# ShapeMaker: Self-Supervised Joint Shape Canonicalization,
Segmentation, Retrieval and Deformation

Yan Di, Chenyangguang Zhang∗, Chaowei Wang∗ Ruida Zhang, Guangyao Zhai, Yanyan
Li,
Bowen Fu, Xiangyang Ji and Shan Gao
Tsinghua University, Northwestern Polytechnical University,
{zcyg22<EMAIL_ADDRESS>
*Authors with equal contributions.

###### Abstract

In this paper, we present ShapeMaker, a unified self-supervised learning
framework for joint shape canonicalization, segmentation, retrieval and
deformation. Given a partially-observed object in an arbitrary pose, we first
canonicalize the object by extracting point-wise affine-invariant features,
disentangling inherent structure of the object with its pose and size. These
learned features are then leveraged to predict semantically consistent part
segmentation and corresponding part centers. Next, our lightweight retrieval
module aggregates the features within each part as its retrieval token and
compare all the tokens with source shapes from a pre-established database to
identify the most geometrically similar shape. Finally, we deform the
retrieved shape in the deformation module to tightly fit the input object by
harnessing part center guided neural cage deformation. The key insight of
ShapeMaker is the simultaneous training of the four highly-associated
processes: canonicalization, segmentation, retrieval, and deformation,
leveraging cross-task consistency losses for mutual supervision. Extensive
experiments on synthetic datasets PartNet, ComplementMe, and real-world
dataset Scan2CAD demonstrate that ShapeMaker surpasses competitors by a large
margin. Codes will be released soon.

##  1 Introduction

In recent years, there has been a notable surge in research interest focused
on generating high-quality 3D models from scans of complex scenes [16, 15, 21,
6, 49, 73, 35]. This technology encourages extensive applications in both
artistic creation [56, 57], robotics [67, 66] and 3D scene perception [41,
69]. Existing methods [54, 41] typically directly utilize deep neural networks
to reconstruct 3D models from imperfect scans. However, the presence of noise
and occlusions poses a significant challenge in accurately capturing fine-
grained geometric structures. To overcome this, Retrieval and Deformation
($\mathbf{R\&D}$) techniques [10, 57, 40, 56, 64, 58, 28, 46] have been
developed. These methods generally involve two key steps: first, identifying
the most geometrically similar source shape from a pre-curated 3D database;
and second, deforming the retrieved shape to achieve precise alignment with
the target input. The $\mathbf{R\&D}$ approach is particularly effective in
producing 3D models enriched with fine details from source shapes.

Figure 1: Illustration of ShapeMaker. Objects obtained from real-world scans
are typically noisy, partial and exhibit various poses, making it challenging
to conduct an effective $R\&D$ process (Red ’X’ on the left). To address this
issue, we propose ShapeMaker that first canonicalizes the objects and then
segments them into semantic parts, facilitating $\mathbf{R\&D}$ processes
(Green ’$\checkmark$’ on the right).

However, existing $\mathbf{R\&D}$ methods usually encounter two primary
challenges that make them susceptible to noise, occlusion and pose variations,
and difficult to be practically utilized. 1) Most $\mathbf{R\&D}$ techniques
[10, 57, 40, 56, 64, 58, 28, 46, 17] operate under the assumption that target
shapes are aligned in a pre-processed canonical space. Typically, these
methods are trained and tested on datasets where shapes have been manually
adjusted to this canonical state. However, when these methods are deployed in
real-world settings, they necessitate either manual alignment of scanned
objects or the use of additional pose estimation networks [18, 71, 72, 48,
14]. Such procedures are not only time-intensive and laborious but also prone
to yielding inconsistent results. This limitation significantly impedes the
direct application of these methods in real-world scenarios. 2) Previous
methods [57, 56] do not design specially for dealing with partially-observed
shapes, making it difficult to handle occluded objects. Although U-RED [17]
considers the partial target shapes as input in the $\mathbf{R\&D}$ process,
it directly encodes the shape as a global embedding, which is not robust when
dealing with significant occlusion.

To address the aforementioned two challenges, in this paper, we present
ShapeMaker, a novel framework that extends traditional $\mathbf{R\&D}$
pipeline to joint self-supervised learning of object canonicalization,
segmentation, retrieval and deformation. Our core contribution lies in that
the four highly-associated processes can be trained simultaneously and
supervise each other via constructing several cross-task consistency losses
(Fig. 1). Specifically, given a partially-observed object scan in an arbitrary
pose, ShapeMaker processes the objects in four steps. First, we follow [30],
which is based on Vector Neurons [13], to extract SE(3)-invariant point-wise
features by progressively separating translation and rotation. We further
follow [8] to normalize the features to disassociate the object scale. Until
here, we successfully obtain affine-invariant point-wise features by
disentangling object’s inherent structure with its pose and size. This
facilitates the Canonicalization of the observed object based on these
intrinsic characteristics (Fig. 2 (B)). Then we predict semantically
consistent part segmentation and corresponding part centers by feeding the
learned features into our Segmentation module (Fig. 2 (C)). Based on the part
segmentation, in the Retrieval module (Fig. 2 (D)), we aggregate features
within each part and collect them together as a comprehensive retrieval token
of the object. For partial objects, we introduce a region-weighted strategy,
which assigns a weight to each part according to the point inside it. Parts
with more points are assigned higher weights during retrieval, which is proved
to be robust to occlusions. We compare the tokens of the target object with
each shape in the pre-constructed database to identify the most geometrically
similar (most similar tokens) source shape. In the final Deformation module
(Fig. 2 (E)), the retrieved source shape is deformed to tightly match the
target object via part center guided neural cage deformation [64].

To summarize, our main contributions are:

  * •

We introduce ShapeMaker, a novel self-supervised framework for joint shape
canonicalization, segmentation, retrieval and deformation, handling partial
target inputs under arbitrary poses. Extensive experiments on the synthetic
and real-world datasets demonstrate that ShapeMaker surpasses existing state-
of-the-art approaches by a large margin.

  * •

We demonstrate that the four highly-associated tasks: canonicalization,
segmentation, retrieval and deformation, can be effectively trained
simultaneously and supervise each other via constructing consistencies.

  * •

We develop the region-weighted retrieval method to mitigate the impact of
occlusions in the $\mathbf{R\&D}$ process.

##  2 Related Works

Neural Shape Representation. The compact representation of 3D shapes in latent
space, based on deep learning, has been a focal point for many researchers.
Some attempts, such as [9, 36, 42, 45, 37, 61, 27, 70], employ neural networks
to construct an implicit function, while others [1, 62, 38, 52, 68] directly
model the shape of objects explicitly using generative models. Another common
architecture in 3D shape representation learning, as seen in [12, 44, 29, 63,
60], is to use an encoder-decoder approach to generate latent representation
vectors for various shapes. Although these methods have demonstrated
impressive representation performance, they often struggle to generate fine-
grained shapes when dealing with occlusion and noise.

CAD Model Retrieval and Deformation. Retrieval and Deformation
($\mathbf{R\&D}$) methods lead another way to recover fine-grained geometric
structures. Previous works directly retrieve the most similar CAD models by
comparing the similarity of expression vectors in either descriptor space [46,
5] or the latent space of neural networks [34, 10, 22, 4]. Considering the
subsequent deformation error, recent efforts introduce deformation-aware
embeddings [56] or proposed new optimization objectives [24] to better capture
the fine structure of deformed target objects. Nevertheless, these methods
yield deteriorated performance when facing partial and pose-agnostic target
shapes in real world. [17] achieves an one-to-many retrieval module for
addressing the issue caused by partial observations, however, it receives
canonicalized target shapes as input, which limit its applicability facing
pose-agnostic target shapes in real world. As the retrieved models often
exhibit some deviation from the target shape, the deformation module is used
to minimize this discrepancy. Traditional approaches [47, 23, 20] aim to fit
the target shape by directly optimizing the deformed shape. Neural network
based techniques attempt to learn a set of deformation priors from a database
of models. They represent deformations as volume warping [32, 25], cage
deformations [64], vertex offsets [58], or flows [28, 65]. These methods
typically constrain two shapes are aligned in the same coordinate system,
making them challenging to apply in real-world scenarios.

SO(3)-Equivariant Methods. Due to the lack of robustness to rotation in
traditional network architectures for point cloud processing, an increasing
body of work [55, 31, 19, 59, 2] has initiated research on SO(3) equivariance.
These efforts are mostly based on steerable convolutional kernels [33]. On the
other hand, another set of works achieves equivariance through pose
estimation. [43] estimates the object’s pose to factor out SO(3)
transformations, achieving approximate equivariance. While [51] learns pose
estimation in a fully unsupervised manner, the equivariant backbone they
employ [50] achieves equivariance primarily through data augmentation, leading
to limited generalization. In this paper, we employ Vector Neural Multi-Layer
Perceptron [13] as the backbone to get neural invariant features for object
canonicalization. It achieves SO(3) equivariance by lifting traditional scalar
neurons to vector neurons.

Figure 2: The pipeline of ShapeMaker. Given a target point cloud obtained from
a single-view scan and a pre-established database (A), ShapeMaker generates
the fine-grained reconstruction result using the joint 4 modules including
Canonicalization (B), Segmentation (C), Retrieval (D) and Deformation (E),
where the first three contains the partial branch for target processing and
the full branch for source processing. Specifically, the target and source
inputs are first canonicalized into the same affine-invariant space (B). Then,
the semantic-consistent region segmentation is yielded from the affine-
invariant features (C). The segmented regions are fed to the region-weight
retrieval module (C) and the part center guided neural cage deformation module
(E) for occlusion-robust $\mathbf{R\&D}$ process. During training, the
partial-full consistency losses (F) are enforced for the two branches.

##  3 Method

Overview. ShapeMaker consists of 4 modules, corresponding to the 4 highly-
associated tasks. Each of the first 3 modules: Canonicalization, Segmentation
and Retrieval modules have two parallel branches, one for complete point cloud
(in orange background) and the other for partial object input (in blue
background). As shown in Fig. 2, given a partial target shape
$S_{tgt}\in\mathbb{R}^{N\times 3}$ in an arbitrary pose, in the
Canonicalization module, we progressively decouples its inherent shape with
rotation $R_{tgt}\in SO(3)$, translation $T_{tgt}\in\mathbb{R}^{3}$ and the
$3\mathrm{D}$ metric size $s_{tgt}\in\mathbb{R}^{3}$ via VN-MLP [13, 30, 8],
yielding the affine-invariant point-wise features $F_{tgt}$. The object can
then be canonicalized via inverse transformation based on intrinsics
$\\{R_{tgt},T_{tgt},s_{tgt}\\}$. In the Segmentation module, $F_{tgt}$ is fed
into a 4-layer MLP network to predict $M$ parts and corresponding part centers
$\\{K_{tgt}^{1},K_{tgt}^{2},...,K_{tgt}^{M}\\}$. The segmentation is
semantically consistent across each category and thus can be matched and
compared for $\mathbf{R\&D}$. In the Retrieval module, inside each region
$M^{i}$, we aggregate the features of all points inside it as its retrieval
token $Q^{i}$. The retrieval token for the object is then represented as
$\mathbf{Q_{tgt}}=\\{Q_{tgt}^{1},Q_{tgt}^{2},...,Q_{tgt}^{M}\\}$. Similarly,
during training, we obtain the intrinsics $\\{R_{src},T_{src},s_{src}\\}$,
part centers $\\{K_{src}^{1},K_{src}^{2},...,K_{src}^{M}\\}$ and retrieval
tokens $\mathbf{Q_{src}}=\\{Q_{src}^{1},Q_{src}^{2},...,Q_{src}^{M}\\}$ via
the branch for complete point cloud. By comparing $\mathbf{Q_{tgt}}$ and
$\mathbf{Q_{src}}$ of each source shape inside the database, we identify the
most geometrically similar source shape $S_{r}$. In the final Deformation
module, $K_{tgt}$ and $K_{src}$ are leveraged to guide the neural cage
deformation [64] to deform the retrieved $S_{r}$ towards $S_{tgt}$, yileding
$S_{src}^{dfm}$.

