Y. LIN ET AL.: IMPROVED 3D-MIRROR SYMMETRY LEARNING FROM A SINGLE IMAGE

1

NeRD++: Improved 3D-mirror symmetry
learning from a single image
Yancong Lin
Silvia L. Pintea
Jan C. van Gemert

Computer Vision Lab,
Delft University of Technology,
Delft, Netherlands

Abstract
Many objects are naturally symmetric, and this symmetry can be exploited to infer
unseen 3D properties from a single 2D image. Recently, NeRD [26] is proposed for
accurate 3D mirror plane estimation from a single image. Despite the unprecedented accuracy, it relies on large annotated datasets for training and suffers from slow inference.
Here we aim to improve its data and compute efficiency. We do away with the computationally expensive 4D feature volumes and instead explicitly compute the feature correlation of the pixel correspondences across depth, thus creating a compact 3D volume. We
also design multi-stage spherical convolutions to identify the optimal mirror plane on thehemisphere, whose inductive bias offers gains in data-efficiency. Experiments on bothsynthetic and real-world datasets show the benefit of our proposed changes for improved
data efficiency and inference speed.

1Introduction

Symmetry exists in nature, in man-made environments, in science and arts. Mirror symmetry,also known as bilateral or reflection symmetry, is an intrinsic property of many objects, andit allows infering the entire object from only partial view. This has been shown to be usefulfor shape completion [5, 13] and single-view 3D reconstruction [22, 23, 24].
Deep learning approaches have achieved astonishing results on estimating mirror symmetries from single-view images, by learning dense features from convolutional networks
instead of relying on local feature matching [5, 18, 26]. In addition to feature learning,
deep networks can incorporate 3D mirror geometry as in NeRD [26], the top-performing
symmetry detection model. However, NeRD [26] builds a computationally 4D feature volumes during learning, resulting in high inference latency. Moreover, its performance also
deteriorates substantially when limited training data is available.
In this paper, we make two improvements over NeRD [26] to increase its data and compute efficiency. Specifically, from learned semantic deep features we calculate a compact 3D
correlation volume for each candidate plane. Correlations measure the similarity between
the features and their mirrored versions, at every location in the featuremap. The optimalplane is characterized by the highest feature correlation. By adding explicit correlation computation into the model, we bypass the expensive 4D feature volumes in NeRD [26], andthus substantially speed up the inference. Our second modification is on data-efficiency by© 2022. The copyright of this document resides with its authors.
It may be distributed unchanged freely in print or electronic forms.

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Y. LIN ET AL.: IMPROVED 3D-MIRROR SYMMETRY LEARNING FROM A SINGLE IMAGE
A planeA spherical point

A camera ray

A point

Asymmetric mirror

Symmetric mirror

Depth

...

Image plane
Mirrored pixels
Origin(a) 3D mirror(b) Correlation calculation
between inputs and mirrors(c) Plane detection
by spherical convolutions

Figure 1: 3D mirror symmetry detection. We identify 3D mirror planes by measuring correlations
between the input and its mirrors over depth. The mirrors are computed by 3D mirror geometry (a) as in
NeRD [26], which localizes the reflections of a given pixel directly on the image plane (b). We measure
the similarity between a pixel and its mirrors by explicitly calculating correlations which indicate towhat extent a point resembles its correspondences. We also adopt multi-stage spherical convolutions
to localize the optimal plane hierarchically on the hemisphere where all candidate planes are sampled(c). We improve both data and compute efficiency over NeRD [26] by introducing (b) and (c).

exploiting geometric priors [10, 11] where we use spherical convolutions on the hemisphere.
The hemisphere is the space containing all the candidate symmetry planes. Rather than using
a huge fully connected layer as in NeRD to locate the optimal symmetry plane, we make useof the geometric prior that the shape of the search-space is a hemisphere: We use sphericalconvolutions as a prior [11]. These two well-chosen inductive biases, are what contributes toboth the data-efficiency and computational efficiency of our model. Fig. 1 summarizes ourapproach.
Our contributions are: (1) improving the data efficiency of NeRD, the top-performing
method on 3D mirror symmetry detection from single images by introducing spherical convolutions; (2) reducing the inference latency significantly (× 20), by calculating explicit
correlations; (3) experimentally demonstrating the added-value of our improvements on two
datasets: ShapeNet [2] and Pix3D [19].

