ISSN 2186-7437

NII Shonan Meeting Report

No. 140

Optimisation Methods inGeometric Vision
Tat-Jun Chin
Anders Eriksson
Yasuyuki Matsushita
January 27–31, 2019

National Institute of Informatics
2-1-2 Hitotsubashi, Chiyoda-Ku, Tokyo, Japan

Optimisation Methods in Geometric Vision
Organizers:
Tat-Jun Chin (The University of Adelaide)
Anders Eriksson (The University of Queensland)
Yasuyuki Matsushita (Osaka University)
January 27–31, 20191

Meeting information

1.1

Technical abstract

Computer vision is concerned with inferring properties of the world from observations in the form of visual images. Such inverse problems typically take shapeas optimization problems, that aim to find the best explanation, for the complex
visual phenomenon that gave rise to a set of noisy and incomplete visual measurements. For computer vision applications to be successful, the underlying
optimization problems must be supported by efficient and dependable solution
methods.
The meeting focuses on a broad subclass of computer vision problems called
geometric vision problems. Roughly, these are problems that exploit fundamental geometrical constraints arising from the image formation process or physicalproperties of the scene (e.g., lighting conditions, characteristics of motions), to
extract information of the scene (e.g., depth, 3D shape, camera trajectory, object identities) from the given visual data. Example geometric vision problems
include structure-from-motion (SfM), simultaneous localization and mapping(SLAM), pose averaging [15], photometric stereo [23], and motion segmentation [8]. Methods for solving geometric vision problems underpin many useful applications, such as 3D reconstruction, robot navigation, object recognition/tracking, and computational photography [21].
Geometric vision is replete with hard optimization problems. By “hard”, wemean that the time needed to solve the optimization problems grows quickly
with the size of the input data. Take, for example, the task of robustly estimating the planar perspective transformation (a.k.a. homography) from outliercontaminated point correspondences between two images. Due to the inherent
intractability of robust homography estimation [7], practitioners often rely on
simple randomized heuristics to find rough approximate solutions, which neither
guarantee optimality nor provide bounds on the approximation error.
The computational difficulty of geometric vision problems is also often compounded by the extremely large size of the input. Take, for example, the taskof bundle adjustment, i.e., calculate 3D points and camera poses that are consistent with a set of images of a scene. In the age of big data, the input image
set is often obtained by “scraping” Internet photo collections, or by conducting

1

long-term surveillance of a scene using a robot. Such input sizes easily overwhelm traditional computing architectures, and distributed or parallel versionsof bundle adjustment must be used [11].

1.2Technical themes

The overall theme for the proposed meeting is recent theoretical and algorithmic
advances on optimization problems in geometric vision. More specific themes
include:• Solvability and approximability of geometric vision problems, e.g., [7, 31];
• Duality in geometric vision problems, e.g., [28, 5, 12];
• Global optimization algorithms for geometric vision, e.g., [7, 12];
• Approximate algorithms including randomized methods, e.g., [22, 20];
• Distributed algorithms for geometric vision problems, e.g., [11];• Incremental algorithms for online geometric optimization, e.g., [31, 20].
Apart from discussing recent progress in geometric optimization through theabove themes, we also aim to chart future research directions and novel application areas. For example:• The role of machine learning in geometric optimization;• Geometric optimization on constrained computing platforms (e.g., smartphones, sensor networks);• Geometric optimization for novel imaging devices (e.g., RGBD cameras,
light field cameras); and• Geometric vision problems from new industries (e.g., self-driving cars,
UAVs).
Following the spirit of Shonan Meetings, we will also consider other related topics, based on the interest of the attendees and the trajectory of the discussions.

1.3

List of participants and program

Please see Appendix A (page 11) and Appendix B (page 12). See also themeeting website [1] for more information.

2

Meeting report

The report is structured around two main themes that were discussed in themeeting (Section 2.1), and an overview of several talks (Section 2.2). In Section 3, a summary of the lessons learned on the organisation of the meeting forthe computer vision community will be presented.

