# SPEAL: Skeletal Prior Embedded Attention Learning for Cross-Source Point
Cloud Registration

Kezheng Xiong1, Maoji Zheng1, Qingshan Xu2, Chenglu Wen1111Corresponding
author., Siqi Shen111footnotemark: 1, Cheng Wang1

###### Abstract

Point cloud registration, a fundamental task in 3D computer vision, has
remained largely unexplored in cross-source point clouds and unstructured
scenes. The primary challenges arise from noise, outliers, and variations in
scale and density. However, neglected geometric natures of point clouds
restrict the performance of current methods. In this paper, we propose a novel
method, termed SPEAL, to leverage skeletal representations for effective
learning of intrinsic topologies of point clouds, facilitating robust capture
of geometric intricacy. Specifically, we design the Skeleton Extraction Module
to extract skeleton points and skeletal features in an unsupervised manner,
which is inherently robust to noise and density variances. Then, we propose
the Skeleton-Aware GeoTransformer to encode high-level skeleton-aware
features. It explicitly captures the topological natures and inter-point-cloud
skeletal correlations with the noise-robust and density-invariant skeletal
representations. Next, we introduce the Correspondence Dual-Sampler to
facilitate correspondences by augmenting the correspondence set with skeletal
correspondences. Furthermore, we construct a challenging novel cross-source
point cloud dataset named KITTI CrossSource for benchmarking cross-source
point cloud registration methods. Extensive quantitative and qualitative
experiments are conducted to demonstrate our approach’s superiority and
robustness on both cross-source and same-source datasets. To the best of our
knowledge, our approach is the first to facilitate point cloud registration
with skeletal geometric priors.

## Introduction

Figure 1: Impossible Triangle of Current Methods. Registration recalls under
different settings are shown: KITTI Odometry (Same-Source), KITTI CrossSource
and the low-overlap test split of KITTI CrossSource. Existing methods fail to
perform as well as SPEAL on all three challenging circumstances.

Point cloud registration is an essential task in graphics, vision, and
robotics. It aims at estimating a rigid transformation to align two partially
overlapping frames of point clouds. Recently, there has been a surge of
interest in learning-based point cloud registration methods. These methods
have made significant progress in addressing the sparsity, partial overlap,
and complex distribution of point clouds in large outdoor scenes (Lu et al.
2021; Huang et al. 2021a; Yew and Lee 2022; Qin et al. 2022). However, the
practical application and advances in point cloud acquisition present more
challenges for point cloud registration, including unstructured scenes and
cross-source data.

In _unstructured scenes_ , the complex natural scenes and objects often make
it difficult to learn discriminative features for registration. This results
in degraded performance of registration algorithms. In the case of _cross-
source data_ , challenges mainly arise from partial overlap, as well as
considerable differences in scale and density, leading to difficulties in
effective feature matching. The combination of noise and outliers from
different sources further downgrades the quality of correspondences. Existing
methods either focus solely on same-source point clouds, or overlook the
intrinsic topological natures of the point clouds. This leads to suboptimal
results for challenging scenarios such as cross-source point cloud
registration and unstructured scenes.

We have observed that skeletons serve as an efficient and robust geometric
representation for point clouds, exhibiting significant potential in various
point cloud understanding tasks (Shi et al. 2021; Lin et al. 2021). They can
effectively encode the geometric intricacy of point clouds. Inspired by this,
we propose a novel transformer-based approach, termed Skeletal Prior Embedded
Attention Learning (SPEAL), to address the aforementioned challenges. Our
method utilizes skeletal geometric priors to learn discriminative features for
accurate and robust correspondences. To the best of our knowledge, our
approach is the first to facilitate point cloud registration with skeletal
geometric priors. Such skeletal geometric priors encourage robust feature
learning by explicitly encoding the intrinsic topological characteristics,
thereby facilitating the correspondences and registration results, as shown in
Fig. 1

Specifically, to incorporate skeletal representations as a geometric prior,
SPEAL comprises three key components: _S keleton Extraction Module (SEM)_, _S
keleton-Aware GeoTRansformer (SAGTR)_, and _C orrespondence Dual-Sampler
(CDS)_. First, with the insights from the medial axis transform (MAT) (Blum
1967), SEM extracts a set of skeleton points with skeletal features from input
point clouds in an unsupervised manner. It is robust to noise and density
variances. Next, SAGTR is designed to learn skeleton-aware discriminative
features, facilitating accurate and robust correspondences. It explicitly
captures the topological natures and effectively learns inter-point-cloud
geometric correlations with our skeletal representations. Finally, CDS samples
reliable correspondences from both superpoints and skeleton points, which
produces reliable coarse correspondences with awareness of the skeletal
structure.

Extensive experiments are carried out on two datasets. One is _KITTI Odometry_
, a large-scale outdoor registration benchmark (Geiger, Lenz, and Urtasun
2012), as well as its cross-source variant named _KITTI CrossSource_ proposed
by us. The other is a large-scale cross-source dataset mostly consisting of
unstructured forest scenes (Weiser et al. 2022). The results demonstrate that
SPEAL is effective and robust for both same-source and cross-source point
cloud registration, as well as for point clouds of unstructured scenes.