###  3.1 Canonicalization

As shown in Fig. 2 (B), the Canonicalization module takes the target point
cloud $S_{tgt}$ as input and disentangle the inherent structure of $S_{tgt}$
with the intrinsics $\\{R_{tgt},T_{tgt},s_{tgt}\\}$, yielding a point-wise
affine-invariant feature $F_{tgt}$. Specifically, we follow VN-MLP [13, 30, 8]
to first decouple translation via VNT [30] and then extract rotation via VNN
[13, 30]. We further follow [8] to normalize the SE(3)-invariant features
obtained above the remove the influence of scaling, yielding $F_{tgt}$ as
follows,

$R_{tgt},T_{tgt},F_{tgt}^{*}=\mathbf{VN\mbox{-}MLP}(S_{tgt})$ (1)
$s_{tgt},F_{tgt}=normalize(F_{tgt}^{*})$ (2)

where $F_{tgt}^{*}$ denotes the SE(3)-invariant features and $F_{tgt}$ denotes
the affine-invariant features. Thereby, the object can be canonicalized with
intrinsics as,

$S^{c}_{tgt}=s_{tgt}R_{tgt}S_{tgt}+T_{tgt}$ (3)

where $S^{c}_{tgt}$ denotes the normalized and canonicalized shape of
$S^{tgt}$. During training, in order to ensure that $F_{tgt}$ fully
encapsulate the geometric information of $S_{tgt}$, we integrate a
supplementary reconstruction branch which takes $F_{tgt}^{*}$ as input and
reconstruct $S^{c}_{tgt}$ in the affine-invariant space [30]. Please refer to
the Supplementary Material for details. For source shape $S_{src}$ from the
database, we follow the same procedures to extract $F_{src}$.

###  3.2 Segmentation

Given the affine-invariant features $F_{tgt}$, we segment the input point
cloud $S_{tgt}$ into $M$ semantically consistent parts. We use a 4-layer MLP
$\Theta_{l}$ (Fig. 2 (C)) to predict a one-hot segmentation label for each
point and use another 4-layer MLP $\Theta_{c}$ to predict $M$ part centers
$\\{K_{src}^{1},K_{src}^{2},...,K_{src}^{M}\\}$. Noteworthy, we don’t need any
ground truth annotations in this segmentation process. Our experiments show
that the network can automatically learn semantically consistent segmentation
solely through consistency supervision from the other three tasks. For source
shape $S_{src}$, we follow a similar process to obtain
$\\{K_{src}^{1},K_{src}^{2},...,K_{src}^{M}\\}$.

###  3.3 Retrieval

The retrieval network aims to identify the model $S_{src}$ from an existing
database that bears the closest resemblance to the target object $S_{tgt}$
after deformation. Traditional methods [17, 57] directly extract the global
features of objects for retrieval, which typically struggles with heavy
occlusion since the global features are susceptible to noise and occlusion,
and prone to producing erroneous retrieval results. In contrast, we employ a
novel region-weighted retrieval method to explicitly encode independent and
semantically consistent regions of the shape. This allows us to accurately
handle partial shapes by identifying the visible regions to retrieve models
most similar to the target.

Specifically, the part segmentation network $\Theta_{l}$ takes $F_{tgt}$ as
input to predict $M$ regions $F_{seg}=[C_{1},C_{2},C_{3},...,C_{N}]^{\top}$ of
$S^{c}_{tgt}$, where $F_{seg}\in\mathbb{R}^{N\times M}$,
$C_{i}\in\mathbb{R}^{M}$ represents the probability of point $i$ belonging to
each part center. Then we use another 4-layer feature aggregator $\Theta_{f}$
(Fig. 2 (D)) to extract the retrieval tokens $Q$ of all parts as follows,

$F_{cls}=F_{seg}^{\top}*\Theta_{f}(F_{tgt}),$ (4)
$Q_{tgt}=F_{cls}/(\sum_{n=1}^{N}F_{seg}^{(n)}).$ (5)

where $Q_{tgt}\in\mathbb{R}^{M\times C}$ contains the $C$-dimensional
retrieval tokens for all the $M$ regions. Here we employ a soft assignment
strategy where each point inside $S_{tgt}$ is estimated $M$ values describing
the probabilities belonging to each of the $M$ parts. Therefore, we first
aggregate features $F_{cls}$ on all points belonging to each part and then
normalize $F_{cls}$ using the sum of probabilities of points in each part, as
in Eq. 4 and Eq. 5. Following a similar strategy, we can obtain $Q_{src}$ for
each source model in the pre-curated database. We just need to compare
$Q_{tgt}$ with the retrieval tokens $Q_{src}$ of all source shapes using the
weighted $\mathcal{L}_{1}$ distance,

$Dis=\omega\sum\mathcal{L}_{1}(Q_{tgt}-Q_{src})$ (6)

where vector $\omega\in\mathbb{R}^{1\times M}$ stores the ratio of point
number of each part with respect to the total point number $N$. Intuitively,
parts with smaller point numbers contribute less in calculating the distance
score, which reduces the influence of noise and occlusion. The source shape
$S_{r}$ with the smallest distance score is identified as the best retrieval.

###  3.4 Deformation

The Deformation module aims to deform the retrieved shape $S_{r}$ to tightly
match the target shape $S_{tgt}$. We utilize the neural cage scaffolding
strategy as in [64, 26]. First, the neural cages $C_{src}$ for $S_{r}$ is pre-
calculated. We utilize the part centers ($K_{tgt}$ and $K_{src}$) to control
the vertice offsets $C_{src2tgt}$ of the neural cage $C_{src}$ to match
$S_{tgt}$. In particular, we employ a neural network $\Theta_{I}$ to predict
an influence vector $I\in\mathbb{R}^{N_{c}\times M}$ for each point concerning
all cage vertices by $I=\Theta_{I}(concat(F_{tgt},F_{src}))$, where $N_{c}$
denotes the number of vertices used in $C_{src}$. $C_{src2tgt}$ is computed
through the influence vectors $I$ and the differences between region centers
(Fig. 2 (E)):

$C_{src2tgt}=C_{src}+\sum_{i=1}^{M}I_{i}(K_{src}^{(i)}-K_{tgt}^{(i)}),$ (7)

Finally, we employ a sparse cage scaffolding strategy [64, 26] to achieve the
deformation field of $S_{src}$. The deformed shape $S_{src2tgt}$ of $S_{src}$
can be expressed as follows:

$S_{src2tgt}=S_{src}+\Psi(C_{src},C_{src2tgt}),$ (8)

where $\Psi$ computes the displacement of each point in $S_{src}$ by
evaluating the differences between $C_{src}$ and $C_{src2tgt}$, thereby
achieving deformation.

###  3.5 ShapeMaker: Joint Training

Our core insight in ShapeMaker is that the four highly-associated tasks:
Canonicalization, Segmentation, Retrieval and Deformation can be trained
simultaneously and supervise each other via introducing cross-task consistency
terms. We mainly introduce two types of losses here, i.e. partial-full
consistency losses and task-oriented loss. For more details, please refer to
the Supplementary Material.

Task-Oriented Loss. In the Canonicalization, we mainly use Chamfer Distance to
constrain the canonicalized $S^{c}_{tgt}$ and $\hat{S^{c}}_{tgt}$ predicted in
the affine-invariant space by the supplementary reconstruction branch, to
enforce the affine-invarianty of $F_{tgt}^{*}$ in Sec. 3.1, by

$\mathcal{L}_{can}=dis_{cham}(S^{c}_{tgt},\hat{S^{c}}_{tgt})+orth(R_{tgt}),$
(9)

with $orth(R_{tgt})$ serving the purpose of enforcing the orthogonality of
matrices.

In the Segmentation, to keep consistency between the part segmentation and the
predicted part center, we jointly train $\Theta_{l}$ and $\Theta_{c}$ with the
following loss, which enforce that each predicted part center $K^{i}_{tgt}$
approximately lies in the center of all points belonging to the part $M_{i}$,

$\mathcal{L}_{seg}=\sum_{m=1}^{M}\|K_{tgt}^{(m)}-(F_{seg}^{\top}*S^{c}_{tgt})^{(m)}\|_{2}$
(10)

To train the Retrieval and Deformation simultaneously, for an input target
$S_{tgt}$, we randomly select a source model $S_{src}$ from the database for
training. Specifically, to eliminate the influence of occlusion, we do not
directly use the global Chamfer Distance of $S_{tgt}$ and $S_{src}$ as ground
truth. Instead, we employ a regional supervision strategy, ensuring that
occluded areas do not contribute to the training of retrieval network. Taking
the $i$-th region as an example, $S_{tgt}^{i}$ represents all points in
$S_{tgt}$ that belong to the $i$-th region. We calculate the average of the
nearest distances $D_{i}$ from each point in $S_{tgt}^{i}$ to the deformed
shape $S_{src}^{dfm}$ to enforce the learning of the regional retrieval tokens
by

$\mathcal{L}_{retrieval}=\frac{1}{M}\sum_{i=1}^{M}MSE(Q_{tgt}^{(i)}-Q_{src}^{(i)},D_{i}).$
(11)

The deformation loss is achieved by directly constraining the Chamfer Distance
between $S_{tgt}$ and $S_{src}^{dfm}$, expressed as:

$\mathcal{L}_{deform}=dis_{cham}(S_{tgt},S_{src}^{dfm})+\|I\|_{2}$ (12)

where we regularize $I$ using the L2 norm.

Partial-Full Consistency Losses. In the first two modules: Canonicalization,
Segmentation, the full branch serves as a guidance to enhance the learning of
the partial branch. Therefore, in each module, we can enforce corresponding
consistency terms between the outputs of the two parallel branches (Fig. 2
(F)).

During the consistency training process, for randomly selected full input
$S_{full}$, we generate a mask $U_{f2p}\in\mathbb{R}^{N}$ to crop it to
simulate the situation of a partial input $S_{partial}=S_{full}U_{f2p}$.

In the Canonicalization module, to enforce the consistency in the affine-
invariant space between the two branches, we apply the same transformation
$U_{f2p}$ to $S_{full}^{c}$ in the affine-invariant space as before and then
use the chamfer distance to constrain its distance to $S_{partial}^{c}$:

$\mathcal{L}_{ccan}=dis_{cham}(S^{c}_{partial},S^{c}_{full}U_{f2p})$ (13)

Similarly, in the Segmentation module, for the consistency constraint of the
part center prediction network $\Theta_{c}$, we directly use the Chamfer
Distance to constrain the region centers detected by the two branches:

$\mathcal{L}_{ccen}=dis_{cham}(K_{partial},K_{full}).$ (14)

In the segmentation network $\Theta_{l}$, we mask the segmentation results of
the full branch $F_{seg}^{(full)}$ and compare them with the results of the
partial branch $F_{seg}^{(partial)}$:

$\mathcal{L}_{cseg}=dis_{cham}(F_{seg}^{(full)}U_{f2p},F_{seg}^{(partial)}).$
(15)

Joint Training. Generally, the joint training of ShapeMaker is divided into
three stages. First, we train the full branch by $\mathcal{L}_{can}$ and
$\mathcal{L}_{seg}$ for construction of Canonicalization and Segmentation
ability. Second, the partial branch is introduced and trained by both the
task-oriented losses for Canonicalization and Segmentation $\mathcal{L}_{can}$
and $\mathcal{L}_{seg}$ and the partial-full consistency loss terms
$\mathcal{L}_{ccan}$, $\mathcal{L}_{ccen}$ and $\mathcal{L}_{cseg}$. Finally,
after training Canonicalization and Segmentation of the both branches,
$\mathcal{L}_{retrieval}$ and $\mathcal{L}_{deform}$ are adopted for joint
Retrieval and Deformation training simultaneously utilizing the both branches
to handle partial target inputs and full source inputs respectively.

##  4 Experiments

###  4.1 Experimental Setup

In this section, we mainly focus on $\mathbf{R\&D}$ experiments, which better
reflects the overall performance of the system. The ablations and analysis
also demonstrate the effectiveness of considering joint Canonicalization and
Part Segmentation.

