=Paper=
{{Paper
|id=Vol-2744/paper21
|storemode=property
|title=Constructing Stereo Images of Error Surfaces in Problems of Numerical Methods Verification
|pdfUrl=https://ceur-ws.org/Vol-2744/paper21.pdf
|volume=Vol-2744
|authors=Sergey Andreev,Alexander Bondarev,Alexander Bondarenko,Vladimir Galaktionov,Nicole Bondareva
}}
==Constructing Stereo Images of Error Surfaces in Problems of Numerical Methods Verification==
https://ceur-ws.org/Vol-2744/paper21.pdf
Constructing Stereo Images of Error Surfaces in
Problems of Numerical Methods Verification*
Sergey Andreev1[0000-0001-8029-1124], Alexander Bondarev1[0000-0001-3681-5212],
Alexander Bondarenko2[0000-0003-4765-6034], Vladimir Galaktionov1[0000-0001-6460-7539]
and Nicole Bondareva1[0000-0002-7586-903X]
1 Keldysh Institute of Applied Mathematics RAS, Moscow, Russia
2 State Res. Institute of Aviation Systems (GosNIIAS), Moscow, Russia
esa@keldysh.ru, bond@keldysh.ru, cod@fgosniias.ru,
vlgal@gin.keldysh.ru, horoshechka1@yandex.ru
Abstract. The tasks of constructing stereo representations of texts and formulas
on an autostereoscopic monitor in stereo presentations designed to display the
results of numerical modeling are an urgent sub-task in developing methods and
algorithms for constructing stereo displays of scientific research results. In this
paper, the construction of stereo images on an autostereoscopic monitor is con-
sidered. The autostereoscopic monitor allows you to watch stereo images with-
out glasses, while ensuring the quality of the stereo image, which is not inferior
to the quality of the stereo image presented using the classic 3D projection ste-
reo system. The possibility of combining several stereo objects with different
parameters on one frame with various parameters allowing to achieve the max-
imum stereo effect is being investigated. This technology has been applied
practically to visualize the problems of verification of numerical methods and
their comparative analysis. Similar solutions for the two-parameter problem are
represented in the form of error surfaces constructed for each numerical method
involved in the comparison. The construction of error surfaces in stereo mode is
implemented for an autostereoscopic monitor based on a multi-view.
Keywords: Stereo Image, Textual Information, Autostereoscopic Monitor,
Multi-View Presentation.
1 Introduction
To date, stereo animation begins to play an important role in displaying the obtained
scientific results in various applied scientific research and industries. This is largely
due to the fact that a situation often arises when a flat image of an object does not
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons Li-
cense Attribution 4.0 International (CC BY 4.0).
* This work has been supported by the RFBR grants 19-01-00402 and 20-01-00358
2 S.Andreev, A. Bondarev, A. Bondarenko, V. Galaktionov, N. Bondareva
have the entirety of information to achieve its goals and limits the analysis to a sche-
matic image or a truncated viewing angle.
Unlike a two-dimensional flat image, three-dimensional stereo allows you to more
effectively explore and model objects with a developed spatial structure or interpret
photographs taken during the experiment from different angles at given times and
combined into animation.
A lot of works has been devoted to this topic. They discuss methods for construct-
ing images for various types of stereo devices, both passive and active types. The
difference between these concepts is that passive ones allow one or another material
to be demonstrated according to a predetermined scenario, however, the viewer is not
able to influence the display process. Active (or interactive) installations mean that
the viewer has the opportunity to work with the demonstration material and directly
influence the demonstration process. General aspects of image construction for vari-
ous types of stereo units are presented in [1].
The specific problems that arise when using a computer system to generate and
visualize a composite multi-screen stereo frame, and methods for solving such prob-
lems are described in sufficient detail in [1-3].
The gradual spread of modern stereo systems has generated great interest in the
development of practical algorithms for stereo presentation of static and animated
images, which is reflected in [4,5-10,15-17]. These works present the tasks of con-
structing stereo images in various fields of research: from the explosion of a Superno-
va to the processes of oil displacement from porous media.
The spread of stereo animations and virtual reality also revealed a rather important
particular structural subproblem. The standard procedure for a report, presentation of
something or a virtual walk implies the presence of not only a visual type of infor-
mation in them, but also a symbolic one - letters, numbers, symbols, metric quantities,
names, numbers, Federal Standards, without which the perception and assimilation of
information by the viewer is significantly complicated.
