 Research Article
 Open Access
 Published:
From 2D Silhouettes to 3D Object Retrieval: Contributions and Benchmarking
EURASIP Journal on Image and Video Processing volume 2010, Article number: 367181 (2010)
Abstract
3D retrieval has recently emerged as an important boost for 2D search techniques. This is mainly due to its several complementary aspects, for instance, enriching views in 2D image datasets, overcoming occlusion and serving in many realworld applications such as photography, art, archeology, and geolocalization. In this paper, we introduce a complete "2D photography to 3D object" retrieval framework. Given a (collection of) picture(s) or sketch(es) of the same scene or object, the method allows us to retrieve the underlying similar objects in a database of 3D models. The contribution of our method includes (i) a generative approach for alignment able to find canonical views consistently through scenes/objects and (ii) the application of an efficient but effective matching method used for ranking. The results are reported through the Princeton Shape Benchmark and the Shrec benchmarking consortium evaluated/compared by a third party. In the two gallery sets, our framework achieves very encouraging performance and outperforms the other runs.
1. Introduction
3D object recognition and retrieval recently gained a big interest [27] because of the limitation of the "2Dto2D" approaches. The latter suffer from several drawbacks such as the lack of information (due for instance to occlusion), pose sensitivity, illumination changes, and so forth. This is also due to the exponential growth of storage and bandwidth on Internet, the increasing needs for services from 3D content providers (museum institutions, car manufacturers, etc.), and the easiness in collecting gallery sets^{1}. Furthermore, computers are now equipped with highly performant, easy to use, 3D scanners and graphic facilities for realtime modeling, rendering, and manipulation. Nevertheless, at the current time, functionalities including retrieval of 3D models are not yet sufficiently precise in order to be available for large usage.
Almost all the 3D retrieval techniques are resource (time and memory) demanding prior to achieve recognition and ranking. They usually operate on massive amount of data and require many upstream steps including object alignment, 3Dto2D projections and normalization. However and when no hard runtime constraints are expected, 3D search engines offer real alternatives and substantial gains in performance, with respect to (only) imagebased retrieval approaches; mainly when the relevant informations are appropriately extracted and processed (see, e.g., [8]).
Existing 3D object retrieval approaches can either be categorized into those operating directly on the 3D content and those which extract "2.5D" or 2D contents (stereopairs or multiple views of images, artificially rendered 3D objects, silhouettes, etc.). Comprehensive surveys on 3D retrieval can be found in [6, 8, 9, 34, 35, 41]. Existing state of the art techniques may also be categorized depending on the fact that they require a preliminary step of alignment or operate directly by extracting global invariant 3D signatures such as Zernike's 3D moments [28]. The latter are extracted using salient characteristics on 3D, "2.5D," or 2D shapes and ranked according to similarity measures. Structurebased approaches, presented in [19, 36, 37, 43], encode topological shape structures and make it possible to compute efficiently, without pose alignment, similarity between two global or partial 3D models. Authors in [7, 18] introduced two methods for partial shapematching able to recognize similar subparts of objects represented as 3D polygonal meshes. The methods in [17, 23, 33] use spherical harmonics in order to describe shapes, where rotation invariance is achieved by taking only the power spectrum of the harmonic representations and discarding all "rotationdependent" informations. Other approaches include those which analyze 3D objects using analytical functions/transforms [24, 42] and also those based on learning [29].
Another family of 3D object retrieval approaches belongs to the frontier between 2D and 3D querying paradigms. For instance, the method in [32] is based on extracting and combining spherical 3D harmonics with "2.5D" depth informations and the one in [15, 26] is based on selecting characteristic views and encoding them using the curvature scale space descriptor. Other "2.5D" approaches [11] are based on extracting rendered depth lines (as in [10, 30, 39]), resulting from vertices of regular dodecahedrons and matching them using dynamic programming. Authors in [12–14] proposed a 2D method based on Zernike's moments that provides the best results on the Princeton Shape Benchmark [34]. In this method, rotation invariance is obtained using the lightfield technique where all the possible permutations of several dodecahedrons are used in order to cover the space of viewpoints around an object.
