Deep vanishing point detection: Geometric priors make dataset variations vanish
Yancong Lin, Ruben Wiersma, Silvia L. Pintea, Klaus Hildebrandt, Elmar Eisemann, and Jan C. van Gemert
Delft University of Technology, the Netherlands

Abstract
Deep learning has improved vanishing point detection
in images. Yet, deep networks require expensive annotateddatasets trained on costly hardware and do not generalize
to even slightly different domains, and minor problem variants. Here, we address these issues by injecting deep vanishing point detection networks with prior knowledge. Thisprior knowledge no longer needs to be learned from data,saving valuable annotation efforts and compute, unlocking
realistic few-sample scenarios, and reducing the impact of
domain changes. Moreover, the interpretability of the priors
allows to adapt deep networks to minor problem variations
such as switching between Manhattan and non-Manhattan
worlds. We seamlessly incorporate two geometric priors:(i) Hough Transform – mapping image pixels to straightlines, and (ii) Gaussian sphere – mapping lines to great
circles whose intersections denote vanishing points. Experimentally, we ablate our choices and show comparable
accuracy to existing models in the large-data setting. We
validate our model’s improved data efficiency, robustness todomain changes, adaptability to non-Manhattan settings.

1. Introduction
Vanishing point detection in images has non-vanishingreal-world returns: camera calibration [1, 9, 21], scene
understanding [18], visual SLAM [13, 33], or even autonomous driving [28]. Deep learning is an excellent approach to vanishing point detection [7, 8, 67, 69], where allgeometric knowledge is learned from large annotated data
sets. Yet, in the real-world, there are several factors thatcomplicate deep learning solutions: (1) Manually annotating large training sets is expensive and error prone; (2)Training models on large data sets require costly computational resources; (3) Practical changes to data collection
cause domain shifts, hampering deep network generalization; (4) Slight changes in the problem setting require acomplete change in deep network architectures. Thus, there
is a need to make deep learning less reliant on data, and its
architectures more robust to variants of the same problem.
In this paper, we add geometric priors to deep vanishing

Figure 1. We add two geometric priors: (i) Hough Transformand (ii) Gaussian sphere mapping, for vanishing points detection.
We transform learned image features to the Hough domain, wherelines are mapped to individual bins. We further project the Houghbins to the Gaussian sphere, where lines become great circles andvanishing points are at the intersection of great circles. Each color
represents a set of image lines related to a vanishing point. Adding
geometric prior knowledge makes our model data-efficient, lessdependent on domain-specifics, and easily adaptable to problem
variations such as detecting a variable number of vanishing points.

point detection. Using geometric priors is data-efficient as
this knowledge no longer needs to be learned from data.
Thus, fewer annotations and compute resources are needed.
Moreover, by relying on priors, the model is less sensitive toparticular idiosyncrasies in the training data and generalizes
better to domains with slightly different data distributions.
Another advantage of a knowledge-based approach is that it
is interpretable, and thus the architecture is easy to adapt toa slightly different problem formulation.
We add two geometric priors, see Fig. 1: (i) the Hough
Transform and (ii) a Gaussian sphere mapping. Our trainable Hough Transform module represents each line as an(offset, angle) pair in line polar coordinates, allowing us
to identify individual lines in Hough space [15]. We subsequently map these lines from Hough space to the Gaussian sphere, where lines become great circles, and vanishingpoints are located at the intersection of great circles [4]. The
benefit of using great circles is that lines are mapped fromthe unbounded image plane to a bounded unit sphere, facilitating vanishing point detection outside the image view.
Both the Hough Transform and the Gaussian sphere mapping are end-to-end trainable, taking advantage of learned
representations, while adding knowledge priors.

This paper makes the following contributions: (1) we
add two geometric priors for vanishing point detection bymapping CNN features to the Hough Transform, and mapping Hough bins to the Gaussian sphere; (2) we validate ourchoices and demonstrate similar accuracy as existing models on the large ScanNet [12] and SceneCity Urban 3D [70];(3) we show that adding prior knowledge increases dataefficiency, improving accuracy for smaller datasets; (4) we
demonstrate our ability to tackle a different problem variant: detecting a varying numbers of vanishing points on
the NYU Depth [45] dataset, where the number of vanishing points varies drastically from 1 to 8; (5) we showthat adding prior knowledge reduces domain shift sensitivity, which we validate by cross-dataset testing.

