Supplementary Material for
SMD-Nets: Stereo Mixture Density Networks
Fabio Tosi1
Yiyi Liao2
Carolin Schmitt2
Andreas Geiger2
1
Department of Computer Science and Engineering (DISI), University of Bologna
2
Autonomous Vision Group, MPI-IS / University of Tübingen1{fabio.tosi5}@unibo.it 2 {firstname.lastname}@tue.mpg.de

Abstract
In this supplementary document, we first provide details of the new high-resolution stereo datasets in Section 1. Next,
we present implementation details regarding training protocols and network architectures in Section 2. Finally, we show
additional qualitative and quantitative results as well as computational cost analysis in Section 3.

1. Dataset Description
In this section, we provide details of the proposed high-resolution stereo datasets, including the synthetic dataset UnrealStereo4K and the real-world active dataset RealActive4K.

1.1. UnrealStereo4K
We create our synthetic dataset, UnrealStereo4K, using the popular game engine Unreal Engine combined with the opensource plugin UnrealCV [9].
Binocular Stereo: Our binocular UnrealStereo4K dataset consists of images of 8 scenes, including indoor and outdoor
environments. We set the baseline to 20cm for indoor environments and 50cm for outdoor environments. We render 1000
stereo pairs at 3840 × 2160 resolution for each scene and divide them into 7720 training pairs, 80 validation pairs and 200 indomain test pairs. We further render 200 out-of-domain test pairs from an unseen scene to evaluate the generalization abilityof our method. For each stereo pair, we randomly sample a camera pose disabling the camera in-plane rotation and performa sanity check to reject invalid camera poses. Specifically, we filter out images according to a minimum depth of 0.25m toavoid viewpoint obstruction. We also filter out invalid stereo pairs if the RGB images are too dark according to their averageintensity. We further check the gradient of the disparity map to keep a stereo pair only if it contains non-trivial geometry
structure. Fig. 1 shows some examples of our binocular UnrealStereo4K dataset corresponding to different rendered scenes.
Active Depth: For the active monocular UnrealStereo4K dataset we render 4 scenes using the intrinsic matrix of the IR
camera on our structured light sensor. As our sensor is designed to be operated in indoor environments with a close working
range, we randomly sample camera poses such that the synthetic active dataset contains larger disparity values. We then warp
the reference dot pattern to each image to simulate the IR camera. We further convert the original RGB image to gray-scaleand combine it with the warped dot pattern to simulate the ambient illumination, following [10]. Both our sensor and thereference dot pattern image are described in Sec. 1.2. The active dataset consists of 3856 training images, 40 validation
images, 100 test images. Fig. 2 shows an example of the synthetic active image, the reference dot pattern and the groundtruth disparity map.
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(a) In-domain scenes(b) Out-of-domain scene
Figure 1: Examples of the Binocular UnrealStereo4K Dataset. We show the left image of each stereo pair on the left andthe corresponding ground truth disparity map on the right.

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Input Image

Reference Dot Pattern

Ground Truth Disparity

Figure 2: Example of the Active UnrealStereo4K Dataset. We warp the reference dot pattern to the reference image andcombine it with the gray-scale image to simulate the IR camera. The reference dot pattern is a constant image of the laser
projector and serves as the right image. The ground truth disparity map is in the coordinate of the reference image.

1.2. RealActive4K
Our real-world active dataset consists of 2570 images of an indoor room captured with a Kinect-like structured light sensor.
The sensor has two main components. The first is a high-resolution monochrome camera (4112 × 3008 pixels) with an IR
filter in front which we refer to as “IR camera”. The second is a powerful IR laser projector with a Diffractive Optical Element(DOE) that casts 200, 000 random points into the scene. We calibrate our system geometrically by estimating the intrinsic
camera matrix, lens distortion and the baseline between camera and projector. Further, we need to capture the reference dot
pattern of the projector. For that purpose, we model the laser projector as a virtual camera that observes a constant dot patternimage which we can calculate from observations of a flat wall. Hereby, we assume the virtual camera to share the intrinsic
matrix of the IR camera for simplicity.
We create the dataset by undistorting and rectifying the raw captures and also adjusting the intensity distribution ofeach image to fit the image statistics of the synthetic active UnrealStereo4K dataset. To obtain pseudo-ground truth asco-supervision during training, i.e. for adapting the features of the stereo network to the real captures, we perform Block
Matching with left-right consistency check. We divide the resulting real-world active dataset into 2500 images for training,
20 for validation and 50 for testing. An example of the real active capture is shown in Fig. 3. For our experiments on realdata, we jointly train on real and synthetic data as described below.

