[Repeat Steps 1–5 for the second imaging device or duplicate and translate
manifold $\protect\CC{}{\widetilde{S}_{1}}{}$ for computing $\protect\CC{}{\widetilde{S}_{2}}{}$.

Overview of the computation steps of Algorithm <ref> to characterize
the $\protect\ACC$ $\protect\CC{}{\widetilde{S}_{1}}{}$ that integrates
viewpoint constraints of the two imaging devices ($s_{1}$, $s_{2}$)
from a range sensor.

§.§.§ Characterization

The $\ACC$ $\CC{}{\widetilde{S}_{1}}{}$, which comprises all constraints
of all imaging devices $s_{t}\in\widetilde{S}$, can be straightforwardly
characterized following the five simple steps given in Algorithm <ref>.
Figure <ref> visualizes the interim manifolds
at each step to ultimately characterize the manifold $\CC{}{\widetilde{S}_{1}}{}$.
Finally, Figure <ref> shows an exemplary extreme viewpoint,
where both imaging devices frames are within their respective $\ACC$s
$\VECJ ps{}{}{s_{1}}\in\CC{}{\widetilde{S}_{1}}{}$ and $\VECJ ps{}{}{s_{2}}\in\CC{}{\widetilde{S}_{2}}{}$
and the feature geometry lies within both frustum spaces.

The space $\CC{}{\widetilde{S}_{2}}{}$ for the second imaging device
can be computed analogously following the same steps. However, the
topology of the resulting $\CC{}{\widetilde{S}_{2}}{}$ will be identical
to $\CC{}{\widetilde{S}_{1}}{}$. Hence, instead of repeating the
steps described in Algorithm <ref> for a second imaging device,
a more efficient alternative is to translate the manifold of $\CC{}{\widetilde{S}_{1}}{}$
to the position of $s_{2}$ at $\VECJ ps{}{}{s_{2}}(\VECJ ts{}{}{s_{2}}=B_{f})$
using the rigid translation

\[
\CC{}{\widetilde{S}_{2}}{}=\text{\emph{translation}}(\CC{}{\widetilde{S}_{1}}{},\VECJ Ts{}{s_{2}}{s_{1}}).
\]

Note that if two imaging devices are the same orientation, the resulting
$\ACC$s are identical, hence $\CC{}{\widetilde{S}_{1}}{}=\CC{}{\widetilde{S}_{2}}{}$.

Characterization of the $\protect\ACC$ for the first sensor in $SE(3)$,
$\protect\CC{}{\widetilde{S}_{1}}{}$ (blue manifold), being delimited
by the $\protect\ACC$ of the second sensor, $\protect\CC{}{s_{2}}{}$(orange
manifold without fill), to acquire a square feature $f_{1}$. The
$\protect\CC{}{\widetilde{S}_{2}}{}$ (orange manifold) analogously
characterizes the $\protect\ACC$ for sensor $s_{2}$ considering
the constraints of $s_{1}$.

Verification of occlusion-free visibility at an extreme viewpoint
${\protect\VECJ ps{}{}{s_{1}}\in\protect\CC{}{\widetilde{S}_{1}}{}}$
and ${\protect\VECJ ps{}{}{s_{2}}\in\text{\ensuremath{\protect\CC{}{\widetilde{S}_{2}}{}}}}$:
rendered scene (left image), depth images of $s_{1}$ (right image
in the upper corner) and $s_{2}$ (right image in the lower corner).

§.§.§ Verification

The joint constrained spaces $\CC{}{\widetilde{S}_{1}}{}$ and $\CC{}{\widetilde{S}_{2}}{}$
for $s_{1}$ and $s_{2}$ were computed according to the steps provided
in Algorithm <ref> for acquiring feature
$f_{1}$. The imaging parameters of both sensors and rigid transformation
between them are given in Table <ref>.

First, the $\ACC$s of both sensors were spanned considering its imaging
parameters and a null orientation of the first sensor, i.e., ${\VECJ r{s,1}{}f{}(\alpha_{s}^{z}=\beta_{s}^{y}=\gamma_{s}^{x}=0{^\circ})}$.
The individual constrained spaces were computed for each sensor considering
the constrained space affected by the feature geometry, i.e., $\CC 3{s_{1}}{}$
and $\CC 3{s_{2}}{}$. Since the frustum space of the second sensor
always lies within the first one, we additionally considered a fictitious
accuracy constraint for the depth of the second sensor, $\CC 5{s_{2}}{}(a_{s_{2}}(z_{min}=500\,\text{mm},z_{max}=700\,\text{mm}))$,
to limit its working distance.

Figure <ref> visualizes the described
scene and resulting manifolds of the constrained spaces. Figure <ref>
displays the frustum spaces, rendered depth images of both sensors,
and the resulting point cloud at an extreme viewpoint. The rendered
images provide a visual verification of our approach demonstrating
that $f_{1}$ is visible from both sensors. Note that the second device
represents an active structured light projector (see in Table <ref>.)

The total computation time of the constrained space was estimated
at $200\,ms$. The computation time depended mainly on the intersection
from Step 5, which corresponded to $100\,ms$. The computation results
just apply to this case. Such analyses are difficult to generalize
since the complexity of Boolean operations depends decisively on the
number of vertices of the manifolds of the occluding objects.

§.§.§ Summary

This subsection introduced the formulation and characterization of
a $\ACC$, which considers the intrinsic configuration of range sensors
comprising at least two imaging devices. Our approach enables the
combination of individual viewpoint constraints for each device and
the characterization of a $\ACC$, which fulfills simultaneously all
viewpoint constraints from all imaging devices.

The strategy proposed in Algorithm <ref> demonstrated to
be valid and efficient to characterize the manifolds of such $\ACC$.
Nevertheless, we do not dispose alternative approaches for its characterization.
For instance, if the frustum spaces are intersected initially, the
resulting frustum space can be used as the base for spanning the rest
of the constraints. However, we consider the steps proposed within
this subsection more traceable, modular, and extendable to consider
further constraints, multisensor systems, or even transferable to
similar problems. For example, a variation of the algorithm could
be applied for maximizing or guaranteeing the registration space between
two different viewpoints, which represents a fundamental challenge
within many vision applications Bauer.2021.

§.§ Robot Workspace

This section outlines the formulation of the robot workspace as a
further viewpoint constraint to be seamlessly and consistently integrated
with the other $\ACC$s.

§.§.§ Formulation

A viewpoint can be considered valid if a sensor pose is reachable
by the robot: hence, it lies within the robot workspace $\SP{}\in\mathcal{W}_{r}$.
The formulation of a constraint can then be straightforwardly formulated
as follows:

\begin{equation}
\CC 8{}{}=\mathcal{W}_{r}=\{\SP{}\in\CC{,8}{}{}\mid\SP{}\in\mathcal{W}_{r}\}.\label{eq:CV8}
\end{equation}

§.§.§ Characterization and Verification

In our work, we assume that the robot workspace is known and can be
characterized by a manifold in the special Euclidean. Considering
this assumption, the constrained space $\CC 8{}{}$ can be seamlessly
intersected with the rest of the viewpoint constraints. Figure <ref>
shows an exemplary scene for acquiring feature $f_{1}$ and the resulting
constrained manifold $\CC 3{}{}\stackrel{}{\bigcap}\CC 8{}{}$, which
considers a robot with a workspace of a half-sphere and a working
distance of $1000\,\text{mm}$–$1800\,\text{mm}$ and the $\ACC$
manifold $\CC 3{}{}$ spanned by the feature geometry.

