Eurographics Workshop on 3D Object Retrieval (2017)
I. Pratikakis, F. Dupont, and M. Ovsjanikov (Editors)

SHREC’17: Deformable Shape Retrieval with Missing Parts
E. Rodolà1 , L. Cosmo2 , O.Litany3,4 , M. M. Bronstein1,3,4 , A. M. Bronstein3,4,5
N. Audebert , A. Ben Hamza , A. Boulch9 , U. Castellani17 , M. N. Do16 , A-D. Duong14 , T. Furuya6 , A. Gasparetto2 , Y. Hong10 , J. Kim10 ,B. Le Saux9 , R. Litman3 , M. Masoumi8 , G. Minello2 , H-D. Nguyen13,15 , V-T. Nguyen13,14 , R. Ohbuchi6 , V-K. Pham13 , T. V. Phan13 ,M. Rezaei8 , A. Torsello2 , M-T. Tran13 , Q-T. Tran13 , B. Truong13 , L. Wan7 , C. Zou11,12
9

8

1 USI Lugano, 2 Università Ca’ Foscari Venezia, 3 Tel Aviv University, 4 Intel Perceptual Computing, 5 Technion, 6 University of Yamanashi,
7 Beijing Jiaotong University, 8 Concordia University, 9 ONERA, The French Aerospace Lab, 10 Sangmyung University, 11 Hengyang Normal University,
12 Simon Fraser University, 13 University of Science, VNU-HCM, 14 University of Information Technology, VNU-HCM, 15 John von Neumann Institute,

VNU-HCM, 16 University of Illinois at Urbana-Champaign, 17 University of Verona

Abstract
Partial similarity problems arise in numerous applications that involve real data acquisition by 3D sensors, inevitably leading
to missing parts due to occlusions and partial views. In this setting, the shapes to be retrieved may undergo a variety oftransformations simultaneously, such as non-rigid deformations (changes in pose), topological noise, and missing parts – a
combination of nuisance factors that renders the retrieval process extremely challenging. With this benchmark, we aim toevaluate the state of the art in deformable shape retrieval under such kind of transformations. The benchmark is organized in
two sub-challenges exemplifying different data modalities (3D vs. 2.5D). A total of 15 retrieval algorithms were evaluated in
the contest; this paper presents the details of the dataset, and shows thorough comparisons among all competing methods.
Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometry and Object
Modeling—Geometric algorithms, languages, and systems

1. Introduction
Shape retrieval concerns the problem of finding 3D shapes from ashape repository, that are as similar as possible to a given query.
It is an active area of research, finding application in several areas of computer vision, graphics, and medical imaging among
several others. A challenging setting of 3D shape retrieval occurs when the shapes to be retrieved are allowed to undergo nonrigid deformations, which may range from simple changes in pose(near-isometric deformations) to more complicated transformations
such as local stretching, intra-class variation, and topological noise.
Even more challenging is the setting in which the objects in question have missing parts in addition to the aforementioned deformations. In this scenario, the query object is a subset of the full modeland the task is to retrieve similar objects with different partiality patterns (or lack thereof), additionally undergoing a non-rigid
transformation; vice versa, the query object might be a full model,
while the shape repository may contain deformed partial views ofobjects of the same class.
With this track, we propose a benchmark to evaluate the performance of partial shape retrieval methods, under the concurrent
presence of:• Non-rigid deformations;• Different amounts and types of partiality;• Topological changes induced by mesh gluing in areas of contact.
c 2017 The Author(s)Eurographics Proceedings c 2017 The Eurographics Association.

Figure 1: Example of shapes from the ‘holes’ (left pair) and ‘range’(right pair) challenges. Shape boundaries are colored in red.

Related benchmarks. Although recent publications in the fieldof shape analysis [DLL∗ 10, WZZ16, CRA∗ 16, CRM∗ 16] suggestthat the community is starting to show interest in the problem
of deformable partial shape retrieval, there is a notable lack ofbenchmarks targeting this task. Similar benchmarks addressing
the related problem of correspondence were recently proposed[CRB∗ 16,LRB∗ 16a], but these only include a relatively small number of shapes – a few hundreds as opposed to the several thousands that would be required to provide a meaningful evaluationof retrieval pipelines. Partial shape retrieval was recently addressedin [PSA∗ 16] for rigid objects (partial views of vases), and a few

