1 Introduction
Encoding and reconstructing 3D shapes is a fundamental problem in computer graphics, computer vision and related fields. Unlike images, which enjoy a canonical representation, 3D shapes are encoded through a large variety of representations, such as point clouds, triangle meshes and volumetric data, to name a few. Perhaps even more importantly, 3D shapes may undergo a diverse set of transformations, ranging from rigid motions to complex nonrigid and articulated deformations, that impact these representations.
The representation issues have become even more prominent in recent years with the advent of learningbased techniques, leading to a number of solutions for learning directly on geometric 3D data [9]. This is challenging as point clouds and meshes lack the regular grid structure exploited by convolutional architectures. In particular, the problem of representations that are well adapted for both shape analysis and especially shape synthesis
remains difficult. For example, several methods for shape interpolation have been proposed by designing deep neural networks, including autoencoder architectures, and interpolating the latent vectors learned by such networks
[34, 1] . Unfortunately, it is not clear if the latent vectors lie in a linear vector space, and thus linear interpolation can lead to unrealistic intermediate shapes.In this paper, we show that 3D shapes can not only be compactly encoded as linear functional operators, using the previously proposed shape difference operators [31], but that this representation lends itself very naturally to learning, and allows us to recover
the 3D shape information, using a novel neural network architecture which we call OperatorNet. Our key observations are twofold: first we show that since shape difference operators can be stored as canonical matrices, for a given choice of basis, they enable the use of a Convolutional Neural Network architecture for shape recovery. Second, we demonstrate that the
functional algebra that is naturally available on these operators can be used to synthesize new shapes, in the context of shape interpolation and shape analogy applications. We argue that because this algebra is welljustified theoretically, it also leads to more accurate results in practice, compared to commonly used linear interpolation in the latent space (see Figure 1).The shape difference operators introduced in [31], have proved to be a powerful tool in shape analysis, by allowing to characterize each shape in a collection as the ‘difference’ to some fixed base shape. These difference operators, in particular, encode precise information about how and where each shape differs from the base, but also, due to their compact representation as small matrices, enable efficient exploration of global variability within the collection. Inspired by the former perspective, purely geometric approaches [7, 12] have been proposed for shape reconstruction from shape differences. Though theoretically welljustified, these approaches rely on solving difficult nonlinear optimization problems and require strong regularization for accurate results, especially when truncated bases are used.
Our OperatorNet, on the other hand, aims to leverage the information encoded at both the pairwise level and the collection level by using the shape collection to guide the reconstruction. It is wellknown that related shapes in a collection often concentrate near a lowdimensional manifold in the shape space [32, 19]. In light of this, the shape difference operators can help to both encode the geometry of the individual shapes, but also help to learn the constrained space of realistic shapes, which is typically ignored by purely geometric approaches.
In addition to demonstrating the representative power of the shape differences in a learning framework, we also extend the original formulation in [31], which only involves intrinsic (i.e., invariant to isometric transformations) shape differences, with a novel extrinsic difference operator that facilitates posedependent embedding recovery. Our formulation is both simpler and more robust compared to previous approaches, e.g. [12], and, as we show below, can more naturally be integrated in a unified learning framework.
To summarize, our contributions are as follows:

We propose a learningbased pipeline to reconstruct 3D shapes from a set of difference operators.

We propose a novel formulation of extrinsic shape difference, which complements the intrinsic operators formulated in [31].

We demonstrate that by applying algebraic operations on shape differences, we can synthesize new operators and thus new shapes via OperatorNet, enabling shape manipulations such as interpolation and analogy.
2 Related Work
Shape Reconstruction
Our work is closely related to shape reconstruction from intrinsic operators, which was recently considered in [7, 12]. In these works, several advanced and purely geometric optimization techniques were proposed that give satisfactory results in the presence of full information [7] or under strong (extrinsic) regularization [12]. These works have also laid the theoretical foundation for shape recovery by demonstrating that shape difference operators, in principle, contain complete information necessary for recovering the shape embedding (e.g. Propositions 2 and 4 in [12]). On the other hand, these methods also highlight the practical challenges raised by this approach to reconstruction: when reconstructing a shape without any knowledge of the collection or “shape space” that it belongs to. In contrast, we show that when using shape difference representations in a learningbased framework, realistic 3D shapes can be recovered efficiently, and moreover that entirely new shapes can be synthesized using the algebraic structure of difference operators, e.g. for shape interpolation and analogy.