2Related work

Planar symmetry detection. A thorough overview on symmetry detection with focus on
2D symmetries, is given in [12]. Further work expands on this by including other typesof symmetries such as medial-axis-like symmetries and by adding synthetic 3D data [4].
More recently, planar symmetry detection with deep networks achieves competitive results[3, 16]. However, for planar symmetry detection objects are typically front-facing, greatly
simplifying the task. In addition, planar symmetry does not encode any 3D perspective
information. Here we differ from these works, as we focus on 3D mirror symmetry fromsingle-view images taken from any perspective.
3D mirror symmetry detection. 3D mirror symmetry is prevalent in both nature and the
man-made environments. There has been excellent research on utilizing geometric transforms for detecting mirror symmetries from 3D inputs [1, 15, 17]. A 3D Hough transformproves effective at detecting mirror symmetry planes from point clouds, in [1]. Alternatively,
planar reflective symmetry transform can find symmetry planes in 3D volumetric data [15].

3

Y. LIN ET AL.: IMPROVED 3D-MIRROR SYMMETRY LEARNING FROM A SINGLE IMAGEz

Mirror plane

Mirror planez
OriginO

Image planey(a) Scale ambiguityx(b) 3D mirror geometry

Figure 2: (a) Scale ambiguity. The two objects (in blue) only differ in scale, but their projections
on the image plane are the same. Therefore, we are unable to determine the scale of the object, or thevalue of the plane offset. In practice, we set b = 1. (b) 3D mirror geometry. 3D points X0 and X00
are symmetric with respect to the given plane ψ defined by n| x + 1 = 0. We represent X0 ∈ R4 and
X00 ∈ R4 in the homogeneous coordinate, where X00 = Mψ X0 as defined in Eq. (1). Mψ ∈ R4×4 is the
3D mirror transformation uniquely determined by the normal direction n [1, 9].

Similarly, we also focus on 3D geometric priors, and specifically on how to improve existingpriors, but instead of relying on 3D data we start from single-view images.
Recently, deep networks have been used for leveraging large datasets for learning 3D
symmetries [5, 18]. Despite being able to detect multiple symmetries, they rely on heavy
post-processing procedures to find the optimal symmetry plane. Moreover, these models
have only been tested on synthetic 3D datasets with voxelized volumes or RGB-D data (i.e.
ShapeNet [2]). In contrast, we propose improvements for end-to-end 3D mirror symmetry
detection, and test on both synthetic and real-world 2D images.
3D mirror symmetry from single-view images. A 2-stage approach can be effective for 3D
mirror symmetry detection from 2D images, by first matching image correspondences andthen applying RANSAC to identify the best symmetry plane [9]. However, this strategy is
no longer applicable in the absence of texture, or on smooth surfaces, or repetitive patterns,
because of incorrect correspondences. Rather than relying on local feature matching, NeRD
[26] makes use of neural networks to learn dense features, making it the top-performing
model. Here, we use NeRD [26] as our starting point and make two essential changes tothe model: we explicitly compute correlations between correspondences and we use sphericconvolutions to correctly localize the optimal symmetry plane on the hemisphere. These two
extensions greatly reduce computations and make our model data-efficient.

3

Revisiting 3D mirror geometry and NeRD

3.1

3D mirror geometry(i) 3D mirror planes. A plane ψ is uniquely defined by its normal direction n ∈ R3 andoffset b ∈ R as n| x + b = 0, where x ∈ R3 denotes points on the plane. However, we areunable to determine b from a single image due to scale ambiguity [14, 26]. This is because
the scene can be moved arbitrarily along the normal direction n and scaled accordingly,
without affecting the image, as shown in Fig. 2(a). Therefore, b is often set to 1 and thenormal direction n of the mirror plane is only the unknown to predict. Moreover, given thata normal direction n is equivalent to a point on a unit hemisphere, we can further define aplane as a spherical point. Thus we can sample planes on a unit hemisphere.