Characterization of the robot workspace as a further $\protect\ACC$
$\protect\CC 8{}{}$ and integration with other $\protect\ACC$s,
e.g., here $\protect\CC 3{}{}$, using a $\protect\ACSG$ Intersection

§.§.§ Discussion

A more comprehensive formulation and characterization of the robot
workspace to consider singularities requires more detailed modeling
of the robot kinematics. Furthermore, our study has not considered
the explicit characterization of the collision-free space of a robot,
which had been the focus of exhaustive research in the last three
decades. We assume that an explicit proof for collision must be performed
in the last step for a selected sensor pose within the $\ACC$. Nevertheless,
we consider that our approach contributes substantially to a significant
problem simplification by delimiting the search space to compute collision-free
robot joint configurations more efficiently.

§.§ Multi-Feature Spaces

Up to this point, our work has outlined the formulation and characterization
of $\ACC$s to acquire just one feature. Within this subsection we
briefly outline the characterization of a $\text{\ensuremath{\ACC}}$,$\CC{}{}F$,
which allows capturing a set of features $F$ and the simultaneous
fulfillment of all viewpoint constraints from all features $f_{m}\in F$
with $m=1,\dots,n$.

§.§.§ Characterization

The characterization of $\CC{}{}F$ can be seamless achieved according
to the two steps described in Algorithm <ref>. In the first
step, the $\ACC$ for all $n$ features, $\CC{}{}{f_{m}}$, are characterized
considering a fixed sensor orientation $\CR{}$ and the individual
constraints $\widetilde{C}(f_{m})$. Then the constrained space $\CC{}{}F$
is synthesized by intersecting all individual constrained spaces.
Figure <ref> shows the characterization of such a space for
the acquisition of two features.

Integration of $\protect\ACC$ for multiple features

* Compute all $n$ $\ACC$s for each feature $\forall f_{m}\in F$ with

\[
\CC{}{}{f_{m}}:=\CC{}{}{}(\CR{},\widetilde{C}(f_{m}))
\]

* Compute the joint $\ACC$ by intersecting all $n$ $\ACC$s:

\[
\CC{}{}F=\CC 9{}F=\stackrel[\begin{aligned}f_{m}\in F\end{aligned}
\]

The characterization of the $\protect\ACC$ spanned by two features
is computed by  intersecting its constrained spaces using the same
sensor orientation.

§.§.§ Verification

To verify the proposed characterization of a constrained space for
acquiring multiple features, we outlined an exemplary use case comprising
two features $\{f_{1},f_{2,}\}\in F$ (see Table <ref>).
We computed the space, $\CC{}{}F$, according to the steps of Algorithm
<ref>, considering an orientation of $\VECJ rs{}{f_{1}}{}(\alpha_{s}^{z}=\beta_{s}^{y}=0{^\circ},\gamma_{s}^{x}=-10{^\circ},)$
relative to the feature $f_{1}$. Figure <ref>
shows the described scene and visualizes the resulting joint space.
The rendered scene and range images of Figure <ref> confirm
the validity of the $\CC{}{}F$, demonstrating that both features
can be simultaneously acquired at two extreme viewpoints within this

Verification of the $\protect\CC{}{}F$ at two extreme viewpoints
$\{\protect\VECJ p{s,1}{}{}{},\protect\VECJ p{s,2}{}{}{}\}\text{\ensuremath{\in\protect\CC{}{}F}}$:
rendered scene (left image) and depth images of $\protect\VECJ p{s,1}{}{}{}$(right
image in the upper corner) and $\protect\VECJ p{s,2}{}{}{}$ (right
image in the lower corner).

§.§.§ Summary and Discussion

Within this subsection, we demonstrated that $\ACC$s from different
features can be seamlessly combined to span a topological space that
guarantees the acquisition of these features satisfying simultaneously
the individual feature viewpoint constraints.

The current study assumes that the sensor orientation can be arbitrarily
chosen and that the features can be acquired jointly by the sensor.
In most applications, such assumptions cannot always be met and the
following fundamental question arises: which features can be acquired
simultaneously and which is an adequate sensor orientation? This questions
fall outside the scope of this paper and yields the motivation of
our ongoing research, which addresses the efficient combination of
$\ACC$s to tackle the superordinated $\AVPP$.

§.§ Constraints Integration Strategy

The integration of viewpoint constraints can be considered to be commutative,
i.e., the order of computation and integration of the constraints
do not affect the characterization of the final constrained space.
However, due to the diverse computation techniques that our our framework
considers, a well-thought-out strategy may contribute to increase
the computational efficiency of the overall process. In this publication,
we outline one possible and simple strategy described in Algorithm
<ref> to integrate all viewpoint
constraints into a single $\ACC$.

The optimal integration of constraints falls outside the scope of
this publication. Moreover, we consider that an optimal and efficient
strategy can be tailored just considering the individual application
and its specific constraints.

Strategy for the integration of viewpoint constraints.

* Consider a fixed sensor orientation $\VECJ rs{fix}{}{s_{1}}$ for
the reference imaging device $s_{1}$.

* Compute the $\ACC$s manifolds

\[
\CC{1-6}{s_{t}}{f_{m}}(\VECJ rs{}{}{s_{t}}(\VECJ rs{fix}{}{s_{1}}))
\]

of imaging device $s_{t}$ for each feature $\forall f_{m}\in F$
considering the sensor orientation of the first device $\VECJ rs{fix}{}{s_{1}}$
and the viewpoint constraints $1-6$.

* Compute the $\ACC$ of all features for sensor $s_{t}$:

\[
\CC{}{s_{t}}F=\stackrel{n}{\bigcap}\CC{1-6}{s_{t}}{f_{m}}.
\]

* Repeat Steps 1–3 for all imaging devices $\forall s_{t}\in\widetilde{S}$.

* Compute the $\ACC$ for all $u$ imaging devices, e.g., for $s_{1}$:

\[
\CC{}{\widetilde{S}_{1}}F=\CC{}{s_{1}}F\stackrel[s_{t}\in\widetilde{S}]{}{\bigcap}\CC 7{s_{1},s_{t}}F.
\]

* Intersect the robot workspace to obtain the final $\ACC$ b, e.g.,
for $s_{1}$:

\[
\CC{}{\widetilde{S}_{1}}F=\CC{}{\widetilde{S}_{1}}F\bigcap\CC 8{}{}.
\]

§ EVALUATION

Within this section, a comprehensive evaluation of the constraint
formulations and their integration is undertaken. First, Subsection
<ref> verifies the formulations of all
regarded viewpoint constraints of our work based on an academic example.
In Subsection <ref>, the framework's
broad generality and applicability is evaluated within an industrial
$\ARVS$ comprising two different sensors.

Overview and description of the viewpoint constraints considered for
the simulated-based analysis.