2.1

Main themes of discussion

The main themes received the most attention during the meeting, though bearing in mind that the list of topics presented and discussed is more diverse thanthese main discussion points (again, an overview of other topics will be given in
Section 2.2).
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Main theme 1: More meaningful interactions between machine learning and geometry
In line with the broader trends in computer vision, a persistent theme in themeeting is the usage of machine learning, particularly deep neural networks(or “deep learning”), to perform tasks that are typically done using geometric techniques prior to the resurgence of neural networks. These tasks include
object pose estimation [18, 13], place and scene recognition [4, 9], 3D reconstruction [24], and simultaneous localisation and mapping (SLAM) [3].
The investigation of deep learning towards the above problems is justified,due to the significant improvements in performance it has enabled in problems
such as image segmentation and object recognition [19]. However, for the geometric problems above, it is unclear under what conditions are deep learning
approaches able to provide significantly better accuracy and performance thantechniques that model the intrinsic geometry of the problems. In fact, some of
the attendees remarked that it is “quite difficult” to develop a deep learning
method that can provide highly accurate object pose estimates.
On the other hand, “pure” geometric techniques are brittle, especially whenpresented with noisy visual inputs from unseen before environments. To enablegeometric techniques to generalise well to different operating conditions, usuallya manual tuning process is required to reselect the algorithm parameters.
During the meeting, Richard Hartley proposed to combine deep learningand geometric techniques in a way that leverages the intrinsic strengths of both
approaches: ability to exploit inherent regularity in scenes by training deep
networks with large amounts of data, and the usage of mathematically justified
models for scene understanding by exploiting the scene geometry.
A potentially fruitful framework along the above lines is to backpropagate
geometric errors into the deep network; the general concept is illustrated in thefollowing diagram.

For example, in the problem of object pose estimation, the deep network canbe used to predict the corners of the 3D bounding box of the target object in
the image, which can then be used to analytically compute the object pose. If
a ground truth pose is available, the pose difference can be used to calculatea residual which is then backpropagated to adjust the network parameters.
Backpropagating “non-standard” error functions has been done, for e.g., in [14].

3Main theme 2: The role of strong duality in computer vision
Computer vision concerns itself with understanding the real world through theanalysis of images. The problems that this analysis give rise to often lead tohighly complicated optimization problems with a mixture of challenging objective functions and constraints, involving high- or even infinite-dimensionalvariables in terms of curves and surfaces. Due to the non-convexity of the overwhelming majority of these problems they are typically exceedingly difficult tosolve globally.
A prominent theme in this meeting was on the role of duality principles
in the field of Computer Vision and related areas. The mathematical conceptof duality lies at the core of a great part of the most efficient optimization
algorithm currently in use. A key reason behind this is the notion of strong
duality, a property in optimization theory establishing the equivalence of a given
minimization problem to that of an associated and convex problem, known as
the dual problem.
There is a well established theory on Lagrangian duality and the existenceof strong duality with respect to convex optimization problems. Unfortunately,much less is currently understood regarding the role of strong duality in nonconvex optimization, hence significant portions of the existing theory can not
be directly applied to this setting. There are however a few notable exceptions
to this rule, problems that are non-convex but for which strong duality does
indeed hold. These are problems that provably admit a large number of local
minima but can still be solved in a way that guarantees a global optima by
invoking duality principles.
In recent years, empirical observations have been made suggesting that, under specific conditions, a broad class of non-convex problems, fundamental toComputer Vision, also belong to this group of exceptions. The consequencesof establishing such a proposition would be far-reaching, both in theory and inpractice. Such a result would imply that the global solution to these primal
non-convex problems are equivalent to the solution of the associated dual convex problem. This allows us to solve exceedingly difficult non-convex problems,with numerous local minima, implicitly through their dual formulations with aguarantee of global optimality and in many instances also in polynomial time.
The discussion were led primarily by Fredrik Kahl, Frank Dellaert, Luca Carlone, Anders Eriksson and Robert Mahony. The topics were mainly centeredaround problems involving pose, with a particular focus on two specific applications, namely rotation Averaging and SLAM, see Figure 1, both fundamental
problems in Computer Vision and Robotics.
Two primary goals of this particular theme and line of research were identified and discussed at the Shonan meeting. They were, firstly to establish newmathematical foundations detailing the role of strong duality in Computer Vision and to further the understanding of the function of strong duality in a broadclass of related optimization problems. Secondly, to develop efficient numericalalgorithms dedicated to solving the resulting dual optimization problems.

