Category-level Part-based 3D Object Non-rigid Registration

Diego Rodriguez

a

, Florian Huber and Sven Behnke

b

Autonomous Intelligent Systems, University of Bonn, Bonn, Germany

Keywords:

Non-rigid Registration, Robot Vision, Shape Spaces.

Abstract:

In this paper, we propose a novel approach for registering objects in a non-rigid manner based on decomposed

parts of an object category. By performing part-based registration, the deforming points match better local

geometric structures of the observed instance. Moreover, the knowledge acquired of an object part can be

transferred to different object categories that share the same decomposed part. This is possible because the

registration is based on a learned latent space that encodes typical geometrical variations of each part inde-

pendently. We evaluate our approach extensively on different object categories and demonstrate its robustness

against outliers, noise and misalignments of the object pose.

1 INTRODUCTION

The non-rigid registration problem aims to model the

deformation between two different feature sets such

as meshes or point clouds. The underlying transfor-

mation between the sets is unknown, which makes

the non-rigid registration problem challenging. These

kind of registrations are used in several applications

such as medical imaging, object reconstruction and

robot vision tasks (Krebs et al., 2017; Zollh

¨

ofer et al.,

2014; Stouraitis et al., 2015). Particularly, non-rigid

registration methods are employed in robot grasping

applications in order to transfer knowledge between

object instances (Stouraitis et al., 2015; Stueckler

et al., 2011; Rodriguez et al., 2018). In this manner,

associated grasping knowledge of an object instance

is adapted to a novel one based on its geometry. The

approach presented in this paper is intended to be used

for grasping transfer knowledge.

Online robot grasping tasks pose several chal-

lenges in terms of perception and planning. One

of these difﬁculties lies in the inference of the non-

observable (from the robot camera perspective) por-

tions of the novel instance, especially because multi-

ple plausible geometries can explain the current ob-

served geometry. Data-driven approaches, as the one

presented in this paper, often learn latent spaces of the

object category for generating plausible shapes based

on the data presented during training (Allen et al.,

2003; Burghard et al., 2013; Rodriguez and Behnke,

a

https://orcid.org/0000-0002-1416-7392

b

https://orcid.org/0000-0002-5040-7525

2018). Thanks to the embedded knowledge in these

latent spaces, the reconstruction can be performed on-

line.

In this paper, we propose a novel approach for reg-

istering instances belonging to the same object cate-

gory that registers decomposed parts independently.

Our approach is inspired by the observation that some

object parts are common across different categories.

For example, a chair and a table both contain legs

with similar geometries and handles often have sim-

ilar shape and function. In this manner, knowledge

acquired from an object category can be transferred

to another. In addition, the registration by parts can

increase the registration accuracy of local structures

by reducing the number of constraints and degrees of

freedom a holistic registration of complex geometries

requires.

The main contribution of this paper is the formula-

tion of a novel approach for part-based non-rigid reg-

istration that can be employed for transferring grasp-

ing knowledge for online robot applications. We eval-

uate our approach on different categories and demon-

strate that our part-based algorithms achieve better re-

sults compared to a holistic-only registration. In ad-

dition, we create a new dataset that contains meshes

and corresponding point clouds together with the re-

spective part decomposition. The part segmentation

is done manually to express semantic concepts and to

ensure the quality of the dataset. The dataset and the

source code of our implementation will be released

upon acceptance.

Rodriguez, D., Huber, F. and Behnke, S.

Category-level Part-based 3D Object Non-rigid Registration.

DOI: 10.5220/0010761800003124

In Proceedings of the 17th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2022) - Volume 4: VISAPP, pages

795-802

ISBN: 978-989-758-555-5; ISSN: 2184-4321

Copyright

c

2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

795

Our approach is especially robust against occlu-

sions because of the embedded knowledge in the

learned shape spaces. Thus, this it is very relevant for

online robot applications where objects are not fully

observable.

2 RELATED WORK

Different approaches have been proposed to address

the non-rigid registration problem according to the

restrictions of the deforming point set to match the

observed one. The most widely used constraints in-

clude: conformal maps (Kim et al., 2011; Zeng et al.,

2010), isometry (Tevs et al., 2009; Ovsjanikov et al.,

2010), thin-plate splines (Chui and Rangarajan, 2003;

Zou et al., 2007), Gaussian ﬁelds (Haehnel et al.,

2003), and motion coherence theory (Myronenko and

Song, 2010). The non-rigid registration has been also

formulated as a probability density estimation prob-

lem (Myronenko and Song, 2010; Horaud et al., 2010;

Ma et al., 2016), where parameters of Gaussian Mix-

ture Models (GMMs) are optimized by means of the

Expectation Maximization (EM) algorithm. The Co-

herent Point Drift (CPD) algorithm (Myronenko and

Song, 2010) makes use of such methods.

