Efficient L₀ resampling of point sets

Highlights

• We present an L0 minimization based framework for point set resampling.

• We propose two algorithms to further speed up L0 point set resampling.

• We achieve state-of-the-art point set consolidation performance.

Abstract

The point data captured by laser scanners or consumer depth cameras are often contaminated with severe noises and outliers. In this paper, we propose a resampling method in an $L_0$ minimization framework to process such low quality data. Our framework can produce a set of clean, uniformly distributed, geometry-maintaining and feature-preserving oriented points. The $L_0$ norm improves the robustness to noises (outliers) and the ability to keep sharp features, but introduces a significant efficiency degradation. To further improve the efficiency of our $L_0$ point set resampling, we propose two accelerating algorithms including optimization-based local half-sampling and interleaved regularization. As demonstrated by the experimental results, the accelerated method is about an order of magnitude faster than the original, while achieves state-of-the-art point set consolidation performance.

Introduction

The emergence of laser scanners and consumer depth cameras has made it possible for rapid and convenient acquisition of point cloud from real-world objects. Due to the inherent defects of hardware and the artifacts when registering multiple point clouds, the acquired point set data are often corrupted with severe noises and outliers, which hinder the subsequent tasks such as surface reconstruction. Many point set resampling techniques have been therefore developed to deal with such imperfect data, aiming at obtaining a set of high quality consolidated points.

We think a high quality consolidated point set should satisfy five criteria: (i) geometry-maintaining, the resampled point set should at least “look like” the original shape; (ii) noise-free, the boring noises and outliers should be removed as much as possible; (iii) feature-preserving, the edges, corners and other sharp features should be well preserved or even enhanced; (iv) uniformly distribution, the resampled points are expected to be evenly distributed in space; (v) reliable normals, a set of reliable normals can facilitate surface reconstruction and points rendering. Although there exists many point set resampling methods, none of them can fulfill all the above five criteria completely in a scheme. The Locally Optimal Projection (LOP) operator (Lipman et al., 2007) is mainly designed for removing noises, which would result in local clusters if the input contains non-homogeneous point density. The variant of LOP, Weighted LOP (WLOP) (Huang et al., 2009), achieves more uniform distribution, but is inadequate in preserving features. Edge-Aware Resampling (EAR) (Huang et al., 2013b) can resample the points in a feature preserving manner, with the guidance of filtered normals. However, it is not suited to handle noises when the features are tempered with large amount of noises and outliers. Neural network based resampling methods, such as Patch-based Progressive Upsampling (PPU) (Wang et al., 2019), can generate a high-resolution point set from a low-resolution input, but it cannot produce plausible results from particularly noisy inputs. Fig. 1 shows the performance of the above methods in a low-quality point set.

In this paper, we propose a point set resampling method in an $L_0$ minimization framework. In the past decade, $L_0$ minimization was introduced to explore the sparsity in the reconstructed signal. And it has shown great ability of noise removing and feature preserving in many computer graphics applications including image smoothing (Xu et al., 2011, Cheng et al., 2014), mesh denoising (He and Schaefer, 2013), spline approximation (Brandt et al., 2015) and volume smoothing (Wang et al., 2015). Our $L_0$ point set resampling method is able to generate a clean, uniform, and feature-preserving set of oriented points that well represent the underlying surface, as illustrated in Fig. 1. Specifically, the fitting to the input points guarantees a well approximation to the underlying geometry; the enforced sparsity of the non-consistency between filtered normals and optimized points, improves the resilience to noise (outliers) and the ability to preserve features; the regularization and density weights contribute to a uniform distribution in the resulting point set; the bilateral normal filtering produces a set of reliable normals.

Although $L_0$ norm brings many benefits, it decreases the overall processing efficiency greatly. As $L_0$ norm is really hard to optimize, the $L_0$ minimization solver (Xu et al., 2011) often takes small steps in the parameter space to approximate solutions carefully and slowly. Besides, the data size of point set is usually very large, which further increases the amount of computation. In this paper, we also provide two algorithms to improve the efficiency of our $L_0$ point set resampling. Firstly, we propose an optimization-based local half-sampling to reduce the number of a sample's neighbors in the input point set. Secondly, we use a interleaved regularization to cut off some repetitive computations. As demonstrated by our experimental results, the accelerated method is about an order of magnitude faster than the original, while achieves nearly the same resampling quality.

In summary, this paper offers contributions in three aspects:

• Firstly, we present an $L_0$ minimization based framework for point set resampling.

• Secondly, we propose two algorithms to further speed up $L_0$ point set resampling.

• Thirdly, we achieve state-of-the-art point set consolidation performance.

The rest of the paper is organized as follows: Section 2 introduces the related work and background, Section 3 presents our point set resampling method using $L_0$ minimization, Section 4 shows two accelerating algorithms, Section 5 demonstrates the experimental results and comparisons, Section 6 discusses the limitations of our proposed method, and finally Section 7 concludes this paper.