View source for Normal distributions transform

The '''normal distributions transform''' ('''NDT''') is a [[point-set registration|point cloud registration]] [[algorithm]] introduced by Peter Biber and Wolfgang Straßer in 2003, while working at [[University of Tübingen]].

The algorithm registers two point clouds by first associating a [[piecewise]] [[normal distribution]] to the first point cloud, that gives the probability of sampling a point belonging to the cloud at a given spatial coordinate, and then finding a transform that maps the second point cloud to the first by maximising the [[likelihood function|likelihood]] of the second point cloud on such distribution as a function of the transform parameters.

Originally introduced for 2D point cloud map matching in [[simultaneous localization and mapping]] (SLAM) and relative position tracking,<ref name="biberAndStraßer">{{harv|Biber|Straßer|2003}}</ref> the algorithm was extended to 3D point clouds<ref>{{harv|Magnusson|2009}}</ref> and has wide applications in [[computer vision]] and [[robotics]]. NDT is very fast and accurate, making it suitable for application to large scale data, but it is also sensitive to initialisation, requiring a sufficiently accurate initial guess, and for this reason it is typically used in a coarse-to-fine alignment strategy.<ref>{{harv|Dong et al.|2020}}</ref><ref>{{harv|Li|Wang|Zhang|2021|pp=21-22}}</ref><ref>{{harv|Cheng et al.|2018|pp=10-11,13}}</ref>

== Formulation ==

The NDT function associated to a point cloud is constructed by partitioning the space in regular cells. For each cell, it is possible to define the mean <math>\textstyle \mathbf{q} = \frac{1}{n} \sum_i \mathbf{x_i}</math> and covariance <math>\textstyle \mathbf{S} = \frac{1}{n} \sum_i \left(\mathbf{x}_i - \mathbf{q}\right) \left(\mathbf{x}_i - \mathbf{q}\right)^\top</math> of the <math>n</math> points of the cloud <math>\mathbf{x}_1, \dots, \mathbf{x}_n</math> that fall within the cell. The probability density of sampling a point at a given spatial location <math>\mathbf{x}</math> within the cell is then given by the normal distribution

:<math>e^{-\frac{1}{2} \left(\mathbf{x} - \mathbf{q}\right)^\top \mathbf{S}^{-1} \left(\mathbf{x} - \mathbf{q}\right)}</math> .

Two point clouds can be mapped by a [[Euclidean transformation]] <math>f</math> with [[rotation matrix]] <math>\mathbf{R}</math> and translation vector <math>\mathbf{t}</math>

:<math>f_{\mathbf{R}, \mathbf{t}}(\mathbf{x}) = \mathbf{R} \mathbf{x} + \mathbf{t}</math>

that maps from the second cloud to the first, parametrised by the rotation angles and translation components.

The algorithm registers the two point clouds by [[mathematical optimization|optimising]] the parameters of the transformation that maps the second cloud to the first, with respect to a loss function based on the NDT of the first point cloud, solving the following problem

:<math>\arg\min_{\mathbf{R}, \mathbf{t}} \left\{ -\sum_i \operatorname{NDT} \left( f_{\mathbf{R}, \mathbf{t}} \left( \mathbf{x_i} \right) \right) \right\}</math>

where the loss function represents the negated likelihood, obtained by applying the transformation to all points in the second cloud and summing the value of the NDT at each transformed point <math>f_{\mathbf{R}, \mathbf{t}}(\mathbf{x})</math>. The loss is piecewise [[continuous function|continuous]] and [[differentiable function|differentiable]], and can be optimised with [[gradient]]-based methods (in the original formulation, the authors use [[Newton's method in optimization|Newton's method]]).

In order to reduce the effect of cell discretisation, a technique consists of partitioning the space into multiple overlapping grids, shifted by half cell size along the spatial directions, and computing the likelihood at a given location as the sum of the NDTs induced by each grid.<ref name="biberAndStraßer" />

== References ==
<references />

== Sources ==
*{{cite conference|title=The normal distributions transform: A new approach to laser scan matching|book-title=Proceedings 2003 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2003)(Cat. No. 03CH37453)|year=2003|volume=3|last1=Biber|first1=Peter|last2=Straßer|first2=Wolfgang}}
*{{cite journal|title=Registration of laser scanning point clouds: A review|year=2018|journal=Sensors|volume=18|pages=1641|issue=5|publisher=Multidisciplinary Digital Publishing Institute|last1=Cheng|first1=Liang|last2=Chen|first2=Song|last3=Liu|first3=Xiaoqiang|last4=Xu|first4=Hao|last5=Wu|first5=Yang|last6=Li|first6=Manchun|last7=Chen|first7=Yanming|doi=10.3390/s18051641 |pmid=29883397 |pmc=5981425 |bibcode=2018Senso..18.1641C |doi-access=free|ref={{harvid|Cheng et al.|2018}}}}
*{{cite journal|title=Registration of large-scale terrestrial laser scanner point clouds: A review and benchmark|year=2020|journal=ISPRS Journal of Photogrammetry and Remote Sensing|volume=163|pages=327–342|publisher=Elsevier|last1=Dong|first1=Zhen|last2=Liang|first2=Fuxun|last3=Yang|first3=Bisheng|last4=Xu|first4=Yusheng|last5=Zang|first5=Yufu|last6=Li|first6=Jianping|last7=Wang|first7=Yuan|last8=Dai|first8=Wenxia|last9=Fan|first9=Hongchao|last10=Hyyppä|first10=Juha|doi=10.1016/j.isprsjprs.2020.03.013 |bibcode=2020JPRS..163..327D |s2cid=216449537|ref={{harvid|Dong et al.|2020}}}}
*{{cite journal|title=A Tutorial Review on Point Cloud Registrations: Principle, Classification, Comparison, and Technology Challenges|year=2021|journal=Mathematical Problems in Engineering|volume=2021|publisher=Hindawi|last1=Li|first1=Leihui|last2=Wang|first2=Riwei|last3=Zhang|first3=Xuping}}
*{{cite thesis|type=Ph.D.|title=The three-dimensional normal-distributions transform: an efficient representation for registration, surface analysis, and loop detection|year=2009|publisher=Örebro universitet|last1=Magnusson|first1=Martin}}

== External links ==
* {{cite web|url=https://www.mathworks.com/help/vision/ref/pcregisterndt.html#mw_d3d9a2dd-32d3-414b-9aa6-4c66c6a03649_head|title=Register two point clouds using NDT algorithm|publisher=MathWorks}}

{{DEFAULTSORT:Normal distributions transform}}

[[Category:Computer vision]]
[[Category:Pattern matching]]