Rodolà et al. / SHREC’17: Deformable Shape Retrieval with Missing Parts

years back in [MPB07, BBC∗ 10] (mostly CAD models, with only
2 classes presenting mild partiality and changes in pose).
By constructing a new, large, specific and challenging datasetwith partial deformable shapes, we aim to provide a valuable
testbed for the most recent top-performing retrieval methods, which
all too often we see “saturate” existing benchmarks that are notspecifically designed with partiality in mind. We hope that thisnew benchmark will foster further interest of the community in thischallenging problem.
2. Dataset
The dataset is composed of transformed versions of shapes belonging to the TOSCA repository [BBK08] and to the SHREC’16
Topology [LRB∗ 16a] benchmark. The former consists of 9 shapecategories (humans and animals) under various changes in pose; thesecond includes different poses of a fixed human subject, additionally undergoing topological transformations in the form of mesh‘gluing’ in areas of self-contact. The dataset we propose is divided
into two subsets, each constituting a different challenge:• ‘holes’: On each model we consider 70 Euclidean farthest samples. A random subset of at least 20% of these samples is usedas seed for an erosion process with random geodesic radius (upto 20% shape diameter per sample). Each resulting partial shapeis remeshed by edge contraction [GH97] to a random amount ofvertices between 300 and the full mesh resolution, followed byapplication of a random similarity transformation.
• ‘range’: A virtual orthographic camera is placed in front of 5
randomly rotated versions of each model, at 4 random elevations
and resolutions. The resulting 2.5D snapshot is triangulated according to a regular grid, with random maximum allowed edge
length (this generates topological shortcuts between parts thatare originally disconnected).
See Fig. 1 for examples. Note that the shapes might consist of multiple connected components in both cases. Each set is further divided into a training set, for which (category-level) ground-truthlabels are provided, and a challenge set. We consider ‘david’ and‘michael’ from the TOSCA repository as belonging to the sameclass, while ‘victoria’ and ‘kid’ constitute separate classes.
In total, the dataset consists of 1216 training / 1078 test shapesfor ‘holes’, with resolution spanning 323 to 56172 vertices, and1082 training / 882 test shapes for ‘range’, with 300 to 7021
vertices. The dataset remains available for download at http://sites.google.com/view/shrec17/.
3. Evaluation measures
The participants were asked to submit a full distance/dissimilarity
matrix D ∈ [0, 1]n×n for each challenge, where n is the numberof shapes in the test set and (di j ) denotes the dissimilarity between shape i and shape j. Participation to both challenges was
not mandatory; up to 3 runs per method were allowed.
For the evaluation we adopt the standard measures for shape retrieval according to the Princeton benchmark protocol [SMKF04]
using code from [LZC∗ 15]. In particular, we show Precision/Recall
curves, and tabled values for nearest neighbor (NN), 1st tier, 2ndtier, and discounted cumulative gain (DCG, at rank equal to n).

4. Methods
The contest saw the participation of 8 research groups, some ofwhich participated with multiple submissions (resulting either fromdifferent retrieval methods, or different runs of the same method).
In addition, the organizers implemented two baselines (Sections 4.1
and 4.2), both of which are extensions of existing approaches, properly modified to account for missing parts.
4.1. Bag of Words (BoW) [L. Cosmo and E. Rodolà]
As a first baseline, we use a simple BoW approach based on thescale-invariant Heat Kernel Signature (siHKS) [KB10], computedon keypoints extracted as the local maxima of the original HKS[SOG09]. The overall pipeline is composed by the following steps:
1. For each shape, salient points are extracted using [SOG09], andsiHKS features are computed with a variant of [KB10];
2. A dictionary is built by descriptor clustering via k-means;
3. Each shape is represented by a frequency histogram, constructedby assigning each feature to the nearest word in the dictionary;
4. The pairwise distance between shape histograms is computed.
The code for this baseline is available on the track website.
Keypoints and descriptors. Salient points of a mesh M are extracted using the method proposed in [SOG09]: given a diffusion
time t and the associated self-diffusivity Ht , we consider a pointx ∈ M as a salient point if Ht (x) > Ht (x0 ) for all x0 in the one-ringneighborhood of x. Boundary vertices are ignored.
The descriptor is then transformed according to:Ḣt = log(Ht+1 ) − log(Ht ) .
Scaling of the original shape corresponds to a shift of Ḣ. Conversely
from the original method [KB10], we
do not use the DFT of Ḣ to attainscale-invariance; instead, we restrict
each descriptor to within a small window starting from the first local minimum/maximum (see inset figure).

(1)

1x

HKS

siHKS

1.5x

HKS
siHKS

BoW representation and matching. Once salient points descriptors are computed for each shape, a dictionary is constructed byclustering all the descriptors with a k-means algorithm, and by considering their centroids as words of a dictionary C = (c0 , . . . , ck ) .
A frequency histogram is then computed for each shape M, assigning each descriptor diM to the nearest word in the dictionary:bM = ∑ earg min j d(d M ,c j ) ,
i

(2)

i

where e j is 1 at position j and 0 at all other indices, and d(ci , c j ) =∑t |cti − ctj | is the cityblock distance.
Finally, the distance between two shapes M and N is computed
using the cosine similarity:
dM,N = 1 − q

b0M bN(b0M bM )(b0N bN ).

(3)

c 2017 The Author(s)Eurographics Proceedings c 2017 The Eurographics Association.

Rodolà et al. / SHREC’17: Deformable Shape Retrieval with Missing Parts

Implementation details. Even if the descriptor used is scaleinvariant, one needs to consider comparable diffusion times to beable to retrieve the relative scale; to this end, the shapes were normalized by the area of their bounding box. Moreover, for increased
robustness to partiality and to avoid boundary effects, small diffusion times must be used for the computation of the HKS. For thisstep we replaced the standard spectral implementation of the heat
kernels [SOG09] with the robust Chebyshev-Padé method introduced in [Pat13]. This proved to be crucial for an accurate evaluation of the descriptors under large amounts of partiality.
The baseline comes in two variants, namely BoW-siHKS (described above) and BoW-HKS, where the logarithm of the HKS is
used instead of siHKS. Diffusion times were sampled logarithmically at 80 samples between 2−16 and 2−6 , and the window sizefor the scale-invariant feature was set to 20 time samples. The dictionary size was set to k = 128.

