Q UATERNION
E QUIVARIANT
C APSULE
N ETWORKS
FOR 3D P OINT C LOUDS
Anonymous authors
Paper under double-blind review
A BSTRACT
We present a 3D capsule architecture for processing of point clouds that is equivariant with respect to the SO (3) rotation group, translation and permutation of the unordered input sets.
The network operates on a sparse set of local reference frames, computed from an input point cloud and establishes end-to-end equivariance through a novel 3D quaternion group capsule layer, including an equivariant dynamic routing procedure.
The capsule layer enables us to disentangle geometry from pose, paving the way for more informative descriptions and a structured latent space.
In the process, we theoretically connect the process of dynamic routing between capsules to the well-known Weiszfeld algorithm, a scheme for solving iterative re-weighted least squares (IRLS) problems with provable convergence properties, enabling robust pose estimation between capsule layers.
Due to the sparse equivariant quaternion capsules, our architecture allows joint object classiﬁcation and orientation estimation, which we validate empirically on common benchmark datasets.
I NTRODUCTION
It is now well understood that in order to learn a compact representation of the input data, one needs to respect the symmetries in the problem domain (Cohen et al., 2019; Weiler et al., 2018a).
Arguably, one of the primary reasons of the success of 2D convolutional neural networks (CNN) is the translation-invariance of the 2D convolution acting on the image grid (Giles & Maxwell, 1987;
Kondor et al., 2018).
Recent trends aim to transfer this success into the 3D domain in order to support many applications such as shape retrieval, shape manipulation, pose estimation, 3D object modeling and detection.
There, the data is naturally represented as sets of 3D points or a point cloud (Qi et al., 2017a;b).
Unfortunately, extension of CNN architectures to 3D point clouds is non-trivial due to two reasons: 1) point clouds are irregular and unstructured, 2) the group of transformations that we are interested in is more complex as 3D data is often observed under arbitrary non-commutative
SO (3) rotations.
As a result, achieving appropriate embeddings requires 3D networks that work on points to be equivariant to these transformations, while also being invariant to the permutations of the point set.
In order to ﬁll this important gap, we propose the quaternion equivariant point capsule network or
QE-Network that is suited to process point clouds and is equivariant to SO (3) rotations compactly parameterized by quaternions (Fig. 2), in addition to preserved translation and permutation equivariance.
Inspired by the local group equivariance (Lenssen et al., 2018; Cohen et al., 2019), we efﬁciently cover SO (3) by restricting ourselves to the sparse set of local reference frames (LRF) that collectively characterize the object orientation.
The proposed capsule layers (Hinton et al., 2011) deduces equivariant latent representations by robustly combining those local LRFs using the proposed Weiszfeld dynamic routing .
Hence, our latent features specify to local orientations disentangling the pose from object existence.
Such explicit storage is unique to our work and allows us to perform rotation estimation jointly with object classiﬁcation.
Our ﬁnal architecture is a hierarchy of
QE-networks, where we use classiﬁcation error as the only training cue and adapt a Siamese version when the relative rotation is to be regressed.
We neither explicitly supervise the network with pose annotations nor train by augmenting rotations.
Overall, our contributions are:
We propose a novel, fully SO (3) -equivariant capsule architecture that is tailored for simultaneous classiﬁcation and pose estimation of 3D point clouds.
This network produces in-
[Figure]
variant latent representations while explicitly decoupling the orientation into capsules, thus attaining equivariance.
Note that equivariance results have not been previously achieved regarding the quaternion parameterization of the 3D special orthogonal group.
By utilizing LRFs on points, we reduce the space of orientations that we consider and hence can work sparsely on a subset of the group elements.
We theoretically prove the equivariance properties of our 3D network regarding the quaternion group.
Moreover, to the best of our knowledge, we for the ﬁrst time establish a connection between the dynamic routing of Sabour et al.
and Generalized Weiszfeld iterations (Aftab et al., 2015).
By that, we theoretically argue for the convergence of the employed dynamic routing.
We experimentally demonstrate the capabilities of our network on classiﬁcation and orientation estimation of 3D shapes.
P RELIMINARIES AND T ECHNICAL B ACKGROUND 2. E QUIVARIANCE
Deﬁnition 1 (Equivariant Map) .
For a G -space acting on X , the map Φ : G × X (cid:55)→ X is said to be equivariant if its domain and co-domain are acted on by the same symmetry group (Cohen &
Welling, 2016; Cohen et al., 2018a):
[Equation]
where g 1 ∈ G and g 2 ∈ G .
Equivalently Φ( T ( g 1 ) x ) =
T ( g 2 ) Φ( x ) , where T ( · ) is a linear representation of the group G .
Note that T ( · ) does not have to commute.
It sufﬁces for T ( · ) to be a homomorphism: T ( g 1 ◦ g 2 )
=
T ( g 1 ) ◦
T ( g 2 ) .
In this paper we use a stricter form of equivariance and consider g 2 = g 1 .
Deﬁnition 2 (Equivariant Network) .
An architecture or network is said to be equivariant if all of its layers are equivariant maps.
Due to the transitivity of the equivariance, stacking up equivariant layers will result in globally equivariant networks e.g., rotating the input will produce output vectors which are transformed by the same rotation (Lenssen et al., 2018; Kondor & Trivedi, 2018).
