# Learning SO(3)-Invariant Semantic Correspondence via Local Shape Transform

Chunghyun Park1 Seungwook Kim111footnotemark: 1 Jaesik Park2 Minsu Cho1
1POSTECH 2Seoul National University
Equal contribution

###### Abstract

Establishing accurate 3D correspondences between shapes stands as a pivotal
challenge with profound implications for computer vision and robotics.
However, existing self-supervised methods for this problem assume perfect
input shape alignment, restricting their real-world applicability. In this
work, we introduce a novel self-supervised Rotation-Invariant 3D
correspondence learner with local Shape Transform, dubbed RIST, that learns to
establish dense correspondences between shapes even under challenging intra-
class variations and arbitrary orientations. Specifically, RIST learns to
dynamically formulate an SO(3)-invariant local shape transform for each point,
which maps the SO(3)-equivariant global shape descriptor of the input shape to
a local shape descriptor. These local shape descriptors are provided as inputs
to our decoder to facilitate point cloud self- and cross-reconstruction. Our
proposed self-supervised training pipeline encourages semantically
corresponding points from different shapes to be mapped to similar local shape
descriptors, enabling RIST to establish dense point-wise correspondences. RIST
demonstrates state-of-the-art performances on 3D part label transfer and
semantic keypoint transfer given arbitrarily rotated point cloud pairs of the
same category, outperforming existing methods by significant margins.

##  1 Introduction

Establishing dense 3D correspondences between different shapes is foundational
to numerous applications across computer vision, graphics, and robotics [28,
22, 9, 41]. One of the primary challenges hindering advancements in this
domain is the difficulty of annotating dense inter-shape correspondences,
which limits the leverage of strongly-supervised learning paradigms.

Recently, self-supervised learning methods have been proposed to address this
issue [21, 3], showing promising directions for 3D correspondence estimation.
Nonetheless, a significant limitation in existing approaches is their
stringent assumption about the alignment of input shape pairs; these methods
strongly assume that the input point cloud pair to establish correspondences
between is precisely aligned. This assumption is rarely met in practice, where
object scans and shape instances can be arbitrarily oriented. We find that the
performance of existing methods degrades significantly when confronted with
rotated input shapes, restricting their real-world applicability (Figure 1).

Figure 1: Semantic correspondence between rotated shapes. We visualize the
semantic correspondence results of the previous SOTA [3] and ours, given two
randomly rotated airplanes from the ShapeNetPart dataset [39]. Green and red
lines indicate the correct and incorrect matches, respectively. For each
method, 100 source points are randomly selected from the source (yellow) for
correspondence visualization222For the details of the inference algorithm,
please refer to Appendix A.. Ours predicts SO(3)-invariant correspondences,
showing superior accuracy and robustness in comparison to the previous SOTA
[3].

To address this challenge, we introduce a novel self-supervised learning
approach, dubbed RIST, designed to reliably determine dense SO(3)-invariant
correspondences between shapes via local shape transform. In essence, RIST
learns to formulate SO(3)-invariant local shape transform for each point in a
dynamic and input-dependent manner. Each point-wise local shape transform maps
the SO(3)-equivariant global shape descriptor of the input shape to a local
shape descriptor, which is passed to the decoder to reconstruct the shape and
pose of the input shape. By training RIST via self- and cross-reconstruction
of input shapes, true semantically corresponding points are trained to yield
similar local shape descriptors, enabling us to determine dense shape
correspondences.

RIST demonstrates state-of-the-art performance on part segmentation label
transfer when evaluated on the ShapeNetPart [39] and ScanObjectNN [35]
datasets. In particular, significant improvements over existing baselines are
observed when our method is applied to randomly oriented shape pair inputs.
Furthermore, our approach also proves to be more effective compared to
existing methods at semantic keypoint transfer when evaluated on the
KeypointNet dataset [40]. This showcases not only the applicability of RIST
across a diverse range of tasks, but also its potential to be utilized for
efficient dense annotation of 3D shapes. These results highlight the efficacy
of RIST in addressing the challenges posed by real-world scenarios where
existing methods fail to perform effectively.

The main contributions of our work are as follows:

  * •

We introduce RIST, a novel self-supervised approach for determining dense
SO(3)-invariant correspondences between arbitrarily aligned 3D objects.

  * •

We propose to formulate the local shape information of each point as a novel
function called local shape transform with dynamic input-dependent parameters,
which effectively maps the global shape descriptor of input shapes to local
shape descriptors.

  * •

RIST achieves state-of-the-art performance on 3D part segmentation label
transfer and 3D keypoint transfer under arbitrary rotations, indicating its
potential for application in a wide range of practical tasks in computer
vision.

##  2 Related Work

Point cloud understanding via self-supervised learning. While traditional
methods for point cloud processing involving hand-crafted features [33, 27]
have shown impressive performance, with the advent of deep learning,
substantial research efforts have been directed towards developing learning-
based algorithms capable of effectively processing and understanding point
clouds [25, 26, 42, 24, 5]. Due to limited large-scale datasets with rich
annotations, self-supervised learning approaches emerged as a viable
alternative. One of the most prominent directions to learn point cloud
representations in a self-supervised manner is learning through self-
reconstruction [38, 43, 23] of the point cloud. Primarily inspired by the
efficacy of point cloud reconstruction as a self-supervised representation
learning scheme, we train RIST to establish 3D correspondences in a self-
supervised manner via self- and cross-reconstruction of point clouds by
leveraging SO(3)-invariant dynamic local shape transform.

Equivariance and invariance to rotation. The conventional method to improve a
neural network’s robustness to rotation is by employing rotation augmentations
during training or inference. However, this tends to increase the resources
required for training and still shows unsatisfactory results when confronted
with an unseen rotation [18, 14]. In recent years, various methods have been
proposed to yield point cloud representations, which are equivariant [6, 32,
29, 2] or invariant [31, 19, 37, 18, 14] to the rotation of the input,
demonstrating enhanced performances under arbitrary input rotations. To
facilitate the rotation-robust establishment of 3D dense correspondences, we
utilize SO(3)-equivariant networks in building RIST, leveraging
SO(3)-equivariant and -invariant representations to guarantee robustness to
rotation by design.

Semantic correspondences under intra-class variations. Finding correspondences
between images or shapes under intra-class variations - manifesting as
differences in shape, size, and orientation within the same category of
objects - poses significant challenges over photometric or viewpoint
variations. This task has been widely studied in the domain of images, where
existing methods make use of sparsely annotated image pair datasets to train
their method in a strongly- or a weakly-supervised manner [4, 13, 34, 11, 15].
However, learning to establish dense yet reliable 3D correspondences between
3D shapes remains challenging, as it is infeasible to label dense
correspondence annotations across point cloud pairs with intra-class
variations. Self-supervised methods have been proposed to address this issue
[21, 3], but they strongly assume that the input point clouds are aligned,
leading to considerable significant degradation when confronted with
arbitrarily rotated point clouds. Additionally, the functional map-based
approach introduced by Huang  _et al_. [10] for non-rigid registration
struggles with topological changes and efficiency. To this end, we propose
RIST to establish reliable 3D dense correspondences irrespective of the input
point clouds’ poses.

Figure 2: Overview: Self-supervised training of RIST. The input point clouds
are independently encoded to SO(3)-equivariant global shape descriptor
$\mathbf{Z}$ and dynamic SO(3)-invariant point-wise local shape transforms
$\\{f_{\theta_{i}}\\}$. The local shape transforms map the global shape
descriptor to local shape descriptors by infusing local semantics and
geometry, which are used as inputs to the decoder for self-reconstruction. For
cross-reconstruction, we apply the local shape transforms formulated from
another point cloud to reconstruct the point cloud, ensuring that the local
shape descriptors successfully capture generalizable local semantics and
geometries. We supervise RIST via penalizing errors in self- and cross-
reconstructions. At inference, we can leverage the local shape transforms for
obtaining local shape descriptors, to identify the dense correspondences.

##  3 RIST for 3D Semantic Correspondence

In this section, we detail the components of RIST, which come together to
facilitate the end-to-end self-supervised training for 3D semantic
correspondence establishment. The objective of 3D semantic correspondence is
as follows; given two different point clouds instances
$\mathbf{P}_{1}\in\mathbb{R}^{N\times 3}$ and
$\mathbf{P}_{2}\in\mathbb{R}^{N\times 3}$ belonging to the same semantic
category, we aim to find all semantically corresponding point pairs
$\\{\mathbf{p}_{i},\mathbf{q}_{i}\\}_{i=1}^{N^{\prime}}$333$N^{\prime}\leq N$;
there could be points with no matches. such that
$\mathbf{p}_{i}\in\mathbf{P}_{1}$ and $\mathbf{q}_{i}\in\mathbf{P}_{2}$. To
achieve this, we claim it is crucial to identify the local shape information
_i.e_. local semantics and geometry, which is generalizable across different
instances within the same category.