4.2. Partial ShapeDNA (DNA) [E. Rodolà and L. Cosmo]
The second baseline is an extension of the ShapeDNA global shapedescriptor described in [RWP06]. This approach is based upon theobservation that, assuming lack of partiality (i.e., in the standard
full shape setting), the spectrum of the Laplace-Beltrami operator
of a shape acts as an isometry-invariant “fingerprint” for recognition. Recently, Rodolà et al. [RCB∗ 16] showed how the removalof shape parts modifies the Laplacian spectrum, providing a strong
prior that was subsequently used for the purpose of shape matching.
In particular, consider a full shape M and a subset N ⊆ M (oran isometrically deformed version of N ), and denote by (λ j )kj=1
and (µi )ki=1 the increasingly ordered sequences of the first k Laplacian eigenvalues of M and N respectively. Then, the equalityµi ≈ λ j holds (approximately) for i ≤ j and i = 1, . . . , r, wherearea(M)r ≈ bk area(N ) c [RCB∗ 16]. See the inset figure below for an illustration. Note that the standard setting where N is a full isometric deformation of M, assumed in the original ShapeDNA method[RWP06], is obtained with i = j and r = k. Litany et al. [LRB∗ 16b]
later showed that the part-to-full analysis of [RCB∗ 16] can be extended with little modifications to the part-to-part setting.
One may use this knowledge for theλtask of partial shape retrieval by com- 0.1µ
paring the spectra (µi ) and (λ j ) at thepairs of indices I = {(i, bi kr c) | i =µr
λk1, . . . , r}, to compute a dissimilarity
score σ(N , M) = ∑(i, j)∈I |µi − λ j |.
0
The approach works under the assump0
25
50
Eigenvalues
tion that the two shapes M, N havethe same scale; however, in this challenge, the test shapes are randomly rescaled so that N ⊆ αM, where α ∈ R is unknown. In particular, this induces a transformation of the eigenvalues µi 7→ α12 µi .
In order to account for the unknown scale, the score σ(N , αM)
is computed for a fixed range of scales α ∈ [10−1 , . . . , 10] sampled
at N = 103 equally spaced values; then, the minimum score is kept
for the given pair of shapes. We refer to Section 6 for further considerations on the lack of a consistent scale across the dataset.
c 2017 The Author(s)Eurographics Proceedings c 2017 The Eurographics Association.

4.3. Deep aggregation of localized statistical features (DLSF)[T. Furuya and R. Ohbuchi]
The algorithm aims at extracting 3D shape descriptors that are robust both against non-rigid deformations and noise, such as irregular holes on the surfaces. It is built upon previous studies on local 3D geometric features [OOFO12] and deep learning-based aggregation of local features [FO16]. Fig. 2 describes the processing
pipeline of DLSF. DLSF takes as its input a set of low-level local
3D geometric features. The low-level feature is inherently invariant
against geometric transformations including translation (1DOF),
uniform scaling (1DOF), and rotation (3DOF) of 3D shapes. DLSF
produces a compact, high-level feature per 3D model for efficientand effective matching among non-rigid 3D models. The algorithm
consists of the following four steps.
Generating oriented point set. Given a polygon-based 3D model,
it is first converted into a 3D oriented point set by sampling the surface using the algorithm by Osada et al. [OFCD02]. This sampling
process gives DLSF a certain invariance against different shape
representations. The algorithm randomly and uniformly samples
points on the surface; each point is associated with the normal vector of the triangle on which the point is sampled. For each model,
3K oriented points are sampled. The oriented point set of the 3Dmodel is uniformly scaled to fit a sphere having unit diameter.
Extracting local statistical features. A set of 100 Spheres-OfInterest (SOIs) is sampled from the oriented point set of the 3Dmodel. Position and radius of each SOI are chosen randomly.
Each SOI is then described by Localized Statistical Feature (LSF)[OOFO12], which is formed as a 625-dimensional vector. The reasons for employing LSF as descriptor for SOIs are two-fold. First,
LSF has invariance against 3D rotation of points within a local
region, a desirable property for attaining robustness to non-rigid
deformations. In the SHREC 2015 non-rigid 3D shape retrieval
track [LZC∗ 15], LSF, combined with an unsupervised local feature
aggregation algorithm, is quite capable in comparing non-rigid 3Dmodels. Secondly, LSF tends to be robust against geometrical andtopological noise caused by irregular holes on the surface. A local feature similar to LSF [RBB09] has successfully been applied
to the task of 3D registration, where 3D models generated by 3Drange scanners present surface cracks and holes.
Aggregating local features. A deep neural network (DNN) is employed to aggregate the set of LSFs into a single feature per 3Dmodel. The DNN consists of two blocks; (1) E-block, performing
Encoding of LSF, and (2) AC-block, performing Aggregation of
the encoded LSFs into a feature per 3D model, followed by Compression of the aggregated feature. The E-block comprises threefully-connected layers, each having 3072, 2048, and 512 neurons,
respectively. The AC block pools, by averaging, the set of encoded
mid-level local features into a single feature per 3D model. The
subsequent three fully-connected layers, with their 1024, 512, and
128 neurons, compress the aggregated feature to produce a compact and salient feature per 3D model. ReLU is used as activation
function for the fully-connected layers.
Comparing aggregated features. Finally, a compact (128dimensional) aggregated feature of the query model is efficiently

Rodolà et al. / SHREC’17: Deformable Shape Retrieval with Missing Parts

Convert tooriented point set

Extract
LSF

Sample
SOIs

Aggregate local features &compress aggregated feature
aggregation
by average
pooling

…

…

…

…

…

…

aggregatedfeature…

…

…

low-level
local features(LSFs)

classification layer(training only)…

…

…

oriented
point sets
inside SOIs

…

…

…

…

oriented
point set

…

…

…

3D
model

Encodelocal featureshigh-level
aggregatedfeaturemid-level
local features
AC-block

E-block

deep neural network for aggregating local features

Figure 2: Processing pipeline of DLSF (Section 4.3). The 3D model is represented as a set of low-level, local 3D geometric features having
invariance against 3D rotation in the SOI. The deep neural network aggregates the local features to a compact and salient feature per 3Dmodel for effective matching among non-rigid 3D models.

compared against the aggregated features of the targets in the
database. Cosine similarity is used for this step.

error incurred when reconstructing one’s descriptor set using theother’s dictionary. No training set is needed in the whole process.