2. T HE Q UATERNION G ROUP H
Deﬁnition 3 (Quaternion) .
A quaternion q is an element of Hamilton algebra H 1 , extending the complex numbers with three imaginary units i , j , k in the form:
[Equation]
(2) with ( q 1 , q 2 , q 3 , q 4 ) T ∈ R 4 and i 2 =
j 2 = k 2 = ijk = − 1 .
q 1 ∈ R denotes the scalar part and v = ( q 2 , q 3 , q 4 ) T ∈ R 3 , the vector part.
The conjugate ¯ q of the quaternion q is given by ¯q :
= q 1 − q 2 i − q 3 j − q 4 k .
A unit quaternion q ∈ H 1 with 1 !
= (cid:107) q (cid:107) :
= q · ¯q and q − 1 = ¯q , gives a compact and numerically stable parametrization to represent orientation of objects on the unit sphere S 3 , avoiding gimbal lock and singularities (Busam et al., 2017).
Identifying antipodal points 1 input : Input points { x 1 , ..., x K } ∈ R K × 3 , local reference frames Q = { q 1 , . . . , q L } ∈ H 1 L , with L multiple of K , activations α =
( α 1 , . .
. , α L ) T , trainable transformations
T = { t i,j } i,j ∈ H 1 L × M 2 output: Updated frames ˆ Q
=
ˆ q 1 , . .
, ˆ q M } ∈ H 1 M , updated activations ˆ α =
(ˆ α 1 , . .
, ˆ α M ) T 3 for All primary (input) capsules i do for All latent (output) capsules j do v
i,j ← q i ◦ t i,j // compute votes 6 for All
) capsules j do ˆ q j ← A j . . .
v j } output capsules for do for i do w i,j ← i ·
(cid:0)
// compute the current weight ˆ q j ← A (cid:0) { v 1 ,j . .
v L,j } , w : ,j
/ see
(5) ˆ α j ← sigmoid (cid:0) − 1 K L (cid:80) 1 δ (ˆ q j , v i,j ) (cid:1) //
recompute activations q and − q with the same element, the unit quaternions form a double covering group of SO (3) .
His closed under the non-commutative multiplication or the Hamilton product:
[Equation]
Deﬁnition 4 (Linear Representation of H 1 ) .
We follow Birdal et al.
and use the T ( q ) (cid:44)   q− q− q− qqq− qq 3 qqq− qq− qqq  .
To be concise we will use capital letters to refer to the matrix representation of quaternions e.g. Q ≡
T ( q ) , G ≡ T ( g ) .
Note that T ( · ) , the injective homomorphism to the orthonormal matrix ring, by construction satisﬁes the condition in Dfn.
(Steenrod, 1951): det( Q ) = 1 , Q (cid:62) =
Q − 1 , (cid:107) Q (cid:107) =
(cid:107) Q i, : (cid:107) =
(cid:107) Q : ,i (cid:107) = 1 and Q − q
I is skew symmetric: Q + Q (cid:62)
= 2 q 1 I .
It is easy to verify these properties.
T linearizes the Hamilton product or the group composition: g ◦ q (cid:44) T ( g ) q (cid:44) Gq .
2. 3D P OINT C LOUDS
Deﬁnition 5 (Point Cloud) .
We deﬁne a 3D surface to be a differentiable 2-manifold embedded in the ambient 3D Euclidean space: M 2 ∈ R 3 and a point cloud to be a discrete subset sampled on
M 2 : X ∈ { x i ∈
M 2 ∩ R 3 } .
Deﬁnition 6 (Local Geometry) .
For a smooth point cloud { x i } ∈ M 2 ⊂ R N × 3 , a local reference frame (LRF) is deﬁned as an ordered basis of the tangent space at x , T x M , consisting of orthonormal vectors: L ( x ) =
[ ∂ 1 , ∂ 2 , ∂ 3 ≡ ∂ 1 × ∂ 2 ] .
Usually the ﬁrst component is deﬁned to be the surface normal ∂ 1 (cid:44) n ∈ S 2 : (cid:107) n (cid:107) = 1 and the second one is picked according to a modality dependent heuristic.
Note that recent trends such as (Cohen et al., 2019) acknowledge the ambiguity and either employ a gauge (tangent frame) equivariant design or propagate the determination of a certain direction until the last layer (Poulenard & Ovsjanikov, 2018).
Here, we will assume that ∂ 2 can be uniquely and repeatably computed, a reasonable assumption for the point sets we consider (Petrelli & Di Stefano, 2011).
For the cases where this does not hold, we will rely on the network’s robustness.
We will explain our method of choice in Sec. 4 and visualize LRFs of an airplane object in Fig.
ϵ − ball 𝐱 𝑖 𝐧 𝑖 {𝐱 𝒊 } ⊂
𝐗 ⊂ ℳ 𝐗 𝒊 ⊂ ℝ Kx {𝐪 𝒊 } 𝒊 ⊂ ℝ KxN lC x 𝐦𝐞𝐚𝐧(⋅) {𝛍 𝒊 } {𝛍 𝒊−𝟏 }ο 𝐗 𝒊 point-to-transform map
Dynamic Routing 𝐭 .