4

Y. LIN ET AL.: IMPROVED 3D-MIRROR SYMMETRY LEARNING FROM A SINGLE IMAGE(ii) 3D mirror transform. Fig. 2(b) shows an illustration of 3D mirror geometry for arandomly sampled plane ψ defined by n| x + 1 = 0. The corresponding 3D mirror transformation Mψ ∈ R4×4 associated to plane ψ : n| x + 1 = 0 is uniquely defined by the normal
direction of the plane n [1, 9]:
I − 2nn| −2n
X00 =
X0 ,
(1)
0
1
{z
}
|
Mψ

where X00 ∈ S and X0 ∈ S are a pair of symmetric 3D points, and S ⊂ R4 is the set of 3D
points on the object surface in homogeneous coordinates.
Given the camera intrinsic matrix K ∈ R4×4 , both X0 and X00 can be projected on theimage plane by x0 = KX0 /d 0 and x00 = KX00 /d 00 , where d 0 and d 00 are the corresponding
depths in the camera space. Thus, the constraint between points x0 and their projections x00
can be derived as:
x00 d 00 = KMP K−1 x0 d 0 ,(2)where x0 = [x0 , y0 , 1, 1/d 0 ] and x00 = [x00 , y00 , 1, 1/d 00 ] indicate the coordinates of the projected
points in the pixel space. Eq. (2) enables us to find the symmetric correspondences of every
pixel at various depths, given a sampled mirror plane, ψ.

3.2

NeRD overview

We briefly recap NeRD, the state-of-the-art model on symmetry detection, which incorporates 3D mirror geometry into learning. NeRD represents our starting point.
(i) Feature learning. NeRD first learns semantic features from RGBA images via a convolutional neural network, which results in a feature map F of size [H ×W ×C], where H, W ,C indicate height, width, and number of channels, respectively.
(ii) 3D mirroring. Given a randomly sampled plane from the hemisphere, ψ, NeRD localizes the symmetric correspondences (x0 , y0 ) for each pixel (x, y) at varying depth d ∈ Diaccording to Eq. (2), where D = {dmin + D−1(dmax − dmin )|i = {0, 1, ..., D − 1}}, dmin anddmax are the minimum and maximum depth values. Subsequently, NeRD concatenates thelearned feature at each pixel (x, y) and its correspondences (x0 , y0 ) across depth d, resulting in a 4D feature volume V of size [32 × D × H × W ]. V is further downsampled by 3D
convolutions and then flattened into a 1D vector of size 16384 for further classification.
(iii) Classification on the hemisphere. To identify the optimal symmetry plane on the hemisphere, NeRD samples a number of candidate planes ψ on the hemisphere hierarchically, andthen employs huge fully connected layers (≥ 8M parameters) to predict the likelihood of acandidate being the symmetry plane. The sampling is repeated 3 times in a coarse-to-finemanner, until the desired resolution is reached.

4

Data-efficient and fast 3D mirror symmetry detection

Built on top of NeRD [26], our model also takes as input an RGBA image and outputs
sampled planes and their associated confidence for being a true mirror plane. We choose
the plane with the highest confidence as our prediction. Although an object may admit

Y. LIN ET AL.: IMPROVED 3D-MIRROR SYMMETRY LEARNING FROM A SINGLE IMAGE

5

Figure 3: Overview. Our model follows [26] and includes three components: (i) feature extraction
and correlation calculation, (ii) 3D mirror, and (iii) plane identification by spherical convolutions. The
model first calculates intra-pixel correlations C using learned features F. Then builds a 3D correlation
volume V for each sampled plane ψ, which is then flattened as a feature descriptor. Here we incorporate explicit correlations to reduce computations compared to [26]. We additionally, adopt sphericalconvolutions on uniformly sampled planes on the hemisphere to locate the optimal plane (depicted ingreen). We highlight in blue our changes with respect to [26].

multiple symmetries, we only predict the principal mirror symmetry. We follow the designof NeRD [26], and decompose our model into three parts, as shown in Fig. 3: (i) feature
extraction and correlation calculation, (ii) 3D mirror, (iii) plane identification by sphericalconvolutions. The 3D mirror follows the original design in [26], while the other two parts
differ, as we make two essential modifications to improve both data and compute efficiency.
We detail these changes below.

