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3D shape representation with spatial probabilistic distribution of intrinsic shape keypoints
 Vijaya K. Ghorpade^{1}Email authorView ORCID ID profile,
 Paul Checchin^{1},
 Laurent Malaterre^{1} and
 Laurent Trassoudaine^{1}
https://doi.org/10.1186/s136340170483y
© The Author(s) 2017
 Received: 15 February 2017
 Accepted: 25 May 2017
 Published: 12 July 2017
Abstract
The accelerated advancement in modeling, digitizing, and visualizing techniques for 3D shapes has led to an increasing amount of 3D models creation and usage, thanks to the 3D sensors which are readily available and easy to utilize. As a result, determining the similarity between 3D shapes has become consequential and is a fundamental task in shapebased recognition, retrieval, clustering, and classification. Several decades of research in ContentBased Information Retrieval (CBIR) has resulted in diverse techniques for 2D and 3D shape or object classification/retrieval and many benchmark data sets. In this article, a novel technique for 3D shape representation and object classification has been proposed based on analyses of spatial, geometric distributions of 3D keypoints. These distributions capture the intrinsic geometric structure of 3D objects. The result of the approach is a probability distribution function (PDF) produced from spatial disposition of 3D keypoints, keypoints which are stable on object surface and invariant to pose changes. Each class/instance of an object can be uniquely represented by a PDF. This shape representation is robust yet with a simple idea, easy to implement but fast enough to compute. Both Euclidean and topological space on object’s surface are considered to build the PDFs. Topologybased geodesic distances between keypoints exploit the nonplanar surface properties of the object. The performance of the novel shape signature is tested with object classification accuracy. The classification efficacy of the new shape analysis method is evaluated on a new dataset acquired with a TimeofFlight camera, and also, a comparative evaluation on a standard benchmark dataset with stateoftheart methods is performed. Experimental results demonstrate superior classification performance of the new approach on RGBD dataset and depth data.
Keywords
 3D descriptors
 Shape signature
 Geodesics
 Weighted graphs
 Object recognition
 Classification
 Machine learning
 Neural networks
1 Introduction
 1.
 2.
 3.
 1.
Local features;
 2.
Global features;
 3.
Distributionbased;
 4.
Spatial map.

Viewbased. A descriptor of each 3D model is constructed from multiple orthographic view directions. Similarity search is done with either shock graph matching [40, 41] or lightfield descriptor dissimilarity [31].

Deformationbased. A pair of 2D/3D shapes is compared by measuring the amount of deformation required to register the shapes exactly. However, Shape fitting [42] or Shape evolution [43] are difficult for 3D shapes.

Point set methods. Here, the descriptor of a shape is given by weighted 3D points. In the first step, the shape is decomposed into its components, and then each component is represented using a weighted point [37]. Curvature of points for example can be a very good measure for weighing [38].

Volumetric error. It is based on calculating a volumetric error between one object and a sequence of offset hulls of the other object [34]. SánchezCruz and Bribiesca [33] presented a method which relates the volumetric error between two voxelized shapes to a transportation distance measuring how many voxels have to move, and how far, to change one shape into another.
Shape analysis approaches can also be flattened into two groups: heat diffusion [15, 17, 44, 45] and nondiffusionbased [21, 24, 46] shape features, according to [47].

A3: angle formed by three random surface points,

D1: distance of a surface point to the center of mass of the model,

D2: distance between two random surface points,

D3: square root of the area of the triangle defined by three random points,

D4: cube root of the volume of the tetrahedron defined by four surface points,
which were chosen for their simplicity to compute and understand, and also because they produce distributions that are invariant to rigid motions and tessellation, insensitive to small perturbation and scale invariant (in case of A3).
However, sensor data is actually 2.5D, and it can only give the partial view information of an object in single acquisition. In order to recognize the object in any pose in real world, it is necessary to have a complete 3D information of the object or multiple complementary views of the same object. As a result, the shape of an object cannot be determined by a single view, and shape distribution for the same object is completely different when viewed from another pose. Interestingly, the 3D keypoints are repeatable and consistent across different views. This wonderful property of 3D keypoints has been realized, the same five shape functions by Osada et al. [21] are analyzed, and it has been found that the geometric distributions of 3D keypoints are unique for individual object. Good 3D keypoints are like anchor points which are stable and holds the object for any rigid deformation/transformation. The shape distribution of them are similar across multiple views, and when trained using machine, learning algorithms can efficiently classify the object’s instances or categories. In previous work, performance evaluation of different 3D keypoint detectors was conducted on RGBD and depth data [48], and it has been observed that ISS3D and SIFT3D are most repeatable keypoints and robust. In this article, the PDFs of geometric/spatial distribution of ISS3D keypoints has been analyzed and successfully exploited to represent an object. Euclidean and topological spacebased norms are considered to build these distributions. KPD (KeyPoint Distribution) term is used to refer the distributions that are based on Euclidean norms and GKPD (Geodesic KeyPoint Distribution) is based on shortest paths on object’s manifold.
GKPD is a combination of feature, graph, and geometrybased methods. It capitalizes the pose invariance and stability of feature detectors. With graph representation of 3D points as nodes and geodesics between them as edges, the GKPD exploits the surface and topological information on the object’s manifold. And lastly, it considers the multiple complementary 2.5D views of the object to be able to detect it in the real world in any pose (a combination of viewbased and point set methods in geometrybased approaches).
