Denoising for 3D Point Cloud Based on Regularization of a Statistical Low-Dimensional Manifold
Abstract
:1. Introduction
- (1)
- A denoising algorithm was designed for Gaussian noise and Laplace noise to denoise the two kinds of noise together;
- (2)
- Using the regularization term of the manifold and the fidelity term of the noise, the basic structure of a denoised point cloud was maintained;
- (3)
- Discrete sampling was used to construct low-dimensional manifolds to avoid a large number of calculations.
2. Related Works
3. Methods
3.1. Low-Dimensional Manifold Model
3.2. Point Cloud Denoising Model
3.2.1. 3D Point Cloud and Noise Model
3.2.2. Statistical Low-Dimensional Manifold Model
3.3. Solution of Point Cloud Denoising Model
3.3.1. Solution Principle
3.3.2. Algorithm Design
Algorithm 1: Denoising for 3D point cloud based on R- |
Input: , , , , . Output: Denoised cloud . 1: Initializing with ; 2: for = 1, 2, … do; 3: Sampling points from as a block center; 4: Find the nearest neighbors of the center of each block to form a surface block; 5: Optimizing objective function: , subject to: ; 6: ; 7:; 8: ; ; 9: converges; 10: end for |
3.3.3. Performance Evaluation
4. Results and Discussion
4.1. Comparative Analysis of Denoising Performance
4.2. Denoising Application for 3D Point Cloud
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
observation image | |
position matrix of block center | |
, , | constant |
number | |
dimension of block manifold | |
distance threshold | |
diagonal matrix | |
coordinate function | |
Gaussian noise | |
count | |
number of nearest fields | |
Laplace noise | |
Laplacian matrix of combined graph | |
global graph Laplace matrix | |
number of samples | |
position of point cloud | |
point cloud with noise | |
number of samples | |
number of samples of ground-truth point cloud | |
number of samples of observed point cloud after denoising | |
block set | |
point of | |
block coordinates | |
position matrix of point cloud | |
midpoint coordinate vector in x-direction | |
midpoint coordinate vector in y-direction | |
midpoint coordinate vector in z-direction | |
index of coordinates, | |
radius of neighborhood | |
number field with size of | |
number field with size of | |
sampling rate of block center | |
length of pixel block | |
width of pixel block | |
sampling matrix | |
average distance | |
multiple threshold of standard deviation | |
image | |
set of ground-truth point cloud | |
set of observed point cloud after denoising | |
ideal model | |
edge weight | |
adjacency matrix | |
pixel | |
coordinate function, = 1,…, | |
constant | |
manifold dimension | |
perturbing noise. | |
reset parameter | |
maximum number of iteration | |
parameter | |
parameter in penalty item | |
mean value of | |
mean value of | |
standard deviation | |
standard deviation of | |
standard deviation of | |
covariance of and | |
edge set | |
metric operator | |
discrete graph | |
manifold | |
manifold size | |
set of point cloud with noise | |
visible surface block | |
pixel index |
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Noise Levels | Means | |||||
---|---|---|---|---|---|---|
Noise | 47.31 | 46.73 | 46.14 | 45.26 | 44.96 | 46.08 |
APSS | 48.12 | 47.52 | 46.87 | 45.96 | 45.48 | 46.79 |
NLD | 47.93 | 46.28 | 46.71 | 45.78 | 45.38 | 46.42 |
FGL | 48.05 | 48.29 | 46.22 | 46.75 | 45.83 | 47.03 |
R-SLDM | 49.78 | 48.77 | 48.26 | 47.36 | 46.98 | 48.23 |
Noise Levels | Means | |||||
---|---|---|---|---|---|---|
Noise | 0.196 | 0.231 | 0.283 | 0.315 | 0.356 | 0.276 |
APSS | 0.187 | 0.216 | 0.267 | 0.289 | 0.313 | 0.255 |
WLOP | 0.179 | 0.208 | 0.257 | 0.273 | 0.308 | 0.245 |
FGL | 0.174 | 0.192 | 0.236 | 0.274 | 0.301 | 0.235 |
R-SLDM | 0.139 | 0.153 | 0.162 | 0.169 | 0.172 | 0.159 |
Noise Levels | Means | |||||
---|---|---|---|---|---|---|
Noise | 0. 944 | 0.921 | 0.913 | 0.895 | 0.856 | 0.906 |
APSS | 0.954 | 0.936 | 0.967 | 0.949 | 0.913 | 0.944 |
WLOP | 0.959 | 0.948 | 0.957 | 0.931 | 0.928 | 0.945 |
FGL | 0.981 | 0.953 | 0.931 | 0.937 | 0.931 | 0.947 |
R-SLDM | 0.979 | 0.983 | 0.962 | 0.961 | 0.952 | 0.967 |
Objects | a | b | c | d | Means |
---|---|---|---|---|---|
Noise | 45.21 | 47.51 | 48.42 | 44.23 | 46.34 |
APSS | 46.63 | 48.63 | 48.98 | 45.63 | 47.47 |
NLD | 46.75 | 47.74 | 48.78 | 46.14 | 47.35 |
FGL | 46.92 | 47.82 | 48,83 | 47.81 | 47.85 |
R-SLDM | 47.83 | 48.64 | 49.74 | 47.73 | 48.49 |
Objects | a | b | c | d | Means |
---|---|---|---|---|---|
Noise | 4.734 | 4.348 | 5.378 | 5.134 | 4.989 |
APSS | 4.257 | 3.789 | 4.898 | 4.568 | 4.378 |
WLOP | 4.191 | 3.695 | 4.788 | 4.414 | 4.272 |
FGL | 3.933 | 2.978 | 3.764 | 3.789 | 3.616 |
R-SLDM | 3.275 | 2.784 | 3.356 | 3.246 | 3.165 |
Objects | a | b | c | d | Means |
---|---|---|---|---|---|
Noise | 0.884 | 0.912 | 0.878 | 0.934 | 0.902 |
APSS | 0.913 | 0.941 | 0.918 | 0.968 | 0.935 |
WLOP | 0.931 | 0.965 | 0.908 | 0.944 | 0.937 |
FGL | 0.956 | 0.953 | 0.928 | 0.954 | 0.950 |
R-SLDM | 0.975 | 0.984 | 0.956 | 0.976 | 0.973 |
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Liu, Y.; Zou, B.; Xu, J.; Yang, S.; Li, Y. Denoising for 3D Point Cloud Based on Regularization of a Statistical Low-Dimensional Manifold. Sensors 2022, 22, 2666. https://doi.org/10.3390/s22072666
Liu Y, Zou B, Xu J, Yang S, Li Y. Denoising for 3D Point Cloud Based on Regularization of a Statistical Low-Dimensional Manifold. Sensors. 2022; 22(7):2666. https://doi.org/10.3390/s22072666
Chicago/Turabian StyleLiu, Youyu, Baozhu Zou, Jiao Xu, Siyang Yang, and Yi Li. 2022. "Denoising for 3D Point Cloud Based on Regularization of a Statistical Low-Dimensional Manifold" Sensors 22, no. 7: 2666. https://doi.org/10.3390/s22072666
APA StyleLiu, Y., Zou, B., Xu, J., Yang, S., & Li, Y. (2022). Denoising for 3D Point Cloud Based on Regularization of a Statistical Low-Dimensional Manifold. Sensors, 22(7), 2666. https://doi.org/10.3390/s22072666