These traditional algorithms, however, have dif-

ﬁculties with partially observed data, because they

do not incorporate any information on the observed

point sets. Shape priors and latent spaces are fre-

quently used to address this issue with different fea-

ture sets such as body shapes (Allen et al., 2003;

Hasler et al., 2009), brain images (Marsland et al.,

2003), faces (Blanz and Vetter, 1999), and collections

of shapes (Nguyen et al., 2011; Huang et al., 2012).

Dense correspondences can be also inferred based on

a compact shape space of similar shapes (Burghard

et al., 2013; Engelmann et al., 2016).

The decomposition of complex object geometries

into parts holds the promise to reduce the registration

error. In (Adeshina and Cootes, 2010), an image-

based approach was developed to ﬁnd correspon-

dences between segment parts of bones and their lo-

cation. Dense correspondences between 3D shapes

have been also estimated by considering independent

parts (Burghard et al., 2013). A hierarchical approach

for registering 2D point sets was proposed in (Xiong

et al., 2018). The method iteratively segments a 2D

point cloud in different parts and ﬁnds the deforma-

tions of each individual part in an ICP (Iterative Clos-

est Point) manner. Similarly, several grasping plan-

ning approaches aim to reduce complexity by trans-

ferring grasp skills between simpler parts or geometri-

cal primitives (cylinders, boxes, spheres) (Aleotti and

Caselli, 2011; Aleotti et al., 2014; Vahrenkamp et al.,

2016).

3 PART-BASED NON-RIGID

REGISTRATION

We propose a learning-based approach for registering

instances belonging to an object category in a non-

rigid manner which considers both the complete ge-

ometry of the objects and their decomposed parts.

We deﬁne a category as a set objects with similar

extrinsic geometry and usage, e.g., drill, spray bot-

tle, etc. Our method deﬁnes a training phase in

which multiple shape spaces are learned based on the

point sets of the training instances. A shape space

is a low-dimensional manifold that describes typi-

cal intra-class geometrical variations. The construc-

tion of these spaces is explained in Section 3.1. The

learned shape spaces are used during inference for

ﬁnding geometries that match best the observed ones.

This search is formulated as an optimization problem

as described in Section 3.2. For a category with p

number of parts, p + 1 shape spaces are learned: one

for each part and an additional one for the entire not

decomposed instances. This additional shape space

is referred as the holistic shape space. Our approach

has two variants: partwise and hol+part. The former

formulates the non-rigid registration as the result of

independent parallel registrations for each object part

(Figure 1). The latter performs ﬁrst a holistic reg-

istration to capture the global features and to initial-

ize the search inside the shape spaces of the decom-

posed parts. Comparisons between both variants are

presented in the evaluation (Section 4).

3.1 Shape Space

The shape spaces presented here are constructed

based on a collection of training 3D point sets. From

the collection, a 3D point set is selected as the canon-

ical model C ∈ R

M×3

= (c

1

, ··· , c

M

)

T

which repre-

sents a nominal instance of the collection, where M

represents the number of 3D points in C. The canoni-

cal model is then registered non-rigidly towards each

of the remaining training point sets T

i

by means of the

Coherent Point Drift (CPD) algorithm (Myronenko

and Song, 2010). Note that all object frames are

aligned before performing the registration. In the mug

category, for example, all axes of the cylinders are

aligned and all handles are placed in the same posi-

tion. The resulting registered point set T

i

is formu-

lated as:

T

i

(C, W

i

) = C + GW

i

, (1)

VISAPP 2022 - 17th International Conference on Computer Vision Theory and Applications

796

Holistic shape space

Shape space drill tops

Figure 1: Category-level part-based registration. Multiple

shape (latent) spaces are learned. Initially, a holistic shape

space of the entire geometries is learned. Then, for each of

the parts, an individual shape space is constructed following

the same algorithm, i.e., by computing deformation ﬁelds

using CPD and by ﬁnding a lower-dimensional space using

PCA-EM. For example, the drill category will establish four

shape spaces, one holistic (green), and one for each of the

parts: shank, base and top (magenta).

where G is a Gaussian kernel matrix deﬁned element-

wise as:

g

i j

:

= G(c

i

, c

j

) = exp

−

1

2β

2

k

c

i

−c

j

k

2

, (2)

and W

i

∈ R

M×3

is a matrix of kernel weights which

can be interpreted as unnormalized deformation ﬁeld

matrix. The deduction of these matrices is out of the

scope of this paper, please refer to (Myronenko and

Song, 2010) for an in-depth analysis.