Viewpoint Constraint     1>p0.5Description     1>p0.2Approach1     Two sensors($s^{1}$, $s^{2}$) with two imaging devices each: $\{s_{1}^{1},s_{2}^{1},s_{3}^{2},s_{4}^{2}\}\in\widetilde{S}$.
The imaging parameters of all devices are specified in Table <ref>.     Linear algebra and geometry2     Relative orientation to the object's frame: $\VECJ rs{}o{}(\alpha_{s}^{z}=\gamma_{s}^{x}=0,\beta_{s}^{y}=6.64{^\circ})$.     Linear algebra and geometry3     A planar rectangular object with three different features $\{f_{1},f_{2},f_{3}\}\in F$
(see Table <ref>)     Linear algebra, geometry, and trigonometry4–5     The workspace of the second imaging device is restricted in the $z$-axis
to the following working distance $z_{s_{2}}>450\,\text{mm}$.     Linear algebra and geometry6     Two objects with the form of an icosahedron ($\kappa_{1}$) and a
octahedron ($\kappa_{2}$) occlude the visibility of the features.     Linear algebra, ray-casting, and $\ACSG$ Boolean Operations7     All constraints must be satisfied by all four imaging devices simultaneously.     Linear algebra and $\ACSG$ Boolean Operations8     Both sensors are attached to a 6-axis industrial robot. The robot
has a workspace of a half-sphere with a working distance of $1000\,\text{mm}$–$1800\,\text{mm}$.     $\ACSG$ Boolean Operation9     All features from the set $G$ must be captured simultaneously.     $\ACSG$ Boolean Operation

§.§ Academic Simulation-Based Analysis

This subsection presents a simple but thorough academic use case that
considers all introduced viewpoint constraints to perform an exhaustive
evaluation of the presented formulations. As stated in the introduction,
with this present scenario we aim to provide a first draft of a much-needed
benchmark for other researchers that can be used as basis for further
development, reproducibility and comparison. The surface models, the
resulting manifolds of the computed $\ACC$s, and the frustum spaces,
can be found attached in the additional material of our publication.

§.§.§ Use-Case Description

The exemplary case regards an $\ARVS$ with two sensors and an object
of interest containing three different features with different sizes
and geometries. Table <ref> gives a detailed
overview of the considered constraints. Besides the imaging parameters
of both range sensors, all other parameters can be assumed to be fictitious
though realistic. The kinematic and imaging models corresponds to
the real $\ARVS$ depicted in Figure <ref>.

Characterization of the $\protect\ACC$ spanned by a set of viewpoint
constraints (see Table <ref>) for a multisensor
scenario to capture a set of features $F$.

Left: Verification scene visualizing the frames and $\protect\ACM$s
of all imaging devices at the extreme sensor pose $\protect\VECJ ps{}{}{s_{1}}\in\protect\CC{}{\widetilde{S}_{1}}F$
($\protect\VECJ ps{}{}{s_{2}}\in\protect\CC{}{\widetilde{S}_{2}}F$,
$\protect\VECJ ps{}{}{s_{3}}\in\protect\CC{}{\widetilde{S}_{3}}F$,
and $\protect\VECJ ps{}{}{s_{4}}\in\protect\CC{}{\widetilde{S}_{4}}F$)
that fulfills all viewpoint constraints. Right: Depth images of all
imaging devices at the corresponding sensor pose.

§.§.§ Results

Following the strategy described in Algorithm <ref>,
the joint $\ACC$s of the four imaging devices, i.e., $\CC{}{\widetilde{S}_{1}}F$,
$\CC{}{\widetilde{S}_{2}}F$, $\CC{}{\widetilde{S}_{3}}F$, and $\CC{}{\widetilde{S}_{4}}{F,}$
were computed for acquiring all features considering all viewpoint
constraints from Table <ref>. Figure <ref>
shows the complexity of the described case comprising three features
and some of the resulting $\ACC$s. The blue manifold $\CC{}{\widetilde{S}_{1}}F$
represents the final constrained space of the first imaging device
$s_{1}$. It can be appreciated that $\CC{}{\widetilde{S}_{1}}F$
is characterized by the intersection of all other $\ACC$s. Moreover,
Fig. <ref> shows that the $\CC{}{\widetilde{S}_{1}}F$
manifold is mainly constrained by the $\ACC$ corresponding to the
second range sensor, i.e., $\CC 7{s_{1},s_{3},s_{4}}F$.

To verify the validity of the computed $\ACC$s, the depth images
and point clouds for all imaging devices at eight extreme viewpoints
were rendered. Figure <ref> shows the corresponding
rendered scene and resulting depth images for each imaging device
at one extreme viewpoint $\VECJ ps{}{}{s_{1}}\in\CC{}{\widetilde{S}_{1}}F$.
The depth images demonstrate that all imaging device can successfully
acquire all features without occlusion simultaneously.

The total computation time for characterizing all $\ACC$s corresponded
to $t(\CC{}{\widetilde{S}_{1}}F,\CC{}{\widetilde{S}_{2}}F,\CC{}{\widetilde{S}_{3}}F,\CC{}{\widetilde{S}_{4}}F)\approx50\,s$.
However, this time comprises other computation steps (e.g., frames
transformation and inverse kinematic operations using ROS-Services),
which distort the effective computation time of the $\ACC$s. A proper
analysis of the computational efficiency of the whole strategy remains
to be further investigated.

§.§.§ Summary

Despite the complexity of the use case, the framework (models, methods,
and integration strategy) presented within this paper demonstrated
its effectiveness in characterize a continuous topological space in
the special Euclidean, where all defined viewpoint constraints could
be fulfilled. The simulated depth-images and point clouds confirmed
that all selected viewpoints within the characterized $\ACC$ satisfied
all regarded constraints. Moreover, the proposed academic example
effectively outlines a simple but sufficient complex scenario to benchmark
our and future viewpoint planning strategies.

§.§ Real Experimental Analysis

To assess the usability and validity of our framework within real
applications, the framework presented in this study was utilized to
generate automatically valid viewpoints for capturing different features
of a car door using real $\ARVS$, i.e., the AIBox from ZEISS.
The AIBox is an industrial measurement cell used to automate different
vision-based quality inspection tasks such as dimensional metrology
and digitization, among others.

§.§.§ System Description

The AIBox is an integrated industrial $\ARVS$, equipped with a structured
light sensor (ZEISS COMET PRO AE), a six-axis industrial robot
(Fanuc M-20ia), and a rotary table for mounting an inspection
object. Moreover, to evaluate the use of our approach considering
a multisensor system, we additionally attached a stereo sensor (rc_visard
65, Roboception) to the structured light sensor. The imaging parameters
of both sensors are given in Table <ref>.
Figure <ref> provides an overview of the reconfigured AIBox
with the stereo sensor. We assume that the inspection object is roughly
aligned, e.g., in Magana.2020b we presented a CNN fully
trained on synthetic data to automate this task using the $\ARVS$

§.§.§ Vision Task Definition

The validation of our framework was performed on the basis of two
vision tasks considering diverse viewpoint constraints. For the first
task, we considered just the ZEISS sensor to acquire the features
$\{f_{1},f_{2}\}\in F_{1}$, which lie on the outside of the door
and can be potentially occluded by the door fixture. For the second
task, we considered both sensors and the acquisition of two features
$\{f_{3},f_{4},f_{5}\}\in F_{2}$ on the inside of the door. The incidence
angle for the first case corresponded to a sensor orientation of $\VECJ r{s_{2}}{}o{}(\alpha_{s}^{z}=\gamma_{s}^{x}=0{^\circ},\beta_{s}^{y}=-15{^\circ})$
and for the second of $\VECJ r{s_{1}}{}o{}(\alpha_{s}^{z}=\beta_{s}^{y}=\gamma_{s}^{x}=0{^\circ})$.
To compensate any kinematic modeling uncertainties, we consider an
overall kinematic error of $\epsilon_{s_{1}}^{x,y,z}=(70.0,70.0,50.0)\,\text{mm}$
for $s_{1}$ and of $\epsilon_{s_{3,4}}^{x,y,z}=(30.0,30.0,30.0)\,\text{mm}$
for $s_{3}$ and $s_{4}$.