Overall, our contributions are threefold:

  * •

We propose a novel learning-based point cloud registration approach, SPEAL,
which is the first to utilize skeletal representations as a geometric prior to
achieve improved performance.

  * •

The proposed SEM is an effective and portable skeleton extractor. Our SAGTR
combined with CDS effectively produces accurate and robust correspondences for
both same-source and cross-source point cloud registration.

  * •

KITTI CrossSource, a novel cross-source point cloud dataset, meets the dire
need (Huang et al. 2021b) of cross-source point cloud registration benchmarks.
This opens up the possibility to bridge the gap between sensor technology and
cross-source applications.

## Related Work

Learning-based Registration Methods. Learning-based registration methods fall
into two categories: correspondence-based methods and direct registration
methods. Correspondence-based methods (Choy, Park, and Koltun 2019; Deng,
Birdal, and Ilic 2018a, b; Gojcic et al. 2019; Yao et al. 2020) first extract
correspondences between two point clouds, and then estimate the transformation
with robust pose estimators. However, traditional robust estimators suffer
from slow convergence and are sensitive to outliers. To address this, deep
robust estimators (Choy, Dong, and Koltun 2020; Bai et al. 2021; Pais et al.
2020; Lee et al. 2021) utilize deep neural networks to reject outliers and
compute the transformation. While these methods require a training procedure,
they improve accuracy and speed. Direct registration methods directly estimate
the transformation between two point clouds in an end-to-end way. Inspired by
Iterative Closest Point (Besl and McKay 1992), some of them (Fu et al. 2021;
Wang and Solomon 2019b, a; Yew and Lee 2020) iteratively build soft
correspondences and then estimate the transformation with SVD. Others (Xu et
al. 2021; Aoki et al. 2019; Huang, Mei, and Zhang 2020) extract a global
feature vector and regress the transformation directly with a neural network.
However, such methods could potentially fail in large-scale scenes.

Transformers in Point Cloud Registration. Originally designed for NLP tasks,
Transformers (Vaswani et al. 2017) have shown remarkable efficacy in computer
vision (Misra, Girdhar, and Joulin 2021; Carion et al. 2020; Dosovitskiy et
al. 2020; Yu et al. 2021b). Recently, transformer-based methods for point
cloud registration have also emerged. Geometric Transformer (Qin et al. 2022)
leverages transformer layers for superpoint matching, while REGTR (Yew and Lee
2022) uses a transformer cross-encoder and a transformer decoder to directly
predict overlap scores. PEAL(Yu et al. 2023) leverages additional overlap
priors from 2D images.

Point Cloud Skeletal Representations. The curve skeleton is a widely-used
skeletal representation due to its simplicity (Huang et al. 2013; Ma, Wu, and
Ouhyoung 2003; Au et al. 2008; Cao et al. 2010). It has shown its potential in
some learning-based methods (Xu et al. 2019; Shi et al. 2021) like keypoint
extraction. However, it is only well-defined for tubular geometries, thus
limiting its expressiveness for point clouds with complex shapes or in large-
scale scenes. The Medial Axis Transform (MAT) (Blum 1967) is another skeletal
representation capable of encoding arbitrary shapes. Some methods (Sun et al.
2015; Yan, Letscher, and Ju 2018; Li et al. 2015) employ simplification
techniques to alleviate the distortion caused by surface noise, but they are
computationally ineffective and require watertight input surfaces. Recent
learning-based efforts (Lin et al. 2021; Wen, Yu, and Tao 2023) use deep
neural networks to predict MAT-based skeletons, thus greatly enhancing the
robustness and computational efficiency. These methods have shown promising
results in various 3D vision tasks, including shape reconstruction and point
cloud sampling (Wen, Yu, and Tao 2023).

## Method

Figure 2: The Overall Pipeline of SPEAL. The backbone extracts superpoints and
multi-level features from $\mathcal{P}$ and $\mathcal{Q}$. Then, SEM and SAGTR
extract skeletal representations and learn discriminative skeleton-aware
features, respectively. Finally, CDS extracts hybrid coarse correspondences
with skeletal priors. The result transformation is computed with LGR.

Problem Statement. Given two point clouds
$\mathcal{P}=\\{\mathbf{p}_{i}\in\mathbb{R}^{3}|i=1,\ldots,N\\}$ and
$\mathcal{Q}=\\{\mathbf{q}_{i}\in\mathbb{R}^{3}|i=1,\ldots,M\\}$, our goal is
to align the two point clouds by estimating a rigid transformation
$\mathbf{T}=\\{\mathbf{R},\mathbf{t}\\}$, where $\mathbf{R}\in SO(3)$ is a 3D
rotation matrix and $\mathbf{t}\in\mathbb{R}^{3}$ is a 3D translation vector.
The transformation can be solved by:

$\min_{\mathbf{R},\mathbf{t}}\sum\nolimits_{(\mathbf{p}_{x_{i}},\mathbf{q}_{y_{i}})\in\mathcal{C}^{\star}}\|\mathbf{R}\mathbf{p}_{x_{i}}+\mathbf{t}-\mathbf{q}_{y_{i}}\|^{2},$
(1)

where $\mathcal{C}^{\star}$ denotes the set of correspondences between two
point clouds $\mathcal{P}$ and $\mathcal{Q}$. In reality,
$\mathcal{C}^{\star}$ is usually unknown. Hence, we need to establish accurate
correspondences $\mathcal{C}$ between two point clouds for a good
transformation.