4Figure 1: Rotation averaging and SLAM, two fundamental problems related to
3D reconstruction, analysis and navigation.

2.2

Overview of selected talks

Discrete-continuous graphical models for robust perception
Assistant Professor Luca Carlone, MIT
Pose graph optimisation (PGO) lies at the core of a robotic perception module. Specifically, PGO is required to enable a robot to navigate in an unknown
environment by iteratively reconciling a sequence of visual motion observations
and keeping track of the trajectory of the robot. Computationally, PGO is inherently an intractable problem due to the manifold structure of the variablesof interest (robot positions and orientations). Moreover, in situations wherelong-term operations are essential, there is a large number of measurements tobe processed by the PGO algorithm.
This talk surveys a class of PGO algorithms that apply convex relaxation andsolve the problem using convex solvers, specifically Semi Definite Programming(SDP) [25]. It has been established that, there is a nonzero bound, such that if
the noise the measurements are no greater than the said bound, then the convex
relaxation is tight. Moreover, good empirical performances of convex relaxation5

methods have been demonstrated. Notably, the SDP technique is faster thanstandard iterative optimisation algorithms [25].
The above theoretical results are based on non-robust formulations of PGO.
However, in real-life settings, outliers (wrong measurements) are inevitable. If
the convex relaxation methods are directly applied on such measurements, thesolutions will be biased. The talk proposes a discrete continuous (DC) framework for PGO, by introducing binary “label” variables into the problem thatenables the assignment of measurements as inliers or outliers. In more detail, if ameasurement is identified as an outlier, its effect on the estimates are discounted.
An interesting connection between the DC-PGO problem and Markov Random
Fields (MRF) was also pointed out. It is likely that algorithmic developments
for the DC-PGO will become a fruitful research direction.

Parametrised complexity in geometric optimisation
Associate Professor Tat-Jun Chin, The University of Adelaide
A number of important and useful geometric problems in computer vision(e.g., robust estimation, point set registration) have been proven to be computationally intractable, not only to solve, but also to approximate [6, 27]. On theother hand, practical applications in real-world settings require these problems
to be solved with some form of guarantee so as to avoid unexpected breakdowns.
Hence, it is not sufficient to rely on heuristic methods to solve these problems,since the quality and/or performance of heuristic methods cannot be bounded.
This talk proposed to investigate the structural properties of the hard geometric problems and develop fixed parameter tractable (FPT) algorithms.
Briefly, FPT algorithms exploit additional insights or structures to a problem,and constrain the exponential growth in runtime to parameters that depend on
the special structures only. Such an endeavour is encapsulated under the fieldof parametrised complexity analysis [10].
The talk illustrates an example based on inlier set maximisation or consensus
maximisation [6], which is a very common problem in computer vision. It isshown how consensus maximisation has a FPT formulation, if an upper boundon the number of outliers in a problem instance is known. It will be interesting
to apply the FPT framework on other intractable problems in computer vision(e.g., point set registration [27]).