§.§.§ Results

For both vision tasks we computed the necessary $\ACC$s aligned to
the strategy presented by Algorithm <ref>.
The $\ACC$s of the first inspection scenario for the camera $\CC{}{\widetilde{S}_{1}}{F_{1}}$
and projector $\CC{}{\widetilde{S}_{2}}{F_{1}}$ of the Comet
Pro AE and its corresponding occlusion spaces are displayed on the
left image of Figure <ref>. To assess
the validity of the characterized $\ACC$s, we chose diverse extreme
viewpoints at the vertices of the $\CC{}{\widetilde{S}_{1}}{F_{1}}$
manifold and performed real measurements. On the right side of Figure
<ref> the real monochrome images of
the camera and the resulting point clouds at two validating viewpoints
are displayed. The 2D images and point clouds prove that both features
can be successfully acquired from these both viewpoints, which confirms
the free sight for the sensor and the illumination of both features
without shadows.

Moreover, on the left of Figure <ref>
the constrained spaces of the first imaging device of each sensor,
i.e., $\CC{}{\widetilde{S}_{1}}{F_{2}}$ and $\CC{}{\widetilde{S}_{3}}{F_{2}}$,
are visualized for the second inspection scenario. Analogously to
the first scenario, two extreme viewpoints at the vertices of the
manifolds were selected to assess the validity of the computed $\ACC$s.
As expected, the real 2D images of all imaging devices and the resulting
point clouds of both sensors at two exemplary extreme viewpoints (shown
on the right of Figure <ref>) demonstrate
that all features can be successfully acquired by the four imaging
devices of both sensors.

§.§.§ Summary

Using an industrial RVS, and regarding real viewpoint constraints,
we were able to validate the formulations, characterization, and application
of $\ACC$s for inspection tasks in an industrial context. These experiments
show the suitability of our framework for an industrial application
on a real $\ARVS$ with multiple range sensors.

Furthermore, our strategy for merging individual $\ACC$s to capture
more than one feature proved to be effective for the regarded vision
tasks. However, a more complex task such as the inspection of all
door features requires a more complex strategy, which considers the
search of features that can be acquired together. This question recalls
the overall $\AVPP$, which falls outside the scope of this publication
and we intend to address in our future work.

Overview of the core components of the reconfigured inspection $\protect\ARVS$

a) Visualization of the characterized $\protect\ACC$s($\protect\CC{}{\widetilde{S}_{1}}{F_{1}}$,$\protect\CC{}{\widetilde{S}_{2}}{F_{1}}$)
to capture features $f_{1}$ and $f_{2}$ by the COMET ProAE.
Right figures: 2D images and corresponding point clouds at two extreme
viewpoints: b) for $\protect\VECJ p{s,1}{}{}{s_{1}}\in\protect\CC{}{\widetilde{S}_{1}}{F_{1}}$
and c) for $\protect\VECJ p{s,2}{}{}{s_{1}}\in\protect\CC{}{\widetilde{S}_{1}}{F_{1}}$.

a): Visualization of the characterized $\protect\ACC$s ($\protect\CC{}{\widetilde{S}_{1}}{F_{2}}$,$\protect\CC{}{\widetilde{S}_{3}}{F_{2}}$)
to capture the feature set $\{f_{3},f_{4},f_{5}\}\in F_{2}$ by the
COMET ProAE and rc_visard 65. Right figures: 2D images
and corresponding point clouds for both sensors at two extreme viewpoints
$\{\protect\VECJ p{s,1}{}{}{s_{1}},\protect\VECJ p{s,2}{}{}{s_{1}}\}\in\protect\CC{}{\widetilde{S}_{1}}{F_{2}}$
(upper figures b) and d): Comet Pro AE, lower figures c) and
e): rc_visard 65).

§ SUMMARY AND OUTLOOK

§.§ Summary

The computation of valid viewpoints considering different system constraints,
named $\AVGP$ in this publication, is considered a complex and unsolved
challenge that lacks a generic and holistic framework for its proper
formulation and resolution. In this paper, we outline the $\AVGP$
as a geometric problem that can be solved explicitly in the special
Euclidean $SE(3)$ using suitable and explicit models of all related
domains of an $\ARVS$ and viewpoint constraints. Within this context,
much of our effort was devoted to the comprehensive and systematic
formulation of the $\AVGP$ and the exhaustive characterization of
domains and viewpoint constraints aligned to the formulation of geometric

The core result of this study is the characterization of $\ACC$s,
which can be understood as topological manifolds that span a space
with infinite viewpoint solutions to acquire one feature or a group
of features considering various viewpoint constraints and modeling
uncertainties. Our approach focuses on providing rather infinite valid
solutions instead of optimal ones. If the entire a priori knowledge
of the RVS can be formalized and integrated into the $\ACC$, then
it we can assume that any viewpoint within it is a local optimum.
Our work shows that a handful of viewpoint constraints can be efficiently
and simply modeled geometrically and integrated in a common framework
to span such constrained spaces. Finally, based on a comprehensive
academic example and a real application, we demonstrate the usability
of such a framework.

§.§ Limitations and Chances

We are aware that the framework proposed in the present study may
have some limitations that may prevent its straightforward application
for other $\ARVS$ or use-cases. First, it must be regarded that our
framework can be classified under the category of model-based approaches.
Therefore, in a first step a priori information to model the components
of the considered $\ARVS$ must be regarded. We consider that an exhaustive
and explicit modeling of the necessary domains is necessary for delivering
solutions that offer a higher generalization for other applications
and systems. For the benefit of generalization, complexity reduction,
and computational efficiency we regarded various simplifications,
which could affect the accuracy of some models and might yield more
conservative, however, more robust solutions.

We firmly believe that the $\AVGP$ can be efficiently solved geometrically.
We demonstrated that many constraints can be explicitly and efficiently
characterized by combining several techniques, including linear algebra,
trigonometry, and geometrical analysis. In the scope of our experiments,
we confirmed that the computation of the $\ACC$s manifolds based
on these approaches ran efficiently in linear times. However, we also
noted that algorithms comprising $\ACSG$ Boolean operations are more
computationally expensive, especially on calculations considering
multiple Boolean operations on the same manifold. Although this limitation
can be minimized by filtering and smoothing algorithms for decimating
manifolds, this characteristic could still be considered insufficient
for some users and applications. Although the shortcomings of $\ACSG$
Boolean techniques regarding their computational efficiency have been
mentioned in some prior works, we also believe that the present available
computational performance and paralleling capabilities of CPUs and
GPUs require a new reevaluation of their overall performance. Additionally,
our work also suggests that combined with efficient imaging processing
libraries, approaches requiring heavy use of $\ACSG$ operations can
be efficiently used within many applications. Nevertheless, a comprehensive
computational efficiency analysis to find a break-even point between
our approach and others remains to be further investigated.

