Equivariant Multi-View Networks

TL;DR

This paper introduces equivariant multi-view networks that perform joint reasoning over multiple views using group convolutions, improving 3D shape retrieval and scene classification by maintaining equivariance throughout the process.

Contribution

It proposes a novel group convolutional approach for multi-view aggregation that preserves equivariance, outperforming traditional view pooling methods.

Findings

Achieved state-of-the-art results in large-scale 3D shape retrieval.

Demonstrated improved panoramic scene classification performance.

Showed that equivariant reasoning enhances global descriptor quality.

Abstract

Several popular approaches to 3D vision tasks process multiple views of the input independently with deep neural networks pre-trained on natural images, achieving view permutation invariance through a single round of pooling over all views. We argue that this operation discards important information and leads to subpar global descriptors. In this paper, we propose a group convolutional approach to multiple view aggregation where convolutions are performed over a discrete subgroup of the rotation group, enabling, thus, joint reasoning over all views in an equivariant (instead of invariant) fashion, up to the very last layer. We further develop this idea to operate on smaller discrete homogeneous spaces of the rotation group, where a polar view representation is used to maintain equivariance with only a fraction of the number of input views. We set the new state of the art in several large scale 3D shape retrieval tasks, and show additional applications to panoramic scene classification.

Tables

Table 1. Table 1: SHREC’17 retrieval results. Top block: aligned dataset; bottom: rotated. We show mean average precision (mAP) and normalized discounted cumulative gain (G). We set the new state of the art by a large margin. Even our 12-view model outperforms the baseline, which shows the potential of keeping equivariance with fewer views.

			micro				macro
Method	score			mAP	G@N			mAP	G@N
RotatNet [20]	67.8			77.2	86.5			58.3	65.6
ReVGG [29]	61.8			74.0	82.8			49.6	55.9
DLAN [13]	57.0			66.3	76.2			47.7	56.3
MVCNN-M-12	69.1			74.9	83.8			63.2	70.3
Ours-12	70.7			77.7	86.3			63.6	70.8
Ours-20	71.4			77.9	86.8			64.9	71.9
Ours-60	71.7			77.8	86.4			65.6	72.3
Ours-R-20	72.2			79.1	87.5			65.4	72.3
DLAN [13]	56.6			65.6	75.4			47.6	56.0
ReVGG [29]	55.7			69.6	78.3			41.8	47.9
MVCNN-M-60	57.5			64.1	75.9			50.9	59.7
Ours-12	58.1			66.4	76.7			49.8	58.6
Ours-20	59.3			66.9	77.0			51.7	60.2
Ours-60	62.1			69.6	79.6			54.6	63.0
Ours-R-60	63.5			71.8	81.1			55.1	63.3

Table 2. Table 2: Aligned ModelNet classification and retrieval. We only compare with published retrieval results. We achieve state of the art retrieval performance even without retrieval-specific model features. This shows that our view aggregation is useful even when global equivariance is not necessary.

	M40 (aligned)			M10 (aligned)
	acc	mAP		acc	mAP
MVCNN-12 [33]	90.1	79.5		-	-
SPNet [40]	92.63	85.21		97.25	94.20
PVNet [41]	93.2	89.5		-	-
SV2SL [14]	93.40	89.09		94.82	91.43
PANO-ENN [31]	95.56	86.34		96.85	93.2
MVCNN-M-12	94.47	89.13		96.33	93.54
Ours-12	94.51	91.82		96.33	95.30
Ours-20	94.69	91.42		97.46	95.74
Ours-60	94.36	91.04		96.80	95.25
Ours-R-12	94.67	93.56		96.78	96.18

Table 3. Table 3: Rotated ModelNet40 classification and retrieval. Note that gap between “Ours” and “MVCNN-M” is much larger than in the aligned dataset, which demonstrates the advantage of our equivariant representation.

	M40 (rotated)
	acc	mAP
MVCNN-80 [33]	86.0	-
RotationNet [20]	80.0	74.20
Spherical CNN [6]	86.9	-
MVCNN-M-60	90.68	78.18
Ours-12	88.50	79.58
Ours-20	89.98	80.73
Ours-60	91.00	82.61
Ours-R-60	91.08	88.57

Table 4. Table 4: Scene classification class-based accuracy.

	single [2]	pano [2]	MV-M-12	Ours-12
acc [%]	33.3	41.0	51.9	53.8

Table 5. Table 5: Ablation study on rotated ModelNet40. Our best performing model is on the top row.

support	layers	views	pretrained	acc	mAP
9	3	60	yes	91.00	82.61
6	3	60	yes	90.63	81.90
3	3	60	yes	89.74	80.49
9	2	60	yes	91.00	81.47
9	1	60	yes	90.88	79.59
9	3	30	yes	89.50	79.20
9	3	10	yes	88.32	74.65
9	3	5	yes	82.77	64.88
9	3	60	no	87.40	70.44

Table 6. Table 6: We perturb the camera pose for each view to gradually break the assumption that they form a group. The model is trained with no perturbations and tested under different levels of perturbations.

std [deg]	acc	mAP
0	93.96	89.74
5	94.20	89.34
15	92.70	86.97
30	88.61	81.16
45	80.98	71.61

Table 7. Table 7: SHREC’17 retrieval results. Top block: aligned dataset; bottom: rotated. Methods are ranked by the micro and macro mAP average (namely, the “score” in the second column). We also show Precision (P), Recall (R), F-score (F1), mean average precision (mAP) and normalized discounted cumulative gain (G), where N is the number of retrieved elements.