Methods for robustified nonlinear least-squares
Professor Christopher Zach, Chalmers University
Given noisy observations of a number of scene points in a number of images,
the goal of bundle adjustment (BA) is to estimate the camera poses and 3D
coordinates of the scene points that are consistent with the observations. Thisis typically casted in a least squares objective function, where the sum of squared
reprojection errors is minimised. To estimate the variables of interest, non-linear
optimisation techniques such as Levenberg-Marquardt are applied.
If the measurements contain outliers, however, the least squares solution will
be biased. A more robust norm such as Tukey’s Biweight or Geman-McCluremust thus be used in the objective function. If the robust norm is smooth, the6resulting robustified BA problem can still be solved directly using standard iterative optimisation algorithms. It has been shown, however, that more effective
techniques exist, namely, iteratively reweighted least squares (IRLS) [16] andlifted optimisation [2]. Moreover, it can be established that both IRLS and lifted
optimisation minimise the same “half-quadratic” objective function, whereby aset of weights (one per measurement) is introduced into the original robustified
BA problem [30]. By explicitly introducing weight variables, the half-quadratic
objective function suffers from far fewer bad local minima.
This talk proposed new ways of tackling the half-quadratic objective function that can potentially deliver better performance than IRLS and lifted optimisation. The first is an “iterated lifting” technique, where a graduated orannealed optimisation approach is introduced in half-quadratic optimisation.
The second is a multi-objective optimisation (MOO) technique, where multiple
versions of the original robustified BA problem, each differing from the otherby the half-quadratic kernel, are jointly optimised using MOO. Initial empiricalresults show that iterated lifting and MOO optimisation are able to converge tobetter solutions than IRLS and regular lifted optimisation.

Semi-calibrated photometric stereo
Professor Yasuyuki Matsushita, Osaka University
High-fidelity shape estimation is a central topic in computer vision. One ofthe promising approaches is photometric stereo that uses photometric information to determine the 3D shape. More specifically, photometric stereo estimatessurface normal of a static scene in a pixel-wise manner from a set of observations
obtained under varying light conditions from a fixed camera. It has been understood that, for a Lambertian surface, its surface normal map can be uniquelydetermined from three observations.
In practice, the intensity observations may contain non-Lambertian components, such as specular reflectance and cast/attached shadows, that can be
regarded as outliers. To deal with the outliers, robust estimation techniques are
employed in the past; for example, `1 -norm minimisation and robust principal
component analysis in the context of photometric stereo [29, 17]. The robustness
against outliers has been improved in the past decade; however, calibration oflight sources (both geometric and photometric calibration) remains a practicalissue due to its laborious process.
This talk introduced a semi-calibrated photometric stereo method, withwhich the need for photometric calibration of light sources can be eliminated.
It has been believed that for a Lambertian photometric stereo, it is needed
to know the light source directions and light source intensities. However, it isshown that the knowledge of light source intensities is unnecessary, but evenunder the condition, a unique shape can be determined. The talk showed that
there exists a linear solution technique to the problem, and it introduced anefficient alternating minimisation strategy to the problem.

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Geometric point light source calibration via structure frommotionDr. Michael Waechter, Osaka University
Estimating the position or direction of a light source accurately is essential
for many computer vision tasks, such as shape from shading, photometric stereo,or reflectance and material estimation. In these tasks, inaccurate light positions
immediately cause errors in their estimates. Despite the importance of accurate
light calibration, it remains laborious as researchers have not yet come up withaccurate and easy to use techniques.
Previously, many geometric point light source calibration procedures involved the use of mirror spheres. Unfortunately, the mirror sphere based methods suffer from the difficulty of determining the sphere boundary and the pointof specular spike location, resulting in unstable accuracy of the calibration.
This talk introduced a new geometric point light source calibration method [26]that uses a plane and pins that are stuck at unknown locations on the plane. By
observing the cast shadows generated by the pinheads, the problem of geometricpoint light source calibration can be solved in a similar manner to Structurefrom-Motion (SfM). Although the pinhead locations are unknown, they can be
simultaneously recovered together with the light source positions/directions. It
showed the new application domain of geometric computer vision.

