As Symmetric As Possible : Shape Completion with
Non-Rigid Registration Leveraging Generalized Cylinder Decomposition
Shuji Oishi1 , Masashi Yokozuka1 , Atsuhiko Banno1
Abstract— To infer a 3D entire shape from its partial observation, a non-rigid registration algorithm that employs embedded
deformations is proposed. We construct a deformation graphon a reference model to discretize the space, and compute acomplex deformation as a collection of affine transformations toalign the reference model toward the given geometric data. To
avoid distortion artifacts during the non-rigid registration, we
introduce constraint ”As symmetric as possible (ASAP)” on thegraph via a generalized cylinder decomposition. ASAP allows
model deformation maintaining its underlying local symmetry,
which leads to plausible shape completion in the area with noobservation. We performed experiments with synthesized data,
and demonstrated that the proposed method successfully restored missing surfaces compared with conventional completion
techniques.

I. INTRODUCTION
Inferring complete 3D shapes of objects from a single viewis important for autonomous robots to grab a structure of thescene. As robots capture geometric data with range sensors
such as RGB-D cameras and laser scanners from a certain
viewpoint, the region behind objects may be occluded. This
incomplete observation creates uncertainty in understanding
the surroundings, and results in, for example, inefficient path
planning and awkward object manipulations. Therefore, theability to infer the entire shape from partial cues allows
robots to better understand the surroundings, leading to morestable task execution.
We humans can easily predict the entire 3D shape basedon its partial observation. To imitate this sophisticated humanvisual perception, a variety of techniques have been developed to infer the underlying geometry, as shall be explainedlater in Section II. These shape completion techniques enable
autonomous robots to perform complicated tasks based evenon limited observations, for example, in robotic grasp planning employing occluded-shape inference[1][2] or in shapeuncertainty-driven tactile exploration[3]. In particular, therecent development of a non-rigid registration[4][5][6][7]makes it possible to generate a detailed complete 3D modelby deforming a reference mesh model onto the given partial
shape. However, a difficulty remains in recovering unseen
surfaces from the partial cues because some parts of the reference model have no target to align with and a meaningful
deformation is impossible.
*This work was supported by JSPS KAKENHI Grant Numbers
16K16084 and 18K18072, and a project commissioned by the New Energy
and Industrial Technology Development Organization (NEDO).
1 Smart Mobility Research Team, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Japan {shuji.oishi,

