# LUP Student Papers

## LUND UNIVERSITY LIBRARIES

### Minimal Deformation Methods for Loop Closure

(2021) In Master's Theses in Mathematical Sciences FMAM05 20202
Mathematics (Faculty of Engineering)
Abstract
This thesis proposes a method for how to find duplicated 3D points in a single Structure from Motion point cloud. Together with related articles, this forms a possible solution to the loop closure problem.

The proposed method works by first selecting candidate pairs of 3D points by comparing BRIEF descriptors of all points. The second step consists of using RANSAC to select the best out of many possible deformations of the map. The deformations are created by finding the minimal increase in reprojection errors while fulfilling the added constraint that a selected candidate pair has to converge. Finding this minimal increase is done by assuming the residual is linear under small changes. The solution is then found by optimizing over the... (More)
This thesis proposes a method for how to find duplicated 3D points in a single Structure from Motion point cloud. Together with related articles, this forms a possible solution to the loop closure problem.

The proposed method works by first selecting candidate pairs of 3D points by comparing BRIEF descriptors of all points. The second step consists of using RANSAC to select the best out of many possible deformations of the map. The deformations are created by finding the minimal increase in reprojection errors while fulfilling the added constraint that a selected candidate pair has to converge. Finding this minimal increase is done by assuming the residual is linear under small changes. The solution is then found by optimizing over the subspace spanned by the smallest eigenvectors to the Hessian for the loss function.

The method was tested on two different datasets. It managed to correctly identify the main duplication in both sets. In the first and simpler of the two datasets, which had 1000 points, all of the 33 pairs identified by the method were verified by hand to be correct. On the second dataset, which had 6000 points, the method found 153 pairs. Seven of the found pairs corresponded with the main split in the map. The rest of the found pairs were points which were already very close together in the original map. This might hint at a possible problem with the original map creation. One which this method could help to solve. (Less)
Please use this url to cite or link to this publication:
author
supervisor
organization
course
FMAM05 20202
year
type
H2 - Master's Degree (Two Years)
subject
publication/series
Master's Theses in Mathematical Sciences
report number
LUTFMA-3436-2021
ISSN
1404-6342
other publication id
2021:E3
language
English
id
9040377
date added to LUP
2021-03-11 17:25:02
date last changed
2021-03-11 17:25:02
```@misc{9040377,
abstract     = {{This thesis proposes a method for how to find duplicated 3D points in a single Structure from Motion point cloud. Together with related articles, this forms a possible solution to the loop closure problem.

The proposed method works by first selecting candidate pairs of 3D points by comparing BRIEF descriptors of all points. The second step consists of using RANSAC to select the best out of many possible deformations of the map. The deformations are created by finding the minimal increase in reprojection errors while fulfilling the added constraint that a selected candidate pair has to converge. Finding this minimal increase is done by assuming the residual is linear under small changes. The solution is then found by optimizing over the subspace spanned by the smallest eigenvectors to the Hessian for the loss function.

The method was tested on two different datasets. It managed to correctly identify the main duplication in both sets. In the first and simpler of the two datasets, which had 1000 points, all of the 33 pairs identified by the method were verified by hand to be correct. On the second dataset, which had 6000 points, the method found 153 pairs. Seven of the found pairs corresponded with the main split in the map. The rest of the found pairs were points which were already very close together in the original map. This might hint at a possible problem with the original map creation. One which this method could help to solve.}},
author       = {{Tegler, Erik}},
issn         = {{1404-6342}},
language     = {{eng}},
note         = {{Student Paper}},
series       = {{Master's Theses in Mathematical Sciences}},
title        = {{Minimal Deformation Methods for Loop Closure}},
year         = {{2021}},
}

```