1.1. Motivations
Due to the compactness of global 3D object descriptors, their performance in capturing the inter/intraclass variabilities are known to be poor in practice [34]. In contrast, local geometric descriptors, even though computationally expensive, achieve relatively good performance and capture inter/intraclass variabilities (including deformations) better than global ones (see Section 5). The framework presented in this paper is based on local features and also cares about computational issues while keeping advantages in terms of precision and robustness.
Our target is searching 3D databases of objects using one or multiple 2D views; this scheme will be referred to as "2Dto3D". We define our probe set as a collection of single or multiple views of the same scene or object (see Figure 2) while ourgallery set corresponds to a large set of 3D models. A query, in the probe set, will either be (i) multiple pictures of the same object, for instance stereopair, user's sketches, or (ii) a 3D object model processed in order to extract several views; so ending with the "2Dto3D" querying paradigm in both cases (i) and (ii). Gallery data are also processed in order to extract several views for each 3D object (see Section 2).
At least two reasons motivate the use of the "2Dto3D" querying paradigm:

(i)
The difficulty of getting "3D query models" when only multiple views of an object of interest are available (see Figure 2). This might happen when 3D reconstruction techniques [21] fail or when 3D acquisition systems are not available. "2Dto3D" approaches should then be applied instead.

(ii)
3D gallery models can be manipulated via different similarity and affine transformations, in order to generate multiple views which fit the 2D probe data, so "2Dto3D" matching and retrieval can be achieved.
1.2. Contributions
This paper is a novel "2Dto3D" retrieval framework with the following contributions.

(i)
A new generative approach is proposed in order to align andnormalize the pose of 3D objects and extract their 2D canonical views. The method is based on combining three alignments (identity and two variants of principal component analysis (PCA)) with the minimal visual hull (see Figure 1 and Section 2). Given a 3D object, this normalization is achieved by minimizing its visual hull with respect to different pose parameters (translation, scale, etc.). We found in practice that this clearly outperforms the usual PCA alignment (see Figure 10 and Table 2) and makes the retrieval process invariant to several transformations including rotation, reflection, translation, and scaling.

(ii)
Afterwards, robust and compact contour signatures are extracted using the set of 2D canonical views. Our signature is an implementation of the multiscale curve representation first introduced in [2]. It is based on computing convexity/concavity coefficients on the contours of the (2D) object views. We also introduce a global descriptor which captures the distributions of these coefficients in order to perform pruning and speed up the whole search process (see Figures 3 and 12).

(iii)
Finally, ranking is performed using our variant of dynamic programming which considers only a subset of possible matches thereby providing a considerable gain in performance for the same amount of errors (see Figure 12).
Figures 1, 2, and 3 show our whole proposed matching, querying, and retrieval framework which was benchmarked through the Princeton Shape Benchmark [34] and the international Shrec'09 contest on structural shape retrieval [1]. This framework achieves very encouraging performance and outperforms almost all the participating runs.
In the remainder of this paper, we consider the following terminology and notation. A probe (query) data is again defined either as (i) a 3D object model (denoted or ) processed in order to extract multiple 2D silhouettes, (ii) multiple sketched contours of the same mental query (target), or (iii) simply 2D silhouettes extracted from multiple photos of the same category (see Figure 2). Even though these acquisition scenarios are different, they allcommonly end up by providing multiple silhouettes describing the user's intention.
Let be a random variable standing for the 3D coordinates of vertices in any 3D model. For a given object, we assume that is drawn from an existing but unknown probability distribution . Let us consider as realizations of , forming a 3D object model. or will be used in order to denote a 3D model belonging to the gallery set while is a generic 3D object either belonging to the gallery or the probe set. Without any loss of generality 3D models are characterized by a set of vertices which may be meshed in order to form a closed surface or compact manifold of intrinsic dimension two. Other notations and terminologies will be introduced as we go through different sections of this paper which is organized as follows. Section 2 introduces the alignment and pose normalization process. Section 3 presents the global and the local multiscale contour convexity/concavity signatures. The matching process together with pruning strategies are introduced in Section 4, ending with experiments and comparison on the Princeton Shape Benchmark and the very recent Shrec'09 international benchmark in Section 5.