yokotsuka-masashi, atsuhiko.banno}@aist.go.jp

In this paper, we propose a method that overcomes the
drawback of non-rigid registration by leveraging symmetry
extracted via the generalized cylinder decomposition[8]. Our
method decomposes a reference model into a set of skeletaland cross-section profiles, and builds a symmetry-aware
deformation graph by analyzing local symmetry based on thedecomposition result. The deformation graph enables an ”As
Symmetric As Possible (ASAP)” constraint to be introduced
in non-rigid registration, which leads to symmetry-awaremodel deformations to infer the missing surface better.
The main contribution of this paper is combining themodel deformation and the symmetry assumption by leveraging the generalized cylinder decomposition for 3D shapecompletion, and hence allow for a more general non-rigid
registration. Note that we are interested in a complete
and detailed generation of a mesh model starting from areference model the shape of which is moderately similar
but not the same as the query, i.e., an object in the same
category. Although conventional non-rigid registration techniques successfully reconstruct the full 3D model from apartial geometric form, they typically require highly similarreference models, which implicitly assumes the target andreference models (almost) describe the same objects. As
another approach, data-driven techniques[9][10][11][2] haveshown the ability to recover complete shapes only frompartial cues; however, a large amount of relevant data isrequired for learning and the reconstruction results tend tohave some undesirable artifacts. In contrast, the proposed
method seeks to deform the reference model to align thepartial observation while retaining the intrinsic geometry byintroducing the ASAP constraint, which leads to high-quality
completion from one example.
The rest of this paper is organized as follows. First,
the previous shape completion approaches are reviewed in
Section II to see the overview. In Section III, we describe
the algorithm of our shape completion method including
non-rigid registration, deformation graph construction via
the generalized cylinder decomposition, and symmetry-awaremodel deformations. In Section IV, we show experimental
results using synthesized data to quantitatively evaluate theperformance of the proposed method, and also compared our
method with other recent techniques.
II. RELATED WORK
Implicit surface: Implicit surface is one of several popularcompletion methods. By representing the object shape asa signed distance function (SDF) and extracting its zeroisosurface by meshing techniques like the marching cubes, asmooth and closed 3D model can be reconstructed. A variety
of techniques such as those based on the compactly supported
radial basis functions[12] is proposed to interpolate scattered point clouds. Concerning robotic applications, Gaussian
process implicit surface (GPIS)[13], Bayesian representationof SDFs, has been employed to incorporate uncertainty innoisy observations[14][3]. However, implicit surface is only
capable of dealing with small defects and not suitable forcompleting areas with no observation.
Symmetric duplication: To construct unseen surfaces, it
may be effective to make some assumption as to the underlying geometry. Symmetry is one useful option to modelorganic objects and artifacts, in which partial observations
are duplicated in another side. Bohg et al.[1] proposed amethod to predict full shapes of manmade objects withsymmetric duplication for reliable object grasping. It finds
the optimal symmetry plane perpendicular to the tabletop on
which the target objects are located by evaluating whether
the mirrored points are consistent with the original points.
A similar approach can be found in [15] followed by holefilling to complete the 3D models.
Speciale et al.[16] also developed a surface denoisingand completion method making use of symmetries in avariational optimization framework for plausible 3D reconstruction. They first detect multiple planar symmetries that
underlie an incomplete 3D model using random sample
consensus (RANSAC) or Hough transform, and introduce
them as a prior in the variation regularizer to reconstructsmooth and symmetric surfaces in a truncated signed distance function. While it leads to simultaneous denoising andcompletion, extracting accurate symmetry axes from limited
observations is not always trivial. Moreover, the assumption
works well on objects whose shapes can be described only
by symmetric duplications of the partial observations.
Model fitting: When prior knowledge such as complete
3D model database is available, an entire shape of the target
is easily recovered by aligning the same or similar 3D modelin the database to the query depth data. This is the so-called”model fitting”, which enables hidden surfaces to be inferredin detail. For example, Moreno et al.[17] developed a robust
RGB-D SLAM that employed real-time object recognitionand pre-constructed 3D model fitting using the iterative
closest points (ICP) algorithm. Choi et al.[18] proposed a
6-DoF object tracking algorithm using an RGB-D camera
parallelizing likelihood evaluations in particle filtering witha GPU for real-time accurate pose estimation. For semantic
mapping, Zeng et al.[19] developed a simultaneous object
detection and localization method employing conditionalrandom field (CRF) methods to model contextual relations
between objects and achieving significantly higher accuracy
in object pose estimation. Model fitting, however, is applicable only to known objects for which prior 3D mesh models
are given, and thus it is necessary to prepare the 3D modelsof all possible objects in advance.
Deformation: Relaxing the limitations of model fitting,deforming the reference model so that the surface wellaligns with the given observation, is one promising ap-

proach. With simultaneous registration and deformation, it
has the capability of generating highly detailed completed 3Dmodels. Amberg et al.[20] proposed a deformation methodthat estimates an affine transformation on each vertex introducing a stiffness term to control the flexibility of thereference model. Similarly, Li et al.[4][5] presented non-rigid
registration algorithms using embedded deformation[21]which efficiently and robustly solved complex deformations
in a non-linear optimization. More recently, GPU-aided
deformation[6] for realtime shape completion, and L1-norm
regularized deformation[7] for robustness in large changes inposture have been proposed.
However, the methods require highly similar reference
models, which implicitly assumes both target and reference
mesh models represent (almost) the same objects. Thisimplicit assumption allows us to neglect deformations ofsurfaces with no observations; vertices in the reference model
having no correspondences are merely displaced to maintainsmoothness and tend to remain in their original shape ofthe reference model. If the partial observation and reference
model describes almost the same object, the reconstruction
result might be fine.
In contrast, our work considers shape completion withmore general non-rigid registration where shapes of thequery and reference model are not the same, i.e., objects
in the same category, and conventional non-rigid registration
techniques might not be suitable for inferring the missing
surfaces. The most similar technique to our method would
be as-conformal-as-possible surface registration proposed by
Yoshiyasu et al.[22]. They introduced an angle-preserving
stiffness term in non-rigid registration to allow changes inlocal scale (similarity transformation) and enabled non-rigid
registration between different size models. However, as is
the case with other techniques, valid constraints are limited
around partial observations.
III. A S S YMMETRIC A S P OSSIBLE D EFORMATIONA. Overview
Whereas the model deformations allow us to reconstructdetailed complete 3D models by aligning and transforming areference model, the hidden surfaces to be estimated tend tohave almost the same shape as the original reference model.
To make plausible deformations there, we also need to pose
valid assumptions as mentioned in Section II to duplicate thedisplacements by analyzing the geometry.
This motivated us to develop a new completion approachthe ”as symmetric as possible (ASAP)” deformation. By
extracting intrinsic symmetry in the reference model andbuilding a symmetry-aware deformation graph, we enjoy thebenefit of symmetric assumption in non-rigid registration(see Fig.1). In this section, we first describe a non-rigid
registration algorithm based on embedded deformation[21],and then introduce the symmetric constraint derived from thegeneralized cylinder decomposition.

