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Efficient Learning on Point Clouds with Basis Point Sets
Sergey Prokudin, Christoph Lassner, Javier Romero
TL;DR
Point clouds are difficult for learning because their unordered structure complicates conventional processing, while occupancy grids are costly and can lose detail. The paper introduces basis point sets, which encode clouds as compact fixed-length distance features usable with standard architectures. BPS matches PointNet accuracy with three orders of magnitude fewer floating-point operations and enables single-pass high-resolution mesh registration, while its fixed basis set remains a method assumption.
Problem
Point clouds are hard to process because they are unstructured, while occupancy grids increase data storage and lose detail through discretization.
Method
Basis point sets encode point clouds as minimal distances to a fixed set of points, producing compact fixed-length inputs for standard machine-learning models.
Results
BPS matches PointNet accuracy on ModelNet40 with three orders of magnitude fewer floating-point operations and supports single-pass high-resolution mesh registration from noisy scans.
Takeaways & Limitations
BPS provides a computationally efficient point-cloud representation that remains competitive in classification and supports real-time, single-pass mesh registration.
Takeaways & Limitations
The basis point set is arbitrary but fixed across the dataset, with its size controlling the trade-off between computational complexity and representation fidelity.
Abstract
from arXiv · showhide
With the increased availability of 3D scanning technology, point clouds are moving into the focus of computer vision as a rich representation of everyday scenes. However, they are hard to handle for machine learning algorithms due to their unordered structure. One common approach is to apply occupancy grid mapping, which dramatically increases the amount of data stored and at the same time loses details through discretization. Recently, deep learning models were proposed to handle point clouds directly and achieve input permutation invariance. However, these architectures often use an increased number of parameters and are computationally inefficient. In this work, we propose basis point sets (BPS) as a highly efficient and fully general way to process point clouds with machine learning algorithms. The basis point set representation is a residual representation that can be computed efficiently and can be used with standard neural network architectures and other machine learning algorithms. Using the proposed representation as the input to a simple fully connected network allows us to match the performance of PointNet on a shape classification task while using three orders of magnitude less floating-point operations. In a second experiment, we show how the proposed representation can be used for registering high-resolution meshes to noisy 3D scans. Here, we present the first method for single-pass high-resolution mesh registration, avoiding time-consuming per-scan optimization and allowing real-time execution.
1. Introduction
Point clouds are rich but difficult for machine learning because they are unstructured and lack fixed size or clear neighborhoods. The paper proposes basis point sets, a compact fixed-length encoding usable with standard models, and evaluates it for classification and mesh registration.
- Motivation: Point clouds vary in cardinality, lack straightforward point correspondences, and have no clear notion of neighborhood for convolution.These properties make direct application of conventional deep-learning operations difficult.
- Proposed Representation: Basis point sets encode each point cloud as minimal distances to a fixed set of points, producing a compact fixed-length vector.The vector length can be adjusted to trade representation fidelity against computational efficiency.
- Applications: The fixed-length encoding can be used with standard machine-learning techniques, including artificial neural networks, for classification and mesh registration.The paper analyzes both point-cloud classification and mesh registration over noisy 3D scans.
- Results: The method matches PointNet accuracy on ModelNet40 while using an order of magnitude fewer parameters and three orders of magnitude fewer floating-point operations.This result is reported for point-cloud shape classification.
- Results: For noisy scans, the method produces aligned high-resolution meshes in a single feed-forward pass with competitive FAUST performance.The approach requires no additional post-processing and can run in real time on a non-GPU laptop.
2. Related Work
Existing 3D representations trade computational efficiency, memory, or shape fidelity against the ability to process point clouds with general learning models. BPS addresses these limitations with a simpler, more efficient encoding that preserves surface information and supports standard architectures.
- Point Clouds: PointNet-like methods process unordered point clouds through per-point networks and pooling, while other approaches organize points with kd-trees or graphs.The paper positions BPS among methods designed for direct point-cloud processing.
- Point Clouds: The proposed encoding is presented as more computationally efficient and simpler to implement than existing point-cloud networks.It does not depend on a specific network architecture or require custom neural-network layers.
- Occupancy Grids: Occupancy grids have cubic complexity, large memory requirements, sparse volumes, and difficulty preserving high-frequency surface details.Even grids of size 128^3 or 256^3 are described as insufficient for accurately representing some high-frequency surfaces.
- Signed Distance Fields: Signed distance fields retain cubic memory requirements and low convolutional efficiency despite storing distance-to-surface information.They partially address surface representation but retain key computational costs of grid-based encodings.
- 2D Projections: Projection-based methods can achieve high accuracy but are limited when 3D shapes contain multiple folds invisible from an external projection.The paper states that its encoding can preserve surface information for objects with arbitrary topology.
3. Method
The method normalizes point clouds, encodes them using a fixed basis point set and nearest-point features, and supports adjustable efficiency–fidelity trade-offs. The encoding can also be computed efficiently and augmented with directional or point attributes.
- Normalization: Each point cloud is normalized to fit a unit ball before encoding.The normalization uses the cloud centroid and maximum distance from that centroid.
- BPS construction: A basis point set is formed by sampling k random points from a ball of radius r.The set is arbitrary but fixed across all point clouds; r and k are method hyperparameters.
- Feature calculation: The feature vector stores the minimal distance from each basis point to the nearest point in the input cloud.This produces a fixed-length representation suitable for learning algorithms.
- Feature calculation: Directional information can alternatively be represented as delta vectors from basis points to their nearest cloud points.Additional nearest-point information, including RGB values and surface normals, can also be stored.
