Content uploaded by Xuequan Lu
Author content
All content in this area was uploaded by Xuequan Lu on Jan 27, 2022
Content may be subject to copyright.
Towards Uniform Point Distribution in Feature-preserving Point Cloud Filtering
Shuaijun Chen
Deakin University
Australia
Jinxi Wang
Northwest A&F University
Yangling, China
Wei Pan
South China University of Technology
China
Shang Gao
Deakin University
Australia
Meili Wang
Northwest A&F University
Yangling, China
Xuequan Lu* (corresponding author)
Deakin University, Australia
xuequan.lu@deakin.edu.au
Abstract
As a popular representation of 3D data, point cloud
may contain noise and need to be filtered before use. Ex-
isting point cloud filtering methods either cannot pre-
serve sharp features or result in uneven point distri-
bution in the filtered output. To address this problem,
this paper introduces a point cloud filtering method that
considers both point distribution and feature preserva-
tion during filtering. The key idea is to incorporate a
repulsion term with a data term in energy minimiza-
tion. The repulsion term is responsible for the point
distribution, while the data term is to approximate the
noisy surfaces while preserving the geometric features.
This method is capable of handling models with fine-
scale features and sharp features. Extensive experi-
ments show that our method yields better results with
a more uniform point distribution in seconds.
Key words: Uniform Point Distribution, Point Cloud Filter-
ing, Feature-preserving
1. Introduction
Researchers have made remarkable achievements in
point cloud filtering in recent years. The newly proposed
methods typically aim at maintaining the sharp features of
the original point cloud while projecting the noisy points
to underlying surfaces. The filtered point cloud data can
then be used for upsampling [12], surface reconstruction
[13,27], skeleton learning [21,22] and computer animation
[24,28], etc.
The existing point cloud filtering methods can be divided
into traditional and deep learning techniques. Among the
traditional class, position-based methods [17,11,29] ob-
tain good smoothing results, while normal-based methods
[27,25] achieve better effects in maintaining sharp edges of
models (e.g., CAD models). Some of these methods incor-
porate repulsion terms to prevent the points from aggregat-
ing but still leave gaps near the edges of geometric features,
which affects the reconstruction quality. Deep learning-
based approaches [31,32,37] require a number of noisy
point clouds with ground-truth models for training and often
achieve promising denoising performance through a proper
number of iterations. These methods are usually based
on local information, and lead to less even distribution in
filtered results even in the presence of a “repulsion” loss
term. It is difficult for these methods to handle unevenly
distributed point clouds and sparsely sampled point clouds
since the patch size is difficult to adjust automatically. Also,
different patch sizes in the point cloud pose a significant
challenge to the learning procedure.
The above analysis motivates us to produce filtered point
clouds with the preservation of sharp features and a more
uniform point distribution.
In this paper, we propose a filtering method that pre-
serves features well while making the points distribution
more uniform. Specifically, given a noisy point cloud with
normals as input, we first smooth the input normals us-
ing Bilateral Filtering [12]. Principal Component Anal-
ysis (PCA) [10] is used for the initial estimation of nor-
mals. Secondly, we update the point positions in a local
manner by reformulating an objective function consisting
of an edge-aware data term and a repulsion term inspired by
[23,25]. The two terms account for preserving geometric
features and point distribution, respectively. The uniformly
distributed points with feature-preserving effects can be ob-
tained through a few iterations. We conduct extensive ex-
periments to compare our approach with various other ap-
proaches, including the position-based learning/traditional
approaches and the normal-based learning/traditional ap-
proaches. The results demonstrate that our method outper-
forms state-of-the-art methods in most cases, both in visu-
alization and quantitative comparisons.
2. Related Work
In this paper, we only review the most relevant work to
our research, including traditional point cloud filtering and
1
deep learning-based point cloud filtering.
2.1. Traditional Point Cloud Filtering
Position-based methods. LOP was first proposed in
[17]. It is a parameterization-free method and does not rely
on normal estimation. Besides fitting the original model,
a density repulsion term was added to evenly control the
point cloud distribution. WLOP [11] provided a novel re-
pulsion term to solve the problem that the original repul-
sion function in LOP dropped too fast when the support
radius became larger. The filtered points were distributed
more evenly under WLOP. EAR [12] added an anisotropic
weighting function to WLOP to smooth the model while
preserving sharp features. CLOP [29] is another LOP-based
approach. It redefined the data term as a continuous repre-
sentation of a set of input points.