2. Pose Estimation
The goal of this step is to make retrieval invariant to 3D transformations (including scaling, translation, rotation, and reflection) and also to generate multiple views of 3D models in the gallery (and possibly the probe^{2}) sets. Pose estimation consists in finding the parameters of the above transformations (denoted resp. , , and ) by normalizing 3D models in order to fit into canonical poses. The underlying orthogonal 2D views will be referred to as the canonical views (see Figure 1).
Our alignment process is partly motivated by advances in cognitive psychology of human perception (see, e.g., [25]). These studies have shown that humans recognize shapes by memorizing specific views of the underlying 3D realworld objects. Following these statements, we introduce a new alignment process which mimics and finds specific views (also referred to as canonical views). Our approach is based on the minimization of a visualhull criterion defined as the area surrounded by silhouettes extracted from different object views.
Let us consider and given a 3D object , our normalization process is generative, that is, based on varying and finding the optimal set of parameters
here denotes the global normalization transformation resulting from the combination of translation, rotation, scaling, and reflection. , , denote, respectively, the "3Dto2D" parallel projections on the , , and canonical 2D planes. These canonical planes are, respectively, characterized by their normals , , and . The visual hull in (1) is defined as the sum of the projection areas ofusing. Let , , here provides this area on each 2D canonical plane.
The objective function (1) considers that multiple 3D instances of the same "category" are aligned (or have the same pose), if the optimal transformations (i.e., ), applied on the large surfaces of these 3D instances, minimize their areas. This makes the normals of these principal surfaces either orthogonal or collinear to the camera axis. Therefore, the underlying orthogonal views correspond indeed to the canonical views^{3} (see Figures 1 and 4) as also supported in experiments (see Figure 10 and Table 2).
It is clear that the objective function (1) is difficult to solve as one needs to recompute, for each possible the underlying visual hull. So it becomes clear that parsing the domain of variation of makes the search process tremendous. Furthermore, no gradient descent can be achieved, as there is no guarantee that is continuous w.r.t., . Instead, we restrict the search by considering few possibilities; in order to define the optimal pose of a given object , the alignment, which locally minimizes the visualhull criterion (1), is taken as one of the three possible alignments obtained according to the following procedure.
Translation and Scaling
and are recovered simply by centering and rescaling the 3D points in so that they fit inside an enclosing ball of unit radius. The latter is iteratively found by deflating an initial ball until it cannot shrink anymore without losing points in (see [16] for more details).
Rotation
is taken as one of the three possible candidate matrices including (i) identity^{4} (i.e., no transformation, denoted none), or one of the transformation matrices resulting from PCA either on (ii) gravity centers or (iii) face normals, of . The two cases (ii), (iii) will be referred to as PCA and normal PCA (NPCA), respectively, [39, 40].
Axis Reordering and Reflection
This step processes only 3D probe objects and consists in reordering and reflecting the three projection planes , in order to generate possible triples of 2D canonical views (i.e., for reordering for reflection). Reflection makes it possible to consider mirrored views of objects while reordering allows us to permute the principal orthogonal axes of an object and therefore permuting the underlying 2D canonical views.
For each combination taken from "scaling translation 3 possible rotations" (see explanation earlier), the objective function (1) is evaluated. The combination that minimizes this function is kept as the best transformation. Finally, three canonical views are generated for each object in the gallery set.
3. Multiview Object Description
Again, we extract the three 2D canonical views corresponding to the projection of an object , according to the framework described earlier. Each 2D view of is processed in order to extract and describe external contours using [2]. Our description is based on a multiscale analysis which extracts convexity/concavity coefficients on each contour. Since the latter are strongly correlated through many views of a given object , we describe our contours using three up to nine views per reordering and reflection. This reduces redundancy and also speeds up the whole feature extraction and matching process (see Figure 5).
In practice, each contour, denoted , is sampled with (2D) points () and processed in order to extract the underlying convexity/concavity coefficients at different scales [2]. Contours are iteratively filtered ( times) using a Gaussian kernel with an increasing scale parameter . Each curve will then evolve into different smooth silhouettes. Let us consider a parameterization of using the curvilinear abscissa as , , let us also denote as a smooth version of resulting from the application of the Gaussian kernel with a scale (see Figure 6).