Fig. 1.

Overview of As Symmetric As Possible (ASAP) deformation graph

B. Surface Deformation with Embedded Deformation
Inspired by [5], [4], and [22], we developed a non-rigid
registration employing the embedded deformation[21] for robust and efficient surface registration. Embedded deformation
discretizes the space by defining a deformation graph on
the reference model (Fig.1(a),(b)) that computes complex
deformations as a collection of transformations. Each nodexi ∈ R3 , i ∈ 1 · · · m of the embedded graph induces a local
affine deformation specified by a 3 × 3 transformation matrix
Ai and a 3 × 1 translation vector ti . Each vertex v j ∈ R3 , j ∈1 · · · n of the reference model is mapped to the new positionv0 j with influences from the surrounding nodes described asfollows:

The additional regularization term Ereg ensures smoothness
in deformation. Actually, Ereg sums two energies, specifically, Ereg = Econsist + Esmooth . The following regularization
Econsist allows us to yield consistent affine transformations
and translations.
2

Econsist = ∑ ∑ Ai (x j − xi ) + xi + ti − (x j + t j ) 2
xi x jIn addition, Esmooth is introduced to ensure that the transformation Ti associated with node xi is compatible with thoseof adjacent nodes as follows:2

Esmooth = ∑ ∑ Ai (x j − xi ) + A j (xi − x j ) 2 .
xi x jv0 j∑ w (v j , xi ) [Ai (v j − xi ) + xi + ti ] ,

=

(1)

xi

2
w (v j , xi ) = |1 − d 2 (v j , xi )/dmax
|3 ,

(2)

where d(v j , xi ) denotes the distance between v j and xi ,and dmax the distance from vi to the farthest neighboring
node. Following [5], we used geodesic distances instead of
Euclidean distances to calculate d(v j , xi ) to avoid distortion
artifacts during non-rigid registration.
The node-wise affine transformations Xi = [Ai , ti ] , i ∈1 · · · m are treated as unknowns to be solved in the nonrigid registration. The local features in the reference model
should be preserved to avoid unnatural deformations, andtherefore all Ai in the similarity transformation should satisfy
orthogonality conditions and have the same norm[22].
Esim = ∑aT1 a2

2+ aT1 a3

2

2

+

2

+ aT3 a3 − aT1 a1+ aT2 a3

2

+ aT2 a2 − aT3 a3

2 

(5)

C. Pairwise Registration
To align the reference model toward the target model,
we define another energy term E f it that shifts graph nodes
toward the corresponding positions. Given correspondences
C = {(xi , qi )|i ∈ 1 · · · s} where qi denotes a vertex of thetarget model corresponding to graph node xi , E f it penalizes distances between the corresponding pairs. Because
the simple point-to-point metric may degrade the reference
model, we project the displacements onto the normals of
the nodes to suppress extreme deformations in the tangential
directions[22]. Additionally, assuming the given correspondences C may contain bad corresponding pairs as is often
the case with ICP, we associate a confidence variable ωiwith each node to evaluate the reliability. Specifically, E f it
is defined accompanied with an energy term Econ f [4] for
regularization as follows:xi

aT1 a1 − aT2 a2

(4)

Efit

=

∑ ωi 2 kxi − (x̃i + µi ñi )k22 ,

(6)

i∈C

(3)
where a1 , a2 , and a3 denote the column vectors of Ai .
Esim plays an important role not only in preferring purerigid motions but also in adjusting local scales to ensure
the reference model aligns with the target model well. Thisis essential in performing non-rigid registration between
models of different sizes.