Though only based on point positions, these approaches
achieved fair smoothing results. Still, since they disregard
normal information, these approaches tend to smear sharp
features such as sharp edges and corners.
Normal-based methods. FLOP [16] added normal in-
formation to the novel feature-preserving projection oper-
ator and preserved the features well. Meanwhile, a new
Kernel Density Estimate (KDE)-based random sampling
method was proposed for accelerating FLOP. MLS-based
approaches [14,15] have also been applied to point cloud
filtering. They relied upon the assumption that the given
set of points implicitly defined a surface. In [1], the au-
thors presented an algorithm that allocated a MLS local ref-
erence domain for each point that was most suitable for its
adjacent points and further projected the points to the un-
derlying plane. This approach used the eigenvectors of a
weighted covariance matrix to obtain the normals when the
input point cloud had no normal information. APSS [7],
RMLS [33], and RIMLS [27] were implemented based on
this, where RIMLS was based on robust local kernel regres-
sion and could obtain better results under the condition of
higher noise. GPF [25] incorporated normal information
to Gaussian Mixture Model (GMM), which included two
terms and performed well in preserving sharp features. A
robust normal estimation method was proposed in [23] for
both point clouds and meshes with a low-rank matrix ap-
proximation algorithm, where an application of point cloud
filtering was demonstrated. To keep the geometry features,
[19] first filtered the normals by defining discrete operators
on point clouds, and then present a bi-tensor voting scheme
for the feature detection step.
Inspired by image denoising, researchers have also in-
vestigated the nonlocal aspects of point cloud denoising.
The nonlocal-based point cloud filtering methods [3,4,36,
2] often incorporated normal information and designed dif-
ferent similarity descriptions to update point positions in
a nonlocal manner. Among them, [3] proposed a similar-
ity descriptor for point cloud patches based on MLS sur-
faces. [4] designed a height vector field to describe the dif-
ference between the neighborhood of the point with neigh-
borhoods of other points on the surface. Inspired by the
low-dimensional manifold model, [36] extends it from im-
age patches to point cloud surface patches, and thus serves
as a similarity descriptor for nonlocal patches. [2] presented
a new multi-patch collaborative method that regards denois-
ing as a low-rank matrix recovery problem. They define the
given patch as a rotation-invariant height-map patch and de-
noise the points by imposing a graph constraint.
Filtering methods that rely on normal information usu-
ally yield good results, especially for point clouds with
sharp features (e.g., CAD models). However, these methods
have a strong dependence on the quality of input normals,
and a poor normal estimation may lead to worse filtering
results.
Our proposed approach falls in the normal-based cate-
gory. Inspired by GPF, we estimate normals of the input
point cloud based on bilateral filtering [12] in order to get
high-quality normal information. Note that if the input point
cloud only contains positional information, PCA is used to
compute the initial normals. The point positions are then
updated in a local manner with the bilaterally filtered nor-
mals [23]. We also add a repulsion term [23] to ensure a
more uniform distribution for filtered points.
2.2. Deep Learning-based Point Cloud Filtering
A variety of deep learning-based methods dealing with
noisy point clouds have emerged [18,6,37,5,34,31,20].
In terms of point cloud filtering, PointProNets [32] intro-
duced a novel generative neural network architecture that
encoded geometric features in a local way and obtained an
efficient underlying surface. However, the generated un-
derlying surface was hard to fill the holes caused by in-
put shapes. NPD [5] redesigned the framework on the ba-
sis of PointNet [30] to estimate normals from noisy shapes
and then projected the noisy points to the predicted refer-
ence planes. Another PointNet-inspired method is called
Pointfilter [37]. It started from points and learned the dis-
placement between the predicted points and the raw input
points. Moreover, this approach required normals only in
the training phase. In the testing phase, only the point po-
sitions were taken as input to obtain filtered shapes with
feature-preserving effects. EC-NET [34] presented an edge-
aware network (similar to PU-NET [35]) for connecting
edges of the original points. This method got promising
results in retaining sharp edges in 3D shapes, but the train-
ing stage required manual labeling of the edges. Inspired
by PCPNet [8], PointCleanNet [31] developed a data-driven
method for both classifying outliers and reducing noise in
raw point clouds. A novel feature-preserving normal esti-
mation method was designed in [20] for point cloud filtering
2
with preserving geometric features. Deep learning-based
filtering methods usually yield good results with more au-
tomation and can often handle point clouds with high den-
sity. That is, low-density shapes as input may lead to poor
filtering outcomes. Also, such methods require to “see”
enough samples during training.
