# BIDCD - Bosch Industrial Depth Completion Dataset

Adam Botach, Yuri Feldman
Technion - Israel Institute of Technology
Haifa, Israel, 3200000
[botach<EMAIL_ADDRESS>Yakov Miron, Yoel Shapiro, Dotan Di Castro
Bosch Center for Artificial Intelligence
Haifa, Israel, 3508409
[Yakov.Miron, Shapiro.Yoel<EMAIL_ADDRESS>
###### Abstract

We introduce BIDCD - the Bosch Industrial Depth Completion Dataset. BIDCD is a
new RGBD dataset of metallic industrial objects, collected with a depth camera
mounted on a robotic manipulator. The main purpose of this dataset is to
facilitate the training of domain-specific depth completion models, to be used
in logistics and manufacturing tasks. We trained a State-of-the-Art depth
completion model on this dataset, and report the results, setting an initial
benchmark. Further, we propose to use this dataset for learning synthetic-to-
depth-camera domain adaptation. Modifying synthetic RGBD data to mimic
characteristics of real-world depth acquisition could potentially enhance
training on synthetic data. For this end, we trained a Generative Adversarial
Network (GAN) on a synthetic industrial dataset and our real-world data.
Finally, to address geometric distortions in the generated images, we
introduce an auxiliary loss that promotes preservation of the original shape.
The BIDCD data is publicly available at https://zenodo.org/communities/bidcd.

##  1 Introduction

$(a)\qquad\qquad\qquad\qquad\ \ \ (b)\qquad\qquad\qquad\qquad\ \ \
(c)\qquad\qquad\qquad\qquad\ \ \ (d)\qquad\qquad\qquad\qquad\ \ \ (e)$
$(f)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(g)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(h)$

Figure 1:  Our contributions. Top-left: Data Collection. (a) Our data
collection setup consists of a Panda arm (Franka-Emika) and two RealSense D415
(Intel) depth cameras, mounted at different angles and distances from the end-
effector. (b) A fused 3D mesh, generated from multiple views. Projecting the
mesh back to the camera view points yields the ground truth estimation.
Bottom: Depth Completion. (f) RGB. (g) Raw depth - the peripheral holes are
out-of-range regions. (h) Depth completion result - the internal holes have
been closed, and the table top has been extrapolated outwards. Top-right:
Synthetic-to-Depth-Camera Domain Adaptation. (a) Synthetic RGB. (b) Synthetic
depth - without any holes or artifacts. (c) Fake depth - the generator adds
internal and peripheral holes.

Robotic manipulation is an important component of the industry 4.0 revolution,
both for logistic tasks such as bin-picking and assembly tasks such as peg-in-
hole. Operation in unstructured environments requires perception, most
commonly implemented with a camera for data acquisition feeding a computer
vision model. Perception models need to provide 3D information for task and
motion planning, which is typically done in six or more Degree-of-Freedom
(DOF). Digital RGB cameras have been used to estimate 3D geometry using
Structure-from-Motion (SfM) [37], single-frame monocular depth estimation
[10], and 3D pose estimation of known objects [32].

In contrast to RGB cameras, distance sensors can provide a direct measurement
of 3D geometry. In outdoor applications, the most notable one being self-
driving cars, multiple different modalities have been utilized for depth
sensing, including LiDAR, Radar, and Ultra Sound (US). For indoor
applications, a highly prevalent modality for 3D acquisition is depth cameras,
which provide RGBD images [9]. These dense depth maps have been successfully
employed for grasp prediction of unknown objects [17, 34]. Depth cameras can
be implemented in several different ways, including stereo vision, structured
light, and Time-of-Flight (ToF), but all these methods suffer from large
errors and loss of information (”holes”) under optical reflections and
deflection. This pitfall is prominent in industrial settings, which routinely
include reflective metallic objects.

The shortcomings of depth sensors have been previously addressed with depth
completion [12, 16, 24, 33, 27] on single frames. Robotic applications often
use mounted cameras, which present the opportunity to utilize multiple views
and applying volumetric fusion methods, such as the Truncated Signed Distance
Function (TSDF) [5, 28]. A preliminary requirement for TSDF is the
registration between the different points of view. However, industrial
applications tend to be confounded with a multitude of artifacts, which may in
turn cause the registration to fail. For this reason and additional
operational considerations, single-frame solutions seem to be a more viable
alternative.

Depth completion can be thought of as a type of guided interpolation. The goal
is to generate a dense depth map using a partial sampling (raw depth), guided
by visual RGB cues such as perspective, occlusions, object boundaries, and
surface normals. Depth completion datasets include RGB and raw-depth inputs
and an estimated dense Ground-Truth (GT) depth map. In driving applications,
the depth is typically obtained with LiDAR [20]. In contrast, depth cameras
are predominant when imaging indoor scenes [22, 6], household objects [2], and
other applications [9]. These two acquisition systems pose distinct challenges
for depth completion, suggesting that they warrant different solutions. LiDAR
provides sparse depth maps where the values of the samples are typically
considered to be sufficiently accurate. Thus, generating a dense depth map is
largely a question of partitioning the influence regions around the depth
samples using RGB cues such as boundaries. Depth cameras, on the other hand,
may yield large connected regions with missing or invalid values, for example
at window panes [22, 6]. Moreover, specular reflections and transparent
recyclable objects [27] may induce large regions of erroneous values that need
to be identified and corrected.

###  1.1 Industrial Object RGBD Dataset

Here, we present an industrial-objects dataset [29] collected with a depth
camera mounted on a robotic manipulator. To generate the ground truth, we
recorded each scene from approximately 60 Points-of-View (POV) and fused them
to produce an estimated GT depth, as described in the methods section. Using
multiple POVs improve the coverage of the scene, mainly by resolving
occlusions and reflections. We configured the cameras to a depth resolution of
0.1 mm and a dynamic range of 0.2-1.2 m, following characteristic range and
accuracy requirements for robotic manipulation. Consequently, certain
peripheral regions of the GT image are left empty (zeros) if they are out-of-
range for all POVs. The GT might also have internal holes stemming from
localized surface reconstruction failures, typically due to poor
observability. Notably, most RGBD datasets were designed to avoid depth
acquisition issues (for example, see [2]). In support of industrial
applications, we created a dataset rich in highly reflective metallic objects.
Our raw depth maps have holes that take on average $~{}8.5\%$ of the image
area, within the pertinent workspace. For details, please refer to Sec. 2.2.2.

We trained a depth completion model on our dataset, to provide a baseline for
this task. Most deep models for depth completion are based on auto-encoders
[11], which are modified to consume RGBD and output a corrected depth map
[12]. In [16], the authors added UNet connections [26] and achieved State-Of-
The-Art (SOTA) performance at that time. In [8], we took the model from [16]
and carried out an extensive architecture search in order to enhance it.

In this work, we trained the NLSPN model from [24]. The authors of the latter
study achieved depth-completion SOTA on the datasets of [20] and [22]. The
NLSPN model uses an auto-encoder to predict several intermediate maps and
passes them to a Convolutional Spatial Propagation Network [4, 3] for
iterative refinement of the depth prediction. Further details can be found in
Sec. 2.3. The results are reported in Sec. 3.2.

###  1.2 Synthetic-to-Real Domain Adaptation

Synthetic RGBD data and annotations can be generated relatively cheaply and
have been successfully used to train deep models for a variety of tasks [23,
19, 18, 7]. However, the inherent domain gap between synthetic and real data
is very likely to impede such efforts. In [30], the authors generated
photorealistic industrial RGBD data, but their flawless depth maps are
radically different from the depth camera outputs we observed in real-world
settings.

Our dataset can also be used in the opposite direction, that is, to learn a
transformation from GT to depth-camera, simulating the appearance of raw
depth. This type of style-transfer could also be applied to synthetic
industrial datasets, such as those described in [14, 30]. The authors of [13]
demonstrated the usefulness of synthetic-to-real-depth domain adaptation for
person segmentation. Here, we trained a newer Generative Adversarial Network
(GAN), taken from [25], using our raw depth maps as the target domain
(”real”). The synthetic data was taken from the dataset presented in [30].

Notably, we encountered an issue of shape loss, where objects in the synthetic
source image vanished in the fake depth map. It appears that the generator can
learn to remove objects and replace them with a hole, or more surprisingly
with the depth values of the work surface on which they are laid. These depth
maps are plausible but contradict the corresponding synthetic RGB image. The
authors of [21] resolved a similar issue with a geometric consistency loss,
which improved the preservation of pertinent features from synthetic RGB
images in the fake (”real”) images. We introduce an alternative auxiliary loss
for preserving the objects in the fake depth maps. For more details, see Sec.
2.4.