Shape Representations for Learning
Our work is also related to the recent techniques aimed at applying deep learning methods to shape analysis. One of the main challenges is defining a meaningful notion on convolution, while ensuring invariance to basic transformations, such as rotations and translations. Several techniques have been proposed based on e.g. geometry images
[33], volumetric [22, 36], pointbased [27] and multiview approaches [28], as well as, very recently intrinsic techniques that adapt convolution to curved surfaces [21, 8] (see also [9] for an overview), and even via toric covers [20], among many others.Despite this tremendous progress in the last few years, defining a shape representation that is compact, lends itself naturally to learning, while being invariant to the desired class of transformations (e.g. rigid motions) and not limited to a particular topology, remains a challenge. As we show below, our representation is wellsuited for learning applications, and especially for encoding and recovering geometric structure information.
Shape Space
Exploring the structure of shape spaces has been an attractive research topic for a long time. Classical PCAbased models, e.g. [2, 14], and more recent shape space models, adapted to specific shape classes such as humans or animals, such as [19, 37]
, or parametric model collections
[32]; all typically leverage the fact that the space of “realistic” shapes is significantly smaller than the space of all possible embeddings. This has also recently been exploited in the context of learningbased shape synthesis applications for shape completion [17], interpolation [4] and point cloud reconstruction [1] among others. These techniques heavily leverage the recent proliferation of large shape collections such as DFAUST [6] and SHAPENET [10] to name a few. At the same time, it is not clear if, for example, the commonly used linear interpolation of latent vectors is welljustified, leading to unrealistic synthesized shapes. Instead, the shape difference operators that we use satisfy a wellfounded multiplicative algebra, which, as we show, can naturally create more realistic synthetic interpolations.3 Preliminaries and Notations
Discretization of Shapes
Throughout this paper, we assume that a shape is given as a triangular mesh , where is the vertex set, and is the set of triangles encoding the connectivity of the vertices.
LaplaceBeltrami Operator
We associate with each shape a discretized LaplaceBeltrami operator, , using the cotangent weight scheme from [23, 26], where is the cotangent weight (stiffness) matrix, and is the diagonal lumped area (mass) matrix. Furthermore, we denote by the diagonal matrix containing the
smallest eigenvalues, and by
the corresponding eigenvectors of
, such that . In particular, the eigenvalues stored in are nonnegative and can be ordered as . The columns of are sorted accordingly, and are orthonormal with respect to the area matrix, i.e. , the identity matrix. It is wellknown that the LaplaceBeltrami eigenbasis provides a multiscale understanding of a shape [16], and allows to approximate the space of functions via a subspace spanned by the first few eigenvectors of .Functional Maps
The functional map framework was introduced in [24] primarily as an alternative representation of maps across shapes. In our context, given two shapes and a pointwise map from to , we can express the functional map from to , as follows:
(1) 
Here, is the area matrix of , and is a binary matrix satisfying if and otherwise. Note that is a matrix, where is the number of basis functions chosen on and . This matrix allows to transport functions as follows: if is a function on expressed as a vector of coefficients , s.t. , then is the vector of coefficients of the corresponding function on , expressed in the basis of .