: 𝐓 𝒊 ⊂ ℝ KxN l C xM l x4 ෣{𝐪 𝒊 } 𝒊 ⊂ ℝ M l x4 ෝ𝛂 𝒊 ⊂ ℝ M l
[Figure]
SO (3) -E QUIVARIANT 3D C APSULE N ETWORKS
Disentangling orientation from representations requires guaranteed equivariances and invariances.
Yet, the original capsule networks of Sabour et al.
(2017) cannot achieve equivariance to general groups.
To this end, Lenssen et al.
(2018) proposed to use a manifold-mean and a special aggregation that makes sure that the trainable transformations get pose-aligned points as input.
We will extend this idea to the non-abelian SO (3) and design capsule networks sparsely operating on a set of LRFs computed on local neighborhoods of points, parameterized by quaternions.
In the following, we ﬁrst explain our novel capusle layers, the main building block of our architecture.
We then show how to stack those layers via a simple aggregation resulting in an SO (3) -equivariant 3D capsule network that yields invariant representations (or activations) as well as equivariant rotations (latent capsules).
3. Q UATERNION E QUIVARIANT C APSULE L AYERS
To construct equivariant layers on the group of rotations, we are required to deﬁne a left-equivariant averaging operator A that is invariant under permutations of the group elements, as well as a distance metric δ that remains unchanged under the action of the group.
For these, we make the following choices:
Deﬁnition 7 (Geodesic Distance) .
The Riemannian (geodesic) distance in the manifold of rotations lead to the following geodesic distance δ ( · ) ≡ d quat ( · ) :
[Equation]
Deﬁnition 8 (Quaternion Mean µ ( · ) ) .
For a set of Q rotations S = { q i } and associated weights w = { w i } ,
the weighted mean operator A ( S , w ) : H 1 n × R n (cid:55)→ H 1 n is deﬁned through the following maximization procedure (Markley et al., 2007):
[Equation]
where M ∈ R 4 × 4 is deﬁned as: M (cid:44) Q (cid:80) i
=1 w i q i q (cid:62) i .
The average quaternion is the eigenvector of
M corresponding to the maximum eigenvalue.
This operation lends itself to both analytic (Magnus, 1985) and automatic differentiation (Laue et al., 2018).
Theorem 1.
Quaternions, the employed mean A ( S , w ) and geodesic distance δ ( · ) enjoy the following properties:
A ( g ◦ S , w ) is left- : A ( g ◦ S , w ) =
g ◦ A ( S , w ) .
QE -Network 𝐗 ⊂ ℳ
QE -Network
QE -Network
QE -Network
QE -Network QE -Network
Downsample
Downsample Intermediate Capsules
Rotation Estimation
Classification 𝐂 𝐗 𝛂 𝐗
Operator A is invariant under permutations:
[Equation]
The transformations g ∈ H 1 preserve the geodesic distance δ ( · ) given in Dfn. 7.
Proof.
The proofs are given in the supplementary material.
We are now ready to construct the group dynamic routing (DR) by agreement that is equivariant thanks to Thm. 1.
The core idea is to route from or assign the primary capsules that constitute the input LRF set, to the latent capsules by an iterative clustering which respects the group structure.
At each step, we assign the weighted group mean to each output capsule.
The weights w ← σ ( x , y ) are inversely propotional to the distance between the vote quaternion and the new quaternion (cluster center).
See Alg.
1 for details.
In the following, we analyze our variant of routing as an interesting case of the afﬁne, Riemannian Weiszfeld algorithm (Aftab et al., 2015; 2014).
Lemma 1.
For σ ( x , y ) =
δ ( x , y ) q − 2 the equivariant routing procedure given in Alg.
1 is a variant of the afﬁne subspace Wieszfeld algorithm
(Aftab et al., 2015; 2014) that is a robust algorithm for computing the L q geometric median.
Proof Sketch.
The proof follows from the deﬁnition of Weiszfeld iteration (Aftab et al., 2014) and the mean and distance operators deﬁned in Sec.
We ﬁrst show that computing the weighted mean is equivalent to solving the normal equations in the iteratively reweighted least squares (IRLS) scheme (Burrus, 2012).
Then, the inner-most loop correspond to the IRLS or Weiszfeld iterations.
We provide the detailed proof in supplementary material.
Note that, in practice one is quite free to choose the weighting function σ ( · ) as long as it is inversely proportional to the geodesic distance and concave (Aftab & Hartley, 2015).
We leave the analyses of the variants of these algorithms as a future work.
The original dynamic routing can also be formulated as a clustering procedure with a KL divergence regularization.
This holistic view paves the way to better routing algorithms (Wang & Liu, 2018).
Our perspective is akin yet more geometric due to the group structure of the parameter space.
Thanks to the connection to Weiszfeld algorithm, the convergence behavior of our dynamic routing can be directly analyzed within the theoretical framework presented by Aftab et al.
Theorem 2.
Under mild assumptions provided in the appendix, the sequence of the DR-iterates generated by the inner-most loop almost surely converges to a critical point.
Proof Sketch.
Proof, given in the appendix, is a direct consequence of Lemma 1 and directly exploits the connection to the Weiszfeld algorithm.