3. Method
Our approach consists of two phases. In phase one, we
smooth the initial normals by Bilateral Filter (refer to [12]
for more details) to ensure the quality of normals. In phase
two, we update point positions with the smoothed normals
to obtain a uniformly distributed point cloud with geomet-
ric features preserved. Figure 1shows an overview of the
proposed approach. We will specifically explain the second
phase in the following section.
3.1. Position Update
We first define a noisy input with Mpoints as P=
{pi}M
i=1,pi∈R3and N={ni}M
i=1,ni∈R3as the
corresponding filtered normals. To obtain local information
from a given point pi, we define a local structure sifor each
point in the point cloud, consisting of the knearest points
to the current point. We employ an edge-aware recovery
algorithm [23] to obtain filtered points by minimizing
D(P, N ) = X
i
X
j∈si
|(pi−pj)nT
j|2+
|(pi−pj)nT
i|2,
(1)
where pidenotes the point to be updated and pjdenotes
the neighbor point in the corresponding set si. Eq. (1) es-
sentially adjusts the angles between the tangential vector
formed by piand pjand the corresponding normal vectors
ni,nj.
Figure 2demonstrates how the points are updated on an
assumed local plane by this edge-aware technique. It can be
seen that the quality of the filtered points depends heavily
on the quality of the estimated normals. Our normals are
generated by bilaterally filtering the original input normals,
given its simplicity and effectiveness.
3.2. Repulsive Force
From Figure 3, it can be seen that the points would con-
tinually move towards the sharp edges at the position up-
date step, thus inducing gaps near sharp edges. This is also
demonstrated by [23] that minimizing D(P, N )inevitably
yields gaps near sharp edges, and the filtered points with
obvious gaps might greatly impact following applications
such as upsampling and surface reconstruction. Thus, we
introduce R(P, N )[25] to better control the distribution of
points.
R(P, N ) = X
i
λi
M
X
j∈si
η(rij )θ(rij ),(2)
Eq. (2) obtains a repulsion force using both
point coordinates and normals, where rij =
(pi−pj)−(pi−pj)nT
jnj
, the term η(r)equals
to −r, and the term θ(r) = e(−r2/(h/2)2)denotes a
smoothly decaying weight function.
3.3. Minimization
By combining Eq. (1) and Eq. (2), our final position
update optimization becomes:
argmin
P
D(P, N ) + R(P, N )(3)
The gradient descent method is employed to minimize
Eq. (3) and obtain the updated point p′
i. The partial deriva-
tive of Eq. (3) with respect to piis:
∂
∂pi
=X
j∈si
njpT
i−njpT
jpinT
j−pjnjT
∂pi
+
λiβij (pi−pj) (I−nT
jnj)
∂pi
,
(4)
where βij denotes θ(rij )
rij
∂η(rij )
∂r
, and Iis a 3×3identity
matrix.
The updated point p′
ican be calculated by:
p′
i=pi+γiX
j∈si
(pj−pi)nT
jnj+nT
ini+
µPj∈siwjβij (pi−pj) (I−nT
jnj)
Pj∈siwjβij
,
(5)
where γiis set to 1
3|si|according to [23], wjdenotes 1 +
Pj∈siθ(∥pi−pj∥), and µis a parameter which aims at
controlling the relative magnitude of the repulsive force.
3.4. Algorithm
The proposed method is described in Algorithm 1. We
first filter the normals using bilateral filtering. By feeding
the filtered normals and raw point positions into Algorithm
1, we can obtain the updated point positions. Depending
on the point number of each model and the noise level, we
choose different kto generate the local patches and perform
several iterations accordingly. Section 4provides the fil-
tered results of different models. Table 1lists our parameter
settings.
3
(a) Noisy input (b) Normal filtering (c) Position update (d) Result
Data term
Repulsion term
1 iteration 5 iterations 15 iterations
(e) Reconstruction
Figure 1. Overview of our approach. (a) Noisy input. Red color denotes points corrupted with noise. (b) Filtered normals. Blue lines
denote filtered results of the initial normals. (c) Our position update method (considering a data term for feature preservation and a
repulsion term for uniform distribution). Multiple iterations are performed to achieve a better filtered result. (d) The filtered point cloud.
(e) The reconstructed mesh based on (d).
Figure 2. The left side represents the original points and the right
side represents the updated points. piand pjdenote the current
point and a neighboring point. ni,njdenote normals of piand
pj, respectively. Here a local plane surface is assumed.