In general, not every functional map matrix arises from a pointwise map, and might include for example soft correspondences, which map a point to a probability density function. All of the tools that we develop below can accommodate such general maps. This is a key advantage of our approach, as it does not rely on all shapes having the same number of points, and only requires the knowledge of functional map matrices, which can be computed using existing techniques
[25, 18].Intrinsic Shape Difference Operators
Finally, to represent shapes themselves, we use the notion of shape difference operators proposed in [31]. Within our setting, they can be summarized as follows: given a base shape , an arbitrary shape and a functional map between them, let (resp. ) be a positive semidefinite matrix, which defines some inner product for functions on (resp. ) expressed in the corresponding bases. Thus, for a pair of functions on expressed as vectors of coefficients , we have
Note that these two inner products are not comparable, since they are expressed in different bases. Fortunately, the functional map plays a role of basis synchronizer. Thus, a shape difference operator, which captures the difference between and is given simply as:
(2) 
where is the MoorePenrose pseudoinverse.
The original work [31] considered two intrinsic inner products, which using the notation above, can be expressed as: , and . These inner products, in turn lead to the following shape differences operators:
Areabased ():  (3)  
Conformal ():  (4) 
These shape difference operators have several key properties. First, they allow to represent an arbitrary shape , as a pair of matrices of size , independent of the number of points, by requiring only a functional map between the base shape and . Thus, the size of this representation can be controlled by choosing an appropriate value of which allows to gain multiscale information about the geometry of , from the point of view of . Second, and perhaps more importantly, these matrices are invariant to rigid (and indeed any intrinsic isometry) transformation of or . Finally, previous works [12] have shown that shape differences in principle contain complete information about the intrinsic geometry of a shape. As we show below these properties naturally enable the use of learning applications for shape recovery.
Functoriality of Shape Differences
Another useful property of the shape difference operators is functoriality, shown in [31], and which we exploit in our shape synthesis applications in Section 7. Given shape differences of shapes and with respect to a base shape , functoriality allows to compute the difference , without functional maps between and . Namely (see Prop. 4.2.4 in [11]):
(5) 
Intuitively, this means that shape differences naturally satisfy the multiplicative algebra: , up to a change of basis ensured by .
This property can be used for shape analogies: given shapes and , to find such that relates to in the same way as relates to (see the illustration in Figure 2), which can be solved by looking for a shape that satisfies: . In our application, we first create an appropriate and then use our network to synthesize the corresponding shape.
Finally, the multiplicative property also suggests a way of interpolation in the space of shape differences. Namely, rather than using basic linear interpolation between and , we interpolate on the Lie algebra of the Lie group of shape differences, using the exponential map and its inverse, which leads to:
(6) 
Here and are matrix exponential and logarithm respectively. Note that, around identity, the linearization provided by the Lie algebra is exact, and we have observed it to produce very accurate results in general.
4 Extrinsic Shape Difference
In theory, with purely intrinsic information one at the best can determine a shape up to isometric transformations, which in our setting means being able to capture the edge lengths of the mesh. Recovering the shape from its edge lengths, while possible in certain simple scenarios, nevertheless often leads to ambiguities, as highlighted in [12]. To alleviate such ambiguities, we propose to augment the existing purely intrinsic shape differences with a novel compatible shape difference operator that gives rise to a more complete characterization of shapes, and in turn boosts our reconstruction.
One basic approach that has been used to combine extrinsic or embeddingdependent information with the multiscale LaplaceBeltrami basis, is simply to project the 3D coordinates, as functions onto the basis, to obtain three vectors of coefficients (one for each coordinates): , where is the matrix of vertex coordinates [16, 15]. Unfortunately representing a shape through , while also multiscale and compact, has several limitations. First this representation is not rotationally invariant, and second, it does not provide information about intrinsic geometry, so that interpolation of coordinate vectors can easily lead to loss of shape area, for example.
Another option, which is more compatible with the shape difference representation and is rotationally invariant, is to encode the inner products of coordinate functions on each shape using the Gram matrix where is again the matrix of coordinate functions. Expressing in the corresponding basis, and using Eq. (2) gives rise to a shape differencelike representation of the shape coordinates. Indeed, the following theorem (see proof in Appendix A) guarantees that the resulting shape difference representation contains the same information, up to rotational invariance, as simply projecting the coordinates onto the basis.
Theorem 1.
Let be the extrinsic inner product encoded in , where is the lumped mass matrix (as normalization factor), then one can recover the projections of the coordinate functions, , on the subspace spanned by from up to rigid transformations. In particular, when is a complete full basis, the recovery of is exact.
