Efficient L0 resampling of point sets
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 minimization framework. In the past decade, 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 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 norm brings many benefits, it decreases the overall processing efficiency greatly. As norm is really hard to optimize, the 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 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:
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Firstly, we present an minimization based framework for point set resampling.
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Secondly, we propose two algorithms to further speed up point set resampling.
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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 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.
Section snippets
Related work
Point set resampling has been extensively studied over the past two decades in computer graphics, and we only review some represented works in recent years. Moving Least Square (MLS) (Alexa et al., 2003) has been successfully used to resample and smooth point cloud in the presence of noise. The core of MLS is to iteratively project points to a locally fit polynomial. As MLS usually assumes the underlying surface is smooth everywhere, it will have problems in handling outliers and sharp
point set resampling
The input to our method is a set of unorganized points , typically unevenly distributed, without normals and containing noises and outliers. The output of our method is an oriented point set consisting of cleaned point positions that represent well the underlying piece-wise smooth surface, and their associated reliable normals .
Accelerating methods
In Algorithm 1, most of the computation time is spent on solving Equation (11). Point set data usually contains a large size of 3D points, so that it can describe the details of a surface. In the filed of point set processing, it is very common that the size of the point set is larger than hundreds of thousands. Hence, the size of the coefficient matrix A is very large, even in the format of sparse matrix. Moreover, Algorithm 1 takes small steps in the optimization space to approach the true
Results and comparisons
In this section, we evaluate the consolidation quality and the processing efficiency of our proposed point set resampling (L0R) and accelerated point set resampling (AL0R). Both L0R and AL0R have only one parameter that need to be specified by users, the normal weight λ. We find that works well for all the testing point set used in this paper, so we keep it fixed. The fast grid-based searching technique is utilized to find original neighbors and self neighbors under a given radius
Limitations
Although our method has shown strengths in handling noises and preserving features, failure cases can still occur when dealing with certain type of thin structures. We show an example in Fig. 12, in which some holes appear in the reconstructed mesh model. The points on the two sides of the vertical plate have exactly opposite normal orientation. Due to the close distances between these points, our resampling would mix them together, subsequently causing errors in the surface reconstruction.
Conclusions
We have proposed a novel point set resampling method in an minimization framework. Our framework can produce a set of clean, uniformly distributed, geometry-maintaining and feature-preserving oriented points from an unorganized and noisy point set. Compared with current point set resampling methods, our method specializes in anti-noise and persevering features based on the norm constraints. To further speed up our method, we also provide two accelerating algorithms. Optimization-based
Declaration of Competing Interest
None.
Acknowledgements
Many thanks to the editor and the anonymous reviewers for their valuable comments. This work was partially supported by the National Natural Science Foundation of China (No. 61802322, No. 61402387 and No. 61602139), the Guiding Project of Fujian Province, China (No. 2018H0037), the Fundamental Research Funds for the Central Universities, China (No. 20720190003), and the Research Funds administered by the Digital Fujian, at the Big Data Institute for Urban Public Safety.
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