			micro						macro
Method	score		P@N	R@N	F1@N	mAP	G@N		P@N	R@N	F1@N	mAP	G@N
RotatNet [20]	67.8		81.0	80.1	79.8	77.2	86.5		60.2	63.9	59.0	58.3	65.6
ReVGG [29]	61.8		76.5	80.3	77.2	74.0	82.8		51.8	60.1	51.9	49.6	55.9
DLAN [13]	57.0		81.8	68.9	71.2	66.3	76.2		61.8	53.3	50.5	47.7	56.3
MVCNN-12 [33]	65.1		77.0	77.0	76.4	73.5	81.5		57.1	62.5	57.5	56.6	64.0
MVCNN-M-12	69.1		83.1	77.9	79.4	74.9	83.8		66.8	68.4	65.2	63.2	70.3
Ours-12	70.7		83.1	80.5	81.1	77.7	86.3		65.3	68.7	64.8	63.6	70.8
Ours-20	71.4		83.6	80.8	81.5	77.9	86.8		66.4	70.1	65.9	64.9	71.9
Ours-60	71.7		84.0	80.5	81.4	77.8	86.4		67.1	70.7	66.6	65.6	72.3
Ours-R-20	72.2		83.6	81.7	82.0	79.1	87.5		66.8	69.9	66.1	65.4	72.3
DLAN [13]	56.6		81.4	68.3	70.6	65.6	75.4		60.7	53.9	50.3	47.6	56.0
ReVGG [29]	55.7		70.5	76.9	71.9	69.6	78.3		42.4	56.3	43.4	41.8	47.9
RotatNet [20]	46.6		65.5	65.2	63.6	60.6	70.2		37.2	39.3	33.3	32.7	40.7
MVCNN-80 [33]	45.1		63.2	61.3	61.2	53.5	65.3		40.5	48.4	41.6	36.7	45.9
MVCNN-M-60	57.5		77.7	67.6	71.1	64.1	75.9		55.7	56.9	53.5	50.9	59.7
Ours-12	58.1		76.1	70.0	72.0	66.4	76.7		54.6	55.7	52.6	49.8	58.6
Ours-20	59.3		76.4	70.5	72.4	66.9	77.0		54.6	58.0	53.7	51.7	60.2
Ours-60	62.1		78.7	72.9	74.7	69.6	79.6		57.6	60.1	56.3	54.6	63.0
Ours-R-60	63.5		78.7	75.0	75.9	71.8	81.1		58.3	60.6	56.9	55.1	63.3

Table 8. Table 8: Aligned ModelNet results. We include classification accuracy and retrieval mAP per class (cls) and per instance (inst).

	M40 (aligned)					M10 (aligned)
	acc inst	acc cls	mAP inst	mAP cls		acc inst	acc cls	mAP inst	mAP cls
Ours-12	94.51	92.49	91.82	88.28		96.33	96.00	95.30	95.00
Ours-20	94.69	92.56	91.42	87.71		97.46	97.34	95.74	95.58
Ours-60	94.36	92.40	91.04	87.30		96.80	96.58	95.25	95.01
Ours-R-20	94.44	92.49	93.19	89.65		97.02	96.97	96.59	96.46

Table 9. Table 9: Rotated ModelNet results. We include ModelNet10 and classification accuracy and retrieval mAP per class (cls) and per instance (inst).

	M40 (rotated)					M10 (rotated)
	acc inst	acc cls	mAP inst cls	mAP cls		acc inst	acc cls	mAP inst	mAP cls
Ours-12	88.50	85.77	79.58	74.64		91.89	91.54	86.93	86.08
Ours-20	89.98	87.65	80.73	75.65		92.60	92.35	87.27	86.65
Ours-60	91.00	89.24	82.61	78.02		92.83	92.80	88.47	88.02
Ours-R-20	91.08	88.94	88.57	84.37		93.05	93.08	92.07	91.99

Table 10. Table 10: Classification accuracy (acc) and retrieval (mAP) comparison against RotationNet. Results for aligned (al) and rotated (rot) datasets, and for the SHREC’17 split of ShapeNet (S17). The mAP for SHREC’17 is the average between micro and macro (score).

	M40 (al)			S17 (al)			S17 (rot)
	acc	mAP		acc	mAP		acc	mAP
RotNet	97.37	93.00		85.39	67.8		77.37	46.6
Ours	94.67	93.56		89.15	72.2		85.93	63.5

Table 11. Table 11: Matterport3D panoramic scene classification extended results.

	avg.	office	lounge	family room	entryway	dining room	living room	stairs	kitchen	porch	bathroom	bedroom	hallway
single [2]	33.3	20.3	21.7	16.7	1.8	20.4	27.6	49.5	52.1	57.4	44.0	43.7	44.7
pano [2]	41.0	26.5	15.4	11.4	3.1	27.7	34.0	60.6	55.6	62.7	65.4	62.9	66.6
MVCNN-M-12	51.9	18.0	16.4	23.8	8.6	46.7	37.1	84.1	73.3	81.0	78.2	81.7	73.8
Ours-12	53.8	27.9	16.4	33.3	11.4	51.1	41.3	80.4	75.8	79.0	72.5	82.9	73.5