As an illustration of Theorem 1, we show in Figure 3 the embeddings recovered from when the number of basis functions in range from 10 to 300.
Representing a shape via its Gram matrix in either the full or the reduced basis has one key limitation, however, since the rank of is at most 3, meaning that the majority of its eigenvalues are zero. This turns out to be an issue in applications, where gaining information about the local geometry of the shape is important, for example in our shape analogies experiments.
To compensate for this rank deficiency, we finalize our construction of the extrinsic inner product by making it Laplacianlike:
(7) 
Where is , i.e., the squared Euclidean distance between points on the shape, weighted by the respective vertex area measure. Since can be regarded as the Laplacian of a complete graph, all but one of its eigenvalues are strictly positive.
It is worth noting that the Gram matrix and the squared Euclidean distance matrix are coherently related and can be recovered from each other as is commonly done in the MultiDimensional Scaling literature [13].
To summarize, given a base shape , another shape and a functional map we encode the extrinsic information of from the point of view of as follows:
(8) 
In Figure 4, we compute and
of the target shape with respect to the base, and color code their respective eigenfunctions associated with the largest eigenvalue, on the shapes to the right. As argued in
[31] these functions capture the areas of highest distortion between the shapes, with respect to the corresponding inner products. Note that the eigenfunction of captures the armpit where the local area changes significantly, while that of captures the hand, where the pose changes are evident.It is worth noting that in [12], the authors also propose a shape difference formulation for encoding extrinsic information, which is defined on the shape offset in order to extract information for surface normal. However, the construction of offset can lead to instabilities, and moreover, this formulation only gives information about local distances, making it hard to recover large changes in pose.
5 Network Details
Problem Setup
Our general goal is to develop a neural network capable of recovering the coordinates functions of a shape, given its representation as a set of shape difference operators. This way we aim to solve the same problem considered in [7, 12]. However, unlike these previous, purely geometric methods, we further leverage a collection of training shapes to learn and constrain the reconstruction to the space of realistic shapes.
Thus, we assume that we are given a collection of shapes, each represented by a set of shape difference operators with respect to a fixed base shape. We also assume the presence of a pointwise map from the base shape to each of the shapes in the collection, which allows us to compute the “ground truth” embedding of each shape. We represent this embedding as three coordinate functions on the base shape. Our goal then is to design a network, capable of converting the input shape difference operators to the ground truth coordinate functions.
At test time, we use this network to reconstruct a target shape given only the shape difference operators with respect to the base shape. These shape difference operators can be obtained using the knowledge of a functional map from the base shape, or synthesized directly for shape analogies or interpolations applications.
Architecture
To solve the problem above we developed the OperatorNet architecture, which takes as input shape difference matrices and outputs coordinate functions representing the original shapes. Our network has two modules: a shallow convolutional encoder and a 3layer dense decoder as shown in Figure 5.
The grid structure of shape differences is exploited by the encoder through the use of convolutions. Note however that translation invariance does not apply to these matrices.
After comparing with multiple depths of encoders, we selected a shallow version as it performed the best in practice, implying that the shape difference representation already encodes meaningful information efficiently. Moreover, as shown in [12] the edge lengths of a mesh can be recovered from intrinsic shape differences through a series of least squares problems, hinting that increasing the depth of the network and thus the nonlinearity might not be necessary with shape differences.
On the other hand, the decoder is selected for its ability to transform the latent representation to coordinate functions for reconstruction and interpolation tasks.
Input Shape Differences
We construct the input shape differences using a truncated eigenbasis of dimension on the base shape, and the full basis on the target one, within all experiments, regardless of the number of vertices on the actually shapes. The functional maps from the base to the targets are induced by the identity maps, since our training shapes are in 11 correspondence. This implies that each of the shapes is represented by three matrices, representing the areabased, conformal and extrinsic shape differences respectively. The independence among the shape differences allows flexibility in selecting the combination of input shape differences. In Section 6, we compare the performance of several combinations. A detailed ablation study with respect to the input channels and network depth is presented in Appendix D.