4.1

Compact correlation volume

Given the learned semantic features F of size [H ×W ×C], where H, W , C indicate height,
width, and number of channels, we calculate intra-pixel correlations. We correlate all pairsof points in the [H ×W ] grid with each other by a dot product over the channel dimension.
This produces a correlation tensor C of size [H ×W × H ×W ]. C encodes the extent to whicha pixel resembles the others. Using Eq. (2), we index the correlation tensor C at C(x, y, x0 , y0 )via bi-linear interpolation, where (x0 , y0 ) are the correspondences of a point (x, y) across acandidate symmetry plane ψ, at various depths d. This results in a compact 3D correlation
volume V of size [D × H × W ], which substantially reduces the computations compared to
the 4D feature volume in NeRD [26]. We aggregate the information over theentire V, and
apply 3D convolutions to downscale V, resulting in an output tensor of size H8 × W8 × D4 .
In practice, we set H, W and C to 64.
The correlation volume V encodes the similarity between each input and its mirrors at
all sampled depths. A higher similarity indicates that the given plane ψ is more likely to bea mirror plane. We flatten the downscaled volume V to a 1D vector. Each sampled plane on
the hemisphere ψ is characterized by one such 1D vector.

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4.2

Y. LIN ET AL.: IMPROVED 3D-MIRROR SYMMETRY LEARNING FROM A SINGLE IMAGE

Spherical convolutions for symmetry plane detection

Each candidate plane is equivalent to a sampled spherical point, and has an associated 1D
feature descriptor obtained from the correlation volume V. Given that the planes lie on
the hemisphere, we take advantage of spherical convolutions to learn the most probablecandidate plane. We use EdgeConv [21] to extract features from the local neighborhood of aplane. Specifically, we treat each sampled spherical point as a node, and compute its top 16
nearest neighbors. We stack 3 layers of EdgeConv, followed by BatchNorm and LeakyReLU
activations, to guarantee a sufficiently large receptive filed.
Following [26], we also sample the planes in a multi-stage sequence by adopting the
Fibonacci lattice [6]. In practice, we sample planes over 3 stages, at multiple scales, in acoarse-to-fine manner. Pi = {nki }Kk=1 ⊂ R3 represents all sampled planes at ith stage, where
n is the normal direction, and K is the number of planes. The sampling at ith stage satisfies
Pi = {nk : arccos(|hnk , n̂i−1 i|) ≤ δi }Kk=1 , where n̂i−1 is the optimal plane from previous stageand δ controls the sampling region.
During training, we minimize the binary cross-entropy loss at each stage and averaged
over the positive and negative samples separately, due to the class imbalance. The positive
samples are the spherical points that are closet to the ground truth at each stage, while theothers are considered negative samples. At test time, we choose the plane with the highest
estimated confidence at the final stage.

5

Experimental analysis

Datasets. We conduct experiments on synthetic ShapeNet [2] and real-world Pix3D [19]datasets. For both datasets the objects are aligned to the canonical space such that the YZ plane is the 3D mirror symmetry plane. On the synthetic ShapeNet dataset, we use thesame subset as in [26] for fair comparison. Images are of size 256x256 px and split in
175,122/500/8,756 training/validation/test sets. For the real-world Pix3D dataset we preprocess the data as in [26]. We first crop the objects inside bounding boxes. And then,we rescale them to 256x256 px, and adjust the camera intrinsic matrix K accordingly. This
results in a dataset of 5,285 and 588 images for training and test respectively.
Evaluation. We follow [26] and evaluate all methods by measuring the angle differenceof the plane normals between the ground-truth and predictions in the camera space. We
calculate the percentage of the predictions that have a smaller angle difference than a given
threshold and compute the area under the angle accuracy (AA) curves.
Implementation details. The x = 0 plane in the object space is the ground truth as it isexplicitly aligned for each object [2]. We set dmin = 0.64, dmax = 1.23, and D = 64 for depth.
We perform spherical convolutions at 3 scales and sample {128, 64, 64} symmetry planes at
each scale. We set the scale factor to be δ = {90.0◦ , 12.86◦ , 3.28◦ }. We train from scratch
for each dataset on Nvidia RTX2080 GPUs with the Adam optimizer [8], for maximum 32
epochs. The learning rate and weight decay are set to be 3 × 10−4 and 1 × 10−7 . We decay
the learning rate by 10 after 24 epochs. To maximize the GPU usage, we set the batch size
to 6.
Baselines. We primarily compare with the state-of-the-art: NeRD [26]. We also implement
a standard baseline using direct regression to estimate the symmetry normal n. For thisbaseline we use a ResNet-50 [7] backbone with an L1 loss. We additionally compare with
Front2Back [24], which detects 3D mirror symmetry using a variant of the iterative closest

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Y. LIN ET AL.: IMPROVED 3D-MIRROR SYMMETRY LEARNING FROM A SINGLE IMAGE

AA on ShapeNet subsets

NeRD

Angle accuracy

80

Ours

70
60
50
40
30
20
100%
(180K)NeRD-@1
NeRD-@3
NeRD-@5

Ours-@1
Ours-@3
Ours-@5

50%
(90K)

10%
(18K)

25%
(45K)

5%
(9K)

2.5%
(4.5K)

1%(1.8K)

Proportion of training samples(a) Angle accuracy on ShapeNet subsets.