Datasets. We evaluate the effectiveness of our joint framework using three
datasets: two synthetic datasets, PartNet [39] and ComplementMe [53], and one
real-world dataset, Scan2CAD [3]. For datasets PartNet and ComplementMe, we
follow the same database splits as in [57], separating their target inputs
into training and testing sets. In our training process, we exclusively employ
mesh models and do not utilize part segmentations as in [57], since the
process of ShapeMaker is fully self-supervised and does not need any
additional annotations. The shapes used in PartNet and ComplementMe datasets
are sourced from ShapeNet [7]. PartNet comprises 1,419 source models in the
database, with 11,433 target models in the training set and 2,861 in the
testing set. In ComplementMe, the numbers are 400, 11,311 and 2,825
respectively. In the synthetic cases, three categories of tables, chairs, and
cabinets are evaluated on both datasets. Scan2CAD [3] is a real-world dataset
developed based on ScanNet [11] with capacity of 14,225 objects. The input
point cloud data on Scan2CAD is generated by reverse-projecting the depth
images. In the real-world cases, we conduct training on the categories of
tables, chairs, and cabinets from PartNet and directly testing on Scan2CAD.

Baselines. Both baseline methods, Uy et al. [57] and U-RED [17], are trained
using the same data partitioning strategy stated above. To ensure fairness in
comparison with ShapeMaker, we augment the training data with pose variations,
keeping other hyperparameters consistent with the original paper. During
testing, we evaluate scenarios where target observations with arbitrary poses
are directly used as input. Additionally, we test scenarios where the inputs
are transformed using an offline pose estimation method [18], simulating the
two-stage route of traditional methods with pre-canonicalizing (Uy et al.[57]
\+ PE and U-RED [17] \+ PE). For experiments on Scan2CAD, we directly use the
baseline models trained on PartNet with the 25% occlusion setting to conduct
zero-shot testing, since real-world ground-truth models are inaccessible for
training.

Evaluation Metrics. We utilize Chamfer Distance (CD) on the magnitude of
$10^{-2}$ to assess both full shape scenarios and partial shape scenarios. We
calculate the metrics following [57] to use the best result among the top 10
candidate objects. The final average metrics are obtained by averaging the
results across all instances.

Implementation Details. During training, we uniformly sample objects to obtain
point clouds with $M=2500$ points to represent shapes. We directly generate
partial point clouds from the corresponding full point clouds by random
cropping for the partial branch inputs. We apply random pose augmentation to
the input shape, specifically with random translations $T_{rand}\in[-0.1,0.1]$
and random rotations $R_{rand}\in[-1,1]$ on three Eulerian angles
respectively. We set the initial learning rate to $1e-3$ and train ShapeMaker
for 200 epochs in every training stage of Sec. 3.5. Regarding the weight of
the loss, in the first stage considering only the full branch,
$\mathcal{L}_{can}$ and $\mathcal{L}_{seg}$ are equally weighted. In the
second stage introducing the partial branch, we primarily emphasize the
partial-full consistency losses, assigning significant weights to
$\mathcal{L}_{ccan}$, $\mathcal{L}_{ccen}$ and $\mathcal{L}_{cseg}$ with
weights set as $5$, $2$, and $2$ respectively, while keeping the remaining
weights default at $1$. In the final stage for joint $\mathbf{R\&D}$, both
$\mathcal{L}_{retrieval}$ and $\mathcal{L}_{deform}$ are equally weighted.

###  4.2 Synthetic Cases

To validate the ability of ShapeMaker tackling the challenge of arbitrary
poses and occlusions, we first use synthetic datasets to simulate this
scenario. We evaluate all methods [57, 17] where object observations with
arbitrary poses are directly used as input. Additionally, we also report
results where inputs are transformed and canonicalized using an offline pose
estimation method [18] for baseline methods (Uy et al. + PE and U-RED + PE).
Moreover, we analyze inference time of ShapeMaker against the $\mathbf{R\&D}$
baselines in the Supplementary Material.

We conduct two types of inputs for evaluation: full inputs using the PartNet
and ComplementMe datasets, and partial input tests using $10\%$, $25\%$, and
$50\%$ occlusion rates on the PartNet dataset. The results of the full input
tests are detailed in Table 1. In PartNet, our ShapeMaker significantly
outperforms the current leading competitors. For the Chamfer Distance on three
categories, ShapeMaker measures at $0.197$, $0.150$, and $0.519$, maintaining
the leading position. Even when the processed PE results are used as input,
the baselines’ results still fall short of ShapeMaker. This demonstrates the
effectiveness of adopting the affine-invariant features in the joint
Canonicalization step. Results from ComplementMe supports the same conclusion,
where ShapeMaker reports significantly better results compared to the baseline
methods. ShapeMaker surpasses the top-performing Uy et al. \+ PE by $85.2\%$.
Such superior results yielded by ShapeMaker demonstrates that the joint
consideration of all four steps improves the matching accuracy a lot.

For evaluation on partial inputs, we control the occlusion rates of partial
point clouds by controlling the position of the cropping planes onto the full
point clouds. The evaluation on partial inputs are presented in Table 2.
Concretely, ShapeMaker outperforms the current top method handling partial
inputs U-RED by $5.018$, $5.666$ and $7.241$ at the occlusion rate of $10\%$,
$25\%$ and $50\%$ respectively. As the occlusion rate increases, the
superiority of the ShapeMaker method grows. Considering PE adopting, the same
trend is exhibited. ShapeMaker surpasses the U-RED + PE by $0.525$, $0.949$,
$2.434$ under three occlusion rates. It demonstrates that the proposed region-
weighted retrieval brings strong robustness of our method against occlusion.

As shown in Fig. 3 and 4, our results shows more geometric resemblance to the
targets compared to other methods. This is attributed to the suitable joint
consideration of the four highly-associated processes, which accurately
decouples the input poses, mapping them to a consistent space for accurate
$R\&D$. The region-weighted retrieval we employ explicitly eliminates the
influence of occluded areas, allowing for a more precise matching with the
source model.

PartNet[39]
---
Method | Chair | Table | Cabinet | Average
Uy et al.[57] | 4.269 | 6.302 | 4.118 | 5.271
Uy et al.[57] \+ PE | 1.507 | 3.006 | 1.070 | 2.219
U-RED[17] | 5.331 | 4.980 | 9.141 | 5.463
U-RED[17] \+ PE | 1.025 | 0.359 | 1.423 | 0.725
Ours | 0.197 | 0.150 | 0.519 | 0.200
ComplementMe[53]
Method | Chair | Table | Cabinet | Average
Uy et al.[57] | 4.018 | 5.480 | – | 4.825
Uy et al.[57] \+ PE | 1.439 | 2.454 | – | 1.999
U-RED[17] | 8.575 | 5.800 | – | 7.044
U-RED[17] \+ PE | 4.954 | 0.847 | – | 2.688
Ours | 0.253 | 0.328 | – | 0.294

Table 1: The Chamfer Distance metrics for joint R$\&$D results on full shapes
under arbitrary poses.

Occlusion | Method | Chair | Table | Cabinet | Average
---|---|---|---|---|---
10% | Uy et al.[57] | 4.372 | 6.395 | 4.179 | 5.365
Uy et al. \+ PE[57] | 1.523 | 2.982 | 1.133 | 2.219
U-RED[17] | 6.025 | 5.375 | 5.269 | 5.640
U-RED[17] \+ PE | 1.207 | 1.012 | 1.669 | 1.147
Ours | 0.676 | 0.481 | 1.212 | 0.622
25% | Uy et al.[57] | 4.654 | 6.927 | 4.750 | 5.795
Uy et al. \+ PE[57] | 1.803 | 3.195 | 1.607 | 2.481
U-RED[17] | 5.196 | 7.215 | 8.164 | 6.442
U-RED[17] \+ PE | 1.684 | 1.387 | 2.795 | 1.625
Ours | 0.878 | 0.643 | 1.071 | 0.776
50% | Uy et al.[57] | 6.070 | 9.322 | 7.929 | 7.841
Uy et al. \+ PE[57] | 3.314 | 5.030 | 4.584 | 4.272
U-RED[17] | 8.696 | 8.387 | 7.613 | 8.455
U-RED[17] \+ PE | 4.722 | 2.015 | 7.903 | 3.628
Ours | 1.197 | 1.079 | 1.872 | 1.194

Table 2: The Chamfer Distance metrics for joint R$\&$D results on partial
shapes under arbitrary poses of PartNet dataset.

###  4.3 Real-world Cases

We test the effectiveness of ShapeMaker on real-world datasets. In such case,
ShapeMaker is trained on the synthetic PartNet with $25\%$ occlusion and
directly tested on the partial scans of the real-world dataset Scan2CAD
without manual pose adjustments. Table 3 displays our results, where our
method significantly outperforms existing competitors. Particularly, compared
to the U-RED, in three categories, the reported Chamfer Distance are reduced
by $92\%$, $96\%$, and $94\%$ respectively. In comparison to Uy et al., the
reported Chamfer Distance are reduced by $91\%$, $98\%$, and $92\%$. Even
compared with the use of PE on the two baseline methods, there remains a
significant leap forward of ShapeMaker. This zero-shot experiments on real-
world further prove that the procedure of joint Canonicalization,
Segmentation, Retrieval and Deformation possesses a strong domain adaptation
ability, showcasing the great potential of ShapeMaker in real-world
applications. The visualizations on Scan2CAD are provided in the supplementary
material.

Figure 3: Qualitative R&D results with full target inputs on PartNet. Figure
4: Qualitative R&D results with partial target inputs on the occlusion rate of
$25\%$ on PartNet.

Method | Chair | Table | Cabinet | Average
---|---|---|---|---
Uy et al.[57] | 4.886 | 7.605 | 8.335 | 6.181
Uy et al.[57] \+ PE | 3.362 | 6.657 | 7.261 | 4.905
U-RED[17] | 5.490 | 5.131 | 10.091 | 5.945
U-RED[17] \+ PE | 2.893 | 3.164 | 5.957 | 3.354
Ours | 0.423 | 0.186 | 0.654 | 0.375

Table 3: The Chamfer Distance metrics for joint R$\&$D results in real-world
Scan2CAD [3].

###  4.4 Ablation Studies

We conduct ablation experiments on PartNet, mainly on two aspects. First, in
the Canonicalization, we investigate the importance of disentangling different
pose intrinsics in Table 4, and demonstrate the effectiveness of joint
considering Canonicalization. Second, in Table 5, we ablate the region-
weighted Retrieval and the part center guided Deformation. Moreover, the
robustness against occlusion is analyzed in Table 2.

Canonicalization Capability. To study the impact of different intrinsics of
object poses in the Canonicalization process, we conduct ablations on
decoupling of translations, rotations and scales. The results are presented in
Table 4. Specifically, in row (1), we make no adjustments to the input poses.
Thanks to the regional-level $R\&D$ process, it shows decent performance.
However, there is still a noticeable gap compared to row (4), indicating the
significance of our proposed joint Canonicalization. In row (2), we solely
decouple the input translations, resulting in a decrease of $14\%$ in reported
metrics. Moving to row (3), upon this foundation, we add decoupling for
rotation, leading to a substantial decrease in reported Chamfer Distance,
averaging at $0.244$. In row (4), we introduce scale decoupling, resulting in
another decrease in the reported metrics. It is evident that accurate
decoupling of rotation is a crucial aspect of the success of the
Canonicalization process. Moreover, it demonstrates that to integrate
Canonicalization and $R\&D$ process is indispensable for the ShapeMaker
process.

Deformation and Retrieval Ability. To validate the effectiveness of our
proposed region-weighted Retrieval and the part center guided neural cage
Deformation, we conduct an ablation study on the PartNet dataset with the
$25\%$ occlusion rate. The results are presented in Table 5. In row (1), we
conduct experiments using global retrieval and global deformation. This means
we directly use an MLP network to extract overall point cloud features as the
retrieval vector [57]. In the deformation network, similarly, we directly use
an MLP network to generate neural cage offsets for deformation [64, 26]. Due
to the lack of extraction of local information, the reported Chamfer Distance
is more than twice of row (4). In row (2), we employ the global retrieval and
the part center guided neural cage deformation. This improvement allows much
more tightly-matched deformation by the retrieved source model, resulting in a
$14\%$ decrease in reported metrics. In row (3), we conduct experiments using
the regional retrieval and the global deformation. The proposed regional-
weighted retrieval handles occluded objects, reducing the impact of occluded
parts. This results in a substantial decrease in Chamfer Distance, down to
$0.973$.

