Normal distributions transform: Difference between revisions

Revision as of 21:06, 20 February 2022

Revision as of 14:31, 27 February 2022

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== 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 \Sigmamathbf{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 \Sigmamathbf{S} \left(\mathbf{x} - \mathbf{q}\right)}</math> .

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