Moreover, we also see room for improvement to increase the efficiency
of some of the algorithms presented. For instance, the computation
of the occlusion space and integration of constraints could also be
improved using an alternative strategy and more efficient algorithms
implemented in low-level programming languages. Additionally, we also
see potential for improving the efficiency of some algorithms. For
instance, the performance of many algorithms could enormously benefit
of computational optimization techniques such as parallelization and
GPU computation. For replication purposes of our work, we encourage
the reader to make a thorough evaluation of the performance of the
state-of-the-art libraries available at the present time, according
to their application needs and system requirements.

§.§ Outlook

We consider the use of $\ACC$s appropriate, but not limited to vision
tasks that rely on features. For example, we showed how our concept
could be extended to applications that generally would not consider
features and demonstrate its application for an object detection problem
with a certain level of spatial uncertainty. Our ongoing work concentrates
on assessing further applications or systems that may benefit from
our approach, e.g., feature-based robot calibration or adaption to
laser sensors. Further studies should still be undertaken in this
direction to verify the usability and explore the limitations of our
framework within other applications and $\ARVS$s.

Recalling that we neglected any sensor parameters that may directly
constrain the $\ACC$, e.g., exposure time, gain, and others, we consider
some other lines of research that integrate such a parameter space
in the $\ACV$. For instance, our ongoing study investigates the combination
of a data-based approach to optimize the exposure times and the use
of $\ACC$s for finding optimized sensor poses.

The most promising future research should be devoted to the overall
problem of the $\AVPP$, which could be reformulated based on the
present study and its findings. Further research that will exploit
this and comprises a holistic strategy for its resolution is already
in progress.

We believe that our work will serve as a solid base and guideline
for further studies to adapt and extend our framework according to
the individual requirements of their concrete applications and $\ARVS$.

§ ACKNOWLEDGMENTS

The framework presented in this paper was developed and thoroughly
evaluated within the scope of the “CyMePro” (Cyber-physical measurement
technology for 3D digitization in the networked production) project,
which was funded by the Bavarian Ministry of Economic Affairs, Regional
Development and Energy (funding code ESB036 / 001).

We thank our research partners AUDI AG and Carl Zeiss Optotechnik
GmbH for the fruitful discussions and their cooperation. Moreover,
we are deeply grateful to our colleagues Daria Leiber for the fruitful
discussions regarding the initial drafting and Benedikt Schmucker
for the critical revision of the present publication.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Springer International Publishing

booktitleSpringer Handbook of Robotics
title3-D Vision for Navigation and Grasping

titleSurvey on the View Planning Problem for Reverse Engineering
and Automated Control Applications
journaltitleComputer-Aided Design

titleViewpoint placement for inspection planning
journaltitleMachine Vision and Applications

In this paper we provide a broad survey of developments in active vision in
robotic applications over the last 15 years. With increasing demand for
robotic automation, research in this area has received much attention. Among
the many factors that can be attributed to a high-performance robotic system,
the planned sensing or acquisition of perceptions on the operating
environment is a crucial component. The aim of sensor planning is to
determine the pose and settings of vision sensors for undertaking a
vision-based task that usually requires obtaining multiple views of the
object to be manipulated. Planning for robot vision is a complex problem for
an active system due to its sensing uncertainty and environmental
uncertainty. This paper describes such problems arising from many
applications, e.g. object recognition and modeling, site reconstruction and
inspection, surveillance, tracking and search, as well as robotic
manipulation and assembly, localization and mapping, navigation and
exploration. A bundle of solutions and methods have been proposed to solve
these problems in the past. They are summarized in this review while enabling
readers to easily refer solution methods for practical applications.
Representative contributions, their evaluations, analyses, and future
research trends are also addressed in an abstract level.

titleActive vision in robotic systems: A survey of recent
journaltitleThe International Journal of Robotics Research

titleA survey of sensor planning in computer vision
journaltitleIEEE transactions on Robotics and Automation

titleA Comprehensive Review of Coverage Path Planning in Robotics
Using Classical and Heuristic Algorithms
journaltitleIEEE Access

The MVP (machine vision planner) model-based sensor planning system for
robotic vision is presented. MVP automatically synthesizes desirable camera
views of a scene based on geometric models of the environment, optical models
of the vision sensors, and models of the task to be achieved. The generic
task of feature detectability has been chosen since it is applicable to many
robot-controlled vision systems. For such a task, features of interest in the
environment are required to simultaneously be visible, inside the field of
view, in focus, and magnified as required. In this paper, we present a
technique that poses the vision sensor planning problem in an optimization
setting and determines viewpoints that satisfy all previous requirements
simultaneously and with a margin. In addition, we present experimental
results of this technique when applied to a robotic vision system that
consists of a camera mounted on a robot manipulator in a hand-eye

titleThe MVP sensor planning system for robotic vision tasks
journaltitleIEEE transactions on Robotics and Automation

titleView planning for automated three-dimensional object
reconstruction and inspection
journaltitleACM Computing Surveys (CSUR)

titleModeling Coverage in Camera Networks: A Survey
journaltitleInternational Journal of Computer Vision

titleOn the optimal placement of cameras for surveillance and the
underlying set cover problem
journaltitleApplied Soft Computing

titlePerformance-oriented view planning for automated object
University of Ottawa

titleAutomatic sensor placement from vision task requirements
journaltitleIEEE Transactions on Pattern Analysis and machine

IEEE Comput. Soc. Press

booktitleProceedings, 1989 International Conference on Robotics
and Automation
titleDetermining the camera and light source location for a
visual task

IEEE Comput. Soc. Press

booktitleProceedings 1992 IEEE Computer Society Conference on
Computer Vision and Pattern Recognition
titleComputing occlusion-free viewpoints

titleComputing occlusion-free viewpoints
journaltitleIEEE Transactions on Pattern Analysis and machine

titleComputing Camera Viewpoints in an Active Robot Work Cell
journaltitleThe International Journal of Robotics Research

titleSolid model acquisition from range imagery
Columbia University

titleConstraint-based sensor planning for scene modeling
journaltitleIEEE Transactions on Pattern Analysis and machine

titleIVIS: An integrated volumetric inspection system
_ an Integrated Volumetric Inspection System:Atta
Inspection Sys
journaltitleComputer Vision and Image Understanding

hen the object being viewed is polyhedral, what is meant by qualitatively
the same is simply that all views within the same aspect of an object must be
able to view the same vertices, edges, and faces on the polyhedron. When the
object is not polyhedral, an aspect is usually defined as a maximal set of
viewpoints from which the topology of the silhouette of the object does not
change, although the definition of topology may vary from application to

titlePlanning for complete sensor coverage in inspection
journaltitleComputer Vision and Image Understanding

Laser scanning;Object inspection;Object reconstruction;Range
camera;rank5;relevant;View planning
This paper presents a multi-phase, model-based approach to view planning
for automated, high fidelity object inspection or reconstruction by means of
laser scanning range sensors. We describe the critical phase, fine modeling,
in detail. Quality objectives and performance measures are defined. Camera
and positioning system performance is modeled statistically. A theoretical
framework is presented. The method is applicable to a broad class of objects
with reasonable geometry and reflectance properties. Sampling of object
surface and viewpoint space is characterized, including measurement and pose
errors. The technique is generalizable for common range cameras and
positioning systems.