Equations

$$\Phi ({\mathcal{T}}_g^{\mathcal{X}}(x))={\mathcal{T}}_g^{\mathcal{Y}}(\Phi (x)),\forall x\in \mathcal{X},g\in G.$$

$$(f\ast h)(y)={\int }_{x\in {\mathbb{R}}^2}f(x)h(y-x)dx.$$

$$(f\ast h)(y)={\int }_{g\in G}f(g)h(g^{-1}y)dg,$$

$$(f\ast h)(y)$$

$$(f\star h)(g)$$

$$f_j^{\ell +1}(y)=\sigma \left(\sum _{i=1}^{c_i}\sum _{g\in G}{f}_i^{\ell }(g)h_{ij}(g^{-1}y)\right),$$

$${}^{\ast }{f}_j^{\ell +1}(y)$$

$${}^{\star }{f}_j^{\ell +1}(g)$$

$$f_j^{\ell +1}(y)=\sigma \left(\sum _{i=1}^{c_i}\sum _{g\in S}{f}_i^{\ell }(y{g}^{-1})h_{ij}(g)\right).$$

$$({\mathcal{T}}_{k}f\ast h)(y)$$

$$={\int }_{g\in G}f(l)h((kl{)}^{-1}y)dl$$

$$={\int }_{g\in G}f(l)h((l^{-1}{k}^{-1}y)dl$$

$$=(f\ast h)(k^{-1}y)$$

$$={\mathcal{T}}_k(f\ast h)(y).$$

$$({\mathcal{T}}_{k}f\ast h)(y)$$

$$={\int }_{g\in G}f(l\eta )h((kl{)}^{-1}y)dl$$

$$={\int }_{g\in G}f(l\eta )h(l^{-1}{k}^{-1}y)dl$$

$$=(f\ast h)(k^{-1}y)$$

$$={\mathcal{T}}_k(f\ast h)(y).$$

$$({\mathcal{T}}_{k}f\star h)(g)$$

$$=(f\star h)(k^{-1}g)$$

$$={\mathcal{T}}_k^{{}^{\prime }}(f\star h)(y).$$

$$h_{ij}(g)=\left\{\begin{array}[]{rcl}1&&{i=j}\quad\text{and}\quad{g=e}\\ 0&&\text{else}\end{array}\right.$$

$$(f\star h{)}_i(k)$$

$$=\{\begin{array}{rlrl}& f_i(k)& & 1\le i\le c_i\\ & 0& & i>c_i\end{array},$$

$$\mathcal{L}=\max (d(z,z_c)-d(z,z_n)+\alpha ,0),$$

Full text

††* Equal contribution.††* http://github.com/daniilidis-group/emvn

1 Introduction

The proliferation of large scale 3D datasets for objects [39, 3] and whole scenes [2, 8] enables training of deep learning models producing global descriptors that can be applied to classification and retrieval tasks.

The first challenge that arises is how to represent the inputs. Despite numerous attempts with volumetric [39, 24], point-cloud [27, 32] and mesh-based [23, 26] representations, using multiple views of the 3D input allows switching to the 2D domain where all the recent image based deep learning breakthroughs (e.g. [15]) can be directly applied, facilitating state of the art performance [33, 20].

Multi-view (MV) based methods require some form of view-pooling, which can be (1) pixel-wise pooling over some intermediate convolutional layer [33], (2) pooling over the final 1D view descriptors [34], or (3) combining the final logits [20], which can be seen as independent voting. These operations are usually invariant to view permutations.

Our key observation is that conventional view pooling is performed before any joint processing of the set of views and will inevitably discard useful features, leading to subpar descriptors. We solve the problem by first realizing that each view can be associated with an element of the rotation group $\mathbf{SO}$(3), so the natural way to combine multiple views is as a function on the group. A traditional CNN is applied to obtain view descriptors that compose this function. We design a group-convolutional network (G-CNN, inspired by [5]) to learn representations that are equivariant to transformations from the group. This differs from the invariant representations obtained through usual view-pooling that discards information. We obtain invariant descriptors useful for classification and retrieval by pooling over the last G-CNN layer. Our G-CNN has filters with localized support on the group and learns hierarchically more complex representations as we stack more layers and increase the receptive field.

We take advantage of the finite nature of multiple views and consider finite rotation groups like the icosahedral, in contrast with  [6, 10] which operate on the continuous group. To reduce the computational cost of processing one view per group element, we show that by considering views in canonical coordinates with respect to the group of in-plane dilated rotations (log-polar coordinates), we can greatly reduce the number of views and obtain an initial representation on a homogeneous space (H-space) that can be lifted via correlation, while maintaining equivariance.

We focus on 3D shapes but our model is applicable to any task where multiple views can represent the input, as demonstrated by an experiment on panoramic scenes.

Figure 1 illustrates our model. Our contributions are:

• •

We introduce a novel method of aggregating multiple views whether “outside-in” for 3D shapes or “inside-out” for panoramic views. Our model exploits the underlying group structure, resulting in equivariant features that are functions on the rotation group.

• •

We introduce a way to reduce the number of views while maintaining equivariance, via a transformation to canonical coordinates of in-plane rotation followed by homogeneous space convolution.

• •

We explore the finite rotation groups and homogeneous spaces and present a discrete G-CNN model on the largest group to date, the icosahedral group. We further explore the concept of filter localization for this group.

• •

We achieve state of the art performance on multiple shape retrieval benchmarks, both in canonical poses and perturbed with rotations, and show applications to panoramic scene classification.

2 Related work

3D shape analysis

Performance of 3D shape analysis is heavily dependent on the input representation. The main representations are volumetric, point cloud and multi-view.

Early examples of volumetric approaches are [3], which introduced the ModelNet dataset and trained a 3D shape classifier using a deep belief network on voxel representations; and [24], which presents a standard architecture with 3D convolutional layers followed by fully connected layers.

Su et al. [33] realized that by rendering multiple views of the 3D input one can transfer the power of image-based CNNs to 3D tasks. They show that a conventional CNN can outperform the volumetric methods even using only a single view of the input, while a multi-view (MV) model further improves the classification accuracy.

Qi et al. [28] study volumetric and multi-view methods and propose improvements to both; Kanezaki et al. [20] introduces an MV approach that achieves state-of-the-art classification performance by jointly predicting class and pose, though without explicit pose supervision.

GVCNN [12] attempts to learn how to combine different view descriptors to obtain a view-group-shape representation; they refer to arbitrary combinations of features as “groups”. This differs from our usage of the term “group” which is the algebraic definition.