Therefore, the main idea of RIST is to dynamically generate a SO(3)-invariant
local shape transform as a function for each point, such that each local shape
transform can map the SO(3)-equivariant global shape descriptor of the input
point cloud to its respective local shape descriptor. In the following, we
elaborate on the network architecture of RIST, in particular how we leverage
SO(3)-equivariant and invariant representations to facilitate the dynamic
formulation of pointwise SO(3)-invariant local shape transforms and the
reconstruction of pose-preserved point clouds (Sec. 3.1). Subsequently, we
introduce our self-supervisory objective function, which trains RIST to self-
and cross-reconstruct the input point clouds in a rotation-equivariant manner
(Sec. 3.2), finally enabling the establishment of 3D dense correspondences
(Sec. 3.3) via corresponding local shape descriptors. Figure 2 illustrates the
outline of the training scheme of RIST.

###  3.1 Network Design of RIST

####  3.1.1 Preliminary: SO(3)-Equivariant Representation

One of the main motivations of RIST is to establish reliable and accurate 3D
dense correspondences given arbitrarily rotated shapes, a setting where
existing work shows to be brittle. This requires our encoder to formulate the
point-wise local shape transforms not only effectively to capture the local
shape semantics and geometry, but also robustly against transformations in the
SO(3) space. To this end, we integrate SO(3)-equivariant networks into RIST to
facilitate robustness to SO(3) transformations of the input. In this work, we
choose VNNs [7] to build our SO(3)-equivariant layers for their simplicity and
efficacy. In VNNs, a single neuron, which is represented by a scalar-list of
values, is lifted to a vector-list feature $\mathbf{V}\in\mathbb{R}^{C\times
3}$, which is essentially a sequence of 3D vectors. The layers of VNNs handle
batches of such vector-list features such that equivariance with respect to
rotation $R\in$ SO(3) is satisfied _i.e_. $f(\mathbf{V}R)=f(\mathbf{V})R$444We
refer the readers to the original paper [7] for further information and
detailed formulations of VNNs.. Notably, we can yield an SO(3)-invariant
output by performing a product of an equivariant vector-list feature
$\mathbf{V}R\in\mathbb{R}^{C\times 3}$ with the transpose of another
consistently equivariant vector-list feature
$\mathbf{U}R\in\mathbb{R}^{C^{\prime}\times 3}$ as follows:
$(\mathbf{V}R)(\mathbf{U}R)^{\top}=\mathbf{V}RR^{\top}\mathbf{U}^{\top}=\mathbf{V}\mathbf{U}^{\top}$.
This serves as a critical functionality when constructing our SO(3)-invariant
local shape transform of our encoder (Sec. 3.1.2).

####  3.1.2 SO(3)-Equivariant Encoder

We design our encoder architecture to take as input a point cloud
$\mathbf{P}\in\mathbb{R}^{N\times 3}$, and simultaneously output an
SO(3)-equivariant global shape descriptor and formulate point-wise
SO(3)-invariant local shape transforms.

SO(3)-equivariant global shape descriptor. Given a point cloud, we first aim
to obtain the SO(3)-equivariant global shape descriptor
$\mathbf{Z}\in\mathbb{R}^{C\times 3}$, which captures the pose and the global
shape characteristics of the input point cloud. We leverage VN-DGCNN [7] as
our encoder architecture, which consists of 4 edge convolutional VN-layers to
capture local semantics at a progressively larger receptive field, and a FPN
[20] to aggregate the multi-level features. Then, we apply the global average
pooling to the aggregated SO(3)-equivariant point-wise features
$\mathbf{V}^{\mathrm{equi}}\in\mathbb{R}^{C\times 3\times N}$ to encode
SO(3)-equivariant global shape descriptor $\mathbf{Z}$ of the input point
cloud. The global shape descriptor can be used subsequently as the input for
our SO(3)-invariant point-wise local shape transform, to be mapped to their
respective local shape descriptors, as shown in Figure 2.

SO(3)-invariant local shape transform. Alongside the extraction of
SO(3)-equivariant global shape descriptors, we also formulate the
SO(3)-invariant local shape transform $f_{\theta_{i}}:\mathbb{R}^{C\times
3}\mapsto\mathbb{R}^{C^{\prime}\times 3}$ for each point
$\mathbf{p}_{i}\in\mathbb{R}^{3}$ of the input point cloud
$\mathbf{P}\in\mathbb{R}^{N\times 3}$. The parameters of each local shape
transform $\theta_{i}\in\mathbb{R}^{C^{\prime}\times C}$ are input-dependent -
thus, dynamic since they are predicted by our encoder for the $i$-th point of
the point cloud. To predict $\theta_{i}$, we first obtain SO(3)-invariant
point-wise features $\mathbf{V}^{\mathrm{in}}\in\mathbb{R}^{C^{\prime}\times
3\times N}$ as described in Sec. 3.1.1. Then, we transform each vectorized
SO(3)-invariant point-wise feature
$\operatorname{vec}(\mathbf{v}^{\mathrm{in}}_{i})\in\mathbb{R}^{3C^{\prime}}$
to the vectorized parameter of the local shape transform
$\operatorname{vec}(\theta_{i})\in\mathbb{R}^{C^{\prime}C}$ by using a multi-
layer perceptron. By reshaping $\operatorname{vec}(\theta_{i})$ to
$\theta_{i}\in\mathbb{R}^{C^{\prime}\times C}$, we finally obtain the dynamic
and SO(3)-invariant local shape transform $f_{\theta_{i}}$ for the point
$\mathbf{p}_{i}$.

The role of these local shape transforms is to map the SO(3)-equivariant
global shape descriptor $\mathbf{Z}\in\mathbb{R}^{C\times 3}$ to their
respective local shape descriptors
$\mathbf{v}^{\prime}_{i}:=f_{\theta_{i}}(\mathbf{Z})\in\mathbb{R}^{C^{\prime}\times
3}$, which is provided as the input to our decoder for reconstruction. Our
self-supervised training scheme encourages the point-wise dynamic local shape
transform to encapsulate the local shape information _e.g_. semantics and
geometry, to enhance the reconstruction performance.

####  3.1.3 SO(3)-Equivariant Decoder

Our decoder aims to reconstruct the initial input shapes using the obtained
SO(3)-equivariant global shape descriptors $\mathbf{Z}$ and the
SO(3)-invariant local shape transforms $\\{f_{\theta_{i}}\\}_{i=1}^{N}$. To
reconstruct the point clouds aligned to their initial poses, we leverage
SO(3)-equivariant layers as the building blocks of our decoder architecture.
We first train our decoder to perform self-reconstruction, using the local
shape descriptors $\mathbf{V}^{\prime}$, _i.e_.
$\mathbf{P}\leftrightarrow\mathbf{P}^{\prime}:=\operatorname{Decoder}(\mathbf{V}^{\prime})=\operatorname{Decoder}(\\{f_{\theta_{i}}(\mathbf{Z})\\}_{i=1}^{N})$.
We also train our decoder to perform cross-reconstruction, where we use the
local shape descriptors obtained using global shape descriptors and local
shape transforms from different point clouds. Specifically, assume we are
given two point clouds $\mathbf{P}_{1},\mathbf{P}_{2}\in\mathbb{R}^{N\times
3}$, with SO(3)-equivariant global shape descriptors
$\mathbf{Z}_{1},\mathbf{Z}_{2}\in\mathbb{R}^{C\times 3}$ and SO(3)-invariant
local shape transforms
$\\{f_{\theta_{i}}^{1}\\}_{i=1}^{N},\\{f_{\theta_{i}}^{2}\\}_{i=1}^{N}$. As
shown in Figure 2, we then can perform cross-reconstruction from
$\mathbf{P}_{2}$ to $\mathbf{P}_{1}$ as follows:
$\mathbf{P}_{1}\leftrightarrow\mathbf{P}^{\prime}_{2\mapsto
1}:=\operatorname{Decoder}(\\{f_{\theta_{i}}^{2}(\mathbf{Z_{1}})\\}_{i=1}^{N})$.
Intuitively, for the above cross-reconstruction to be carried out
successfully, the local shape transforms for points of a true correspondence
should hold similar dynamic parameters, mapping global shape descriptors to
similar local shape descriptors. By training RIST to cross-reconstruct point
clouds, we are supervising local shape transforms to map corresponding points
between shapes to similar local shape descriptors, which encode local
semantics and geometry that are generalizable across different instances
within a category.

###  3.2 Self-Supervised Objective

Due to the lack of annotated datasets for dense 3D inter-shape
correspondences, we train RIST in a self-supervised manner by penalizing
inaccurate shape reconstructions. First, we supervise RIST for self-
reconstruction using the following loss:

$\mathcal{L}_{\mathrm{SR}}=\lambda_{\mathrm{MSE}}\operatorname{MSE}(\mathbf{P},\mathbf{P}^{\prime})+\lambda_{\mathrm{EMD}}\operatorname{EMD}(\mathbf{P},\mathbf{P}^{\prime}),$
(1)

where $\operatorname{MSE}$ is the Mean Squared Error, $\operatorname{EMD}$
stands for the Earth Mover’s Distance, and both $\lambda_{\mathrm{MSE}}$ and
$\lambda_{\mathrm{EMD}}$ are weight coefficients. In essence, we are trying to
minimize the difference between the input and reconstructed point cloud. We
also supervise RIST for cross-reconstruction as follows:
$\mathcal{L}_{\mathrm{CR}}=\lambda_{\mathrm{CD}}\operatorname{CD}(\mathbf{P}_{1},\mathbf{P}^{\prime}_{2\mapsto
1})$, where $\operatorname{CD}$ stands for the Chamfer distance, and
$\lambda_{\mathrm{CD}}$ is a weight coefficient. Finally, our total loss
$\mathcal{L}_{\mathrm{total}}$ is defined as:
$\mathcal{L}_{\mathrm{total}}=\mathcal{L}_{\mathrm{SR}}+\mathcal{L}_{\mathrm{CR}}$.
We omit the $\operatorname{CD}$ loss from self-reconstruction, as we can
directly use the input point cloud to provide supervision using the
$\operatorname{MSE}$ loss. We also omit the $\operatorname{EMD}$ loss from
cross-reconstruction, as $\operatorname{EMD}$ tends to overlook the fidelity
of detailed structures [36], which is crucial in cross-reconstruction of
shapes under intra-class variations.