Interestingly, the registration is uniquely captured

by the matrix W

i

since C is the same across all train-

ing instances, i.e., G depends only on the canonical

model C. Moreover, the dimensionality of W

i

also

remains constant for all registrations and more impor-

tantly it is ordered, such that a k-row corresponds to

a 3D deformation vector of the k-point in C. This

allows us to ﬁnd a lower-dimensional latent space by

ﬁnding the principal components of all these deforma-

tion ﬁeld matrices W

i

. This latent space is calculated

by the Principal Component Analysis - Expectation

Maximization (PCA-EM) algorithm.

The top of Figure 1 shows a holistic learned shape

space of a drill category. Note that the presented

principal components correlate qualitatively with the

width and the height of the instances.

3.2 Part-based Registration

The input of our approach is a point set belonging to

an object category, for instance, a point cloud com-

ing from real sensory data (e.g., a RGB-D camera).

As result of the registration, the canonical point set is

deformed to match the geometry of a novel observed

instance. In this manner, the observed object is recon-

structed. In addition, a deformation ﬁeld is inferred

which can be used to transform points and 3D frames

associated to the canonical model.

Novel category-like shapes can be generated by

interpolating and extrapolating in the shape spaces. In

this manner, the non-rigid registration of shapes, ei-

ther of the entire object or of an object part, is formu-

lated as an optimization problem that minimizes the

closest distance between the observed point set O and

the deformed one T (C, Wm(x)). The energy function

to minimize is formulated as:

E(x, θ) =

M

∑

m=1

min

n

k

O

n

− Θ(T (C

m

, W

m

(x)), θ)

k

2

.

(3)

where θ deﬁnes the parameters of a rigid transforma-

tion incorporated to account for misalignments of the

observed object pose and Θ is a function that applies

this rigid transformation. A good initialization of the

object pose is however required because of many lo-

cal minima of the optimization problem. In Equation

(3), the notation A

n

refers to the point n of matrix A.

Note that by the incorporation of the rigid term, the

optimization becomes non-linear. This optimization

problem is solved by using the ceres optimizer

1

.

In order to improve the registration accuracy of

complex local structures, our approach includes the

registration of each object part independently. There-

fore, initially, the objects are decomposed into sim-

pler parts. The decomposition is done manually by

experts to express semantic concepts and to ensure the

quality of the segmentation. Automatic approaches

such as (Araslanov et al., 2016; Palafox et al., 2021)

are alternatives to perform the part decomposition.

The partwise variant registers each part indepen-

dently. In this manner, for an object with three parts,

three different shape spaces will be learned and three

different optimization problems will be solved, one

for each part. Note that the rigid transformation with

parameters θ added in the energy function (Eq. 3)

helps to align the point sets. For instance, for the drill

category shown in Figure 1, three shape spaces are

required: base, top part, and shank.

The hol+part variant of our method is composed

of two steps. In the ﬁrst step, an initial registration

takes place that registers the full geometry of the ob-

ject in a holistic manner, i.e., the registration of the

canonical model towards the observed one without

any part decomposition. In the second step, a partwise

1

http://ceres-solver.org

Category-level Part-based 3D Object Non-rigid Registration

797

Holistic-only registration

Part-based registration

Figure 2: Holistic and partwise registration of a watering

can with three parts: sprout (red), tank (green) and han-

dle(gold). Note how the registration accuracy of the local

structures (e.g., the handle) is improved by the part-based

registration. The blue points represent the deforming point

set.

registration is carried out, i.e., each individual part is

registered. The holistic registration aims to capture

global structures such as the dimensions of the object,

while the objective of partwise registration is to im-

prove registration accuracy of local structures.

The holistic registration serves as an initialization

of the optimization for each of the parts. In this man-

ner, Eq.(3) ﬁnds an optimum latent vector x

h

and the

parameters of a local registration θ

h

. The values of θ

h

are passed directly to the optimization of each of the

parts. However, the latent vector x

h

cannot be passed

directly, since each shape space represents the defor-

mation ﬁelds of different shapes. The latent vector

x

h

maps to a deformation ﬁeld W

h

. This matrix is

separated in multiple matrices according to the part

each deformation vector belongs to. By doing this,

we avoid to perform an additional optimization that

ﬁnds the latent vector of each part that matches the

results of the holistic registration. On the other hand,

this matrix separation assumes the contribution of de-

formation vectors of different parts as negligible. Be-

cause the holistic registration serves as initialization,

the ﬁnal result is given by the deformation vectors de-

ﬁned by each of the individual parts.