It is worth noting that recent learningbased shape matching techniques enable efficient (functional) maps estimation. In particular, we adapt the framework of
[30] and evaluate OperatorNet trained with computed shape differences in Section 6.Datasets
We train OperatorNet on two types of datasets: humans and animals.
For human shapes, our training set consists of 9440 shapes sampled from the DFAUST dataset [6] and 8000 from the SURREAL dataset [35], which is generated with the model proposed in [19]. The DFAUST dataset contains 4D scan of 10 human characters subject to a various of motions. On the other hand, the SURREAL dataset injects more variability to the body types.
For animals, we used the parametric model proposed in SMAL [37] to generate 1800 animals of 3 different species, including lions, dogs, and horses. The meshes of the humans (resp. animals) are simplified to 1000 vertices (resp. 1769 vertices).
Loss Function
OperatorNet reconstructs coordinate functions of a given training shape. Our shape reconstruction loss operates in two steps. First, we estimate the optimal rigid transformation to align the ground truth coordinate functions and the reconstructed ones using the Kabsh algorithm [3] with ground truth correspondences. Second, we estimate the mean squared error between the aligned reconstruction and the ground truth.
(9) 
Here is the function that computes the optimal transformation between and
. We align the computed reconstructions to the ground truth embedding, so that the quality of the reconstructed point cloud is invariant to rigid transformations. This is important since the shape difference operators are invariant to rigid motion of the shape, and thus the network should not be penalized, for not recovering the correct orientation. On the other hand, this loss function is differentiable, since we use a closedform expression of
, given by the SVD, which enables backpropagation in neural network training.6 Evaluation
In this section, we provide both qualitative and quantitative evaluations of the results from OperatorNet, and compare them to the geometric baselines.
Evaluation Metrics
We denote by and the groundtruth and the reconstructed meshes respectively. First we denote by , where is the rotationallyinvariant distance defined in Eq. (9) and is the coordinate functions of . For a comprehensive, unbiased evaluation and comparison, we introduce the following two new metrics: (1) , i.e., the relative error of mesh volumes; (3) , where is the length of edge .
Baselines
Two baselines are considered: (1) the intrinsic reconstruction method from [7], in which we evaluate with the ‘ShapefromLaplacian’ option and use full basis in both the base shape and the target shape; (2) the reconstruction method from [12], where the authors construct offset surfaces of the shapes to take account of extrinsic geometry, we evaluate with the same basis truncation as our input. Moreover, the latter also gives a version of purely intrinsic reconstruction. Beyond that, we also consider the nearest neighbourhood retrieval from the training set with respect to distances between shape difference representations.
Test Data
We retain 800 shapes from DFAUST dataset as the test set, which contain 10 subcollections (character + action sequence, each consists of 80 shapes) that are isolated from the training/validation set. For the efficiency of baseline evaluation, we further sample 5 shapes via furthest point sampling regarding the pairwise Hausdorff distance from each of the subcollection, resulting in a set of 50 shapes, that covers significant variability in both styles and poses in the original test set.
Quantitative Results
We list all the scores regarding the metrics defined above in Table 1. First of all, OperatorNet using both intrinsic and extrinsic shape differences achieved the smallest reconstruction error (i.e., ), and the purely intrinsic version is next to the best. OperatorNet trained on shape differences from computed functional maps achieve competing performances showing that our method is efficient even in the absence of ground truth onetoone correspondences. Note also that all versions of OperatorNet significantly outperform the other baselines.
Regarding the volume and edge recovery accuracy, either the complete or the intrinsiconly version of OperatorNet achieves second to the best result. It is worth noting that, since the nearest neighbourhood search in general retrieves the right body type, the volume is well recovered. On the other hand, full Laplacian of the target shape is provided as input for the ShapefromLaplacian baseline, thus it is expected to preserve well the intrinsic information.
