Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Point Cloud Synthesis Using Inner Product Transforms

Ernst Röell, Bastian Rieck

TL;DR

We introduce the Inner Product Transform (IPT) to encode point clouds as 2D descriptors, enabling a two-stage pipeline: image-based IPT synthesis followed by an image-to-point-cloud reconstruction via a lightweight IP-Encoder. The IPT is provably injective and admits a metric-based distance, with a practical surrogate enabling end-to-end learning and stable latent spaces. Empirically, the approach achieves competitive reconstruction and generation performance on ShapeNet while delivering vastly faster training and inference than many baselines, and shows robustness to out-of-distribution IPTs. This framework opens avenues for efficient, topology-informed point-cloud generation and potential extensions to higher-dimensional data and graph-structured objects.

Abstract

Point cloud synthesis, i.e. the generation of novel point clouds from an input distribution, remains a challenging task, for which numerous complex machine learning models have been devised. We develop a novel method that encodes geometrical-topological characteristics of point clouds using inner products, leading to a highly-efficient point cloud representation with provable expressivity properties. Integrated into deep learning models, our encoding exhibits high quality in typical tasks like reconstruction, generation, and interpolation, with inference times orders of magnitude faster than existing methods.

Point Cloud Synthesis Using Inner Product Transforms

TL;DR

We introduce the Inner Product Transform (IPT) to encode point clouds as 2D descriptors, enabling a two-stage pipeline: image-based IPT synthesis followed by an image-to-point-cloud reconstruction via a lightweight IP-Encoder. The IPT is provably injective and admits a metric-based distance, with a practical surrogate enabling end-to-end learning and stable latent spaces. Empirically, the approach achieves competitive reconstruction and generation performance on ShapeNet while delivering vastly faster training and inference than many baselines, and shows robustness to out-of-distribution IPTs. This framework opens avenues for efficient, topology-informed point-cloud generation and potential extensions to higher-dimensional data and graph-structured objects.

Abstract

Point cloud synthesis, i.e. the generation of novel point clouds from an input distribution, remains a challenging task, for which numerous complex machine learning models have been devised. We develop a novel method that encodes geometrical-topological characteristics of point clouds using inner products, leading to a highly-efficient point cloud representation with provable expressivity properties. Integrated into deep learning models, our encoding exhibits high quality in typical tasks like reconstruction, generation, and interpolation, with inference times orders of magnitude faster than existing methods.

Paper Structure

This paper contains 31 sections, 8 theorems, 17 equations, 9 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Methods
  4. Inner Product Transforms
  5. Properties.
  6. The IP-Encoder
  7. Model architecture.
  8. The IPT as an optimisation problem.
  9. Experiments
  10. Reconstructing Point Clouds
  11. Architecture and experimental setup.
  12. Results.
  13. Generating Novel Point Clouds
  14. Architecture and experimental setup.
  15. Results.
  16. ...and 16 more sections

Key Result

Theorem 1

Given two point clouds $X, Y$ with $X \neq Y$, we have $\mathrm{IPT\xspace}(X) \neq \mathrm{IPT\xspace}(Y)$.

Figures (9)

  • Figure 1: Given a point cloud on the left, we compute its Inner Product Transform (IPT). For generative tasks, we train a generative model (middle) to reconstruct and generate the distribution of IPTs. The (possibly-generated) IPT is then passed through the encoder model to obtain the reconstructed (or novel) point cloud. Our pipeline is decoupled, permitting any generative image model to be used to generate point clouds.
  • Figure 2: Examples of reconstructed\ref{['sfig:encoder']} and generated\ref{['sfig:generated']} point clouds using our IP-Encoder model for three classes in the ShapeNet dataset.
  • Figure 3: Linear interpolation between IPTs results in a smooth interpolation in the point-cloud domain. Given the IPT of two airplanes, we linearly interpolate between the pixel values and pass each step through the IP-Encoder to obtain a prediction of the intermediary point cloud. Although the IP-Encoder has not been specifically trained on such data, it is able to produce meaningful reconstructions, since it has learned the distribution of shapes. We provide two examples of the interpolation process, one with high pairwise similarity (top), the other with low pairwise similarity (bottom).
  • Figure S.4: An overview of the IPT calculation for $32$ directions and resolution of $32$. For a given direction vector $\xi$, we filter the point cloud with hyperplanes. The partially filtered point cloud is shown in \ref{['sfig:Filtration']} and points included in the filtration are coloured blue. For each of the $32$ directions, sampled uniformly from the sphere, the respective curves along each direction are shown in \ref{['sfig:Curves']} and the partially-completed curve from \ref{['sfig:Filtration']} is highlighted. Note that neighbouring curves are not necessarily related, requiring us to treat each curve as its own signal. Each of the $32$ curves is discretised in $32$ steps and stacked to form an image representation of the point cloud \ref{['sfig:Image']} of size $32\times 32$. The row corresponding to the full curve from \ref{['sfig:Filtration']} and each of the rows corresponds to the curve in the same index in \ref{['sfig:Curves']}. For complex geometries such as the ShapeNet data, we empirically observe that at least$128$ directions are required, using a resolution of $128$.
  • Figure S.5: A visual comparison between down- and upsampling using our IP-Downsampler \ref{['app:downsample_downsampler_appendix']} and uniform subsampling \ref{['app:downsample_uniform_appendix']}. The downsampled point clouds using the IP-Downsampler are equidistantly spread, in contrast to uniform subsampling. Compare to uniform subsampling, we achieve higher upsampling quality. Our model has a bias towards equidistant point clouds and leads to a better representation of the underlying shape.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • Theorem 2
  • proof
  • Lemma 2
  • proof
  • Lemma 2
  • proof
  • ...and 2 more