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Integrating Product Coefficients for Improved 3D LiDAR Data Classification (Part II)

Patricia Medina, Rasika Karkare

TL;DR

This work integrates measure-theoretic product coefficients with nonlinear representation learning to advance 3D LiDAR point-cloud classification. By augmenting spatial features with dyadic-product coefficients and replacing PCA with a PyTorch autoencoder, the approach achieves higher classification accuracy and F1-macro scores, particularly when paired with KNN or RF classifiers. The study demonstrates that richer, nonlinear feature representations better preserve class structure and neighborhood relationships, significantly improving performance over PCA-based baselines. The findings suggest practical benefits for geospatial tasks and point to future enhancements via advanced nonlinear models and larger-scale datasets.

Abstract

This work extends our previous study on enhancing 3D LiDAR point-cloud classification with product coefficients \cite{medina2025integratingproductcoefficientsimproved}, measure-theoretic descriptors that complement the original spatial Lidar features. Here, we show that combining product coefficients with an autoencoder representation and a KNN classifier delivers consistent performance gains over both PCA-based baselines and our earlier framework. We also investigate the effect of adding product coefficients level by level, revealing a clear trend: richer sets of coefficients systematically improve class separability and overall accuracy. The results highlight the value of combining hierarchical product-coefficient features with autoencoders to push LiDAR classification performance further.

Integrating Product Coefficients for Improved 3D LiDAR Data Classification (Part II)

TL;DR

This work integrates measure-theoretic product coefficients with nonlinear representation learning to advance 3D LiDAR point-cloud classification. By augmenting spatial features with dyadic-product coefficients and replacing PCA with a PyTorch autoencoder, the approach achieves higher classification accuracy and F1-macro scores, particularly when paired with KNN or RF classifiers. The study demonstrates that richer, nonlinear feature representations better preserve class structure and neighborhood relationships, significantly improving performance over PCA-based baselines. The findings suggest practical benefits for geospatial tasks and point to future enhancements via advanced nonlinear models and larger-scale datasets.

Abstract

This work extends our previous study on enhancing 3D LiDAR point-cloud classification with product coefficients \cite{medina2025integratingproductcoefficientsimproved}, measure-theoretic descriptors that complement the original spatial Lidar features. Here, we show that combining product coefficients with an autoencoder representation and a KNN classifier delivers consistent performance gains over both PCA-based baselines and our earlier framework. We also investigate the effect of adding product coefficients level by level, revealing a clear trend: richer sets of coefficients systematically improve class separability and overall accuracy. The results highlight the value of combining hierarchical product-coefficient features with autoencoders to push LiDAR classification performance further.
Paper Structure (9 sections, 1 theorem, 8 equations, 6 figures, 5 tables)

This paper contains 9 sections, 1 theorem, 8 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Prior Work and Key Results from Our First Study
  3. The Data
  4. Product coefficients
  5. Product coefficients
  6. Real world applications
  7. Computing product coefficients on Lidar
  8. Classification experiments
  9. Conclusion and Future Research Directions

Key Result

lemma 1

Let $X$ be a dyadic set with binary set system $B$ whose non-leaf sets are $B_n$.

Figures (6)

  • Figure 1: Plot representing LiDAR point cloud of a tree from a Washington state park. Exact coordinates are available. We consider only three classes (with the three more visible ones being ground, canopy and trunk). Data was provided by Prof. Monica Moskal, Remote Sensing and Geospatial Analysis Lab (RSGAL), University of Washington.
  • Figure 2: A pulse can be reflected off a tree's trunk, branches, and foliage as well as reflected off the ground. This originates one of the data features (number of returns) This figure was used for the first time in one of the author's paper (see medina2019heuristic)
  • Figure 3: illustration of the first level of a dyadic set or binary tree for a set $X$. On the right we have how this translate into an interval
  • Figure 4: The diagram represents the binary tree for a general set $S.$ The diagram includes the notation used in the computation of the seven product coefficients. There are $2^0, 2^1$ and $2^2$ product coefficients per level.
  • Figure 5: When computing product coefficients per point $(x_i, y_i, z_i)$, consider a sphere $S+i$of radius 2. We first slice the sphere in along the $x$-axis and compute the first product coefficient $a_S$, then slice the sphere along the $y-$ axis and compute two product coefficients. One for the left child $L(S_i)$ and the other for the right child $R(S_i).$ Last, we slice along the $z-$ axis and compute the the last four product coefficients.
  • ...and 1 more figures

Theorems & Definitions (2)

  • definition 1
  • lemma 1: Dyadic Product Formula Representation