(b) Examples of prediction errors.

Figure 4: Exp 1: Data efficiency. (a) Quantitative comparison between our model and NeRD whentraining on various ShapeNet subsets. The difference of the two models is limited when the training
data is ample (e.g., 18K ≥). However, our model outperforms NeRD when training on limited data
(e.g. ≤ 9K images). (b) Qualitative comparison on the 1% subset. We project detected symmetries
on the image plane, where green is the true symmetry plane, and red indicates the prediction error.
We use the ground truth scale for plotting, to resolve the scale ambiguity. Our model makes preciser
predictions than NeRD when training on the 1% subset.

point approach. However, Front2Back requires prior depth maps, and has only been tested on
ShapeNet. We also compare with RotCon [25] which proposes a continuous representationfor estimating 3D rotation. We use the L1 loss for training and report its performance on
both datasets. We further consider DISN [23] and NCOS [20]. DISN learns 6D rotation
representation for estimating camera poses on ShapeNet. We recover the normal of themirror plane from camera poses and report the performance of their pre-trained models on
ShapeNet. NCOS defines a normalized object coordinate space (NOCS) and identifies 6D
representations of camera poses. We use NOCS to estimate object orientation in ShapeNet.

5.1

Exp 1: Data efficiency

We evaluate the data efficiency of our model by reducing the number of training samples to{50%, 25%, 10%, 5%, 2.5%, 1%} on the ShapeNet dataset, which has approximately 200K
training images in total. We train all models from scratch and compare the AA scores at
3◦ and 5◦ on the complete test set. We compare our model with NeRD which holds stateof-the-art result in Fig. 4(a), and display a few examples of detected symmetry planes inFig. 4(b). The two models have a comparable amount of parameters, thus removing theimpact of parameters. In general, our model shows superiority over NeRD with the decreaseof training samples, and this advantage accentuates on subsets with fewer than 10K images.
When training on 100% − 10% subsets we observe minimal differences, while on 10% − 1%subsets we observe a drastic difference (up to 10% in AA). This indicates that our model ismore effective at learning from limited data. Notably, our model can achieve similar results
to NeRD with only half of the training data, as seen on the 2.5% and 5% subsets, thus
demonstrating the data efficiency of our model.

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Y. LIN ET AL.: IMPROVED 3D-MIRROR SYMMETRY LEARNING FROM A SINGLE IMAGE
NeRD
Ours
DISN
RotConFront2Back

NOCS

ResNet1.0

Datasets

Percentage

0.8

ShapeNet[2]AA@1◦ AA@3◦

Metrics

0.6

ResNet [7]RotCon [25]DISN [23]NOCS [20]0.4
0.2

Front2Back [24]0.0

NeRD [26]
Ours
0

1

2

3

4

20.8
18.7
9.3
0.6
9.3
57.3
55.7

AA@5◦

55.7
54.6
26.1
7.9
26.1
75.4
75.5

69.5
69.4
34.1
17.3
34.1
80.7
82.0

5

Angular errors(a) Percentage of predictions per angular error.

(b) Angle accuracy.

Figure 5: Exp 2.1: Comparison on synthetic ShapeNet. Comparison with existing baselines ResNet[7], RotCon [25], DISN [23], NOCS [20], Front2Back [24] and the recent NeRD [26]. Our modelshows competitive results with the top-performing model.
RotConResNet

NeRD

Ours

Percentage

0.8
0.6
0.4
0.2
0.0

0

5

10

15

Datasets

Speed

Metrics

(FPS)

AA@1◦

Pix3D[19]AA@5◦

AA@10◦

ResNet [7]Rotcon [25]
NeRD [26]
Ours

66
1.4
25

1.5
2.2
22.7
27.5

12.3
14.1
46.0
53.0

23.6
25.8
55.8
62.8

20

Angular errors(a) Percentage of predictions per angular error.