The problems of constructing stereo images for geometries and fields of physical
quantities have been developed in sufficient detail, but quite a few works have been
devoted to the development of specific practical approaches and algorithms for con-
structing stereo representations of texts and formulas with sufficient expressiveness
and the necessary stereo effect. One of them, which cannot be ignored, is the work
[11] describing a study conducted in Japan of the perception of stereo images of fonts
on the screens of stereoscopic mobile devices.
This work is a continuation of a series of research papers [1-3,7,12-14,18,19] con-
ducted at the Keldysh Institute of Applied Mathematics RAS based on the available
stereo units of two types.
The first type of device is a 3D projection stereo system for displaying stereo
presentations, educational applications, graphics and films. It is an example of a clas-
sical stereoscopic system using two projectors, a screen and linear polarization glass-
es. Studies [12–13] describe in detail studies on the presentation of textual infor-
mation on this type of stereo unit using a linear stereo base. In these works, test in-
scriptions were constructed with variations of the font, background, font embossment,
the whole angle of rotation of the inscription, and shift along the linear stereo base.
Constructing Stereo Images of Error Surfaces in Problems of Numerical… 3
The parameters that provided the maximum effect were found and the basic require-
ments for the fonts used and a number of conditions were identified, the satisfaction
of which is necessary to achieve the optimal result.
The second type of device is a Dimenco DM654MAS autostereoscopic monitor.
Autostereoscopic monitors provide stereo images without having to track the position
of the observer. Typically, such monitors make it possible to observe stereo images,
providing several fixed segments in the space for observation. The viewer can move
between segments, getting the opportunity to view the displayed object in 3D from
different angles. The principle of operation of the autostereoscopic monitor is the use
of parallax partitions or Fresnel lenses installed behind the protective glass of the
screen, which gives it one of the most important advantages: displaying the image
does not require special glasses or other devices from the viewer.
An autostereoscopic monitor is capable of demonstrating a visualization object us-
ing two methods: either using a composite frame containing views of the visualization
object at different angles that form a certain viewing sector - this is called a multi-
view, or using depth maps.
In multi-view image construction, nine frames are combined into one image ac-
cording to the principle of a 3 × 3 matrix. In the first and last (ninth) frames, the de-
sired object, for example, the inscription is in its extreme positions, and in the inter-
mediate frames it is rotated sequentially by a given angle.
After solving the problem of creating stereo texts on their own, as separate frames
in a stereo presentation or stereo animation, another important sub-task was identi-
fied: combining images and text information in one frame. This problem is described
in [18] in detail. The developed methods and approaches for constructing stereo im-
ages made it possible to apply them to the construction of error surfaces for the analy-
sis of the comparative accuracy of OpenFOAM solvers for the oblique shock wave
problem with variation of the Mach number and angle of attack. To build stereo im-
ages, a multi-view method was used on an autostereoscopic monitor.
2 The construction of stereo images on an autostereoscopic
monitor using multi-view method
An autostereoscopic monitor has the ability to demonstrate a visualization object us-
ing a composite frame containing views of the visualization object at different angles,
which together form a certain viewing sector. Usually, these are nine views. These
nine views form eight stereopairs ([1 | 2], [2 | 3], [3 | 4], ..., [8 | 9]), however, the
viewer can observe only one of the stereopairs in one position, depending on the loca-
tion in defined angular sector of observation. When the observer moves from sector to
sector, he receives information about the object using all nine views, that is, as if
looking at the visualization object from different angles.
When building a multi-view stereo image, nine frames are combined into one im-
age according to the principle of a 3 × 3 matrix. In the first and last (ninth) frames, the
represented object is in its extreme positions. In the first frame, the object is in its
4 S.Andreev, A. Bondarev, A. Bondarenko, V. Galaktionov, N. Bondareva
rightmost position, and in the last frame, in the leftmost position. Then in the middle
frames, respectively, from the second to the eighth, with certain equal intervals, it
moves to the left until it reaches the extreme left position. In this case, in each frame,
the object is progressively rotated by the same angle and shifted by a predetermined
interval. This allows you to achieve the most effective result when creating a stereo
image of the object.