Econf

=

∑ 1 − ωi 2

2

,

(7)

i∈C

where x̃i and ñi are the current position and normal ofnode x̃i respectively, and µi is set as µi = (qi − x̃i ) · ñi . Notethat each ωi is treated as an unknown as well as the affine
transformations to naturally eliminates bad correspondencesthat cause extreme deformations.

Fig. 2.

Symmetry constraint derived from generalized cylinders

D. Symmetry constraint via generalized cylinder decomposition
To better infer the missing surface, we pose an additional
constraint concerning the symmetricity by leveraging thegeneralized cylinder. Cylindrical parts are ubiquitous inmany natural and artificial objects. Therefore, the generalized cylinder, which is composed of pairs of skeletal andcross-section profile curves, covers a broad scope of object
shapes[23][24][25]. We decompose the reference 3D model
into generalized cylinders to analyze its symmetry, and build
a symmetry-aware embedded graph so that displacements
toward the partial observation affect the other side of thereference model to maintain the symmetric structure, thereby
recover missing parts in more detail.
First, a reference mesh model is decomposed into a set ofgeneralized cylinders following [8]. The generalized cylinder decomposition[8] progressively builds local to non-local
cylinders and decides on a globally optimized combination ofcylinders in a manner of the exact cover problem. It provides
consistent skeletal and cross-section profile curves (see Fig.
1(c)), and enables us to analyze local symmetry.
As a next step, we introduce symmetry-aware edges in theembedded deformation graph based on the decomposition result. The embedded deformation graph is constructed throughfarthest-point sampling and K-nearest neighbors (m = 400
and K = 4 in all experiments) to approximate the reference
model by a coarse mesh model. To preserve the symmetryof the reference model, we additionally connect symmetricnodes. Specifically, we use a skeleton point for a central
symmetry, or extract an axial symmetry passing through theskeleton point that minimizes symmetry distance [26] on
every 2D profile curve, and form symmetric point pairs in
the generalized cylinders (Fig.2). The symmetric edges arethen added in the deformation graph by approximating thepoint pairs with the closest nodes (Fig.1(d)).
Finally, for symmetry-aware deformation, we replace
Equation (6) with the following ASAP energy term:no2Easap = ∑ ωi 2 kxi − (x̃i + µi ñi )k22 + x j − (x̃ j + µi ñ j ) 2 ,
i∈C

(8)

where (xi , x j ) are connected with a symmetric edge. ASAP
allows the reference model to deform itself maintaining its
underlying symmetry by duplicating displacements µi on
another side; this leads to plausible shape completion in
the area with no observation. Note that if the symmetricnode x j has its own position constraint, the latter symmetricconstraint in Equation (8) is ignored to avoid inconsistent
deformation.
E. Optimization
We define an objective function in the non-rigid registration by summing up the energy terms:
E= αsim Esim + αreg Ereg + αasap Easap + αcon f Econf . (9)The number of optimization variables is 13m, corresponding
to the affine transformations and confidences in the embedded deformation graph. We minimize Equation (9) by solving
the nonlinear least-squares problem with the Gauss-Newton
algorithm using Cholesky factorization. By alternating theenergy minimization and correspondence determination, we
gradually deform the reference model.
The optimization weights are initially set as αsim = 100,αreg = 100, αasap = 10, and αcon f = 10, and αreg is halved
at every iteration until αreg < 10. We then remove theconfidence variables ω and replace Easap as the following
energy term to pose the vertex-wise constraints; we thenperform an iterative optimization again to capture the detailsof the target model according to vertex correspondences C ∗ .
∗
Easap
= ∑

i∈C ∗

n
o
2
kvi − (ṽi + µi ñi )k22 + v j − (ṽ j + µi ñ j ) 2 ,(10)
IV. E XPERIMENTS

A. Results
We performed experiments to evaluate the performance ofthe proposed method. 3D models used in the experiments
were obtained from a benchmark[27]. We first selected threemodels belonging to different categories as reference models
to be deformed (Fig.3(a)). To evaluate the completion results

Fig. 3.