We use simple convexity/concavity coefficients as local descriptors for each 2D point on (). Each coefficient is defined as the amount of shift of between two consecutive scales and . Put differently, a convexity/concavity coefficient denoted is taken as , here denotes the norm.
Runtime
Even though multiview feature extraction is offline on the gallery set, it is important to achieve this step in (near) real time for the probe data. Notice that the complexity of this step depends mainly on the number of silhouettes and their sampling. Table 1 shows average runtime for alignment and feature extraction, in order to process one object, and for different numbers of silhouettes. These experiments were achieved on a standard 1 Ghz (G4) PowerPC including 512 MB of Ram and 32 MB of VRam.
4. CoarsetoFine Matching
4.1. Coarse Pruning
A simple coarse shape descriptor is extracted both on the gallery and probe sets. This descriptor quantifies the distribution of convexity and concavity coefficients through 2D points belonging to different silhouettes of a given object. This coarse descriptor is a multiscale histogram containing 100 bins as the product of scales of the Gaussian kernel (see Section 3) and quantification values for convexity/concavity coefficients. Each bin of this histogram counts, through all the viewpoint silhouettes of an object, the frequency of the underlying convexity/concavity coefficients. This descriptor is poor in terms of its discrimination power, but efficient in order to reject almost all the false matches while keeping candidate ones when ranking the gallery objects w.r.t. the probe ones (see also processing time in Figure 9).
4.2. Fine Matching by Dynamic Programming
Given are two objects , , respectively, from the probe and the gallery sets and the underlying silhouettes/curves , . A global scoring function is defined between , as the expectation of the matching pseudodistance involving all the silhouettes , as
here is the number of silhouettes per probe image (in practice, or , see Section 5).
Silhouette matching is performed using dynamic programming. Given two curves , , a matching pseudodistance, denoted DSW, is obtained as a sequence of operations (substitution, insertion, and deletion) which transforms into [43]. Considering the samples from , and the underlying local convexity/concavity coefficients , the DSW pseudodistance is
here denotes the norm, and is the dynamic programming matching function, which assigns for each curvilinear abscissa in its corresponding abscissa in . Given the distance matrix with , the matching function is found by selecting a path in . This path minimizes the number of operations (substitution, deletion, and insertion in order to transform into ) and preserves the ordering assumption (i.e., if is matched with then should be matched only with , ). We introduce a variant of the standard dynamic programming; instead of examining all the possible matches, we consider only those which belong to a diagonal band of , that is, is allowed to take only small values (see Figures 7 and 8).
Dynamic programming pseudodistance provides a good discrimination power and may capture the intraclass variations better than the global distance (discussed in Section 4.1). Nevertheless, it is still computationally expensive but when combined with coarse pruning the whole process is significantly faster and also precise (see Figure 9 and Table 2). Finally, this elastic similarity measure allows us to achieve retrieval while being robust to intraclass object articulations/deformations (observed in the Shrec Watertight set) and also to other effects (including noise) induced by handdrawn sketches (see Figures 14, 15, 16, and 17).
Runtime
Using the coarsetofine querying scheme described earlier, we adjust the speedup/precision tradeoff via a parameter . Given a query, this parameter corresponds to the fraction of nearest neighbors (according to our global descriptor) used in order to achieve dynamic programming. Lower values of make the retrieval process very fast at the detriment of a slight decrease of precision and vice versa. Figure 9 shows runtime performance with respect to on the same hardware platform (with 9 views).
5. Experiments
5.1. Databases
In order to evaluate the robustness of the proposed framework, we used two datasets. The first one is the Watertight dataset of the Shrec benchmark while the second one is the Princeton Shape Benchmark, widely used in the 3D contentbased retrieval community.
Shrec Watertight Dataset
This dataset contains 400 "3D" objects represented by seamless surfaces (without defective holes or gaps). The models of this database have been divided into classes each one contains objects. The 3D models were taken from two sources: the first one is a deformation of an initial subset of objects (octopus, glasses,), while the second one is a collection of original 3D models (chair, vase, four legs,).