Figure 3. The movement of the filtered points around sharp edges.
Blue points denote an underlying surface, yellow and green points
indicate two neighboring points that need to be moved, respec-
tively. (a-b) show the movement of pjwith fixed pi. (c-d) show
the movement of piwith fixed pj. This reveals that points are
moving toward the sharp edges and concentrating there, leading to
gaps around the sharp edges.
Algorithm 1 Towards uniform point distribution in feature-
preserving point cloud filtering
Input: Noisy point set P, corresponding filtered normals
N.
Output: Uniformly distributed set of filtered points P′.
Initialize: iteration t, repulsion term µ, local patch si
for Each iteration do
for Each point pido
construct a local patch si;
update point position via Eq. (5);
end for
end for
4. Experimental Results
The proposed method is implemented in Visual Stu-
dio 2017 and runs on a PC equipped with i9-9750h and
RTX2070. Most examples in this paper are executed in 7
seconds. The most time-demanding is the object from Fig-
ure 11 that takes 28 seconds.
4.1. Parameter Setting
The parameters include the local neighborhood size k,
the coefficient of repulsion force µ, and the number of iter-
ations t. Considering that the number of points affects the
range of neighbors significantly, in order to find the appro-
priate kneighbors for different models, here we determine
the size of kin the range [15, 45] (k= 30 by default) ac-
cording to the point number of each model. To make the
distribution of points more even while preserving the fea-
tures, we use the parameter µto balance the magnitude of
4
the repulsive force among points and tto control the num-
ber of iterations. For the models with sharp edges, we set
a relatively large µwith a low number of iterations, specif-
ically µ= 0.3,t= 10 or t= 5. For models with smooth
surfaces (e.g., non-CAD model), we use a smaller value of
µand a higher number of iterations twith µ= 0.1,t= 30.
Table 1gives all the parameters of the models used in the
experiments.
Table 1. Parameter settings for different models.
Parameters k µ t
Figure 430 0.3 5
Figure 530 0.3 5
Figure 630 0.3 5
Figure 730 0.3 3
Figure 830 0.3 5
Figure 930 0.3 5
Figure 10 30 0.3 10
Figure 11 30 0.3 5
Figure 12 30 0.3 5
4.2. Compared Approaches
The proposed method is compared with the state-
of-the-art techniques which include the non-deep learn-
ing position-based method CLOP [29], non-deep learning
normal-based methods GPF [25] and RIMLS [27], and deep
learning-based methods TotalDenoising (TD) [9], Point-
CleanNet (PCN) [31] and Pointfilter (PF) [37]. We em-
ploy the following rules for a fair comparison: (a) We first
normalize and centralize the noisy input. (b) As GPF and
RIMLS all require high-quality normals, we adopt the same
Bilateral Filter [12] to obtain the same input normals for
each model. (c) We try our best to tune the main param-
eters of each method to produce their final visual results.
(d) For the deep learning-based methods, we use the results
of the 6th iteration for TD and iterate three times for both
PCN and PF. (e) For visual comparison, we use EAR [12]
for upsampling and achieve a similar number of upsampling
points with the same model. As for surface reconstruction,
we adopt the same parameters for the same model.
4.3. Evaluation Metrics
Before discussing the visual results, we introduce two
common evaluation metrics for analyzing the performance
quantitatively. Suppose the ground-truth point cloud and
the filtered point cloud are respectively defined as: S1=
{xi}|S1|
i=1 , S2={yi}|S2|
i=1 . Notice that the number of ground-
truth points |S1|and filtered points |S2|may be slightly dif-
ferent.
1) Chamfer Distance:
eCD (S1, S2) = 1
|S1|X
x∈S1
min
y∈S2
∥x−y∥2
2+
1
|S2|X
y∈S2
min
x∈S1
∥y−x∥2
2,
(6)
2) Mean Square Error:
eMSE(S1, S2) = 1
|S1|
1
|NN(y)|X
x∈S1
X
y∈NN (y)
∥x−y∥2
2,
(7)
where NN(y)denotes the nearest neighbors in S1for
point yin S2. Here we set |NN (y)|= 10 similar to
[37], which means we search 10 nearest neighbors for
each point yin the predicted point set S2.
