Anisotropic Multi-Scale Graph Convolutional Network for Dense Shape Correspondence Mohammad Farazi Wenhui Zhu Zhangsihao Yang and Yalin Wang

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Anisotropic Multi-Scale Graph Convolutional Network for Dense Shape

Correspondence

Mohammad Farazi, Wenhui Zhu, Zhangsihao Yang, and Yalin Wang

Arizona State University

Tempe, Arizona

{mfarazi,wzhu59,zyang195,ylwang}@asu.edu

Abstract

This paper studies 3D dense shape correspondence, a

key shape analysis application in computer vision and

graphics. We introduce a novel hybrid geometric deep

learning-based model that learns geometrically meaning-

ful and discretization-independent features. The proposed

framework has a U-Net model as the primary node feature

extractor, followed by a successive spectral-based graph

convolutional network. To create a diverse set of ﬁlters,

we use anisotropic wavelet basis ﬁlters, being sensitive to

both different directions and band-passes. This ﬁlter set

overcomes the common over-smoothing behavior of con-

ventional graph neural networks. To further improve the

model’s performance, we add a function that perturbs the

feature maps in the last layer ahead of fully connected lay-

ers, forcing the network to learn more discriminative fea-

tures overall. The resulting correspondence maps show

state-of-the-art performance on the benchmark datasets

based on average geodesic errors and superior robustness

to discretization in 3D meshes. Our approach provides new

insights and practical solutions to the dense shape corre-

spondence research.

1. Introduction

In recent years, learning-based non-rigid shape corre-

spondence approaches have been revolutionized with the

success of geometric deep learning on unstructured data

[7]. These approaches come with various learning scenar-

ios, from unsupervised to supervised learning methods on

3D mesh or point cloud representations [18, 14, 16, 25, 37].

While modeling dense shape correspondence problem can

differ in terms of loss functions and problem formulations,

the learned descriptors, ideally, have to be invariant to near-

isometric deformations and discretization of 3D shapes.

Unfortunately, many spatial-based graph neural network

models, relying on neighbor feature aggregation, fail to gen-

Discretization-Free Feature Learning

𝒗 ∈ 𝑹𝟑

𝒗 ∈ 𝑹𝟏𝟐𝟖

Anisotropic Wavelet Graph Convolution

Correspondence

map

Local Awareness

Global Awareness

Spatial-based

Spectral-based

Local Awareness

Global Awareness

Spatial-based

Spectral-based

Figure 1. Our hybrid model learns discretization-free features us-

ing a U-Net structure based on a spatial-based point cloud model,

KPconv. [29]. Then, to learn more globally and locally aware

features, we use anisotropic wavelet based ﬁlters for our graph

convolution layer to capture intrinsic information among different

band-passes. Anisotropic wavelets also help our ﬁlters to be sen-

sitive to direction.

eralize to meshes with different sizes and discretizations as

they overﬁt to the mesh connectivity [28, 32, 13].

Lately, studies in [32, 12, 28, 13] have addressed the dis-

cretization invariance problem. The core to their approach

is the idea of using the Laplace Beltrami Operator (LBO)

eigen-basis and heat diffusion. The omnipresence of LBO

can be used in many forms like the pre-computation of 3D

shape wavelet ﬁlters [32], heat diffusion as a spatial com-

munication among the features [28], or general functional

map settings [26]. However, unlike state-of-the-art graph

convolutional networks [22, 33, 11, 30, 16], spectral mod-

els usually have inferior performance in single-resolution

mesh setting in terms of perfect matched points and average

geodesic errors. This can be seen as a trade-off between lo-

arXiv:2210.09466v2 [cs.CV] 11 Nov 2022

cal and global feature learning awareness. To remedy this,

introducing a more diverse set of ﬁlters in different band-

passes and hybridization with spatial-based models can be

a promising solution.

In this study, we propose a hybrid model (Figure 1) that

incorporates the merits of both spatial and spectral-based

approaches to overcome the limitations mentioned above.

Our model learns in an end-to-end setting, starting with a

point cloud U-Net block to learn geometric features and

feed them to a spectral-based graph convolutional network

to learn robust and discriminative features. To overcome

the over-smoothing nature of graph convolutional networks,

we adopt graph wavelet kernel as diverse ﬁlters to capture

intrinsic information from different band-pass lenses. To

make our ﬁlters even more diverse, we also propose to use

the anisotropic LBO to make our kernels directionally sen-

sitive. Finally, as the last layer of our network, we creatively

use a layer to perturb the feature map to force the network

to learn more discriminative features. As a result, it signiﬁ-

cantly improves our system performance.