3 Methods of combining image and text information
After solving the problem of creating stereo labels on their own, as separate frames in
a stereo presentation or stereo film, an equally important sub-task arises - combining
the image and text information in one frame. Currently, in the case of demonstration
of scientific results to observers, very many objects require accompanying infor-
mation located directly on the same frame as the image. In many cases, the signature
and the object cannot be separated into different frames, since they make up a single
logical display of information. For example, when depicting a coordinate system, one
cannot fail to mark a designation for each of the coordinates. An example is Fig. 1.
This figure is not informative, like a graph, because it does not carry accurate data and
is not bound to coordinates. This is a simple three-dimensional model, which is a
schematic three-dimensional graph.
Fig. 1. Three-dimensional model of the coordinate system in volume.
The constructed nine-view image gave a stereo effect, which was recognized by ob-
servers as satisfactory. However, the presented figure does not carry an informative
load, since it is not accompanied by the necessary additional information, and the
viewer does not even have an idea about the coordinate system.
Fig. 2 shows a more complete image. The coordinates are signed on it, and the
letters indicating the coordinates also provide a stereo effect.
Constructing Stereo Images of Error Surfaces in Problems of Numerical… 5
Fig. 2. Three-dimensional model of the coordinate system in volume with signed coordinates.
To construct this stereo image, an image matrix was used (Fig. 2), to which volume
letters were added and sequentially shifted and rotated on each frame by the same
interval experimentally calculated earlier. As a result, a stereo image with a volume
signature was obtained, the stereo effect of which was pronounced. Along with this,
an additional effect was revealed when the viewer from different positions saw that
the letters are on different planes each time. For example, the location of the letter Z
was perceived by observers either behind the horizontal axis or in front of it, depend-
ing on the viewer's location in a particular observation sector in front of the screen of
the autostereoscopic monitor.
Fig. 3 shows one of the results of previous studies - a multi-view image of the sim-
ulation results of a supersonic flow around a cone at an angle of attack with the corre-
sponding inscription [18]. The task was to combine in one stereo image the objects of
different depths and sizes and text information (in this case, the cone and the inscrip-
tion at the figure). Here the image of the simulated cone and separately the inscrip-
tions to it are combined. Each of them is rotated by its own experimentally revealed
angle and is shifted by a certain distance, also different. As shown in the figure, a
matrix of images is further compiled, which in turn comprise a single stereo image. In
the end, the inscription was located on top of the cone, but behind its tip, which in
turn was perceived by observers as protruding from the screen by several centimeters.
Fig. 3. Image of the simulation results of a supersonic flow around a cone with the correspond-
ing signature [18].
A similar problem was posed for constructing stereo mappings of error surfaces with
corresponding inscriptions.
6 S.Andreev, A. Bondarev, A. Bondarenko, V. Galaktionov, N. Bondareva
4 The construction of stereo for the problems of comparing the
accuracy of numerical methods
Stereoimaging is a useful and effective tool for deep understanding and presentation
of the results of solving mathematical modeling problems. Construction of stereo
images for comparative verification of numerical methods turned out to be a very
effective visualization tool. In order to assess the accuracy of the numerical method
for a specific problem, it is necessary to compare the numerical results with the refer-
ence solution (exact, calculated or experimental solution). The visual representation
of the numerical calculation error allows for a comprehensive comparative assessment
of numerical methods. In the absence of a reference solution, a certain estimate of the
accuracy can be carried out on the basis of an ensemble of numerical solutions ob-
tained using several independent numerical methods. Similar numerical solutions are
presented in [19-21].
Below we present examples of constructed stereoimages for the presentation of the
calculation results for the comparative assessment of the accuracy of numerical meth-
ods [19-21].
The first example presents the results of constructing error surfaces for oblique
shock wave problem. These results are presented for four OpenFOAM solvers with
variations in the Mach number from 2 to 4 and variations in the angle of attack from 6
to 20 degrees [19]. It should be noted that error surfaces for the class of problems of
the numerical methods accuracy comparative analysis were constructed in [19] for the
first time. The left part of Fig. 4 shows four error surfaces for four OpenFOAM solv-
ers - rhoCentralFoam (rCF), pisoCentralFoam (pCF), sonicFoam (sF) and QGDFoam
(QGDF) with the corresponding text labels and notations. To construct a single stereo
image, the same techniques were used as in [18] for the image of the cone. The con-
structed multi-view image is displayed on the right side of Fig. 4.