###  3.3 SO(3)-Invariant Correspondence

In this section, given two randomly rotated point clouds $\mathbf{P}_{1}$ and
$\mathbf{P}_{2}$, we elaborate on how our RIST establishes the 3D dense
correspondence from $\mathbf{P}_{1}$ to $\mathbf{P}_{2}$. As shown in Figure
2, we first encode the SO(3)-equivariant global shape descriptor of
$\mathbf{P}_{1}$, $\mathbf{Z}_{1}\in\mathbb{R}^{C\times 3}$, and the
SO(3)-invariant local shape transform functions of $\mathbf{P}_{2}$,
$\\{f_{\theta_{i}}^{2}\\}_{i=1}^{N}$. Then, we cross-reconstruct
$\mathbf{P}_{1}$ as follows: $\mathbf{P}^{\prime}_{2\mapsto
1}:=\operatorname{Decoder}(\\{f_{\theta_{i}}^{2}(\mathbf{Z_{1}})\\}_{i=1}^{N})$.
Finally, we define the 3D dense correspondence from $\mathbf{P}_{2}$ to
$\mathbf{P}_{1}$ as the nearest point pairs among all possible pairs between
$\mathbf{P}_{1}$ and $\mathbf{P}_{2\mapsto 1}^{\prime}$. Since both encoder
and decoder are SO(3)-equivariant, the cross-reconstructed point cloud
$\mathbf{P}_{2\mapsto 1}^{\prime}$ is aligned to $\mathbf{P}_{1}$. As a
result, our RIST can predict 3D dense correspondences between randomly rotated
point clouds, while previous approaches [3, 21] experience a high rate of
failure.

##  4 Experiments

Training Method Airplane Cap Chair Guitar Laptop Motorcycle Mug Table Average
w/o Rotations FoldingNet [38] 17.8 34.7 22.5 22.1 36.2 12.6 50.0 34.6 28.8
AtlasNetV2 [8] 19.7 31.4 23.6 22.7 36.0 13.1 49.7 35.2 28.9 DPC [17] 22.7 37.1
25.6 31.9 35.0 17.5 51.3 36.8 32.2 CPAE [3] 21.0 38.0 26.0 22.7 34.9 14.7 51.4
35.5 30.5 RIST (ours) 52.1 54.5 58.3 74.1 56.5 48.6 75.0 41.3 57.6 w/
Rotations FoldingNet [38] 22.5 33.2 24.0 31.0 35.9 13.5 49.9 37.0 30.9
AtlasNetV2 [8] 21.1 32.7 25.2 28.8 35.5 14.5 49.9 41.0 31.1 DPC [17] 24.6 38.5
25.6 40.2 34.9 19.3 51.8 37.3 34.0 CPAE [3] 17.0 36.6 24.5 39.4 37.4 15.8 51.9
36.7 32.4 RIST (ours) 51.2 57.0 55.0 73.5 60.6 48.5 72.2 44.4 57.8

Table 1: Average IoU (%) of part label transfer for eight categories in the
ShapeNetPart dataset [39]. Ours consistently outperforms previous approaches
[38, 8, 17, 3] both with and without rotation augmentations during training,
achieving the state-of-the-art IoU. We also provide results of the other
classes in Appendix I.

We present evaluations of RIST on the tasks of 3D part segmentation label
transfer and 3D semantic keypoint transfer, following prior work [21, 3]. For
both tasks, each method is trained with and without rotation augmentations,
and tested on arbitrarily rotated inputs to validate its rotation robustness
in predicting semantic correspondences between the rotated inputs. Note that
previous approaches [38, 8, 21, 3] used aligned point clouds or slightly
rotated point clouds with a subset of SO(3), not the full SO(3), for testing -
which is unrealistic in practice, as described in Appendix H.

Datasets. We use the ShapeNetPart [39] and ScanObjectNN [35] datasets to
evaluate RIST on the task of 3D part segmentation label transfer, which
requires 3D dense semantic correspondence. The ShapeNetPart dataset [39]
consists of 16,880 synthetic 3D data from 16 categories. The ScanObjectNN
dataset [35] contains real-world scanned data, and provides part label
annotations for the chair category. Following the previous work [3], we use
the same pre-processed KeypointNet [40] dataset for the 3D semantic keypoint
transfer task. Since both KeypointNet and ShapeNetPart are based on the
ShapeNet dataset [1], we use the eight overlapping categories between the
ShapeNetPart and KeypointNet [40] datasets to evaluate each method on both
tasks without fine-tuning the method on each dataset. For all tasks, we follow
the experiment setting of the previous work [3].

Baseline methods. Throughout the evaluation section, we mainly compare RIST
against CPAE [3], the state-of-the-art self-supervised method to establish 3D
dense correspondence by exploiting an intermediate UV canonical space. When
open-sourced pre-trained models or codes are applicable, we also compare RIST
with AtlasNetV2 [8], FoldingNet [38] and DPC [17]. AtlasNetV2 proposes to
represent shapes as the deformation and combination of learnable elementary 3D
structures, which can be extended to 3D correspondence establishment.
FoldingNet introduces a folding-based decoder to ‘fold’ a canonical 2D grid
into the 3D object surface, where the canonical 2D grid can be applied to
identify cross-shape correspondences. DPC predicts 3D dense correspondence
between non-rigidly deformed 3D humans, meaning that it can be a powerful
baseline for both 3D part segmentation label transfer and 3D keypoint transfer
tasks as well.

Implementation details. We use VN-DGCNN [7] as our SO(3)-equivariant encoder,
and VN-based multi-layer perception as our SO(3)-equivariant decoder. For a
fair comparison, we set the dimension, $C$, of SO(3)-equivariant global shape
descriptor $\mathbf{Z}\in\mathbb{R}^{C\times 3}$ as 170 ($\approx 512/3$)
since CPAE [3] uses 512-dimensional global shape descriptors. Following the
training setup of CPAE [3], we use $\lambda_{\mathrm{MSE}}$,
$\lambda_{\mathrm{EMD}}$, and $\lambda_{\mathrm{CD}}$ as 1000, 1, and 10,
respectively. RIST is implemented in PyTorch, and is optimized with the Adam
[16] optimizer at a constant learning rate of 1$e^{-3}$.

Figure 3: Qualitative results of part label transfer on the ShapeNetPart
dataset [39]. We visualize the label transfer results via learned
correspondences of each method with the ground truth labels of targets. Note
that the input shapes were arbitrarily rotated at evaluation, differently for
both the source and targets of each row, but have been aligned in the above
figure for better visibility of part label transfer results. RIST shows to
outperform CPAE [3] consistently, showing a high resemblance to ground truth
results.

Method w/o Rotations w/ Rotations FoldingNet [38] 23.2 23.3 AtlasNetV2 [8]
23.6 24.1 DPC [17] 23.9 23.9 CPAE [3] 24.4 23.9 RIST (ours) 39.6 37.9

Table 2: Average IoU (%) of part label transfer for the chair category in the
ScanObjectNN dataset [35]. Ours shows the best IoU both with and without
rotation augmentations during training.

###  4.1 Part Segmentation Label Transfer

We compare RIST with the state of the art in 3D part segmentation label
transfer on ShapeNetPart [39] and ScanObjectNN [35]. For both datasets, we use
the average of instance-wise IoU scores as the evaluation metric.

ShapeNetPart [39]. The quantitative results are presented in Table 1, where
RIST outperforms CPAE [3], DPC [17], AtlasNetV2 [8], and FoldingNet [38] on
all classes by a large margin with and without rotation augmentations during
training. We also provide the qualitative results of the part segmentation
label transfer experiments on ShapeNetPart in Figure 3. Attributing to the
SO(3)-invariant nature of correspondences established by RIST, we are able to
transfer part labels significantly more accurately given randomly rotated
shape pairs.

ScanObjectNN [35]. We evaluate each method trained on synthetic chair data of
ShapeNet [1] on the real chair data of ScanObjectNN [35], which is partial and
more challenging than ShapeNetPart [39], without fine-tuning. As shown in
Table 2, RIST consistently outperforms previous approaches [38, 8, 17, 3] both
with and without rotation augmentations during training. The qualitative
results presented in Figure 4 show that RIST can predict rotation-robust 3D
semantic correspondence between real and partial chair shapes, while CPAE [3]
fails.