Overview and Notations. Our work leverages skeletal priors in an end-to-end
neural network to facilitate correspondences. The pipeline is shown in Fig. 2,
following the hierarchical correspondence paradigm. To extract multi-level
features for point clouds, we leverage the KPConv-FPN backbone (Lin et al.
2017; Thomas et al. 2019). The points at the coarsest level of the backbone
are _superpoints_ , denoted as $\hat{\mathcal{P}}$ and $\hat{\mathcal{Q}}$.
Their associated features are
$\hat{\mathbf{F}}^{\mathcal{P}}\in\mathbb{R}^{|\hat{\mathcal{P}}|\times
d_{t}}$ and
$\hat{\mathbf{F}}^{\mathcal{Q}}\in\mathbb{R}^{|\hat{\mathcal{Q}}|\times
d_{t}}$. Then, our proposed SEM, SAGTR and CDS are used to extract reliable
and accurate coarse correspondences with skeletal priors. Finally, we employ
the Point Matching Module and Local-to-Global Registration (Qin et al. 2022)
to obtain dense correspondences and estimate the final rigid transformation.

### Skeleton Extraction Module

The Skeleton Extraction Module aims to approximate the Medial Axis Transform
(MAT) by leveraging a convex combination of input points, which provides a
well-defined skeletal representation for arbitrary shapes in an unsupervised
manner. Inspired by existing methods (Lin et al. 2021), it overcomes the
computational expense and sensitivity to surface noise of traditional MAT
computation.

Specifically, for all points in
$\hat{\mathcal{P}}\in\mathbb{R}^{|\hat{\mathcal{P}}|\times 3}$ and their
features
$\hat{\mathbf{F}}^{\mathcal{P}}\in\mathbb{R}^{|\hat{\mathcal{P}}|\times
d_{t}}$, SEM aims to extract $N_{s}$ skeleton points
$\mathcal{S}^{\mathcal{P}}\in\mathbb{R}^{N_{s}\times 3}$, their skeletal
features $\hat{\mathbf{F}}^{\mathcal{P}}_{s}\in\mathbb{R}^{N_{s}\times
d_{t}}$, and their radii $\mathcal{R}^{\mathcal{P}}\in\mathbb{R}^{N_{s}\times
1}$ . We extract skeletons for $\mathcal{Q}$ in the same way. To this end, we
employ a multi-layer perceptron (MLP) to predict the weights
$\mathbf{\mathcal{W}}\in\mathbb{R}^{|\hat{\mathcal{P}}|\times N_{s}}$. The MLP
is shared across $\hat{\mathcal{P}}$ and $\hat{\mathcal{Q}}$. Then, the
skeleton points $\mathcal{S}^{\mathcal{P}}$ are obtained as the convex
combination(Lin et al. 2021) of input points $\hat{\mathcal{P}}$:

$\mathcal{S}^{\mathcal{P}}=\mathbf{\mathcal{W}}^{T}\hat{\mathcal{P}}\ s.t.\
j\\!=\\!1,\ldots,N_{{s}},\
\sum\nolimits_{i=1}^{|\hat{\mathcal{P}}|}\mathcal{W}(i,j)=1.$ (2)

The weighting scheme enhances the robustness of skeleton extraction by
effectively filtering out noise and outliers. Similarly, we extract their
skeletal features by
$\hat{\mathbf{F}}^{\mathcal{P}}_{s}=\mathbf{\mathcal{W}}^{T}\hat{\mathbf{F}}^{{\mathcal{P}}}$.

To predict the radius of each skeleton point, we first compute the closest
distance for an input point $\hat{\mathbf{p}}$ to all skeleton points as
follows:

$d(\hat{\mathbf{p}},\mathcal{S}^{\mathcal{P}})=\min\nolimits_{\mathbf{s}\in\mathcal{S}^{\mathcal{P}}}\|\hat{\mathbf{p}}-\mathbf{s}\|_{2}.$
(3)

The distances for all input points are then summarized in a vector
$\mathcal{D}^{\mathcal{P}}\in\mathbb{R}^{|\hat{\mathcal{P}}|\times 1}$. Next,
the radii of all the skeleton points are computed through a linear combination
of their closest distances from all the input points, i.e.,
$\mathcal{R}^{\mathcal{P}}=\mathcal{W}^{T}\mathcal{D}^{\mathcal{P}}$. This
approximation is based on the observation that the predicted weights for a
skeleton point $\mathbf{s}$ are significant only for the input points that in
a local neighborhood of $\mathbf{s}$, and diminish to 0 for the input points
far away from $\mathbf{s}$.

The skeleton extraction is a fundamentally different task from the point cloud
registration. Therefore, the module is separately supervised by the _skeleton
loss_ (Lin et al. 2021), and we block the gradient flow from this module to
the backbone for a more stable training process and better performance (See
supplementary materials).