Experimental results and error comparisonsquantitatively, we synthesized incomplete geometric dataassuming partial observation (Fig.3(b)). Specifically, each
incomplete model was used as a query and a reference
model from the same category was deformed with nonrigid registration techniques to infer the entire shapes. The
completion results were evaluated by calculating residual
errors between the original and completed models based onvertex correspondences via nearest neighbor searches.
Note that the initial registration is performed manually
so that the positions and rotations are roughly aligned,
which can be replaced with conventional rigid registrations.
Correspondences between the reference and target models
are computed using closest point methods like ICP whileeliminating invalid ones when the normal difference exceeds
40[deg], or the position difference exceeds 20% of the longest
side of the bounding box of the reference model. In addition,
we assume that the reference models in this experiment have
local point-symmetric structures, and built point-symmetryaware embedded graphs for ASAP.
Fig. 3(c)-(f) shows the completed 3D models with nonrigid registration techniques including the proposed method.

Quantitative comparisons are also shown in Table I. Overall,
the method of [20] aligned reference models well toward
the query shapes, however, the other sides of the modelsignificantly distorted. The non-rigid alignment technique[4] based on the embedded deformation did not work wellbecause the method assumes isometric registration while theshapes of reference and target models in this experiments
were substantially different. Although the approach of [22]provided better results by allowing similarity transformations
for different size models, the other side of the deformedmodels still remained as in the original reference models.
In contrast, our method restored the entire 3D models fromthe partial observation by making the graph nodes cooperateeach other to retain the symmetry of the structure, whichleads to better shape predictions.
B. Limitations
In the human case, the performance of the proposed
method got worse compared with in the other cases. Thismay come from the poor description of the underlying
symmetry as a result of the rough assumption of pointsymmetry. Therefore, we perform an additional experimentFig. 4.

Completion with an axial symmetry-aware graph

TABLE IQ UANTITATIVE COMPARISONS USING 3D MESH MODELS FROM [27]. T HE ERRORS INDICATE AVERAGE AND MAX DISTANCES [%] FROM THE
COMPLETE TARGET MODELS TO THE DEFORMED REFERENCE MODELS RELATIVE TO THE LONGEST SIDES OF THE BOUNDING BOXES .
Category
tumblermug
human

Referencemodel 40model 27
model 2RMS errors (average / max)Targetmodel 25
model 33
model 13

Amberg et al.[20]9.32 / 33.8
6.03 / 26.9
0.557 / 4.22

with an axial-symmetry-aware embedded graph. As shown in
Fig. 4 and Table I, the graph represents the local symmetry
in the reference model better, especially in the torso, andimproved the completion performance. This implies that
adaptive symmetry extraction is essential to obtain valid
symmetry constraints.
In addition, as mentioned in [22], non-rigid shape matching should be used to register different size models. In the
mug case, none of the registration techniques successfullyaligned with the handle because of difficulty in obtaining
valid correspondences via closest point method. Moreover,
ASAP can largely shrink or inflate the partial shapes of
the reference model when it couldn’t get appropriate correspondences (See red circles in Fig. 3). Therefore, non-rigid
matching like [28] should be used in the initial registration,
and as well in the iterative optimization if needed, to boost
the performance of the proposed method.
V. C ONCLUSION
We proposed a method to infer an entire 3D shape frompartial geometric data with a new non-rigid registration.
By representing the reference model as a symmetry-aware
deformation graph, we developed an effective non-rigid registration algorithm that tried to preserve the local symmetry
and restore plausible surfaces.
For better performance in non-rigid registration of different size objects, we will improve the correspondence searchand the symmetry extraction. In addition, while we currently
focus only on local symmetry extracted via generalized cylinder decomposition, some objects also have global symmetry.

Li et al.[4]5.22 / 25.4
3.12 / 14.8
1.27 / 4.97

Yoshiyasu et al.[22]2.37 / 15.1
3.65 / 18.1
0.508 / 5.23

Our method
1.32 / 9.08
2.10 / 11.8
0.708 / 9.36 (point-sym.)0.625 / 4.87 (axial-sym.)

Therefore, we will address shape inference using global andlocal symmetry simultaneously for better completion results.
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