2 Related work
Then, Wasserstein metric related to the Monge Kantorovich optimal transport problem was used as similarity measure to compare φ(Ω) of different shapes. On the other hand, Ion et al. [63] considered geodesic eccentricity, i.e., quantile Q _{ x }(1), to construct a histogrambased descriptor, which actually calculates maximum geodesic distances corresponding to boundary points.
where d c _{ j } is the area of the triangle having centroid c _{ j } and C is the total area of the manifold \(\mathbb {M}\).
Gal et al. [66] proposed poseoblivious shape signature, a 2D histogram which is a combination of two scalar functions defined on the boundary surface of the 3D shape: localdiameter measures the diameter of the 3D shape in the neighborhood of each shape and centricity, which is similar to D2, considers average geodesic distance instead of Euclidean distance. Osada et al. [21] defined five shape functions which basically calculate Euclidean distances between the random points. They used their distributions to retrieve similar shapes, thanks to a couple of statistical dissimilarity measures.
3 Background
3.1 Minimal paths
3.2 3D keypoints
The second flaw, shape variation due to pose changes, can be tackled by considering efficient 3D keypoint detectors. The 3D keypoints or interest points are stable, repeatable, and consistent across different views. Good 3D points are like anchor points which are stable and hold the shape of the object for any rigid deformation or transformation [48]. Constructing shape functions on 3D keypoints instead of all the points on the surface seems a good bet.
4 Approach
There are four stages in the proposed approach (see Fig. 4). Firstly, the point cloud is preprocessed to obtain noisefree object representation and then detect ISS3D keypoints for every single view of all the objects. Then, each view of the object is transformed into a graph and the shortest paths between keypoints are calculated. Every view is then represented as probability distributions of geodesics. In the last stage, distribution learning and classification are carried out.
4.1 Point cloud filtering
The first step in the approach is to filter the point cloud from systematic and nonsystematic noise. This has been achieved by using statistical outlier filters and jump edge filters [71]. The noisefree point cloud is then utilized in the next steps to construct graphs and distributions.
4.2 Intrinsic shape signatures
 1.In the first step, each point p _{ i } is weighted with a value inversely proportional to the number of points in its spherical neighborhood of radius r _{ density }.$$ {w_{i}}= \frac{1}{\left\\left\{{p_{j}}:\left{p_{j}}{p_{i}}\right<{r_{density}}\right\}\right\} $$(8)
This criterion actually helps for uniform weighing of the points, as some points are sparsely distributed.
 2.A weighted scatter or covariance matrix for p _{ i } using all its neighbors p _{ j } within a distance r _{ frame } is calculated:$$ {COV(p_{i})}=\frac{\sum_{\leftp_{j}p_{i}\right<r_{frame}}{w_{j}}\left({p_{j}}{p_{i}}\right)\left({p_{j}}{p_{i}}\right)^{\intercal}}{\sum_{\leftp_{j}p_{i}\right<r_{frame}} {w_{j}}} $$(9)
 3.
The eigen values and eigen vectors are computed in the decreasing order of magnitude.
 4.
A 3D coordinate system is constructed using p _{ i } as the origin, and e ^{1}, e ^{2}, and their cross product e ^{1} ⊗ e ^{2} as the x−, y−, and z− axes respectively. This reference system is actually the intrinsic coordinate system, F _{ i }, which is a characteristic of the local object shape and indifferent to viewpoint. However, the basis specifies the vector of its axes in the sensor coordinate system, hence view dependent, and directly encodes the pose transform between the sensor coordinate system and the local objectoriented intrinsic frame. As the eigen vector of the scatter matrix computes a direction in the 3D space based on the amount of point position variations, its orientation actually has 180° ambiguity. Due to this, each axis has two possible orientations, and therefore, the intrinsic reference frame at a basis point has four variants as shown in Fig. 8.
Only those points with successive eigen values \(\left ({\left \{\lambda ^{1}_{i},\lambda ^{2}_{i},\lambda ^{3}_{i}\right \}}\right)\) below a certain threshold are retained after eigen analysis. The geometric distribution of these points on the surface is robust against view changes and acts as anchor points. The geodesics between them maintain a consistency and are of constant magnitude. A group of ISS3D keypoints K∈S⊂Ω is detected, and a set of \({\left \lbrace {d_{\varOmega }}({x}, {y})\right \rbrace }_{x \in {K} \subset \varOmega, y, \in {K} \subset \varOmega, \forall x \neq y} \subset \mathbb {R^{+}}\) geodesics and L2 norms are calculated between them. It should be noted that only ISS keypoints are considered, not the complete ISS feature/descriptor, for building PDFs.
4.3 Graph making
Extracting geodesics on a single manifold (or mesh representation) of an object is trivial. However, polygonal mesh representation of object leads to S _{1},S _{2},…,S _{ n }⊂Ω ((S _{1}∪S _{2} ∪⋯∪S _{ n })≃Ω) submeshes, on which calculating minimal paths is not easy. So, the point cloud is represented as a simple undirected graph without selfloop and edge labels. Given a point cloud V=(v _{1},v _{2},…,v _{ n }), a weighted undirected graph G=(V,E,W) is constructed. E is the set of edges, and W are weights on the edges, such that for edge \({E_{(v_{i},v_{j})}},\) weight \({W_{(v_{i},v_{j})}}={L}^{2}_{(v_{i},v_{j})}\) (where L ^{2} is Euclidean L2 norm between two vertices). However, G is not a complete graph, where every vertex is connected to every other vertex (otherwise the shortest path is equal to the Euclidean distance itself). But it is constructed in a way that there exists a path to go from one node to any other node such that no node (especially ISS3D keypoint) is isolated to be traversed. This is achieved by designing local complete graph.