The DNN is trained with a two-step process. The E-block is pretrained first by using a large set of labeled SOIs. Then, the entire
network is trained by using a set of labeled 3D models.

In this method, the computation of the HKS [SOG09] descriptors is modified on the following aspects: (1) The descriptors arecalculated on the largest connected component for a disconnectedshape, while some descriptors of the boundary vertices and their 1ring neighbors are excluded; (2) The diffusion time scales are adaptively set for each shape, rather than some fixed values. The modified descriptors are called I-HKS descriptors. To compute thesedescriptors for a shape, the eigenvalues λ0 , . . . , λ99 and the corresponding eigenfunctions φ0 , . . . , φ99 of the Laplace-Beltrami operator are used. After setting tmin = 4 ln 10/λ1 and tmax = 4 ln 10/λ99
according to [SOG09], the diffusion time scales are chosen fromtstart = tmin to tend = tmin + (tmax − tmin )/10. The time scales arethen formulated as:

Pre-training E-block. To learn expressive local feature better
suited for accurate 3D shape retrieval, the E-block is pre-trained byusing a large number of labeled SOIs so that it could predict object
categories of the SOIs. To do so, SOIs are sampled at random position and scale from the labeled training 3D models. The label foreach SOI is identical to the label for the 3D model from which the
SOI is sampled. A total of 3M labeled SOIs are collected from thetraining set. The classification layer is appended at the output endof the E-block. Parameters in the E-block, i.e., connection weights
among neurons, are randomly initialized and cross-entropy loss is
minimized by using AdaGrad [DHS11] with initial learning rateequal to 0.1. A 20% dropout is performed for all the hidden layers.
The pre-training is iterated for 10 epochs.

ti = 10lg tstart +lg tend −lg tstart
(i−1)
n−1

,

i = 1, . . . , n.

(4)

Next, sparse dictionary learning [MBPS10] is utilized to compute a dictionary for each shape. For a shape SA , after its I-HKS
set F A = { f Aj | j = 1, . . . , NA } is computed, its dictionary DA can beobtained by solving the constrained optimization problem

Training whole network. After supervised pre-training of the Eblock, the entire network is trained including both the E-block and
the AC-block by using the labeled 3D models from the training set.
The parameters in E-block inherit as their initial values the result of
the pre-training. The parameters in the AC-block are randomly initialized. Training of the entire DNN is done by minimizing crossentropy loss by using AdaGrad with initial learning rate equal to
0.1 and 20% dropout. The training is iterated for 100 epochs.

where γ Aj consists of sparse coefficients and T is a sparsity threshold.

DLSF has per-query runtime of 0.063s, of which 0.061s areneeded for feature extraction, and 0.002s for comparison (measured
on a machine with Intel Core i7-6700 CPU, Nvidia GeForce GTX
1080 GPU and 64GB DRAM).

The shape similarity measure is slightly different to [WZZ16].
Given two shapes SA and SB , if using SA ’s dictionary D A tosparsely code SB ’s I-HKS f Bj , the reconstruction error is expressedas:

DA = arg min
D̃
DA

1 NA
∑ k f Aj − D A γ Aj k22
NA j=1E( f Bj , D A ) = min k f Bj − D A γ Bj k22
4.4. Sparse reconstruction (SR) [L. Wan et al.]Wan et al. [WZZ16] propose to measure the similarity between
two shapes based on sparse reconstruction of shape descriptors.
For each shape, its local descriptors and sparse dictionary are computed. The similarity between two shapes is then defined by thes.t. kγγAj k0 ≤ T,

s.t. kγγBj k0 ≤ T.

(5)

(6)

Therefore, the average reconstruction error for coding the I-HKS
set F B is formulated as:
F B, DA) =
E(F

1 NB
∑ E( f Bj , D A ).
NB j=1(7)c 2017 The Author(s)Eurographics Proceedings c 2017 The Eurographics Association.

Rodolà et al. / SHREC’17: Deformable Shape Retrieval with Missing Parts

Similarly, if using SB ’s dictionary D B to represent SA ’s I-HKS set
F A , D B ). The distanceF A , the average reconstruction error is E(Fbetween SA and SB is finally defined as:
F B , D A ), E(FF A , D B )).
Dist(SA , SB ) = min(E(F(8)

All the parameter settings are the same as [WZZ16].
This method assumes that a partial shape has a large connectedcomponent which still keeps most of the corresponding full shape.
So, if a partial shape is broken into many pieces or misses too muchsurface information, this method may be not suitable.

4.5. Geodesic multi-resolution framework (GMR)[M. Masoumi et al.]The framework of [MLH16, MH17] is based on the eigensystemof the Laplace-Beltrami operator, which is invariant to isometric transformations. A geodesic multi-resolution descriptor is defined by incorporating the vertex area into the definition of spectral
graph wavelet [LB13] in a bid to capture more geometric information and, hence, further improve its discriminative ability. A Mexican hat wavelet is utilized as a generating kernel, which considers
all frequencies equally important as opposed to the cubic spline
kernel [LB13]. Furthermore, in order to capture the spatial relations between features, the geodesic multi-resolution descriptor isweighted by a geodesic exponential kernel.
The framework consists of four main steps. The first step isto represent each 3D shape in the dataset D by a spectral graphwavelet signature matrix S, where S = (s1 , . . . , sm ) ∈ R p×m , si isthe p-dimensional local descriptor at vertex i, and m is the number of mesh vertices. In the second step, the area-weighted spectral
graph wavelet signatures si are mapped to high-dimensional midlevel feature vectors using the soft-assignment coding step of the
BoF model, resulting in a k × m matrix U = (u1 , . . . , um ) whose
columns are the k-dimensional mid-level feature codes. In the third
step, the k × k geodesic multi-resolution matrix F is computed using the mid-level feature codes matrix and a geodesic exponential
kernel as:|