Input Image

Block Matching Disparity

Figure 3: Example of the RealActive4K Dataset. On the left is the captured active image and the right shows the Block
Matching disparity map used as pseudo-ground truth for training.

2. Implementation Details
We present details regarding the sampling strategy, stereo super-resolution, architecture designs and training/inference
choices in this section.
32.1. Sampling Strategy
In Fig. 4, we show the effect of different dilation kernel sizes ρ×ρ applied to the boundary mask of a ground truth disparity
map. We sample points based on the dilated boundary masks when the Depth Discontinuity Aware (DDA) sampling strategy
is used during training.

Ground Truth Disparity

Boundary Mask (ρ = 0)

Boundary Mask (ρ = 10)

Boundary Mask (ρ = 20)

Figure 4: Boundary Masks on the UnrealStereo4K Dataset using different size of dilation kernels. During training, given
the total number of training points N , we randomly and uniformly select N/2 points from the domain of all pixels belonging
to depth discontinuity regions (white) and N/2 points uniformly from the continuous domain of all remaining pixels (black).

2.2. Stereo Super-Resolution
Our continuous formulation allows us to exploit ground truth at higher resolution than the input I, which we refer to asstereo super-resolution. Specifically, let us assume that a ground truth disparity map at resolution W 0 × H 0 pixels is available
for the input image I with size W × H, where W 0 = sW , H 0 = sH and s >= 1. Given a 2D location x sampled from thecontinuous image domain [0, W − 1] × [0, H − 1], we retrieve its disparity at sx from the ground truth disparity map using
nearest neighbor interpolation. We scale the interpolated disparity down by a factor of s such that the disparity range aligns
with the input image.

2.3. Architecture
We now provide additional architectural details regarding the proposed SMD-Nets on different 2D and 3D stereo backbones. In particular, we follow the original implementation of three stereo networks: U-Net [3], PSM [1] and HSM [12].
While in the main paper we describe a stereo backbone that generates a feature map at W × H × D resolution for simplicity,
our model is able to exploit features from different resolutions/layers in practice. In our experiments, we adopt a different
combination of feature maps depending on the specific backbone ψθ in order to capture both local and global context.
PSM: For the PSM [1] stereo backbone, we use feature maps of the reference image computed using Spatial Pyramid
Pooling (SPP) modules as well as stereo matching cost probabilities extracted from stacked hourglass 3D convolutions on theconstructed cost volume. More specifically, we perform bilinear interpolation on convolutional features of size 41 W × 14 H ×32
and the matching cost probabilities of dimension 41 W × 14 H ×64. Such design choice results into D = 96, the input dimensionof our SMD Head. Notice that the softargmax operation on the cost volume, adopted as the last step of the original PSM
network to obtain the final disparity value, is not needed as the proposed SMD Head directly regresses the parameters of thedisparity distribution.
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HSM: For the HSM [12] stereo backbone, we use multi-level features extracted from the reference image by the Feature
Pyramid Encoder, which consists of an encoder-decoder architecture with skip connections and multiple SPP modules. Similarly to PSM, we concatenate features with stereo matching cost probabilities computed on a single low resolution cost
1
1
1
1
W × 16
H × 128, 32
W × 32
H × 128
volume. Specifically, we interpolate features at 12 W × 12 H × 32, 18 W × 18 H × 64, 16
1
1
resolution and matching cost probabilities of dimension 8 W × 8 H × 16. This results into D = 368.
U-Net: For the U-Net [3] stereo and monocular backbone, we simply concatenate features of the last 4 levels of the decoder
in order to exploit coarse-to-fine information. In particular, we bilinear interpolate feature maps of size W × H × 16,
1
1
1
1
1
1
2 W × 2 H × 32, 4 W × 4 H × 64 and 8 W × 8 H × 128. This results into D = 240.