2. Related work
Geometry-based vanishing point detection. Vanishingpoints occur at intersections of straight lines. Lines canbe found by contour detection [71] or a dual point-to-line
mapping [30]. The common approach, however, is using an
explicit straight-line parameterization in the Hough Transform [40, 47, 51]. We exploit this straight-line parameterization as prior knowledge in a Hough Transform module.
Combining lines to vanishing points can be done by measuring the probability of a group of lines passing through thesame point [57], voting schemes [19,64], or hypothesis testing by counting the number of inlier lines such as J-Linkage[59] used in [17, 58]. Other approaches see vanishing point
detection as a grouping problem by applying line clustering [3, 43, 53], expectation-maximization [1, 14, 27, 54], or
branch-and-bound [5, 6, 23]. While these methods work
well, they do not exploit prior knowledge of the 3D world.
A strong geometric prior for vanishing points is modeled by the Gaussian-sphere [4, 10]. A line in an image
represents a great circle on the Gaussian-sphere and the intersections of great circles on the sphere denote vanishingpoints, detected as local maxima [4, 51]. Mapping lines to
the Gaussian sphere shifts the problem from the unbounded
image plane to the constrained parameter space defined by
the Gaussian sphere [10, 41, 56]. Constraining the search
space is a form of regularization, and is particularly beneficial for limited-data deep learning. We exploit this priorknowledge by incorporating the Gaussian sphere mapping.
Learning-based vanishing point detection.
Vanishing point detection can be learned from large annotateddatasets [7, 8, 66, 67]. It is effective to split the problem
in separate stages: line detection, inverse gnomonic projection, network training and post-processing, as in Kluger et
al. [25]. Conic convolutions on hemisphere points, provided further improvements, in Zhou et al. [69]. In contrast, rather than focusing on accurate large-scale deep models, we consider challenging real-world scenarios, such as:

limited training samples, cross-dataset domain switch, andnon-Manhattan world.
Robustness to domain shifts. Classical solutions to vanishing point detection [17, 35, 55] are built exclusively onprior-knowledge. Such methods are data-free, and thus designed to work on any domain. Yet, they cannot take advantage of expressive deep-feature learning for vanishingpoint detection [38,69]. On the other hand, deep models arenotoriously sensitive to distribution shifts between training
and test [29, 39]. Active research on this includes: domain
adaptation [50, 61, 65], domain generalization [68], multidomain learning [36, 52], etc. Such solutions entail significant changes to the deep network model, adding complexity for practical real-world applications. Hence, we focus
on a single method which does not requiring large modelchanges for robustness to minor domain shifts. Our goal
is to combine the robustness of knowledge-based methods,
with the power of deep representation learning.
Manhattan versus non-Manhattan world. The Manhattan world assumes exactly 3 vanishing points. This assumption has been proven useful for orthogonal vanishing point
detection [2, 5, 44, 63]. However, the Manhattan assumption does not hold in several real-world scenarios such as
non-orthogonal walls and wireframes in man-made structures. Vanishing point detection in non-Manhattan worldis done by robust multi-model fitting [26], horizon line
detection [66], branch-and-bound with a novel mine-andstab strategy [32], Bingham mixture model fitting [31] or
non-maximum suppression on the Gaussian sphere [38].
In our work, we refrain from adding explicit orthogonality constraints, which makes our method applicable to nonManhattan scenarios as well. We rely on the Hough Transform and the Gaussian sphere to map pixel-wise representations to the entire hemisphere. And using a clustering algorithm we detect multiple vanishing points simultaneously.

3. Geometric priors for VP detection
General outline of our approach. Fig. 2 depicts the overall
structure of our model. We build on two geometric priors:(i) Hough Transform, and (ii) Gaussian sphere mapping. A
CNN learns image features, which are then mapped to a line
parameterization via Hough Transform. We project the features of parameterized lines to the Gaussian sphere where
spherical convolutions precisely localize vanishing points.
(i) Hough Transform. Similar to [69], we use a singlestack hourglass network [46] to extract image features, F
to be mapped into Hough space [37], HT . The HT spaceparameterizes image lines in polar coordinates using a setof discrete offsets ρ and discrete angles θ, defining a 2D
discrete histogram. In practice, a set of pixels (x(i), y(i))
along a line indexed by i, vote for a line parameterization to[184x180]x128[184x180]x128[32768]x64[32768]x64[128x128]x128[32768]x1[512x512]x3

