CVX based SLAM algorithms paper read

CVX based ICP paper read

• 1 Banch and Bound
• 1.1 Practical Global Optimization for Multiview Geometry
• 1.2 Branch-and-Bound Methods for Euclidean Registration Problems
• 2. Lagrangian Duality
• 2.1 Convex Global 3D Registration with Lagrangian Duality
• 2.2 Solving Quadratically Constrained Geometrical Problems using Lagrangian Duality
• 2.3 Duality-based Verification Techniques for 2D SLAM
• 2.4 Lagrangian Duality in 3D SLAM: Verification Techniques and Optimal Solutions

It is a summary of a few point cloud matching algorithms based on convex optimzation theory.

1 Banch and Bound

1.1 Practical Global Optimization for Multiview Geometry

article researchgate

• Agarwal S , Chandraker M K , Kahl F , et al. Practical Global Optimization for Multiview Geometry[C]// Computer Vision - ECCV 2006, 9th European Conference on Computer Vision, Graz, Austria, May 7-13, 2006, Proceedings, Part I. Springer, Berlin, Heidelberg, 2006.

This article talks about using Banch and Bound algorithm for three types of Multiview Geometry problems:

• Camera pixels triangualtion
• Camera Resectioning, which is to estimate camera matrix P.
• Projection from $\mathbf{P}^{n}$Pn to $\mathbf{P}^{m}$Pm, which could be homography estimation and use 3d point matches to estiamte relative transformation.

This article uses :

• Fractinal Programming to reform these problems
• Using Convex Envelope of these Fractinal problems as a lower bound for Banch and Bound
• Use three type of nosie models :
  ** Ordinary Gaussian noise model :
  $p(x|x') = \frac{1}{2 \pi \sigma^{2}} \exp( - \frac{ \| x - x' \|^{2}_{2}}{2\sigma^{2}})$p(x∣x′)=2πσ21​exp(−2σ2∥x−x′∥22​​)
  ** A special case for Laplace nosie
  $p(x|x') = \frac{1}{2 \pi \sigma} \exp( - \frac{ \| x - x' \|_{2}}{\sigma})$p(x∣x′)=2πσ1​exp(−σ∥x−x′∥2​​)
  ** Bivariate Laplace distribution
  $p(x|x') = \frac{1}{4 \sigma} \exp( - \frac{ \| x - x' \|_{1}}{\sigma})$p(x∣x′)=4σ1​exp(−σ∥x−x′∥1​​)

Here is its result on these three problems with different noise models.

• it shows that the tridational method (Bundle Adjustment method) for point cloud match is very senstive to noise, and easily trapped in local minimal. While BNB method find a global optimal is apparent much better.
• And it shows that L2 Guassian noise model shows better result for camera resectioning (the least reprojection error). While L1 Laplacian model shows better result for ICP type problems.
• And from the Table2, we can see a real L1 norm will conerge much faster. (It may be true that the 21 centry is the centry for L1 norm)

1.2 Branch-and-Bound Methods for Euclidean Registration Problems

paper link

• Olsson C , Kahl F , Oskarsson M . Branch-and-Bound Methods for Euclidean Registration Problems[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2009, 31(5):783-794.

This is a article focus on the 3D points registration problems, which needs already matched point pairs.

• The lower bounds are obtained by Convex relaxations of $\mathbf{SO}(3)$SO(3), which is the rotation part. It uses quaternion expression of the rotation. And apply a convex envelop over $q_{i}q_{j}$qi​qj​ .

• It shows a much better result compared to linear algebra methods (DLT), and have a competable result compared to DLT + bundle adjustment.
• It is much much more stable compared to the other methods.
• While even though they didn’t show, I think the comptation time it needs is much more than the other two methods.

2. Lagrangian Duality

2.1 Convex Global 3D Registration with Lagrangian Duality

articel link

• Briales J , Gonzalez-Jimenez J . Convex Global 3D Registration with Lagrangian Duality[C]// IEEE Conference on Computer Vision & Pattern Recognition. IEEE, 2017.

This article focus on Lagrangian Duality of ICP problems, and in practice it shows strong duality holds, and it shows a competable result compared with the article of 1.2(Branch-and-Bound Methods for Euclidean Registration Problems).

• It use the matrix representation of rotation.
• It has a very detail summary of the ICP methods (point-to-point, point-to-line, and point-to-plane).
• It reform the problem into QCQP (nonconvex quadratically constrained quadratic programs) (much more about QCQP could be found here ), so the problem can be reformed into SDP (semidefinite program) via Lagrangian Duality (or via SDP relaxation).

I strongly recommond to read its supplemenraty material, it is a very good learning material for QCQP and ICP.

The results are mainly compared with the article of 1.2(Branch-and-Bound Methods for Euclidean Registration Problems). It shows that strong duality holds in practice (no proof offered), and this method shows a much more short CPU runtime (as we don’t need to search).

2.2 Solving Quadratically Constrained Geometrical Problems using Lagrangian Duality

article link

This content of this article is already contained in the upper article 1.2 and 1.1.

2.3 Duality-based Verification Techniques for 2D SLAM

article link

• Carlone L , Dellaert F . Duality-based verification techniques for 2D SLAM[J]. Proceedings IEEE International Conference on Robotics & Automation, 2015, 2015:4589-4596.

It reform the problem into QCQP and use SDP relaxation to find bound. And this article mainly focus on a evulation of SLAM result, using a lower bound and a upper bound. Does not offer a BNB (or other) algorithm to be aimed at solving the problem.

• The loss function is difference of poses.
• It use matrix representation of rotation, and use Chordal distance(element wise squared norm) as error term.
• Reform the problem into QCQP and find SDP relaxations.
• Find a lower bound using exact SDP relaxation and a looser bound using a simplfied SDP.
• Find upper bound by projecting the duality result into feaible set.

Better see the next article for more, which a more detailed version of this work.

2.4 Lagrangian Duality in 3D SLAM: Verification Techniques and Optimal Solutions

article link

• Carlone L , Rosen D M , Calafiore G , et al. Lagrangian Duality in 3D SLAM: Verification Techniques and Optimal Solutions[C]// 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2015.

Mostly the same as the article before (2.3 Duality-based Verification Techniques for 2D SLAM) except that it in extended to 3D SLAM problems.

Here we see the analysis of results.

The bounds are good, and these two articles are very good material for a application of relaxation of QCQPs.