Both variants, i.e., the partwise and the hol+part,

are evaluated in Section 4. Figure 2 shows the results

of the non-rigid registration performed in a holistic-

only and in a part-based manner. Note especially for

the handle how the registration by parts improve the

registration accuracy.

Our approach works directly on point sets but it

can also be used to register meshes. An additional

afﬁnity matrix G(T

m

, T) is introduced for this pur-

pose. This matrix maps the deformations from a point

set T to the corresponding mesh vertices T

m

. The

point set T of a mesh T

m

is deﬁned by ray-casting

operations and a voxel ﬁlter. This results in a point

cloud of uniform density. The deformed mesh ver-

tices C

m

0

of the canonical mesh will be then deﬁned

Figure 3: Canonical models of the evaluated categories:

drill, mug, spray bottle, camera and watering can with the

corresponding part decomposition.

as:

C

m

0

= C

m

+ G(C

m

, C) ∗ W

i

. (4)

4 EVALUATION

We tested the category-level part-based registration

approach on ﬁve different categories: drill (17), cam-

era (13), spray bottle (19), watering can (9), and mug

(24). The number of instances for training the shape

spaces are given inside parenthesis. Figure 3 presents

the canonical model of each category with their part

decomposition. The meshes were collected from

online databases: Sketchfab

2

, GrabCad

3

, 3DWare-

house

4

, and from the object meshes recently released

in (Wang et al., 2019). For constructing the shape

spaces, CPD is parametrized with λ = 2, β = 2 and 5

latent dimensions.

Our approach is evaluated with fully and partially

observed shapes. All the testing instances are pre-

sented for the ﬁrst time to our registration method.

The partial views are generated by ray-casting a sin-

gle view of the object from 42 different observation

poses on a tessellated sphere, emulating what a robot

Fully observed

Partially observed

Observed mesh

Segmented cloud

Holistic-only

Hol+part

Figure 4: Results of the non-rigid registration performed on

the spraybottle and camera categories by the hol+part vari-

ant of our approach, compared with a holistic registration.

Observe how the quality of the local structures is improved

by the part-based registration. The canonical deforming

model is shown in blue. For clarity of the registered im-

ages, the observed instances are colored red.

2

https://sketchfab.com/

3

https://grabcad.com/library

4

https://3dwarehouse.sketchup.com/

VISAPP 2022 - 17th International Conference on Computer Vision Theory and Applications

798

0.01 0.02

8.0

8.2

8.4

8.6

8.8

Reg. Error - Drills

Hol

Part

Hol+Part

0.05 0.10 0.15 0.20

8.0

8.5

9.0

9.5

10.0

0.2 0.4 0.6

25

50

75

100

0.025 0.050 0.075 0.100

50

100

150

0.01 0.02

Noise [m]

6.5

7.0

7.5

8.0

Reg. Error - Spraybottles

0.05 0.10 0.15 0.20

Outlier

6

7

8

9

0.2 0.4 0.6

Rotation [rad]

10

20

30

40

50

0.025 0.050 0.075 0.100

Translation [m]

20

40

60

80

100

Figure 5: Average registration errors on full views of the drill and spray bottle categories with respect to different levels of

noise, outliers, rotation and translation.

perceives in a real application. We compare the part-

wise and the hol+part variants of our method against

the holistic registration that do not consider decom-

posed parts. Qualitative results of fully and partially

observed instances are presented in Figure 4. Note

how local structures are better captured by the regis-

tration on the part level.

In order to perform a quantitative evaluation, we

deﬁne the following registration error:

E(T, C) =

1

M

M−1

∑

m=0

min

n

k

T

n

− C

m

k

2

, (5)

which aggregates a normalized distance to the closest

point from the ground truth points T. The numerical

comparison between partwise, hol+part and holistic

on complete ground truth observed instances is pre-

sented in Table 1. Observe that the hol+part variant

outperforms the holistic-only and the partwise regis-

tration variant for fully observed models for all cate-

gories. This means that the registration by parts im-

proves the overall accuracy and that an initial global

registration is beneﬁcial for initializing the subse-

quent partwise one with fully observed instances.

A similar evaluation is performed on partially ob-

served instances whose results are presented in Ta-

Table 1: Average registration error on the evaluated cate-

gories with fully observed instances.

Holistic Partwise Hol+Part

Camera 3.96 3.96 3.67

Drill 7.19 7.06 7.04

Mug 5.21 5.41 5.20

Spray bottle 7.60 7.22 7.05

Watering can 18.72 15.61 15.55

Table 2: Average registration error on the evaluated cate-

gories with 42 partial views.