###  1.3 Contributions

We illustrate our contributions in Fig. 1 and list them here.

  * •

Data collection - we randomized over 300 scenes of industrial objects and
dozens of POVs, collecting a total of $\sim$33k RGBD images using a wrist-
mounted (eye-in-hand) depth camera. The challenging nature of this setup
yielded many holes in the raw depth images ($\sim 8.5\%$ of the image area).

  * •

Depth ground-truth - the RGBD images were processed with a customized pipeline
for filtering, registration, and Point-Cloud (PCD) fusion. The fused mesh was
back-projected to the original POV to yield an estimated ground-truth depth
map.

  * •

Depth completion - we trained NLSPN [24] on our dataset, setting an initial
benchmark score.

  * •

Domain adaptation - harnessing our dataset, we trained an existing GAN model
[25] to enhance the photorealism of a synthetic dataset taken from [30].
Moreover, to address shape loss and the vanishing of objects in the generated
depth maps, we introduce an auxiliary loss (see (2)).

##  2 Methods

In this section, we describe how we created the dataset and present the
experiments performed for the depth completion task and synthetic-to-real
domain adaptation.

###  2.1 Data Collection

radius [mm] | elevation [deg] | azimuth [deg]
---|---|---
349.7 - 416.8 | 44.4 - 78.2 | -151.3 - 153.4
Table 1:  POV geometric characteristics. The table provides spherical
coordinates, two orientation angles, and the distance to the work surface. The
coordinate ranges are given with their 10th and 90th percentiles.

We randomized scenes of industrial objects scattered within a rectangular
workspace on top of a table. Each scene consists of up to 10 objects,
predominantly metallic items such as screws, cylinders, and heat sinks. The
data was collected with two RealSense D415 depth cameras, mounted on a Panda
robotic arm from Franka-Emika. The motion path included 30-70 randomized
Points-of-View (POV) with the end-effector poses scattered on a hemisphere,
while taking into consideration kinematic limitations of the manipulator. The
end-effector was kept oriented towards a shared focal point on the table
surface. In Table 1, we describe the POV camera-axis elevation and azimuth,
and the distance of POV from table surface taken along the camera view axis.

Due to the reflective nature of the scenes, the raw depth includes many holes
at highlights and object edges. The reflections also cause misleading
artifacts with erroneous depth values, as demonstrated in the middle column in
Fig. 2. Another contributing factor is partial occlusion, i.e. ”shadows”. On
average, the raw depth holes within the workspace take up $\sim 8.5\%$ of the
image area.

###  2.2 Ground Truth Generation

For each raw depth map, the ground truth is a corresponding dense depth map
containing correct distances, approximately within $\pm 2mm$ (see Fig. 2).
Pixels with a distance outside the dynamic range of the camera are designated
as invalid (zero values). To estimate the ground-truth, we applied to each
scene the pipeline described below. For clarity, we break it down into three
stages, starting from the single POVs raw inputs, followed by multi-view
fusion into a 3D mesh, and back to the separate POVs GT.

$(a)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(b)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(c)$

Figure 2:  Depth Ground Truth. Each row depicts a different sample. From left
to right: (a) RGB, (b) raw-depth, (c) GT-depth. In column (b), note how
artifacts in the out-of-range regions may appear very close (purple) or very
far (yellow). On column (c), we demonstrate how depth-filtering removes
artifacts and multi-view fusion adds missing information, and how projecting
the pre-defined workspace boundaries onto the depth frames eliminates
irrelevant information. On the bottom row column (c), the GT depth has a small
inner hole of less than $1\%$ of the image area.

####  2.2.1 Depth Filtering

Each depth frame was processed individually to remove artifacts, while
retaining as much valid information as possible. In addition to providing
”raw-depth” and ”gt-depth” (ground truth), our dataset includes an
intermediate result denoted as ”cleaned-depth”. The pipeline is depicted in
Algo. 1 and consists of the following steps:

  1. 1.

Workspace limits \- we project the workspace boundaries onto the raw depth and
remove all values outside its perimeter. Tall objects positioned near the
outer rim may get trimmed because their top protrudes the visual borderline.
Hence, we enlarge the actual workspace by a certain margin when applying this
step.

  2. 2.

Morphological operations \- looking at the binary mask of the depth values,
the objects and tabletop typically form one large blob and false artifacts
form separate small blobs. As the main blob may touch artifact blobs, we
perform thinning and opening to separate them. We then threshold the blob area
to filter out small blobs.

  3. 3.

Outlier removal \- the depth image is first partitioned into a coarse grid and
the median depth value is then calculated for each cell. We define a range
limit, and depth values exceeding the allowable range from the cell’s median
are removed. The reasonable range of depth values within a grid cell depends
on grid parameters, POV elevation angle, and object height from the table. We
used a fixed range threshold, chosen to accommodate the largest possible range
within the data.

1

2

Input: $D_{raw}(720\times 1280),\times$ N; $POV\textrm{, homogeneous matrices
}H_{i}$; $K_{cam}\textrm{, camera intrinsics}$; $WSB,\textrm{work space
boundaries}$; $th_{area},\textrm{minimal area threshold}$;
$grid,\textrm{coarse image partitioning}$;

3

4for  _$H_{i}\leftarrow POV$_ do

5

// Workspace limits

6 $mask_{ws}\leftarrow WSB\textrm{ projection with }H_{i},K_{cam}$

7 initialize $D_{ws}$

8 $D_{ws}[mask_{ws}]=D_{raw,i}[mask_{ws}]$

9

// Morphological operations

10 $mask_{cln}\leftarrow D_{ws}>0$

11 apply thinning and binary-open on $mask_{cln}$

12 apply bwlabel on $mask_{cln}\Rightarrow blobs$

13 initialize $D_{cln,i}$

14 for  _$blb\leftarrow blobs$_ do

15 if  _$blb >th_{area}$_ then

16 $D_{cln,i}[blb]=D_{ws}[blb]$

17

18

19

// Outlier removal

20 for  _$cell\leftarrow grid$_ do

21 $m=median(D_{cln}[cell])$

22 $mask_{out}=D_{cln,i}[cell]\notin m\pm margin$

23 $D_{cln,i}[cell][mask_{out}]=0$

24

25

26

27 return $D_{cln}$, cleaned depth maps of scene

28

Algorithm 1 Depth Filtering

1

2

Input: $RGB$; $D_{cln}$, cleaned depth maps of scene; $POV$, homogeneous
transform matrices $H_{i}$; $th_{dist}$, upper limit on transform distances;
$th_{clust}$, minimal cluster size threshold

3

4initialize $posegraph$ with N nodes, and no edges

5 for  _$pov_{i}\leftarrow POV$_ do

6 $node_{i}\leftarrow pov_{i}$

7

8

// Create edges

9 for  _$i\leftarrow 1\cdots N$_ do

10 $RGBD_{cln,i}\leftarrow\left(RGB_{i},D_{cln,i}\right)$

11 for  _$j\leftarrow i+1\cdots N$_ do

12 $RGBD_{cln,j}\leftarrow\left(RGB_{j},D_{cln,j}\right)$

13 $H_{kin}\leftarrow H_{i}^{-1}\cdot H_{j}$

14 $H_{icp}\leftarrow ICP\left(RGBD_{cln,i},RGBD_{cln,j}\right)$

15 $H_{delta}\leftarrow H_{kin}^{-1}\cdot H_{icp}$

16 $t_{delta}\leftarrow H_{delta}\textrm{ translation vector}$

17 if  _$|t_{delta}| <th_{dist}$_ then

18 $edge_{ij}\leftarrow H_{icp}$

19

20

21

// Bundle adjustment

22 apply global optimization on $posegraph$

23

// TSDF-Volume integration

24 $mesh\leftarrow TSDF\left(RGBD_{cln}\right)$

25

26 $TriangleClusters\leftarrow\textrm{apply clustering on }mesh$

27 for  _$clst\leftarrow TriangleClusters$_ do

28 if  _$clst <th_{clust}$_ then

29 $\textrm{remove }clst\textrm{ from }mesh$

30

31

32

33 return $mesh$, 3D model of scene surface

34

Algorithm 2 Multi-View Fusion

####  2.2.2 Multi-View Fusion

After pre-processing the depth channels, we fused the RGBD frames using the
”multiway registration” procedure from [35]. For convenience, we outline the
steps in Algo. 2. The RGBD frames are organized as a pose-graph, where each
pose node corresponds to a POV pose and the robot kinematics are used to
initialize it. Each edge of the pose-graph, i.e. relative transform between a
pair of POVs, is refined with an extension of Iterative-Closest-Point [1].
This stage is prone to errors. While the authors in [35] recommend pruning
edges with low confidence scores, we found this to be insufficient. We,
therefore, replaced replaced the exclusion criteria with a limit on the
acceptable ICP update, and prune out transforms deviating beyond a certain
distance from the nominal kinematic transforms. The pose-graph then undergoes
global-optimization with the Levenberg-Marquardt algorithm to resolve
conflicting relative-transforms (edges). The final pose-graph is used to cast
the RGBDs into a shared Point-Cloud (PCD). The PCD is then triangulated to
generate a mesh, using [5]. To remove the remaining artifacts, we applied
clustering to the mesh vertices and removed small clusters.