Point-cloud based methods [27] achieve intermediate performance between volumetric and multi-view, but are much more efficient computationally. While meshes are arguably the most natural representation and widely used in computer graphics, only limited success has been achieved with learning models operating directly on them [23, 26].

In order to better compare 3D shape descriptors we will focus on the retrieval performance. Recent approaches show significant improvements on retrieval: You et al. [41] combines point cloud and MV representations; Yavartanoo et al. [40] introduces multi-view stereographic projection; and Han et al. [14] implements a recurrent MV approach.

We also consider more challenging tasks on rotated ModelNet and SHREC’17 [29] retrieval challenge which contains rotated shapes. The presence of arbitrary rotations motivates the use of equivariant representations.

Equivariant representations

A number of workarounds have been introduced to deal with 3D shapes in arbitrary orientations. Typical examples are training time rotation augmentation and/or test time voting [28] and learning an initial rotation to a canonical pose [27]. The view-pooling in [33] is invariant to permutations of the set of input views.

A principled way to deal with rotations is to use representations that are equivariant by design. There are mainly three ways to embed equivariance into CNNs. The first way is to constrain the filter structure, which is similar to Lie generator based approach  [30, 17]. Worral et al. [38] take advantage of circle harmonics to have both translational and 2D rotational equivariance into CNNs. Similarly, Thomas et al. [35] introduce a tensor field to keep translational and rotational equivariance for 3D point clouds.

The second way is through a change of coordinates; [11, 18] take the log-polar transform of the input and transfer rotational and scaling equivariance about a single point to translational equivariance.

The third way is to make use of an equivariant filter orbit. Cohen and Welling propose group convolution (G-CNNs) with the square rotation group [5], later extended to the hexagon [19]. Worrall and Brostow [37] proposed CubeNet using Klein’s Four-group on 3D voxelized data. Winkels et al. [36] implement 3D group convolution on Octahedral symmetry group for volumetric CT images. Cohen et al. [7] very recently considered functions on the icosahedron, however their convolutions are on the cyclic group and not on the icosahedral as ours. Esteves et al. [10] and Cohen et al. [6] focus on the infinite group $\mathbf{SO}$(3), and use the spherical harmonic transform for the exact implementation of the spherical convolution or correlation. The main issue with these approaches is that the input spherical representation does not capture the complexity of an object’s shape; they are also less efficient and face bandwidth challenges.

3 Preliminaries

We seek to leverage symmetries in data. A symmetry is an operation that preserves some structure of an object. If the object is a discrete set with no additional structure, each operation can be seen as a permutation of its elements.

The term group is used in its classic algebraic definition of a set with an operation satisfying the closure, associativity, identity, and inversion properties. A transformation group like a permutation is the “missing link between abstract group and the notion of symmetry”  [25].

We refer to view as an image taken from an oriented camera. This differs from viewpoint that refers to the optical axis direction, either outside-in for a moving camera pointing at a fixed object, or inside-out for a fixed camera pointing at different directions. Multiple views can be taken from the same viewpoint; they are related by in-plane rotations.

Equivariance

Representations that are equivariant by design are an effective way to exploit symmetries. Consider a set $\mathcal{X}$ and a transformation group $G$. For any $g\in G$, we can define group action applied on the set, $\mathcal{T}^{\mathcal{X}}_{g}:\mathcal{X}\rightarrow\mathcal{X}$, which has property of homomorphism, $\mathcal{T}^{\mathcal{X}}_{g}\mathcal{T}^{\mathcal{X}}_{h}=\mathcal{T}^{\mathcal{X}}_{gh}$. Consider a map $\Phi:\mathcal{X}\rightarrow\mathcal{Y}$. We say $\Phi$ is equivariant to $G$ if

[TABLE]

In the context of CNNs, $\mathcal{X}$ and $\mathcal{Y}$ are sets of input and feature representations, respectively. This definition encompasses the case when $\mathcal{T}^{\mathcal{Y}}_{g}$ is the identity, making $\Phi$ invariant to $G$ and discarding information about $g$. In this paper, we are interested in non-degenerate cases that preserve information.

Convolution on groups

We represent multiple views as a functions on a group and seek equivariance to the group, so group convolution (G-Conv) is the natural operation for our method. Let us recall planar convolution between $f,h\colon\mathbb{R}^{2}\mapsto\mathbb{R}$, which is the main operation of CNNs:

[TABLE]

It can be seen as an operation over the group of translations on the plane, where the group action is addition of coordinate values; it is easily shown to be equivariant to translation. This can be generalized to any group $G$ and $f,h\colon G\mapsto\mathbb{R}$,

[TABLE]

which is equivariant to group actions from $G$.

Convolution on homogeneous spaces

For efficiency, we may relax the requirement of one view per group element and consider only one view per element of a homogeneous space of lower cardinality. For example, we can represent the input on the 12 vertices of the icosahedron (an H-space), instead of on the 60 rotations of the icosahedral group.

A homogeneous space $\mathcal{X}$ of a group $G$ is defined as a space where $G$ acts transitively: for any $x_{1},x_{2}\in\mathcal{X}$, there exists $g\in G$ such that $x_{2}=gx_{1}$.

Two convolution-like operations can be defined between functions on homogeneous spaces $f,h\colon\mathcal{X}\mapsto\mathbb{R}$:

[TABLE]

where $\eta\in\mathcal{X}$ is an arbitrary canonical element. We denote (4) “homogeneous space convolution” (H-Conv), and (5) “homogeneous space correlation” (H-Corr). Note that convolution produces a function on the homogeneous space $\mathcal{X}$ while correlation lifts the output to the group $G$.

We refer to [22, 4] for expositions on group and homogeneous space convolution in the context of neural networks.