Occlusion Robustness. We test shapes at different occlusion levels by altering
the occlusion ratio in the input. For each specific occlusion ratio, we
deliberately crop a portion of the complete point cloud to simulate occlusion.
We test scenarios with occlusion ratios of $10\%$, $25\%$, and $50\%$, and the
results are presented in Table 2. Observably, as the occluded regions
increased, the Chamfer Distance significantly rises for the baseline methods.
Taking the U-RED + PE as an example, its reported metrics increase from
$1.147$ to $3.628$, doubling in value, as the occluded area expands. In
contrast, our method increases by less than $1$-fold, which exhibits strong
robustness against occlusion.

| Trans. | Rot. | Scal. | Chair | Table | Cabinet | Average
---|---|---|---|---|---|---|---
(1) |  |  |  | 0.571 | 0.502 | 1.233 | 0.590
(2) | ✓ |  |  | 0.468 | 0.442 | 1.096 | 0.506
(3) | ✓ | ✓ |  | 0.213 | 0.200 | 0.674 | 0.244
(4) | ✓ | ✓ | ✓ | 0.197 | 0.150 | 0.519 | 0.200

Table 4: Ablations on the Canonicalization process, we demonstrate the
effectiveness of the joint Canonicalization by ablating different pose
intrinsics. Here, Trans. denotes the decoupling of translation, Rot.
represents the rotation, and Scal. signifies the scale.

| Gl. R. | Re. R. | Gl. D. | Re. D. | Chair | Table | Cabinet | Average
---|---|---|---|---|---|---|---|---
(1) | ✓ |  | ✓ |  | 1.672 | 1.446 | 1.800 | 1.570
(2) | ✓ |  |  | ✓ | 1.539 | 1.305 | 1.641 | 1.431
(3) |  | ✓ | ✓ |  | 1.042 | 0.874 | 1.223 | 0.973
(4) |  | ✓ |  | ✓ | 0.878 | 0.643 | 1.071 | 0.776

Table 5: Ablations of the $R\&D$ process. GL. R. denotes the global feature
based retrieval [57], Re. R. represents the proposed region-weighted
retrieval, GL. D. signifies direct neural cage deformation using global
features [64, 26], and Re. D. denotes the adopted regional part center guided
neural cage deformation.

##  5 Conclusion

In this paper, we present ShapeMaker, a unified self-supervised learning
framework for joint shape canonicalization, segmentation, retrieval and
deformation. Given a partially-observed object in an arbitrary pose, we first
canonicalize the object by extracting point-wise affine-invariant features.
Then, the affine-invariant features are leveraged to predict semantically
consistent part segmentation and corresponding part centers. Afterwards, the
lightweight region-weighted retrieval module aggregates the features within
each part as its retrieval token and compare all the tokens with source shapes
from a pre-established database to identify the most geometrically similar
shape. Finally, we deform the retrieved shape in the deformation module to
tightly fit the input object by harnessing part center guided neural cage
deformation. Extensive experiments on synthetic datasets PartNet,
ComplementMe, and real-world dataset Scan2CAD demonstrate that ShapeMaker
surpasses competitors by a large margin. In the future, we plan to further
applicate I-RED to various downstream tasks like robotic grasping.
Limitations. are discussed in the Supplementary Material.