Princeton Shape Benchmark
This dataset contains 907 "3D" objects organized in 92 classes. This dataset offers a large variety of objects for evaluation.
For the two datasets, each 3D object belongs to a unique class among different semantic concepts with strong variations including human, airplane, chair, and so forth. For instance, the human class contains persons with different poses and appearances "running, seating, walking, etc.", globally the two databases are very challenging.
5.2. Evaluation Criteria
We evaluated our method using recallprecision. Precision is defined as the fraction of relevant retrieved objects over the number of displayed 3D models while recall is defined as the fraction of relevant retrieved objects over the total number of relevant 3D models in the dataset. A plot that approaches the 11 corner indicates better retrieval results. In addition to recall/precision plot, we use several quantitative statistics to evaluate the results.

(i)
The nearest neighbor (NN). It represents the fraction of the first nearest neighbors which belong to the same class as the query.

(ii)
The firsttier (FT) and the secondtier (ST). These measures give the percentage of objects in the same class as the query that appear in the best matches. For a given class containing objects, is set to for the firsttier measure while is set to for secondtier (ST).

(iii)
Finally, we use the discounted cumulative gain (DCG) measure which gives more importance to wellranked models. Given a query and a list of ranked objects, we define for each ranked object a variable equal to if its class is equal to the class of the query and otherwise. The DCG measure is then defined as.
(4)
We take the expectation of these measures on the entire database, that is, by taking all the possible object queries.
5.3. Performance and Discussion
Alignment
Figure 10 shows the performance of our alignment method presented in Section 2 on the Watertight dataset. For that purpose, we define a ground truth, by manually aligning "3D" models^{5} ( categories each one has objects) in order to make their canonical views parallel to the canonical planes , and (see Figure 11, and also Figure 4). The error is then defined as the deviation (angle in degrees or radians) of the automatically aligned objects w.r.t. the underlying ground truth (see Figure 11, ).
Different alignment methods were compared including the classic (PCA), normal PCA (NPCA), and our method. We also show the alignment error of the initial (not) aligned database (None). The plot in Figure 10 shows a comparison of the percentage of 3D objects in the database, which are automatically and correctly aligned up to an angle w.r.t. the underlying 3D models in the ground truth.
Table 2 illustrates the statistics defined earlier. We clearly see that our new alignment method gives better results compared to the classical PCA and NPCA. Again our pose estimation method makes it possible to extract several canonical 2D views and for each one we compared results using either three or nine 2D views per object (see results in Figure 13 and rows 4 and 5 of Table 2). Regarding the influence of the number of views, the performances increase for the two datasets.
CoarsetoFine Retrieval
In order to control/reduce the runtime to process and match local signatures, we used our pruning approach based on the global signature discussed in Section 4.1. The parameter allows us to control the tradeoff between robustness and speed of the retrieval process. A small value of gives realtime (online) responses with an acceptable precision while a high value requires more processing time but gives better retrieval performance. Figure 12 shows the NN, FT, ST, and DCG measures for different pruning thresholds . Table 2 shows different statistics for , and .
Table 3 shows also the performance of matching using dynamic programming versus adhoc naive matching (i.e., through the curvilinear abscissa, in (2). Dynamic programming outperforms the naive matching by allowing to be equal to () in contrast to naive matching (); this clearly makes dynamic programming more flexible in order to handlelocal deformations (see again Table 3).
5.4. Benchmarking and Comparison
Shrec Watertight Dataset
First, comparisons of our approach with respect to different methods/participants are available and were generated by a third party in the Shrec'09 Structural Shape Retrieval contest (see Table 4). This dataset contains 200 objects and results were evaluated on 10 queries. The performance of this shape retrieval contest were measured using 1st (10 objects) and 2nd (20 objects) tier precision and recall, presented as the Fmeasure. This is a global measure which provides us with the overall retrieval performance.
We submitted in this benchmark four runs.

(i)
Run 1 (MCC 1): 9 silhouettes and pruning threshold . The average runtime for each query is 0.03 s.

(ii)
Run 2 (MCC 2): 9 silhouettes and pruning threshold . The average runtime for each query is 9.4 s.