### Skeleton-Aware GeoTransformer

The registration of cross-source point clouds poses significant challenges,
including noise, density differences, and scale variances. Skeleton points
exhibit consistency and robustness against these challenges. Therefore, we
propose the SAGTR module to encode the structure of point clouds. It comprises
two key components: _Skeleton-aware Geometric Self-Attention_ and _Skeleton-
aware Cross-Attention_. They are interleaved for $N_{t}$ times to further
extract non-skeletal and skeletal hybrid features
$(\mathbf{H}^{\mathcal{P}},\mathbf{H}_{s}^{\mathcal{P}})$ and
$(\mathbf{H}^{\mathcal{Q}},\mathbf{H}_{s}^{\mathcal{Q}})$. These features
encode inter-point-cloud and intra-point-cloud correlations and skeletal
geometric priors. They contribute to accurate and robust coarse
correspondences.

Skeleton-Aware Geometric Self-Attention. In the following, we describe the
computation for $\hat{\mathcal{P}}$, and the computation for
$\hat{\mathcal{Q}}$ is exactly the same. Given an input feature matrix
$\mathbf{X}\in\mathbb{R}^{L\times d_{t}}$ ($L=|\hat{\mathcal{P}}|+N_{s}$ is
the length of the input sequence), the output feature matrix
$\mathbf{Z}\in\mathbb{R}^{L\times d_{t}}$ is the weighted sum of all projected
input features:

$\mathbf{z}_{i}=\sum\nolimits_{j=1}^{{L}}a_{i,j}(\mathbf{x}_{j}\mathbf{W}^{V}),$
(4)

where $a_{i,j}$ is the weight coefficient computed by a row-wise softmax on
the attention score $e_{i,j}$, and $e_{i,j}$ is computed as:

$e_{i,j}=(\mathbf{x_{i}}\mathbf{W}^{Q})(\mathbf{x_{j}}\mathbf{W}^{K}+\mathbf{r}^{p}_{i,j}\mathbf{W}^{P}+\mathbf{r}^{s}_{i,j}\mathbf{W}^{S})^{T}/\sqrt{d_{t}}.$
(5)

Here, $\mathbf{r}^{s}_{i,j}$ and $\mathbf{r}^{p}_{i,j}$ are _Skeleton-Aware
Structure Embedding_ and _Point-Wise Structure Embedding_ respectively. We
follow (Qin et al. 2022) to compute $\mathbf{r}^{p}_{i,j}$, which encodes non-
skeletal geometric structures between superpoints. $\mathbf{r}^{s}_{i,j}$
encodes skeletal latent geometric information of point clouds, which will be
described next.
$\mathbf{W}^{Q},\mathbf{W}^{K},\mathbf{W}^{V},\mathbf{W}^{P},\mathbf{W}^{S}\in\mathbb{R}^{d_{t}\times
d_{t}}$ are the respective projections for queries, keys, values, point-wise
structure embedding and skeleton-aware structure embedding. Fig. 3 shows the
computation of Skeleton-aware Geometric Self-attention.

Figure 3:  The structure (left) and computational graph (right) of skeleton-
aware geometric self-attention.

We design a novel approach, termed _Skeleton-Aware Structure Embedding_ , to
encode skeletal latent structural information in the geometric space. The
insight is to leverage the transformation invariance and robustness in the
local geometric structure formed by the skeleton points. This embedding
includes _skeleton-wise distance embedding_ and _skeleton-wise angular
embedding_. They respectively capture distance and angle information of the
local geometric structure formed by skeleton points around superpoints.

Figure 4: The computation of skeleton-aware structure embedding.

Specifically, given two superpoints
$\hat{\mathbf{p}}_{i},\hat{\mathbf{p}_{j}}\in\hat{\mathcal{P}}$, their kNN
skeleton points are
$\mathbf{s}^{i}_{1},\ldots,\mathbf{s}^{i}_{k}\in\mathcal{K}^{s}_{i}$ and
$\mathbf{s}^{j}_{1},\ldots,\mathbf{s}^{j}_{k}\in\mathcal{K}^{s}_{j}$,
respectively. Based on them, as shown in Fig. 4, the computation of skeleton-
wise structure embedding $\mathbf{r}^{s}_{i,j}$ is twofold: 1) _Skeleton-Aware
Distance Embedding._ For each superpoint $\mathbf{p}_{j}$, we first compute
$\rho^{s}_{j}=\sum_{\mathbf{s}^{j}_{x}\in\mathcal{K}_{j}^{s}}d(\mathbf{s}^{j}_{x},\mathbf{p}_{j})$
, where
$d(\mathbf{s}^{j}_{x},\mathbf{p}_{j})=\|\mathbf{s}^{j}_{x}-\mathbf{p}_{j}\|_{2}$
denotes the distance between $s_{i}$ and $p_{j}$ in the Euclidean space. Then,
the skeleton-aware distance embedding $\mathbf{d}^{s}_{j}$ is computed by
applying a sinusoidal function on
$(\rho^{s}_{i}-\rho^{s}_{j})/\sigma^{s}_{d}$. 2) _Skeleton-Aware Angular
Embedding._ For each skeleton point $s^{j}_{x}\in\mathcal{K}^{s}_{j}$, we
first compute the angle
$\theta^{x}_{i,j}=\angle(\mathbf{s}^{j}_{x}-\mathbf{p}_{j},\mathbf{p}_{i}-\mathbf{p}_{j})$.
Based on the angles, the skeleton-wise angular embedding
$\mathbf{a}^{s}_{i,j,x}$ is computed by applying a sinusoidal function on
$\theta^{x}_{i,j}/\sigma^{s}_{a}$. Herein, $\sigma^{s}_{d}$ and
$\sigma^{s}_{a}$ control the sensitivity on skeleton-wise distances and angles
respectively. The final skeleton-aware structure embedding
$\mathbf{r}^{s}_{i,j}$ is the aggregation of the skeleton-aware angular
embedding $\mathbf{a}^{s}$ and the skeleton-aware distance embedding
$\mathbf{d}^{s}$:

$\mathbf{r}^{s}_{i,j}=\mathbf{d}^{s}_{i,j}\mathbf{W}^{D}+\mathrm{mean}_{x}\\{\mathbf{a}^{s}_{i,j,x}\mathbf{W}^{A}\\},$
(6)

where $\mathbf{W}^{D}$ and $\mathbf{W}^{A}$ are trainable weights.

To help the successive cross-attention layers to capture the geometric
structure with skeletal priors, our skeleton-aware self-attention layers also
produce the skeleton-aware positional encoding $\mathbf{E}^{{}^{\prime}}$ by
applying the attention scores on the skeleton-aware structure embedding
$\mathbf{r}^{s}_{i,j}$:

${\mathbf{E}}^{{}^{\prime}}_{i,k}=\sum\nolimits_{j=1}^{{L}}a_{i,j}\cdot
r_{i,j,k}^{s}.$ (7)

Skeleton-Aware Cross-Attention. Several existing works (Qin et al. 2022; Yew
and Lee 2022) have utilized the cross-attention mechanism for inter-point-
cloud feature exchange. However, they either lack positional encoding or fail
to explicitly consider the geometric structure of point clouds, leading to
suboptimal performance. To address this, we propose the skeleton-aware cross-
attention to explicitly learn the correlation of point clouds with skeletal
priors, as is shown in Fig. 5.

Figure 5:  The structure (left) and computational graph (right) of skeleton-
aware cross-attention.

Given feature maps with their skeleton-aware positional encoding
$(\mathbf{X}^{\mathcal{P}},\mathbf{E}^{{}^{\prime}}_{\mathcal{P}})$ and
$(\mathbf{X}^{\mathcal{Q}},\mathbf{E}^{{}^{\prime}}_{\mathcal{Q}})$ for
$\hat{\mathcal{P}}$ and $\hat{\mathcal{Q}}$ respectively. A skeleton-aware
cross-attention layer first adds the positional encoding to features to
produce skeleton-aware features $\mathbf{X}^{{}^{\prime}\mathcal{P}}$ and
$\mathbf{X}^{{}^{\prime}\mathcal{Q}}$. Then, the output for
$\hat{\mathcal{P}}$ are computed with the features of $\hat{\mathcal{Q}}$:

$\mathbf{z}_{i}^{\mathcal{P}}=\sum\nolimits_{j=1}^{|\hat{\mathcal{Q}}|}a_{i,j}(\mathbf{x}_{j}^{{}^{\prime}\mathcal{Q}}\mathbf{W}^{V}).$
(8)

Similarly, the weights $a_{i,j}$ are computed by a row-wise softmax on the
attention score $e_{i,j}$:

$e_{i,j}=(\mathbf{x}_{i}^{{}^{\prime}\mathcal{P}}\mathbf{W}^{Q})(\mathbf{x}_{j}^{{}^{\prime}\mathcal{Q}}\mathbf{W}^{K})^{T}/\sqrt{d_{t}}.$
(9)

The same cross-attention implementation goes for $\hat{\mathcal{Q}}$. In
contrast to skeleton-aware geometric self-attention that captures the intra-
point-cloud transformation-invariant geometric structure, the cross-attention
here captures the inter-point-cloud geometric corrections and consistency. The
hybrid features obtained from SAGTR are therefore discriminative enough for
matching.

### Correspondence Dual-Sampler

With discriminative features, it is vital to extract accurate coarse
correspondences. Geometric Transformer (Qin et al. 2022) only matches the
superpoints. Despite its efficiency, superpoints are sparse and may be
unrepeatable, leading to outlier correspondences. Existing efforts strive to
tackle this issue with sophisticated sampling strategies (Li et al. 2023) or
overlap priors from 2D images (Yu et al. 2023). Unfortunately, their required
complicated sampling and extra 2D images result in suboptimal computational
efficiency, which hinders the application of such methods. To this end, we
propose CDS to effectively augment the correspondence set with our effective
skeletal representation, leading to a more accurate and robust hybrid coarse
correspondence set.