Figure 2. Overview: The model starts from in input image, and predicts vanishing points on the Gaussian hemisphere by relying on two
geometric priors: (i) Hough Transform, and (ii) Gaussian sphere mapping. We use a convolutional network to learn features which are thenmapped to Hough space, where each bin is a line. We filter the Hough space and project Hough bins to the Gaussian hemisphere and apply
spherical convolutions to find vanishing points. We indicate the size of the learned features above, where the last dimension is the numberof channels. We sample 32,768 points on the hemisphere using the Fibonacci lattice [20], resulting in features maps of size 32,768. Ourmodel learns to classify spherical points as vanishing points or not using a binary cross-entropy loss. There is no intermediate supervision.

which they all belong:X
HT (ρ, θ) =F (ρ cos θ − i sin θ, ρ sin θ + i cos θ) (1)i

The Hough Transform module starts from an [H×W ] feature map F and outputs an [Nρ ×Nθ ] Hough histogram HT ,where Nρ and Nθ are the number of sampled offsets andangles in Hough Transform. We set H=128, W =128,
Nρ =184, and Nθ =180. This results in [Nρ ×Nθ ] possible line parameterizations. We find the local maxima inthe Hough domain by performing a 1D convolutions over
the offsets. This removes the noisy responses in the Hough
space, as in Fig. 2. We refer the readers to [37] for details.
(ii.1) Gaussian sphere mapping. The Gaussian sphere isa unit sphere centered at the camera origin, O. Vanishingpoints on the sphere are represented as normalized 3D line
directions δ.
Starting from a bin in the Hough domain (ρAB , θAB ),−−→corresponding to a line direction in the image plane AB,
we want to map this to the Gaussian sphere. Two image
points A and B sampled from a line represented by its HT
bin (ρAB , θAB ), together with the camera center O, form aplane ψ as depicted in Fig. 3(a). The plane ψ is describedby its normal vector:−→ −−→OA × OB→−n = (nx , ny , nz ) = −→ −−→ .
OA × OB(2)−
This normal vector →n is the only information we need to−−→map the image line direction AB to the Gaussian sphere.
The spherical coordinates (α, β) describe a point on theGaussian sphere, where α is the azimuth defined as the angle from the z-axis in the xz plane, and β is the elevation

representing the angle measured from the xz plane towards
the y-axis, as shown in Fig. 3(a). The intersection between
the plane ψ and the Gaussian sphere is a great circle. This
great circle represents the projection of the image line direc−−→tion AB on the Gaussian sphere. Intersections of multiple
great circles are potential vanishing points, see Fig. 3(b).
We compute the projection of the image line direction−−→
AB, by estimating the elevation β as a function of the az−
imuth α and the normal vector →n [51]:−β(α, →n ) = tan−1−nx sin α − nz cos α
,
ny

(3)

where we uniformly sample α in the range [0, π).
Because the Gaussian sphere is symmetric we only needa hemisphere. We sample N points on the Gaussian hemisphere using a Fibonacci lattice [20] and then project lines,
corresponding to bins in the Hough space, to these N sampled sphere points. For each line parameterization in Hough−
space (ρ, θ), we first compute its normal vector →n . We,then, estimate its corresponding (α, β) spherical coordinates using Eq. (3). We subsequently assign each (α, β)pair to its nearest neighbor in the sampled points from the
Fibonacci lattice, by computing their cosine distance. To
parallelize this process, we precompute the projection of all
Hough line parameterizations onto the sampled sphere locations. This mapping is stored in an [Nρ ×Nθ ×M ] tensor,
where [Nρ ×Nθ ] is the number of line parameterizations in
Hough space and M is the number of sampled azimuth angles α. We set N =32,768 and M =1,024.
(ii.2) Spherical convolutions on the hemisphere. We employ spherical convolutions to predict vanishing points. We
treat the points sampled on the hemisphere as a point cloudand use EdgeConv [62] to convolve over the hemisphere.
EdgeConv operates on a k-nearest neighbor graph on the

y

A

U

B

z

O
V
x

(a) The Gaussian sphere(b) Vanishing points on the Gaussian sphere

Figure 3. Gaussian sphere representations for vanishing points [4, 51]: (a) The Gaussian sphere is a unit sphere located at the camera
center, O. Points on the sphere are encoded by two angles: (α, β) the azimuth and the elevation, respectively. A line segment AB in
the image plane together with the camera center O forms a plane ψ, highlighted in blue. To define the mapping from the image to the→sphere, we only need to know the normal −n to the plane ψ. (b) Image lines are projected as great circles on the sphere. The intersectionof multiple great circles on the sphere represents a vanishing point.