Op.Net (Int+Ext)  0.014  0.045  
Op.Net (Int)  2.41  0.013  0.046 
Op.Net (Ext)  1.25  0.017  0.046 
Op.Net (Comp)(Ext)  3.86  0.021  0.052 
Op.Net (Comp)(Int+Ext)  6.22  0.022  0.053 
SfL [7]  48.8  0.081  
FuncChar [12](Int)  65.1  0.356  0.118 
FuncChar [12] (Int+Ext)  28.4  0.028  0.110 
NN  25.5  0.043 
Qualitative Results
We demonstrate the reconstructed shapes from OperatorNet and the aforementioned baselines in Figure 6, the red shape in each row therein is the respective groundtruth target shape. The base shape in this experiment (also the base shape we compute shape difference on) is shown in Figure 4, which is in the rest pose. The geometric baselines in general perform worse when the poses changes significantly with respect to the base (see the top two rows in Figure 6), but gives relatively better result when the difference is mainly on the style (see the last row).
Our method, on the other hand, produces consistently good reconstruction in all the cases. It is also worth noting that, as expected, OperatorNet using all 3 types of shape differences gives both the best quantitative and qualitative results.
Finally, we present a qualitative verification of the generalization power of OperatorNet in Appendix B.
7 Applications
In this section, we present all of our results using OperatorNet trained with both intrinsic and extrinsic shape differences, which are induced by groundtruth maps.
7.1 Shape Interpolation
Given two shapes, we first interpolate the regarding shape differences using the formulation in Eq.(6) (an alternative is to interpolate shape differences linearly, we provide comparison to such in Appendix C), and then synthesize intermediate shapes by inferring the interpolated shape differences with OperatorNet.
We compare our method against several baselines. First, a PointNet autoencoder is trained with the encoder architecture from [27] and with our decoder. Two versions of PointNet are trained: one autoencoder with spatial transformers and one without. The autoencoder without spatial transformers performs better at reconstruction and interpolation, it is therefore selected for the comparisons. Another autoencoder based on PointNet++ [29] is similarly trained.
Nearest neighbour (NN) interpolation retrieves the nearest neighbour of the interpolated shape differences in the training set and uses the corresponding labels for interpolation.
Moreover, we also compare to the interpolation result from a recent work [5], where a GAN is trained to generate realistic human shapes. Note that, as stated in [5], the interpolation is done as follows: first one picks two randomly generated latent vectors , which, via the GAN gives arise to two shapes . Then, the interpolation between the two shapes are achieved as , where . In particular, we randomly generate 1000 shapes using their trained model and pick that are nearest to the red shapes in the last row of Figure 7 for interpolation comparison.
In Figure 7, the first row shows the interpolation between the two end shapes by [4], which is not developed evenly, for instance, the arms change abruptly during the three middle shapes, while there is little change on that region afterwards. As shown in the second and the third rows, both the results of the two autoencoders suffer obvious area distortions in the arms (see a detailed comparison in Figure 1). The fourth row of Figure 7 indicates the NN approach fails to deliver a continuous deformation sequence. In contrast, interpolation using OperatorNet is continuous and respects the structure and constraints of the body, suggesting that shape differences encode efficiently the structure.
7.2 Shape Analogy
Our second application is to construct semantically meaningful new shapes based on shape analogies. Given shapes , our goal is to construct a new shape , such that is to as is to .
Following the discussion in Section 3, the functoriality of shape differences allows an explicit and mathematically meaningful way of constructing the shape difference of , given that of and . Namely, Then, with our OperatorNet, we reconstruct the embedding of the unknown by feeding to the network.
We compare our results to that of the PointNet autoencoder. For the latter, we reconstruct by decoding the neuralnetwork’s latent code with respect to the additive formula, i.e., , where is the latent code of shape (and similarly for ).
In Figure 10, we show shape analogies of pose transfer (top row) and style transfer (bottom row) via OperatorNet and PointNet autoencoder on human shapes. It is evident that our results are both more natural and intuitive.
We also show analogies among animals in Figure 11, where we present both pose transfer (top row) and style transfer (bottom row) and comparison to the results of PointNet.