####  2.2.3 Back Projection and Hole Filling

To generate GT depth maps, the mesh is projected back to each POV, as
described in Algo. 3. At this stage, the GT depth maps still contain internal
holes, which can be filled using RGB-guided interpolation. The authors of [2]
considered Bi-Lateral Filtering (BLF) [31] and the method from [15] and chose
to apply the latter. For our dataset, BLF is more suitable, mainly due to the
large peripheral out-of-range regions. As shown in (1), we implement a BLF for
RGB distances and pixel-coordinate distances.

$\begin{split}&\hat{D}[i,j]=\frac{1}{Z[i,j]}\cdot\sum_{uv}D[u,v]\cdot
w[u,v]\\\ &Z[i,j]=\sum_{uv}w[u,v]\\\ &w[u,v]=G_{RGB}[u,v]\cdot
G_{Euc}[u,v]\cdot V[u,v]\\\
&G_{RGB}[u,v]=\exp\left(-0.5\cdot\frac{||RGB_{ij}-RGB_{uv}||^{2}}{\sigma_{RGB}^{2}}\right)\\\
&G_{Euc}[u,v]=\exp\left(-0.5\cdot\frac{||(i,j)-(u,v)||^{2}}{\sigma_{Euc}^{2}}\right)\end{split}$
(1)

Where $D$ denotes depth, $V$ is a binary mask of depth validity, the
coordinates $uv$ span the filter’s support around the central pixel at $ij$,
”Euc” is used to denote Euclidean distance in $uv$ coordinates, and
$\sigma_{RGB}$ & $\sigma_{Euc}$ are configuration parameters.

The filter was applied only to invalid pixels, keeping valid values as-is. To
reduce extrapolation into out-of-range regions, we required a minimum
occupancy of $50\%$ of valid values within the filter’s support.

1

2

Input: $mesh$; $RGB$; $K_{cam}\textrm{, camera intrinsics}$; $POV\textrm{,
homogeneous matrices }H_{i}$; $S\textrm{, support of BLF as square binary
kernel}$

3

4for  _$i\leftarrow N$_ do

5 $D_{GT,i}\leftarrow mesh\textrm{ projection with }H_{i}\textrm{ and
}K_{cam}$

6 $mask_{v}\leftarrow D_{GT}==0$

7 $mask_{o}\leftarrow mask_{v}*S<0.5\cdot\sum S$

8 $mask_{BLF}=mask_{v}\cdot mask_{o}$

9 $D_{BLF}\leftarrow BLF\left(D_{GT,i},RGB_{i}\right)$

10 $D_{GT,i}[mask_{BLF}]\leftarrow D_{BLF}[mask_{BLF}]$

11

12 return $D_{GT}$, scene ground truth depth maps

13

Algorithm 3 Back Projection and Hole Filling

####  2.2.4 Manual Inspection and Retries

All the GT depth maps were manually inspected to ensure high fidelity. In
seldom cases, we vetted a scene but removed up to $10\%$ of the POV due to
persistent artifacts. Scenes that failed curation were reprocessed with
stricter parameters, i.e. with more aggressive filtering at the cost of higher
information loss. We depict this in Fig. 3, where we compare the proportion of
invalid GT values for 10 random scenes (total of 650 POV) processed with the
strict and lenient configurations. For this subset, the raw-depth holes within
the projected workspace (”inner”) constitute, on average, $8.5\%$ of the image
area.

Figure 3:  Configuration comparison. We depict the proportional-area of
invalid Ground-Truth values, under two different pipeline configurations. The
”lenient” configuration retains more information, while the ”strict”
configuration rejects more artifacts. The total size of holes includes
peripheral out-of-range regions, which are of less interest. Inner holes are
only those within the projected workspace (see Sec. 2.2.1).

###  2.3 Depth Completion

We implemented the model presented in [24], consisting of two stages. The
first stage is an auto-encoder that predicts an initial depth map, affinities,
and confidence maps. The second stage uses the intermediate outputs to predict
the final depth map via iterative non-local spatial propagation. The
propagation is akin to convolution with adaptive weights and kernels,
determined by confidence and affinities, respectively. In our implementation,
we omitted the confidence prediction to simplify the model. We expect this
removal to have a negligible impact since the ablation study in [24] showed
that it reduced errors only by $\sim 0.5\%$. The resolution of the input RGBD
frames is 720x1280 pixels, with a 9:16 aspect ratio. The NLSPN model was
trained on images reduced to $60\%$ in each dimension, i.e. 432x768 pixels.

###  2.4 Synthetic-to-Real Depth GAN

We applied domain adaptation to synthetic data from [30], to appear more like
our depth maps. We trained the GAN model from [25], an improvement of the
model in [36]. The generator inputs are synthetic RGBD and it provides the
discriminator with a modified depth map. The synthetic and real datasets are
unpaired, i.e. the fake and real images fed to the discriminator depict
different scenes, taken from different POVs. We convert the synthetic RGB to
grayscale, and feed it to the discriminator without modification in
conjunction with the fake depth map. We confirmed with an ablation trial (not
shown here) that the additional grayscale information improves geometric
consistency. The conversion to grayscale reduces the appearance gap between
the synthetic and real datasets. Otherwise the discriminator might latch on to
extraneous features, such as the work surface color.

We faced a challenge of vanishing objects in the fake depth (see Fig. 5). The
generator learned to ”remove” objects and replace them with a hole, or the
underlying surface that they were placed on. We mitigated this by adding a
marginal preservation loss $L_{p}$, as shown in (2). We use $D$ to denote
depth, and $syn$ for synthetic. The loss penalizes the generator for modifying
the input depth unless the modification is larger than the margin threshold.
The rationale behind the preservation loss is that holes and artifacts in raw
depth images are usually substantially different from their immediate
surroundings. The undesired object removal modifies the synthetic depth by
object height, which is approximately two orders of magnitude smaller than the
desired changes.

$\begin{split}\Delta&=\left|D_{syn}-D_{fake}\right|\\\ margin&=20^{th}\
percentile\ of\ \Delta\\\
L_{p}[i,j]&=\begin{cases}\Delta[i,j]&\Delta[i,j]<margin\\\
0&otherwise\end{cases}\\\ \end{split}$ (2)

$(a)\qquad\qquad\qquad\qquad\qquad\quad(b)\qquad\qquad\qquad\qquad\qquad\quad(c)\qquad\qquad\qquad\qquad\qquad\quad(d)$

Figure 4:  Depth completion with NLSPN. Each row depicts a different sample.
Left-to-right: (a) RGB, (b) Raw-Depth, (c) Depth Prediction, and (d) Ground-
Truth. Notice how the model is able to fill shadows and holes on objects. On
the peripheral out-of-range regions the artifacts are subdues, and the table
top is correctly extended.

$(a)\qquad\qquad\quad(b)\qquad\qquad\quad(c)\qquad\qquad\quad(d)\qquad\qquad\quad(a)\qquad\qquad\quad(b)\qquad\qquad\quad(c)\qquad\qquad\quad(d)$

Figure 5:  Synthetic-to-Real Depth Domain Adaptation. We display two samples
on each row. Left to right: (a) synthetic RGB, (b) synthetic depth, (c) fake
depth w/o preservation loss, (d) fake depth w/ preservation loss $L_{p}$. The
generator successfully adds peripheral and internal holes. As expected, the
results w/ $L_{p}$ on column (d) appear more realistic.

##  3 Results

###  3.1 Industrial RGBD Dataset

We were able to collect and generate GT for 319 different scenes of industrial
objects. Each scene was acquired from $63.6\pm 8.1$ POVs (mean $\pm$ 1 std),
resulting in over 33k RGBD frames. Our dataset is organized by scenes. For
each POV, we provide the RGB and raw-depth inputs and the corresponding GT-
depth. As shown in Fig. 2, the GT depth maps have empty peripheral regions
which take up $\sim 20\%$ of the image area and might include small internal
holes taking up less than $1\%$.

