigning a giant to one of these evolutionary phases. Less
striking changes of asteroseismic data are expected from the changing internal
structure of a giant be it as the star evolves along the RGB or as the He-core
of a CHeB star exhausts its He. Such changes may prove to be the basis for
identifying the precise location of a RGB or a CHeB giant in its evolution. To
quantify the evolution of giants in the $\Delta\Pi\rm_{1}$ $-$ $\Delta\nu$
plane, we make use of the Modules for Experiments in Stellar Astrophysics
(MESA; Paxton et al., 2011, 2013, 2015, 2018).

We started with the MESA model constructed by Schwab (2020). We activated the
output columns for asteroseismic parameters and also decreased the step
interval to get fine details about the core He-flash phase.555The MESA model
constructed by Schwab (2020) is publicly available on Zenodo at doi:
https://doi.org/10.5281/zenodo.3960434 Then, we evolved models of solar-
metallicity stars of masses 0.8, 1.0, and 1.2 $M_{\odot}$ until the centre
mass fraction of 4He drops below 10-6 to conclude the CHeB phase.

The evolutionary tracks for the red-giant phases – RGB and CHeB – of these
stars are shown in Figure 4. Arrows indicate the global direction of evolution
of giants along the tracks. In the top panel, the RGB branch corresponds to
the tracks with $\Delta\nu$ $\lesssim 40$ $\mu$Hz which are nearly identical
for $M\lesssim 1.8M_{\odot}$ (Stello et al., 2013). For masses $M\geq
1.9M_{\odot}$, evolutionary tracks (not shown in this panel) provide an
increasing $\Delta\Pi\rm_{1}$ at a given $\Delta\nu$ (Stello et al., 2013).
Prior to entry to the RGB, the tracks in the top panel are separated by mass.
Evolution up the RGB is shown in more detail in the middle panel where the
$\Delta\nu$ range matches that shown in Figure 3. Location of the luminosity
bump is indicated in the top panel and revealed in the middle panel by the
‘S-type’ loops at $\Delta\nu\simeq 6$ $\mu$Hz. Locating the luminosity bump
among giants in the asteroseismic sample is relevant to the exploration of Li
among giants because Li production via thermohaline instabilities at the
luminosity bump has been proposed (Charbonnel & Zahn, 2007). The RGB and CHeB
MESA tracks are shown for solar metallicity models but are insensitive to
factor of two changes in metallicity and unchanged when initial rotation
$\Omega_{\rm ZAMS}$/$\Omega_{\rm crit}$ is changed between zero and 0.4.

The CHeB phase begins in the bottom left of the lower panel where the RGB
track ends with readjustment to the He-core flash (Figure 4). Thanks to the
small step size, we can also see the positions of subsequent mini He-flashes.
These subsequent He-flashes are weaker compared to the initial strong He flash
because re-ignition of He occurs but less violently (Mocák et al., 2010). For
all three model stars of mass 0.8, 1.0 and 1.2 $M_{\odot}$, temporal evolution
from start to end of the CHeB phase is illustrated by Figure 5. A base-10 log
scale is used for the time axis considering the short duration of the He-
flashing region compared to the total duration of the CHeB phase. Figure shows
the variation of star’s asteroseismic parameters, luminosity and $T\rm_{eff}$.
All the three model stars undergoes mini-He core flashes following the initial
strong He-flash. Temporal evolution for all the three masses is very similar
in form. By far most of the time in the core He-burning phase is spent on the
path from about ($\Delta\nu,\Delta\Pi_{1})$ = (4.5,300) to (1.5,100), as shown
clearly in Figure 5.

The long-duration portions of the tracks for the three masses cover the
majority of the observations for CHeB stars in Figure 3. The He core flashes
last only for about 1 Myr and, hence, very few CHeB stars will be observed
during this phase. For the major episode of CHeB, $\Delta\Pi\rm_{1}$ ranges
from about 150 to 320 sec. This is broadly consistent with observations in
Figure 3. Indeed, just three CHeB giants with $\Delta\nu\simeq 7$ $\mu$Hz are
obvious outliers. This trio may be secondary RC giants (Bedding et al., 2011;
Stello et al., 2013).

Figure 6: $A$(Li) as a function of $\Delta\nu$ for our sample stars. Used
labels are same as in Figure 2.