3. E QUIVARIANT 3D P OINT C APSULE N ETWORK A RCHITECTURE
The essential ingredient of our architecture, QE-Network, is shown in Fig.
We also provide a corresponding pseudocode in our suppl.
material. Given a local patch of points with coordinates x i ⊂ R K × 3 and rotations parametrized as quaternions q i ⊂ H 1 L , we compute the quaternion average as the initial pose candidate.
This candidate is used to bring the receptive ﬁeld in a canonical [Table] orientation.
Since the points in the local receptive ﬁeld lie in continuous R 3 , training a discrete set of pose transformations t i,j based on local coordinates is not possible.
Instead, we employ a pointto-transform network t ( · ) : R 3 → R M × N c × 4 , where M is the number of output capsules and
N c =
LK is the number of input capsules per input point.
The network is shared over all input points to compute the transformations t i,j = ( t ( p 1 ) , ..., t ( p K )) i,j , which are used to calculate the votes for dynamic routing as v i,j = q i ◦ t i,j . The network t () consists of fully-connected layers, which regress the transformations, similar to common operators for continuous convolutions (Sch¨utt et al., 2017; Wang et al., 2018; Fey et al., 2018). It is the continuous alternative to directly optimizing transformations lying in a grid kernel, as it is done in the original dynamic routing by Sabour et al.
(2017) for the 2D image case.
Note that t () predicts quaternions by unit-normalizing the regressed output.
Although Riemannian layers of B ´ ecigneul & Ganea (2018) or spherical predictions of Liao et al.
(2019) can improve the performance, the simple strategy works reasonably for our case.
After computing the votes, we iteratively reﬁne the output pose (weighted average of votes) by routing by agreement as shown in Alg. 1.
In order to gradually increase the receptive ﬁeld, we stack QE-networks creating a deep hierarchy, pooling the points and the LRFs before each layer.
Note that we are allowed to do so thanks to the properties of equivariance.
In particular, we input N = 64 patches to our architecture that is composed of two QE-networks.
We call the centers of these patches pooling centers .
In the ﬁrst layer, each of those centers are linked to their immediate vicinity leading to K = 9 -star local connectivity from which we compute the 64 × 64 × 4 intermediary capsules.
The input LRFs of the ﬁrst layer is sampled from pre-calculated LRF-set and the input activation is set to 1.
The LRFs in the second layer l = 2 are the output capsules of the ﬁrst layer, l = 1 and are routed to the output capsules that are as many as the number of classes C , M 2 = C . The activation of the second layer is updated by the output of the ﬁrst layer as well.
This construction is shown in Fig.
Speciﬁcally, for l = 1 , we use K = 9 , N lc = 1 , M l = 64 and for l = 2 , K = 64 , N lc = 64 , M l = C = 40 .
This way, in this last layer all the pooling centers act as a single patch ( K = 64 ).
A single QE-network acts on this patch to create the ﬁnal C × 4 capsules and C activations. More details is reported in
Alg. 3 of appendix.
E XPERIMENTAL E VALUATIONS
Implementation Details We implement our network in PyTorch and use the ADAM optimizer (Kingma & Ba, 2014) with a learning rate of 0 .
The point-transformation mapping network is implemented by two FC-layers composed of 64 hidden units.
We set the initial activation of the input LRF to 1 . 0 .
In each layer, we use 3 iterations of DR.
For classiﬁcation we use the spread loss (Sabour et al., 2018) and the rotation loss is identical to δ ( · ) .
Surface normals are computed by local plane ﬁts (Hoppe et al., 1992).
We compute the second axis of the LRF, ∂ 2 , by FLARE (Petrelli & Di Stefano, 2012), that uses the normalized projection of the point within the periphery of the support showing the largest distance, onto the tangent plane of the center: ∂ 2 = p max − p (cid:107) p max − p (cid:107) .
Note that using other LRFs such as SHOT (Tombari et al., 2010) or the more modern GFrames of Melzi et al.
(2019) is possible.
We found FLARE to be sufﬁcient for our experiments.
Prior to all operations, we ﬂip all the LRF quaternions such that they lie on the northern hemisphere : { q i ∈ S 3 : q wi > 0 } .
3D Shape Classiﬁcation.
We use ModelNet40 dataset of (Wu et al., 2015; Qi et al., 2017b) to assess our classiﬁcation performance.
Each shape is composed by 10 K points.
We assign the LRFs [Table] to a subset of the uniformly sampled points, N =
(Birdal & Ilic, 2017).
We train the networks without any rotation augmentation (NR) and put them to test under arbitrary SO (3) rotations (AR).
Our results are shown in Tab.
1 along with that of PointNet (PN)
(Qi et al., 2017a), PointNet++ (PN++) (Qi et al., 2017a), KD-treeNet
(Li et al., 2018a), Point2Seq (Liu et al., 2019b), Spherical
CNNs (Esteves et al., 2018), PRIN (You et al., 2018) and PPF-FoldNet (PPF) (Deng et al., 2018a).
We also present a version of our algorithm ( Var ) that avoids the canonicalization within the QEnetwork.
This is a non-equivariant network that we still train without data augmentation.
While this version gets comparable results to the state of the art for the NR/NR case, it cannot handle random
SO (3) variations (AR).