We also provide the intermediate results: a mask of the projected workspace
and the cleaned-depth (see Algo. 1). To support 3D approaches, we publish a
parallel 3D dataset 111anonymized for review containing camera intrinsic
parameters, POVs given as homogeneous matrix, and the colored meshes
calculated in Algo. 2.

We applied the ground-truth generation pipeline to previously collected data
of single items, which we provide separately at [29], as BIDCD-SI. This
dataset consists of single-object scenes, collected with a rig of four static
depth cameras. It consists of slightly over 400 scenes and approximately 4000
samples.

###  3.2 Depth Completion

| RMSE | MAE
---|---|---
Raw | 122.1 | 49.0
NLSPN | 25.3 | 8.9
Table 2:  Depth completion errors. The errors are measured in millimeters,
given as RMSE and MAE. Raw refers to acquisition errors and NLSPN to depth-
completion.

We summarize the test results of the NLSPN model in Table 2 and show examples
of predictions in Fig. 4. The errors are calculated against the GT and only
where the GT is valid (non-zero). We report Root-Mean-Square-Error (RMSE) and
Mean-Absolute-Error (MAE); the latter is less sensitive to extreme values. The
NLSPN model can reduce the acquisition errors approximately by a factor of
five. Further work will be necessary to achieve the practical requirements of
$\sim$1 mm accuracy.

###  3.3 Synthetic-to-Real Depth Domain Adaptation

We trained a GAN to generate fake depth maps, similar to our real-world depth
maps, conditioned on synthetic RGBD images. We modified [25] to consume RGBD
rather than RGB images and to generate depth rather than RBG. In addition, we
modified the generator to pass a grayscale image of the synthetic image as-is,
without modifying it. The synthetic RGBD images were taken from [30]. The
discriminator was tasked with classifying the depth maps as ”real” and the
generated depth maps as ”fake”.

In Fig. 5, we present results of the generative model for four samples. It
depicts the RGBD inputs from [30] and two generated fake depth maps, w/ and
w/o the preservation loss $L_{p}$ (see (2)). The predictions on column (c)
(w/o $L_{p}$) displays several issues, for example large implausible holes at
the table top. The predictions w/ the preservation loss $L_{p}$ are shown on
column (d). The auxillary loss managed to improve geometric consistency with
the synthetic RGB and depth.

For more examples, see Fig. S3 in the appendix.

To evaluate this model quantitatively, we input GT images from our dataset as
”synthetic” to the generator and compare the $Raw_{fake}$ to the $Raw_{real}$,
i.e. the corresponding raw depth. The fake depth is not expected to be
identical to the original raw depth, but we do expect some similarity (e.g.,
size and positioning of holes). We take the absolute difference as error,
$err=|Raw_{real}-Raw_{fake}|$, and use its median value $e_{med}$ to estimate
the merit of our preservation loss $L_{p}$. Applying $L_{p}$ reduces $e_{med}$
from 46 to 31, a considerable decrease of $\sim 30\%$.

##  4 Conclusions

To our knowledge our dataset is the first public RGBD dataset for industrial
objects. To ensure high variability, we used a few dozens of objects with
approximately 10 different backgrounds and randomized Points-of-Views.

We trained the NLSPN depth-completion model on our data and report its errors.
From our experience with grasping models, we asses that the achieved accuracy
would be adequate for the more robust models such as [34], but might be
insufficient for more sensitive models such as [17].

We performed domain adaptation, synthetic-to-depth-camera, as a means to
ameliorate training on synthetic RGBD data. We demonstrate this with a GAN
from [25] and introduce an auxiliary preservation-loss to improve its
performance.

We hope that our contribution will help push forwards robotic manipulation in
industrial applications.

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##  Appendix A Supplementary Figures

Figure S1: Depth Ground Truth. Each row (left-to-right) depicts an example of
RGB, raw-depth, and GT-depth. The raw depth of intricate parts, such as the
heat sink and the aluminium profiles is fragmented and missing. The GT
pipeline can retrieve most of the information, with some loss of fine detail.

Figure S2: Depth completion with NLSPN. Left-to-right: RGB, Raw-Depth,
Prediction, and Ground-Truth. Each row is a different sample. Notice how the
model is can fill shadows, holes on objects, and highlights on the work
surface. It also partially compensates for the depth-camera dynamic range
limits and extrapolates the work surface.

Figure S3:  Synthetic-to-Real Depth Domain Adaptation. Top-down: consecutive
frames,emphasizing consistency. Left to right: synthetic RGB, synthetic depth,
fake depth w/ preservation loss.


# Invariant cones for strange attractors of Lozi, Hénon and Belykh type maps

D. A. Grechko
Volga State University of Water Transport
Nizhny Novgorod, 603950 Russia
Lobachevsky State University of Nizhny Novgorod
Nizhny Novgorod, 603022 Russia
<EMAIL_ADDRESS>
&V. N. Belykh
Volga State University of Water Transport
Nizhny Novgorod, 603950 Russia
Lobachevsky State University of Nizhny Novgorod
Nizhny Novgorod, 603022 Russia
<EMAIL_ADDRESS>
N. V. Barabash
Volga State University of Water Transport
Nizhny Novgorod, 603950 Russia
Lobachevsky State University of Nizhny Novgorod
Nizhny Novgorod, 603022 Russia
<EMAIL_ADDRESS>

###### Abstract

We consider strange attractors of two dimensional generalized map with one
nonlinearity [1, 2] such that Lozi, Hénon and Belykh maps are particular cases
of it. We describe technique of invariant expanding and contracting cones
creation for study of hyperbolic properties. Theorems of singular hyperbolic
attractors for new modifications of Lozi, Hénon and Belykh-type maps are
presented.

_Keywords_ dynamical systems $\cdot$ nonlinear maps $\cdot$ attractors $\cdot$
hyperbolicity $\cdot$ bifurcations

##  1 Introduction

In this paper we consider map $f:\;\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ of
the form [1, 2]

$f:\quad\begin{array}[]{l}\bar{x}=x+y+g(x),\\\
\bar{y}=\lambda\big{(}y+g(x)\big{)},\end{array}$ (1)

where $0<\lambda<1$ and $g(x)$ is a piecewise smooth function. This map arised
as a model of phase-locked loop (PLL) [2, 3]. Namely, the simplest version of
PLL models has the form of ODE system

$\begin{array}[]{l}\dot{x}=u+\omega_{1}g(x),\\\
\dot{u}=-\lambda_{0}u+\omega_{2}g(x),\end{array}$ (2)

where $\omega_{1,2}$, $\lambda_{0}$ are positive paramters, and $g(x)$ is
sine-like periodic function serving the output of PLL phase detector.

The discrete time model of PLL is similar to result of applying Euler method
with discretely defined time derivatives in (2)

$\begin{array}[]{l}x_{i+1}=x_{i}+hu_{i}+h\omega_{1}g(x_{i}),\\\
u_{i+1}=u_{i}-h\lambda_{0}u_{i}+h\omega_{2}g(x_{i}).\end{array}$ (3)

The variable $u$ and parameters $\lambda_{0}$, $\omega_{1}$, $\omega_{2}$
change in (3)

$hu=y,\quad\lambda=1-\lambda_{0}h,\quad
h\omega_{1}=a,\quad\omega_{2}h^{2}=\lambda a$ (4)

leads to map (1).

Now we demonstrate that this map has a universal character.

###  1.1 Sine-Gordon equation

For the steady-state equation of Sine-Gordon type

$(1-\lambda)\dfrac{\partial u}{\partial x}+\dfrac{\partial^{2}u}{\partial
x^{2}}=\sin u$

we consider the following discrete version

$(1-\lambda)(u_{j}-u_{j-1})+u_{j+1}-2u_{j}+u_{j-1}=\sin u_{j}$

or equation

$u_{j+1}-u_{j}-\lambda(u_{j}-u_{j-1})=\sin u_{j}.$ (5)

If we set $u_{j}=x_{j}$, $\lambda(u_{j}-u_{j-1})=y_{j}$ then equation (5)
implies that

$\begin{array}[]{l}x_{j+1}=x_{j}+y_{j}+\sin x_{j},\\\
y_{j+1}=\lambda(y_{j}+\sin x_{j}),\end{array}$

and we obtain the dynamical system generated by map (1) with $g(x)=\sin x$.