Fig. 4. Image of the surface deviation from the exact solution for four OpenFOAM solvers with
variation of the Mach number and angle of attack for the oblique shock wave problem.
Constructing Stereo Images of Error Surfaces in Problems of Numerical… 7
Fig. 4 shows that the best accuracy in the class of problems is provided by the rCF
and pCF solvers, for which the error surfaces almost coincide. The developed tech-
nology allows one to create stereo images for these surfaces separately. The results
are presented in Fig. 5.
Fig. 5. Image of error surface comparison for pCF and rCF solvers.
Fig. 5 shows that the divergence of error surfaces is most pronounced for the
smallest Mach number and the largest angle of attack. The constructed stereo image
allows the researcher to thoroughly study the accuracy properties of solvers not on
one single problem, but on the class of problems defined by the ranges of variation of
the defining parameters (Mach number and angle of attack).
The next example of stereo image is constructed for the results, described in [21].
In this research the authors used advantage of the ensemble of numerical solutions
obtained by independent numerical algorithms. To obtain the approximation error, the
differences between numerical solutions are treated in the frame of the Inverse Prob-
lem that is posed in the variational statement with the zero order regularization. In this
work the authors [21] analyse the ensemble of numerical results that is obtained by
five OpenFOAM solvers for the inviscid compressible flow around a cone at zero
angle of attack. Fig. 6 presents the exact error of the pressure for the flowfield, com-
puted by rCF. Fig. 7 presents the error of the pressure estimated by the Inverse Prob-
lem for the same flowfield.
For both cases the stereo images were created. Stereoimaging allows the research-
ers to study the internal structures of error in the flowfield in question.
The results obtained allow us to conclude that the previously developed technology
for constructing stereoscopic presentations of the results of numerical studies can be
used with benefit in problems of comparative assessment of the accuracy of numerical
methods.
8 S.Andreev, A. Bondarev, A. Bondarenko, V. Galaktionov, N. Bondareva
Fig. 6. The exact error of the pressure (for the flowfield, computed by rCF).
Fig. 7. The error of the pressure estimated by the Inverse Problem (for the flowfield, computed
by rCF).
5 Conclusion
This work is a continuation of a works series devoted to the implementation of the
project to build stereo presentations of the results solving mathematical modeling
problems. The results of numerical experiments on the presentation of textual infor-
mation on an autostereoscopic monitor that allows the construction of stereo images
using depth and multi-view maps are presented. The problem of combining images
and textual information in one frame for a multi-view presentation is considered. As a
result of practical experiments, the most clear and expressive stereo effect was
achieved.
The construction of stereo frames is carried out in the previously developed mode
of combining in one stereo frame the main object of the visualization and the corre-
sponding text labels and symbols. The constructed stereo frames provide the re-
searcher with the possibility of a deep and thorough visual analysis of the results.
Constructing Stereo Images of Error Surfaces in Problems of Numerical… 9
Developed technology for constructing stereoscopic presentations of the results of
numerical studies can be used with benefit in problems of comparative assessment of
the accuracy of numerical methods.
6 Acknowledgments
This work was supported by RFBR grants 19-01-00402 and 20-01-00358.
References
1. Andreev, S.et al: Generation of Stereo-Presentations in Photorealistic Rendering and Sci-
entific Visualization. Keldysh Institute preprints (61), 1-16 (2010). [in Russian]
2. Andreev, S.et al: Synthesis of photorealistic three-dimensional images in modern presenta-
tion systems. Software&Systems (3), 37-40 (2007). [in Russian]
3. Andreev, S., Filina, A. Using stereo presentations for visualization of scientific calcula-
tions results. Scientific Visualization 4(1), 12-21 (2012).
4. Vandanov, V. et al: Hardware-software complex of 3D presentations based on a virtual
studio and virtual environment. In: Proceedings of the 1-st international conference «3D
visualization of scientific, technical and social reality. Cluster technologies of modeling»
pp.73-77. Izhevs. (2009). [in Russian]
5. Mezhenin, A., Tozik, V.: 3D Visualization using the stereo image effect. In: Proceedings
of the 2-nd international conference «3D visualization of scientific, technical and social re-
ality. Cluster technologies of modeling» Izhevsk (2010). [in Russian].