(b) Angle accuracy.

Figure 6: Exp 2.2: Comparison on real-world Pix3D dataset. Our model performs the best on
this challenging real-world dataset, as the knowledge of 3D mirror no longer needs to be learned frommassive data. Moreover our model is ×20 faster than NeRD at inference time.

5.2

Exp 2: Comparison with state-of-the art

Exp 2.1: Comparison on synthetic data. In Fig. 5 we compare with the state-of-the-art
on ShapeNet. NeRD [26] has the best AA at 1◦ , when large training sets are available. Ourmodel is competitive to NeRD. The prediction error of both is less than 1◦ in 80% of thetest cases. The ResNet baseline using direct regression can only reach approximately 50%
AA at 1◦ , indicating that naive convolutions lack the ability to exploit the mirror symmetry,even with ample training data. We also notice that end-to-end approaches outperform models
relying on heavy post-processing, such as Front2Back [24].
Exp 2.2: Comparison on real-world data. To further validate the effectiveness of ourmodel, we also test on the real-world Pix3D dataset [19], as shown in Fig. 6. It is worth
noting that the prediction error on Pix3D is relatively larger than on ShapeNet. On one
hand, there is limited training data: 5, 000 images in total, which is significantly less thanShapeNet. On the other hand, the camera configuration differs from image to image, thus
making it hard to make precise predictions. Our model outperforms all the other models
consistently, thus demonstrates the superiority of our design. NeRD lags behind due to ahigh demand for training data. Moreover, our mode is ×20 faster than NeRD at inferencetime, as showing in Fig. 6(b). Please see the supplementary material for qualitative results.

Y. LIN ET AL.: IMPROVED 3D-MIRROR SYMMETRY LEARNING FROM A SINGLE IMAGE

3D Mirror

Correlation Sphericalvolumesconvolutions

9

AA@1◦ AA@5◦

a 7
7
7
0.8
9.5
b 3
3
7
15.8
44.4
c 3
7
3
8.0
31.5
d 3
3
3
22.1
53.5
Table 1: Exp 3: Ablation studies. We quantitatively verify the added value of 3D mirror geometry,
3D correlation volumes, and spherical convolutions on the ShapeNet 1% subset. All these design
choices are essential for the performance of our model.

5.3

Exp 3: Ablation studies

To verify the contribution of each component in our design, we conduct ablation studies, as
shown in Tab. 1. All models are trained on the ShapeNet 1% subset. Model (a) is a simple
baseline using direct regression and shows inferior results to the others in detecting 3D mirror
geometry. We replace the spherical convolutions in our design (d) with 1 × 1 convolutions
in model (b). Comparing (b) and (d), we find that spherical convolutions improve the results
significantly. In model (c), we replace the convolutions over the 3D cost volumes V by taking
the max over the depth dimension. By doing so, we only obtain the correspondence withthe highest correlation across different depths for each pixel, thus removing the 3D spatial
information. However, model (c) substantially underperforms model (d), thus validating thenecessity of 3D cost volumes. The ablation studies justify the added value of the 3D mirror,
correlation volumes, and spherical convolutions.

6

Conclusions and drawbacks

This paper analyzes improvements for 3D mirror symmetry detection from single-view perspective images. We explicitly incorporate feature correlations and spherical convolutions
into the state-of-the-art 3D mirror detection [26]. This provides the model with improved
data efficiency and computation efficiency. Extensive experiments on both synthetic andreal-world datasets demonstrate the benefits of our proposed changes when compared tostate-of-the-art.
One of the drawbacks of this work is that if the found point correspondences are notsufficiently similar in appearance, the mirror plane detection will be erroneous. A second
drawback is the incapability in detecting multiple symmetry planes, and this is restrictive as
certain objects may display multiple symmetries, such as local symmetries, translational androtational symmetries. In addition, mirror symmetries sometimes cannot be characterized bya single plane, such as intrinsic symmetries commonly seen in non-rigid deformable objects:
e.g. human bodies. Extending the current work for detecting multiple types of symmetries,
such as rotation symmetry and translation symmetry is a viable future direction. Anotherpromising future research direction is exploring the usage of 3D mirror in single-view 3D
reconstruction, such as weakly-supervised depth estimation and shape completion.

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