###  1.2 Standard map

For $\lambda=1$, $g(x)=k\sin x$ map (1) becomes area preserving Chirikov
standard map [4] playing essential role in theory of the conservative chaos.

###  1.3 Zaslavsky map

Zaslavsky map [5, 6] has the form

$\begin{array}[]{l}\bar{z}=\eta(z+a\sin\theta),\\\
\bar{\theta}=\theta+\omega+z+a\sin\theta,\end{array}$

where $a$, $\eta$ and $\omega$ are parameters. If we set $\theta=x$,
$z=y-\omega\eta$, $g(x)=\omega(1-\eta)+a\sin x$, $\eta=\lambda$ then Zaslavsky
map takes the form (1).

###  1.4 Lozi-Hénon map

Consider famous 2-D map

$\begin{array}[]{l}\bar{x}=y+1-aU(x),\\\ \bar{y}=bx,\end{array}$ (6)

which is Hénon map [7, 8] for $U(x)=x^{2}$ and the Lozi map [9] for
$U(x)=|x|$. Variable and parameter change in (6)

$(x,y)\rightarrow(x,y-bx),\quad\lambda=-b,$

leads the Lozi-Hénon map to the form (1) with the following function

$g(x)=1-(1+\lambda)x-aU(x).$

In this paper which is based on the results from [1] we present sufficient
conditions of hyperbolicity for map (1) (section 2). As an example of applying
these conditions we consider the Lozi map written in the form of map (1)
(section 3). In section 4 we present analysis of nonlinear piecewise smooth
hybrid Lozi-Hénon map. Finally in section 5 we consider the original form of
Belykh map and present new features of Belykh attractor in the case of
periodic function $g(x)$.

##  2  Conditions of hyperbolicity

Invariant expanding and contracting cones play an essential role in the theory
of hyperbilocity [13, 14, 12, 21], being the main tool of proven hyperbolic
properties of certain maps [10, 15, 1, 16, 19, 17, 18]. In this section we
present detailed technique of cones construction briefly presented in [1].

###### Definition 1

Assume that a smooth map $F:\;G\rightarrow G$, where $G$ is an open subset of
$\mathbb{R}^{2}$, has the same in each point of $G$ invariant cones $K^{u}$
and $K^{s}$ [1] such that $DFK^{u}\subset K^{u}$ and $DF^{-1}K^{s}\subset
K^{s}$, where linearization $DF$ is expanding in $K^{u}$ and contracting in
$K^{s}$ for any point of $G$. Then map $F$ is called hyperbolic.

Note that despite of usual notation of hyperbolicity [12, 11] this definition
has a strong restriction that cones $K^{u}$ and $K^{s}$ are one and the same
in each point of region $G$.

Consider map (1) in region $G=G^{-}\cup G^{+}$ where $G^{-(+)}$ corresponds to
pieces of function $g(x)$ monotonicity with negative (positive, respectively)
derivative $g^{\prime}(x)$. In other words, we consider piecewise smooth map
(1) with bounded away from zero derivative $g^{\prime}(x)$

$g^{\prime}(x)\triangleq d(x),\quad|d(x)|\geq\varepsilon>0,$

for each point of $G$.

###### Theorem 1

Let the conditions

$\begin{array}[]{ll}d(x)\geq\varepsilon>0&\textrm{for }x\in G^{+},\\\ \\\
d(x)<-2(1+\lambda)&\textrm{for }x\in G^{-},\end{array}$ (7)

hold. Then map (1) is hyperbolic in region $G$ covered by the next invariant
unstable and stable cones

$K^{u}=\left\\{0<\frac{u_{2}}{u_{1}}<\lambda\right\\},\quad
K^{s}=\left\\{-\frac{1}{1-\lambda}<\frac{u_{1}}{u_{2}}<\frac{1}{1+\lambda}\right\\},$
(8)

where $u_{1,2}$ are local coordinates in each point of $G$.

Proof. To construct cones $K^{u,s}$ we consider variational equations as a
linearized map

$L:\quad\bar{\textbf{u}}=A(x)\textbf{u},$ (9)

where

$\textbf{u}=\begin{pmatrix}u_{1}\\\
u_{2}\end{pmatrix},\quad\bar{\textbf{u}}=\begin{pmatrix}\bar{u}_{1}\\\
\bar{u}_{2}\end{pmatrix},\quad A(x)=\begin{pmatrix}1+d(x)&1\\\ \lambda
d(x)&\lambda\end{pmatrix}.$ (10)

Eigenvalues $\mu_{1,2}(x)$ and eigenvectors
$\textbf{V}_{1,2}(x)=\big{(}1,\alpha_{1,2}(x)\big{)}$ of matrix $A(x)$ take
the form

$\begin{array}[]{l}\mu_{1,2}(x)=\frac{1}{2}\left(1+d(x)+\lambda\pm\sqrt{(1+d(x)+\lambda)^{2}-4\lambda}\right),\\\
\\\
\alpha_{1,2}(x)=\frac{1}{2}\left(-1-d(x)+\lambda\pm\sqrt{(1+d(x)+\lambda)^{2}-4\lambda}\right).\end{array}$
(11)

Under conditions (7) of Theorem 1 eigenvalues (11) satisfy the following
conditions

$\begin{array}[]{ccl}\mu_{1}(x)>1,&0<\mu_{2}(x)<1,&\textrm{for}\;d(x)>0,\\\
\\\
-1<\mu_{1}(x)<0,&\mu_{2}(x)<-1,&\textrm{for}\;d(x)<-2(1+\lambda),\end{array}$
(12)

which imply that each point of $G$ is of saddle type.

From formulas (11) satisfying conditions (12) it follows that the eigenvectors
define the next directions

$\begin{array}[]{ll}\textbf{V}_{1}\big{|}_{G^{+}}\;\textrm{and}\;\textbf{V}_{2}\big{|}_{G^{-}}\;\textrm{--
unstable directions},\\\ \\\
\textbf{V}_{2}\big{|}_{G^{+}}\;\textrm{and}\;\textbf{V}_{1}\big{|}_{G^{-}}\;\textrm{--
stable directions}.\end{array}$ (13)

The angle boundaries of eigenvectors are defined by edge values of coordinates
$\alpha_{1,2}(x)$. Directions of the angle turn are defined by derivatives of
$\alpha_{1,2}(x)$ with respect to $d(x)$

$\alpha_{1,2}^{\prime}(x)=\frac{1}{2}\left(-1\pm\frac{1+d(x)+\lambda}{\sqrt{(1+d(x)+\lambda)^{2}-4\lambda}}\right)$

which gives inequalities

$\begin{array}[]{rrl}\alpha_{1}^{\prime}(x)>0,&\alpha_{2}^{\prime}(x)<0&\textrm{for}\;d\geq\varepsilon>0,\\\
\\\
\alpha_{1}^{\prime}(x)<0,&\alpha_{2}^{\prime}(x)>0&\textrm{for}\;d<-2(1+\lambda).\end{array}$

Due to (11) we obtain the next intervals of monotonic change of $\alpha_{1,2}$
as functions of $d(x)$

$\begin{array}[]{lll}\begin{array}[]{l}\alpha_{1}(x)\in(0,\lambda),\\\
\alpha_{2}(x)\in(-1+\lambda,-\infty),\end{array}&\textrm{for}\;d(x)\in(0,\infty),\\\
\\\ \begin{array}[]{l}\alpha_{1}(x)\in(\infty,1+\lambda),\\\
\alpha_{2}(x)\in(\lambda,2\lambda),\end{array}&\textrm{for}\;d(x)\in(-\infty,-2(1+\lambda))\end{array}$
(14)

Thus, the family of all possible eigenvectors $\textbf{V}^{u}$ and
$\textbf{V}^{s}$ in region $G$ for $d(x)>0$ and $d(x)<-2(1+\lambda)$ are
bounded by the next edge vectors (see Fig. 1)

$\begin{array}[]{ll}\textbf{V}_{bot}^{u}=(1,0),&\textbf{V}_{up}^{u}=(1,2\lambda),\\\
\\\
\textbf{V}_{left}^{s}=(-1,1-\lambda),&\textbf{V}_{right}^{s}=(1,1+\lambda).\end{array}$
(15)

As a result using these edge vectors we obtain the form of unstable and stable
invariant cones (8). $\Box$

Figure 1 illustrates iterates of linearized map $L$ along a trajectory of $f$
(eq. (1)) in $G$. Due to Theorem 1 any eigenvector $\textbf{V}^{u}(x)$
($\textbf{V}^{s}(x)$) belongs to unstable (stable) invariant cone $K^{u}$
($K^{s}$). Therefore, any trajectory in region $G$ is of saddle type.