Figure 4: Qualitative results of part label transfer on ScanObjectNN [35].
Note that both source and target point clouds were arbitrarily rotated at
evaluation, but have been aligned in the figure for better visibility of part
label transfer results. The results show that RIST reasonably predicts the
semantic correspondences between arbitrarily rotated and partial real point
clouds. Figure 5: Percentage of Correct Keypoints (PCK) for the 12 categories
of the KeypointNet dataset [40] with and without rotation augmentations during
training. RIST consistently outperforms previous approaches on all classes and
thresholds in both settings.

###  4.2 3D Semantic Keypoint Transfer

Following the previous work [3], we compute the distances from the transferred
$M$ keypoints to the ground truth keypoints, and report PCK (Percentage of
Correct Keypoints) of our transferred keypoints, which is computed by:

$\displaystyle\text{PCK}=\frac{1}{M}\sum_{m=1}^{M}\mathbbm{1}[\|\mathbf{k}_{m}-\hat{\mathbf{k}}_{m}\|\leq\tau],$
(2)

where $\tau$, $\mathbf{k}_{m}$, and $\hat{\mathbf{k}}_{m}$ are a distance
threshold, $m$-th ground truth keypoint, and $m$-th transferred keypoint,
respectively. The results on the KeypointNet dataset [40] are illustrated in
Figure 5 for varying distance thresholds $\tau$. It can be seen that RIST
consistently outperforms baseline methods with and without rotation
augmentations during training for all classes, by up to 10$\times$ on certain
classes and thresholds. This substantiates RIST’s superior efficacy at
establishing dense 3D correspondences between varying shapes. However, for
certain classes such as Bathtub or Table, the performance is noticeably low,
outperforming baseline methods only by a tight margin. We speculate this to be
due to the prevalent rotational symmetry of those classes, making it
especially challenging to establish accurate 3D correspondences under
arbitrary rotations. The qualitative results of RIST in comparison to baseline
methods are presented in Figure 6. It can also be seen that RIST can identify
more accurate keypoint correspondences compared to CPAE [3] under arbitrary
rotations, confirming the results of Figure 5.

Figure 6: Keypoint transfer results for airplane and motorcycle categories of
KeypointNet [40]. Each row contains a shape pair, each with ground-truth
keypoints and the keypoint transfer results. Note that the input shapes were
arbitrarily rotated at evaluation, but have been aligned in the above figure
for better visibility of keypoint transfer results. RIST shows to transfer the
keypoints more accurately.  Figure 7: Ablation study on losses (the leftmost)
and the components of model architecture (the others); self-reconstruction
loss (SR), cross-reconstruction loss (CR), equivariant decoder (ED), local
shape transform (LST), and local feature (LF). When excluding the equivariant
decoder, local shape transforms, or local features, the default is to use an
SO(3)-variant decoder, UV coordinates [3], or global features for encoding,
respectively.

###  4.3 Ablation Study and Analyses

We perform an ablation study to justify the design choice of RIST, and
evidence the efficacy of each component.

Self- and cross-reconstruction. We train RIST in a self-supervised manner via
penalizing errors in self- and cross- reconstruction of input point clouds. We
conduct an ablation study on RIST’s reconstruction, providing comparative
results for scenarios with and without its use. The results are illustrated in
the first graph of Figure 7. It can be seen that with and without rotation
augmentations during training, incorporating both self- and cross-
reconstruction yields the best results. Removing self-reconstruction results
in a much dramatic drop in performance; we conjecture this is because without
self-reconstruction, the dynamic local shape transform (Sec. 3.1.2) fails to
capture the required locality of its own point cloud in the first place, being
unsuitable to establish correspondences.

Encoder outputs and SO(3)-equivariance. RIST uses VNNs [7] as the
SO(3)-equivariant layers to facilitate 3D dense correspondence establishment
between arbitrarily rotated point cloud pairs, leveraging local shape
transform to map global shape descriptors to local shape descriptors which
encode the pointwise semantics and local geometry. We perform an ablation
study to demonstrate the efficacy of local shape transforms and
SO(3)-equivariant and -invariant representations in RIST on the motorcycle
class of the KeypointNet dataset [40]. We start our comparison from the
architecture of CPAE [3], given that they also employ an encoder-decoder
architecture to self-supervise their network via shape reconstruction. The
results are presented in the two rightmost graphs of Figure 7, showing the
evaluation results with and without rotation augmentations during training in
order. It can be seen that our design choice of using equivariant encoders and
decoders shows consistent improvements over using an SO(3)-variant
counterpart. Also, using UV coordinates as proposed in CPAE [3] performs worse
compared to our dynamic local shape transform, evidencing the comparatively
better efficacy of transforming each point to their local shape descriptors
via our dynamic SO(3)-invariant shape transform. While using local features as
inputs to the encoder shows varied trends across with and without augmentation
settings, using the point-wise local feature as input is a key component that
facilitates the learning of point-wise dynamic local shape transform that is
essential in establishing the 3D correspondences in RIST.

##  5 Conclusion

We’ve introduced RIST, a novel self-supervised learner for dense 3D semantic
matching across shapes of the same category, even with arbitrary rotations.
Its robustness stems from our innovative use of SO(3)-equivariant and
-invariant representations, enabling dynamic local shape transforms that
preserve rotation equivariance. These transforms map global descriptors to
local ones, facilitating the establishment of dense correspondences. Our
method outperforms existing ones on tasks like part label and keypoint
transfer, enhancing applicability in computer vision and robotics, e.g., AR/VR
and texture mapping. Future research could focus on improving robustness under
common corruption.

Acknowledgement. This work was supported by IITP grants (No.2021-0-02068: AI
Innovation Hub (50%), No. 2022-0-00290: Visual Intelligence for Space-Time
Understanding and Generation (40%), No.2019-0-01906: Artificial Intelligence
Graduate School Program at POSTECH (5%), NO.2021-0-01343, Artificial
Intelligence Graduate School Program at Seoul National University (5%)) funded
by the Korea government (MSIT). Seungwook was supported by the Hyundai-Motor
Chung Mong-koo Foundation.

Supplementary Material

In this supplementary material, we provide a detailed explanation of RIST and
additional experiment results.

##  Appendix A Inference Algorithm of RIST

In this section, we provide a detailed algorithm for the inference process of
RIST for a better understanding of RIST. As shown in Algorithm 1, given a pair
of shapes, we cross-reconstruct one shape and find the nearest neighbor for
each point of the cross-reconstructed shape from the other shape, following
the inference algorithm of the previous work [3].

Algorithm 1 : Inference

Input: A pair of shapes $(\mathbf{P},\mathbf{Q})$,
$\operatorname{Encoder}(\cdot)$, $\operatorname{Decoder}(\cdot)$

Output: Correspondence $\mathcal{C}$ for all $\mathbf{p}\in\mathbf{P}$

1:$\mathbf{Z}_{p}$, $\\{f^{p}_{\theta_{i}}\\}$ $\leftarrow$
$\operatorname{Encoder}(\mathbf{P})$

2:$\mathbf{Z}_{q}$, $\\{f^{q}_{\theta_{i}}\\}$ $\leftarrow$
$\operatorname{Encoder}(\mathbf{Q})$

3:$\mathbf{Q}^{\prime}$ $\leftarrow$
$\operatorname{Decoder}(\\{f^{p}_{\theta_{i}}(\mathbf{Z}_{q})\\})$
$\triangleright$ Cross-recon. from $\mathbf{P}$ to $\mathbf{Q}$

4:$\mathcal{C}\leftarrow\\{\\}$ $\triangleright$ Initialization

5:for {$i\leftarrow 1$ to $|\mathbf{P}|$} do

6: $\mathbf{p}\leftarrow\mathbf{P}_{i}$

7:
$\mathbf{q}\leftarrow\operatorname{NearestNeighborSearch}(\mathbf{Q}^{\prime}_{i},\mathbf{Q})$

8: $\mathcal{C}\leftarrow\mathcal{C}~{}\cup~{}(\mathbf{p},\mathbf{q})$

9:end for

##  Appendix B Ablation Study on Losses

In this section, we conduct an ablation study of the components of our self-
reconstruction loss (Eq. 1) on the motorcycle category of the ShapeNetPart
dataset [39]. As shown in Table A1, our choice (b) shows the best performance
among models trained with the seven loss variants from (a) to (g).

$\operatorname{MSE}$ $\operatorname{EMD}$ $\operatorname{CD}$ IoU ($\%$) (a) ✓
✓ ✓ 46.0 (b) ✓ ✓ 48.5 (c) ✓ ✓ 47.0 (d) ✓ ✓ 46.4 (e) ✓ 45.8 (f) ✓ 46.4 (g) ✓
44.9

Table A1:  Ablation study on the self-reconstruction loss. The model trained
with ((b): $\operatorname{MSE}$ and $\operatorname{EMD}$) shows the best
performance, justifying our choice for the self-reconstruction loss.

##  Appendix C Multi-class Training

In this section, we provide the experiment results of previous approaches [38,
8, 17, 3] and ours trained with multiple classes (airplane and chair) in the
ShapeNetPart dataset [39]. As shown in Table A2, RIST outperforms previous
approaches [38, 8, 17, 3] by a large margin.