6. Mikhaylyuk, M., Huraskin, I.: Synthesis of stereo images for virtual reality systems using
an optical tracking system. Software&Systems (3) 10-14 (2006).
7. Andreev, S. [et al]: Modelling and visualisation of blade assembly with complicated shape
for power turbine. Scientific Visualization 7(4) 1-12 (2015).
8. Torgashev, M., Timokhin, P.: The technology of stereo video files’ synthesis for the sys-
tem of 3D real-time visualization. Software Products and Systems (3), 74-80 (2012).
9. Torgashev, M.: Implementation of stereo mode for various devices for real time display-
ing. Software Products and Systems (2), 23-29 (2010).
10. Mikhaylyuk, M., Maltsev, A., Timokhin, P.: The methods of 3D stereo visualization of da-
ta obtained in simulation of unstable oil displacement from porous media. Proceedings of
Scientific Research Institute for System Analysis RAS 8(2), 125-129 (2018). [in Rus-
sian].
11. Hiromu, I. [et al.]: Visibility Experiment and Evaluation of 3D Character Representation
on Mobile Displays. In: Stephanidis C. (Ed.): Posters, Part II, HCII 2011, CCIS 174, pp.
46–51 (2011).
12. Andreev, S., Bondareva, N.: Constructing a representation of textual information in stereo
presentations. In: Proceedings of the 28-th International Conference of Computer Graphics
and Vision GraphiCon-2018, pp. 86-89. Tomsk (2018). [in Russian]
13. Andreev, S., Bondarev, A., Galaktionov, V., Bondareva, N.: The problems of stereo ani-
mations construction on modern stereo devices. Scientific Visualization 10(4) 40 – 52
(2018). https://doi.org/10.26583/sv.10.4.04
14. Andreev, S., Bondareva, N., Denisov, E.: Stereo Presentations Problems of Textual infor-
mation on an Autostereoscopic Monitor. Scientific Visualization 11(4) 90 – 101 (2019). ,
https://doi.org/10.26583/sv.11.4.08
10 S.Andreev, A. Bondarev, A. Bondarenko, V. Galaktionov, N. Bondareva
15. Kalinichenko, A., Sveshnikova, N., Yurin, D.: Epipolar geometry and evaluation of its re-
liability from the results of the restoration of the three-dimensional scene by factorization
algorithms. In: Proceedings of the 16-th International Conference of Computer Graphics
and Vision GraphiCon-2006 (2006). [in Russian].
16. 3D Stereoscopic Photography. http://3dstereophoto.blogspot.com/. last accessed 2020/08
22.
17. Zhdanov, A., Zhdanov, D., Bogdanov, N., Potemin, I., Galaktionov, V., Sorokin, M.: Dis-
comfort of Visual Perception in Virtual and Mixed Reality Systems. Programming and
Computer Software 45(4) 147–155 (2019). https://doi.org/10.1134/S036176881904011X
18. Andreev, S. , Bondarev, A., Bondareva, N.: Stereoscopic construction of textual infor-
mation in presentations of research results on an autostereoscopic monitor. Scientific Vis-
ualization 12(1), 132 – 139 (2020). https://doi.org/10.26583/sv.12.1.12
19. Bondarev, A., Kuvshinnikov, A.: Parametric study of the accuracy of OpenFOAM solvers
for the oblique shock wave problem. IEEE The Proceedings of the 2019 Ivannikov
ISPRAS Open Conference 2019, 108–112 (2019).
https://doi.org/10.1109/ISPRAS47671.2019.00023
20. Alekseev, A., Bondarev, A., Kuvshinnikov, A.: On uncertainty quantification via the en-
semble of independent numerical solutions. Journal of Computational Science 42, 101114
(2020). https://doi.org/10.1016/j.jocs.2020.101114
21. Alekseev, A., Bondarev, A., Kuvshinnikov, A.: A posteriori error estimation via differ-
ences of numerical solutions. In: Krzhizhanovskaya, V.V., et al. (eds.): ICCS 2020,
LNCS, vol. 12143, pp. 508–519. Springer, Cham (2020). https://doi.org/10.1007/978-3-
030-50436-6_37