## References

  * [1] Panos Achlioptas, Olga Diamanti, Ioannis Mitliagkas, and Leonidas Guibas.  Learning representations and generative models for 3d point clouds.  In International conference on machine learning, pages 40–49. PMLR, 2018.
  * [2] Brandon Anderson, Truong Son Hy, and Risi Kondor.  Cormorant: Covariant molecular neural networks.  Advances in neural information processing systems, 32, 2019.
  * [3] Armen Avetisyan, Manuel Dahnert, Angela Dai, Manolis Savva, Angel X Chang, and Matthias Nießner.  Scan2cad: Learning cad model alignment in rgb-d scans.  In Proceedings of the IEEE/CVF Conference on computer vision and pattern recognition, pages 2614–2623, 2019.
  * [4] Armen Avetisyan, Angela Dai, and Matthias Nießner.  End-to-end cad model retrieval and 9dof alignment in 3d scans.  In Proceedings of the IEEE/CVF International Conference on computer vision, pages 2551–2560, 2019.
  * [5] Frederic Bosche and Carl T Haas.  Automated retrieval of 3d cad model objects in construction range images.  Automation in Construction, 17(4):499–512, 2008.
  * [6] Anh-Quan Cao and Raoul de Charette.  Monoscene: Monocular 3d semantic scene completion.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3991–4001, 2022.
  * [7] Angel X Chang, Thomas Funkhouser, Leonidas Guibas, Pat Hanrahan, Qixing Huang, Zimo Li, Silvio Savarese, Manolis Savva, Shuran Song, Hao Su, et al.  Shapenet: An information-rich 3d model repository.  arXiv preprint arXiv:1512.03012, 2015.
  * [8] Yunlu Chen, Basura Fernando, Hakan Bilen, Matthias Nießner, and Efstratios Gavves.  3d equivariant graph implicit functions.  In European Conference on Computer Vision, pages 485–502. Springer, 2022.
  * [9] Zhiqin Chen and Hao Zhang.  Learning implicit fields for generative shape modeling.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 5939–5948, 2019.
  * [10] Manuel Dahnert, Angela Dai, Leonidas J Guibas, and Matthias Nießner.  Joint embedding of 3d scan and cad objects.  In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 8749–8758, 2019.
  * [11] Angela Dai, Angel X Chang, Manolis Savva, Maciej Halber, Thomas Funkhouser, and Matthias Nießner.  Scannet: Richly-annotated 3d reconstructions of indoor scenes.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 5828–5839, 2017.
  * [12] Angela Dai, Charles Ruizhongtai Qi, and Matthias Nießner.  Shape completion using 3d-encoder-predictor cnns and shape synthesis.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 5868–5877, 2017.
  * [13] Congyue Deng, Or Litany, Yueqi Duan, Adrien Poulenard, Andrea Tagliasacchi, and Leonidas J Guibas.  Vector neurons: A general framework for so (3)-equivariant networks.  In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 12200–12209, 2021.
  * [14] Yan Di, Fabian Manhardt, Gu Wang, Xiangyang Ji, Nassir Navab, and Federico Tombari.  So-pose: Exploiting self-occlusion for direct 6d pose estimation.  In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 12396–12405, 2021.
  * [15] Yan Di, Henrique Morimitsu, Shan Gao, and Xiangyang Ji.  Monocular piecewise depth estimation in dynamic scenes by exploiting superpixel relations.  In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 4363–4372, 2019.
  * [16] Yan Di, Henrique Morimitsu, Zhiqiang Lou, and Xiangyang Ji.  A unified framework for piecewise semantic reconstruction in dynamic scenes via exploiting superpixel relations.  In 2020 IEEE International Conference on Robotics and Automation (ICRA), pages 10737–10743. IEEE, 2020.
  * [17] Yan Di, Chenyangguang Zhang, Ruida Zhang, Fabian Manhardt, Yongzhi Su, Jason Rambach, Didier Stricker, Xiangyang Ji, and Federico Tombari.  U-red: Unsupervised 3d shape retrieval and deformation for partial point clouds.  arXiv preprint arXiv:2308.06383, 2023.
  * [18] Yan Di, Ruida Zhang, Zhiqiang Lou, Fabian Manhardt, Xiangyang Ji, Nassir Navab, and Federico Tombari.  Gpv-pose: Category-level object pose estimation via geometry-guided point-wise voting.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6781–6791, 2022.
  * [19] Carlos Esteves, Christine Allen-Blanchette, Ameesh Makadia, and Kostas Daniilidis.  Learning so (3) equivariant representations with spherical cnns.  In Proceedings of the European Conference on Computer Vision (ECCV), pages 52–68, 2018.
  * [20] Vignesh Ganapathi-Subramanian, Olga Diamanti, Soeren Pirk, Chengcheng Tang, Matthias Niessner, and Leonidas Guibas.  Parsing geometry using structure-aware shape templates.  In 2018 International Conference on 3D Vision (3DV), pages 672–681. IEEE, 2018.
  * [21] Georgia Gkioxari, Jitendra Malik, and Justin Johnson.  Mesh r-cnn.  In Proceedings of the IEEE/CVF international conference on computer vision, pages 9785–9795, 2019.
  * [22] Can Gümeli, Angela Dai, and Matthias Nießner.  Roca: Robust cad model retrieval and alignment from a single image.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4022–4031, 2022.
  * [23] Qi-Xing Huang, Bart Adams, Martin Wicke, and Leonidas J Guibas.  Non-rigid registration under isometric deformations.  In Computer Graphics Forum, volume 27, pages 1449–1457. Wiley Online Library, 2008.
  * [24] Vladislav Ishimtsev, Alexey Bokhovkin, Alexey Artemov, Savva Ignatyev, Matthias Niessner, Denis Zorin, and Evgeny Burnaev.  Cad-deform: Deformable fitting of cad models to 3d scans.  In Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part XIII 16, pages 599–628. Springer, 2020.
  * [25] Dominic Jack, Jhony K Pontes, Sridha Sridharan, Clinton Fookes, Sareh Shirazi, Frederic Maire, and Anders Eriksson.  Learning free-form deformations for 3d object reconstruction.  In Computer Vision–ACCV 2018: 14th Asian Conference on Computer Vision, Perth, Australia, December 2–6, 2018, Revised Selected Papers, Part II 14, pages 317–333. Springer, 2019.
  * [26] Tomas Jakab, Richard Tucker, Ameesh Makadia, Jiajun Wu, Noah Snavely, and Angjoo Kanazawa.  Keypointdeformer: Unsupervised 3d keypoint discovery for shape control.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 12783–12792, 2021.
  * [27] Wonbong Jang and Lourdes Agapito.  Codenerf: Disentangled neural radiance fields for object categories.  In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 12949–12958, 2021.
  * [28] Chiyu Jiang, Jingwei Huang, Andrea Tagliasacchi, and Leonidas J Guibas.  Shapeflow: Learnable deformation flows among 3d shapes.  Advances in Neural Information Processing Systems, 33:9745–9757, 2020.
  * [29] Jincen Jiang, Xuequan Lu, Lizhi Zhao, Richard Dazaley, and Meili Wang.  Masked autoencoders in 3d point cloud representation learning.  IEEE Transactions on Multimedia, 2023.
  * [30] Oren Katzir, Dani Lischinski, and Daniel Cohen-Or.  Shape-pose disentanglement using se (3)-equivariant vector neurons.  In European Conference on Computer Vision, pages 468–484. Springer, 2022.
  * [31] Risi Kondor, Zhen Lin, and Shubhendu Trivedi.  Clebsch–gordan nets: a fully fourier space spherical convolutional neural network.  Advances in Neural Information Processing Systems, 31, 2018.
  * [32] Andrey Kurenkov, Jingwei Ji, Animesh Garg, Viraj Mehta, JunYoung Gwak, Christopher Choy, and Silvio Savarese.  Deformnet: Free-form deformation network for 3d shape reconstruction from a single image.  In 2018 IEEE Winter Conference on Applications of Computer Vision (WACV), pages 858–866. IEEE, 2018.
  * [33] Leon Lang and Maurice Weiler.  A wigner-eckart theorem for group equivariant convolution kernels.  arXiv preprint arXiv:2010.10952, 2020.
  * [34] Yangyan Li, Hao Su, Charles Ruizhongtai Qi, Noa Fish, Daniel Cohen-Or, and Leonidas J Guibas.  Joint embeddings of shapes and images via cnn image purification.  ACM transactions on graphics (TOG), 34(6):1–12, 2015.
  * [35] Jiachen Liu, Pan Ji, Nitin Bansal, Changjiang Cai, Qingan Yan, Xiaolei Huang, and Yi Xu.  Planemvs: 3d plane reconstruction from multi-view stereo.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 8665–8675, 2022.
  * [36] Lars Mescheder, Michael Oechsle, Michael Niemeyer, Sebastian Nowozin, and Andreas Geiger.  Occupancy networks: Learning 3d reconstruction in function space.  In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 4460–4470, 2019.
  * [37] Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoorthi, and Ren Ng.  Nerf: Representing scenes as neural radiance fields for view synthesis.  Communications of the ACM, 65(1):99–106, 2021.
  * [38] Kaichun Mo, Paul Guerrero, Li Yi, Hao Su, Peter Wonka, Niloy Mitra, and Leonidas J Guibas.  Structurenet: Hierarchical graph networks for 3d shape generation.  arXiv preprint arXiv:1908.00575, 2019.
  * [39] Kaichun Mo, Shilin Zhu, Angel X Chang, Li Yi, Subarna Tripathi, Leonidas J Guibas, and Hao Su.  Partnet: A large-scale benchmark for fine-grained and hierarchical part-level 3d object understanding.  In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 909–918, 2019.
  * [40] Liangliang Nan, Ke Xie, and Andrei Sharf.  A search-classify approach for cluttered indoor scene understanding.  ACM Transactions on Graphics (TOG), 31(6):1–10, 2012.
  * [41] Yinyu Nie, Xiaoguang Han, Shihui Guo, Yujian Zheng, Jian Chang, and Jian Jun Zhang.  Total3dunderstanding: Joint layout, object pose and mesh reconstruction for indoor scenes from a single image.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 55–64, 2020.
  * [42] Jeong Joon Park, Peter Florence, Julian Straub, Richard Newcombe, and Steven Lovegrove.  Deepsdf: Learning continuous signed distance functions for shape representation.  In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 165–174, 2019.
  * [43] Charles R Qi, Hao Su, Kaichun Mo, and Leonidas J Guibas.  Pointnet: Deep learning on point sets for 3d classification and segmentation.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 652–660, 2017.
  * [44] Yuchen Rao, Yinyu Nie, and Angela Dai.  Patchcomplete: Learning multi-resolution patch priors for 3d shape completion on unseen categories.  Advances in Neural Information Processing Systems, 35:34436–34450, 2022.
  * [45] Edoardo Remelli, Artem Lukoianov, Stephan Richter, Benoit Guillard, Timur Bagautdinov, Pierre Baque, and Pascal Fua.  Meshsdf: Differentiable iso-surface extraction.  Advances in Neural Information Processing Systems, 33:22468–22478, 2020.
  * [46] Adriana Schulz, Ariel Shamir, Ilya Baran, David IW Levin, Pitchaya Sitthi-Amorn, and Wojciech Matusik.  Retrieval on parametric shape collections.  ACM Transactions on Graphics (TOG), 36(1):1–14, 2017.
  * [47] Olga Sorkine and Marc Alexa.  As-rigid-as-possible surface modeling.  In Symposium on Geometry processing, volume 4, pages 109–116. Citeseer, 2007.
  * [48] Yongzhi Su, Yan Di, Guangyao Zhai, Fabian Manhardt, Jason Rambach, Benjamin Busam, Didier Stricker, and Federico Tombari.  Opa-3d: Occlusion-aware pixel-wise aggregation for monocular 3d object detection.  IEEE Robotics and Automation Letters, 8(3):1327–1334, 2023.
  * [49] Jiaming Sun, Yiming Xie, Linghao Chen, Xiaowei Zhou, and Hujun Bao.  Neuralrecon: Real-time coherent 3d reconstruction from monocular video.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 15598–15607, 2021.
  * [50] Weiwei Sun, Wei Jiang, Eduard Trulls, Andrea Tagliasacchi, and Kwang Moo Yi.  Acne: Attentive context normalization for robust permutation-equivariant learning.  In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 11286–11295, 2020.
  * [51] Weiwei Sun, Andrea Tagliasacchi, Boyang Deng, Sara Sabour, Soroosh Yazdani, Geoffrey Hinton, and Kwang Moo Yi.  Canonical capsules: Unsupervised capsules in canonical pose.  arXiv preprint arXiv:2012.04718, 1, 2021.
  * [52] Yongbin Sun, Yue Wang, Ziwei Liu, Joshua Siegel, and Sanjay Sarma.  Pointgrow: Autoregressively learned point cloud generation with self-attention.  In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 61–70, 2020.
  * [53] Minhyuk Sung, Hao Su, Vladimir G Kim, Siddhartha Chaudhuri, and Leonidas Guibas.  Complementme: Weakly-supervised component suggestions for 3d modeling.  ACM Transactions on Graphics (TOG), 36(6):1–12, 2017.
  * [54] Maxim Tatarchenko, Stephan R Richter, René Ranftl, Zhuwen Li, Vladlen Koltun, and Thomas Brox.  What do single-view 3d reconstruction networks learn?  In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 3405–3414, 2019.
  * [55] Nathaniel Thomas, Tess Smidt, Steven Kearnes, Lusann Yang, Li Li, Kai Kohlhoff, and Patrick Riley.  Tensor field networks: Rotation-and translation-equivariant neural networks for 3d point clouds.  arXiv preprint arXiv:1802.08219, 2018.
  * [56] Mikaela Angelina Uy, Jingwei Huang, Minhyuk Sung, Tolga Birdal, and Leonidas Guibas.  Deformation-aware 3d model embedding and retrieval.  In Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part VII 16, pages 397–413. Springer, 2020.
  * [57] Mikaela Angelina Uy, Vladimir G Kim, Minhyuk Sung, Noam Aigerman, Siddhartha Chaudhuri, and Leonidas J Guibas.  Joint learning of 3d shape retrieval and deformation.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 11713–11722, 2021.
  * [58] Weiyue Wang, Duygu Ceylan, Radomir Mech, and Ulrich Neumann.  3dn: 3d deformation network.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1038–1046, 2019.
  * [59] Maurice Weiler, Mario Geiger, Max Welling, Wouter Boomsma, and Taco S Cohen.  3d steerable cnns: Learning rotationally equivariant features in volumetric data.  Advances in Neural Information Processing Systems, 31, 2018.
  * [60] Haozhe Xie, Hongxun Yao, Xiaoshuai Sun, Shangchen Zhou, and Shengping Zhang.  Pix2vox: Context-aware 3d reconstruction from single and multi-view images.  In Proceedings of the IEEE/CVF international conference on computer vision, pages 2690–2698, 2019.
  * [61] Qiangeng Xu, Zexiang Xu, Julien Philip, Sai Bi, Zhixin Shu, Kalyan Sunkavalli, and Ulrich Neumann.  Point-nerf: Point-based neural radiance fields.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 5438–5448, 2022.
  * [62] Guandao Yang, Xun Huang, Zekun Hao, Ming-Yu Liu, Serge Belongie, and Bharath Hariharan.  Pointflow: 3d point cloud generation with continuous normalizing flows.  In Proceedings of the IEEE/CVF international conference on computer vision, pages 4541–4550, 2019.
  * [63] Shuo Yang, Min Xu, Haozhe Xie, Stuart Perry, and Jiahao Xia.  Single-view 3d object reconstruction from shape priors in memory.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3152–3161, 2021.
  * [64] Wang Yifan, Noam Aigerman, Vladimir G Kim, Siddhartha Chaudhuri, and Olga Sorkine-Hornung.  Neural cages for detail-preserving 3d deformations.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 75–83, 2020.
  * [65] Michela Zaccaria, Fabian Manhardt, Yan Di, Federico Tombari, Jacopo Aleotti, and Mikhail Giorgini.  Self-supervised category-level 6d object pose estimation with optical flow consistency.  IEEE Robotics and Automation Letters, 8(5):2510–2517, 2023.
  * [66] Guangyao Zhai, Xiaoni Cai, Dianye Huang, Yan Di, Fabian Manhardt, Federico Tombari, Nassir Navab, and Benjamin Busam.  Sg-bot: Object rearrangement via coarse-to-fine robotic imagination on scene graphs.  arXiv preprint arXiv:2309.12188, 2023.
  * [67] Guangyao Zhai, Dianye Huang, Shun-Cheng Wu, HyunJun Jung, Yan Di, Fabian Manhardt, Federico Tombari, Nassir Navab, and Benjamin Busam.  Monograspnet: 6-dof grasping with a single rgb image.  In 2023 IEEE International Conference on Robotics and Automation (ICRA), pages 1708–1714. IEEE, 2023.
  * [68] Guangyao Zhai, Evin Pinar Örnek, Shun-Cheng Wu, Yan Di, Federico Tombari, Nassir Navab, and Benjamin Busam.  Commonscenes: Generating commonsense 3d indoor scenes with scene graphs.  arXiv preprint arXiv:2305.16283, 2023.
  * [69] Cheng Zhang, Zhaopeng Cui, Yinda Zhang, Bing Zeng, Marc Pollefeys, and Shuaicheng Liu.  Holistic 3d scene understanding from a single image with implicit representation.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 8833–8842, 2021.
  * [70] Chenyangguang Zhang, Yan Di, Ruida Zhang, Guangyao Zhai, Fabian Manhardt, Federico Tombari, and Xiangyang Ji.  Ddf-ho: Hand-held object reconstruction via conditional directed distance field.  arXiv preprint arXiv:2308.08231, 2023.
  * [71] Ruida Zhang, Yan Di, Zhiqiang Lou, Fabian Manhardt, Federico Tombari, and Xiangyang Ji.  Rbp-pose: Residual bounding box projection for category-level pose estimation.  In European Conference on Computer Vision, pages 655–672. Springer, 2022.
  * [72] Ruida Zhang, Yan Di, Fabian Manhardt, Federico Tombari, and Xiangyang Ji.  Ssp-pose: Symmetry-aware shape prior deformation for direct category-level object pose estimation.  In 2022 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 7452–7459. IEEE, 2022.
  * [73] Zihan Zhu, Songyou Peng, Viktor Larsson, Weiwei Xu, Hujun Bao, Zhaopeng Cui, Martin R Oswald, and Marc Pollefeys.  Nice-slam: Neural implicit scalable encoding for slam.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 12786–12796, 2022.


# Absolute frequency measurement of the $6s^{2}~{}^{1}S_{0}\to
6s6p~{}^{3}P_{1}$ $F=3/2\to F^{\prime}=5/2$ ${}^{\text{201}}$Hg transition
with background-free saturation spectroscopy

Adam Linek  Piotr Morzyński  and Marcin Witkowski  Institute of Physics,
Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University,
Grudzia̧dzka 5, PL-87-100 Toruń, Poland<EMAIL_ADDRESS>
###### Abstract

We report the development of a method for eliminating background-induced
systematic shifts affecting precise measurements of saturation absorption
signals. With this technique, we measured the absolute frequency of the
$6s^{2}~{}^{1}\text{S}_{0}\to 6s6p~{}^{3}\text{P}_{1}$ transition in
${}^{201a}\text{Hg}$ ($F=3/2\to F^{\prime}=5/2$) to be $1181541111051(83)$
kHz. The measurement was referenced with an optical frequency comb
synchronized to the frequency of the local representation of the UTC. This
specific atomic line is situated on the steep slope of the Doppler background
at room temperature, which results in a frequency systematic shift. We
determined the dependence of this shift on the properties of both the spectral
line and the background of the measured signal.

††journal: oe

##  1 Introduction

Saturation spectroscopy is a powerful Doppler-free technique providing
detailed information about the electronic structure of atoms through very
precise frequency measurements of the atomic transitions. The method is the
basis for various laser frequency stabilization and locking techniques [1, 2,
3, 4, 5, 6, 7]. The saturation spectrum’s high-frequency resolution is
achieved thanks to a significant reduction of the Doppler broadening. However,
the spectral lines obtained with the Doppler-free saturation spectroscopy are
often situated on a non-flat Doppler background, which limits the precision of
the measurements. This limitation is because the signal’s maximum no longer
corresponds to the atomic transition frequency. This frequency shift is
usually estimated from the fit to the measured line and background profiles.
Such treatment is sufficient enough for high-quality data and fitting models.
However, the low signal-to-noise ratio measurements result in high
uncertainties of the fitting parameters, which makes this method inefficient.

In this paper, we present a novel method stemming from saturation spectroscopy
but modified by employing a new way to deal with the undesirable background-
induced systematic shift of the atomic transition frequency. The technique is
fairly insensitive to temperature changes, which entail changes in the Doppler
profile and, thus, in the shape of the measured signal’s background. Using
this method, we measured the absolute frequency of the
$6s^{2}~{}^{1}\text{S}_{0}\to 6s6p~{}^{3}\text{P}_{1}$ transition in
${}^{201a}\text{Hg}$ ($F=3/2\to F^{\prime}=5/2$) with a room-temperature
mercury vapor cell. This line lies on the steep slope of the Doppler profile
at room temperature (Fig. 1), which made it possible to exploit the method’s
advantages fully. The uncertainty budget of the measurement includes all
significant frequency shifting effects such as AC-Stark, Zeeman shift, and
pressure-dependent collisional shift.