Figure 1: Unstable $K^{u}$ and stable $K^{s}$ invariant cones (red and blue
triangles, respectively). Edge vectors of cones are $\textbf{V}^{u}_{up}$,
$\textbf{V}^{u}_{bot}$ (red arrows) for $K^{u}$, and $\textbf{V}^{s}_{left}$,
$\textbf{V}^{s}_{right}$ (blue arrows) for $K^{s}$. In depicted case, while
point $(x_{1},y_{1})$ is mapped by $f$ (dashed black curves) from $G^{+}$ to
$G^{-}$, its eigenvector $\textbf{V}^{u}(x_{1})\in K^{u}$ (black bold arrow)
is expanded under linear map $L$ (see Eq. (9)). In the next step, point
$(x_{2},y_{2})$ returns to $G^{+}$ under the action of $f$, and its unstable
eigenvector $\textbf{V}^{u}(x_{2})\in K^{u}$ (gray bold arrow) expands under
$L$ rotated by $\pi$ (due to negative multipliers, see Eq. (11)). In stable
cone $K^{s}$ all eigenvectors are mapped with contraction (not shown).

Using Theorem 1 we obtain the next statement.

###### Theorem 2

Let $G$ be a trapping domain, $fG\subset G$, of map $f$ (eq. (1)), satisfying
conditions of Theorem 1. Then map $f$ has singular hyperbolic attractor.

Proof. As $G$ is trapping domain, so map (1) has an attractor
$\mathcal{A}=\bigcap_{k=1}^{\infty}f^{k}G$. Then from Theorem 1 it follows
that attractor $\mathcal{A}$ is hyperbolic. The singularity of this attractor
appears when points of attractor $\mathcal{A}$ lie at discontinuity lines
$\left\\{x=\textrm{const},y\in\mathbb{R}^{1}\right\\}$ at which derivative
$d(x)$ changes the sign. $\Box$

##  3 Continuous piecewise linear function and Lozi attractor

We start to demonstrate application of Theorem 1 for the Lozi map written in
the form (1) with piecewise linear function

$g(x)=1-a|x|-(1+\lambda)x,\quad\lambda=-b,$ (16)

where $\lambda>0$ corresponds to orientation preserving case.

Note that results of this sections are well-known (see [15] and refs.
therein). Here we present simple proof of the next statement.

###### Theorem 3

Let parameters $\lambda$ and $a$ satisfy the next two conditions

$a>1+\lambda,\quad\lambda<1,$ (17)

and

$H(\lambda,a)\triangleq
a(2\lambda-a+2)\sqrt{a^{2}-4\lambda}+a(2\lambda^{2}-6\lambda-a^{2}+2a)+4\lambda^{2}\geq
0.$ (18)

Then the Lozi map (1), (16) has singular hyperbolic Lozi attractor (see Fig.
2).

Proof. The derivative of function (16)

$d(x)=-a\,\textrm{sign}\,x-1-\lambda$ (19)

satisfies conditions of hyperbolicity (7) in one and the same parameter domain
(17) for $G^{-}(x>0)$ and $G^{+}(x<0)$.

Note that in this case in each point of $G$ expanding cone $K^{u}$ (8)
contains two constant eigenvectors $\textbf{V}_{1}\big{|}_{x>0}$ and
$\textbf{V}_{2}\big{|}_{x<0}$, and contracting cone $K^{s}$ contains two
constant eigenvectors $\textbf{V}_{1}\big{|}_{x<0}$ and
$\textbf{V}_{2}\big{|}_{x>0}$.

Thereafter to prove the existence of Lozi attractor it is sufficient to find
the part of parameter domain (17) corresponding to existence of trapping
region $G_{t}$, $fG_{t}\subset G_{t}$.

Map (1), (16) has two saddle type fixed points
$O_{1}\left(\frac{1}{1+\lambda-a},0\right)$ and
$O_{2}\left(\frac{1}{1+\lambda+a},0\right)$ with multipliers $\mu_{1,2}$

$\mu_{1,2}=\frac{1}{2}\left(-a\,\textrm{sign}\,x\pm\sqrt{a^{2}-4\lambda}\right),$

satisfying conditions (12). Eigenvectors
$\textbf{V}_{1,2}=(1,\alpha_{1,2}(x))$, where coordinates (11) take the form

$\alpha_{1,2}(x)=\frac{1}{2}(a\,\textrm{sign}\,x+2\lambda\pm\sqrt{a^{2}-4\lambda}).$

As far as the eingenvectors are constant in each region, map (1), (16) has
unstable and stable invariant foliations in regions $x<0$ and $x>0$ parallel
to separatrices $W_{1,2}$ of fixed points $O_{1}$ and $O_{2}$

$\begin{array}[]{cl}W_{1,2}\big{|}_{x<0}:y=\alpha_{1,2}\left(x+\frac{1}{a-1-\lambda}\right),\\\
W_{1,2}\big{|}_{x>0}:y=\alpha_{1,2}\left(x-\frac{1}{a+1+\lambda}\right).\end{array}$
(20)

We construct continuation of unstable separatrix of point $O_{1}$ using (20)
as its first part. This part $W_{1}|_{x<0}$ intersects discontinuity line
$\\{x=0,y\in\mathbb{R}^{1}\\}$ in the point
$M_{1}(0,\frac{\alpha^{-}_{1}}{a-1-\lambda})$, where
$\alpha^{-}_{1}=\alpha_{1}\big{|}_{x<0}$. Then continuation of
$W_{1}\big{|}_{x<0}$ to region $x>0$ is line segment $M_{1}M_{2}$, where point
$M_{2}=f\big{|}_{x<0}M_{1}=(x_{2},y_{2})$,
$x_{2}=\frac{\alpha^{-}_{1}}{a-1-\lambda}+1$, $y_{2}=\lambda x_{2}$.

The next continuation of $W_{1}$ is line segment $M_{2}M_{3}$, where
$M_{3}=f\big{|}_{x>0}(M_{2})$, $M_{3}=(x_{3},y_{3})$,
$x_{3}=y_{2}+1-(a+\lambda)x_{2}$, $y_{3}=\lambda(y_{2}+1-(a+1+\lambda)x_{2})$.
For parameter region (17), $M_{3}$ lies in region $x<0$ due to $x_{3}=\lambda
x_{2}+1-(a+\lambda)x_{2}=1-a-\frac{a\alpha_{1}^{-}}{a-1-\lambda}<0$.

Introduce segment $I=\\{x=x_{3},\alpha_{2}(x_{3}+\frac{1}{a-1-\lambda})\leq
y\leq y_{3}\\}$, connecting $M_{3}$ and unstable separatrix
$W_{2}\big{|}_{x<0}$.

The area $G_{t}$ bounded by $W_{1}\big{|}_{x<0}\cup M_{1}M_{2}\cup
M_{2}M_{3}\cup I$ is trapping region, $fG_{t}\subset G_{t}$, under condition
(18), which is redone inequality $\alpha_{2}(x_{3}+\frac{1}{a-1-\lambda})\leq
y_{3}$.

Hence, common parameter region (17), (18) corresponds to Lozi attractor.
$\Box$

###### Remark 1

For $H(\lambda,a)=0$ point $M_{3}$ reaches stable separatrix
$W_{2}\big{|}_{x<0}$ and becomes a point of homoclinic orbit. After this
homoclinic bifurcation for $H(\lambda,a)<0$ point $M_{3}$ lies under
$W_{2}\big{|}_{x<0}$, thus area $G_{t}$ is no longer trapping region.

Fig. 3 illustrates that breakdown of Lozi attractor occurs as a result of
either hyperbolicity condition loss or via homoclinic bifurcation.

Figure 2: Parameters’ plane $(\lambda,d)$ of the Lozi map (1), (16).
Parameters from yellow region corresponds to existence of invariant domain
$G_{t}:fG_{t}\subset G_{t}$. This region is bounded by bifurcation of
homoclinic orbit to saddle $O_{1}$ (blue curve), defined by equation
$H(\lambda,d)=0$ in (18). Corresponding phase picture see in Fig. 3. At black
dashed line $a=1+\lambda$ fixed point $O_{2}$ becomes saddle (see conditions
(12)). Grey region corresponds to hyperbolic phase space of the map.
Intersection of yellow and gray regions (dashed area) corresponds to the
existence of strange Lozi attractor. Figure 3: Homoclinic orbit in the Lozi
map (1), (16) for $\lambda=0.2$ and $a=1.905$. Unstable and stable manifolds
$W_{1,2}$ of saddle $O_{1}$ are red and blue lines, respectively. Black broken
line is the continuation of $W_{1}$, which forms homoclinic orbit.