titleModel-based view planning
journaltitleMachine Vision and Applications

* Invalid format of field 'urldate'

titleView and sensor planning for multi-sensor surface
_2016_Surf._Topogr. _Metrol._Prop._4_02400
journaltitleSurface Topography: Metrology and Properties

booktitle2017 IEEE/RSJ International Conference on Intelligent
Robots and Systems (IROS)
titleSampling-based coverage motion planning for industrial
inspection application with redundant robotic system

B-spline surface;Inspection automation;model-based planning;Object
space exploration;View planning
Machine Vision and Applications,

titleFeature-Driven Viewpoint Placement for Model-Based Surface
achine Vision and Applications

3D object positioning;3D range imaging system;approximate
model;automatic optical inspection;CAD;computer aided facilities
layout;computer vision;feature inspection representation;feature
visibility;feature-based object models;Finite impulse response
filter;GASP;general automatic sensor planning;generalized sensor model;image
segmentation;image sensors;imaging sensors;inspection;inspection
scripts;measurement reliability;model-based planning;object location;optimal
sensor placements;Position measurement;quantitative
assessment;rank5;relevant;robust outlier removal;Robustness;Sensor
systems;Software performance;Software systems;surface fitting;surface
segmentation;viewpoint optimality
titleModel-based planning of optimal sensor placements for
journaltitleIEEE transactions on Robotics and Automation

titleA solution to the next best view problem for automated
surface acquisition
journaltitleIEEE Transactions on Pattern Analysis and machine

booktitleJoint Pattern Recognition Symposium
titleFeature and viewpoint selection for industrial car

booktitleScandinavian Conference on Image Analysis
titleA versatile model-based visibility measure for geometric

titleOff-line view planning for the inspection of mechanical
journaltitleInternational Journal on Interactive Design and
Manufacturing (IJIDeM)

titleSensor planning system for fringe projection scanning of
sheet metal parts

titleAutomated inspection planning of free-form shape parts by
laser scanning
journaltitleRobotics and Computer-Integrated Manufacturing

The objective of this paper is to describe a new method to determine the
three-dimensional (3D) digitizing strategy for a computer aided design (CAD)
known mechanical part by using a plane laser sensor. The Research Centre for
Automatic control in Nancy, France has initiated a project to create an
automatic 3D digitizing system. Previous papers focused on the visibility
determinations using spherical geometry in a three axis environment. Here,
the latest developments are presented. They enable one to improve this
approach by taking into account a five axis environment. Using a new approach
based on the Minkovsky operations to calculate the visibility of the
different face of the B-Rep model of the part, the minimum set of directions
required to entirely digitize the part is computed.

title3D Digitizing Strategy Planning Approach Based on a CAD
journaltitleJournal of Computing and Information Science in

* Invalid format of field 'urldate'

family=Tekouo Moutchiho,

titleA New Programming Approach for Robot-based Flexible
Inspection Systems

Technical University of Munich

booktitleProceedings of the International Workshop on Distributed
Smart Cameras
titleA Look-up Table Based Approach for Solving the Camera
Selection Problem in Large Camera Networks

ACM Press

booktitleProceedings of the seventeenth annual symposium on
Computational geometry - SCG '01
titleA randomized art-gallery algorithm for sensor placement
New York, New York, USA

Algorithms;Animation;Artificial Intelligence;Automated;automatic
optical inspection;automatic sensor
placement;automation;characterization;Christofides algorithm;Computational
geometry;Computer-Assisted;Distributed computing;feature extraction;genetic
algorithm;genetic algorithms;Image Enhancement;Image Interpretation;image
sensors;Imaging;industrial robots;Machine vision;minimax techniques;min-max
criterion;model-based robot vision system;Models;object feature
resampling;Pattern Recognition;Photogrammetry;Robot sensing systems;robot
vision;Robot vision systems;Robotics;Sensor phenomena;sensor planning;Sensor
systems;Theoretical;Three-Dimensional;Transducers;viewpoint planner
titleAutomatic sensor placement for model-based robot vision
journaltitleIEEE transactions on systems, man, and cybernetics.
Part B, Cybernetics : a publication of the IEEE Systems, Man, and Cybernetics

titleAutomated camera layout to satisfy task-specific and floor
plan-specific coverage requirements
journaltitleComputer Vision and Image Understanding

titleSemiautomatic Model-Based View Planning for Active
Triangulation 3-D Inspection Systems
journaltitleIEEE/ASME Transactions on Mechatronics

titleCoverage path planning with targetted viewpoint sampling for
robotic free-form surface inspection
journaltitleRobotics and Computer-Integrated Manufacturing

A novel method is proposed in this paper for automatic acquisition of
three-dimensional (3-D) models of unknown objects by an active vision system,
in which the vision sensor is to be moved from one viewpoint to the next
around the target to obtain its complete model. In each step, sensing
parameters are determined automatically for incrementally building the 3-D
target models. The method is developed by analyzing the target's trend
surface, which is the regional feature of a surface for describing the global
tendency of change. While previous approaches to trend analysis are usually
focused on generating polynomial equations for interpreting regression
surfaces in three dimensions, this paper proposes a new mathematical model
for predicting the unknown area of the object surface. A uniform surface
model is established by analyzing the surface curvatures. Furthermore, a
criterion is defined to determine the exploration direction, and an algorithm
is developed for determining the parameters of the next view. Implementation
of the method is carried out to validate the proposed method.

titleVision sensor planning for 3-D model acquisition
journaltitleIEEE transactions on systems, man, and cybernetics.
Part B, Cybernetics : a publication of the IEEE Systems, Man, and Cybernetics

titleVolumetric Next-best-view Planning for 3D Object
Reconstruction with Positioning Error
journaltitleInternational Journal of Advanced Robotic Systems

titleEfficient next-best-scan planning for autonomous 3D surface
reconstruction of unknown objects
journaltitleJournal of Real-Time Image Processing

titleMulti-Sensor Next-Best-View Planning as Matroid-Constrained
Submodular Maximization
journaltitleIEEE Robotics and Automation Letters

Springer International Publishing

booktitleSpringer Handbook of Robotics

family=Puente León,

Springer Berlin Heidelberg

titleMachine Vision
Berlin, Heidelberg

titleA genetic algorithm for the set covering problem
journaltitleEuropean Journal of Operational Research

titleA General Method for Sensor Planning in Multi-Sensor
Systems: Extension to Random Occlusion

journaltitleInternational Journal of Computer Vision

approximation theory;baseline greedy algorithm;Cameras;computational
complexity;decision theory;function approximation;greedy algorithms;intuitive
approach;learning (artificial intelligence);Markov processes;MDP;novel score
function;optimisation;Optimization;optimization problem;Planning;RL
algorithms;RL framework;SCOP;Solid modeling;Three-dimensional displays;view
planning problem;VPP
We present a Reinforcement Learning (RL) solution to the view planning
problem (VPP), which generates a sequence of view points that are capable of
sensing all accessible area of a given object represented as a 3D model. In
doing so, the goal is to minimize the number of view points, making the VPP a
class of set covering optimization problem (SCOP). The SCOP is NP-hard, and
the inapproximability results tell us that the greedy algorithm provides the
best approximation that runs in polynomial time. In order to find a solution
that is better than the greedy algorithm, (i) we introduce a novel score
function by exploiting the geometry of the 3D model, (ii) we device an
intuitive approach to VPP using this score function, and (iii) we cast VPP as
a Markovian Decision Process (MDP), and solve the MDP in RL framework using
well-known RL algorithms. In particular, we use SARSA, Watkins-Q and TD with
function approximation to solve the MDP. We compare the results of our method
with the baseline greedy algorithm in an extensive set of test objects, and
show that we can outperform the baseline in almost all cases.