Figure 1: The saturation spectroscopy of the 1S0$\to$3P${}_{1}~{}$transition
in the room-temperature mercury vapor cell. Five Doppler-broadened profiles of
the mercury isotopes with sub-Doppler Lamb dips are visible. At room
temperature, the Doppler-broadened profiles are of $\sim 1$ GHz widths, much
more than the isotopic shifts of some of the Lamb dips. The overlapping leads
to situating the sub-Doppler spectral lines at the slopes, while the one for
the 201aHg (shown in the inset) is the steepest.

The study of spectral lines of mercury has been a subject of interest for over
a century [8, 9, 10, 11, 12, 13, 14, 15, 16]. Although the mercury’s
intercombination 1S0$\to$3P${}_{1}~{}$line has been the subject of numerous
spectroscopic studies (this applies mainly to isotope 198Hg), the $F=3/2\to
F^{\prime}=5/2$ ${}^{\text{201}}$Hg transition, to our knowledge, has not been
measured directly. There were, however, reported results [11] of its isotope
shift relative to 198Hg (measured in [17]), from which one can infer the
frequency value to be $1181541128(17)$ MHz. This value agrees with our result
$1181541111051(83)$ kHz within the uncertainty. The advent of high-power diode
lasers and a non-linear frequency conversion paved the way for more precise
spectroscopic measurements of mercury [18] and enabled the exploration of new
areas of fundamental research based on the mercury atom [19, 20, 21, 22, 23,
24]. Recently, the isotopically resolved Hg spectrum of the
${}^{1}S_{0}\to~{}^{3}P_{1}$ intercombination transition was used for
developing a new method for measuring the gaseous elemental mercury
concentration in the air [25]. Our measurement enriches this spectrum with a
new value, which will increase the method’s accuracy.

##  2 Experimental setup

The experimental setup scheme is shown in Fig. 2. The laser system (Toptica
TA-FHG Pro) is based on a $1016$ nm laser diode in an external resonator
configuration (ECDL). The power of the infrared light is amplified up to $2$ W
with the tapered amplifier (TA). The frequency is doubled twice, resulting in
the ultraviolet 254 nm laser beam with a power of up to $120$ mW. The typical
laser beam intensity used in the experiment is $140$ mW/cm2.

Figure 2: Scheme of the experimental setup consisting of a laser source
(Toptica TA-FHG Pro), an optical frequency comb, and a saturation spectroscopy
setup. The infrared 1016 nm laser beam generated by an ECDL is referenced to
the optical frequency comb. The frequency of IR light is doubled twice to
obtain the UV light at a wavelength of $254$ nm. The spectroscopy setup
consists of a spectroscopic mercury cell, an AOM2 in double-pass
configuration, and two independent power stabilization systems for the pumping
and probing beams.

To stabilize and measure the frequency of the ECDL, an optical frequency comb
is used. After amplification and frequency shifting by an acousto-optic
modulator (AOM1), a part of the infrared light is spatially superimposed with
the comb output beam and sent to an avalanche photodiode (APD) for the
frequency beat detection. The beat note signal is mixed with the RF signal
from the generator (GEN) so that the frequency of the resulting signal is
down-converted and adapted to the operating range of the RF counter. Its
readings are used to determine the absolute frequency of infrared, and thus
ultraviolet, laser light. In addition, the mixed RF signal is provided to a
digital phase detector, which outputs a DC signal proportional to the phase
difference between the mixed RF signal and a reference $20.5$ MHz signal. The
DC output signal is provided to the PID controller (Toptica FALC110), which
generates a signal modulating the ECDL’s current and its piezo voltage. With
this approach, the infrared laser light is frequency-narrowed to about $100$
kHz and locked to the optical frequency comb mode. To determine the comb mode
number, a small fraction of the infrared light is guided to the wavelength
meter, whose uncertainty ($60$ MHz) is a few times smaller than the comb
repetition frequency ($250$ MHz).

Ultraviolet light is sent to the spectroscopy system. Using the saturation
effect of the atomic transition [26], a sub-Doppler Lamb dip is observed.
While the spectral line is free from the Doppler broadening, its width is a
few times larger than the natural linewidth due to power broadening,
transient-time broadening, and pressure broadening. The spectroscopy signal is
used as an input of the frequency stabilization system, which tunes the UV
frequency to the center of the atomic line. The frequency shifting is realized
with an acousto-optic modulator (AOM2), whose RF driving signal is generated
by a µC-operated direct digital synthesizer (DDS). The UV frequency-shifted
beam is separated with a Glan-Taylor prism (GTP1) and split into another two
beams by using the prism (GTP2). These two beams act as the pumping and
probing beams in the spectroscopy system. Their power ratio can be chosen with
a half-wave plate in front of the GTP2. We applied both beams’ independent
power-lock systems with PD1 and PD2 to ensure good power stability. For
typical experimental conditions, the relative power stability is at the level
of $8.3$ ppm and $6.5$ ppm for the pumping and probing beam, respectively. To
eliminate the influence of the Doppler background, we periodically block the
pumping beam, which disables the saturation of the atomic transition and
measure the background level, which is then digitally subtracted. Pumping and
probing beams are sent through AR-coated wedged fused silica windows to a 1 mm
thick cylindrical absorption cell containing Hg vapors. The cell is attached
to a servo that allows its rotation, thus changing the length of the optical
path of the probe beam inside the cell. The saturated absorption spectroscopy
signal is detected with a photodiode (PD3) and sent to the microcontroller,
where the UV frequency correction is calculated (see section 3).

To account for the influence of the Zeeman effect and collision-induced
shifts, the spectroscopic cell was placed inside a spectroscopic chamber shown
in Fig. 3.

Figure 3: Quarter-section of the spectroscopic chamber design. Two aluminum
chambers surround the spectroscopic. The temperature stabilization is realized
with two Peltier modules below the inner chamber. The excessive heat is
removed from the system by coolant flowing through a copper plate. The cell is
surrounded by eight offset-compensated 3-axis magnetometers enabling magnetic
field and magnetic gradient measurements. The chamber is equipped with six
coils in three independent Helmholtz arrangements to compensate for a stray
magnetic field. UV light passes through the chamber via four AR-coated wedged
fused silica windows.

The chamber enables thermal stabilization at the level of $0.1$$~{}^{\circ}$C.
The temperature reading is based on a measurement of a Pt100’s resistance with
the use of an analog-digital converter. The model used for translating the
resistance into the temperature extends its uncertainty to $0.5~{}^{\circ}$C.
The chamber is equipped with three pairs of coils in Helmholtz configuration
to compensate for a stray magnetic field. Eight 3-axis offset-compensated
magnetometers distributed around the cell enable both magnetic field strength
(with the uncertainty of a few µT) and magnetic gradient measurements.

The whole spectroscopic chamber consists of two aluminum chambers: the inner
and the outer one depicted in Fig. 3 in grey and silver, respectively. The
volume between the inner and outer chambers is pumped out to improve thermal
insulation. The inner chamber is filled with nitrogen, preventing unwanted
water condensation or freezing during chamber cooling.

The temperature stabilization is realized with two Peltier modules below the
inner chamber. The excessive heat is removed through the water-cooled copper
plate. The temperature of the inner chamber is measured using a Pt100
temperature sensor attached to its wall. This provides feedback to the PI
controller.

##  3 Method

To account for the background-induced frequency shift, we improved a digital
lock method (DLM), which has proven to work very well for symmetric spectral
lines [27, 28, 18]. The scheme of this method is shown in Fig. 4.

Figure 4: Scheme of the DLM. The frequency of the laser light is shifted
cyclically by $\Delta f$ towards lower and higher values corresponding to the
opposite slopes of the sub-Doppler spectral line obtained with the pumping
beam enabled. After each frequency jump, the spectroscopic signal is measured,
and the difference between two consecutive values $\varepsilon=I_{1}-I_{2}$ is
used to calculate the frequency correction by which the $f_{\text{AOM}}$ is
shifted after each cycle.

The DLM is based on cyclical switching of the laser frequency between two
values $f_{\text{UV}}+f_{\text{AOM}}-\Delta f$ and
$f_{\text{UV}}+f_{\text{AOM}}+\Delta f$, which are related to opposite slopes
of a measured line. After each frequency jump, the spectroscopy signal $I$ is
measured by a photodiode. The difference between two consecutive values
$(\varepsilon=I_{1}-I_{2}=I_{f_{\text{UV}}+f_{\text{AOM}}-\Delta
f}-I_{f_{\text{UV}}+f_{\text{AOM}}+\Delta f})$ is used to calculate the
frequency correction $f_{corr}$

$f_{corr}=\left(I_{f_{\text{UV}}+f_{\text{AOM}}-\Delta
f}-I_{f_{\text{UV}}+f_{\text{AOM}}+\Delta f}\right)\cdot\text{gain},$ (1)

by which the $f_{\text{AOM}}$ is shifted after each cycle. The µC calculates
the frequency correction after every two frequency jumps. For the symmetric
line, the laser frequency lying exactly at the center of the measured profile
results in $\varepsilon=0$, thus also $f_{corr}=0$. With this loop, the laser
frequency $\left(f_{\text{UV}}+f_{\text{AOM}}\right)$ follows the spectral
line’s maximum position.

The DLM works very well for symmetrical profile. However, it may lie on a
slope when a spectral line shares a Doppler-broadened absorption profile with
other spectral lines, such as those from other isotopes. The steeper the
slope, the more systematic error will be introduced by the DLM (we describe
this effect in detail in section 5). To solve this issue, we developed the
modified digital lock method (mDLM). The scheme of the mDLM is depicted in
Fig. 5.

Figure 5: Scheme of the mDLM. At the beginning of the cycle, the pumping beam
is disabled, and the background-only measurement (dashed red line) is
performed at $f_{\text{UV}}+f_{\text{AOM}}\pm\Delta f$, giving
$I_{BG,1},I_{BG,2}$. Then, the pumping beam is switched on and the
spectroscopic signal of the sub-Doppler spectral line (solid red line) is
measured for the same frequencies, giving $I_{1}$ and $I_{2}$. The differences
$\Delta_{1}=I_{1}-I_{BG,1}$ and $\Delta_{2}=I_{2}-I_{BG,2}$ are used to
calculate the frequency correction by which the $f_{\text{AOM}}$ is shifted.

As a significant improvement, we introduced the on-the-fly background-
eliminating scheme based on disabling the saturation of the atomic transition.
It is managed by repeating the frequency jumping cycle with the blocked
pumping beam. Once the pumping beam is disabled, the Lamb dip disappears, and
only the Doppler-broadened absorption profile remains. The cycle of two
frequency jumps and signal measurements is repeated, but this time it
corresponds only to the background levels, subtracted from the previously
obtained spectral line measurements. As a result, we get two values
$I_{BG,1}$, $I_{BG,2}$ related to the opposite background-free slopes and use
them to calculate the frequency correction $f_{corr}$

$f_{corr}=\left(\Delta_{1}-\Delta_{2}\right)\cdot\text{gain},$ (2)

where

$\begin{split}\Delta_{1}&=I_{1}-I_{BG,1}=I^{\text{line+bg}}_{f_{\text{UV}}+f_{\text{AOM}}-\Delta
f}-I^{\text{bg}}_{f_{\text{UV}}+f_{\text{AOM}}-\Delta f},\\\
\Delta_{2}&=I_{2}-I_{BG,2}=I^{\text{line+bg}}_{f_{\text{UV}}+f_{\text{AOM}}+\Delta
f}-I^{\text{bg}}_{f_{\text{UV}}+f_{\text{AOM}}+\Delta f}.\end{split}$

The $I^{\text{line+bg}}$ and $I^{\text{bg}}$ correspond to the values measured
in the presence and absence of the pumping beam, respectively.

Switching off the pumping beam is accomplished with a mechanical shutter
controlled by a TTL signal from the µC. The full-closing time of the shutter
is $5$ ms. The time frame of the entire measurement cycle of the mDLM is shown
in Fig. 6

Figure 6: The time frame of the mDLM. At the beginning of the cycle, the
shutter is closed, and the background level measurements are performed
consecutively for the left and right slopes of the spectral line. Next, the µC
sends the opening signal to the shutter controller. The full-opening time is
$5$ ms. Once the shutter is fully opened, the spectral line measurements are
performed, and the frequency correction is calculated and applied. Afterward,
the shutter is closed, and the cycle is repeated.