Finite rotation groups

Since our representation is a finite set of views that can be identified with rotations, we will deal with finite subgroups of the rotation group $\mathbf{SO}$(3). A finite subgroup of $\mathbf{SO}$(3) can be the cyclic group $\mathcal{C}_{k}$ of multiples of $2\pi/k$, the dihedral group $\mathcal{D}_{k}$ of symmetries of a regular $k$-gon, the tetrahedral, octahedral, or icosahedral group [1].

Our main results are on the icosahedral group $\mathcal{I}$, the 60-element non-abelian group of symmetries of the icosahedron (illustrated in the supplementary material). The symmetries can be divided in sets of rotations around a few axes. For example, there are 5 rotations around each axis passing through vertices of the icosahedron or 3 rotations around each axis passing through its faces centers.

Equivariance via canonical coordinates

Some configurations produce views that are related by in-plane rotations. We leverage this to reduce the number of required views by obtaining rotation invariant view descriptors through a change to canonical coordinates followed by a CNN.

Segman et al. [30] show that changing to a canonical coordinate system allows certain transformations of the input to appear as translations of the output. For the group of dilated rotations on the plane (isomorphic to $\mathbf{SO}(2)\times\mathbb{R}^{+}$), canonical coordinates are given by the log-polar transform.

Since planar convolutions are equivariant to translation, converting an image to log-polar and applying a CNN results in features equivariant to dilated rotation, which can be pooled to invariant descriptors on the last layer [11, 18].

4 Method

Our first step is to obtain $\lvert G\rvert$ views of the input where each view $x_{i}$ is associated with a group element $g_{i}\in G$111Alternatively, we can use $\lvert\mathcal{X}\rvert$ views for a homogeneous space $\mathcal{X}$ as shown in 4.3.. Each view is fed to a CNN $\Phi_{1}$, and the 1D descriptors extracted from the last layer (before projection into the number of classes) are combined to form a function on the group $y\colon G\mapsto\mathbb{R}^{n}$, where $y(g_{i})=\Phi_{1}(x_{i})$. A group convolutional network (G-CNN) $\Phi_{2}$ operating on $G$ is then used to process $y$, and global average pooling on the last layer yields an invariant descriptor that is used for classification or retrieval. Training is end-to-end. Figure 1 shows the model.

The MVCNN with late-pooling from [20], which outperforms the original [33], is a special case of our method where $\Phi_{2}$ is the identity and the descriptor is $y$ averaged over $G$.

4.1 View configurations

There are several possible view configurations of icosahedral symmetry, basically consisting of vertices or faces of solids with the same symmetry. Two examples are associating viewpoints with faces/vertices of the icosahedron, which are equivalent to the vertices/faces of its dual, the dodecahedron. These configurations are based on platonic solids, which guarantee a uniform distribution of viewpoints. By selecting viewpoints from the icosahedron faces, we obtain 20 sets of 3 views that differ only by 120 deg in plane rotations; we refer to this configuration as $20\times 3$. Similarly, using the dodecahedron faces we obtain the $12\times 5$ configuration.

In the context of 3D shape analysis, multiple viewpoints are useful to handle self-occlusions and ambiguities. Views that are related by in-plane rotations are redundant in this sense, but necessary to keep the group structure.

To minimize redundancy, we propose to associate viewpoints with the 60 vertices of the truncated icosahedron (which has icosahedral symmetry). There is a single view per viewpoint in this configuration. This is not a uniformly spaced distribution of viewpoints, but the variety is beneficial. Figure 3 shows some view configurations we considered.

Note that our configurations differ from both the 80-views from [33] and 20 from [20] which are not isomorphic to any rotation group. Their 12-views configuration is isomorphic to the more limited cyclic group.

4.2 Group convolutional networks

The core of the group convolutional part of our method is the discrete version of (3). A group convolutional layer with $c_{i},c_{j}$ channels in the input and output and nonlinearity $\sigma$ is then given by

[TABLE]

where $f_{i}^{\ell}$ is the channel $i$ at layer $\ell$ and $h_{ij}$ is the filter between channels $i$ and $j$, where $1\leq j\leq c_{j}$. This layer is equivariant to actions of $G$.

Our most important results are on the icosahedral group $\mathcal{I}$ which has 60 elements and is the largest discrete subgroup of the rotation group $\mathbf{SO}$(3). To the best of our knowledge, this is the largest group ever considered in the context of discrete G-CNNs. Since $\mathcal{I}$ only coarsely samples $\mathbf{SO}$(3), equivariance to arbitrary rotations is only approximate. Our results show, however, that the combination of invariance to local deformations provided by CNNs and exact equivariance by G-CNNs is powerful enough to achieve state of the art performance in many tasks.

When considering the group $\mathcal{I}$, inputs to $\Phi_{2}$ are $60\times n$ where $n$ is the number of channels in the last layer of $\Phi_{1}$ (n=512 for ResNet-18). There are $c_{i}\times c_{j}$ filters per layer each with up to the same cardinality of the group.

We can visualize both filters and feature maps as functions on the faces of the pentakis dodecahedron, which is the dual polyhedron of the truncated icosahedron. It has icosahedral symmetry and 60 faces that can be identified with elements of the group. The color of the face associated with $g_{i}\in\mathcal{I}$ reflects $f(g_{i})$, which is vector valued. Figure 2 shows some equivariant feature maps learned by our method.

4.3 Equivariance with fewer views

As illustrated in Figure 3, the icosahedral symmetries can be divided in sets of rotations around a few axes. If we arrange the cameras such that they lie on these axes, images produced by each camera are related by in-plane rotations.

As shown in Section 3, converting one image to canonical coordinates can transform in-plane rotations in translations. We’ll refer to converted images as “polar images”. Since fully convolutional networks can produce translation-invariant descriptors, by applying them to polar images we effectively achieve invariance to in-plane rotations [11, 18], which makes only one view per viewpoint necessary. These networks require circular padding in the angular dimension.