4.4. Visual Comparisons
Point clouds with synthetic noise. To show the denois-
ing effect of our method, we conduct experiments on mod-
els with synthetic noise, which are Gaussian noise at levels
of 0.5% and 1.0%, respectively. Compared to other state-of-
the-art methods, our visual results outperform them in terms
of both smoothing and feature-preserving aspects. The re-
sults benefit from the fact that the position update consid-
ers normal information makes the filtered points distribute
more evenly.
Meanwhile, we also observe the traits of other methods
in the experiments. CLOP always obtains good results in
terms of smoothing. However, since it is a position-based
method, it may blur sharp features. While GPF adds a gap-
filling step after projecting the points onto the underlying
surface, it is still difficult to maintain a uniform distribution,
especially when points are near sharp edges. This method
may also make those models with less sharp features abnor-
mally sharp. RIMLS yields promising results in both noise
removal and feature preservation. Still its filtered points are
often unevenly distributed, which affects the performance
in following applications such as upsampling and surface
reconstruction.
The learning-based method TD also yields good smooth-
ing results, but it does not seem to maintain the fine features
of the model well. PCN typically produces less sharp fea-
tures, and PCN can hardly obtain good smoothing effects
under a relatively high level of noise. PF does not need nor-
mal information at the test stage, but it can still achieve a
good feature-preserving effect while denoising. However,
when the noisy points are sparse, this method cannot ex-
tract the information from the sparse point cloud, leading to
distortion of the filtered points.
5
(a) Noisy input (b) CLOP (c) GPF (d) RIMLS (e) TD (f) PCN (g) PF (h) Ours
Figure 4. Results on the Bunnyhi model corrupted with 0.5% synthetic noise. The second row gives the surface reconstruction results.
(a) Noisy input (b) CLOP (c) GPF (d) RIMLS (e) TD (f) PCN (g) PF (h) Ours
Figure 5. Results on Rockerman corrupted with 0.5% synthetic noise. The second row gives the corresponding upsampling results.
(a) Noisy input (b) CLOP (c) GPF (d) RIMLS (e) TD (f) PCN (g) PF (h) Ours
Figure 6. Results on Icosahedron corrupted with 1.0% synthetic noise. The second row gives the corresponding upsampling results.
6
(a) Noisy input (b) CLOP (c) GPF (d) RIMLS (e) TD (f) PCN (g) PF (h) Ours
Figure 7. Results on Dodecahedron corrupted with 0.5% synthetic noise. The first row gives the corresponding upsampling results and the
reconstruction meshes are shown at the bottom.
(a) Noisy input (d) RIMLS
(e) TD (f) PCN (g) PF (h) Ours
(b) CLOP (c) GPF
Figure 8. Results on kitten corrupted with 1.0% synthetic noise. upsampling is included.
Since the normal information is taken into account for
our method, it can keep the sharp features well. Impor-
tantly, the better uniform point distribution makes it stand
out in point cloud filtering and following applications like
upsampling and surface reconstruction.
From the first row of Figures 4,5and 6, it can be easily
seen that our method obtains the most uniform point dis-
tribution. Figures 5,6and 8give the upsampling results
of the three different models after filtering. As seen in the
second rows of Figure 5and Figure 6, the sharp edges are
maintained well during denoising. The enlarged box in Fig-
ure 8also shows the effect of our filtering method, where
the shape of the kitten’s ears is maintained quite well. The
results of surface reconstruction are presented in Figures 4
and 7. As can be seen from the enlarged box, we maintain
the bunny’s mouth and nose features in Figure 4very well.
Figure 7also shows the filtered results of our method on a
simple geometric model with sharp edges. Our method is
the best in terms of maintaining details and sharp edges.
Point clouds with raw scan noise. In addition to
the synthetic noise, we also perform experiments on raw
scanned point clouds. The results of our method and ex-
isting methods on different raw point clouds are given in
Figures 9,10,11 and 12. The filtered results of ours and
other approaches given in Figure 9show that our method
performs better in terms of smoothing and preserving de-
tails. As can be seen from the enlarged box, most methods
make the mouth of the model blurry or disappear after de-
noising. Note that although the model we use here has the
same shape as PF [37], the filtered results may be different
since our sampling points are sparser than theirs.
Figure 10 shows the filtered results on a raw scanned
model named BuddhaStele. The first row gives the results
after upsampling, and the second row provides the results of
7
(a) Noisy input (b) CLOP (c) GPF (d) RIMLS (e) TD (f) PCN (g) PF (h) Ours
Figure 9. Results on one raw scanned model named Nefertiti. Upsampling is included.