We separately construct the non-skeletal correspondence set $\mathcal{C}$ and
skeletal correspondence set $\mathcal{C}_{s}$ by feature matching: We first
compute the Gaussian correlation matrix
$\mathbf{S}\in\mathbb{R}^{|\hat{\mathcal{P}}|\times|\hat{\mathcal{Q}}|}$ for
the normalized features $\mathbf{F}^{\mathcal{P}}$ and
$\mathbf{F}^{\mathcal{Q}}$, and then use a dual-normalization operation (Sun
et al. 2021; Rocco et al. 2018) to suppress ambiguous matches. Finally, we
select at most $N_{c}$ largest entries for each correspondence set.

Since skeleton points lying on non-overlap regions may introduce outlier
correspondences, we introduce the _Spectral Denoising_ procedure to filter
$\mathcal{C}_{s}$ with a spectral matching algorithm (Leordeanu and Hebert
2005): We firstly compute a compatibility matrix based on the 3D spatial
consistency of $\mathcal{C}_{s}$. Then, we iteratively remove components
conflicting with the item of the maximum principal eigenvector until either
the principal eigenvector becomes zero or $|\mathcal{C}_{s}|$ equals the
minimum number of the main cluster. The main cluster, denoted as
$\mathcal{C}_{s}^{{}^{\prime}}$, is the final skeletal correspondence set.

Finally, we resample the least confident $N_{s}$ entries of $\mathcal{C}$ with
top $N_{k}$ correspondences in $\mathcal{C}_{s}^{{}^{\prime}}$ to obtain the
hybrid correspondence set $\mathcal{C}^{{}^{\prime}}$, thereby improving the
accuracy and robustness of the hybrid coarse correspondence set by replacing
potential outliers with more reliable correspondences.

### Losses

We use a registration loss and a skeleton loss to supervise SPEAL. The
registration loss consists of _Overlap-aware Circle Loss_ ($\mathcal{L}_{oc}$)
and _Point Matching Loss_ ($\mathcal{L}_{p}$) from Geometric Transformer (Qin
et al. 2022):

$\displaystyle\mathcal{L}$ $\displaystyle=\mathcal{L}_{oc}+\mathcal{L}_{p}.$
(10)

The skeleton loss in Lin et al. (2021) is used to supervise the SEM. It is the
weighted sum of _Sampling Loss_ $\mathcal{L}_{\text{s}}$, _Point-to-sphere
Loss_ $\mathcal{L}_{\text{r}}$ and _Radius Regularizing Loss_
$\mathcal{L}_{\text{p2s}}$:

$\displaystyle\mathcal{L}_{\text{skeleton}}$
$\displaystyle=\mathcal{L}_{\text{s}}+\lambda_{1}\mathcal{L}_{\text{p2s}}+\lambda_{2}\mathcal{L}_{\text{r}},$
(11)

where $\lambda_{1}$ and $\lambda_{2}$ are hyperparameters to balance the
losses. Please refer to the supplementary material for more details.

## Experiments

### Datasets and Experimental Setup

Same-Source Dataset. The KITTI Odometry dataset (Geiger, Lenz, and Urtasun
2012) serves as a widely-used dataset for odometry and SLAM evaluation. It can
also be employed to test same-source point cloud registration. This dataset
comprises 11 sequences of LiDAR point clouds. We follow the existing practices
(Qin et al. 2022; Huang et al. 2021a) to use sequences 00-06 for training,
sequences 07-08 for evaluation and sequences 09-10 for testing.

Cross-Source Datasets. Currently, there are few cross-source datasets of
large-scale outdoor scenes available for registration tasks. This hinders the
development of cross-source registration methods. Therefore, we have developed
a novel dataset222The dataset will be made publicly available termed _KITTI
CrossSource_ derived from KITTI Odometry. Our proposed dataset includes 11
sequences of LiDAR point clouds and reconstructed point clouds generated from
stereo images using MonoRec (Wimbauer et al. 2021). We improve the
reconstruction quality with a filter-and-combine strategy. Please refer to the
supplementary material for details.

The GermanyForest3D dataset is derived from an existing large-scale forest
scene dataset (Weiser et al. 2022). It contains cross-source point cloud data
acquired in 12 forest plots in south-west Germany under leaf-on and leaf-off
conditions. Each plot provides Airborne Laser Scanning (ALS), Terrestrial
Laser Scanning (TLS) and UAV-borne Laser Scanning (ULS) point clouds. In this
paper, we use ALS and ULS scans to evaluate the cross-source registration
performance. In experiments, we use 10 plots for training, 1 for validation
and 1 for testing.

Data Preprocessing. For the GermanyForest3D dataset, the point clouds of each
plot are subdivided into $30\mathrm{m}\times 30\mathrm{m}\times 30\mathrm{m}$
blocks to make them suitable for the registration task. For all datasets, the
Iterative Closest Point (ICP) algorithm from the Open3D library (Zhou, Park,
and Koltun 2018) is used to refine the noisy ground truth transformation,
following previous works (Qin et al. 2022; Lu et al. 2021). The point clouds
are downsampled with a voxel size of $0.3\mathrm{m}$.