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Lessons for potential future meetings

As a conclusion to this report, we would like to report some of the lessons wehave learned as organisers of this Shonan Meeting.
First, it is worth highlighting that this was likely to be the first Shonan Meeting that was organised and attended exclusively by members of the computer
vision community1 . Therefore, there was not a significant body of experienceto draw from in the organisation and planning of the meeting.
There are a few aspects of the program (Appendix B) that could be improved. Chiefly, the adopted program was inspired by typical computer vision
meetings, where attendees are encouraged give a presentation (oral or poster)
to describe his/her own current favourite research topic. While the carefully
selected list of invitees ensured strong commonalities in the presentations anddiscussions (e.g., Lagrangian duality and global optimality in geometric optimisation, interaction between geometry and deep learning), there was not a strong
coalescence around a few clear-cut research questions/themes on which unified
discussions and progress could be made.
In future instances of Shonan Meetings (or other meeting series in the styleof Dagstuhl) by and for the computer vision community, we suggest to• Identify a handful of “Area Chairs” for key topics, perhaps even during themeeting proposal stage. The role of an Area Chair could include the selectionof presenters and the contents of the presentations, and leading the discussions
during the breakout sessions.
1 These are the researchers who regard conferences such as Computer Vision and Pattern

Recognition (CVPR) and journals such as IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI) to be their flagship publication venues.

8• Deemphasise the CVPR tradition to give each participant a speaking slot.
Rather, encourage the participants to contribute productively and passionately to the identified research themes via discussions and breakout sessions.
Another aspect for improvement is gender diversity in the list of participants.
While the invitation list included female researchers, the gender of the actual
participants in the meeting was heavily biased towards males.
Despite the areas of improvements above, there are also very encouraging
signs from the meeting. Chiefly, almost all of the attendees agree that the
Shonan Meeting series represents an excellent alternative to the traditional conferences and workshops targeted by the computer vision community. With the
explositive growth of commercial interest and investment in AI, the research
directions in computer vision are often driven by immediate industry demands.
In this context, events in the format of Shonan Meetings offer a welcomed opportunity to put longer-term scientific agendas back in the focus of researchers.
On the other hand, we believe that Shonan Meetings could benefit fromgreater participation from the computer vision community, which has been successful in recent years to attract the next generation of computing researchers,
as well as in identifying commercially impactful problems and applications.

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SLAM. In IEEE Computer Vision and Pattern Recognition (CVPR), 2018.
[4] S. Brahmbhatt, J. Gu, K. Kim, J. Hays, and J. Kautz. Geometry-aware learningof maps for camera localization. In CVPR, 2018.
[5] L. Carlone, G. Calafiore, C. Tommolillo, and F. Dellaert. Planar pose graph optimization: duality, optimal solutions, and verification. IEEE Trans. on Robotics,
2016.
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and Pattern Recognition (CVPR), 2016.
[12] A. Eriksson, C. Olsson, F. Kahl, and T.-J. Chin. Rotation averaging andstrong duality. In IEEE Conference on Computer Vision and Pattern Recognition(CVPR), 2018.