Figure 4. Spherical convolutions on the hemisphere. We use
EdgeConv [62] for precise vanishing point localization on theGaussian sphere. The concatenation of the previous feature maps
is fed into the final layer to produce a prediction.

points. It learns to represent local neighborhoods by applying a non-linear function to the neighbors’ features, andthen aggregates those features with a symmetric operator.
The neighbors’ features are localized by subtracting the features of the centroid. Like [62], we take the per-feature
maximum over the neighbors to aggregate edge features.
As shown in Fig. 4, the spherical part of our model contains 5 EdgeConv modules [62]. Each EdgeConv module transforms neighboring features with a fully connected
layer, a BatchNorm layer [22] and a LeakyReLU activation.
We use N =32,768 nodes on the hemisphere and computethe 16 nearest neighbors for each node. We concatenate thefeatures maps from previous layers and feed them into thelast EdgeConv layer to produce the final prediction.
Model training and inference. We train the model using
the binary cross-entropy loss. For each annotated vanishingpoint, we label its nearest neighbor in the sampled points
as +1 and the others as 0. Because the number of positive
samples is considerably lower than the negative samples,
we compute two separate average losses over the positive
and the negative samples, and then sum these. There is nointermediate supervision or guidance.
During inference, we use DBSCAN [16,49] to cluster all
points on the Gaussian sphere based on the cosine distance.

Figure 5. Multi-scale sampling on the hemisphere. We sample points at three scales for detecting vanishing points in the
Manhattan world, as in [69]. Blue indicates sampling at the firstscale, while green indicates fine-grained sampling at the following
scales. The red crosses are the predictions at each scale.

The eps parameter of DBSCAN [49] is set to be 0.005. The
point with the highest confidence in each cluster is the prediction. We rank all predictions by confidence.
Multi-scale sampling with the Manhattan assumption.
For the Manhattan world, we know beforehand that there
are only 3 orthogonal vanishing points, therefore in this scenario, we can use a multi-scale sampling strategy to reduce
computation, as in [69]. Here, we sample points and apply spherical convolutions at 3 scales: δ ≈ {90◦ , 13◦ , 4◦ }and N = {512, 128, 128}, where δ controls the samplingradius and N indicates the number of sampled points respectively. Fig. 5 displays the multi-scale sampling. The
spherical convolution networks share the same architecture
while processing different number of samples. We provide
details in the supplementary material.

4. Experiments
Datasets. We evaluate on three datasets following the Manhattan world assumption: SU3 (SceneCity Urban 3D) [70],ScanNet [12], YUD [14], as well as the NYU Depth [45]dataset which does not follow the Manhattan world assumption. The SU3 dataset contains 23K synthetic images,

0.9

Ablation study: AA Curves on ScanNet

HT (# of angles)

Sphere (# of points)

0.8

90

180

8K

16K

32K

77.1

79.3

73.3

77.4

79.3

◦

AA@3

Percentage

0.7
0.6
0.5
0.4

1. LSD + Sphere-NMS
2. LSD + Sphere-Convs
3. Canny-edge + HT-Conv + Sphere-Convx4
4. Conv + HT+NMS + Sphere-Convx4
5. Conv + HT-Conv + Sphere-Convx4 (Ours)
6. Conv + HT-Conv + Sphere-Convx2
7. Conv + HT-Conv + Sphere-Convx1
8. Conv + HT-Conv + Convx1

0.3
0.2
0.1
0.0
0.0

2.5

5.0

7.5

10.0

12.5

15.0

Angle difference

17.5

20.0

Figure 6. Exp 1: Model choices. We show the effects of the twogeometric priors quantitatively on the ScanNet-1% subset. Adding
HT layers and spherical convolutions outperforms the baselines,
thus demonstrating the effectiveness of geometric priors.