##  4 Microcontroller and direct digital synthesizer

A home-made digital device (Fig. 7) consisting of a direct digital synthesizer
Analog Devices AD9959 evaluation board (DDS) and a Kamami ZL26ARM evaluation
board with an STM32F107 microcontroller (µC) were used to precisely control
the frequency of the RF signal supplied to the acousto-optical modulator and
to implement the frequency correction algorithm described in section 3. The
reference clock for DDS is a 100 MHz signal coming from UTC (AOS) [29, 30].
The DDS includes four output RF channels digitally tunable from 0 MHz to 210
MHz with around 100 mHz resolution. All DDS parameters are set by the µC
through the SPI interface, allowing the output frequency to be switched in a
microsecond which is negligible compared to the cycle time. The opening and
closing of the shutter are controlled by the TTL signal from the µC digital
output. The signal is sent to the TTL input of the shutter driver.

Figure 7: Scheme of the digital device for background-free spectroscopy
control. The microcontroller executes the frequency correction algorithm and
communicates with peripheral devices: ADC - analog-digital converter for
measuring the signal from the photodetector, DDS - direct digital synthesizer
for precise frequency control of AOMs, digital ports for sending TTL signals,
e.g., to the shutter. Ethernet communication acts as a user interface with the
device.

The signal from the photodetector (PD3 in Fig. 2) is measured using a 12-bit
ADC converter built in the µC. The reading from the photodiode is averaged
over the multiple measurements (about a thousand points per 1 ms) to improve
the signal-to-noise ratio.

The digital background-free locking algorithm is implemented in a µC program
written in C language. The program can also be switched to the background-free
or background-included scan mode to observe the line profile and determine the
initial lock parameters. All parameters of the µC operation are set through
commands sent via the Ethernet network. The µC can also save user-selected
parameters and measurements and share them via Ethernet.

##  5 Background-induced systematic shift

If the separation of the spectral lines is small compared to their Doppler
widths, the absorption profiles overlap, and the associated Lamb dips are no
longer situated at the profile center but on the slope. Consequently, the peak
of the Lamb dip is frequency-shifted according to the atomic resonant
frequency. To analyze this shift, we model the measured atomic transition as a
Lorentz profile with a linear background. The latter is justified since the
width of the Doppler-broadened absorption profile is much larger than the
linewidth of the atomic transition. The observed spectral line intensity
profile can be then written as

$I(f)=\frac{A}{1+\left[\frac{2\left(f-f_{0}\right)}{w}\right]^{2}}+a(f-f_{0})+b,$
(3)

where $A$ and $w$ are the amplitude and the width of the Lorentz profile,
respectively, $f-f_{0}$ is the detuning from the atomic resonant frequency,
and $a$, together with $b$, is the linear slope parameters. In the case of the
DLM, the value $f_{\text{DLM}}$ at which the light frequency is stabilized
does not correspond to the resonant frequency $f_{0}$ of the atomic transition
but is shifted instead. Moreover, this shift depends on the jumps magnitude
$\Delta f$ in the frequency correction cycles. To calculate the frequency
shift ($f_{\text{DLM}}-f_{0}$) as a function of $\Delta f$, one can solve the
equation

$I(f+\Delta f)=I(f-\Delta f),$ (4)

which can be reduced to

$\alpha_{5}(f-f_{0})^{4}+\alpha_{3}(f-f_{0})^{2}+\alpha_{2}(f-f_{0})+\alpha_{1}=0,$
(5)

where

$\begin{split}\alpha_{5}&=16,\\\ \alpha_{3}&=8w^{2}-32\Delta f^{2},\\\
\alpha_{2}&=-\frac{8w^{2}A}{a},\\\ \alpha_{1}&=w^{4}+8w^{2}\Delta
f^{2}+16\Delta f^{4}.\end{split}$

The dependence of the frequency $f_{\text{DLM}}$ satisfying the Eq. (4) on
$\Delta f$ is elaborate but for $\Delta f<w$ can be approximated as

$f_{\text{DLM}}(\Delta f)=\beta_{4}(\Delta f)^{4}+\beta_{2}(\Delta
f)^{2}+\beta_{1},$ (6)

where $\beta_{i}$ consist of the profile and slope parameters.

The background introduces a systematic error that cannot be avoided within
DLM, even with infinitesimally small frequency jumps $\Delta f$. The mDLM
solves this issue as the spectral line with the slope background (Fig. 8)

Figure 8: Sub-Doppler spectral line profile obtained with the saturation
spectroscopy of the 1S0$\to$3P${}_{1}~{}$transition in 201aHg. This line is
situated on the slope arising from sharing the Doppler-broadened absorption
profile with neighboring 199aHg and 204Hg spectral lines. The results are
averaged over fifteen scans and fit according to Eq. 3. The bottom part of the
figure shows the residuals. Figure 9: Sub-Doppler spectral line profile
obtained with the saturation spectroscopy of the
1S0$\to$3P${}_{1}~{}$transition in 201aHg with linear background subtracted
on-the-fly by the µC. Subtracting the background makes the spectral line seen
as symmetric. The results are averaged over fifteen scans and fit with the
Lorentzian profile. The bottom part of the figure shows the residuals.

is seen to be symmetric due to on-the-fly background subtracting (Fig. 9). To
validate the equation (6), the dependence of the frequency at which the laser
light was stabilized as a function of the frequency jump $\Delta f$ was
measured for DLM and mDLM (Fig. 10) with the 1S0$\to$3P${}_{1}~{}$201aHg
transition.

Figure 10: Dependence of the relative frequency at which the laser light was
stabilized on $\Delta f$ in reference to mDLM average value (solid red line).
Square points (DLM) are fit according to Eq. (6) (dashed black line) and to
the solution of Eq. (4) (dashed red line). The mDLM results (cross points) do
not depend on the $\Delta f$.

For the DLM, unlike the mDLM case, the frequency value shifts rapidly with the
frequency jump. Moreover, even for a small $\Delta f$, the discrepancy
$\Delta_{\text{DLM-mDLM}}$ between results obtained with both methods
persists. In general, this deviation depends on the ratio $a/A$ and the
linewidth $w$. Typically, $\Delta_{\text{DLM-mDLM}}$ is much smaller than the
linewidth. Hence, for the Lorentzian profile, the deviation can be
approximated as

$\Delta_{\text{DLM-mDLM}}\approx\frac{aw^{2}}{8A}.$ (7)

For our specific experimental conditions, the discrepancy shown in Fig. (10)
is of $280$ kHz.

##  6 µC software parameters

In the mDLM, the frequency correction $f_{corr}$ is calculated by the µC
according to equation (2). The gain has to be adjusted accordingly to ensure
the frequency correction is fast enough to keep up with the frequency drifts
of the spectral line and the laser light.

Figure 11: Dependence of the relative frequency at which the laser light was
stabilized on the gain in reference to the mDLM average value (solid line).
For high gain, the frequency corrections are too extensive, causing big
uncertainties. Too low gains make the system unable to keep up with the
frequency drifts of the laser light.

Fig. 11 shows the dependence of the frequency at which the laser light was
stabilized on the gain. For high gain, the frequency correction $f_{corr}$ is
too extensive, which leads to high uncertainties. Too small gain makes the
system unable to keep up with the frequency drifts, which results in
scattering the results in the tens of kHz range.

As shown in Fig. 10, in the mDLM the frequency at which the laser light
stabilizes does not depend on the frequency jump $\Delta f$. Despite this, the
parameter $\Delta f$ should be adjusted according to the spectral linewidth
$w$. For the optimal frequency jump, the sensitivity of the mDLM is the
highest. This is the case for the steepest parts of the slopes being probed.
It corresponds to the maximum of the first derivative of the spectral line
profile. For the Lorentzian profile, the optimal $\Delta f=w/\sqrt{12}$. For
this specific value, the deviation of the laser frequency from the spectral
line’s resonance results in the largest response of the correction system,
shown in Fig. 12.

Figure 12: Dependence of the laser light stabilizing frequency uncertainty on
the frequency jump. The first derivative of a Lorentz profile is fit to the
collected data. The maximum of the fit corresponds to the $\Delta
f=w/\sqrt{12}$ (dashed line), at which the mDLM sensitivity is the highest.

##  7 Absolute frequency

To validate the mDLM performance, we used it to determine the absolute
frequency of the 1S0$\to$3P${}_{1}~{}$transition in 201aHg ($F=3/2\to
F^{\prime}=5/2$), located on the steepest slope of the Doppler profile among
all Hg isotopes at room temperature. The determined absolute frequency is
$1181541111051(83)$ kHz. To obtain this value, the optical frequency comb was
used and the error budget was measured. The total uncertainty was calculated
according to the following formula

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}u(f_{\text{abs}})=\sqrt{u_{\text{stat}}^{2}+u(\Delta_{\text{AC-
Stark}})^{2}+u(\Delta_{\text{Hg-
Hg}})^{2}+u(\Delta_{\text{Hg-H}_{2}})^{2}+\Sigma^{3}_{i=1}u(\Delta_{\text{Zeeman}-i})^{2}},}$
(8)

where $u_{\text{stat}}=3.9$ kHz is the statistical uncertainty of the line
position measured under known and controllable experimental conditions,
$\Delta_{\text{AC-Stark}}$ is the AC-Stark frequency shift, $\Delta_{\text{Hg-
Hg}}$ and $\Delta_{\text{Hg-H}_{2}}$ are the pressure shifts, and
$\Delta_{\text{Zeeman}-i}$ are Zeeman shifts of the $x$, $y$, and $z$-axis,
respectively. All the components are described in the following subsections
and the values are summarized in Table 1.

###  7.1 AC-Stark shift

The energy levels of the atoms are shifted under the influence of the electric
field of the laser beam. To estimate the contribution of this effect, we
measured the position of the spectral line for different UV light power (Fig.
13). A linear fit weighted by the points’ uncertainties yields the AC-Stark
shift correction factor of $-148.6(9.4)$ kHz/mW.

The uncertainties are mainly related to the accuracy of the power meter used
(Thorlabs S130VC) and the power measuring method. The total uncertainty is
deduced to be $5.4\%$.

Figure 13: Dependence of the spectral line position on the total power of the
UV beams. The solid red line is a weighted fit to the results. The AC-Stark
shift is deduced from the slope to be $-148.6(9.4)$ kHz/mW.

###  7.2 Pressure shift

The pressure-dependent collisions between Hg atoms and residual gases affect
the atomic transition frequency. Since the vapor pressure of Hg depends on the
temperature, the spectroscopic cell temperature was actively stabilized. The
average temperature of the cell during measurements was
$19.97(50)~{}^{\circ}$C, which corresponds to $0.1708(75)$ Pa vapor pressure
[31]. The temperature uncertainty is limited by the accuracy of the analog-
digital converter used in the Pt100-based temperature measurement scheme.

We used the same approach as presented in [18] to estimate a collision-induced
shift. The collisions between mercury atoms shift the transition frequency by
$-22(22)$ kHz/Pa. The contribution from the collisions with residual gases was
estimated, assuming molecular hydrogen as a dominant component, which is
consistent with the fused silica cell residual gases composition [32]. H2
pressure at room temperature is less than $0.13$ Pa, corresponding to the
spectral line shift of $-35(35)$ kHz/Pa.

###  7.3 Zeeman shift

The 201Hg isotope (abundance of $13.18\%$ [33]) is a fermion with non-zero
nuclear spin ($I=3/2$), which leads to hyperfine splitting of the
${}^{3}P_{1}$ state into three states ($F=5/2,3/2,1/2$) [34]. The $F=3/2\to
F^{\prime}=5/2$ transition is the measured one. The ground state
${}^{1}S_{0},F=3/2$ splitting is negligible in a weak magnetic field as the
nuclear Landé factor is much smaller than the orbital and the electron ones.
The excited state ${}^{3}P_{1},F=5/2$ splits into six substates characterized
by magnetic quantum number $m_{F}$. To a first approximation, the linear
Zeeman effect splits the sublevels symmetrically and therefore has no
contribution to the absolute frequency shift. On the other hand, the quadratic
Zeeman effect does not cause splitting but can produce a non-negligible shift
of the hyperfine levels.