Overall, our contributions are summarized as follows:

(1) To our knowledge, it is the ﬁrst geometric deep learning

framework that learns both spatial-based (via U-net) and

spectral-based (via anisotropic wavelet graph convolution

network) geometric features in a data-driven fashion. The

learned features are sensitive to subtle geometric changes

and robust to 3D discretization differences.

(2) We employ anisotropic wavelet ﬁlters to address the

common over-smoothing drawback of conventional graph

convolutional networks. Instead of only low-pass ﬁlter-

ing, we design graph wavelet functions with different band-

passes. Our work achieves a diverse set of ﬁlters sensitive

to various directions and band-passes and effectively learns

rich intrinsic geometric features.

(3) We creatively apply a simple feature perturbation func-

tion in our last layer. It signiﬁcantly boosts our model’s per-

formance in both average geodesic errors and convergence

speed. The remarkable result may enrich our understand-

ing of geometric learning strategy designs and inspire new

architectures in the geometric deep learning ﬁeld.

Our extensive experimental results verify the effective-

ness of our method. We hope this work contributes to shape

correspondence research and sheds new light on general ge-

ometric deep learning mechanisms to maximize their learn-

ing power.

2. Related Works

Shape correspondence computation methods generally

fall into three main categories: (1) traditional hand-crafted

descriptors deﬁned in spatial and spectral domains, (2)

optimization-based methods rooted in the spectrum of

shapes with functional maps being the pillar, and (3), more

recent geometric deep learning techniques, notably graph

neural networks.

Starting with hand-crafted descriptors, they mainly fall

into spatial and spectral methods. The spatial-based de-

scriptors are usually based on statistics of locally deﬁned

features [27]. These methods, mostly, suffer from gener-

alizability regarding changes in surface discretization and

capturing global information of the shape [32]. This behav-

ior is similar to spatial-based graph neural network frame-

works that rely on learning small receptive ﬁelds around

each vertex, making feature learning less globally aware.

Spectral-based methods heavily rely on the LBO spec-

trum. Importantly, these intrinsic descriptors are isometry-

invariant, making them robust to arbitrary spatial transfor-

mations. The methods proposed in [3, 8, 10] belong to

the spectral-based category. Most recently, wavelet-based

spectral-based descriptors have been proposed in [32, 21],

exploiting a multi-scale setting to diversify the set of learned

ﬁlters. However, our study does not rely on hand-crafted

features as they are sub-optimal in learning-based models.

The second category of shape correspondence tech-

niques is based on optimizing a dense map between shapes.

Particularly, the vanguard of such methods is the success-

ful functional map framework [26]. More recently, the au-

thors in [13, 12, 23] proposed unsupervised and supervised

learning schemes on top of the functional map for a more

robust and accurate correspondence. In [13], authors pro-

posed a U-Net structure to learn point-wise features to cir-

cumvent the pre-computation of hand-crafted features re-

sulting in geometrically meaningful features independent of

meshing structure. In [23], authors used a deep residual

network model to enhance the functional maps in creating

a soft correspondence map. One of the most recent works

on using a functional map based on learned features is the

work in [36]. They employed a continuous geodesic convo-

lution in an end-to-end fashion. In another study [12] with

functional map setting, authors introduced an unsupervised

framework for learning orientation-preserving features for

functional map computation, which is also robust to dis-

cretization changes in the mesh. The last category belongs

to the state-of-the-art methods rooted in geometric deep

learning, a new realm of deep learning in unstructured data

like point clouds and meshes. In [5], authors proposed win-

dowed Fourier transform in a supervised-learning setting to

learn local shape descriptors for shape matching problems

Later, they used anisotropic LBO to learn multi-kernel ﬁl-

ters to further improve the accuracy and diversity of learned

features [21]. Though mainly used for node and graph clas-

siﬁcation settings, graph neural network models have also

been used for 3D dense shape correspondence. Frame-

works in [16, 25, 6] are among methods that hold state-of-

the-art performance in graph classiﬁcation, 3D shape seg-

mentation, and shape matching. Another model based on

anisotropic LBO is proposed in [22] and their experiment

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AnisotropicMulti-ScaleGraphConvolutionalNetworkforDenseShapeCorrespondenceMohammadFarazi,WenhuiZhu,ZhangsihaoYang,andYalinWangArizonaStateUniversityTempe,Arizonafmfarazi,wzhu59,zyang195,ylwangg@asu.eduAbstractThispaperstudies3Ddenseshapecorrespondence,akeyshapeanalysisapplicationincomputervisionandg...

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Anisotropic Multi-Scale Graph Convolutional Network for Dense Shape Correspondence Mohammad Farazi Wenhui Zhu Zhangsihao Yang and Yalin Wang

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