Method Airplane Chair Average FoldingNet [38] 20.9 23.9 22.4 AtlasNetV2 [8]
21.1 24.6 22.9 DPC [17] 22.7 25.6 24.2 CPAE [3] 16.6 14.8 15.7 RIST (ours)
34.4 34.7 34.6

Table A2:  Part label transfer with multi-classes training.

##  Appendix D Generalization to Unseen Classes

In this section, we evaluate the generalization ability of previous approaches
[38, 8, 17, 3] and ours to unseen classes. Specifically, we train each method
on the airplane category in the ShapeNetPart dataset [39] and test it on the
chair category. As shown in Table A3, RIST shows a competitive result with an
unseen category, outperforming previous approaches [38, 8, 3] except DPC [17].

FoldingNet [38] AtlasNetV2 [8] DPC [17] CPAE [3] RIST 24.8 23.0 28.2 15.6 27.3

Table A3: Generalization results for the part label transfer.

##  Appendix E Inference on Aligned Shapes

In this section, we provide the results of previous approaches [38, 8, 17, 3]
and ours evaluated on the ShapeNetPart [39], ScanObjectNN [35], and
KeypointNet [40] datasets, but with aligned test shapes, as shown in Table A4,
Table A5, and Figure A1, respectively. Note that under the aligned setting,
the input shape pairs are perfectly aligned both at train and test time -
which is an unrealistic setting in practice. For each method, we also include
the results with rotated shapes to show the performance difference between
aligned and rotated settings. It can be seen that while the drop in
performance for previous approaches [38, 8, 17, 3] from the aligned to the
rotated setting is drastic, the difference is negligible in RIST,
demonstrating the robustness of our SO(3) correspondence establishment scheme
against arbitrary rotations. While RIST is not always competitive on all
settings, it is impractical to expect perfectly aligned shapes in real-world
situations; on the realistic setting of SO(3) evaluation, RIST consistently
shows the best results.

Inference Method Airplane Cap Chair Guitar Laptop Motorcycle Mug Table Average
Aligned FoldingNet [38] 56.5 54.9 63.1 73.1 81.9 21.5 75.5 54.0 60.1
AtlasNetV2 [8] 51.7 44.7 63.3 65.0 84.0 41.5 84.2 59.3 61.7 DPC [17] 60.5 65.8
65.3 74.4 88.0 53.3 85.4 66.4 69.9 CPAE [3] 61.3 61.6 72.6 78.9 89.9 55.4 86.5
72.5 72.3 RIST (ours) 52.1 54.4 58.3 74.1 56.7 48.7 75.6 41.3 57.7 Rotated
FoldingNet [38] 17.8 34.7 22.5 22.1 36.2 12.6 50.0 34.6 28.8 ($\downarrow$
31.3) AtlasNetV2 [8] 19.7 31.4 23.6 22.7 36.0 13.1 49.7 35.2 28.9
($\downarrow$ 32.8) DPC [17] 22.7 37.1 25.6 31.9 35.0 17.5 51.3 36.8 32.2
($\downarrow$ 37.7) CPAE [3] 21.0 38.0 26.0 22.7 34.9 14.7 51.4 35.5 30.5
($\downarrow$ 41.8) RIST (ours) 52.1 54.5 58.3 74.1 56.5 48.6 75.0 41.3 57.6
($\downarrow$ 0.1)

Table A4: Average IoU (%) of part label transfer for eight categories in the
ShapeNetPart dataset [39] on aligned and rotated shapes. Note that each method
is trained without rotation augmentation. RIST shows the most negligible
performance drop ($0.1\%$ in IoU) with rotated shapes, while previous
approaches [38, 8, 17, 3] show large performance drops (at least $30\%$ in
IoU).

Inference FoldingNet [38] AtlasNetV2 [8] DPC [17] CPAE [3] RIST (ours) Aligned
33.6 34.8 36.3 33.8 39.6 Rotated 23.2 ($\downarrow$ 10.4) 23.6 ($\downarrow$
11.2) 23.9 ($\downarrow$ 12.4) 24.4 ($\downarrow$ 9.4) 39.6 ($-$)

Table A5: Average IoU (%) of part label transfer for the chair category in the
ScanObjectNN dataset [35] on aligned and rotated shapes. Note that each method
is trained without rotation augmentation. RIST does not show any performance
drop with rotated shapes, while previous approaches [38, 8, 17, 3] show large
performance drops (at least $9\%$ in IoU).  Figure A1: Percentage of Correct
Keypoints (PCK) for the 12 categories of the KeypointNet dataset [40] on
aligned and rotated shapes. Note that each method is trained without rotation
augmentation. While previous approaches [38, 8, 17, 3] are vulnerable to
rotations, RIST shows a negligible performance drop with rotated shapes.

We further evaluate CPAE [3] when integrated together with two 3D shape
alignment methods. We present results when using PCA or VN-SPD [12] as the
alignment method in Figure A2. PCA yields only marginal performance
improvements, likely due to sign and order ambiguities. In contrast, using the
SOTA learning-based alignment method, VN-SPD, produces competitive results
with RIST, but its inconsistency in aligning shapes to a canonical orientation
limits performance as also reported in Katzir _et al_. [12], sacrificing the
efficiency.

Figure A2: Comparision of RIST with the combination of CPAE [3] and aligning
methods; PCA and VN-SPD [12]. Note that RIST shows competitive results to the
combined method of CPAE [3] and VN-SPD [12], which is a SE(3)-equivariant
orientation predictor and requires additional parameters (17.4M).

##  Appendix F Comparision with 3D Keypoint Estimators

We compare RIST with SC3K [45], a recent self-supervised method for coherent
3D keypoint estimation. However, since this 3D keypoint estimation method
cannot be evaluated on 3D keypoint matching, we instead used the Dual
Alignment Score (DAS)555A metric for 3D keypoint estimation task SC3K [45]
used. of RIST for an empirical comparison with SC3K. Additionally, to
facilitate comparison on the part label transfer task, we extended the number
of keypoints estimated by SC3K to match the total number of points in a point
cloud _e.g_., 2048.

Method Airplane Car Chair SC3K [45] 81.3 73.8 86.2 RIST (ours) 82.4 76.9 81.8

Table A6: Dual Alignment Score of SC3K [45] and RIST. During the evaluation,
we use 10 keypoints for both SC3K and RIST.

Method Airplane Car Chair SC3K [45] 22.3 23.0 24.7 RIST (ours) 51.2 48.0 55.0

Table A7: Part label transfer results of SC3K [45] and RIST. Note that we
train SC3K [45] with 2048 keypoints.

As shown in Tables A6 and A7, RIST exhibits competitive DAS results compared
to SC3K, although it is not trained for 3D keypoint estimation, and
significantly outperforms SC3K on the part label transfer task.

##  Appendix G Evaluation with Pseudo-Ground Truth

Figure A3: 3D semantic correspondence results of RIST using DIT [44] as
pseudo-ground truth on ShapeNet [1]. We use the official checkpoints of DIT to
generate pseudo-ground truth of 3D semantic correspondence for rotated 3D
shapes.

We utilize DIT [44] to establish pseudo-ground truth on ShapeNet [1] for a
direct evaluation of RIST’s dense semantic correspondence capabilities for
airplane, car, and chair classes, using official checkpoints. As shown in
Figure A3, the results show a similar trend of part label transfer results,
showing that RIST outperforms previous approaches.

##  Appendix H Implementation Details of SO(3)

In this section, we explain the implementation details of uniformly sampling
random rotations and highlight the differences from the previous approach [3].
Cheng  _et al_. [3] samples rotation angles from $\mathcal{N}(0,0.2^{2})$ and
then clamps them to $[-\frac{1}{2}\pi,\frac{1}{2}\pi]$, which limits the range
of rotation. In our work, we follow Shoemake _et al_. [30] to uniformly sample
to cover full SO(3), which is more challenging.

##  Appendix I Part Label Transfer Results on More Classes

We initially presented evaluations only for the classes of ShapeNetPart [39]
that are shared with those of KeypointNet [40]. In Table A8, we further
present part label transfer results on the remaining classes of ShapeNetPart
[39].

Method Bag Car Ear. Knife Lamp Pistol Rocket Skate. CPAE [21] 43.2 20.3 33.4
36.3 31.1 26.8 27.7 52.0 RIST (ours) 50.8 48.0 36.3 57.9 35.9 54.7 34.4 54.4

Table A8: Part label transfer results on ShapeNetPart [39]. RIST consistently
outperforms the previous state-of-the-art method on the remaining classes of
ShapeNetPart [39].

##  Appendix J Matching with Local Shape Transform

In this section, We experiment with a variant of RIST (RIST${}_{\text{LST}}$),
which matches 3D shapes using similarity between SO(3)-invariant Local Shape
Transform (LST). As shown in Table A9, our current scheme of comparing point
positions yields noticeably better results on ShapeNetPart [39], meaning that
our trained decoder is better adept at handling topology-varying structures

Method Airplane Chair CPAE 17.0 24.5 RIST${}_{\text{LST}}$ 48.6 50.3 RIST 51.2
55.0

Table A9: Part label transfer results of RIST${}_{\text{LST}}$. Note that we
use randomly rotated 3D shapes for the evaluation.