Note that PPF uses the point-pair-feature (Birdal & Ilic, 2015) encoding and hence creates invariant input representations.
For the scenario of NR/AR, our equivariant version outperforms all the other methods, including equivariant spherical CNNs (Esteves et al., 2018) by a signiﬁcant gap of at least 5% even when (Esteves et al., 2018) uses the mesh.
The object symmetries in this dataset are responsible for a signiﬁcant portion of the errors we make.
It is worth mentioning that we also trained TFNs (Thomas et al., 2018) for that task, but their memory demand made it infeasible to scale to this application.
Number of Parameters.
Use of LRFs helps us to restrict the rotation group to certain elements and thus we can use networks with signiﬁcantly less parameters (as low as 0 .
44 M ) compared to others as shown in Tab. 1.
Number of parameters in our network depends upon the number of classes, e.g. for ModelNet10 we have 0 .
047 M parameters.
Rotation estimation in 3D point clouds.
Our network can estimate both the canonical and relative object rotations without pose-supervision.
To evaluate this desired property, we used the well classiﬁed shapes on ModelNet10 dataset, a sub-dataset of Modelnet40 (Wu et al., 2015).
We generate multiple instances per shape by transforming the instance with ﬁve arbitrary SO (3) rotations.
As we are also affected by the sampling of the point cloud, we resample the mesh ﬁve times and generate different pooling graphs across all the instances of the same shape.
Our QE-architecture can estimate the pose in two ways: 1) by directly using the output capsule with the highest activation, 2) by a siamese architecture that computes the relative quaternion between the capsules that are maximally activated as shown in Fig.
Both modes of operation are free of the data augmentation and we give further schematics of the latter in our appendix Fig.
Our results against the baselines including a naive averaging of the LRFs (Mean LRF) and principal axis alignment (PCA) are reported in Tab. 2 as the relative angular error (RAE).
We further include results of PointNetLK Aoki et al.
and IT-Net (Yuan et al., 2018), two state of the art 3D networks that iteratively aligns two point sets.
These methods are in nature similar to iterative closest point (ICP) algorithm (Besl &
McKay, 1992) but 1) do not require an initialization (ﬁrst iteration estimates the pose), 2) learn data driven updates.
Methods that use mesh inputs such as Spherical CNNs (Esteves et al., 2018) cannot be included here as the random sampling of the same surface would not affect those.
We also avoid methods that are just invariant to rotations (and hence cannot estimate the pose) such as Tensorﬁeld Networks (Thomas et al., 2018).
Finally, note that , IT-net (Yuan et al., 2018) and PointLK need to train a lot of epoches (e.g. 500) with random SO (3) rotation augmentation in order to get the models that cover the full SO (3) , whereas we train only for ∼ 100 epochs.
We include more details about the baselines in the appendix under Fig. 8.
RAE between the ground truth and the prediction is computed as the relative angle in degrees: d ( q 1 , q 2 ) /π .
Note that resampling and random rotations render the job of all methods difﬁcult. [Figure]
However, both our version that tries to ﬁnd a canonical alignment and the siamese variant which seeks a relative rotation are better than the baselines.
As unsupervised pose estimation of symmetric objects is a challenging task we also report results on the non-symmetric subset (No Sym).
[Table]
Robustness against point and LRF resampling.
Density changes in the local neighborhoods of the shape are an important cause of error for our network.
Hence, we ablate by applying random resamplings (patch-wise dropout) to the objects in ModelNet10 dataset and repeating the pose estimation and classiﬁcation as described above.
The ﬁrst part (LRF-10K) of Tab. 3 shows our ﬁndings against gradual increases of the number of patches.
Here, we sample 2K LRFs from the 10K LRFs computed on an input point 10K. 100% dropout corresponds to 2K points in all columns.
On second ablation, we reduce the amount of points on which we compute the LRFs, to 2K and 1K respectively.
As we can see from the table, our network is robust towards the changes in the LRFs as well as the density of the points.
R ELATED W ORK
Deep learning on point sets. The capability to process raw, unordered point clouds within a neural network is introduced by the prosperous PointNet (Qi et al., 2017a) thanks to the point-wise convolutions and the permutation invariant pooling functions. Many works have extended PointNet primarily to increase the local receptive ﬁeld size (Qi et al., 2017b; Li et al., 2018b; Shen et al., 2018;
Wang et al., 2019). Point-clouds are generally thought of as sets. This makes any permutationinvariant network that can operate on sets an amenable choice for processing points (Zaheer et al., 2017; Rezatoﬁghi et al., 2017).
Unfortunately, common neural network operators in this category are solely equivariant to permutations and translations but to no other groups. Equivariance in Neural Networks. The early attempts to achieve invariant data representations usually involved data augmentation techniques to accomplish tolerance to input transformations (Maturana & Scherer, 2015; Qi et al., 2016; 2017a). Motivated by the difﬁculty associated with augmentation efforts and acknowledging the importance of theoretically equivariant or invariant representations, the recent years have witnessed a leap in theory and practice of equivariant neural networks (Bao & Song, 2019; Kondor & Trivedi, 2018).