Moreover, we present a set of more challenging shape analogies that transfer gender across human shapes in Figure 9. Namely, we fix , and test analogies with different (the 6 male shapes in red). Note that and are two characters in comparable poses and styles but of different genders, ideally the right analogies, , should be a ‘female’ version of with similar poses and styles.
We present the analogies from OperatorNet in the first results column, and those from PointNet in the second results column (both in blue). It is obvious that PointNet, though works in some cases, in general produces less semantically meaningful analogies than ours (see discrepancies in the red dotted boxes).
8 Conclusion & Future Work
In this paper we have introduced a novel learningbased technique for recovering shapes from their difference operators. Our key observation is that shape differences, stored as compact matrices lead themselves naturally to learning, and allow to both recover the underlying shape space in a collection, and encode the geometry of individual shapes. We also introduced a novel extrinsic shape difference operator and showed its utility for shape reconstruction and other applications such as shape interpolation and analogies.
Currently our approach is only wellsuited to shapes represented as triangle meshes. Future work is required in extending our framework to both learning (optimal) inner products from data, and adapting it to other representations, such as point clouds or triangle soups.
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Appendix A Proof of Theorem 1
Proof.
Since is known to be of rank 3, and is symmetric, we have, by SVD:
where,
are respectively the top 3 singular vectors and singular values of
. Therefore, we have , where is a rigid transformation matrix satisfying . In other words, we recover from , where is equivalent to up to rigid transformations. And to recover the projection of in , we simply compute .To conclude, our recovery is done in two steps: first we use SVD to recover as , and in the second step, we compute for the projection of in the space spanned by , and is equivalent to up to rigid transformations. ∎
Appendix B Verification of the Generalization power of OperatorNet
To demonstrate the generalization power of OperatorNet, we show in Figure 12 our reconstructions of test shapes from the SURREAL dataset. For comparison, we also retrieve the shapes in the training set, whose shape differences are the nearest to the ones of the test shapes. In each of the figures, the top row presents the groundtruth test shapes; the middle row shows reconstructions from OperatorNet; the bottom row demonstrates the shapes retrieved from the training set via nearest neighbourhood search regarding shape differences.
It is evident that OperatorNet accurately reconstruct the test shapes, which deviate from the shapes in the training set significantly, suggesting that our network generalizes well in unseen data.
Appendix C Comparison of Interpolation Schemes for Shape Differences
In the following experiment we note that, since the shape differences are represented by matrices, it is also possible to interpolate shape differences linearly, i.e., . However, as we argue in Section 3, the multiplicative property of shape differences suggests that it is more natural to interpolate the difference operators following Eq. (6). To illustrate this point, we show in Figure 14 interpolated sequences with respect to the two schemes above – the multiplicative one in the top row and the linear one in the bottom row. It is visually evident that the former leads to more continuous and evenly deformed sequence. Moreover, we compute the distance between consecutive shapes in both sequences and plot the distributions in the bottom panel of Figure 13 as a quantitative verification.
Appendix D Ablation Study on Network Design
We investigate multiple architectures for OperatorNet. In Table 2 we compare the reconstruction performance over different combinations of input shape differences, and different depths of encoders.
We report the performance of 4 different convolutional encoders from 1 to 4 layers deep by doubling the number of neurons every layer.
Two trends are observed in Table 2: first, we always achieve the best performance when all three types of shape differences are used, for all depth of the network; second, fixing the combination of input shape differences, the network performs better as its depth gets shallower.
Putting these two observations together, we justify our final model, which has one single layer convolutional encoder and uses all three types of shape differences as input.
Encoder architecture  Area  Ext  Conf  A+E  A+C  E+C  A+E+C 

Conv. 8  8.61  4.29  3.78  3.82  3.41  2.56  2.46 
Conv. 816  9.08  4.54  4.28  4.65  3.93  3.10  3.05 
Conv. 81632  9.90  5.54  4.91  5.59  4.88  3.71  3.55 
Conv. 8163264  11.16  6.39  5.93  6.89  5.42  4.35  4.24 
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