When associating only a single view per viewpoint, the input is on a space of points instead of a group of rotations222They are isomorphic for the $60\times 1$ configuration.. In fact, the input is a function on a homogeneous space of the group; concretely, for the view configurations we consider, it is on the icosahedron or dodecahedron vertices.

We can apply discrete versions of convolution and correlation on homogeneous spaces as defined in Section 3:

[TABLE]

The benefit of this approach is that since it uses 5x (3x) fewer views when starting from the $12\times 5$ ($20\times 3$) configuration, it is roughly 5x (3x) faster as most of the compute is done before the G-CNN. The disadvantage is that learning from polar images can be challenging. Figure 4 shows one example of polar images produced from views.

When inputs are known to be aligned (in canonical pose), an equivariant intermediate representation is not necessary; in this setting, we can use the same method to reduce the number of required views, but without the polar transform.

4.4 Filter localization

G-CNN filters are functions on $G$, which can have up to $\lvert G\rvert$ entries. Results obtained with deep CNNs during the past few years show the benefit from limited support filters (many architectures use $3\times 3$ kernels throughout). The advantages are two-fold: (1) convolution with limited support is computationally more efficient, and (2) it allows learning of hierarchically more complex features as layers are stacked. Inspired by this idea, we introduce localized filters for discrete G-CNNs333Localization for the continuous case was introduced in [10].. For a filter $h\colon G\mapsto\mathbb{R}$, we simply choose a subset $S$ of $G$ that is allowed to have nonzero filter values while $h(G-S)$ is set to zero. Since $S$ is a fixed hyperparameter, we can compute (6) more efficiently:

[TABLE]

To ensure filter locality, it is desirable that elements of $S$ are close to each other in the manifold of rotations. The 12 smallest rotations in $\mathcal{I}$ are of $72$ deg. We therefore choose $S$ to contain the identity and a number of $72$ deg rotations.

One caveat of this approach is that we need to make sure $S$ spans $G$, otherwise the receptive field will not cover the whole input no matter how many layers are stacked, which can happen if $S$ belongs to a subgroup of $G$ (see Figure 5). In practice this is not a challenging condition to satisfy; for our heuristic of choosing only $72$ deg rotations we only need to guarantee that at least two are around different axes.

5 Experiments

We evaluate on 3D shape classification, retrieval and scene classification, and include more comparisons and an ablation study in the supplementary material. First, we discuss the architectures, training procedures, and datasets.

Architectures

We use a ResNet-18 [15] as the view processing network $\Phi_{1}$, with weights initialized from ImageNet [9] pre-training. The G-CNN part contains 3 layers with 256 channels and 9 elements on its support (note that the number of parameters is the same as one conventional $3\times 3$ layer). We project from 512 to 256 channels so the number of parameters stay close to the baseline. When the method in Section 4.3 is used to reduce the number of views, the first G-Conv layer is replaced by a H-Corr.

Variations of our method are denoted Ours-X, and Ours-R-X. The R suffix indicate retrieval specific features, that consist of (1) a triplet loss444Refer to the supplementary material for details. and (2) reordering the retrieval list so that objects classified as the query’s predicted class come first. Before reordering, the list is sorted by cosine distance between descriptors. For SHREC’17, choosing the number N of retrieved objects is part of the task – in this case we simply return all objects classified as the query’s class.

For fair assessment of our contributions, we implement a variation of MVCNN, denoted MVCNN-M-X for X input views, where the best-performing X is shown. MVCNN-M-X has the same view-processing network, training procedure and dataset as ours; the only difference is that it performs pooling over view descriptors instead of using a G-CNN.

Training

We train using SGD with Nesterov momentum as the optimizer. For ModelNet experiments we train for 15 epochs, and 10 for SHREC’17. Following [16], the learning rate linearly increases from 0 to $lr$ in the first epoch, then decays to zero following a cosine quarter-cycle. When training with 60 views, we set the batch size to 6, and $lr$ to $0.0015$. This requires around 11 Gb of RAM. When training with 12 or 20 views, we linearly increase both the batch size and $lr$.

Training our 20-view model on ModelNet40 for one epoch takes $\approx 353s$ on an NVIDIA 1080 Ti, while the corresponding MVCNN-M takes $\approx 308s$. Training RotationNet [20] for one epoch under same conditions takes $\approx 1063s$.

Datasets

We render $12\times 5$, $20\times 3$ and $60\times 1$ camera configurations (Section 4.1) for ModelNet and the ShapeNet SHREC’17 subset, for both rotated and aligned versions. For the aligned datasets, where equivariance to rotations is not necessary, we fix the camera up-vectors to be in the plane defined by the object center, camera and north pole. This reduces the number of views from $12\times 5$ to 12 and from $20\times 3$ to 20. For the rotated datasets, all renderings have 60 views and follow the group structure. Note that the rotated datasets are not limited to the discrete group and contain continuous rotations from $\mathbf{SO}(3)$. We observe that the $60\times 1$ configuration performs best so those are the numbers shown for “Ours-60”. For the experiment with fewer views, we chose 12 from $12\times 5$ and 20 from $20\times 3$ that are converted to log-polar coordinates (Section 4.3). For the scene classification experiment, we sample 12 overlapping views from panoramas. No data augmentation is performed.

5.1 SHREC’17 retrieval challenge

The SHREC’17 large scale 3D shape retrieval challenge [29] utilizes the ShapeNet Core55 [3] dataset and has two modes: “normal” and “perturbed” which correspond to “aligned” and “rotated” as we defined in Section 5.2. The challenge was carried out in 2017 but there has been recent interest on it, especially on the “rotated” mode [6, 10, 21].