(iii)
Run 3 (MCC 3): 9 silhouettes and pruning threshold . The average runtime for each query is 36.2 s.

(iv)
Run 4 (MCC 4): 3 silhouettes and pruning threshold . The average runtime for each query is 3.1 s.
We can see in Table 4 that the third run of our method (shown in bold) outperforms the others for the firsttier and is equivalent to the C SID CMVD 3 for the secondtier (see Table 7 for the significance of method acronyms). The results for the second run are similar to the BF SIFT 1 and to the C SID CMVD 1 methods.
Princeton Shape Benchmark Dataset
Table 5 shows a comparison of the four runs of our approach on the Princeton Shape Benchmark; these runs outperform the other participating methods (described in [4, 10, 34]).
HandDrawn Sketches and Photos
Finally, we compared our approach with respect to two querying schemes including (i) 2 handdrawn sketches per mental category^{6} or (ii) silhouettes from multiview real pictures. In both scenarios, gallery data are processed in the same way as inTable 4 (MCC 4),that is, by aligning 3D objects using our pose estimation method and processing them in order to extract 3 views. The results on the two databases, in Figures 14 to 17, Table 6 and Figure 18, show very encouraging performances on real data (sketches and real pictures) and clearly open very promising directions for further extensions and improvements.
6. Conclusion
We introduced in this paper a novel and complete framework for "2Dto3D" object retrieval. The method makes it possible to extract canonical views using a generative approach combined with principal component analysis. The underlying silhouettes/contours are matched using dynamic programming in a coarsetofine way that makes the search process efficient and also effective as shown through extensive evaluations.
One of the major drawbacks of dynamic programming resides in the fact that it is not a metric, so one cannot benefit from lossless acceleration techniques which provide precise results and efficient computation. Our extension is to tackle this issue by introducing new matching approaches that allow us to speedup the search process while keeping high precision.
Endnotes
1. Even though in a chaotic way because of the absence of consistent alignments of 3D models.
2. Obviously, normalization is achieved on the probe set only when queries are 3D models. As for the 2D photo or the sketch scenarios, one assumes that at least three silhouettes are available corresponding to three canonical views.
3. Again, this is in accordance with cognitive psychology of human perception (defined, e.g., in [25]).
4. The initial object pose is assumed to be the canonical one.
5. http://perso.enst.fr/~sahbi/file/Watertight_AlignmentGroundTruth.zip.
6. The user will imagine a category existing in the Watertight gallery set and will draw it.
References
 1.
Hartveldt J, Spagnuolo M, Axenopoulos A, et al.: SHREC'09 track: structural shape retrieval on watertight models. Proceedings of Eurographics Workshop on 3D Object Retrieval, March 2009, Munich, Germany 7783.
 2.
Adamek T, O'Connor NE: A multiscale representation method for nonrigid shapes with a single closed contour. IEEE Transactions on Circuits and Systems for Video Technology 2004,14(5):742753. 10.1109/TCSVT.2004.826776
 3.
Ankerst M, Kastenmüller G, Kriegel HP, Seidl T: Nearest neighbor classification in 3D protein databases. Proceedings of the 7th International Conference on Intelligent Systems for Molecular Biology (ISMB '99), August 1999, Heidelberg, Germany 3443.
 4.
Ansary TF: Model retrieval using 2d characteristic views, Ph.D. thesis. 2006.
 5.
Bellman R: Dynamic programming. Science 1966,153(3731):3437. 10.1126/science.153.3731.34
 6.
Biasotti S, Giorgi D, Marini S, Spagnuolo M, Falcidieno B: A comparison framework for 3D object classification methods. Proceedings of the International Workshop on Multimedia Content Representation, Classification and Security (MRCS '06), 2006, Lecture Notes in Computer Science 4105: 314321.
 7.
Biasotti S, Marini S, Spagnuolo M, Falcidieno B: Subpart correspondence by structural descriptors of 3D shapes. Computer Aided Design 2006,38(9):10021019. 10.1016/j.cad.2006.07.003
 8.
Del Bimbo A, Pala P: Contentbased retrieval of 3D models. ACM Transactions on Multimedia Computing, Communications and Applications 2006,2(1):2043. 10.1145/1126004.1126006
 9.