While laying out the fundamentals of the group convolution, G-CNNs (Cohen & Welling, 2016) guaranteed equivariance with respect to ﬁnite symmetry groups.
Similarly, Steerable CNNs (Cohen & Welling, 2017) and its extension to 3D voxels (Worrall & Brostow, 2018) considered discrete symmetries only.
Other works opted for designing ﬁlters as a linear combination of harmonic basis functions, leading to frequency domain ﬁlters (Worrall et al., 2017; Weiler et al., 2018b).
Apart from suffering from the dense coverage of the group using group convolution, ﬁlters living in the frequency space are less interpretable and less expressive than their spatial counterparts, as the basis does not span the full space of spatial ﬁlters.
Achieving equivariance in 3D is possible by simply generalizing the ideas of the 2D domain to 3D by voxelizing 3D data.
However, methods using dense grids (Chakraborty et al., 2018; Cohen & Welling, 2017) suffer from increased storage costs, eventually rendering the implementations infeasible.
An extensive line of work generalizes the harmonic basis ﬁlters to SO (3) by using e .
g ., a spherical harmonic basis instead of circular harmonics (Cohen et al., 2018b; Esteves et al., 2018;
Cruz-Mota et al., 2012).
In addition to the same downsides as their 2D, these approaches have in common that they require their input to be projected to the unit sphere (Jiang et al., 2019), which poses additional problems for unstructured point clouds.
A related line of research are methods which deﬁne a regular structure on the sphere to propose equivariant convolution operators (Liu et al., 2019a; Boomsma & Frellsen, 2017)
To learn a rotation equivariant representation of a 3D shape, one can either act on the input data or on the network.
In the former case, one either presents augmented data to the network (Qi et al., 2017a;
Maturana & Scherer, 2015) or ensures rotation-invariance in the input (Deng et al., 2018a;b; Khoury et al., 2017).
In the latter case one can enforce equivariance in the bottleneck so as to achieve an invariant latent representation of the input (Mehr et al., 2018; Thomas et al., 2018; Spezialetti et al., 2019).
Further, equivariant networks for discrete sets of views (Esteves et al., 2019b) and crossdomain views (Esteves et al., 2019a) have been proposed.
Here, we aim for a different way of embedding equivariance in the network by means of an explicit latent rotation parametrization in addition to the invariant feature.
Marcos et al.
(2017) developed Vector Field Networks , which was followed by the 3D Tensor Field
Networks (TFN) (Thomas et al., 2018) that are closest to our work.
Based upon a geometric algebra framework, the authors did achieve localized ﬁlters that are equivariant to rotations, translations and permutations.
Moreover, they are able to cover the continuous groups.
However, TFN are designed for physics applications, is memory consuming and a typical implementation is neither likely to handle the datasets we consider nor can provide orientations in an explicit manner. Capsule Networks. The idea of capsule networks was ﬁrst mentioned by Hinton et al. (2011), before Sabour et al.
(2017) proposed the dynamic routing by agreement , which started the recent line of work investigating the topic.
Since then, routing by agreement has been connected to several well-known concepts, e.g. the EM algorithm Sabour et al.
(2018), clustering with KL divergence regularization Wang & Liu (2018) and equivariance (Lenssen et al., 2018).
They have been extended to autoencoders (Kosiorek et al., 2019) and GANs Jaiswal et al.
Further, capsule networks have been applied for speciﬁc kinds of input data, e.g. graphs (Xinyi & Chen, 2019), 3D point clouds (Zhao et al., 2019) or medical images (Afshar et al., 2018).
C ONCLUSION AND D ISCUSSION
In this work, we have presented a new framework for achieving permutation invariant and SO (3) equivariant representations on 3D point clouds.
Proposing a variant of the capsule networks, we operate on a sparse set of rotations speciﬁed by the input LRFs thereby circumventing the effort to cover the entire SO (3) .
Our network natively consumes a compact representation of the group of 3D rotations - quaternions, and we have theoretically shown its equivariance.
We have also established convergence results for our Weiszfeld dynamic routing by making connections to the literature of robust optimization.
Our network is among the few for having an explicit group-valued latent space and thus naturally estimates the orientation of the input shape, even without a supervision signal.
Limitations.
In the current form our performance is severely affected by the shape symmetries.
The length of the activation vector depends on the number of classes and for achieving sufﬁciently descriptive latent vectors we need to have a signiﬁcant number of classes.
On the other side, this allows us to perform with merit on problems where the number of classes are large.
Although, we have reported robustness to those, the computation of LRFs are still sensitive to the point density changes and resampling.
LRFs themselves are also ambiguous and sometimes non-unique.
Future work.
Inspired by Cohen et al.
and Poulenard & Ovsjanikov (2018)
our feature work will involve establishing invariance to the direction in the tangent plane.
We also plan to apply our network in the broader context of 3D object detection under arbitrary rotations and look for equivariances among point resampling.
A P ROOF OF P ROPOSITION
Before presenting the proof we recall the three individual statements contained in Prop. 1:
A ( g ◦ S , w ) is left-equivariant: A ( g ◦ S , w ) =
g ◦ A ( S , w ) .
Operator A is invariant under permutations:
[Equation]
A ( { q 1 , . . . , q Q } , w )
[Equation]
The transformations g ∈ H 1 preserve the geodesic distance δ ( · ) .