F = UKU ,(9)

where U is a k × m matrix of geodesic multi-resolution codes (i.e.
mid-level features), and K = (κi j ) is a m × m geodesic exponential
kernel matrix whose elements are given by!
di2j
κi j = exp −,(10)εwith di j denoting the geodesic distance between vertices vi andv j , and ε is a positive, carefully chosen parameter that determines the width of the kernel. Matrix F is reshaped into a k2 dimensional global descriptor xi . In the fourth step, the geodesic
multi-resolution vectors xi of all n shapes in the dataset are arranged into a k2 × n data matrix X = (x1 , . . . , xn ). Finally, a query x
is compared to all data points in X using the `1 -distance. The lower
the value of this distance, the more similar the shapes.
The experiments were conducted on a desktop computer withan Intel Core i5 3.10 GHz CPU and 8 GB RAM; the algorithms
c 2017 The Author(s)Eurographics Proceedings c 2017 The Eurographics Association.

were implemented in MATLAB. In this setup, a total of 201 eigenvalues and associated eigenfunctions of the LBO were computed.
The resolution parameter was set to R = 2 (i.e., the spectral graphwavelet signature matrix is of size 5 × m, where m is the numberof vertices) and the kernel width to ε = 0.1. The parameter of thesoft-assignment coding is computed as α = 1/(8µ2 ), where µ is themedian size of the clusters in the vocabulary.

4.6. SnapNet [A. Boulch et al.]The objective of the approach is to learn a classifier that will produce similar outputs for the same shapes with different poses. The
training set is composed of 10 different classes. For a new shape,
the classification vector obtained with this new shape is used as a
description of the shape.
Training dataset. The training dataset isgenerated by taking snapshots around the
3D model (see inset) [Gra14]. In order tocreate visually consistent snapshots, thepoint cloud is remeshed using [MRB09].
The snapshots are 3-channel images: thefirst channel encodes the distance to thecamera (i.e., depth map), the second is the normal orientation to
the camera direction, and the third channel is an estimation of thelocal noise in the point cloud (ratio of eigenvalues of the neighborhood covariance matrix).
The adopted CNN is a VGG16 [SZ14] with a final fullyconnected layer with 10 outputs. The weights are initialized withthe model trained on the ILSVRC12 contest; the network is thenfine-tuned using a step learning rate policy.
Distance computation. The classifier is then applied to images andproduces images classification vectors vim . For each model, a prediction vector VM is computed based on the images :VM =∑im∈M vim.
|| ∑im∈M vim ||2

The distance matrix X contains the pairwise `2 distances between the VM . Each line is then normalized using a soft max :Xi, j =

exp(Xi, j ).
∑ j exp(Xi, j )Note that matrix X is not symmetric. Finally, a symmetric distancematrix is defined D = XT X. The values of D are clipped according
to the 5th and 50th percentiles and then re-scaled in [0, 1].
The method was implemented Python and C++, using the deeplearning framework PyTorch. The experiments were performed ona CPU Intel Xeon(R) E5-1620 3.50GHz. The training part was operated on a Nvidia Titan X Maxwell GPU and the test part (predictions) on a Nvidia GTX 1070. Generating the snapshot took around
10 seconds per model. The training took around 8 hours. The prediction vectors were generated in 2 seconds per model and the dissimilarity matrix was computed in less than 10s.

Rodolà et al. / SHREC’17: Deformable Shape Retrieval with Missing Parts

In the training step, a total of 1216 × 42 = 51072 (‘holes’) and
1082×42 = 45444 (‘range’) depth images (of size 224×224) wereused as training data, where 10% of these were used for validation. The training used momentum with a decay of 0.9, learning
rate of 0.01, and every epoch was decayed using an exponential
rate of 0.96 with dropout ratio set to 0.1. All training data waszero-centered and then scaled by its standard deviation as a preprocessing step; batch size was set to 64. Training for ‘holes’ wasstopped after 38 epochs, where the validation accuracy reached84.49% with no further growth. Training for ‘range’ was stopped
after 34 epochs, with 86.47% validation accuracy and no further
growth.
In the test step, for a given 3D shape, a 10-dim. feature vector is extracted to represent the model. As in the training step, 42
depth images are taken from the 3D shape and input to the trained
GoogLeNet, which outputs 10 prediction values per depth image,
making for a total of 42 × 10 prediction values. These are averagedon each dimension to generate a 10-dim. vector, which is taken as
the final feature vector for the test shape. The final score between
two 3D shapes is given by the Euclidean distance of the corresponding feature vectors.

Figure 3: Pipeline of the 2VDI-CNN method (Section 4.7).

4.7. 2D-view depth image and CNN-based method(2VDI-CNN) [Y. Hong and J. Kim]
This method makes use of 2D-view depth images [PTK06,LGS10]
and a convolutional neural network (CNN); Fig. 3 shows the general pipeline.
A given 3D shape is represented by 42 depth images which areused for training. In order to take depth images, the center (averageof vertices coordinates) of the 3D shape is translated to the origin
and the maximum polar distance of the vertices is scaled to one.
Then, 42 depth images are taken on the vertices of a Pentakis icosidodecahedron (subdivided icosahedron), which is a convex polyhedron with 80 triangular faces and 42 vertices.
The training shapes are labeled to 10 categories, and each depth
image is assigned the label of the corresponding class. These depth
images and corresponding labels are put into a CNN for training. The chosen architecture is GoogLeNet [SLJ∗ 15] (also referred
to as Inception v1), the winning architecture on ImageNet 2014.
Although there are other architectures that perform better than
GoogLeNet, such as Inception v4 [SIVA16] and ResNet [HZRS16],
the 4GB GPU used for the contest set a hardware limit on the typeof architecture. GoogLeNet’s last softmax layer was changed to
output 10-way prediction values.