###  3.4 Li as a function of $\Delta\nu$ and $\Delta\Pi\rm_{1}$ in giants

Our comparison of observed Li abundances for RGB and CHeB giants with their
asteroseismic data is made in two steps. First we compare the Li abundances
with the $\Delta\nu$ measurements (Figure 6) and then with the
$\Delta\Pi\rm_{1}$ data (Figure 7). This separation is made because not every
giant in our sample has both $\Delta\Pi\rm_{1}$ and $\Delta\nu$ available.

$A$(Li) as a function of $\Delta\nu$ for 187 giants is shown in Figure 6. The
figure resembles the luminosity versus $A$(Li) plots with the RGB-tip towards
the top-right corner. As previously, RGB and CHeB giants are represented by
filled squares and circles, respectively, as indicated by the key. For the RGB
stars, the $\Delta\nu$ range about 0 to 18 $\mu$Hz spans the range covered by
Figure 4 with stars spanning the luminosity bump (LB) and a few stars nearing
the RGB tip ($\Delta\nu\simeq 1\mu$Hz). Many of the RGB giants along almost
the entire RGB have the $A$(Li) $\simeq 1.5-2.0$, as expected for post dredge-
up giants. In this small sample, there is no indication that the LB denotes a
change in $A$(Li) measurements except for a slight increase in the measured
$A$(Li) for $\Delta\nu$ $\leq 6\mu$Hz. RGB giants with $A$(Li) upper limits
may have resulted from extraordinary Li destruction on the main sequence or in
the transition to the RGB. Figure 6 shows that the upper limits for RGB giants
trend to their lowest $A$(Li) as $\Delta\nu$ declines and, thus, as the giants
approach the RGB tip and the limit $A$(Li) $\sim-1.5$.

Giants belonging to CHeB phase including secondary RC stars are shown as
filled circles in Figure 6 and are concentrated around 4 $\mu$Hz. The few
higher values of $\Delta\nu$ likely arise from secondary RC stars, all with
detectable Li. Several stars with lower $\Delta\nu$ may be identified as more
advanced in the CHeB. Almost every CHeB giant is Li-rich, i.e., $A$(Li) $\geq
1.5$ but a few stars with lower Li abundances including a sample with $A$(Li)
upper limits as low as -0.5.

Figure 7: Li abundances in RGB and core-He-burning (CHeB) giants as a function
of their gravity-mode period spacing ($\Delta\Pi\rm_{1}$). Used labels are
same as in Figure 2.

Li abundances as a function of $\Delta\Pi\rm_{1}$ for our sample stars are
shown in Figure 7. The few RGB giants have $\Delta\Pi\rm_{1}$ between about 60
and 90 sec about as expected from Figure 4. The CHeB giants span the range in
$\Delta\Pi\rm_{1}$ about as expected from Figure 4 except absence of CHeB
giants with very low $\Delta\nu$ and high $\Delta\Pi\rm_{1}$. The dearth of
CHeB giants with $\Delta\Pi\rm_{1}$ $<$ 200 in our observational sample is due
to very fast evolution of giants in this phase (Figure 5). In general, figure
is suggesting a decreasing trend in A(Li) with increase in $\Delta\Pi\rm_{1}$.

##  4 Discussion and Conclusions

In this paper, we assemble data from the literature on asteroseismology and Li
abundances for giants. Our final sample of 187 giants consists of 44 RGB, 140
CHeB and three giants with unclassified evolutionary phase.

First, to investigate the mass dependence of Li enrichment in giants, we
estimated masses for our sample stars from asteroseismic scaling relations as
well as from the PARAM which provides isochrones based Bayesian estimates
using asteroseismic data. Estimated masses for the CHeB giants from
asteroseismic scaling relation extend down to about 0.5 $M_{\odot}$, which is
not the case for mass estimates from the PARAM. The presence of such low-mass
giants in the CHeB phase seems implausible on account of very long main-
sequence lifetimes compared to the Galaxy’s age. However, it might be
explained by large mass loss in the giant phase or an issue with the
asteroseismic scaling relations. Nevertheless, after comparison with isochrone
based mass estimates from the PARAM, our first impression is that there is an
issue with mass estimates from scaling relation for giants in the CHeB phase.
A more detailed study with a larger sample is needed to further resolve this
issue.