Proof.
We will prove the propositions in order.
We
start
by
transforming
each element and replace
q i
by ( g ◦ q i ) of the cost in Eq (5):
[Equation]
where M i =
w i q i q (cid:62) i and p =
G (cid:62) q .
From orthogonallity of G it follows p =
G − 1 q = ⇒
g ◦ p = q and hence g ◦ A ( S , w ) =
A ( g ◦ S , w ) .
The proof follows trivially from the permutation invariance of the symmetric summation operator over the outer products in Eq (9).
It is sufﬁcient to show that | q (cid:62) 1 q
[Equation]
|
[Equation]
g [Equation] (cid:62)
[Equation]
g
[Equation]
q 2 ) | for any g ∈ H 1 :
[Equation]
where g ◦ q ≡ Gq .
The result is a direct consequence of the orthonormality of G .
B P ROOF OF L EMMA
We will begin by recalling some preliminary deﬁnitions and results that aid us to construct the connection between the dynamic routing and the Weiszfeld algorithm.
Deﬁnition 9 (Afﬁne Subspace) .
A d -dimensional afﬁne subspace of R N is obtained by a translation of a d -dimensional linear subspace V ⊂ R N such that the origin is included in S :
[Equation]
Simplest choices for S involve points, lines and planes of the Euclidean space.
Deﬁnition 10 (Orthogonal Projection onto an Afﬁne Subspace) .
An orthogonal projection of a point x ∈ R N onto an afﬁne subspace explained by the pair ( A , c ) is deﬁned as:
[Equation]
c denotes the translation to make origin inclusive and A is a projection matrix typically deﬁned via the orthonormal bases of the subspace.
Deﬁnition 11 (Distance to Afﬁne Subspaces) .
Distance from a given point x to a set of afﬁne subspaces { S 1 , S 2 . . .
S k } can be written as Aftab et al.
[Equation]
Lemma 2.
Given that all the antipodal counterparts are mapped to the northern hemisphere, we will now think of the unit quaternion or versor as the unit normal of a four dimensional hyperplane h , passing through the origin:
[Equation]
(19) q d is an added term to compensate for the shift.
When q d =
the origin is incident to the hyperplane.
With this perspective, quaternion q i forms an afﬁne subspace with d = 4 , for which the projection operator takes the form:
[Equation]
Proof.
We consider Eq (20) for the case where c
= 0 and A
= ( I − qq (cid:62) ) .
The former follows from the fact that our subspaces by construction pass through the origin.
Thus, we only need to show that the matrix A
= I − qq (cid:62) is an orthogonal projection matrix onto the afﬁne subspace spanned by q .
To this end, it is sufﬁcient to validate that A is symmetric and idempotent:
A (cid:62) A
= AA
=
A 2 = A .
Note that by construction q (cid:62) q is a symmetric matrix and hence A itself.
Using this property and the unit-ness of the quaternion, we arrive at the proof:
[Equation]
It is easy to verify that the projections are orthogonal to the quaternion that deﬁnes the subspace by showing proj S ( q ) (cid:62) q = 0 :
[Equation]
Also note that this choice corresponds to tr ( qq (cid:62) )
= (cid:80) d +
[Equation]
=
[Equation]
α
[Equation]
=
[Equation]
Lemma 3.
The quaternion mean we suggest to use in the main paper Markley et al.
(2007) is equivalent to the Euclidean Weiszfeld mean on the afﬁne quaternion subspaces.
Proof.
We now recall and summarize the L q -Weiszfeld Algorithm on afﬁne subspaces Aftab et al.
(2015), which minimizes a q -norm variant of the cost deﬁned in Eq (18):
[Equation]
Deﬁning M i
=
I − A
i , Alg. 2 summarizes the iterative procedure.
Note that when q = 2 , the algorithm reduces to the computation of a non-weighted mean ( w i = 1 ∀ i ), and a closed form solution exists for Eq (30) and is given by the normal equations:
[Equation]
Compute the weights w t = { w ti } :
[Equation]
= (cid:107) M i
( x t − c i ) (cid:107) q −
[Equation]
=
[Equation]
k (29)
Solve: x t +1 = arg min
[Equation]
(cid:88)
[Equation]
For the case of our quaternionic subspaces c =
and we seek the solution that satisﬁes:
[Equation]
[Equation] It is well known that the solution to this equation under the constraint (cid:107) x (cid:107) = 1 lies in nullspace of M = 1 k k (cid:80) i =1 M i and can be obtained by taking the singular vector of M that corresponds to the largest singular value.
Since M i is idempotent, the same result can also be obtained through the eigendecomposition:
[Equation]
(33) which gives us the unweighted Quaternion mean Markley et al.
C P ROOF OF T HEOREM
Once the Lemma 1 is proven, we only need to apply the direct convergence results from the literature.
Consider a set of points Y = { y 1 . . .
y K }
where K > 2 and y i ∈ H 1 .
Due to the compactness, we can speak of a ball B ( o , ρ ) encapsulating all y i .
We also deﬁne the D = { x ∈ H 1 | C q ( x ) <
C q ( o ) } , the region where the loss decreases.
We ﬁrst state the assumptions that permit our theoretical result.