##  Appendix K Alignment of Qualitative Results

In this section, we provide both unaligned and aligned qualitative results for
a better understanding of how our qualitative results were drawn. As shown in
Figure A4, both source and target shapes are randomly rotated at the inference
time. Note that we use the part segmentation labels transferred by RIST for
the target shape in Figure A4.

Figure A4: Visualization for aligning qualitative results of part label
transfer on the ShapeNetPart dataset [39].

##  Appendix L Qualitative Results of Part Label Transfer

We provide additional qualitative results on the ShapeNetPart dataset [39]
that were not included in our manuscript due to space constraints, as shown in
Figures A5 and A6.

Figure A5: Qualitative results of part label transfer on the motorcycle class
in the ShapeNet part dataset [39]. Note that the input shapes were arbitrarily
rotated, differently for each target column, but have been aligned for better
visibility of part label transfer results. RIST shows to outperform CPAE [3]
consistently, showing a high resemblance to ground truth results.

Figure A6: Qualitative results of part label transfer on the airplane class in
the ShapeNet part dataset [39]. Note that the input shapes were arbitrarily
rotated, differently for each target column, but have been aligned for better
visibility of part label transfer results. RIST shows to outperform CPAE [3]
consistently, showing a high resemblance to ground truth results.

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coherence_prl_suppl

# Coherent radiation of electrons in an intense laser pulse

E. G. Gelfer<EMAIL_ADDRESS>ELI Beamlines facility, The Extreme Light
Infrastructure ERIC, Dolni Brezany 252 41, Czech Republic  A. M. Fedotov
<EMAIL_ADDRESS>National Research Nuclear University MEPhI, Moscow, 115409,
Russia  O. Klimo  ELI Beamlines facility, The Extreme Light Infrastructure
ERIC, Dolni Brezany 252 41, Czech Republic  FNSPE, Czech Technical University
in Prague, Prague, Czech Republic  S. Weber  ELI Beamlines facility, The
Extreme Light Infrastructure ERIC, Dolni Brezany 252 41, Czech Republic

###### Abstract

We study the conditions for coherent radiation of a group of electrons driven
by a strong pulsed electromagnetic plane wave. The required conditions are
first analyzed for a couple of neighboring electrons having the same
velocities, then generalized to their macroscopic number and the range of
parameters is identified to optimize the effect. We show that for a solid
density plasma coherence affects the frequencies up to hundreds of keV, and
the low-frequency part of the spectrum can be enhanced by many orders of
magnitude. Our analytical findings are tested with 3D particle-in-cell
simulations of a dense electron bunch passing through a laser pulse, clearly
demonstrating how the coherence can essentially modify the observed radiation
spectrum.

An electron quivering in a laser field acquires relativistic energy if the
dimensionless field strength $a_{0}$ obeys

$a_{0}=\frac{eE_{0}}{m\omega_{L}c}\gtrsim 1,$ (1)

where $E_{0}$ is the electric field amplitude, $\omega_{L}$ is the field
frequency, $-e$ and $m$ are the electron charge and mass, $c$ is the speed of
light. Radiation of the electron in such a regime is called the nonlinear
inverse Thomson/Compton scattering (NTS/NCS) depending on whether or not the
recoil of the electron due to emissions of individual photons can be neglected
[1]. For brevity, we omit the word “inverse” in the following.

Laser driven NTS and NCS serve as a source of bright X– and gamma rays [2, 3,
4, 5, 6, 7, 8, 9, 10, 11] with important applications in nuclear spectroscopy
[12, 13, 14, 15], medicine [16, 17], X-ray radiography [18, 2], positron
production via quantum electrodynamical (QED) cascades [19, 20, 21, 22, 23,
11], etc., see the review [24].

The theory of NTS by a single electron in a monochromatic plane wave is well
established [25, 26, 27, 28]. However, when a laser irradiates a target
constituted of many particles, the emitted radiation can be coherently
enhanced. While the energy of incoherent radiation scales with the number of
particles $N$ as $\mathcal{E}^{(N)}_{incoh}\sim N$, the coherent scaling would
be $\mathcal{E}^{(N)}_{coh}\sim N^{2}$. This possibility was explored and
indeed confirmed for two nonrelativistic orbiting particles [29], synchrotron
radiation by electrons in circular accelerators [30, 31, 32, 33, 34], self-
amplified spontaneous emission by electrons in undulators [35, 36, 37, 38],
production of light scalars due to the beyond-the-standard-model interactions
[39] and the linear Thomson scattering [40]. Here we investigate, how the
coherence manifests itself in the radiation of an electron bunch interacting
with an intense laser pulse.

In order to radiate coherently, the particles should move in an external field
along nearby trajectories with close initial coordinates and momenta. The
distance between the particles is prescribed by their density, in the current
paper we focus solely on its impact on coherence assuming equal initial
momenta. The effect of momentum spread (due to e.g. a nonzero temperature or
uncertainty principle) will be neglected but was discussed in Ref. [41].

The coherency enhances radiation at lower frequencies stronger than at higher
ones, hence modifies the radiation spectrum. For NTS this was pointed out in
[42]. Many-particle radiation spectra in accelerators [31] and laser fields
[42] at low frequencies $\omega$ often scale as $\omega^{-8/3}$ in contrast to
$\omega^{1/3}$ as for a single particle 111The considered frequencies are
assumed still higher than the frequency of variation of the external field..
The same scaling [44, 45] was found for the relativistic oscillating mirror
mechanism of high harmonics generation by reflecting laser pulses from solid
targets [46, 47, 44, 48]. However, it was argued [49, 45, 50, 51, 52, 53] that
the resulting spectra of high harmonics can differ if the competing mechanisms
of high harmonics generation become important.

The major tool for theoretical studies of laser–plasma interactions is a
particle-in-cell (PIC) numerical simulation. In modern PIC codes the quantum
and incoherent part of radiation by charged particles is implemented as
emission of individual photons via Monte-Carlo event generators [42]. Between
the emission events particles are moving along the classical trajectories thus
forming currents, which produce classical radiation. This radiation, however,
is resolvable on the grid only for wavelengths significantly larger than the
size of the grid cells.

As the coherent enhancement affects mostly the low frequency part of the
spectrum, it was ignored in the context of NTS and NCS, where typically the
focus is on higher frequencies around the synchrotron peak. These frequency
ranges are supposed to be well separated from one another [42], as needed to
prevent a double counting of radiation co-implemented in PIC simulations by
the two different approaches. Here we revise this assumption and identify the
parameters ensuring the coherent enhancement of the classical radiation of
electrons interacting with an electromagnetic wave.

For simplicity we consider circular polarization. Let us start with a
monochromatic plane wave with the 4-potential
$A^{\mu}=mca_{0}/e\\{0,0,-\sin\phi,\cos\phi\\}$ and wave vector
$k_{L}^{\mu}=\omega_{L}/c\\{1,1,0,0\\}$. Here $\phi=\omega_{L}(t-x/c)$ is the
phase and $\omega_{L}$ is the frequency of the wave. According to [27, 26],
energy spectrum of the emitted radiation reads

$d\mathcal{E}_{k}=-j_{\mu}^{*}(k)j^{\mu}(k)\frac{d\mathbf{k}}{4\pi^{2}},$ (2)

where $k^{\mu}=\\{\omega/c,\mathbf{k}\\}$ is the radiated wave 4-vector
($\mathbf{k}=\omega/c\,\mathbf{n}$), $\omega$ and
$\mathbf{n}=\\{\cos\theta,\sin\theta\cos\varphi,\sin\theta\sin\varphi\\}$ are
the frequency and the direction of the emitted radiation and $j^{\mu}(k)$ is
the Fourier transform of the 4-current.

The latter can be represented in the form [26]

$j^{\mu}(k)=\frac{ec}{\omega_{L}p_{-}}e^{-i\Phi}\int\limits_{-\infty}^{\infty}d\phi\,\pi^{\mu}(\phi)e^{-\frac{ick_{\mu}}{\omega_{L}p_{-}}\int_{0}^{\phi}\pi^{\mu}(\phi^{\prime})d\phi^{\prime}},$
(3)

where the kinetic momentum $\pi^{\mu}(\phi)$ reads

$\pi^{\mu}(\phi)=p^{\mu}-\frac{e}{c}A^{\mu}(\phi)+k_{L}^{\mu}\left(\frac{ep_{\nu}A^{\nu}}{\omega_{L}p_{-}}-\frac{e^{2}A_{\nu}A^{\nu}}{2c\omega_{L}p_{-}}\right),$
(4)

$p^{\mu}$ is the constant of motion corresponding to the electron momentum
without an external field and $p_{-}=p^{0}-p_{x}$.

The phase in Eq. (3) does not contribute to the radiation spectrum (2) of a
single particle but controls the coherent emission of several particles (see
below). It reads

$\Phi=\phi_{0}+\tilde{\Phi},\quad\tilde{\Phi}=-\frac{ck_{\mu}}{\omega_{L}p_{-}}\int\limits_{0}^{\phi_{0}}\pi^{\mu}(\phi^{\prime})d\phi^{\prime},$
(5)

where $x_{0}^{\mu}$ and $\phi_{0}=k_{L\mu}x_{0}^{\mu}$ are the initial
coordinates of the electron and the corresponding phase of the wave.