2.4. Training
Here, we report the training details for each dataset and different models. We implement our framework in PyTorch [8]and use Adam with β1 = 0.9 and β2 = 0.999 as optimizer [5]. We train all models using a single NVIDIA V100 GPU.
We scale the ground truth disparity to [0, 1] for each dataset for numerical stability. We predict µ1 , µ2 in the range of [0, 1]
accordingly. π is in [0, 1] by definition. We also predict b1 , b2 in [, 1], where we set  = 1e−2 . Therefore we adopt a sigmoid
function as activation of the last layer of our SMD Head and clip b1 and b2 to be larger or equal to .
UnrealStereo4K: For the UnrealStereo4k dataset, we use the entire training set (7720 stereo pairs) for training and evaluate
on both in-domain (200 stereo pairs) and out-domain (200 stereo pairs) test sets. We scale the ground truth disparity to [0, 1]
dividing by Dmax for numerical stability, where we set Dmax = 256. We train all networks end-to-end from scratch for
200 epochs using a batch size 12 for HMS and U-Net backbones, and 60 epochs with batch size 4 for PSM due to memory
limitations. We experimentally find that PSM converges faster and thus requires less training epochs.
During training, we use random crops of size 512 × 384 from input images at 960 × 540 while keeping the ground truthat the original full resolution 3840 × 2160. From each crop we sample 50, 000 training points, regardless of the samplingstrategy. Moreover, we perform chromatic augmentations on the fly, including random brightness, gamma and color shifts
sampled from uniform distributions ([0.5, 1.5] for brightness, [0.8, 1.1] for gamma and [0.8, 1.2] for contrast) to both thereference and the target images. We further augment the training procedure by flipping the two images horizontally andvertically while adapting the ground truth disparities (left and right) accordingly.
For the active monocular dataset we need to adjust some training details since the projected dots get indistinguishable
from one another or dissolve to noise when using a low-resolution input (e.g. 960 × 540). Therefore, we use random cropsof 1024 × 768 pixels from input images of size 2056 × 1504, set Dmax to 786 and choose batch size 4 to stay in memory
limits. We also do not use any chromatic augmentations on the monochrome input images. Furthermore, we follow [10]and remove intensity variations caused by different reflection properties of the materials in the scene. Specifically, we apply
local contrast normalization (LCN) to the monochrome input images and concatenate both the original and the LCN image
as input left image for the network. The right image is given by the pre-calibrated constant reference dot pattern image. See
Fig. 2 for an example. Apart from those adjustments, the training and testing procedure is the same as for the binocular stereo
dataset.
KITTI 2015: For KITTI, we fine-tune the models pre-trained on UnrealStereo4K on 160 stereo pairs of the KITTI 2015
training set [7]. For this, we use random crops of size 512 × 256 from the original KITTI resolution while scaling the
ground truth using Dmax = 256 as scale factor. Since the provided ground truth disparities are sparse, we rely on the naı̈ve
random sampling strategy to train our models. Similarly to UnrealStereo4K, we set 12 and 4 as batch size for HSM and PSM
respectively and train them for 1000 and 500 epochs. Then, we validate all models on the remaining 40 images of the KITTI
2015 training set. Similarly to [2], we use the model based on the PSM backbone with the best performance on validation set
for submission to the official online benchmark.
RealActive4K: We fine-tune active depth models jointly on the real-world RealActive4K captures and the synthetic imagesof the active monocular UnrealStereo4K data. This allows our network to (i) accurately reason about object boundaries
present in the synthetic data, (ii) adapt its features to both synthetic and real inputs and (iii) learn from a larger and morediverse set of training images compared to the relatively small amount of real images in the RealActive4K dataset.
Since the pseudo-ground truth obtained from Block Matching does not provide reliable samples near disparity edges,
we avoid any possibly wrong disparity estimates by eroding the image areas with positive Block Matching predictions by aconservative margin. Then we use the naı̈ve random sampling strategy on the real-world captures. To contrast this lack ofsamples near disparity boundaries, we use DDA sampling on all synthetic input images. We fine-tune all networks for 100
epochs.
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Left Image

Right Image

µ1

µ2

π

1−π

b1

b2

h

dˆ

Figure 5: Bimodal Output Representation. We visualize a stereo pair of the proposed UnrealStereo4K dataset, the estimatedfive-dimensional vector (π, µ1 , b1 , µ2 , b2 ), the uncertainty h computed by evaluating the differential entropy of the continuous
ˆ
mixture distribution and the final disparity map d.
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2.5. Inferenceˆ which is scaled back to
During testing, we select the mode with the highest density value as the predicted disparity d,
the original disparity range by multiplying Dmax . Given an (arbitrary) output resolution, we leverage our model to predict
disparity values at each pixel. In Fig. 5 we visualize all the estimated quantities of our bimodal formulation.

3. Additional Experimental Results
In this section, we first present additional quantitative results using the HSM backbone, for both the ablation study of theoutput representation and the evaluation on the KITTI 2015 dataset. Next, we provide extensive qualitatively evaluations on
all datasets considered in the main paper. We further analyze the computational cost of our method.

