Correction: Prokop et al. Low Overlapping Point Cloud Registration Using Line Features Detection. Remote Sens. 2020, 12, 61

There was an error in the original publication [1]. The text of the manuscript differed with the project code, available at: https://github.com/Milos9304/LowOverlapPCRegistration (accessed on 1 June 2025), in the metric function used to evaluate closeness of points on the parabolic cylinder. While the original manuscript suggested using the close-form integral of arc-length function, this choice in practical implementation leads to numerical instabilities, and an approximation of the integral, satisfying the properties of a metric function, was found to provide more reliable results.

A correction has been made to Section 3. Point Cloud Registration Using Line Features Detection, 3.5.1. Finding First Translational Parameter, 4th paragraph, by insertion:

“which has a closed-form expression (see Appendix A.1)

However, due to numerical inefficiency and round-off errors related to calculation of the inverse hyperbolic sine, sinh⁻¹(·), the use of Equation (7) in the actual implementation is discouraged and an approximation should be used in practice (see Appendix A.1) for details.”

Another insertion has been performed in Appendix A. Parabolic Cylinder, 1st paragraph:

“Appendix A.1. The Arc Length on the Surface of Parabolic Cylinder and Its Approximation

We begin with the arc length formula, initially introduced in Equation (6):

Given the parabolic cylinder $z=-K^2{y}^2+H$ (Equation (3)) we compute the derivative ${\displaystyle \frac{dz}{dy}}=-2K^{2}y$ and thus ${\left({\displaystyle \frac{dz}{dy}}\right)}^2=4K^4{y}^2$. Substituting into the arc length formula:

To evaluate this integral, we use the standard result ([30], p. 1035):

In our case, we substitute $a^2=1$, $x=2K^{2}y$, so we let:

Rewriting the integral:

Using the standard integral result (Equation (7)):

Substitute $u=2K^{2}y$ back in and we get the final, close formula form:

Despite the availability of the exact analytical arc length expression (Equation (7)), in our implementation we do not employ it due to its numerical and computational drawbacks. Specifically, evaluating the inverse hyperbolic sine function, ${sinh}^{-1}\left(\cdot \right)$, at high precision introduces instability for small $|y^A-y^B|$, where the function becomes highly sensitive to floating-point errors. This was empirically observed in our experiments, where results for closely spaced line intersections showed significant fluctuations and degraded reliability in fitness scoring. Instead, our code, that was used to run experiments in this paper, used a modified metric to evaluate “closeness” of the points $y^A,y^B$, defined by the integral whose closed-form expression avoids inverse trigonometric or logarithmic components and is computationally simpler and more numerically stable. While this function no longer corresponds to the true arc length (Equation (A3)), it retains monotonicity and smoothness, and its weighting naturally penalizes larger distances. This is desirable in our scoring context, as it discourages the matching of distant or unrelated features while preserving correct ordering locally. Moreover, in the limit as $|y^A-y^B|\to 0$, the expression becomes asymptotically equivalent to the true arc length, ensuring that local comparisons remain consistent with the underlying geometry. Indeed, the close formula of Equation (A2) can be evaluated the following way: Given the indefinite integral of Equation (A2) we substitute

Solving for $dy$, we get

Thus, the integral becomes

Integrating:

Substituting back $u=1+4K^4{y}^2$, we get the closed-form expression: which practically serves viable, and numerically more stable, alternative to the parabolic arclength function.”

In the original publication, ref. 30 was not cited.

The authors state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.