- BPS selection strategies: Uniform random sampling in the unit ball provides a good trade-off among efficiency, universality, and surface reconstruction results.An extensive 3D grid can instead maximize performance at increased computational complexity.
- Complexity: The encoding has complexity O(n log n + k log n) per point cloud when using ball-tree nearest-neighbor searches.The paper uses Euclidean distances and notes that kNN search can be integrated into an end-to-end learning pipeline.
4. Analysis
The analysis compares BPS with alternative point-cloud encodings for surface reconstruction and machine learning, finding that BPS preserves detail efficiently while supporting flexible basis arrangements and models.
- Surface reconstruction: BPS stores continuous global distance information, improving surface-detail representation over binary occupancy grids and local distance fields.The representation uses distances to basis points rather than binary occupancy indicators or truncated local distances.
- Surface reconstruction: 103 random ModelNet40 shapes are encoded and reconstructed using raw points, occupancy grids, and BPS under a fixed description length.Quality is evaluated with normalized bidirectional Chamfer distance between original and reconstructed point clouds.
- Surface reconstruction: BPS produces less than half the encoding error of occupancy grids for point clouds up to roughly 10^4 points, while approaching subsampled point-cloud accuracy.Unlike raw points, BPS yields an ordered, fixed-length vector suitable for machine-learning methods.
- Basis point selection strategies: Random uniform and HCP basis selection provide the best reconstruction results, while regular grids enable convolution operations and learned translation- and rotation-invariant features.The paper mainly uses random uniform ball sampling for its simplicity and efficiency.
- Machine-learning evaluation: The experiments then evaluate encoding and basis-selection strategies for machine-learning applicability, including occupancy grids, TDFs, and global or local BPS features.With a fixed architecture, global BPS significantly outperforms its local counterparts on ModelNet40 classification.
5. Learning with Basis Point Sets
The experiments evaluate BPS representations for classification and single-pass mesh registration. BPS supports competitive accuracy while reducing computational cost and enabling real-time alignment from noisy scans.
- 3D Shape Classification: Using global distances as BPS features improves accuracy over occupancy flags and is less affected by grid-size reduction.Reducing grid size causes a 5% accuracy drop for occupancy-based methods but only a -0.7% change for BPS.
- 3D Shape Classification: Directional information hurts plain fully connected networks but improves 3D-convolutional models.
- 3D Shape Classification: BPS features match PointNet performance with a simple fully connected network and match PointNet++ with a more efficient 3D-convolutional model.The fully connected model trains in several minutes on a single GPU.
- Single-Pass Mesh Registration from 3D Scans: The mesh-registration network maps BPS-encoded point clouds directly to the vertex positions of a common template.Its training data are generated from SMPL bodies with varied shapes, poses, and added noise.
6. Conclusion and Future Work
The paper concludes that BPS provides compact, fixed-length point-cloud representations with computational advantages across applications. It also identifies alternative BPS arrangements and rotation-invariant representations as future directions.
- Conclusion: BPS provides compact fixed-length point-cloud representations that can preprocess inputs for varied machine-learning models.
- Conclusion: Experiments across two applications show orders-of-magnitude processing-time advantages while retaining competitive accuracy.
- Future Work: Future work includes alternative BPS arrangements and corresponding convolutions for improved efficiency and rotation-invariant learning.
Supplementary Material
The supplied passage identifies the Max Planck Institute for Intelligent Systems.
- Supplementary Material: The Max Planck Institute for Intelligent Systems is named in the supplementary material.
1. Encoding time
The supplementary analysis measures BPS encoding runtime across point-cloud and basis-point counts. It finds real-time performance for the main application range and exposes an efficiency–fidelity trade-off through the basis-point count.
- Encoding Time: The runtime benchmark varies point-cloud size from 10^1 to 10^5 points and the number of basis points k.It uses a random subset of 10^3 ModelNet40 CAD models.
- Encoding Time: BPS encoding is super real-time for point clouds with n < 10^5 and k < 10^4.Combined with the deep network, this enables real-time mesh registration from raw scans.
- Encoding Time: Varying k tailors BPS encoding to a platform and application through an efficiency–fidelity trade-off.
2. Space and time complexity
The paper evaluates BPS efficiency through encoding and inference-time comparisons. Its fully connected model achieves sub-millisecond inference, while encoding remains real-time for the main application settings.
- Inference times are compared across point-cloud processing networks using available implementations and the authors' models.
- Encoding is real-time for point clouds with n < 10^5 and BPS with k < 10^4.
- Sub-millisecond inference time is achieved by the fully connected model on CPU and GPU.
3. Training details
The training details describe the architectures, optimization procedures, and schedules used for the ModelNet40, Conv3D, and FAUST models.
- The training description covers the most important elements of the network architectures and training strategy.
- ModelNet40 models: The ModelNet40 MLP uses two 400-unit fully connected hidden layers, ReLU activations, dropout, and a final softmax layer.
- ModelNet40 models: The Conv3D model uses four convolutional blocks, max-pooling after each pair, and two 512-unit fully connected layers.
- FAUST model: The FAUST model uses fully connected layers of size 1024 and trains with SGD using momentum of 0.9 for 1000 epochs.
4. Results on DYNAMIC FAUST
BPS processes DYNAMIC FAUST frames independently without retraining or smoothness post-processing, while remaining sensitive to large outlier regions far from the body.
- Thousands of DYNAMIC FAUST point clouds are aligned using the FAUST network without retraining or fine-tuning.
- Each frame is processed independently, and the video uses no smoothness post-processing.
- Strong outliers from the floor and scanner reduce alignment accuracy, especially when many points lie far from the body.
- Large distant outlier regions drastically change the representation because of point-cloud size normalization.