Correction: Lowe, T.; Pinskier, J. Tree Reconstruction Using Topology Optimisation. Remote Sens. 2023, 15, 172

Thomas Lowe; Joshua Pinskier

doi:10.3390/rs15112739

and

Robotics and Autonomus Systems Group, CSIRO Data61, Pullenvale, QLD 4069, Australia

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Remote Sens.2023, 15(11), 2739;https://doi.org/10.3390/rs15112739

This article belongs to the Special Issue 3D Point Clouds in Forest Remote Sensing II

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Text Correction

There was an error in the original publication [1]. Appendix A was not up-to-date when published.

A correction has been made to Appendix A (The corrected Appendix A shows below):

Appendix A. Sensitivity

From Equation (2) the compliance for the smoothed densities

\bar{x}

can be expressed as a sum over all elements i:

\begin{matrix} c (\bar{x}) = \sum_{i} \tilde{E} u_{i}^{T} k_{i} u_{i} \end{matrix}

(A1)

where

\tilde{E}

is defined in Equation (6) as the product of two terms. Using the product rule, the differential with respect to each smoothed element density

{\bar{x}}_{i}

can then be written as the sum of two terms:

\begin{matrix} \frac{\partial c (\bar{x})}{\partial {\bar{x}}_{i}} = a_{i} u_{i}^{T} k_{i} u_{i} + \sum_{j} b_{i j} u_{j}^{T} k_{j} u_{j} \end{matrix}

(A2)

where

\begin{matrix} a_{i} = - A {({\bar{x}}_{i})}^{α} p {\bar{x}}_{i}^{p - 1} (E_{S} - E_{V}) \end{matrix}

(A3)

and

\begin{matrix} b_{i j} = - \frac{α A {({\bar{x}}_{j})}^{α - 1} (E_{V} + {\bar{x}}_{j}^{p} (E_{S} - E_{V}))}{\max (r_{i j}, 1)} \end{matrix}

(A4)

The sensitivity with respect to the unsmoothed design variable

x

is then:

\begin{matrix} \frac{\partial c (\bar{x})}{\partial x_{i}} = \frac{\sum_{j \in N_{e}} W_{i j} {\hat{x}}_{i} (β) \frac{\partial c (\bar{x})}{\partial {\bar{x}}_{j}}}{\sum_{j \in N_{e}} W_{i j}} \end{matrix}

(A5)

where

{\hat{x}}_{i} (β) = β \frac{1 - {tanh}^{2} (β (\tilde{x_{i}} - 0.5))}{2 tanh (0.5 β)}

(A6)

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

Reference

Lowe, T.; Pinskier, J. Tree Reconstruction Using Topology Optimisation. Remote Sens. 2023, 15, 172. [Google Scholar] [CrossRef]

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Correction: Lowe, T.; Pinskier, J. Tree Reconstruction Using Topology Optimisation. Remote Sens. 2023, 15, 172

Appendix A. Sensitivity

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