4.3.1 Local complete graph
4.3.2 Shortest paths
4.4 KPD: keypoint distribution
4.5 Shape distribution vs KPD
4.6 Geodesic keypoint distribution

GKPD1: geodesic distance from the graph centroid to all the keypoints,

GKPD2: geodesic distance between two keypoints,

GKPCD2: weighted shortest path magnitude between two keypoints.
GKPCD2 is the GKPD2version for a graph weighted by curvature factor C, and while backpropagating Euclidean distances are integrated instead of actual edge weights, curvatureinduced edge weights do not follow symmetry property. Each object view is then represented as a probability distribution function of GKPD1, GKPD2, and GKPCD2 values.
4.7 Hybrid keypoint functions
The combination of the Euclidean and topological KPFs has also been experimented, as the authors hypothesize that hybrid KPFs capture both the geometrical structure of the object as well as the surface information. However, individual KPFs are not strong enough discriminative for each object category. This hypothesis is tested on object classification accuracy.
5 Learning and classification
For classification, ensemble method and neural networks are considered. Both of them are supervised learning versions, where we present set of training examples of the form {(x _{1},y _{1}),…,(x _{ m },y _{ m })} for some unknown function y=f(x). The x _{ i } are typically vectors of the form 〈x _{ i,1},x _{ i,2},…,x _{ i,n }〉, also called the features of x _{ i } and x _{ ij } refer to the jth feature of x _{ i }. In this approach, x _{ ij } represents the probability value of KPF or shape function in jth bin. Concatenating the feature vectors of Euclidean and geodesic distribution is also experimented, to get a more robust and distinct feature vector as this leads to more discriminative feature space for each class. The y values are typically drawn from a discrete set of classes {1,…,K} in the case of classification. Given a set S of training examples, a learning algorithm outputs a classifier, h _{ i } from the hypothesis space. In ensemble methods, a set of classifiers’ individual decisions are combined in weighted or unweighted voting fashion to classify new examples. The ensembles are quite often more accurate than the individual classifiers that make them up. In neural networks, the function f(x) is composition of other functions g _{ i }(x) which are also composition of other functions like a network structure with nonlinear weighted sum as the type of composition in general, \({f}(\mathbf {x})=\mathbf {K}\left (\sum _{i}{w_{i}}{g_{i}}(\mathbf {x})\right)\), K is an activation function. Two popular libraries which implement these two learning methods: XGBoost and Keras, have been considered.
5.1 Gradient Boosting
Gradient boosting produces a prediction model in the form of an ensemble of weak prediction models. XGBoost (eXtreme Gradient Boosting) library [78] is based on boosted trees by Gradient Boosting Machine (GBM) [79] and is a highly sophisticated algorithm robust against all kind of data irregularities. It has many advantages including regularized boosting, parallel processing, tree pruning, handling missing values, and builtin crossvalidation. For the evaluation, multi:softmax model with default booster parameters and treespecific parameters is used. Although, several experiments were carried out on hyperparameters to achieve the best performance. The dataset PDFs are split into training and test sets with the last column being the class label. The features being the probability values in particular range (bin) in KPF. The performance greatly depends on the bin size of the histograms (PDF) and can lead to overfitting with increasing the number of features as they become redundant and does not convey extra information.
5.2 Neural networks
Keras deep learning library [80] which is capable of running on top of either Theano [81] or TensorFlow numerical libraries [82] has been used. It is a minimalist, highly modular neural network library as the authors say. A Sequential model is adopted having linear stack of 2D layers (Dense). The output class values are one hot encoded adhering to good practice protocols. The baseline neural network constructed is a simple fully connected network with one hidden layer containing 200 neurons. Rectifier activation function is used for hidden layer and sigmoid for the output layer. For reproducibility of the results, seed number is set to seven and 10fold crossvalidation with shuffling.