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[13] M. Feng, S. Hu, M. H. Ang, and G. H. Lee. 2D3D-MatchNet: learning to match
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[14] B. Fernando and S. Gould. Discriminatively learned hierarchical rank pooling
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[18] A. Kendall, M. Grimes, and R. Cipolla. Posenet: A convolutional network forreal-time 6-dof camera relocalization. In CVPR, 2015.
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convolutional neural networks. In Advances in Neural Information Processing
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[20] H. M. Le, T.-J. Chin, and D. Suter. An exact penalty method for locally convergent maximum consensus. In IEEE Conference on Computer Vision and Pattern
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[21] Y. Matsushita, E. Ofek, W. Ge, X. Tang, and H.-Y. Shum. Full-frame video
stabilization with motion inpainting. In IEEE Trans. on Pattern Analysis andMachine Intelligence (PAMI), 2006.
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[23] J. Park, S. N. Sinha, Y. Matsushita, Y.-W. Tai, and I. S. Kweon. Robust multiview photometric stereo using planar mesh parameterization. In IEEE Trans. on
Pattern Analysis and Machine Intelligence (PAMI), 2016.
[24] R. Ranftl and V. Koltun. Deep fundamental matrix estimation. In European
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Intl. Workshop on the Algorithmic Foundations of Robotics (WAFR), 2016.
[26] H. Santo, M. Waechter, M. Samejima, Y. Sugano, and Y. Matsushita. Light
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Appendix A — List of participants
1. Alessio Del Bue, Italian Institute of Technology
2. Ali Bab-Hadiashar, RMIT University
3. Anders Eriksson, Queensland University of Technology
4. Carl Olsson, Chalmers Institute of Technology
5. Christopher Zach, Chalmers University of Technology
6. Danda Pani Paudel, ETH Zurich
7. David Suter, Edith Cowan University
8. Florian Bernard, Max-Planck-Institute for Informatics
9. Frank Dellaert, Georgia Tech
10. Fredrik Kahl, Chalmers University of Technology
11. Gim Hee Lee, National University of Singapore
12. Hongdong Li, Australian National University
13. Jamie Sherrah, Digital Animal Interactive Inc. (FTSY)
14. Jesus Briales, Facebook
15. Kenichi Kanatani, Okayama University
16. Laurent Kneip, Shanghai Tech
17. Luca Carlone, MIT
18. Michael Brown, York University
19. Michael Waechter, Osaka University
20. Ping Tan Simon, Fraser University
21. Richard Hartley, Australian National University
22. Robert Mahony, Australian National University
23. Simon Lucey, Carnegie Mellon University
24. Sudipta Sinha, Microsoft Research
25. Tarek Hamel, Universite Cote d’Azur
26. Tat-Jun Chin, The University of Adelaide
27. Viorela Ila, University of Sydney
28. Yasuyuki Matsushita, Osaka university
29. Yinqiang Zheng, NII
30. Yongduek Seo, Sogang University

11Appendix B — Meeting program
Sunday 27 January 2019
1500-1900 Check-in
1900-2100 Welcome banquet
Monday 28 January
0730-0900 Breakfast
0900-0915 Welcoming address - Tat-Jun Chin, Anders Eriksson, Yasuyuki Matsushita
0915-1000 Keynote 1 - Fredrik Kahl
1000-1015 Michael Brown
1015-1030 Christopher Zach
1030-1100 Coffee break
1100-1115 Robert Mahony
1115-1130 Yongduek Seo
1130-1200 Breakout session 1
1200-1330 Lunch
1330-1345 Sudipta Sinha
1345-1400 Yinqiang Zheng
1400-1415 Michael Waechter
1415-1430 Viorela Ila
1430-1500 Breakout session 2
1500-1530 Coffee break
1530-1545 Hongdong Li
1545-1600 David Suter
1600-1630 Breakout session 3
1630-1800 Free time
1800-1930 Dinner
Tuesday 29 January
0730-0900 Breakfast
0900-0915 Program briefing
0915-1000 Keynote 2 - Frank Dellaert
1000-1015 Ping Tan
1015-1030 Jamie Sherrah
1030-1100 Cofee break
1100-1115 Ali Bab-Hadiashar
1115-1130 Gim Hee Lee
1130-1200 Breakout session 4
1200-1315 Lunch
1315-1330 Group photo121330-1345 Kenichi Kanatani
1345-1400 Jesus Briales
1400-1415 Carl Olsson
1415-1445 Breakout session 51445-1515 Coffee break1515-1530 Florian Bernard
1530-1545 Alessio Del Bue
1545-1600 Danda Pani Paudel
1600-1630 Breakout session 6
1630-1800 Free time
1800-1930 Dinner
Wednesday 30 January
0730-0900 Breakfast
0900-0915 Program briefing
0915-1000 Keynote 3 - Richard Hartley
1000-1015 Tarek Hamel
1015-1030 Laurent Kneip
1030-1100 Coffee break
1100-1115 Simon Lucey
1115-1130 Luca Carlone
1130-1200 Breakout session 7
1200-1330 Lunch
1300-2045 Excursion and dinnerThursday 31 January
0730-0930 Breakfast and check-out
0930-0945 Yasuyuki Matsushita
0945-1000 Anders Eriksson
1000-1015 Tat-Jun Chin1015-1100 Coffee break
1100-1130 Breakout session 81130-1200 Conclusion and wrap-up
1200-1330 Lunch and end of meeting13