which are split into 80%, 10% and 10% for training, validation and testing respectively. The ScanNet has more than
200K real-world images, among which 189,916 examples
are used for training. The “ground truth” VPs are estimatedfrom surface normals as in [69], thus being less precise thanother datasets. In the NYU Depth dataset, the number ofvanishing points varies from 1 to 8 across images, making
it more challenging. The NYU Depth dataset has 1,449 images, approximately ×200 and ×20 smaller than the ScanNet dataset and the SU3 dataset, respectively, further increasing the difficulty of training CNN models. We additionally demonstrate the effect of geometric priors on thesmall-scale YUD dataset with only 102 images. Detailed
comparisons are in the supplementary material. Unlessspecified otherwise, we use the ground-truth focal length
on SU3, ScanNet and YUD for the Manhattan assumption.
Evaluation. On the SU3, ScanNet and YUD datasets (Manhattan assumption), we evaluate the angle difference between the predicted and the ground-truth vanishing points
in the camera space, as in [26, 38, 69]. We then estimate
the percentage of the predictions that have a smaller angle
difference than a given threshold and compare the angle accuracy (AA) under different thresholds, as in [38, 69]. We
use the ground-truth focal length to exploit the orthogonal
constraint. On the NYU Depth dataset we follow [26] andfirst rank detected vanishing points by confidence, and thenuse the bipartite matching [11] to calculate the angular errors for the top k predictions. After matching, we generate
the recall curve and measure the area under the curve (AUC)
up to a threshold, e.g. 10◦ .
Baselines. We compare our model with J-Linkage [17],Contrario-VP [55], Quasi-VP [34, 35], NeurVPS [69] andCONSAC [26] on SU3, ScanNet and YUD. On the nonManhattan NYU Depth dataset we only compare with J-

Table 1. Quantization analysis on SU3-10% subset. Denser
samplings improve performance. In practice, we uniformly sample 180 angles from [0, π) for HT, and 32K points on the sphere.

Linkage, T-Linkage [42], CONSAC and VaPid [38], as
the other models rely on the Manhattan assumption. J/TLinkage, Contrario-VP and Quasi-VP are non-learning
methods, employing line segment detection [60]. NeurVPS
and our model are end-to-end trainable, while CONSAC
needs line segments as inputs. We follow the official implementations and use the default hyperparameters to reproduce all results. We do not consider the baselines [31,38,64]
due the lack of code/results on certain datasets.
Implementation details. We implement our model in Pytorch [48], and provide the code online 1 . Our models aretrained from scratch on Nvidia RTX2080Ti GPUs with theAdam optimizer [24]. The learning rate and weight decay
are set to be 4 × 10−4 and 1 × 10−5 , respectively. To
maximize GPU usage, we set the batch size to 4 and 16
when using multi-scale sampling. On the SU3 and NYU
Depth datasets, we train the model for a maximum of 36
epochs, with the learning rate decreases by 10 after 24 training epochs. On the ScanNet dataset, we train for 10 epochs
and decay the learning rate by 10 after 4 epochs. On the
YUD datset we pre-train on SU3.

4.1. Exp 1: Evaluating model choices
We evaluate on a subset of ScanNet containing 1% of
the data, and provide the results in Fig. 6. Model (1) is
a non-learning baseline using a classic line segment detector (LSD) [60] and non-maximum suppression (NMS) on
the sphere. Model (2) replaces the NMS with sphericalconvolutions, but still shows inferior result as LSD fails
to detect reliable line segments. Model (3) combines a
Canny-edge detector, Hough Transform and spherical convolutions. Comparing (3-5) indicates the added value oflearning semantics from images, rather than using classicedge detectors. Comparing (4-5) shows the effectivenessof backpropagating through Hough Transform. Comparing(5-8) exemplifies the added value of spherical convolutions.
Our method combines both classical and deep learning approaches into an end-to-end trainable model.
We also evaluate the impact of quantizations numericallyon the synthetic SU3-10% subset, which contains precise
VP annotations, thus making quantization a crucial factor.
As shown in Tab. 1, fine-grained sampling is essential for abetter result.
1 https://github.com/yanconglin/VanishingPoint_
HoughTransform_GaussianSphereDatasets

SU3 [70]

ScanNet [12]Params

FPS

AA@3

AA@5

AA@3

AA@5

AA@10

AA@3

AA@5◦

AA@10◦

J-Linkage† [17]