6 Datasets
7 Results, evaluation, and discussion
7.1 Evaluation
Classification accuracy rate in percent with shape keypoint functions
Keypoint functions  Washington RGBD  SR4KD  

Multiclass classification  Neural networks  Multiclass classification  Neural networks  
KPD2  93.093  93.380 ± 0.730  90.643  82.570 ± 3.340 
KPD3  87.535  87.350 ± 1.630  90.059  83.710 ± 4.010 
KPA3  85.212  80.220 ± 1.090  83.041  71.180 ± 5.790 
GKPD2  91.228  90.030 ± 0.960  91.328  88.380 ± 4.410 
GKPD1  88.110  84.100 ± 0.890  79.532  78.980 ± 5.310 
GKPCD2  86.900  83.530 ± 0.810  81.870  81.300 ± 1.940 
KPD2 + KPA3  97.9801  97.15 ± 1.46  95.3215  97.15 ± 1.46 
KPD2 + GKPD2  96.961  97.07 ± 0.49  92.3977  94.76 ± 3.30 
KPD2 + GKPCD2  97.9926  97.06 ± 3.49  95.9064  93.39 ± 2.90 
KPD2 + GKPD2 + GKPCD2  97.824  97.32 ± 0.43  94.7638  94.99 ± 2.96 
KPA3 + KPD2 + GKPD2 + GKPDC2  98.4863  98.0448 ± 1.83  97.274  96.58 ± 2.38 
Comparison of classification rate on Washington RGBD Objects database
Methods  Classification accuracy in %  

RGBD  Only D  
Random Forest [50]  79.600 ± 4.001  66.8 ± 2.5 
Lai et al. [50]  81.9 ± 2.8  53.1 ± 1.7 
Nonlinear SVM  83.8 ± 3.5  64.7 ± 2.2 
IDL [87]  85.4 ± 3.20  70.200 ± 2.001 
CNNRNN [88]  86.8 ± 3.3  78.9 ± 3.8 
Bo et al. [89]  87.5 ± 2.9  — 
Schwarz et al. [90]  89.4 ± 1.3  — 
KPD2 + GKPD CD2  97.980  95.906^{a} 
Proposed method  98.486  97.274^{a} 
7.2 Parameter tuning
In order to obtain the optimal results, couple of, if not several parameters, must be set for both multiclass and neural network learning algorithms. A series of meticulous experiments were conducted, by tuning these hyperparameters. Some of the parameters were set with trial and error methodology, as they do not have proper equation with accuracy.
7.2.1 Multiclass classification
7.2.1.1 Learning parameter (α or eta)
7.2.1.2 Tree depth (maxdepth)
The maximum depth of the tree in GBM is used to control overfitting as higher depth values will allow model to learn relations very specific to a particular sample. Usually, it takes a range of values from 1−10. An optimal value of 12 is found to be the best for the SR4KD data (with α=0.20 from previous experiment) (Fig.21 b).
7.2.1.3 Number of rounds
Number of rounds helps the model to learn from previous errors. It usually does not affect much the performance, as can be seen in Fig. 21 c. The rest of the booster parameters (Gamma(minimum loss reduction for split), subsample(=1), random column samples for each tree and lambda) are default values, as they do not affect the performance much. Both multi:softmax and multi:softprob have been tried.
As it is observed in Fig. 21, hyperparameter tuning has little effect once the optimal feature size has been found.
7.2.2 Neural nets
Most of the parameters in Deep Learning or Neural Networks are tuned with trial and error, as they does not seem to have an equation with performance or rather being chancy.
7.2.2.1 Number of neurons
7.2.2.2 Hidden layers
The hidden layers are gradually increased, with constant neurons in each layer (input =200, hidden =100). The number of layers has minuscule impact on performance (Figs. 22 b and 23 b). One single hidden layer seems to be optimum, which is what has been evaluated in this work.
7.2.2.3 Dropout
Dropout is a regularization technique for neural network models proposed by Srivastava et al. [86], it is a technique where randomly selected neurons are ignored during training. They are droppedout randomly. This means that their contribution to the activation of downstream neurons is temporally removed on the forward pass and any weight updates are not applied to the neuron on the backward pass. A series of experiments is conducted, by dropping out the neurons in the first and single hidden layer. It can be seen that, as the number of dropped neurons is increased, the performance decreases; however, a percentage of 10 seems to give the best results.
8 Conclusions
In this paper, an innovative approach for shape representation has been presented. The new method simplifies object shape representation in the form of simple probability distribution functions which can be easily and quickly computed and which are robust against real word pose variances. It is paramount to recognize objects which undergo pose transformation in the real world, unlike other approaches which perform well on synthetic datasets but not on real sensor data. The 3D keypoints act as stable anchors on the surface of the object and remain intact even if the object undergoes rigid transformation. These properties of keypoints have been exploited, their geometric and spatial distribution are analyzed through some keypoint functions, and it has been observed that their distribution is consistent even after view changes. A new dataset of objects from a SwissRanger TimeofFlight camera has been created for experimenting object classification and applied some of the best of the machine learning methods and neural networks with hyperparameter tuning. Superior classification results are obtained compared to the other stateoftheart methods on the same dataset. Future research could open up a better GKPD CD2, by considering an effective backpropagation strategy and better featureselection techniques.
9 Endnote
Declarations
Funding
This work is supported by the French government research program Investissements d’Avenir through the RobotEx Equipment of Excellence (ANR10 EQPX44) and the IMobS3 Laboratory of Excellence (ANR10LABX1601), by the European Union through the program Regional com petitiveness and employment 2007–2013 (ERDF  Auvergne region), and by the Auvergne region.
Availability of data and materials
The dataset IPTOFD can be found at ftp://ftp.ip.univbpclermont.fr/iptofd.
Authors’ contributions
VKG, PC and LT conceived the idea. VKG deisgned and performed the experiments and LM helped with dataset making. VKG and PC wrote the paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
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Authors’ Affiliations
References
 J Schöning, G Heidemann, in Proc. of the 11th Joint Conf. on Computer Vision, Imaging and Computer Graphics Theory and Applications, (2016), pp. 192–197. doi:10.5220/0005784801920197.
 S Tang, A Godil, in Proc. SPIE, vol. 8290, 8290, (2012), pp. 82,900N–82,900N–15. doi:10.1117/12.912153.
 JWH Tangelder, RC Veltkamp, Multimedia tools and applications. 39(3), 441 (2007). doi:10.1007/s1104200701810.