3.1. Quantitative Results
Output Representation: Tab. 1 is provided as a complement to Tab. 1 in the main paper for an extensive evaluation. Morespecifically, we report results on the binocular UnrealStereo4K stereo dataset using different output representations, adopting
HSM [12] as backbone. Note that our bimodal representation outperforms both the direct regression and the unimodal
formation using the HSM backbone. In combined with Tab. 1 in the main paper, the results suggest that our SMD Head canbe advantageously combined with a wide class of stereo backbones to improve the disparity prediction at object boundaries.
Dim.
1
2
5

Avg
1.85
2.63
1.40

SEE3
σ(1)
37.59
43.49
26.78

σ(2)
20.49
27.12
12.65

Avg
1.73
2.51
1.26

SEE5
σ(1)
35.55
41.66
24.70

σ(2)
19.27
25.98
11.40

EPE
Avg
σ(3)
1.11
5.47
2.07
8.90
1.00
4.42

Table 1: Output Representation analysis on the binocular UnrealStereo4K test set. “Dim.” refers to the output dimensionof the SMD Head where 1 indicates the point estimate d, 2 the unimodal output representation (µ, b) [4] and 5 our bimodal
formulation (π, µ1 , b1 , µ2 , b2 ).

KITTI 2015: Tab. 2 is provided as a complement to Tab. 5 in the main paper. In particular, we report results on the KITTI
2015 validation set adopting HSM [12] as stereo backbone. Here, we compare our model with the original HSM [12] stereo
network and HSM [12] trained following [2]. Consistent with results using the PSM backbone, our method built on the HSM
backbone leads to better SEE and EPE compared to both the original HSM backbone and [2].
Method
HSM [12]
HSM [12] + CE + SM [2]
HSM [12] + Ours

Avg
1.37
1.46
1.18

SEE3
σ(1)
29.13
31.48
20.30

σ(2)
14.80
15.63
9.71

Avg
1.25
1.33
1.04

SEE5
σ(1)
25.83
27.86
17.23

σ(2)
13.63
14.49
8.86

EPE
Avg σ(3)0.79 3.25
0.85 3.55
0.74 2.67

Table 2: Comparison on the KITTI 2015 Validation Set using boundaries extracted from instance segmentation masks
to evaluate at depth discontinuities (given the sparse KITTI ground truth, depth discontinuities cannot be derived from theground truth depth maps).

3.2. Qualitative Results
Output Representation: In Fig. 6 we complement Tab. 1 in the main paper with point cloud visualizations on the UnrealStereo4K dataset. Specifically, we show results obtained using different backbones on passive stereo, monocular andactive depth estimation tasks. For each backbone, we compare standard disparity regression (i.e., disparity point estimate), a
unimodal Laplacian distribution and our bimodal Laplacian mixture distribution.
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U-Net [3]PSM [1]
HSM [12]

Disparity Regression
Unimodal(a) Binocular Stereo

BimodalReference Image

Disparity Regression
Unimodal(b) Monocular Depth

BimodalReference Image

Disparity Regression(c) Active Depth

Bimodal

HSM [12]

HSM [12]

U-Net [3]Reference Image

UnimodalFigure 6: Point Cloud Comparison of Output Representations on the UnrealStereo4K dataset. We show outcomes ofdifferent 2D and 3D backbones on (a) binocular stereo, (b) monocular depth and (c) active depth. Note that our bimodal
representation notably alleviates bleeding artifacts near object boundaries compared to both disparity regression and theunimodal formulation.

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PSM [1] + Ours

PSM [1] + CE + SM [2]

PSM [1] + SM [2]

PSM [1] + BF [6]

PSM [1]

UnrealStereo4K: We present Fig. 7 as a complement to Fig. 5 in the main paper. Specifically, Fig. 7 compares the predicteddisparity maps of different strategies which aim to address the over-smoothing problem.

Figure 7: Qualitative Results on UnrealStereo4K. The left column shows the predicted disparity maps while the rightdepicts the corresponding error maps. We zoom-in a patch in all images to better perceive details near depth boundaries.

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KITTI 2015: We also report qualitative results on stereo pairs belonging to the KITTI 2015 dataset [7]. More specifically,in Fig. 8 we show disparity maps predicted by our mixture density formulation using PSM [1] as backbone on testing imagesof the online KITTI 2015 benchmark. Fig. 9, instead, depicts point cloud comparisons of the same model with respect to theoriginal PSM [1] network and [2]. It can be observed how our model allows to notably alleviate bleeding artifacts comparedto the original PSM [1] network and to obtain more accurate results in distant regions compared to [2].