—
—
—
0.2 M
22 M
7M
5M
7M

1.0
0.6
29.0
3.0
0.5
5.5
23.0
5.5

82.0
64.8
75.9
86.3
93.9
84.0
84.8
81.7

87.2
72.2
80.7
90.3
96.3
90.2
90.7
88.7

15.7
12.0
14.7
15.8
24.0
24.8
22.9
22.2

27.3
21.4
25.3
24.6
41.8
42.1
39.8
38.8

43.0
35.3
39.4
36.0
64.4
63.7
62.4
59.9

60.8
58.6
58.6
61.7
52.4
60.7
59.5
59.1

71.8
70.7
61.0
73.6
64.0
74.3
72.6
72.6

81.5
81.8
74.0
84.4
77.8
86.3
85.4
84.6

Quasi-VP [34]CONSAC† [26]
NeurVPS [69]
Ours
Ours∗
Ours†

◦

◦

◦

YUD [14]

Metrics
Contrario-VP† [55]

◦

◦

◦

Table 2. Exp 2: Manhattan world. Angular accuracy on SU3, ScanNet and YUD datasets. Ours achieves the best results on the the
YUD dataset, and is competitive on the larger ScanNet and SU3 datasets. Ours∗ adopts the multi-scale sampling strategy, thus being
significantly faster. † assumes unknown focal length, thus making the Manhattan assumption no longer applicable. Ours† shows a constantdecrease over Ours across datasets, indicating the usefulness of the orthogonal constraint. Supplementary material provides qualitativevisualizations. We conclude that adding priors does not reduce accuracy in the large scale setting.

1.0

0.6
0.4
0.2
0.0
0.0

Contrario-VP
1.0

0.6
0.4

1.0

1.5

2.0

Angle difference

2.5

3.0

0.0
0.0

0.6
0.4
0.2

0.2

0.5

NeurVPS-22M

AA Curve on YUD dataset

0.8

0.8

Percentage

Percentage

0.8

CONSACAA Curve on ScanNet dataset

Percentage

1.0

J-Linkage

Quasi-VP

Ours-7M

AA Curve on SU3 dataset

2.5

5.0

7.5

10.0

12.5

15.0

Angle difference

17.5

20.0

0.0

0

2

4

6

Angle difference

8

10

Figure 7. Exp 2: Manhattan world. AA curves on on the ScanNet, SU3 and YUD datasets containing 3 orthogonal vanishing points.
Learning-based approaches outperform methods relying purely on line segments and grouping, validating the power of representation
learning. Our model shows comparable results to the best performing NeurVPS on ScanNet, while using 3× less parameters. On thesmaller YUD dataset, our model slightly exceeds state-of-the-art. Generally, with ample data, our approach is comparable to others.

4.2. Exp 2: Validation on large datasets
We validate that adding prior knowledge does not deteriorate accuracy when there is plenty of data. We compare to five state-of-the-art baselines [17, 26, 34, 55, 69] on
the ScanNet, SU3 and YUD datasets. On the ScanNet andSU3 datasets, we train all learning models from scratch on
the full training split. On the YUD dataset, we use thepre-trained models on SU3 without fine-tuning. For CONSAC and J-Linkage, we select top 3 nearest neighbors to theground truth. We also measure the inference speed on a single RTX2080 GPU. Multi-scale sampling Ours∗ achieves
23 FPS, a large speedup over the vanilla design, as we utilize the orthogonality for efficient sampling.
Tab. 2 shows the AA scores on the ScanNet, SU3 andYUD datasets, while Fig. 7 depicts AA curves for varying
angle differences. The SU3 dataset is easier as most images contain strong geometric cues (e.g. sharp edges andcontours); this is no longer the case in the ScanNet dataset.

The prediction error on the more realistic ScanNet dataset issignificantly larger for all methods. On the ScanNet dataset,
NeurVPS and our model are visibly better than methods relying on predefined line segments as inputs. The main advantage of NeurVPS and our model is their ability to learn
useful feature representations directly from images. On theSU3 dataset, NeurVPS exceeds the other methods in the
low-error region (from 0◦ to 1◦ ). J-Linkage, Quasi-VP andCONSAC have similar results, and all of them stabilize at
1◦ . On SU3 our model is less accurate in 0◦ -1◦ , yet it compensates at ≥ 1◦ . Our inferior performance in 0◦ -1◦ range
results from the quantization errors in Hough Transformand the Gaussian sphere mapping. On the small-scale YUD
dataset [14], our model achieves comparable accuracy without fine-tuning, and exceeds the other methods in the ≥ 2◦
area, indicating the generalization ability of our model inthe small data regime. We conclude that our model using
prior knowledge performs similar to existing solutions.