For the sake of simplicity, we assume that the initial momentum of the
particle is aligned along with the direction of propagation of the wave,
$\mathbf{p}\parallel\mathbf{k}_{L}$. A single particle NTS spectrum in a plane
wave is discussed, e.g., in Refs. [25, 26, 27, 28, 54]. For convenience, we
also provide a compact derivation in Supplementary Material [55]. Here it is
enough to emphasize its following features in the setup under consideration:
(i) It respects axial symmetry and is emitted in a narrow ring around the
polar angle

$\theta_{0}=2\arctan\xi,\quad\xi=\frac{p_{-}}{mc\sqrt{1+a_{0}^{2}}}.$ (6)

(ii) Its power spectrum $P_{s}^{(1)}$ consists of discrete harmonics at
$\omega_{s}=s\omega_{L}(1+\xi^{2})/2$, $s=1,2,\ldots$ having the maximum at

$\omega_{m}\sim\omega_{L}(1+\xi^{2})a_{0}^{3}$ (7)

(corresponding to $s\sim a_{0}^{3}$) and exponential fade out for
$\omega_{s}\gg\omega_{m}$. (iii) For $a_{0}\gg 1$ the spectrum scales at
$\omega_{s}\ll\omega_{m}$ as $\propto\omega_{s}^{1/3}$,

$P^{(1)}1(\omega_{s})\approx\frac{\alpha
e^{2}\omega_{L}^{2}}{c}(1+\xi^{2})^{5/3}\left(\frac{\omega_{s}}{\omega_{L}}\right)^{1/3},$
(8)

where the constant
$\alpha=\frac{3^{2/3}}{4\pi^{2}}[\Gamma^{2}(1/3)/6+\Gamma^{2}(2/3)/2^{1/3}]\approx
0.14$.

Next consider two electrons quivering in a circularly polarized plane wave.
The total 4-current $J^{\mu}(k)=j_{1}^{\mu}(k)+j_{2}^{\mu}(k)$, where
$j_{1,2}^{\mu}(k)$ for individual electrons are defined as in (3). Let us
assume that their initial momenta are equal
$\mathbf{p}_{1}=\mathbf{p}_{2}\equiv\mathbf{p}$ and represent their initial
displacement as $\Delta x_{0}^{\mu}=\\{0,\mathbf{R}\\}$. Then
$|j_{1}(k)|^{2}=|j_{2}(k)|^{2}=|j^{\mu}(k)|^{2}$ and the squared total current
consists of an incoherent and interference terms

$J_{\mu}^{*}(k)J^{\mu}(k)=2|j^{\mu}(k)|^{2}\left(1+\cos{\Delta\Phi}\right).$
(9)

According to (5) the phase shift consists of two terms
$\Delta\Phi=\Delta\phi_{0}+\Delta\tilde{\Phi}$. Note that $\Delta\tilde{\Phi}$
(coming from the shift in $\tilde{\Phi}$) was not accounted for in Ref. [41].
This term diminishes the coherence when the displacement $\mathbf{R}$ has a
longitudinal (parallel to $\mathbf{k}_{L}$) component.

Figure 1: Emission spectrum of two electrons in a circularly polarized plane
wave. Blue circles (green squares) represent the two-particle emission
spectrum with account for the interference term with electrons initially
separated by displacement $\mathbf{R}\bot\mathbf{k}_{0}$ for $R=R_{\perp}$
($R=10R_{\perp}$) with $\omega=\omega_{m}$, respectively [see Eq. (10)]. For
comparison red solid/dashed curve represents the doubled/quadrupled single
electron spectrum [see (LABEL:Ps1f) in [55]]. $a_{0}=5$, $\xi=10$.

Let us look closer at the interference term. If the displacement of the
particles is perpendicular to the direction of wave propagation
$\mathbf{R}\bot\mathbf{k}_{L}$ then $\Delta\phi_{0}=\omega
R\sin\theta\cos\varphi/c$ and $\Delta\tilde{\Phi}=0$, so that integration over
the azymuthal angle $\varphi$ with account for axial symmetry of the single
particle radiation results in

$\int\limits_{0}^{2\pi}d\varphi\cos{\Delta\Phi}=2\pi J_{0}(\omega
R\sin\theta/c).$

Due to (i) above, only angles close to (6) contribute to the subsequent
integral over the polar angle $\theta$. Therefore after integration of Eq. (9)
over the solid angle the interference term cannot be ignored against the
incoherent one if $\omega R\sin\theta_{0}/c\lesssim 1$. Using (6) this can be
written as

$R\lesssim R_{\bot}\equiv\frac{c}{\omega}\frac{1+\xi^{2}}{2\xi}.$ (10)

We illustrate in Fig. 1 the relevance of the criterion Eq. (10) for
discriminating the coherence by comparing the two-electron radiation spectra
for $R=R_{\perp}$ and $R=10R_{\perp}$ with the doubled and quadrupled single
electron spectrum. Since $R_{\perp}$ in Eq. (10) depends on the frequency, we
evaluate it at $\omega=\omega_{m}$. Clearly, if Eq. (10) is satisfied then the
interference indeed remains constructive, meaning radiation is coherent for
all frequencies up to $\omega=\omega_{m}$. It is also seen that once Eq. (10)
is violated the coherence is eventually lost.

The general case of an arbitrary placement of the particles in the wave is
treated the same way (see [55] for details). In particular, if
$\mathbf{R}\parallel\mathbf{k}_{L}$ then instead of Eq. (10) we obtain

$R\lesssim R_{\parallel}\equiv\frac{c}{\omega}\frac{\xi^{2}}{2(1+\xi^{2})}.$
(11)

The dependence of the coherency criterion on the direction of the displacement
vector $\mathbf{R}$ can be (almost) understood in terms of the Doppler effect
and Lorentz contraction. Indeed, consider the reference frame $K^{\prime}$
where both particles are on average at rest and (assuming $a_{0}\gg 1$)
radiate almost in a plane perpendicular to $\mathbf{k}_{L}$. The velocity of
$K^{\prime}$ with respect to the lab frame $K$ is $V=(1-\xi^{2})/(1+\xi^{2})$.
In $K^{\prime}$ the particles radiate coherently if the distance $R^{\prime}$
between them is shorter than the radiation wavelength $\sim c/\omega^{\prime}$
[29]. Since the wavevector $\mathbf{k}^{\prime}$ of radiation in $K^{\prime}$
is almost perpendicular to $\mathbf{k}_{L}$, due to the Doppler effect [27] we
have $\omega\approx\gamma\omega^{\prime}$, where
$\gamma=1/\sqrt{1-V^{2}}=(1+\xi^{2})/2\xi$.

In the transverse case $\mathbf{R}\bot\mathbf{k}_{L}$ we have $R=R^{\prime}$,
so that $R^{\prime}<c/\omega^{\prime}$ entails Eq. (10). But in the parallel
case $\mathbf{R}\parallel\mathbf{k}_{L}$ we have rather $R^{\prime}=\gamma R$,
hence arrive at $R_{\parallel}\sim c/\omega$, which coincides with Eq. (11)
only for $\xi\gtrsim 1$. The latter apparent discrepancy is explained by that
the argument based solely on the Doppler effect and Lorentz contraction does
not involve contribution $\tilde{\Phi}$ in Eq. (5), which vanishes identically
in the transverse case but remains in general, though for $\xi\gg 1$ we have
$\Delta\tilde{\Phi}\ll\Delta\phi_{0}$.

In real applications the electrons are exposed to a laser pulse rather than to
an infinite plane wave. This causes broadening of the harmonics making the
spectrum continuous [54]. Therefore it is natural to pass from the power
spectrum (8) to the spectral density $d\mathcal{E}/d\omega$. They are related
by [27]

$\left.\frac{d\mathcal{E}}{d\omega}\right|_{\omega=\omega_{s}}=\frac{4T}{\omega_{L}(1+\xi^{2})^{2}}P(\omega_{s}),$
(12)

where $T$ is the duration of the laser pulse.

Finally let us turn to the discussion of the emission spectrum of a
macroscopic electron bunch. A reasonable order-of-magitude estimate can be
obtained by multiplying the one particle spectrum (8) [see also LABEL:Ps1f in
[55]] by the factor $NN_{coh}$, where $N=nSl$ is the total number of electrons
($n$ is the electron density, $S$ and $l$ are cross section and thickness of
the bunch), $N_{coh}$ is the number of electrons inside a coherence volume
$\simeq R_{\parallel}R_{\bot}^{2}$ surrounding each electron. According to
(10) and (11) such a volume can be estimated as

$N_{coh}\sim\max\left\\{\eta\frac{n}{n_{c}}\left(\frac{\omega}{\omega_{L}}\right)^{-3}(1+\xi^{2}),1\right\\},$
(13)

where $\eta=c/32\pi\omega_{L}r_{e}$, $r_{e}=e^{2}/mc^{2}\approx 2.8\cdot
10^{-13}$ cm is the classical electron radius, and $n_{c}=m\omega_{L}^{2}/4\pi
e^{2}$ is the plasma critical density. For $1\mu$m laser wavelength we have
$\eta\approx 5.7\cdot 10^{5}$ and $n_{c}\approx 1.1\cdot 10^{21}$ cm-3.