##  4 Attractor of nonlinear hybrid Lozi-Hénon map

Consider map (6) in the case of function

$U(x)=1-a(l|x|+(1-l)x^{2}),$ (21)

which is the Lozi map for $l=1$ and Hénon map for $l=0$. As before we
transform map (6) to map (1) with function

$g(x)=1-al|x|-(1+\lambda)x-a(1-l)x^{2}.$ (22)

The condition of hyperbolicity (7) become the same for $G=G^{-}\cup G^{+}$
condition for $l>0$

$al>1+\lambda,$ (23)

and condition for Hénon map $l=0$

$|x|>\frac{1+\lambda}{2a}.$ (24)

Note that due to (23) when $l$ tends to zero when Hénon part dominates in
$g(x)$ parameter $a$ in (23) increases to infinity.

Condition (23) eliminates extremums of $g(x)$ with $g^{\prime}(x)=0$ and force
$g(x)$ to have maximum at point $(x=0,y=1)$ with discontinuous derivative.

Map (1), (22) has saddle points $O_{1,2}$ $(x_{1,2},0)$, where

$x_{1,2}^{s}=\frac{1}{2a(1-l)}\left[-1-\lambda\pm\left(al-\sqrt{(al\mp(1+\lambda))^{2}+4a(1-l)}\right)\right].$
(25)

Here separatrices of saddle $O_{1}$ are non-linear in initial region $x<0$ and
defined by nonlinear functions

$W_{1,2}\big{|}_{G^{-}}:\;y=\varphi_{1,2}(x-x_{1}^{s}).$ (26)

###### Theorem 4

There exists a value $l^{*}\in(0,1)$ such that for $l^{*}<l<1$ hybrid Lozi-
Hénon map has strange hyperbolic attractor if condition (23) and inequality

$-y_{3}\leq(1-\lambda)(x_{3}-x_{1}^{s})$ (27)

are valid.

Proof. Let condition of hyperebolicity (23) is valid. Consider sufficient
conditions for existence of trapping region.

In parameter domain (23) cones (8) give the next bounds of separatrices

$\begin{array}[]{c}0<\varphi_{1}(x-x_{1}^{s})<\lambda(x-x_{1}^{s}),\\\
\varphi_{2}(x-x_{1}^{s})<-(1-\lambda)(x-x_{1}^{s}).\end{array}$ (28)

Hence, unstable separatrix $W_{1}\big{|}_{x<0}$ intersects the line
$\\{x=0,y\in\mathbb{R}^{1}\\}$ in point $M_{1}(0,y_{1})$, where

$0<y_{1}<\bar{y}_{1}=-\lambda x_{1}^{s}.$ (29)

The image $f\big{|}_{x<0}(M_{1})=M_{2}(x_{2},y_{2})$ has the coordinates

$x_{2}=y_{1}+1,\quad y_{2}=\lambda(y_{1}+1),$ (30)

and the first piece of continuation of separatrix $W_{1}\big{|}_{x<0}$ to
region $x>0$ is curve $\wideparen{M_{1}M_{2}}$ connecting $M_{1}$ and $M_{2}$.
The next piece is curve $M_{2}M_{3}$ connecting $M_{2}$ with its image
$M_{3}=f\big{|}_{x>0}(M_{2})$, $M_{3}(x_{3},y_{3})$, where

$\begin{array}[]{l}x_{3}=1-al(y_{1}+1)-a(1-l)(y_{1}+1)^{2},\\\
y_{3}=\lambda-\lambda(al+1)(y_{1}+1)-\lambda a(1-l)(y_{1}+1)^{2},\end{array}$
(31)

From (31) we obtain the next inequalities

$\begin{array}[]{c}x_{3}<x_{3}\big{|}_{y_{1}=0}=1-a<0,\\\
y_{3}<y_{3}\big{|}_{y_{1}=0}=-\lambda a<0.\end{array}$ (32)

implying that $M_{3}\in G^{-}(x<0)$ and the second piece of separatrix
continuation $\wideparen{M_{2}M_{3}}$ intersects region $x<0$.

Similarly to the case of Lozi map $l=1$ we introduce segment
$I_{l}=\left\\{x=x_{s},\varphi_{2}(x_{3}-x_{1}^{s})<y<y_{3}\right\\}$. As
image $f\big{|}_{x<0}I_{l}\in\left\\{x_{3}<0\right\\}$, so area $G_{l}$
bounded by closed curve $\partial
G_{l}=W_{1}\big{|}_{x<0}\cup\wideparen{M_{1}M_{2}}\cup\wideparen{M_{2}M_{3}}\cup
I_{l}$ is trapping region of map (1), (22).

Using formulas in (25)-(31) we obtain that the region of parameters (27) is
not empty and exists only for $0<a<a^{*}(l,\lambda)$. Then the condition of
hyperbolicity (23) and condition of trapping domain existence (27) are
compatible for some value $l^{*}$ defined by equation
$a^{*}(\lambda,l)l=1+\lambda$. $\Box$

###### Remark 2

Obviously Theorem 1 does not work in the case of Hénon map for $l=0$ due to
breach of hyperbolicity (23) for the whole domain $G=G^{-}\cup G^{+}$. However
condition (27) for trapping domain is valid for $l=0$, and condition of
hyperbolicity (24) can be used for nonwandering subset fully located in domain
(24).

##  5  Discontinuous function and Belykh attractor

Belykh map [20] and its multidimensional generalizations [22, 23, 24]. were
considered in terms of ergodic theory in order to prove existence of
continuous invariant measure [20, 21, 22, 23, 24, 25, 26]. Here we consider
dynamical properties of Belykh attractor and present its new features for the
map $f$ of cylinder $S^{1}\times R^{1}$.

We introduce sawtooth 2-periodic function comming from PLL modeling

$g(x)=a(x-(2k-1))\quad\textrm{for}\quad x\in(2k-2,2k),\quad k\in\mathbb{Z}.$
(33)

In the case $x\in\mathbb{S}^{1}$ we can rewrite $g(x)$ in the form

$g(x)=a(x-1),\quad x(\textrm{mod}\;2).$

First we consider the map (1) at two periods $(k=0,1)$ of function $g(x)$
which takes the form

$g(x)=\left\\{\begin{array}[]{cl}a(x+1),\quad x<0,\\\ \\\ a(x-1),\quad x\geq
0.\end{array}\right.$ (34)

Here for convenience we consider $x\in\mathbb{R}^{1}$ instead $x\in[-2,2]$.

Matrix $A$ in (9) having the form $A=\begin{pmatrix}1+a&1\\\ \lambda
a&\lambda\end{pmatrix}$ is constant for any $x\in\mathbb{R}^{1}$, and as
$d(x)\equiv a>0$ and trace $s\triangleq 1+a+\lambda$ of matrix $A$, so due to
(11) eigenvalues $\mu_{1}>1$, $0<\mu_{2}<1$ and eigenvectors
$\textbf{V}_{1,2}=(1,\alpha_{1,2})$ coordinates $\alpha_{1,2}$ satisfy
conditions

$0<\alpha_{1}<1,\quad\alpha_{2}<-(1+\lambda)<0.$ (35)

As far as constant eigenvectors $\textbf{V}_{1,2}$ are the same for any $x$
map $f$ has unstable $\Phi_{c}^{u}$ and stable $\Phi_{c}^{s}$ invariant
foliations

$\begin{array}[]{cl}\Phi_{c}^{u}=\\{x,y\,|\,y=\alpha_{1}(x+c_{1})\\},\\\ \\\
\Phi_{c}^{s}=\\{x,y\,|\,y=\alpha_{2}(x+c_{2})\\},\end{array}$ (36)

where constants $c_{1,2}\in\mathbb{R}^{1}$. Note that cones $K^{u}$ and
$K^{s}$ are superfluous in the case of map (1), (34) because each of them
contain one constant vector everywhere.

The map $f$ (1), (34) has two similar fixed points $O_{1,2}(\pm 1,0)$ of
saddle type with unstable $W_{1,2}^{u}$ and stable $W_{1,2}^{s}$ separatrices
(see Figure 4)

$\begin{array}[]{cl}W_{1,2}^{u}=\\{x,y\,|\,y=\alpha_{1}(x\pm 1)\\},\\\ \\\
W_{1,2}^{s}=\\{x,y\,|\,y=\alpha_{2}(x\pm 1)\\},\end{array}$ (37)

lying in invariant foliations (36).