Table 1 shows the results. N is the number of retrieved elements, which we choose to be the objects classified as the same class as the query. The Normalized Discounted Cumulative Gain (NDGC) score uses ShapeNet subclasses to measure relevance between retrieved models. Methods are ranked by the mean of micro (instance-based) and macro (class-based) mAP. Several extra retrieval metrics are included in the supplementary material. Only the best performing methods are shown; we refer to [29] for more results.

Our model outperforms the state of the art for both modes even without the triplet loss, which, when included, increase the margins. We consider this our most important result, since it is the largest available 3D shape retrieval benchmark and there are numerous published results on it.

5.2 ModelNet classification and retrieval

We evaluate 3D shape classification and retrieval on variations of ModelNet [39]. In order to compare with most publicly available results, we evaluate on “aligned” ModelNet, and use all available models with the original train/test split (9843 for training, 2468 for test). We also evaluate on the more challenging “rotated” ModelNet40, where each instance is perturbed with a random rotation from $\mathbf{SO}$(3).

Tables 2 and 3 show the results. We show only the best performing methods and refer to the ModelNet website555http://modelnet.cs.princeton.edu for complete leaderboard. Classification performance is given by accuracy (acc) and retrieval by the mean average precision (mAP). Averages are over instances. We include class-based averages on the supplementary material.

We outperform the retrieval state of the art for both ModelNet10 and ModelNet40, even without retrieval-specific features. When including such features (triplet loss and reordering by class label), the margin increases significantly.

We focus on retrieval and do not claim state of the art on classification, which is held by RotationNet [20]. While ModelNet retrieval was not attempted by [20], the SHREC’17 retrieval was, and we show significantly better performance on it (Table 1).

5.3 Scene classification

We have shown experiments for object-centric configurations (outside-in), but our method is also applicable to camera-centric configurations (inside-out), which is demonstrated on the Matterport3D [2] scene classification from panoramas task. We sample multiple overlapping azimuthal views from the panorama and apply our model over the cyclic group of 12 rotations, with a filter support of 6. Table 4 shows the results; the full table with accuracies per class and input samples are in the supplementary material.

The MV approach is superior to operating directly on panoramas because (1) it allows higher overall resolution while sharing weights across views, and (2) views match the scale of natural images so pre-training is better exploited. Our MVCNN-M outperforms both baselines, and our proposed model outperforms it, which shows that the group structure is also useful in this setting. In this task, our representation is equivariant to azimuthal rotations; a CNN operating directly on the panorama has the same property.

5.4 Discussion

Our model shows state of the art performance on multiple 3D shape retrieval benchmarks. We argue that the retrieval problem is more appropriate to evaluate shape descriptors because it requires a complete rank of similarity between models instead of only a class label.

Our results for aligned datasets show that the full set of 60 views is not necessary and may be even detrimental in this case; but even when equivariance is not required, the principled view aggregation with G-Convs is beneficial, as direct comparison between MVCNN-M and our method show. For rotated datasets, results clearly show that performance increases with the number of views, and that the aggregation with G-Convs brings huge improvements.

Interestingly, our MVCNN-M baseline outperforms many competing approaches. The differences with respect to the original MVCNN [33] are (1) late view-pooling, (2) use of ResNet, (3) improved rendering, and (4) improved learning rate schedule. These significant performance gains were also observed in [34], and attest to the representative potential of multi-view representations.

One limitation is that our feature maps are equivariant to discrete rotations only, and while classification and retrieval performance under continuous rotations is excellent, for tasks such as continuous pose estimation it may not be. Another limitation is that we assume views to follow the group structure, which may be difficult to achieve for real images. Note that this is not a problem for 3D shape analysis, where we can render any arbitrary view.

6 Conclusion

We proposed an approach that leverages the representational power of conventional deep CNNs and exploits the finite nature of the multiple views to design a group convolutional network that performs an exact equivariance in discrete groups, most importantly the icosahedral group. We also introduced localized filters and convolutions on homogeneous spaces in this context. Our method enables joint reasoning over all views as opposed to traditional view-pooling, and is shown to surpass the state of the art by large margins in several 3D shape retrieval benchmarks.

7 Acknowledgments

We are grateful for support through the following grants: NSF-IIP-1439681 (I/UCRC), NSF-IIS-1703319, NSF MRI 1626008, ARL RCTA W911NF-10-2-0016, ONR N00014-17-1-2093, ARL DCIST CRA W911NF-17-2-0181, the DARPA-SRC C-BRIC, and by Honda Research Institute.

Supplementary material: Equivariant Multi-View Networks

We divide this material in “Extended experiments”, “Proofs”, “Implementation details”, and “Visualization”. Numbering and citations follow the main text.

Appendix A Extended experiments

We include an ablation experiment and further results for ModelNet, SHREC’17 large scale retrieval, comparison with RotationNet [20], and Matterport3D scene classification.

A.1 Ablation

We run an experiment to compare effects of (1) filter support size, (2) number of G-Conv layers, and (3) missing views. We evaluate on rotated ModelNet40 with “Ours-60” model as baseline that has with 9 elements in the support, 3 G-Conv layers and all 60 views.

When considering less than 60 views, we introduce view dropout during training where a random number (between 1 and 30) of views is selected for every mini-batch. This improves robustness to missing views. During test, a fixed number of views is used. Table 5 shows the results. As expected, we can see some decline in performance with fewer layers and smaller support, which reduces the receptive field at the last layer. Our method is shown to be robust to missing up to 50% of the views, with noticeable drop in performance when missing 80% or more.

In the second ablation experiment, we investigate the performance as the assumption that the views can be associated with group elements is broken. We perturb the camera pose for each view with randomly sampled rotations given some standard deviation. The model is trained once on ModelNet40 (aligned) and tested under different levels of perturbations. Table 6 shows the results.

A.2 SHREC’17

We show all metrics for the SHREC’17 large scale retrieval challenge in Table 7.