 IK Kazmi, L You, JJ Zhang, in Computer Graphics, Imaging and Visualization (CGIV), 2013 10th Int. Conf, (2013), pp. 1–10. doi:10.1109/CGIV.2013.11.
 C Singh, Pooja, Optics and lasers in engineering. 50(5), 655 (2012). doi:10.1016/j.optlaseng.2011.11.012.
 CH Wei, Y Li, WY Chau, CT Li. Pattern Recogn. 42(3), 386 (2009). doi:doi:10.1016/j.patcog.2008.08.019.View ArticleGoogle Scholar
 M Yang, K Kpalma, J Ronsin, in Pattern Recognition, ed. by PY Yin. A Survey of Shape Feature Extraction Techniques (INTECH, 2008), pp. 43–90. https://hal.archivesouvertes.fr/hal00446037.
 AM Bronstein, MM Bronstein, R Kimmel, Generalized multidimensional scaling: a framework for isometryinvariant partial surface matching. Proc. Natl. Acad. Sci. 103(5) (1168).Google Scholar
 QX Huang, B Adams, M Wicke, LJ Guibas, NonRigid Registration Under Isometric Deformations. Comput. Graph. Forum. 27(5), 1449–1457 (2008). doi:http://dx.doi.org/10.1111/j.14678659.2008.01285.x.View ArticleGoogle Scholar
 Y Sahillioğlu, Y Yemez, Computer Graphics Forum (2013). doi:10.1111/cgf.12007.
 M Ovsjanikov, Q Mérigot, V Pătrăucean, L Guibas, in Proc. of the Eleventh Eurographics/ACMSIGGRAPH Symp. on Geometry Processing, SGP ’13 (Eurographics AssociationAirelaVille, 2013), pp. 1–11. doi:10.1111/cgf.12167.
 S Biasotti, B Falcidieno, D Giorgi, M Spagnuolo, Mathematical Tools for Shape Analysis and Description (Morgan & Claypool, 2014). doi:10.2200/S00588ED1V01Y201407CGR016.
 L Zhang, MJ da Fonseca, A Ferreira, Survey on 3D shape descriptors. Tech. rep., Technical Report, DecorAR (FCT POSC/EIA/59938/2004) (2007).Google Scholar
 AM Bronstein, MM Bronstein, R Kimmel, SIAM J. Sci. Comput. 28(5) (2006). doi:10.1137/050639296.
 AM Bronstein, MM Bronstein, R Kimmel, M Mahmoudi, G Sapiro. Int. J. Comput. Vision. 89(2), 266 (2010). doi:10.1007/s1126300903016.View ArticleGoogle Scholar
 H Tabia, H Laga, D Picard, PH Gosselin, in IEEE Conf. on Computer Vision and Pattern Recognition, (2014), pp. 4185–4192. doi:10.1109/CVPR.2014.533.
 Y Fang, M Sun, K Ramani, in CVPR Workshops, (2011), pp. 9–16. doi:10.1109/CVPRW.2011.5981684.
 Y Fang, M Sun, M Kim, K Ramani, in Computer Vision and Pattern Recognition (CVPR), IEEE Conf. on, (2011), pp. 2145–2152. doi:10.1109/CVPR.2011.5995695.
 JWH Tangelder, RC Veltkamp, in Shape Modeling Applications.Proc, (2004), pp. 145–156. doi:10.1109/SMI.2004.1314502.
 CB Akgül, B Sankur, Y Yemez, F Schmitt. IEEE Trans. Pattern Anal. Mach. Intell. 31(6) (2009). doi:10.1109/TPAMI.2009.25.
 R Osada, T Funkhouser, B Chazelle, D Dobkin. ACM Trans. Graph. 21(4), 807 (2002). doi:10.1145/571647.571648.View ArticleGoogle Scholar
 M Ankerst, G Kastenmüller, HP Kriegel, T Seidl, 3D Shape Histograms for Similarity Search and Classification in Spatial Databases (Springer Berlin Heidelberg, Berlin, Heidelberg, 1999). doi:10.1007/3540484825_14.View ArticleGoogle Scholar
 DV Vranic, D Saupe, in Proceedings of the EURASIP Conference on Digital Signal Processing for Multimedia Communications and Services (ECMCS 2001), ed. by K Fazekas. 3D Shape Descriptor Based on 3D Fourier Transform (Budapest, 2001), pp. 271–274.Google Scholar
 D Saupe, DV Vranic, in Proc. of the 23rd DAGMSymp. on Pattern Recognition (SpringerVerlagLondon, 2001), pp. 392–397.Google Scholar
 J Sivic, A Zisserman, in Computer Vision, 2003. Proceedings. Ninth IEEE Int. Conf. on, (2003), pp. 1470–1477 vol. 2. doi:10.1109/ICCV.2003.1238663.