Denote points $M_{1}(x_{1},y_{1})=W_{1}^{u}\cap\\{x=0\\}$, where
$x_{1}=0,y_{1}=\alpha_{1}$ and
$M_{2}(x_{2},y_{2})=W_{2}^{s}\cap\left\\{x>0,y=\alpha_{1}(x+1)\right\\}$,
where

$x_{2}=\frac{s-2\lambda}{\sqrt{s^{2}-4\lambda}},\quad
y_{2}=\alpha_{1}(x_{2}+1).$

###### Theorem 5

For

$a\leq 1-\lambda$ (38)

map (1), (34) has singular hyperbolic Belykh attractor.

Proof. The image
$f\big{|}_{x<0}(M_{1})=\big{(}\bar{x}_{1},\bar{y}_{1})=(\alpha_{1}+a,\lambda(\alpha_{1}+a)\big{)}$,
does not reach point $M_{2}$ when $\bar{x}_{1}<x_{2}$, i.e. for

$\alpha_{2}+a<\frac{s-2\lambda}{\sqrt{s^{2}-4\lambda}}.$ (39)

Symmetrically $f\big{|}_{x>0}(\bar{M}_{1})$ does not reach point $\bar{M}_{2}$
for (39). Inequality (39) takes the form

$(2-s)(s+\sqrt{s^{2}-4\lambda})>0,$ (40)

which is valid under condition (38). Hence parallelogram
$P=O_{1}M_{2}O_{2}\bar{M}_{2}$ is invariant, $fP\subset P$, and there exists
an attractor $\mathcal{A}\subset P$.

Folliations (36) cross domain $P$ for $c_{1,2}\in[-1,1]$ with stretch
$\mu_{1}>1$ along $\Phi_{c}^{u}$ and shrink $\mu_{2}<1$ along $\Phi_{c}^{s}$.
Hence each orbit of attractor $\mathcal{A}$ is uniformly hyperbolic with
Lyapunov exponents $h_{1}=\ln\mu_{1}>0$ and $h_{2}=\ln\mu_{2}<0$. Singularity
of attractor $\mathcal{A}$ is due to the fact that a set of parameters
$(a,\lambda)$ corresponding to $\mathcal{A}\cap\\{x=0\\}\neq\emptyset$ is a
set of “border collision” bifurcations. For example, period $p$ orbit
disappears when its point reaches discontinuity line $\\{x=0\\}$ under a
parameter change. $\Box$

Figure 4: Phase portrait of Belykh map (1), (34) for bifurcational case
$\lambda=a=0.5$ (see condition (38)) when point $M_{1}$ maps to stable
manifold $W_{2}^{s}$, $fM_{1}=M_{2}$. This is thr edge case when parallelogram
$P$ (yellow) forms trapping region $fP\subset P$. This parallelogram maps into
two trapezoids (areas bounded with dashed lines): upper $fP\big{|}_{x<0}$ and
bottom $fP\big{|}_{x>0}$. Blue points are singular hyperbolic Belykh
attractor.

###### Remark 3

In basis of eigenvectors map (1), (34) can be written in the form

$\begin{array}[]{lll}\bar{u}_{1}=\mu_{1}(u_{1}+1)+1,&\bar{u}_{2}=\mu_{2}(u_{2}+1)+1,&v<0,\\\
\\\
\bar{u}_{1}=\mu_{1}(u_{1}-1)-1,&\bar{u}_{2}=\mu_{2}(u_{2}-1)-1,&v>0,\end{array}$
(41)

where $v=u_{1}+ku_{2}$, $0<k<1$. In the literature [21], Belykh map is met
just in the form (41).

###  5.1 Periodic case

Consider the case of 2-periodic function $g(x)$ (33) which becomes bounded
function.

###### Lemma 1

Annulus $R=\\{x,y\;|\;x\in\mathbb{S}^{1},|y|<\frac{a}{1-\lambda}\\}$ is
absorbing domain of map $f$ (1), (33).

Proof. Function $g(x)=a(x-1)$ is bounded at $\mathbb{S}^{1}$: $|g(x)|<a$ at
the period $0<x<2$. Then $|\bar{y}|<\lambda(|y|+a)$ and variable $|y|$
decreases at each iterate for $\lambda(|y|+a)<|y|$, i.e. for
$|y|\geq\frac{a}{1-\lambda}$. The latter implies that annulus $R$ is absorbing
domain.$\Box$

###### Theorem 6

For $a>1-\lambda$, $0<\lambda<1$ map (1), (33) has singular hyperbolic
attractor consisting of three components: oscillating $\mathcal{A}_{o}\subset
P$, rotating $\mathcal{A}_{r}\subset R\setminus P$ and rotating with
oscillations $\mathcal{A}_{ro}\subset R$.

Proof. From Lemma 1 it follows that $fR\subset R$ and the map has an attractor
$\mathcal{A}$. Due to existence of folliations $\Phi_{c}^{u}$ and
$\Phi_{c}^{s}$ the attractor is singular hyperbolic. For $a>1-\lambda$ image
$fM_{1}$ overjumps point $M_{2}$ and $fP\setminus P\neq\emptyset$ is triangle
$\Delta_{1}=M_{2}fM_{1}M_{4}$ (or trapezoid) for $y>0$ and symmetrical
$\bar{\Delta}_{1}$ for $y<0$. Image $f^{k}\Delta$ beginning from some $k$
starts to enwind the cylinder forming rotating attractor $\mathcal{A}_{r}$ and
intersecting the triangle $\Delta_{2}=M_{1}M_{3}M_{2}$. Obviously
$f^{k}\Delta_{2}\subset P$, where $k\geq 1$, i.e. trajectories from
$\Delta_{2}$ reach region $P$. Almost all trajectories $f^{k}\Delta_{2}$,
$k>k^{*}$, leave region $P$ and form attractor $\mathcal{A}_{ro}$. $\Box$

Figure 5: Phase portrait of Belykh map (1) for periodic function $g(x)$ (33)
under conditions of Theorem 6 (with parameters $\lambda=0.8$, $a=0.6$).
Attractors $\mathcal{A}_{o}$, $\mathcal{A}_{r}$ and $\mathcal{A}_{ro}$ are
light gray dots. Parallelogram $P$ (yellow) maps into two trapezoids (dashed
contours): upper $fP\big{|}_{x<0}$ and bottom $fP\big{|}_{x>0}$. Red triangle
$\Delta_{1}$ plays the role of “output gate” from $P$ for trajectories of
$A_{r}o$, and blue triangle $\Delta_{2}$ plays the role of “input gate” to
$P$. Coordinates of points: $M_{1}(0,0.4)$, $M_{2}(0.5,0.6)$, $M_{3}(0,1.2)$
$fM_{1}(1,0.8)$, $\tilde{M}_{1}(0,-0.4)$, $\tilde{M}_{2}(-0.5,-0.6)$,
$f\tilde{M}_{1}(-1,-0.8)$. Points $\tilde{M}$ are odd symmetric to points $M$.

Corollary. For $a>0$, $0<\lambda<1$ map $f$ has singular hyperbolic attractor.
Indeed for $a\leq 1-\lambda$ that is Belykh attractor and for $a>1-\lambda$
three attractors $\mathcal{A}_{o}$, $\mathcal{A}_{r}$ and $\mathcal{A}_{ro}$
from Theorem 5.

###### Remark 4

Attractor $\mathcal{A}_{o}$ is a Cantor set similar to that of Smale’s
horseshoe. Unlike attractors $\mathcal{A}_{r}$ and $\mathcal{A}_{ro}$ it’s
basin haz zero Lebesgue measure.

###### Remark 5

The model of continuous time PLL (2) has simple dynamics. Indeed, Lyapunov
function

$V=\frac{u^{2}}{2}-\omega_{2}\int\limits_{0}^{x}g(\xi)d\xi,$

for system (2), (33) with derivative

$\dot{V}=-\alpha u^{2}-\omega_{1}\omega_{2}g^{2}(x)<0\quad\textrm{for}\;x\neq
2k-1,u\neq 0,$

Hence system (2) is globally asymptotically stable so that all trajectories,
besides stable separatrices, are attracted by the point $(x=u=0)$ \- glued
stable equilibrium (see Figure 6).

Under transition from system (2) to map (1), (33) this zero point immediately
becomes strange attractor which “amplitude” is of order $\alpha_{1}+a\backsim
h$ (see (4)).

Figure 6: Phase portrait of phase-locked loop system (2) for $\nu$=0.

##  6 Acknowledgements

This work was supported by the Russian Science Foundation under grant No.
22-21-00553.

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