AA @10 on ScanNet subsets

20
100%
10% 5% 2.5% 1%(200K)(20K) (10K) (5K) (2K)
Proportion of training samples

AA @10 on SU3 subsets
100
95
90
85
80
75
70100% 50% 25% 10% 5% 2.5% 1%(20K)(10K) (5K) (2K) (1K) (0.5K)(0.2K)
Proportion of training samples(a) Data efficiency on ScanNet(b) Data efficiency on SU3

50

Percentage

Percentage

60
40
30

Ours-7M
NeurVPS-7M
NeurVPS-22M

Quasi-VP
J-Linkage

CONSAC
Contrario-VP

Figure 8. Exp 3.(a): Reduced data. We report AA @10◦ on various subsets of the ScanNet and SU3 datasets, and indicate the number ofparameters in the legend. On the ScanNet dataset, we outperform other methods on the 10%, 5% and 2.5% subsets. Our model degrades
gracefully when reducing the training samples from 20K to 1K on the SU3 subset, while NeurVPS has a drastic drop in accuracy. CONSAC
achieves top results on SU3 due to pre-extracted line segments, but fails on ScanNet because of inaccurate line detection. There is a similar
trend for the baselines relying on line segment detection. Our model predictions are stable having small variances (±0.50 and ±0.43 on
the 1% subsets of ScanNet and SU3 respectively) across 3 repetitions. This experiment validates the data efficiency of our model.

4.3. Exp 3: Challenging scenarios
4.3.1

Datasets

Exp 3.(a): Reduced data

We evaluate data efficiency by reducing the number oftraining samples to {10%, 5%, 2.5%, 1%} on the ScanNet
dataset, resulting in approximately 20K, 10K, 5K and 2K
training images. Similarly, we also sample the SU3 dataset
into {50%, 25%, 10%, 5%, 2.5%, 1%} subsets. We train alllearning models from scratch using the default hyperparameters on each subset.
In Fig. 8 we compare the AA scores at 10◦ with stateof-the-art methods. We use our vanilla design without themulti-scale sampling speedup. The first thing to notice isthat non-learning methods are robust to data reduction. Yet,
non-learning methods cannot take any training data into account, and thus they do not perform as well when more data
is available, as we validated in the previous experiment. On
the ScanNet dataset, our model visibly exceeds the othermethods on the 10%, 5% and 2.5% subsets. In comparison, NeurVPS suffers from large accuracy decreases on
small training data subsets. When decreasing the number ofsamples to 2K (1% subset), we still achieve competitive accuracy when compared to the non-learning methods, while
NeurVPS fails to make reasonable predictions due to thelack of data. This shows the capability of our model to learn
from limited data, thanks to the added geometric priors.
The NeurVPS model has ×3 more parameters than ourmodel due to its fully-connected layer with 16M parameters. For fairness, we also consider ‘NeurVPS-7M’ with
reduced fully-connected layers, having a similar number of

NYU Depth [45]top-k = #gt

top-k = #pred

AUC

◦

@5

◦

@10

@5◦

@10◦

J-Linkage [17]T-Linkage [42]CONSAC [26]CONSAC [26]+DLSD [37]VaPiD [38]
Ours

49.30
43.38
49.46

61.28
58.05
65.00

54.48
47.48
54.37

68.34
64.59
69.89

46.78

61.06

49.94

65.96

55.92

69.10
69.57

57.19

71.62

Table 3. Exp 3(b): Non-Manhattan scenario. We report AUC
scores on the NYU Depth dataset. Here “top-k = #gt” indicates thek most confident predictions where k is the number of annotatedinstances [26], while for “top-k =#pred” all predictions are used
for evaluation. Our model exceeds state-of-the-art when detecting
a varying number of vanishing points.

parameters with our model. Both NeurVPS variants perform similar on various subsets. On the SU3 dataset theaccuracy of NeurVPS decreases significantly when reducing the training dataset size, despite its superiority on thelarge training subsets. In comparison, our model degrades
gracefully when training data decreases from 20K to 1K.
Notably, on the 1% subset, with only 200 images for training, we are still able to achieve comparable performancewith non-learning methods.