GoogLeNet was built by TFLearn (http://tflearn.org)
on top of Tensorflow. Depth images were taken with MATLAB 3D Model Renderer (http://www.openu.ac.il/home/hassner/projects/poses/). The method was implemented on
a 3.60 GHz i7-4790 CPU, 8GB DDR3 RAM, GeForce GTX 960
4GB rig. On this system, the average runtimes for taking depth images and generating the predictions for 10 shapes is 3.25s and 0.31s(on GPU) respectively. Calculating the similarity between a queryand 882 shapes takes only 0.005s. Better performance could possibly be achieved by tuning the hyperparameters for training and byusing better CNN architectures.
4.8. Non-parametric spectral model (NPSM)[A. Gasparetto et al.]The method described in [GMT15] is a supervised approach for theconstruction of a generative model based on the spectral decomposition of the Laplace-Beltrami operator. The idea is to define astatistical framework that models a shape as two independent generative models for the eigenvector and the eigenvalue componentsof the spectral representation of the Laplacian. In particular, it is assumed that the spectral embedding space of the eigenvector part isa set of independent observations which follows an unknown distribution. The underlying distribution is estimated in a non-parametricway, through kernel density
estimation. In particular, the posteriorM Φ
probability P Φ |Θ can be computed by solving the problem:
n

max

N

−

max (Nhd )−n ∏ ∑ eO∈O(d) S∈{±1}d

Φ 2
kOSφM
i −θ j k
2h2

,

(11)

i=1 j=1

where d is the embedding dimension. For the results produced forthis contest, d = 50, hence the first 50 smallest eigenvalues (andcorresponding eigenvectors) have been used to build the models
and to infer them.
The two most important benefits brought by this point-wise apc 2017 The Author(s)Eurographics Proceedings c 2017 The Eurographics Association.

Rodolà et al. / SHREC’17: Deformable Shape Retrieval with Missing Parts

proach regard the robustness (and somehow, the scalability) of thealgorithm and its acquired invariance to vertex correspondences.
Two alignment steps are defined and applied to the eigenvector
model in order to take care of the residual rotations between eigenvector matrices. The first one deals with the non-uniqueness of theretrieved eigenvectors, problem which often takes the name of sign
ambiguity. A heuristic-based method is used to solve this problem,
even if it does not guarantee the identification of all the sign flips
that should be performed.
The second alignment step deals with the mixing-eigenspaces
problem. The problem concerns the retrieval of eigenvectors that
correspond to very close eigenvalues, which could result in a swap
between them. To solve it, a rotation matrix which aligns the eigenvector matrices is introduced into the model computation process.
Such rotation is computed by casting the problem as an Orthogonal Procrustes problem, in which the orthonormal transformation
which maximizes a certain probability (defined in terms of ParzenRosenblatt kernel density estimator) is sought.
On the other hand, in order to define a descriptor that is robust to
small non-isometric perturbations, the eigenvalues are assumed to
be log-normally distributed:d1
P(Λ |Θ ) = (2π) ∏eλii=1 σiMΛd2−(ln λi −µi )2
2σ2i

.

(12)

Finally, the posterior probability of a mesh with respect to a certain class is computed as the combination of posterior probabilities
of both models as follows:P(M|Θ) = P(ΛM |ΘΛ )P(ΦM |ΘΦ ) .

(13)

The posterior probabilities of a mesh with respect to each class ofthe training set are treated as components of a feature vector whichcharacterizes a mesh. The distance (in this contest, Chebyshev’s
distance) between the feature vectors of two meshes represents thescore between them.
4.9. A statistical model of Riemannian metric variation(RMVM) [A. Gasparetto and A. Torsello]
The method proposed in [GT15] consists in a supervised techniqueto learn a statistical model build on the Riemannian metric variations on deformable shapes based on the spectral decomposition of
the Laplace-Beltrami operator. Similarly to NPSM (Section 4.8),
the method employs a statistical framework that models a shape as
two independent models for the eigenvectors and for the eigenvalues. The eigenvector matrices of a set of discrete representations(i.e., meshes representing the shape in different poses) are assumed
to be points on the manifold of special orthogonal matrices SO(n).
Here the model is assumed to follow a Γ-distribution over the manifold geodesic distances from a manifold centroid Φ0dg (φ, φ0 ) ≈ 2n − Tr φT φ0 + O(Θ4i ) ,(14)
where Θi are the angles of the residual rotation φT φ0 .
The eigenvalues are assumed to be log-normally distributed for the same stability considerations presented by
c 2017 The Author(s)Eurographics Proceedings c 2017 The Eurographics Association.

Aubry et al. [ASC11]. The shape centroid is computed as follows:!
NargmaxTr

φ0 ,Ri ∈SO(p)∑ Ri φTi φ0

,

(15)

i

where the rotation matrix Ri is introduced in order to align the
eigenvectors of the Laplacian of a mesh i, since its embedding isdefined up to an isometry. This is solved by separately optimizingfor φ0 and Ri in an iterative process! !
NΦ0 = argmaxTrΦ0 ∈O(n)∑ Ri ΦTiN

Ri =argmaxRi ∈SO(n)

(16)

Φ0

i

Tr

!

!