 AM Bronstein, MM Bronstein, M Ovsjanikov, FeatureBased Methods in 3D Shape Analysis (Springer London, London, 2012). doi:10.1007/9781447140634_5.View ArticleGoogle Scholar
 M Körtgen, GJ Park, M Novotni, R Klein, in The 7th Central European Seminar on Computer Graphics, (2003).Google Scholar
 A Johnson, SpinImages: A Representation for 3D Surface Matching, (Pittsburgh, 1997).Google Scholar
 M Hilaga, Y Shinagawa, T Kohmura, TL Kunii, in Proc. of the 28th Annual Conf. on Computer Graphics and Interactive Techniques. SIGGRAPH ’01 (ACMNew York, 2001), pp. 203–212. doi:10.1145/383259.383282.Google Scholar
 A Brennecke, T Isenberg, in Simulation and Visualization, (2004), pp. 299–310.Google Scholar
 DY Chen, XP Tian, YT Shen, M Ouhyoung, On Visual Similarity Based 3D Model Retrieval. Comput. Graph. Forum. 22(3), 223–232 (2003). doi:http://dx.doi.org/10.1111/14678659.00669.View ArticleGoogle Scholar
 R Ohbuchi, K Osada, T Furuya, T Banno, in Shape Modeling and Applications. IEEE Int. Conf. on, (2008), pp. 93–102. doi:10.1109/SMI.2008.4547955.
 H SánchezCruz, E Bribiesca. Image Vis. Comput. 21(12), 1027 (2003). doi:doi:10.1016/S02628856(03)001197.View ArticleGoogle Scholar
 M Novotni, R Klein, in Proceedings International Conference on Shape Modeling and Applications. A geometric approach to 3D object comparison (IEEE Computer SocietyWashington, 2001), pp. 167–175. doi:10.1109/SMA.2001.923387.View ArticleGoogle Scholar
 R Basri, L Costa, D Geiger, D Jacobs, in PhysicsBased Modeling in Computer Vision, Proc. of the Workshop on, (1995), p. 135, doi:10.1109/PBMCV.1995.514678.
 I Cohen, N Ayache, P Sulger, Tracking points on deformable objects using curvature information (Springer Berlin Heidelberg, Berlin, Heidelberg, 1992). doi:10.1007/3540554262_51.View ArticleGoogle Scholar
 TK Dey, J Giesen, S Goswami, in Algorithms and Data Structures: 8th Int. Workshop, WADS 2003, Ottawa, Ontario, Canada, July 30  August 1, 2003, ed. by F Dehne, JR Sack, and M Smid (Springer Berlin HeidelbergBerlin, Heidelberg, 2003), pp. 25–36. doi:10.1007/9783540450788_3.
 JWH Tangelder, RC Veltkamp, in Shape Modeling International, 2003, (2003), pp. 119–129. doi:10.1109/SMI.2003.1199608.
 DG Lowe. Int. J. Comput. Vision. 60(2), 91 (2004). doi:10.1023/B:VISI.0000029664.99615.94.View ArticleGoogle Scholar
 D Macrini, A Shokoufandeh, S Dickinson, K Siddiqi, S Zucker, in Object recognition supported by user interaction for service robots, 3. Viewbased 3D object recognition using shock graphs (IEEE, 2002), pp. 24–28. doi:10.1109/ICPR.2002.1047786.
 CM Cyr, BB Kimia. Int. J. Comput. Vision. 57(1), 5 (2004). doi:10.1023/B:VISI.0000013088.59081.4c.View ArticleGoogle Scholar
 D Terzopoulos, D Metaxas. IEEE Trans. Pattern Anal. Mach. Intell. 13(7), 703 (1991). doi:10.1109/34.85659.View ArticleGoogle Scholar
 D DeCarlo, D Metaxas. IEEE Trans. Pattern Anal. Mach. Intell. 20(11) (1186). doi:10.1109/34.730554.
 J Sun, M Ovsjanikov, L Guibas, in Proceedings of the Symposium on Geometry Processing. A Concise and Provably Informative Multiscale Signature Based on Heat Diffusion (Eurographics AssociationAirelaVille, 2009), pp. 1383–1392. http://dl.acm.org/citation.cfm?id=1735603.1735621.Google Scholar
 M Aubry, U Schlickewei, D Cremers, in Computer Vision Workshops (ICCV Workshops), IEEE Int. Conf. on, (2011), pp. 1626–1633. doi:10.1109/ICCVW.2011.6130444.
 V Jain, H Zhang. Comput. Aided Des. 39(5), 398 (2007). doi:10.1016/j.cad.2007.02.009.View ArticleGoogle Scholar
 Y Fang, J Xie, G Dai, M Wang, F Zhu, T Xu, E Wong, in 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). 3D deep shape descriptor, (2015), pp. 2319–2328. doi:10.1109/CVPR.2015.7298845.
 VK Ghorpade, P Checchin, L Malaterre, L Trassoudaine, in Int. Conf. on Control, Automation, Robotics and Vision (ICARCV) (Phuket, Thailand, 2016).Google Scholar
 Swissranger, Heptagon. http://hptg.com/industrial/. Accessed 22 Aug 2016.
 K Lai, L Bo, X Ren, D Fox, in Robotics and Automation (ICRA), IEEE Int.Conf. on, (2011), pp. 1817–1824. doi:10.1109/ICRA.2011.5980382.
 LC Monteverde, CR Ruiz, Z Huang, A shape distribution for comparing 3D models (Springer Berlin Heidelberg, Berlin, Heidelberg, 2006). doi:10.1007/9783540694236_6.View ArticleGoogle Scholar
 CY Ip, D Lapadat, L Sieger, WC Regli, in Proc. of the Seventh ACM Symp. on Solid Modeling and Applications. SMA ’02 (ACMNew York, 2002), pp. 273–280. doi:10.1145/566282.566322.View ArticleGoogle Scholar
 CY Ip, WC Regli, L Sieger, A Shokoufandeh, in Proc. of the Eighth ACM Symp. on Solid Modeling and Applications. SM ’03 (ACMNew York, 2003), pp. 322–327. doi:10.1145/781606.781659.View ArticleGoogle Scholar
 Y Liu, H Zha, H Qin, in IEEE Int. Conf. on Shape Modeling and Applications (SMI’06), (2006), pp. 16–16. doi:10.1109/SMI.2006.41.