A.3 ModelNet

Since some methods show ModelNet40 results as averages per class instead of the more common average per instance, we include exteded tables with these metrics. We also present results on rotated ModelNet10. Tables 8 and 9 shows the results for the aligned and rotated versions, respectively.

A.4 Comparison with RotationNet

We provide further comparison against RotationNet [20]. While RotationNet remains SoTA on aligned ModelNet classification, our method is superior on all retrieval benchmarks. We also outperform RotationNet on more challenging classification taks: rotated and aligned ShapeNet, and rotated ModelNet. Table 10 shows the results.

A.5 Scene classification

We show examples of original input and our 12 overlapping views in Figure 6. The complete set of results in the same format as [2] is shown in Table 11.

Appendix B Proofs

We demonstrate the equivariance of operations, and show that MVCNN [33] is a special case of our method.

B.1 Equivariance of G-Conv, H-Conv, H-Corr

We demonstrate here the equivariance of the main operations used in our method: group convolution, homogeneous space convolution and correlation. We assume a compact group $G$ and one of its homogeneous spaces $\mathcal{X}$. $\mathcal{T}_{k}$ is the action of $k\in G$. Let us start with G-Conv (3), where $f,h\colon G\mapsto\mathbb{R}$:

[TABLE]

For H-Conv (4), where $f,h\colon\mathcal{X}\mapsto\mathbb{R}$, we have:

[TABLE]

Finally, for H-Corr (5), where $f,h\colon\mathcal{X}\mapsto\mathbb{R}$, we have:

[TABLE]

Note that, in this case, $\mathcal{T}^{{}^{\prime}}_{k}$ is not equal $\mathcal{T}_{k}$ because inputs and outputs are in different spaces.

B.2 MVCNN is a special case

Now we show that our model can replicate MVCNN [33] by fixing the filters $h_{ij}(g)$, where $i,j$ denote the output and input channel and $g$ denotes the element in group:

[TABLE]

Combining with group correlation (the result can also be achieved by group convolution), we show that

[TABLE]

where $c_{i}$ is the number of input channels. In this way, the input is “copied” into the output and the our model produces the exact same descriptor as an MVCNN with late pooling after the last layer.

Appendix C Implementation details

We include details about our triplet loss implementation and about our procedure for visualization of discrete rotation groups and their homogeneous spaces.

C.1 Triplet loss

We implement a simple triplet loss. During training, we keep a set containing the descriptors for the last seen instance of each class, $Z=\{z_{i}\}$, where $i$ is the class label. For each entry in the mini-batch, let $c$ be the class and $z$ its descriptor. We take the descriptor in $Z$ of the same class as a positive example ($z_{c}$), and chose the hardest between all the others in the set as the negative: $z_{n}=\text{argmin}_{z_{i}\in Z,\,i\neq c}(d(z_{i},z))$, where $d$ is a distance function. The contribution of this entry to the loss is then,

[TABLE]

where $\alpha$ is a margin. We use $\alpha=0.2$ and $d$ is the cosine distance. Note that this method is only used in the “Ours-R” variations of our method.

C.2 Feature map visualization

Our features are functions on a subgroup of the rotation group $\mathbf{SO}$(3). Since $\mathbf{SO}$(3) is a 3-manifold (which can be embedded in $\mathbb{R}^{5}$), visualization is challenging. As we operate on the discrete subgroup of 60 rotations, we choose a solid with icosahedral symmetry and 60 faces as a proxy for visualization – the pentakis dodecahedron, which is the dual of the truncated icosahedron (the “soccer ball” with 60 vertices).

We associate the color of each face with the feature vector at that element of the group. Since the vector is high-dimensional (usually 256 or 512-D), we use PCA over all feature vectors in a layer (or groups of channels in a layer) and project it into the 3 principal components that can be associated with an RGB value. The same idea is applied to visualize functions on the homogeneous spaces, where the dodecahedron and icosahedron are used as proxies.

Appendix D Visualization

We visualize the icosahedral group properties and include more examples of our equivariant feature maps.

D.1 Icosahedral group

The icosahedral group $\mathcal{I}$ is the group of symmetries of the icosahedron, which consists of 60 rotations, as visualized in Figure 8. We show how the equivariance manifests as permutation when rendering multiple views of an object according to the group structure in Figure 7. Figure 9 shows the Cayley Table for $\mathcal{I}$; note that the color assigned for each group element matches the color in Figure 7.

D.2 Feature maps

We visualize more examples of our equivariant feature maps in Figures 10, 11, 12. Each figure shows 8 different input rotations, the first 5 are from a subgroup of rotations around one axis with 72 deg spacing, the other 3 are from other subgroup with 120 deg spacing. We show the axis of rotation in red. The first column is a view of the input, the second is the initial representation on the group or H-space, and the other 3 are features on each G-CNN layer.

Our method is equivariant to the 60-element discrete rotation group even with only 12 or 20 input views. In Figure 10 we take only 12 input views, giving initial features on the H-space represented by faces of the dodecahedron. Note that the 5 first rotations in this case are in-plane for the views corresponding to the axis of rotation. Due to our procedure described in Section 4.3, this gives an invariant descriptor which can be visualized as the face with constant color. Similarly, in Figure 11, we take 20 views and the invariant descriptor can be seen in the last 3 rotations.

Equivariance is easily visualized on faces neighboring the axis of rotation. For the dodecahedron, we can see cycles of 5 when the axis is on one face and cycles of 3 when the axis is on one vertex. For the icosahedron, we can see cycles of 3 when the axis is on one face and cycles of 5 when the axis is on one vertex. For the pentakis dodecahedron (Figure 12), we can see groups of 5 cells that shift one position when rotation is of 72 deg and groups of 6 cells that shift two positions when rotation is of 120 deg.