Metrics. Following previous practices (Qin et al. 2022; Lu et al. 2021; Huang
et al. 2021a), we evaluate the registration performance using following
metrics: _Relative Rotation Error_ (RRE), _Relative Translation Error_ (RRE),
and _Registration Recall_ (RR). We use a RRE threshold and a RTE threshold to
compute RR for all datasets ($\textrm{RRE}<0.5\degree$ and
$\textrm{RTE}<0.3\textrm{m}$ for GermanyForest3D and $\textrm{RRE}<5\degree$
and $\textrm{RTE}<2\textrm{m}$ for KITTI datasets). Additionally, we measure
the quality of correspondences with Inlier Ratio (IR), which is the fraction
of extracted correspondences whose residuals are below a certain threshold
under the ground-truth transformation.

Implementation Details. To train SPEAL, we use Adam (Kingma and Ba 2014)
optimizer with an initial learning rate of $1e-4$ and a weight decay of
$1e-6$. We train SPEAL for $200$ epochs with a batch size of $1$ on a NVIDIA
RTX 3090 GPU.

Baselines. We compare our method with state-of-the-art methods of three
classes: (a) Traditional methods, including RANSAC (Fischler and Bolles 1981)
and FGR (Zhou, Park, and Koltun 2016). (b) Transformer-based methods,
including CoFiNet (Yu et al. 2021a), Predator(Huang et al. 2021a), PCAM (Cao
et al. 2021), REGTR (Yew and Lee 2022) and Geometric Transformer (abbreviated
as GeoTrans.) (Qin et al. 2022). (c) Other learning-based methods, including
FCGF (Choy, Park, and Koltun 2019), DGR (Choy, Dong, and Koltun 2020) and
HRegNet (Lu et al. 2021).

### Cross-Source Results

Method | KITTI CrossSource | KITTI Odometry
---|---|---
RRE($\degree$) | RTE(m) | RR($\%$) | RRE($\degree$) | RTE(m) | RR($\%$)
RANSAC | 6.14 | 9.46 | 0.8 | 0.54 | 0.13 | 91.9
FGR | – | – | – | 0.96 | 0.93 | 39.4
FCGF | – | – | – | 0.30 | 0.095 | 96.6
DGR | – | – | – | 0.37 | 0.320 | 98.7
HRegNet | 2.19 | 0.84 | 69.3 | 0.29 | 0.120 | 99.7
CoFiNet | 1.99 | 0.81 | 67.6 | 0.41 | 0.085 | 99.8
Predator | 5.06 | 2.59 | 34.2 | 0.27 | 0.068 | 98.8
PCAM | 4.07 | 2.40 | 45.9 | 0.79 | 0.12 | 98.0
GeoTrans. | 1.87 | 0.63 | 96.8 | 0.24 | 0.068 | 99.8
SPEAL | 1.41 | 0.58 | 97.3 | 0.23 | 0.069 | 99.8
Table 1: Cross-source (the proposed KITTI CrossSource) and same-source (KITTI
Odometry) registration results on the KITTI datasets. ”–” indicates the method
is not applicable to the dataset.

KITTI CrossSource. The quantitative results are reported in Table 1. Our
method achieves state-of-the-art performance on this dataset. For traditional
methods, FGR is not applicable to this dataset and our approach outperforms
RANSAC by a large margin. HRegNet is a recent SOTA for outdoor large-scale
scenes. However, it presents suboptimal performance on this dataset, showing
considerable performance decay for cross-source data. In contrast, our SPEAL
is more accurate in terms of all metrics, and has a $28\%$ higher RR than
HRegNet. Among transformer-based methods, our method surpasses GeoTrans. by a
large margin, showing the effectiveness of the integrated skeletal priors.

Overlap | RRE ($\degree$) ↓ | RTE ($\mathrm{m}$) ↓ | RR($\%$) ↑
---|---|---|---
$\leq 30\%$ | $>30\%$ | $\leq 30\%$ | $>30\%$ | $\leq 30\%$ | $>30\%$
RANSAC | 112.0 | 91.1 | 24.66 | 17.4 | – | –
FGR | 41.86 | 28.3 | 14.32 | 8.49 | – | –
FCGF | 1.54 | 0.53 | 0.49 | 0.19 | 8.7 | 56.6
DGR | 1.06 | 0.38 | 0.36 | 0.10 | 32.8 | 76.2
HRegNet | 1.16 | 1.40 | 0.141 | 0.238 | 24.7 | 41.7
REGTR | 3.11 | 2.48 | 0.89 | 0.73 | 11.3 | 23.9
GeoTrans. | 0.328 | 0.176 | 0.097 | 0.053 | 88.1 | 96.5
SPEAL(ours) | 0.296 | 0.165 | 0.088 | 0.048 | 91.7 | 99.3
Table 2: Cross-source registration results on the GermanyForest3D dataset. ”–”
indicates that the method is not applicable to the dataset.

GermanyForest3D. This cross-source dataset is with large scale and
unstructured scenes, which are challenging for registration. The evaluation
results under different overlap ratios are shown in Table 2. Traditional
methods show suboptimal performance, and RANSAC even fails to register low
overlap point clouds. Learning-based methods overall perform better. However,
current methods still suffer from considerable performance decay especially
under low overlap condition. SPEAL outperforms all the others by a large
margin, showing outstanding robustness introduced by skeletal priors in low
overlap and unstructured condition.