 CB Akgul, B Sankur, F Schmitt, Y Yemez, in Shape Modeling and Applications (SMI’07). IEEE Int. Conf. on, (2007), pp. 3–12. doi:10.1109/SMI.2007.27.
 R Ohbuchi, T Otagiri, M Ibato, T Takei, in Computer Graphics and Applications. Proc. 10th Pacific Conf. on, (2002), pp. 265–274. doi:10.1109/PCCGA.2002.1167870.
 R Ohbuchi, T Minamitani, T Takei, in Theory and Practice of Computer Graphics.Proc, (2003), pp. 97–104. doi:10.1109/TPCG.2003.1206936.
 Y Liu, J Pu, H Zha, W Liu, Y Uehara, in 3D Data Processing, Visualization and Transmission (3DPVT). Proc. 2nd Int. Symp. on, (2004), pp. 896–903. doi:10.1109/TDPVT.2004.1335410.
 H Rea, J Corney, D Clark, N Taylor, in 3D Data Processing, Visualization and Transmission (3DPVT). Proc. 2nd Int. Symp. on, (2004), pp. 167–174. doi:10.1109/TDPVT.2004.1335191.
 P Jiantao, L Yi, X Guyu, Z Hongbin, L Weibin, Y Uehara, in 3D Data Processing, Visualization and Transmission (3DPVT 2004). Proc. 2nd Int. Symp. on, (2004), pp. 95–101. doi:10.1109/TDPVT.2004.1335181.
 J Rabin, G Peyré, LD Cohen, in Proc. of 11th European Conf. on Computer Vision, Heraklion, Crete, Greece, Part V, ed. by K Daniilidis, P Maragos, and N Paragios (Springer Berlin HeidelbergBerlin, Heidelberg, 2010), pp. 771–784. doi:10.1007/9783642155550_56.
 AB Hamza, H Krim. IEEE Trans. Image Process. 15(8), 2249 (2006). doi:10.1109/TIP.2006.875250.View ArticleGoogle Scholar
 A Ion, NM Artner, G Peyré, SBL Mármol, WG Kropatsch, LD Cohen, in Search in 3D 2008. 3D Shape Matching by Geodesic Eccentricity (IEEEAnchorage, 2008), pp. 1–8. https://hal.archivesouvertes.fr/hal00419826.Google Scholar
 G Peyré, LD Cohen. Int. J. Comput. Vision. 69(1), 145 (2006). doi:10.1007/s1126300668593.View ArticleGoogle Scholar
 A Borovkov, Mathematical Statistics. Estimation of Parameters, Testing of Hypotheses, (Nauka, 1984).Google Scholar
 R Gal, A Shamir, D CohenOr. IEEE Trans. Vis. Comput. Graph. 13(2), 261 (2007). doi:10.1109/TVCG.2007.45.View ArticleGoogle Scholar
 G Peyré, M Péchaud, R Keriven, LD Cohen, Foundations and Trends in Computer Graphics and Vision. 5(3–4), 197 (2010). doi:10.1561/0600000029.
 LD Cohen, R Kimmel, Computer Vision and Pattern Recognition, Proc.CVPR ’96, IEEE Computer Society Conf. on, (1996). doi:10.1109/CVPR.1996.517144.
 VK Ghorpade, LD Cohen, in 2015 IEEE International Conference on Image Processing (ICIP), (2015), pp. 3009–3013. doi:10.1109/ICIP.2015.7351355.
 G Peyré, L Cohen, Geodesic computations for fast and accurate surface remeshing and parameterization, (Birkhäuser Basel, Basel, 2005). doi:10.1007/3764373849_18.
 VK Ghorpade, P Checchin, L Trassoudaine, in Mobile Robots (ECMR), 2015 European Conf. on, (2015), pp. 1–6. doi:10.1109/ECMR.2015.7324208.
 C Schmid, R Mohr, C Bauckhage. Int. J. Comput. Vision. 37(2), 151 (2000). doi:10.1023/A:1008199403446.View ArticleGoogle Scholar
 S Salti, F Tombari, LD Stefano, in 3D Imaging, Modeling, Processing, Visualization and Transmission (3DIMPVT), 2011 Int. Conf. on, (2011), pp. 236–243. doi:10.1109/3DIMPVT.2011.37.
 PJ Besl, ND McKay. IEEE Trans. Pattern Anal. Mach. Intell. 14(2), 239 (1992). doi:10.1109/34.121791.View ArticleGoogle Scholar
 Y Zhong, in Computer Vision Workshops (ICCV Workshops), IEEE 12th Int. Conf. on (IEEE, 2009), pp. 689–696.Google Scholar
 EW Dijkstra, A Note on Two Problems in Connexion with Graphs. Numer. Math.1(1), 269–271 (1959). doi:http://dx.doi.org/10.1007/BF01386390.MathSciNetView ArticleMATHGoogle Scholar
 J Siek, LQ Lee, A Lumsdaine, Boost Random Number Library (2000). http://www.boost.org/libs/graph/. Accessed 24 Feb 2017.