Reference Image

PSM [1] + Ours

Error Map

+Ours

+CE [2]

PSM [1] Ref. Image

Figure 8: Qualitative Results on KITTI 2015 Test Set. Error maps are downloaded from the online KITTI 2015 benchmark.

Figure 9: Point Cloud Comparison on KITTI 2015 Validation Set. The first row shows the reference image and the rest
shows the point cloud comparison. Note that [2] produces bleeding artifacts at far regions (e.g., black car at far distance in the
middle column) while our method predicts sharp object boundaries at both near and far regions. Please zoom in for details.
10Middlebury v3: We also show qualitative results concerning generalization capability on the Middlebury v3 dataset [11]in Fig. 10. In particular, we compare the original PSM [1] network to our bimodal mixture representation using PSM as
backbone. It can be clearly perceived that our strategy recovers more accurate and sharper details on boundary regions.

Reference Image

PSM [1]

PSM [1] + Ours

Figure 10: Generalization on Middlebury v3 [11] using the original PSM [1] and our SMD-Net, both trained on stereo
pairs of our UnrealStereo4K training set. Following the common pratice for the Middlebury dataset, we adopt the colormap’jet’ for disparity visualization.

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RealActive4K: Fig. 11 demonstrates more qualitative results on the RealActive4K dataset. We show the projected disparity
map that we use for co-supervision during training and a point cloud comparison of our bimodal output representation andstandard disparity regression.

Reference Image

Pseudo GT

Disparity Regression (L1)

BimodalFigure 11: Generalization on RealActive4K. Note that despite the pseudo-ground truth disparity not providing information
on disparity edges, our model is able to predict sharp object boundaries for real-world input images while at the same time
inferring plausible disparity values in regions where the classical method used for supervision fails entirely.

3.3. Computational Cost Analysis
Training Time and GPU Memory: We compare the training time and memory footprint of our SMD-Nets to the original
backbones and [2] in Tab. 3. Specifically, we evaluate the memory consumption on a single NVIDIA V100 GPU with 32
GB GPU RAM and the average time of a single iteration (forward + backward) during training. As can be seen, the crossentropy loss (CE) [2] noticeably increases the memory footprint and training time on both, PSM and HSM backbones. The
main reason is that the CE loss requires the network to trilinearly interpolate a matching cost probability distribution to thefull resolution W × H × Dmax with Dmax denoting the maximum disparity. In contrast, our simple bimodal representation
predicts a compact five-dimensional vector at every point and thus is more efficient to train. Note that on the PSM backbone
our bimodal formulation requires only a marginal increase of the memory and training time compared to the original PSM,
while it is more expensive on the HSM backbone. This is due to the different feature dimension D we use for the different
backbones (D = 96 for PSM and D = 368 for HSM). We remark that our method performs favorably with a relatively small
D. We leave the selection of a good D to balance performance and training time for future research.
MethodPSM [1]
PSM [1] + CE [2]
PSM [1] + Ours

Memory (GB)
23.80
31.73
24.78

Time (s)
2.854
4.835
2.889

Method
HSM [12]
HSM [12] + CE [2]
HSM [12] +Ours

Memory (GB)
17.65
27.11
27.54

Time (s)
0.578
7.096
0.983

Table 3: Memory Usage and Time Consumption During Training. We present the memory footprint in Gigabytes (GB)and the average time for one iteration (forward + backward) in seconds.
Inference Time: We now analyze the average time for estimating a single disparity map using the proposed SMD-Nets.
Tab. 4 compares the inference time of our SMD-Net at different output resolutions based on the HSM backbone. Note thatthe inference time of the backbone remains the same as the input resolution is fixed. Though the evaluation time of the SMD
Head increases wrt. the output resolution, our method is still able to estimate a disparity map at 3840 × 2160 resolution inabout 3.6 seconds. Tab. 5 shows the inference time of predicting an output disparity map at 3840 × 2160 resolution using
12different backbones. Here the inference time depends on the input dimension D of the SMD Head, where a smaller D leads
to faster inference.
Output Resolution
960 × 540
1920 × 1080
3840 × 2160

T (Stereo Backbone)
0.030
0.030
0.030

T (SMD Head)
0.206
0.856
3.524

Table 4: Inference Time using the HSM backbone at different output resolutions. T (·) denotes the average time in seconds.

Stereo Backbone
U-Net [3]PSM [1]
HSM [12]T (Stereo Backbone)
0.012
0.575
0.030

T (SMD Head)
3.070
2.402
3.524

Table 5: Inference Time at 3840 × 2160 output resolution using different stereo backbones. T (·) denotes the average timein seconds.

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