booktitle2017 IEEE Conference on Computer Vision and Pattern
Recognition (CVPR)
titleA Reinforcement Learning Approach to the View Planning

Springer New York

booktitleAutonomous Robot Vehicles
titleSpatial Planning: A Configuration Space Approach
New York, NY

seriesThe Springer International Series in Engineering and
Computer Science, Robotics
titleRobot Motion Planning
Boston, MA

Cambridge University Press

titlePlanning algorithms

utomated planning
San Francisco, Calif.

seriesCognitive technologies
titleEssentials of constraint programming

booktitleICRA workshop on open source software
titleROS: an open-source Robot Operating System

titleConcept of a learning knowledge-based system for programming
industrial robots
journaltitleProcedia CIRP

booktitle2020 25th IEEE International Conference on Emerging
Technologies and Factory Automation (ETFA)
titleKnowledge-Based Service-Oriented System for the Automated
Programming of Robot-Based Inspection Systems

titleMesh arrangements for solid geometry
journaltitleACM Transactions on Graphics

journaltitleACM Transactions on Graphics

family=Unity Technologies,

Unity Technologies

Siemens AG

titleRay casting for modeling solids
journaltitleComputer Graphics and Image Processing

titleAn Introduction to ray tracing

titleOn the shape of a set of points in the plane
journaltitleIEEE Transactions on Information Theory

Eurographics Association

We show that surface reconstruction from oriented points can be cast as a
spatial Poisson problem. This Poisson formulation considers all the points at
once, without resorting to heuristic spatial partitioning or blending, and is
therefore highly resilient to data noise. Unlike radial basis function
schemes, our Poisson approach allows a hierarchy of locally supported basis
functions, and therefore the solution reduces to a well conditioned sparse
linear system. We describe a spatially adaptive multiscale algorithm whose
time and space complexities are proportional to the size of the reconstructed
model. Experimenting with publicly available scan data, we demonstrate
reconstruction of surfaces with greater detail than previously achievable.

booktitleProceedings of the Fourth Eurographics Symposium on
Geometry Processing
seriesSGP '06
titlePoisson Surface Reconstruction
Goslar, DEU

titleRegistration strategy of point clouds based on
region-specific projections and virtual structures for robot-based inspection

booktitle2020 25th IEEE International Conference on Emerging
Technologies and Factory Automation (ETFA)
titlePoseNetwork: Pipeline for the Automated Generation of
Synthetic Training Data and CNN for Object Detection, Segmentation, and
Orientation Estimation

Description of General Requirements

General Requirement      Description1. Generalization       The models and approaches used should be abstracted and generalized
in the best possible level so that they can be used for different
components of an RVS and can be applied to solve a broad range of
vision tasks.2. Computational Efficiency     The methods and techniques used should strive a low level of computational
complexity. Whenever possible, analytical and linear models should
be preferred over complex techniques, such as stochastic and heuristic
algorithms. Nevertheless, considering offline scenarios the trade-off
between computing a good enough solution within an acceptable amount
of time should be individually assessed.3. Determinism      Due to traceability and safety issues within industrial applications
deterministic approaches should be prioritized. 4. Modularity and Scalability      The approaches and models should consider in general a modular structure
and promote their scalability. 5. Limited A-priori Knowledge      The parameters required to implement the models and approaches should
be easily accessible for the end-users. Neither in-depth optics nor
robotic knowledge should be required.

Overview of index notations for variables

Notation      1>p0.36Index Description    $x\coloneqq$variable, parameter, vector, frame or transformation
$d\coloneqq$$\ARVS$ domain, i.e., sensor, robot,
feature, object, environment or d
element of a list or set
$\fr rbxdn$     $n\coloneqq$related domain, additional notation or depending variable
$r\coloneqq$base frame of the coordinate system $B_{r}$ or space
of feature $f$
$b\coloneqq$origin frame of the coordinate system $B_{b}$
Notes     The indexes $r$ and $b$ just apply for pose vectors, frames and
transformations.Example     The index notation can be better understood consider following examples:

* $d$: Let the geometry of a feature be described by a surface point
$\VECJ gf{}{}{}\in\mathbb{R}^{3}$

* $d$: If the feature comprises more surface points, then let the point
with the index $2$ be denoted by $\VECJ g{f,2}{}{}{}\in\mathbb{R}^{3}$.

* $r$: Assuming that the position of a surface point $\VECJ gf{}{}{}$
is described in the coordinate system of the feature, $B_{f}$, then
it follows: $\VECJ gf{}f{}$. In case that the base coordinate
frame has the same notation as the domain itself, i.e. $r=d$, then
just the index for the domain is given: $\VECJ gf{}{}{}=\VECJ gf{}f{}$.

* $b$: In case that the frame of the surface point is given in the
coordinate reference system of the object $B_{o}$, then following
notation applies: $\VECJ gf{}o{}=\VECJ gf{}of$.

List of most common symbols

Symbol     1>p0.3Description    General$c$     Viewpoint constraint$\widetilde{C}$      Set of viewpoint constraints$f$     Feature$F$     Set of features$s_{t}$     $t$ imaging device of sensor $s$$v$     Viewpoint$\SP{}$     Sensor pose in $SE(3)$    Spatial Dimensions$B$     Frame$\VECJ t{}{}{}{}$     Translation vector in $\mathbb{R}^{3}$$\VECJ r{}{}{}{}$     Orientation matrix in $\mathbb{R}^{3x3}$$\boldsymbol{V}$     Manifold vertex in $\mathbb{R}^{3}$    Topological Spaces$\CC{}{}{}$     $\ACC$ for a set of viewpoint constraints $\widetilde{C}$ $\CC i{}{}$     i $\ACC$ of the viewpoint constraint $c_{i}$$\CF$     Frustum space

Overview of features and occlusion objects used for verification steps
and simulation-based analysis

Feature     1>p0.1$f_{0}$     1>p0.1$f_{1}$,$f_{1}^{*}$     1>p0.12$f_{2}$     1l$f_{3}$     1l$\kappa_{1}$     1l$\kappa_{2}$Topology     Point     Slot     Circle     Half-Sphere     Icosahedron     OctaederGeneralized Topology     -     Square     Square     Cube     -     -Dimensions in $\text{mm}$     $l_{f_{o}}=0$     $l_{f_{1}}=50$, $h_{f_{1}^{*}}=30$     $l_{f_{2}}=20$     ${l_{f_{2}}=40}$, $h_{f_{2}}=40$     edge length: $\approx14.0$     edge length: $\approx20.0$Translation vector in object's frame $\fr{}{}to{}=(x_{o},y_{o},z_{o})^{T}$
in $\text{mm}$     $\left(\begin{array}{c}
\end{array}\right)$     $\left(\begin{array}{c}
\end{array}\right)$     $\left(\begin{array}{c}
\end{array}\right)$     $\left(\begin{array}{c}
\end{array}\right)$     $\left(\begin{array}{c}
\end{array}\right)$     $\left(\begin{array}{c}
\end{array}\right)$Rotation in Euler Angles in object's frame $\fr{}{}ro{}(\gamma_{s}^{x},\beta_{s}^{y},\alpha_{s}^{z})$
in ${^\circ}$     $(0,0,0)$     $(0,0,0)$     $(0,20,0)$     $(0,0,0)$     $(0,0,0)$     $(0,0,0)$

Geometrical analysis for rotation around z axis.