 T Chen, C Guestrin, in Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA, August 1317, 2016. KDD’16 (ACMNew York, 2016), pp. 785–794. doi:10.1145/2939672.2939785.Google Scholar
 JH Friedman. Ann. Stat, 1189–1232 (2001).Google Scholar
 F Chollet (2015). Keras. https://github.com/fchollet/keras. Accessed 24 Feb 2017.
 R AlRfou, A Alain, C Almahairi, D Angermueller, N Bahdanau, F Ballas, J Bastien, A Bayer, A Belikov, Y Belopolsky, A Bengio, J Bergeron, V Bergstra, JB Bisson, N Snyder, N Bouchard, X BoulangerLewandowski, A Bouthillier, O de Brébisson, PL Breuleux, K Carrier, J Cho, P Chorowski, T Christiano, MA Cooijmans, M Côté, A Côté, YN Courville, O Dauphin, J Delalleau, G Demouth, S Desjardins, L Dieleman, M Dinh, V Ducoffe, S Dumoulin, D Ebrahimi Kahou, Z Erhan, O Fan, M Firat, X Germain, I Glorot, M Goodfellow, C Graham, P Gulcehre, I Hamel, JP Harlouchet, B Heng, S Hidasi, A Honari, S Jain, K Jean, M Jia, V Korobov, A Kulkarni, P Lamb, E Lamblin, C Larsen, S Laurent, S Lee, S Lefrancois, N Lemieux, Z Léonard, JA Lin, C Livezey, J Lorenz, Q Lowin, PA Ma, O Manzagol, RT Mastropietro, R McGibbon, B Memisevic, V van Merriënboer, M Michalski, A Mirza, C Orlandi, R Pal, M Pascanu, C Pezeshki, D Raffel, M Renshaw, A Rocklin, M Romero, P Roth, J Sadowski, F Salvatier, J Savard, J Schlüter, G Schulman, IV Schwartz, D Serban, S Serdyuk, É Shabanian, S Simon, S Spieckermann, J Ramana Subramanyam, J Sygnowski, G Tanguay, J van Tulder, S Turian, P Urban, F Vincent, H Visin, D de Vries, DJ WardeFarley, M Webb, K Willson, L Xu, L Xue, S Yao, S Zhang, Y Zhang, Theano: A Python framework for fast computation of mathematical expressions. arXiv eprints. abs/1605.02688: (2016). http://arxiv.org/abs/1605.02688.Google Scholar
 M Abadi, A Agarwal, P Barham, E Brevdo, Z Chen, C Citro, GS Corrado, A Davis, J Dean, M Devin, S Ghemawat, I Goodfellow, A Harp, G Irving, M Isard, Y Jia, R Jozefowicz, L Kaiser, M Kudlur, J Levenberg, D Mané, R Monga, S Moore, D Murray, C Olah, M Schuster, J Shlens, B Steiner, I Sutskever, K Talwar, P Tucker, V Vanhoucke, V Vasudevan, F Viégas, O Vinyals, P Warden, M Wattenberg, M Wicke, Y Yu, X Zheng, TensorFlow: largescale machine learning on heterogeneous systems (2015). Software available from https://www.tensorflow.org/. Accessed 24 Feb 2017.
 SwissRanger. SwissRanger, Heptagon (2016).Google Scholar
 RB Rusu, S Cousins, in 2011 IEEE International Conference on Robotics and Automation. 3D is here: Point Cloud Library (PCL), (2011), pp. 1–4. doi:10.1109/ICRA.2011.5980567.
 Xtion PRO LIVE  specifications. https://www.asus.com/3DSensor/Xtion_PRO_LIVE/specifications/. Accessed 23 Aug 2016.
 N Srivastava, G Hinton, A Krizhevsky, I Sutskever, R Salakhutdinov. J. Mach. Learn. Res. 15(1), 1929 (2014). http://dl.acm.org/citation.cfm?id=2627435.2670313. Accessed 24 Feb 2017.MathSciNetGoogle Scholar
 K Lai, L Bo, X Ren, D Fox, in Robotics and Automation (ICRA), IEEE Int. Conf. on, (2011), pp. 4007–4013. doi:10.1109/ICRA.2011.5980377.
 R Socher, B Huval, B Bath, CD Manning, AY Ng, in Proceedings of the 25th International Conference on Neural Information Processing Systems. Convolutionalrecursive Deep Learning for 3D Object Classification (Curran Associates Inc.USA, 2012), pp. 656–664. http://dl.acm.org/citation.cfm?id=2999134.2999208.Google Scholar
 L Bo, X Ren, D Fox, in Experimental Robotics: The 13th International Symposium on Experimental Robotics, ed. by JP Desai, G Dudek, O Khatib, and V Kumar. Unsupervised Feature Learning for RGBD Based Object Recognition (Springer International PublishingHeidelberg, 2013), pp. 387–402. doi:http://dx.doi.org/10.1007/9783319000657_27.View ArticleGoogle Scholar
 M Schwarz, H Schulz, S Behnke, in 2015 IEEE International Conference on Robotics and Automation (ICRA). RGBD object recognition and pose estimation based on pretrained convolutional neural network features, (2015), pp. 1329–1335. doi:10.1109/ICRA.2015.7139363.