Distribution of Li-rich giants as a function of mass is of particular
interest. It can provide clues about processes leading to Li enrichment in
giants. For example, if Li enrichment is associated with core He-flash only
then all the Li-rich giants are supposed to have masses lower than the maximum
mass expected to have suffered a core He-flash, i.e. $M\leq$ 2.25 $M_{\odot}$.
However, in our sample, we have a significant number of Li-rich CHeB giants
with masses higher than 2.25 $M_{\odot}$ (Figure 2). This suggests that whilst
the core He-flash may be the dominant process leading to Li enrichment in low-
mass giants, there is a secondary process leading to Li enrichment in the more
massive stars.

To study evolution of Li in giants, one of the primary task is to untangle
giants of different evolutionary phases (like, RGB, CHeB, etc). This can be
done either based on stars’ position in HR diagram or using asteroseismology.
In this study our goal is to understand changes in asteroseismic parameters
(especially $\Delta\Pi\rm_{1}$ and $\Delta\nu$) of giants as they evolve. For
this we used the MESA models. Thanks to the consideration of very small step
size, models reveal the very minute and vital information regarding the
variation of seismic parameters during the red giant phase and suggest the
presence of subsequent mini He-flashes along with the initial strong core He-
flash (Figure 4). However, all of this activity occurs in a brief evolutionary
phase before the giant assumes its steady state phase of core He-burning.

Based on observational data, we explored the variation of A(Li) as a function
of seismic parameter $\Delta\nu$ and $\Delta\Pi\rm_{1}$. From the distribution
of A(Li) as a function of $\Delta\nu$, which is similar to the distribution of
A(Li) as a function of luminosity, we found no indication of Li enrichment
near the luminosity bump. Also, A(Li) trends to $\sim$ -1.5 dex as $\Delta\nu$
approaches zero, i.e. near the RGB tip.

Evolution of Li as a function of $\Delta\Pi\rm_{1}$ is of particular interest.
Data suggest a decreasing trend in A(Li) with an increase in asymptotic period
spacing $\Delta\Pi\rm_{1}$ for CHeB giants (Figure 7). In CHeB giants, the
presence of higher A(Li) at lower $\Delta\Pi\rm_{1}$ points towards Li
enrichment at the early stage of the CHeB phase. Also, the dearth of CHeB
giants with $\Delta\Pi\rm_{1}$ $<$ 200 can be explained by fast evolution of
giants in this phase (Figure 5). A more detailed study of stellar models will
benefit understanding the evolution of Li near the RGB tip. One way to further
this study is to get asymptotic period spacing $\Delta\Pi\rm_{1}$, which is
available for only 64, for all the 187 giants in our sample.

Many giants in the RGB and CHeB phases have now been probed for their
asteroseismic data but lack spectroscopic measurements of Li and other
elemental abundances. Expansion of our small samples of asteroseismic and Li
data on RGB and CHeB giants would be of great interest for several reasons.
For example, CHeB giants and RGB giants at the luminosity bump partially
overlap in the HR diagram and spectroscopic separation is difficult but
asteroseismic data provides a certain identification of these two evolutionary
phases. Expansion of our sample would finally answer the question about the
extent of surface Li enrichment for giants evolving through the luminosity
bump. In this regard, data from the ongoing Large Sky Area Multi-Object Fibre
Spectroscopic Telescope (LAMOST)-Kepler/K2 survey will be beneficial. In a
similar spirit, expansion of the CHeB sample should allow examination of the
Li enrichment as a function of stellar mass and also of the age of a CHeB
star. Much remains to be done by observers and theoreticians.

## Acknowledgments

We thank the anonymous referee for the thorough reviews of our manuscript. We
also thank the editors for constructive and helpful suggestions. We
acknowledge the use of the Delphinus and Fornax server at IIA, Bengaluru, for
evolving our model stars. We also acknowledge the use of the MESA models
constructed by Schwab (2020). This research has made use of the Simbad
database (Wenger et al., 2000), operated at the Centre de Données
Astronomiques de Strasbourg (CDS; Genova et al., 2000), France, and available
at http://cdsweb.u-strasbg.fr. This work has also made use of NASA’s
Astrophysics Data System (ADS) available at https://ui.adsabs.harvard.edu. The
ADS is operated by the Smithsonian Astrophysical Observatory under NASA
Cooperative Agreement NNX16AC86A.

## ORCID iDs

Deepak: https://orcid.org/0000-0003-2048-9870
David L. Lambert: https://orcid.org/0000-0003-1814-3379

## Data Availability

The data underlying this article are publicly available on Zenodo at
https://doi.org/10.5281/zenodo.4519625.

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