Bustos B, Keim D, Saupe D, Schreck T, Vranić D: An experimental comparison of featurebased 3D retrieval methods. In Proceedings of the 2nd International Symposium on 3D Data Processing, Visualization, and Transmission, September 2004, Thessaloniki, Greece. IEEE Computer Society; 215222.
 10.
Chaouch M, VerroustBlondet A: Enhanced 2D/3D approaches based on relevance index for 3Dshape retrieval. Proceedings of IEEE International Conference on Shape Modeling and Applications (SMI '06), June 2006, Matsushima, Japan 36.
 11.
Chaouch M, VerroustBlondet A: A new descriptor for 2D depth image indexing and 3D model retrieval. Proceedings 14th IEEE International Conference on Image Processing (ICIP '07), September 2006 6: 373376.
 12.
Chen D: Threedimensional model shape description and retrieval based on lightfield descriptors, Ph.D. thesis. Department of Computer Science and Information Engineer, National Taiwan University, Taipei, Taiwan; June 2003.
 13.
Chen D, Ouhyoung M: A 3d model alignment and retrieval system. Proceedings of the International Workshop on Multimedia Technologies, December 2002 14361443.
 14.
Chen DY, Tian XP, Shen YT, Ouhyoung M: On visual similarity based 3D model retrieval. Computer Graphics Forum 2003,22(3):223232. 10.1111/14678659.00669
 15.
Ansary TF, Daoudi M, Vandeborre JP: A Bayesian 3D search engine using adaptive views clustering. IEEE Transactions on Multimedia 2007,9(1):7888.
 16.
Fischer K, Gärtner B: The smallest enclosing ball of balls: combinatorial structure and algorithms. International Journal of Computational Geometry and Applications 2004,14(45):341378.
 17.
Funkhouser T, Min P, Kazhdan M, et al.: A search engine for 3D models. ACM Transactions on Graphics 2003,22(1):83105. 10.1145/588272.588279
 18.
Funkhouser T, Shilane P: Partial matching of 3d shapes with prioritydriven search. Proceedings of the 4th Eurographics Symposium on Geometry Processing, June 2006, Cagliari, Italy 131142.
 19.
Hilaga M, Shinagawa Y, Kohmura T, Kunii TL: Topology matching for fully automatic similarity estimation of 3D shapes. Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH '01), August 2001, Los Angeles, Calif, USA 203212.
 20.
Horn BKP: Extended Gaussian images. Proceedings of the IEEE 1984,72(12):16711686.
 21.
Jin H, Soatto S, Yezzi AJ: Multiview stereo reconstruction of dense shape and complex appearance. International Journal of Computer Vision 2005,63(3):175189. 10.1007/s1126300568767
 22.
Kang SB, lkeuchi K: Determining 3d object pose using the complex extended Gaussian image. Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, June 1991, Maui, Hawaii, USA 580585.
 23.
Kazhdan MM, Funkhouser TA, Rusinkiewicz S: Rotation invariant spherical harmonic representation of 3d shape descriptors. In Proceedings of the Eurographics/ACM SIGGRAPH Symposium on Geometry Processing (SGP '03), June 2003, Aachen, Germany. Eurographics Association; 156165.
 24.
Laga H, Takahashi H, Nakajima M: Spherical wavelet descriptors for contentbased 3D model retrieval. Proceedings of IEEE International Conference on Shape Modeling and Applications (SMI '06), June 2006, Matsushima, Japan 15.
 25.
Leek EC: Effects of stimulus orientation on the identification of common polyoriented objects. Psychonomic Bulletin and Review 1998,5(4):650658. 10.3758/BF03208841
 26.
Mahmoudi S, Daoudi M: 3D models retrieval by using characteristic views. Proceedings of the 16th International Conference on Pattern Recognition (ICPR '02), August 2002, Quebec, Canada (2):457460.
 27.
NIST : Shape retrieval contest on a new generic shape benchmark. 2004, http://www.itl.nist.gov/iad/vug/sharp/benchmark/shrecGeneric/Evaluation.html
 28.