Scaling factors for the vertices of the constrained space $\protect\VECJ Vk{\protect\CC 3{}{}}{}{}$
considering a sensor rotation around the $x$-axis or $y$-axis relative
to the feature frame

4>p0.35|Rotation around y-Axis     4>p0.35Rotation around x-Axis30.1$k$ Vertex of $\VECJ Vk{\CC 3{}{}}{}{}$     4c|$\VECJ rs{}f{}(\alpha_{s}^{z}=\gamma_{s}^{x}=0,\beta_{s}^{y}\neq0)$     4c$\VECJ rs{}f{}(\alpha_{s}^{z}=\beta_{s}^{y}=0,\gamma_{s}^{x}\neq0)$2-9 3-9 4-9 5-9 6-9 7-9 8-9 9-9
2c|$\Delta_{k}^{x}$     $\Delta_{k}^{y}$     $\Delta_{k}^{z}$     $\Delta_{k}^{x}$     2c|$\Delta_{k}^{y}$     $\Delta_{k}^{z}$    $\beta_{s}^{y}<0$     $\beta_{s}^{y}>0$                    $\gamma_{s}^{x}<0$     $\gamma_{s}^{x}>0$     1     $\lambda^{x}$     $\rho^{x}$     $\frac{l_{f}}{2}$          $\frac{l_{f}}{2}$     $\lambda^{y}$     $\rho^{y}$     2     $\rho^{x}$     $\lambda^{x}$     $\frac{l_{f}}{2}$          $\frac{l_{f}}{2}$     $\lambda^{y}$     $\rho^{y}$     3     $\sigma^{x}$     $\rho^{x}$     $\frac{l_{f}}{2}+\varsigma^{x,y}$          $\frac{l_{f}}{2}+\varsigma^{y,x}$     $\rho^{y}$     $\sigma^{y}$     4     $\rho^{x}$     $\sigma^{x}$     $\frac{l_{f}}{2}+\varsigma^{x,y}$     20.08$\rho^{z,y}$     $\frac{l_{f}}{2}+\varsigma^{y,x}$     $\rho^{y}$     $\sigma^{y}$     20.08$\rho^{z,x}$5     $\lambda^{x}$     $\rho^{x}$     $\frac{l_{f}}{2}$          $\frac{l_{f}}{2}$     $\rho^{y}$     $\lambda^{y}$     6     $\rho^{x}$     $\lambda^{x}$     $\frac{l_{f}}{2}$          $\frac{l_{f}}{2}$     $\rho^{y}$     $\lambda^{y}$     7     $\sigma^{x}$     $\rho^{x}$     $\frac{l_{f}}{2}+\varsigma^{x,y}$          $\frac{l_{f}}{2}+\varsigma^{y,x}$     $\sigma^{y}$     $\rho^{y}$     8     $\rho^{x}$     $\sigma^{x}$     $\frac{l_{f}}{2}+\varsigma^{x,y}$          $\frac{l_{f}}{2}+\varsigma^{y,x}$     $\sigma^{y}$     $\rho^{y}$

[1. Vertex of $\protect\VECJ V1{\protect\CC 3{}{}}{}{}$]

[2. Vertex of $\protect\VECJ V2{\protect\CC 3{}{}}{}{}$]

[3. Vertex of $\protect\VECJ V3{\protect\CC 3{}{}}{}{}$]

[4. Vertex of $\protect\VECJ V4{\protect\CC 4{}{}}{}{}$]

Derivation of the geometrical relationships for each vertex of the
$\protect\ACC$ $\protect\CC 3{}{}$ considering considering a sensor
rotation of $\protect\VECJ rs{}f{}(\beta_{s}^{y}>0,\alpha_{s}^{z}=\gamma_{s}^{x}=0)$
using the Extreme Viewpoint Interpretation.

Exemplary flexible characterization of the viewpoint constraints (e.g.,
kinematic errors and sensor accuracy) using different scaling vectors
for each vertex.

Computation of View Rays for Occlusion Space

* Considering the simplification of the feature topology (cf. <ref>)
let a set of view rays denoted by $\VECJ{\tau}{f,c,l,m}{}{}{}\in\Sigma_{c},c=\{0,1,2,3,4\}$
be shot at each feature corner point $\VECJ g{f,c}{}{}{}\in\mathbb{R}^{3}$
with following direction vectors $\VECJ{\sigma}{c,l,m}{}{}{}(\sigma_{c,l}^{x},\sigma_{c,m}^{y})\in\mathbb{R}^{3}$:

\[
\VECJ{\tau}{f,c,m,n}{}{}{}=\VECJ g{f,c}{}{}{}+\VECJ{\sigma}{c,m,n}{}{}{}(\sigma_{m}^{x},\sigma_{n}^{y})
\]

* The direction vectors are characterized by a grid of equidistant rays,
which can be expressed by means of the aperture angles $\sigma_{m}^{x}$
and $\sigma_{n}^{y}$:

\[
-\frac{\sigma_{max}^{x}}{2}\leq\sigma_{m}^{x}\leq\frac{\sigma_{max}^{x}}{2}\text{ and }-\frac{\sigma_{max}^{y}}{2}<\sigma_{n}^{y}<\frac{\sigma_{max}^{y}}{2}.
\]

The maximal aperture angles $\sigma_{max}^{x}$ and $\sigma_{max}^{y}$
can simply correspond to the FOV angles of the sensor. An efficient
alternative is to consider the aperture angles $\CC{}{}{}$, which
already comprises the FOV angles and other constraints. The total
number of rays depends on the chosen step size $d^{\sigma}$ for computing
the equidistant view rays:

\[
\begin{aligned}l\in[1,\dotsc,\frac{\sigma_{max}^{x}-\sigma_{min}^{x}+1}{d^{\sigma}}] & ,\\
m\in[1,\dotsc,\frac{\sigma_{max}^{y}-\sigma_{min}^{y}+1}{d^{\sigma}}] & .
\end{aligned}
\]

Occlusion $\protect\ACC$ in SE(3) (red manifold) and the occlusion
free $\protect\ACC$ (blue manifold) to acquire a square feature $f_{1}$
considering an occlusion body $\kappa$ (icosahedron in orange).

Characterization of the $\protect\ACC$, $\protect\CC{}{}F$, in $SE(3)$
to acquire a set of features $\{f_{1},f_{2,}\}\in F$ being characterized
by the intersection of the individual $\protect\ACC$s $\protect\CC{}{}{f_{1}}$
and $\protect\CC{}{}{f_{2}}$.