To estimate the Zeeman shift, the spectral line position was independently
measured for different magnetic fields for each axis. The external magnetic
field of the Helmholtz coils was varied over a range that did not split the
spectral line measurably. The results corresponding to the vertical ($x$-axis,
Fig. 14) and horizontal ($z$-axis, Fig. 15), both perpendicular to the UV
beams, manifest significant quadratic and linear Zeeman effects.

Figure 14: Dependence of the spectral line position on the magnetic field in
the $x$-axis (vertical, perpendicular to the UV beams). The solid red curve is
a weighted parabola fit to the results. The quadratic and linear Zeeman shifts
are deduced to be $-0.0211(13)$ kHz/µT2 and $0.07(16)$ kHz/µT, respectively.
Figure 15: Dependence of the spectral line position on the magnetic field in
the $z$-axis (horizontal, perpendicular to the UV beams). The solid red curve
is a weighted fit to the results. The quadratic and linear Zeeman shifts are
deduced to be $0.0170(15)$ kHz/µT2 and $0.89(22)$ kHz/µT, respectively.

For the axial magnetic field component (Fig. 16), a very large shift, which
exhibits only a linear dependence in the measured range, is observed.

Figure 16: Dependence of the spectral line position on the magnetic field in
the $y$-axis (along UV beams). The solid red line is a weighted fit to the
results. The linear Zeeman shift is deduced to be $-18.677(53)$ kHz/µT

The magnetic field was measured by eight 3-axis offset-compensated
magnetometers located symmetrically around the spectroscopic cell. Each
reading has an uncertainty related to the accuracy of the offset calibration.
Each point in the graphs corresponds to the average value of the
magnetometers’ readings for a given axis. The variance of the readings
specifies the uncertainties of the points.

To account for the Zeeman effect, the magnetic field and its gradient were
measured during the absolute frequency measurement. The uncertainty-weighted
sensors’ readings were averaged, resulting in six magnetic field values, one
for each side of the spectroscopic cell. The uncertainty of these results was
deduced from the calculated internal and external variances, where the larger
of the two was chosen. The average of pairs of points along the same axis
determines the value of the magnetic field at the center of the spectroscopic
cell, which was calculated to be $43.6(3.6)$ µT, $-11.6(4.3)$ µT, and
$-7.46(0.77)$ µT for x, y, and z-axis, respectively.

Table 1: Systematic shifts and their uncertainties determined for the
experimental conditions, i.e., UV beams total power of $0.440(24)$ mW,
spectroscopic cell temperature of $19.97(50)~{}^{\circ}$C, and magnetic field
of $43.6(3.6)$ µT, $-11.6(4.3)$ µT and $-7.46(0.77)$ µT for x, y, and z-axis
respectively. All results are in kHz. Effect
---
AC-Stark | Pr. shift Hg-Hg | Pr. shift Hg-H2 | Zeeman x-axis | Zeeman y-axis | Zeeman z-axis
65.4(5.4) | 3.8(3.8) | 4.6(4.6) | 37.1(9.8) | -217(80) | 5.7(1.7)

##  8 Conclusions

In summary, we demonstrated a novel method to stabilize the frequency of laser
light on the position of a spectral line. The method is especially beneficial
in the case of the atomic lines located on the slope of the absorption
profile, which induces a systematic frequency shift. We derived a formula to
calculate the background-induced shift, which was then compared with the
experimental results, showing full agreement. To demonstrate the performance
of our method, we performed a measurement of the absolute frequency of the
1S0$\to$3P${}_{1}~{}$201aHg improving the accuracy by three orders of
magnitude. Our result agrees with the previously measured value within the
uncertainty.

## Funding

Narodowe Centrum Nauki (2021/42/E/ST2/00046, 2021/41/B/ST2/00681); European
Metrology Programme for Innovation and Research (20FUN01 TSCAC).

## Acknowledgments

This project 20FUN01 TSCAC has received funding from the EMPIR programme co-
financed by the Participating States and from the European Union’s Horizon
2020 research and innovation programme. This work was supported by the
National Science Centre, Poland Project no. 2021/42/E/ST2/00046. A. Linek is
supported by the National Science Centre, Poland, Project no.
2021/41/B/ST2/00681.

The measurements were performed as a part of research done at the National
Laboratory FAMO (KL FAMO) in Toruń, Poland. KL FAMO in Toruń is supported by
the subsidy of the Ministry of Science and Higher Education. The authors would
like to thank Roman Ciuryło for fruitful discussions. The authors also thank
Mode-Locked Technology Sp. z o.o. for providing the digital phase detector for
laser frequency stabilization.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are available in the open
data repository [35].

## References

  * [1] R. K. Raj, D. Bloch, J. J. Snyder, G. Camy, and M. Ducloy, “High-Frequency Optically Heterodyned Saturation Spectroscopy Via Resonant Degenerate Four-Wave Mixing,” Phys. Rev. Lett. 44, 1251–1254 (1980).
  * [2] G. C. Bjorklund, “Frequency-modulation spectroscopy: a new method for measuring weak absorptions and dispersions,” Opt. Lett. 5, 15–17 (1980).
  * [3] J. E. Debs, N. P. Robins, A. Lance, M. B. Kruger, and J. D. Close, “Piezo-locking a diode laser with saturated absorption spectroscopy,” Appl. Opt. 47, 5163–5166 (2008).
  * [4] E. van Ooijen, G. Katgert, and P. van der Straten, “Laser frequency stabilization using Doppler-free bichromatic spectroscopy,” Applied Physics B 79, 57–59 (2004).
  * [5] G. Wasik, W. Gawlik, J. Zachorowski, and W. Zawadzki, “Laser frequency stabilization by Doppler-free magnetic dichroism,” Applied Physics B 75, 613–619 (2002).
  * [6] L. Krzemień, K. Brzozowski, A. Noga, M. Witkowski, J. Zachorowski, M. Zawada, and W. Gawlik, “Laser frequency stabilization by magnetically assisted rotation spectroscopy,” Optics Communications 284, 1247–1253 (2011).
  * [7] M. Aldous, J. R. C. Woods, A. Dragomir, R. Roy, and M. Himsworth, “Carrier frequency modulation of an acousto-optic modulator for laser stabilization.” Optics express 25 11, 12830–12838 (2017).
  * [8] H. Kayser and C. Runge, “Ueber die Spectra der Elemente der zweiten Mendelejeff’schen Gruppe,” Annalen der Physik 279(6), 385–409 (1891).
  * [9] C. E. Moore, _Atomic Energy Levels_ , vol. 3 (U.S. Government Printing Office, 1958).
  * [10] C. V. Stager and R. H. Kohler, “Hyperfine structure of Hg199 and Hg201 in the 3P1 state,” Bull. Am. Phys. Soc. Ser. II 5(4), 274 (1960).
  * [11] W. G. Schweitzer, “Hyperfine structure and isotope shifts in the 2537-Å line of mercury by a new interferometric method,” J. Opt. Soc. Am. 53(9), 1055–1072 (1963).
  * [12] J. Reader, C. H. Corliss, W. L. Wiese, and G. A. Martin, _Wavelengths and transition probabilities for atoms and atomic ion_ (U.S. Government Printing Office, 1980).
  * [13] R. H. Kohler, “Detection of double resonance by frequency change: application to Hg201,” Phys. Rev. 121(4), 1104–1111 (1961).
  * [14] E. B. Saloman, “Wavelengths, energy level classifications, and energy levels for the spectrum of neutral mercury,” J. Phys. Chem. Ref. Data 35(4), 1519–1548 (2006).
  * [15] C. J. Sansonetti and D. Veza, “Doppler-free measurement of the 546 nm line of mercury,” Journal of Physics B: Atomic, Molecular and Optical Physics 43, 205003 (2010).
  * [16] A. E. Kramida, “Re-optimized energy levels and Ritz wavelengths of (198)Hg I,” J. Res. Natl. Inst. Stand. Technol. 116(2), 599–619 (2011).
  * [17] W. G. Schweitzer, “Wavelength Passed by Hg198 Zeeman-Split Absorption Filter,” J. Opt. Soc. Am. 53(9), 1250–1252 (1963).
  * [18] M. Witkowski, G. Kowzan, R. Munoz-Rodriguez, R. Ciuryło, P. S. Żuchowski, P. Masłowski, and M. Zawada, “Absolute frequency and isotope shift measurements of mercury 1S0-3P1 transition,” Opt. Express 27, 11069–11083 (2019).
  * [19] M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark, “Search for new physics with atoms and molecules,” Rev. Mod. Phys. 90, 025008 (2018).
  * [20] C. Bouchiat, “Parity violation in atomic processes,” Journal of Physics G: Nuclear Physics 3(2), 183–197 (1977).
  * [21] K. V. P. Latha, D. Angom, B. P. Das, and D. Mukherjee, “Probing CP violation with the electric dipole moment of atomic mercury,” Phys. Rev. Lett. 103, 083001 (2009).
  * [22] M. D. Swallows, T. H. Loftus, W. C. Griffith, B. R. Heckel, E. N. Fortson, and M. V. Romalis, “Techniques used to search for a permanent electric dipole moment of the 199Hg atom and the implications for CP violation,” Phys. Rev. A 87, 012102 (2013).
  * [23] B. Graner, Y. Chen, E. G. Lindahl, and B. R. Heckel, “Reduced limit on the permanent electric dipole moment of 199Hg,” Phys. Rev. Lett. 116, 161601 (2016).
  * [24] E. J. Angstmann, V. A. Dzuba, and V. V. Flambaum, “Relativistic effects in two valence-electron atoms and ions and the search for variation of the fine-structure constant,” Phys. Rev. A 70, 014102 (2004).
  * [25] A. Srivastava and J. T. Hodges, “Development of a High-Resolution Laser Absorption Spectroscopy Method with Application to the Determination of Absolute Concentration of Gaseous Elemental Mercury in Air,” Anal. Chem. 90, 8781–6788 (2018).
  * [26] D. W. Preston, “Doppler-free saturated absorption: Laser spectroscopy,” Am. J. Phys. 64 (1996).
  * [27] P. Morzyński, P. Wcisło, P. Ablewski, R. Gartman, W. Gawlik, P. Masłowski, B. Nagórny, F. Ozimek, C. Radzewicz, M. Witkowski, R. Ciuryło, and M. Zawada, “Absolute frequency measurement of rubidium 5S-7S two-photon transitions,” Opt. Lett. 38, 4581–4584 (2013).
  * [28] P. Morzyński, P. Wcisło, P. Ablewski, R. Gartman, W. Gawlik, P. Masłowski, B. Nagórny, F. Ozimek, C. Radzewicz, M. Witkowski, R. Ciuryło, and M. Zawada, “Line shape measurements of rubidium 5S-7S two-photon transition,” J. Phys. Conf. Ser. 548, 012023 (2014).
  * [29] J. Azoubib, J. Nawrocki, and W. Lewandowski, “Independent atomic timescale in poland—organization and results,” Metrologia 40, S245–S248 (2003).
  * [30] Z. Jiang, A. Czubla, J. Nawrocki, W. Lewandowski, and F. Arias, “Comparing a gps time link calibration with an optical fibre self-calibration with 200 ps accuracy,” Metrologia 52 (2015).
  * [31] M. L. Huber, A. Laesecke, and D. G. Friend, “Correlation for the vapor pressure of mercury,” Ind. Eng. Chem. Res. 45, 7351–7361 (2006).
  * [32] W. Palosz, “Residual gas in closed systems-I: development of gas in fused silica ampoules,” J. Cryst. Growth 267, 475–483 (2004).
  * [33] J. K. Böhlke, J. R. de Laeter, P. D. Bièvre, H. Hidaka, H. S. Peiser, K. J. R. Rosman, and P. D. P. Taylor, “Isotopic compositions of the elements, 2001,” J. Phys. Chem. Ref. Data 34, 57–67 (2005).
  * [34] T. Bernd, “Optical mercury magnetometry for fundamental physics experiments,” Dissertation, Technische Universität München, München (2018).
  * [35] M. Witkowski, A. Linek, and P. Morzyński, “Absolute frequency measurement of the 6s2 1S0 → 6s6p 3P1 F = 3/2 → F’ = 5/2 201Hg transition with background-free saturation spectroscopy,” (2022). https://doi.org/10.18150/PJIBRO.