Novotni M, Klein R: 3D Zernike descriptors for content based shape retrieval. In Proceedings of the 8th Symposium on Solid Modeling and Applications, 2003, Seattle, Wash, USA. ACM Press; 216225.
 29.
Ohbuchi R, Kobayashi J: Unsupervised learning from a corpus for shapebased 3D model retrieval. Proceedings of the ACM International Multimedia Conference and Exhibition, October 2006, Santa Barbara, Calif, USA 163172.
 30.
Ohbuchi R, Nakazawa M, Takei T: Retrieving 3d shapes based on their appearance. In Proceedings of the 5th ACM SIGMM International Workshop on Multimedia Information Retrieval (MIR '03), 2003 Edited by: Sebe N, Lew MS, Djeraba C. 3945.
 31.
Osada R, Funkhouser T, Chazelle B, Dobkin D: Matching 3d models with shape distributions. In Proceedings of the International Conference on Shape Modeling and Applications (SMI '01), May 2001, Los Alamitos, Calif, USA. Edited by: Werner B. IEEE Computer Society; 154166.
 32.
Papadakis P, Pratikakis I, Perantonis S, Theoharis T, Passalis G: SHREC'08 entry: 2D/3D hybrid. Proceedings of IEEE International Conference on Shape Modeling and Applications (SMI '08), June 2008 247248.
 33.
Saupe D, Vranic DV: 3d model retrieval with spherical harmonics and moments. In Proceedings of the 23rd DAGMSymposium on Pattern Recognition, 2001, Lecture Notes in Computer Science. Volume 2191. Edited by: Radig B, Florczyk S. Springer; 392397.
 34.
Shilane P, Min P, Kazhdan M, Funkhouser T: The Princeton shape benchmark. Proceedings of the Shape Modeling International (SMI '04), 2004, Washington, DC, USA 167178.
 35.
Tangelder JWH, Veltkamp RC: A survey of content based 3D shape retrieval methods. Proceedings of the Shape Modeling International (SMI '04), June 2004 145156.
 36.
Tierny J, Vandeborre JP, Daoudi M: 3d mesh skeleton extraction using topological and geometrical analyses. Proceedings of the 14th Pacific Conference on Computer Graphics and Applications, October 2006, Taipei, Taiwan 8594.
 37.
Tung T, Schmitt F: The augmented multiresolution Reeb graph approach for contentbased retrieval of 3D shapes. International Journal of Shape Modeling 2005,11(1):91120. 10.1142/S0218654305000748
 38.
Vranić DV: An improvement of rotation invariant 3Dshape descriptor based on functions on concentric spheres. Proceedings of IEEE International Conference on Image Processing, 2003 3: 757760.
 39.
Vranic DV: 3D model retrieval, Ph.D. thesis. University of Leipzig; 2004.
 40.
Vranić DV, Saupe D, Richter J: Tools for 3Dobject retrieval: KarhunenLoeve transform and spherical harmonics. In Proceedings of the 4th IEEE Workshop on Multimedia Signal Processing, September 2001, Budapest, Hungary Edited by: Dugelay JL, Rose K. 293298.
 41.
Zaharia T, Prêteux F: 3D versus 2D/3D shape descriptors: a comparative study. Imaging Processing: Algorithms and Systems III, January 2004, San Jose, Calif, USA, Proceedings of SPIE 5298: 4758.
 42.
Zarpalas D, Daras P, Axenopoulos A, Tzovaras D, Strintzis MG: 3D model search and retrieval using the spherical trace transform. EURASIP Journal on Advances in Signal Processing 2007, 2007:14.
 43.
Tierny J, Vandeborre JP, Daoudi M: Invariant high level reeb graphs of 3D polygonal meshes. Proceedings of the 2rd International Symposium on 3D Data Processing, Visualization, and Transmission (3DPVT '06), June 2007, Chapel Hill, NC, USA 105112.
Acknowledgment
This work was supported by the European Network of Excellence KSpace and the French National Research Agency (ANR) under the AVEIR Project, ANR06MDCA002.
Author information
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Received
Revised
Accepted
Published
DOI
Keywords
 Dynamic Programming
 Visual Hull
 Discount Cumulative Gain
 Discount Cumulative Gain
 Canonical View