### Same-Source Results

Table 1 also lists the quantitative results on the same-source dataset KITTI
Odometry. Compared with recent state-of-the-arts, our method achieves
comparable performance in terms of RTE and RR, and outperforms all the other
methods in terms of RRE. This result indicates that our method is also
effective for same-source registration, while achieving state-of-the-art
performance for cross-source registration.

### Analysis

(a) Gorund Truth

(b) Pose (SPEAL)

(c) Pose (GeoTrans.)

(d) Superpoint Correspondences (GeoTrans.)

(e) Superpoint Correspondences (SPEAL)

(f) Skeletal Correspondences (SPEAL)

Figure 6: Qualitative results on KITTI CrossSource. Red and green denotes
outlier and inlier correspondences, respectively.

Effectiveness of the Skeletal Priors. To qualitatively verify the
effectiveness of the skeletal representation, we visualize the dual-class
correspondences from the CDS module, including superpoints and skeleton
points. The qualitative results are shown in Fig. 6. In addition to the
challenges of partial overlap and density differences, this scan also presents
the challenge of unstructured objects. However, SPEAL is still able to extract
right correspondences with the help of skeletons, while the current state-of-
the-art, GeoTrans., completely fails. SPEAL achieves an IR of $36.8\%$, which
is nearly $10\times$ higher than GeoTrans. It is worth noting that with our
spectral denoising step, the skeletal correspondences achieve an IR of $75\%$,
demonstrating the effectiveness of the spectral denoising step.

(a) Registration recalls with different RRE and RTE thresholds

(b) Correspondence and registration results under different overlap ratios

Figure 7: Quantitative results on robustness of SPEAL on the KITTI CrossSource
dataset.

Robustness. Fig. 7(a) displays registration recalls with different RRE and RTE
thresholds in KITTI CrossSource. SPEAL consistently outperforms the other
methods. In particular, it achieves a registration recall of $86.04\%$ in the
challenging low overlap of $30\%\sim 50\%$, which is $9.3\%$ higher than the
second-best method. In addition, Fig. 7(b) compares registration recalls and
inlier ratios under different overlap with GeoTrans. Our method consistently
achieves higher inlier ratios under all overlap ratios. This demonstrates the
superior quality of correspondences generated by our method with skeletal
priors. The results prove that our SPEAL is robust to various threshold
settings, and demonstrates outstanding performance in low overlap condition.

### Ablation Studies

Overall Effectiveness. We conduct ablation studies to assess the effectiveness
of SPEAL on GermanyForest3D. We compare different configurations of SPEAL,
including: (a) vanilla geometric self-attention and vanilla cross attention,
(b) skeleton-aware self-attention and vanilla cross attention, (c) skeleton-
aware self-attention and skeleton-aware cross attention. In addition, we also
compare with (d) the method without the CDS module, which only samples the
coarse correspondences from superpoints. The results in Table 3 demonstrate
the effectiveness of our design.

Model | RRE($\degree$)↓ | RTE($\mathrm{m}$)↓ | RR($\%$)↑
---|---|---|---
(a) vanilla | 0.19 | 0.054 | 96.2
(b) cross attn. w/o SPE | 0.18 | 0.051 | 97.4
(c) w/ SSE & SPE | 0.18 | 0.049 | 97.1
(d) corr. w/o CDS | 0.19 | 0.051 | 96.3
(e) SPEAL (Ours) | 0.16 | 0.048 | 99.3
Table 3: Ablation studies on SPEAL. Spectral Denoising | RRE($\degree$)↓ | RTE($\mathrm{m}$)↓ | RR($\%$)↑
---|---|---|---
Sup. Corr. | Skel. Corr.
|  | 0.17 | 0.051 | 98.1
✓ |  | 0.20 | 0.063 | 97.8
✓ | ✓ | 0.19 | 0.062 | 98.1
| ✓ | 0.16 | 0.048 | 99.3
Table 4: Ablation studies on the CDS module. Sup. Corr. and Skel. Corr. denote
the superpoint correspondences and skeletal correspondences, respectively.

Spectral Denoising in CDS. To validate the effectiveness of spectral
denoising, we also ablate the CDS module. We compare different schemes for
applying the spectral denoising. The results in Table 4 demonstrate that the
spectral denoising step is only necessary for the skeletal correspondences and
leads to inferior performance in other configurations.

## Conclusion

In this paper, we have proposed SPEAL, a novel point cloud registration method
that leverages a MAT-based skeletal representation to capture the geometric
intricacies, thereby facilitating registration. Our method introduces SEM to
extract the skeleton points and their skeletal features. Furthermore, we
design SAGTR and CDS which explicitly integrate skeletal priors to ensure
robust and accurate correspondences. Extensive experiments demonstrate that
SPEAL is effective for both same-source and cross-source point cloud
registration.

Acknowledgement. This work was supported in part by the National Key R&D
Program of China under Grant 2021YFF0704600, the Fundamental Research Funds
for the Central Universities (No. 20720220064).

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