∑ ΦTi Φ0 Ri

(17)

i

where both optimizations can be solved exactly through Singular
Value Decomposition.
The two statistical models are combined to compute the posterior
probability of a mesh to belong to a certain class:!
p( j) =∏ log N (xi j ; µi , σi ) Γ(k, θ)(gd ) .
j

(18)

i

The main drawback of this method concerns the manifold inwhich the eigenvector matrices lie. Indeed, to compute the eigenvector centroid, the eigenvector matrices must share a common intrinsic space. Hence, the meshes are assumed to have both the same
number of vertices and a vertex-to-vertex correspondence (at least
for the training set). Unfortunately, this is not the case for this contest. The issue is addressed by (sub)sampling the meshes to a common number of vertices, and approximating the correspondence
map by casting the problem as an assignment problem. The sameprocess, which is explained in detail in the original work [GT15],
is applied to the meshes to be classified.
From a practical point of view, the embedding dimension d is set
to 50, i.e., the 50 smallest eigenvalues (and corresponding eigenvectors) are used to build the models and to infer them. The score
between two meshes is computed as the Euclidean (RMVM-Euc)and Spearman’s (RMVM-Spe) distance between the score vectors
whose components are the probability density computed with respect to each class of the training set.
4.10. Bag of Words framework with RoPS (BoW+RoPS)[M-T. Tran et al.]In this method, the similarity of two query objects qi and q j is notmeasured directly, but via an intermediate domain consisting of the
training set. Similar objects are retrieved per-query from the training set; then, the two queries are compared indirectly (see Fig. 4).
Let R(qi ) and R(q j ) be the rank lists of training objects corresponding to the query objects qi and q j , respectively. Let nK (q, c)
be the number of training objects in category c appearing among
the top K of rank list R(q). Define sK (q, c) = nK (q, c)/K to represent the score of a query object q being in category c. Thisway, each query q is encoded as a normalized 20-dim. vectorv(q) = (sK (q, 1), sK (q, 2)..., sK (q, 20)). To achieve high precision,
only the top 1 or 2 best score values are kept in the feature vector

Rodolà et al. / SHREC’17: Deformable Shape Retrieval with Missing Parts
Query
model

Top-K Voting
20-dimensionalscore vector

FeatureExtraction
RandomSampling,
RoPS………
Top-K retrieved models

Retrieval
Inverted Index Tree
L1 measure

4.11. Supervised bag-of-features (SBoF) [R. Litman et al.]Quantization,
Feature Encoding
FeatureDescriptors

C++ and run on a 2GHz Intel Xeon CPU E5-2620, 1GB RAM.
The retrieval process was written in Matlab R2012b, where feature
quantization and distance calculation were performed on a 2.2GHz
Intel Xeon CPU E5-2660, 12 GB RAM. The average time to calculate model features is 1-2s, whereas it takes on average 0.1s tocompare a test object against the entire training set.

Soft Assignment
tf,idf weighting

Histogram of
3D visual words

Figure 4: Encoding a 3D object using BoW (Section 4.10).

v(q); other elements in v(q) are zeroed. Finally, the similarity of qi
and q j is defined as the dot product of v(qi ) and v(q j ).
The framework for 3D object retrieval [PSA∗ 16] is further enriched using RoPS features [GSB∗ 13] to retrieve similar objects
in the training set corresponding to a query object. The method
is based on the Bag-of-Words (BoW) scheme for visual object retrieval [SZ03, NNT∗ 15].
First, each 3D object is normalized to fit in a unit cube. Then,
uniform random samples of 5% ≤ pSampling ≤ 20% of the total
number of vertices are taken for each object. RoPS [GSB∗ 13] isused as a local feature, restricted to a Euclidean sphere with radiusr ∈ [0.01, 0.2] around each sampled vertex.
The extracted features are used to build a codebook with size
equal to 10% of the total number of features in the corpus, using
Approximate K-Means. Soft-assignment [PCI∗ 08] with 3 nearest
neighbors is used to reduce quantization error. As a query can bepart of a training object and vice versa, an `1 asymmetric distance[ZJS13] is used to measure the dissimilarity of each pair of objects.

This is an implementation of the method proposed in [LBBC14],
with some slight modifications made specifically for this challenge. The first and major change is the selection of point descriptors, where the original intrinsic descriptors were replacedby the extrinsic SHOT [STDS14] descriptor, due to its good performance in other recent challenges dealing with partial shapes[RCB∗ 16, CRB∗ 16]. The second change is due to the higher dimensionality of the SHOT descriptor, where a bigger dictionary ofsize 256 is adopted as opposed to the original 32.
The rest of the hyper-parameters were selected in a manner similar to [PSR∗ 14], by cross validation over the training set. First, as apre-processing step, all shapes from the ‘holes’ set were downsampled to 15K faces. Sparse coding was done over unit `2 norm SHOT
descriptors, with regularization value set to λ = 0.25. Each shape’s
sparse codes were pooled into a single histogram using averagepooling, and compared using the `1 metric. The triplet loss margin
was selected to be µ = 0.2.
The method was implemented in MATLAB, with parts in C++
for the computation of SHOT features. Dictionary learning took
about 5h on a 12-core 2.4GHz XEON processor with 48GB ofmemory. At test time, shapes were ranked according the the `1similarity of their 256 histogram descriptor, which took less than
100ms.

For the ‘holes’ challenge: since shapes have irregular mesh,they are subdivided to reduce significant difference in vertex density between different parts of the surface. A random sampling
pSampling = 10% and support radius r = 0.05 are used for RoPS,
and the codebook size is set to 180K. Three runs are provided:

5. Results• Run1: The top K = 1 in the rank list is considered.
• Run2: The top K = 9 are considered, with a voting scheme forthe best category.
• Run3: The top K = 9 are considered, with a voting scheme forthe two best categories.