Holistic Partwise Hol+Part

Camera 0.241 0.20 0.24

Drill 0.67 0.64 0.68

Mug 0.21 0.22 0.22

Spray bottle 0.64 0.65 0.68

Watering can 1.99 1.84 1.90

ble 2. Note that when objects are not complete, the

hol+part variant does not perform well compared to

the other two methods. With partial views, some parts

might have very few points and their contribution to

the holistic registration is low, for this reason, using

the result of an initial holistic registration might lead

to a bad initialization for the subsequent partwise reg-

istration.

We also evaluate the robustness of our approach

against different degrees of noise, outliers, rotation

and translation. We initially add noise to the position

of the observed point set. This noise is sampled from

a Gaussian distribution with zero mean. The value

of the standard deviation is increased gradually from

0.005 to 0.025. Secondly, outliers are incorporated

to the observed instances, i.e., the position of the ob-

served points is not altered but random points sam-

pled from a Uniform distribution inside the bounding

box of the object are added. The ratio of number of

outliers goes from 0.04 to 0.2. We also analyze the

effect of having errors on the observed object pose.

We consider rotation and translation separately. We

rotate the observed instance in each of the global x-,

y- and z-axis consecutively with the same angle. The

direction of rotation of each axis is randomized. The

angle values goes from 0.13 to 0.65 rad with a step of

Category-level Part-based 3D Object Non-rigid Registration

799

Noise

Outliers

Rotation

Translation

Figure 6: Registration results of the holistic-only and part-based registration on partial views on four different categories with

noise, outliers, rotation and translation. The aim is to deform the canonical model (blue) onto the observed instances (red).

The ﬁrst three columns show the observed models with the segmented parts. For clarity in the registered images, the observed

instances are completely colored red. Observe how the registration accuracy of the lens of the camera, the handle of the mug

and the trigger of the spray bottle is improved by the part-based registration. For the translation, the canonical model is also

presented to indicate the observed object pose.

0.01 0.02

0.8

1.0

Reg. Error - Drills

Hol

Part

Hol+Part

0.05 0.10 0.15 0.20

0.65

0.70

0.75

0.80

0.2 0.4 0.6

2

4

6

0.025 0.050 0.075 0.100

2.5

5.0

7.5

10.0

0.01 0.02

Noise [m]

0.6

0.7

0.8

0.9

Reg. Error - Spraybottles

0.05 0.10 0.15 0.20

Outlier

0.50

0.55

0.60

0.65

0.70

0.2 0.4 0.6

Rotation [rad]

1

2

3

4

0.025 0.050 0.075 0.100

Translation [m]

2

4

6

Figure 7: Average registration errors on partial views of the drill and spray bottle categories with respect to different levels of

noise, outliers, rotation and translation. Each point represents the mean error over 42 partial views.

VISAPP 2022 - 17th International Conference on Computer Vision Theory and Applications

800

0.13 rad. Finally, the observed instance is translated

in all three axes from 0.02 to 0.1 m. Examples of the

altered observed instances with the noise, outliers, ro-

tation and translation together with their correspond-

ing holistic and hol+part registrations are presented

in Figure 6. Note how local geometries such as the

lens of the camera, the handle of the mug and the trig-

ger of the spray bottles are better captured with the

hol+part registration.

Numerical results of the drill and spray bottle cat-

egories with fully observed instances are presented in

Figure 5. Figure 7 reports the average registration er-

rors of partial views. Only two categories are pre-

sented but the remaining three behave similarly. The

hol+part variant continues outperforming the other

two methods with noise and outliers on fully observed

instances. However, for rotation and translation, the

partwise variant is more robust, which implies that

misalignments on a global level of the object are more

difﬁcult to reﬁne. Moreover, on partial views the part-

wise variant continues outperforming the other two

methods and is more robust against noise, outliers ro-

tation and translation.

5 CONCLUSION

In this paper, we have presented a novel part-based

non-rigid registration method that improves the accu-

racy of local structures by registering individually de-

composed parts of the object. The method is based

on learned shape spaces of each of the decomposed

parts. We have shown in the experimental section that

a holistic registration followed by a partwise registra-

tion (hol+part) is very effective for registering fully

observed objects, while a partwise registration per-

forms better with partial views. We demonstrated the

robustness of our approach against noise, outliers and

misalignments in rotation and translation. In the fu-

ture, we plan to investigate hierarchies and dependen-

cies between parts of more complex objects. More-

over, we will evaluate the strengths and limitations of

the part-based registration when applied to grasping

skill transfer methods.

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