These assumptions are required by the works that establish the convergence of such Weiszfeld algorithms Luenberger et al.
Aftab & Hartley (2015); Aftab et al.
H1.
y 1 . . .
y K should not lie on a single geodesic of the quaternion manifold.
H2.
D is bounded and compact.
The topological structure of SO (3) imposes a bounded convexity radius of ρ < π/ 2 . H3.
The minimizer in Eq (30) is continuous.
H4.
The weighting function σ ( · ) is concave and differentiable.
H5.
Initial quaternion (in our network chosen randomly) does not belong to any of the subspaces.
Note that H5 is not a strict requirement as there are multiple ways to circumvent (simplest being
a re-
initialization).
Under these assumptions, the sequence produced by Eq (30) will converge to a critical point unless x t =
y
i for any t and i Aftab et al.
For q = 1 , this critical point is on one of the subspaces speciﬁed in Eq (19) and thus is a geometric median.
Note that due to the assumption H2 , we cannot converge from any given point.
For randomly initialized networks this is indeed a problem and does not guarantee practical convergence.
Yet, in our experiments we have not observed any issue with the convergence of our dynamic routing.
As our result is one of the few ones related to the analysis of DR, we still ﬁnd this to be an important ﬁrst step.
For different choices of q : 1 ≤ q ≤ 2 , the weights take different forms.
In fact, this IRLS type of algorithm is shown to converge for a larger class of weighting choices as long as the aforementioned conditions are met.
That is why in practice we use a simple sigmoid function.
D O UR S IAMESE A RCHITECTURE AND T HE A LGORITHM
For estimation of the relative pose with supervision, we beneﬁt from a Siamese variation of our network.
In this case, latent capsule representations of two point sets X and Y jointly contribute to the pose regression as shown in Fig.
[Figure]
We show additional results from the computation of local reference frames and the multi-channel capsules deduced from our network in Fig. 6.
Finally, the overall algorithm of our network is summarized under Alg. 3.
1 input : Input points of one patch { x 1 , ..., x K } ∈ R K × 3 , primary (input) capsules Q = { q 1 , . .
, q L } ∈ H 1 L , with L = N c × K , N c is the number of primary capsule channels, activations α
= ( α 1 , . .
. , α L ) T 2 output: Updated frames ˆ Q = { ˆ q 1 , . . . , ˆ q M } ∈ H 1 M , updated ˆ α = (ˆ α
ˆ α M
T do x (cid:48)
i ← µ (cid:0) Q ( n c ) (cid:1) − 1 ◦ x i // rotate point in a canonical orientation t ← t ( x (cid:48) i ) //
Point to Transform, t ( · ) : R 3 → R M ×
∈
Alg. 3 summarizes the overall pipeline of our QE-net depicted in Fig. We use multiple layers in a hierarchical architecture. In the ﬁrst layer, the input primary capsules are represented by LRFs computed with FLARE algorithm Petrelli & Di Stefano (2012). Therefore, the number of input capsule channels N c in the ﬁrst layer is equal to 1 .
Its activation is also defaulted to 1. The output of a former layer is propagated to the input of the latter, creating the hierarchy. [Figure] [Figure]
E A DDITIONAL D ETAILS ON E VALUATIONS
Details on the evaluation protocol. For Modelnet40 dataset used in Tab. 1, we used the ofﬁcial split with 9,843 shapes for training and 2,468 different shapes for testing. For rotation estimation in Tab. 2, we used the ofﬁcial Modelenet10 dataset split with 3991 for training and 908 shapes for testing. 3D point clouds (10K points) are randomly sampled from the mesh surfaces of each shape Qi et al.
(2017a;b).
The objects in training and testing dataset are different, but they are from the same categories so that they can be oriented meaningfully. During training, we did not augment the dataset with random rotations. All the shapes are trained with single orientation (well-aligned). We call this trained with NR .
During testing, we randomly generate multiple arbitrary SO (3) rotations for each shape and evaluate the average performance for all the rotations. This is called test with AR .
This protocol is used in both our algorithms and the baselines.
Confusion of classiﬁcation in ModelNet.
We now report the confusion matrix in the task of classiﬁcation on the all the objects of ModelNet10.
The classiﬁcation and rotation estimation affects one another.
As we can see from Fig.
7, the ﬁrst ﬁve categories that exhibit less rotational symmetry has the higher classiﬁcation accuracy than their rotationally symmetric counterparts.
Distribution of errors reported in Tab. 2.
We now provide more details on the errors attained by our algorithm as well as the state of the art.
To this end, we report, in Fig.
8 the histogram of errors that fall within quantized ranges of orientation errors.
It is noticeable that our Siamese architecture behaves best in terms of estimating the objects rotation. For completeness, we also included the results of the variants presented in our ablation studies: Ours-2kLRF, Ours-1kLRF.
They evaluate the model on the re-calculated LRFs in order to show the robustness towards to various point densities. We have also modiﬁed IT-Net and PointNetLK only to predict rotation because the original works predict both rotations and translations.
Finally, note here that we do not use data augmentation for training our networks (see AR), while both for PointNetLK and for IT-Net we do use augmentation.
[Figure]
toward the 60 ◦ the worse the performance of the method.