Reading the P/R curves, the best performing method for the‘holes’ challenge is DLSF (Section 4.3), followed closely by
BoW+RoPS (Section 4.10). In terms of NN, the best performing methods are DLSF, 2VDI-CNN (Section 4.7), and SBoF (Section 4.11). DLSF, 2VDI-CNN, and BoW+RoPS also exhibit thebest scores in the 1-Tier, 2-Tier, and DCG measures.

For the ‘range’ challenge: topological noise is removed bydeleting triangles with abnormally long edge. Random sampling
pSampling = 10% is used. Due to low mesh quality, a support radiusr = 0.1 is used for RoPS. Since the grid-like topology induces alarge amount of vertices, the codebook size is increased to 1.2M.
Three runs are provided as above, with Run2 and Run3 using K = 5
instead of K = 9.
The codebook training module was implemented in Python 2.7
and run on a 2.4 GHz Intel Xeon CPU E5-2620 v3, 64 GB RAM.
The 3D feature extraction and description module was written inPrecision/Recall curves for all methods on the two challenges areshown in Fig. 5; NN, 1-Tier, 2-Tier and DCG scores are reported inTables 1 and 2.

The ‘range’ challenge received less submissions (9 against the
15 for ‘holes’, baselines included). In terms of P/R, the best performing methods are BoW+RoPS and 2VDI-CNN, which also receive top scores in the 1-Tier, 2-Tier, and DCG measures. The highest NN scores are attained again by 2VDI-CNN and SBoF. Overall, we note a consistent behavior for methods that participated inboth challenges. In particular, SBoF has a higher NN score than
BoW+RoPS in both challenges, despite the latter having a significant advantage in terms of P/R.
c 2017 The Author(s)Eurographics Proceedings c 2017 The Eurographics Association.

Rodolà et al. / SHREC’17: Deformable Shape Retrieval with Missing Parts

NN
0.969
0.811
0.643
0.639
0.519
0.515
0.377
0.178
0.130

1-Tier
0.906
0.317
0.910
0.908
0.326
0.915
0.268
0.184
0.183

2-Tier
0.977
0.510
0.962
0.964
0.537
0.959
0.485
0.371
0.366

DCG
0.980
0.769
0.962
0.965
0.736
0.960
0.699
0.640
0.640

0.8

Table 1: Retrieval accuracy for the ‘range’ challenge, sorted decreasingly by NN score. Best results are reported in bold.
NN
1.000
0.906
0.815
0.710
0.607
0.597
0.578
0.392
0.380
0.347
0.251
0.241
0.186
0.117
0.078

1-Tier
0.971
0.818
0.326
0.370
0.918
0.877
0.261
0.226
0.894
0.222
0.228
0.225
0.172
0.172
0.163

2-Tier
0.999
0.937
0.494
0.566
0.970
0.963
0.436
0.402
0.965
0.395
0.410
0.395
0.343
0.349
0.348

DCG
0.998
0.954
0.780
0.790
0.968
0.956
0.725
0.679
0.955
0.676
0.676
0.676
0.642
0.641
0.632

0.6

0.4

0.2

0

0

0.6

0.8

1

0.8

1

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

Recall2VDI-CNN
BoW+RoPS-3
DLSF
NPSM
SBoF

6. Discussion and conclusions

BoW+RoPS-1
BoW-HKS (b)DNA (b)
RMVM-EucSR

BoW+RoPS-2
BoW-siHKS (b)GMR
RMVM-Spe
SnapNetFigure 5: Precision/Recall curves for all methods in the two challenges. In the legend, ‘(b)’ identifies the baselines.

‘holes’ dataset, due to the random similarity transformation being
applied to each shape. This has a detrimental effect on all methods
relying on spectral (Laplacian-based) quantities and heat diffusion.
Defining local, deformation- and scale-invariant features remains
an open challenge tackled by few.

Given the non-rigid nature of the dataset, it is interesting tonote that the best performing methods only rely on extrinsic, notdeformation-invariant quantities. We conjecture that this is in partdue to the piecewise-rigid transformations, and the presence of similar shapes in the training and test sets. In particular, the only purely
intrinsic spectral approach (DNA) achieved the worst scores in bothchallenges. We further note the lack of intrinsic CNN-based methods such as [BMRB16].

References

Finally, a notable difficulty is the lack of a consistent scale in the[BMRB16]

c 2017 The Author(s)Eurographics Proceedings c 2017 The Eurographics Association.

0.4

challenge: ‘holes’

1

Table 2: Retrieval accuracy for the ‘holes’ challenge, sorted decreasingly by NN score. Best results are reported in bold.

Out of the 11 participating methods (without counting differentruns), 6 were supervised learning techniques and the remaining 5
were non learning-based. Despite coming from separate researchgroups, we note a surprising consistency in the general approach:the adoption of BoW-like frameworks on one hand, and the use ofsynthetic 2D views on the other. In both cases, we observe a wide
variance in the accuracy among the different submissions, suggesting that the old adagio “the devil is in the details” is as valid forlearning-based approaches as it is for axiomatic modeling. At thesame time, we did not observe a significant difficulty bias acrossdifferent shape categories or deformations.

0.2

RecallPrecision

MethodDLSF
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BoW-siHKS (baseline)
BoW+RoPS-3
BoW+RoPS-1
BoW-HKS (baseline)
RMVM-Euc
BoW+RoPS-2
NPSM
RMVM-SpeSR
GMR
SnapNetDNA (baseline)

challenge: ‘range’

1

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BoW+RoPS-1
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c 2017 The Author(s)Eurographics Proceedings c 2017 The Eurographics Association.