According to Eq. (13), the coherent amplification ($N_{coh}>1$) affects the
frequencies in the range

$\omega<\omega_{coh}\sim\eta^{1/3}\omega_{L}\left(\frac{n}{n_{c}}\right)^{1/3}(1+\xi^{2})^{1/3}.$
(14)

As a result, the coherently amplified low-frequency part of the spectrum
($\omega\ll\omega_{m}$) behaves as

$\left.\frac{d(\mathcal{E}/mc^{2})}{d(\omega/\omega_{L})}\right|_{coh}\sim\frac{\alpha}{\pi}\eta\frac{SlT\omega_{L}^{4}}{c^{3}}(1+\xi^{2})^{\frac{2}{3}}\left(\frac{n}{n_{c}}\right)^{2}\left(\frac{\omega}{\omega_{L}}\right)^{-\frac{8}{3}},$
(15)

for $\omega<\omega_{coh}$, which should be contrasted to the usual incoherent
spectrum

$\left.\frac{d(\mathcal{E}/mc^{2})}{d(\omega/\omega_{L})}\right|_{incoh}\sim\frac{\alpha}{\pi}\frac{SlT\omega_{L}^{4}}{c^{3}}(1+\xi^{2})^{-\frac{1}{3}}\frac{n}{n_{c}}\left(\frac{\omega}{\omega_{L}}\right)^{\frac{1}{3}},$
(16)

which remains for $\omega>\omega_{coh}$. As usual, the coherent radiation
energy scales as $\propto n^{2}$ rather than $\propto n$. Besides, Eq.(15)
differs from (16) also by factors $1+\xi^{2}\gtrsim 1$ and $\eta\gg 1$. The
resulting shapes of the spectrum are also fundamentally different (scaling as
$\propto\omega^{-8/3}$ in Eq. (15) $\omega\ll\omega_{m}$ instead of
$\propto\omega^{1/3}$ in Eq. (16)).

Figure 2: Radiation spectra for $n=0.1n_{c}$ (red curves) and $n=0.01n_{c}$
(blue curves): solid lines – results of 3D numerical simulations using PIC
code SMILEI, dashed lines – analytical estimate Eq. (15), dotted lines –
analytical estimate Eq. (16); inset – estimates for $n=0.1n_{c}$ with and
without account for coherent amplification in a wider frequency range
including the peak at $\omega=\omega_{m}$.
$a_{0}=5,p_{-}=10mc,l=0.5\lambda_{L},S=0.25\lambda_{L}^{2},\lambda_{L}=1\mu$m.

To check these predictions we performed three-dimensional (3D) particle-in-
cell (PIC) simulations of an electron bunch colliding head on with a laser
pulse. Though modern PIC-codes are capable of describing radiation emitted in
a quantum regime [42], its implementation so far totally ignores all the
collective effects, including possible coherent amplification. Being also
interested in a low-frequency range where the classical description works
well, we therefore analyzed the output of the built-in Maxwell solver.

The results of simulations with the PIC code SMILEI [56] are presented in Fig.
2, the parameters of simulations are given in the caption. They correspond to
$\xi\approx 2>1$, for which radiation is mostly emitted along the $x$ axis.
From simulation data we extract the field values at the distant plane
$x=18\lambda_{L}$ (where $\lambda_{L}$ is the laser wavelength), perform their
fast Fourier transform and combine them into the Fourier transformed Poynting
vector. Further details of our numerical approach are presented in [55]. The
approximate match of the solid (simulation) and dashed (estimated
analytically) curves for each of the two densities differing by an order of
magnitude is a good indication of the relevance of the scalings (15) in both
density ($\propto n^{2}$) and frequency ($\propto\omega^{-8/3}$).

This approach, however, has severe limitations, which currently make it
challenging to simulate the effect of coherent amplification directly for
arbitrary parameters. First, the Maxwell solver resolves only frequencies
obeying $\omega\ll d\omega_{L}$, where $d$ is the number of grid cells per
wavelength. Second, in a PIC code the particles are combined into
macroparticles densely populating grid cells and radiating coherently within
each cell (this, in particular, means that so computed radiation can scale as
$\propto\omega^{-8/3}$ even for $\omega>\omega_{coh}$).

(a)0.5cm0.8cm (b)0.9cm0.8cm

(a)

(b)

Figure 3: Radiation enhancement factor $N_{coh}(\omega;n)$ (13) for a slow
electron bunch [$\xi=1$] (a) and an ultrarelativistic electron bunch
[$\xi=8\cdot 10^{3}$] (b); dashed line separates the regions of coherent
[$N_{coh}>1$] and incoherent [$N_{coh}=1$] radiation.

Therefore, in looking for the parameters for which the coherent amplification
can be revealed in laser–plasma interactions, we rely mostly on extrapolation
of our estimates. The coherency enhances radiation by factor
$N_{coh}(\omega,n;\xi)$ estimated in Eq. (13). Its dependence on the radiated
frequency $\omega$ and density $n$ (with the upper limit $10^{2}$
corresponding to the solid state density) is illustrated in Fig. 3 for two
values of $\xi$. The first value $\xi=1$ [panel (a)] corresponds to the case
of the initially slow electrons. As shown in the figure, in this case the
coherency affects the frequencies up to keV range, and in the low frequency
limit the enhancement factor $N_{coh}$ reaches the value above $10^{8}$. The
second value $\xi\approx 8\cdot 10^{3}$ [panel (b)] corresponds to
ultrarelativistic $10$ GeV electrons (this energy level has been recently
achieved with laser wakefield acceleration technique [57, 58]) colliding with
a laser pulse of intensity $I=10^{19}$ W/cm2 [$a_{0}\approx 5$]. Here the
coherency extends to the frequencies above 100 keV with $N_{coh}\sim 10^{13}$
for $n\sim n_{c}$ and $N_{coh}\sim 10^{15}$ for $n\sim 100n_{c}$ in the low
frequency limit. Note, however, that while qualitatively our estimate should
be valid, an accurate quantitative description of the radiation of such
energetic electrons should be done using the quantum approach, which is out of
the scope of this paper.

The coherency can affect even the whole spectrum up to its maximum (7) if
$\omega_{coh}\gtrsim\omega_{m}$ or explicitly

$a_{0}\lesssim\eta^{1/9}\left(\frac{n}{n_{c}}\right)^{1/9}(1+\xi^{2})^{-2/9}.$
(17)

In particular, for slow particles ($\xi\lesssim 1$) and $n\sim n_{c}$ we
obtain $a_{0}\lesssim 5$, which corresponds to the intensity $I\lesssim
10^{19}$ W/cm2.

Even if the high frequency part of the spectrum is unaffected by the
coherency, the total energy of the coherently enhanced radiation
$\mathcal{E}_{coh}\sim\int_{\omega_{L}}^{\omega_{coh}}(d\mathcal{E}/d\omega)d\omega$
can be still higher than of incoherent one
$\mathcal{E}_{incoh}\sim\int_{\omega_{coh}}^{\omega_{m}}(d\mathcal{E}/d\omega)d\omega$
because of the huge values of $N_{coh}$, see Fig. 3. Indeed,

$\frac{\mathcal{E}_{coh}}{\mathcal{E}_{incoh}}\sim\eta\frac{n}{n_{c}}\frac{(1+\xi^{2})^{-2/3}}{a_{0}^{4}}.$
(18)

For example, assuming the electrons are initially slow [$\xi\lesssim 1$] and
$a_{0}\sim 10$ ($I\sim 10^{20}$ W/cm2), the energy emitted into the coherently
enhanced spectral range surpasses that into the incoherent range already at
$n\sim 10^{-2}n_{c}$ [$n\sim 10^{19}$ cm-3].

In the present paper for the first time we investigated in details the effect
of coherency on NTS. In particular, we derived the bounds on electron
separation for the coherency of their classical radiation in an intense laser
field. We demonstrated that the coherence can significantly modify the
radiation spectrum of a macroscopic number of electrons interacting with an
intense laser pulse. The low frequency part is enhanced, and the maximal
frequency of the coherently enhanced radiation depends on the particle
density. These findings are essential for correct description of experiments
with dense electron bunches at modern laser facilities. In particular,
according to our preliminary simulations, the enhancement of radiated energy
can also enhance the radiation friction force acting on electrons in a dense
bunch as compared to that in vacuum or in rare plasma [27, 59, 1, 60, 61, 62,
63, 64, 65, 66, 67]. This implies that radiation friction in dense electron
bunches might become essential at considerably lower laser intensities than
commonly accepted.

The authors are grateful to Myron Sukhanov for the help with numerical
simulations and to Mickael Grech, Daniel Seipt and Nina Elkina for valuable
discussions. E.G.G. and S.W. were supported by the project ADONIS (Advanced
research using high intensity laser produced photons and particles)
CZ.02.1.01/0.0/0.0/16_019/0000789 from European Regional Development Fund.
S.W. and O.K. were also supported by High Field Initiative
(HiFI)(CZ.02.1.01/0.0/0.0/15_003/0000449) from European Regional Development
Fund. A.M.F. was supported by the MEPhI Program Priority 2030 and by the
Russian Science Foundation (Grant No. 20-12-00077, interpretation of the
coherence enhancement condition). The numerical simulations were performed
using the code SMILEI and the resources of the ELI ERIC SUNRISE cluster.

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