AUC on NYU Depth dataset(number of VPs: 1-8)

0.9Synthetic - real-world data
Train

0.8
0.7

RecallOurs

AA@3◦ 15.2
AA@5◦ 25.9
AA@10◦ 39.5

0.5
0.4

YUD [14]

NeurVPSCONSAC

Ours

NeurVPSCONSAC

11.1
20.3
35.5

10.1
17.3
27.2

60.7
74.3
86.3

53.8
65.6
79.7

61.7
73.6
84.4

Real-world data

0.3

CONSAC+LSDCONSAC+Deep LSD
J-Linkage
T-Linkage
Ours

0.2
0.1
0

2

4

6

Angle difference

8

Train

10

Figure 9. Exp 3.(b): Non-Manhattan scenario. We plot the recall curve on the non-Manhattan NYU dataset. Our method outperforms state-of-the-art, illustrating the ability of the model todetect a varying number of vanishing points. See supplementarymaterial for qualitative visualizations.

4.3.2

ScanNet [12]Models0.6

0.0

SU3 [70]

Test

NYU Depth [45]Test

ScanNet [12] (AA)ModelsOurs

CONSACOurs

CONSACOurs

CONSAC

◦

33.6

30.3

83.2

82.7

71.4

75.0

@10

YUD [14] (AA)YUD+ [14] (AUC)

Table 4. Exp 3.(c) Cross-dataset domain switch. “Train” and“Test” specify the training and test datasets. CONSAC uses preextracted lines, thus being accurate on YUD/YUD+. However,its accuracy is lower on ScanNet due to the lack of reliable lines.
In comparison, Ours is more accurate on both ScanNet and YUD
without tuning. Our geometric priors improve the transferabilityof the model across datasets.

Exp 3.(b): Non-Manhattan scenario

We compare with state-of-art methods in a more realistic
non-Manhattan scenario with limited annotated data. CONSAC [26] uses the line segment detection in [60]. We also
consider a variant of CONSAC with the more recent line
segment detector in [37].
Fig. 9 and Tab. 3 display the recall curve and the AUC
values on the NYU Depth dataset, respectively. Our model
consistently outperforms state-of-the-art baselines, and theimprovement is more pronounced for larger angular differences. Although achieving the second-best result, VaPiD[38] assumes a constant number of instances and requires
non-maximum suppression, which often results in over- andunder-prediction. Our model outperforms existing methods
by exploiting geometric priors, while not limiting the number of vanishing points detected.

4.4. Exp 3.(c): Cross-dataset domain switch
We conduct cross-dataset test on multiple datasets, as
displayed in Tab. 4. We compare with NeurVPS and CONSAC, which achieve top accuracy on individual datasets.
When generalizing from synthetic dataset to the real-world(e.g. from SU3 to YUD), our model shows comparative results to CONSAC, which relies on prior line segment detection, making it robust to domain shifts. We observe a similar trend on real-world datasets (e.g. from NYU to YUD).
However, on the challenging ScanNet dataset, Ours exceeds
CONSAC, indicating the advantage of learning semantics
over using pre-extracted lines. In contrast, NeurVPS does
not transfer well to another dataset. This validates the robustness of the two priors in tackling domain shifts.

5. Conclusion and limitations
This paper focuses on vanishing point detection relying on well-founded geometric priors. We add two geometric priors as building blocks in deep neural networks
for vanishing point detection: Hough Transform and Gaussian sphere mapping. We validate experimentally the added
value of our geometric priors when compared to state-ofthe-art Manhattan methods, and show their usefulness on realistic/challenging scenarios: with reduced samples, in the
non-Manhattan world where the challenge is to predict avarying number of vanishing points without the orthogonality assumption, and across datasets.
Limitations. Despite of these improvements, our model
also has several limitations. We pre-compute offline themapping from images to the Hough bins and to the Gaussian sphere by fixing the size of the Hough histogram, as
well as the Fibonacci sampling. However, these samplings
introduce quantization errors which set an upper bound on
accuracy. This is the primary reason for the limited accuracy on the SU3 dataset in the low-error region. A future
research avenue is exploring an analytical mapping fromimage pixels to the Gaussian sphere. In addition, our model
still relies on hundreds of fully labeled samples for training.
One might consider testing the added geometric priors in an
unsupervised or weakly-supervised setting.

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