(a) Noisy input
(b) CLOP (c) GPF (d) RIMLS (e) TD (f) PCN (g) PF (h) Ours
Figure 10. Results on the raw scanned model named BuddhaStele. The first row gives the corresponding upsampling results and the
reconstruction meshes are shown at the bottom.
surface reconstruction using Screened Poisson [13]. From
the details such as the stairs in the model, it can be seen
that our method still outperforms the other methods. In Fig-
ure 11, our method maintains the sharp edges well on this
model. As seen in the zoom-in box, other state-of-the-art
methods either distort the sharp edges or smooth them out.
Figure 12 shows filtered results on a raw scanned model
named David. Our method preserves feature better during
filtering. Seeing details from the zoom-in box, our approach
maintains the facial features better than others.
4.5. Quantitative Comparisons
We also make a quantitative comparison of the two eval-
uation metrics introduced in the previous section. Note that
since there is no corresponding ground-truth model for the
point clouds with raw scanned noise, we choose the models
with synthetic noise for quantitative evaluation. The results
under the Chamfer Distance metric are given in Table 2. De-
spite the fact that deep learning-based methods are trained
on a large number of point clouds, our method still outper-
forms all deep learning-based methods and even achieves
the lowest quantitative results among most models. In terms
of the other evaluation metric MSE, our method still out-
performs most deep learning methods and holds the lowest
quantitative error among most models, as shown in Table 3.
These quantitative results remain consistent with the vi-
sual results, demonstrating that our method generally out-
performs existing methods both visually and quantitatively.
We consider it is because our method provides a better uni-
form distribution of the filtered points and can handle both
sparsely and densely sampled point clouds. In the case of
sparse sampling, some deep learning-based methods are un-
able to obtain meaningful local geometric information from
the sparse points of local neighbors. It is also worth noting
8
(a) Noisy input (b) CLOP (c) GPF (d) RIMLS (e) TD (f) PCN (g) PF (h) Ours
Figure 11. Results on the raw scanned model named Realscan. Upsampling is included.
(a) Noisy input (d) RIMLS
(e) TD (f) PCN (g) PF (h) Ours
(b) CLOP (c) GPF
Figure 12. Results on the raw scanned model named David. Upsampling is included.
that although RIMLS achieves comparable results to ours in
some of the visual results, its error values are greater than
those of our method in most cases due to its uneven point
cloud distribution.
4.6. Ablation Study
Parameters. We first perform experiments on a point
cloud containing 7682 points with different values of k.
From Figure 13, the best value of kis 30. This is also the
default value for this parameter. It is easy to know that the
range of kis highly related to the density of the point cloud.
With a fixed value of k, when the model has a sparser dis-
tribution, the local range delimited by kwill become larger,
which may lead to an excessive range that should not be
treated as local information, resulting in a less desired out-
come. For point clouds with denser distribution, the lo-
cal neighbors’ krange becomes smaller, meaning that the
kneighborhood contains only a smaller range of local in-
formation, further leading to an uneven distribution of the
point cloud. Generally, we use a larger kfor point clouds
with denser points to ensure an appropriate number of local
neighbors.
As µis related to the number of iterations t, we give
filtered results for different values of µunder certain itera-
tions. Figure 14 demonstrates the filtered point clouds ob-
tained for different µvalues when t= 30 and k= 30. We
can see from this figure that as µincreases, the distribution
of the point cloud becomes more uniform, but a too-large
µwill make the model turn into chaos again. Figure 14(b)
shows the filtered results with a low value of µwhen i=30
and k=30. As we can see, a smaller µis better for maintain-
ing the feature edges of the model.
We also conduct experiments under different iterations.
Figure 15 indicates that with an increasing number of iter-
ation, the distribution of the filtered point cloud becomes
more uniform. However, Figure 20(d) shows that if the iter-
9
Table 2. Quantitative evaluation results of the compared methods and our method on the synthetic point clouds in Figures 4,5,6,7and 8.
Note that * represents deep learning methods. Chamfer Distance (×10−5) is used here. The best method for each model is highlighted.
Methods Figure 4Figure 5Figure 6Figure 7Figure 8Avg.
CLOP [29] 7.84 25.35 26.46 23.83 6.73 16.70
GPF [25] 16.19 31.85 21.52 18.35 15.54 17.58
RIMLS [27] 3.72 5.22 15.70 10.98 4.16 7.12
TD* [9] 23.88 13.20 24.43 19.22 11.86 16.15
PCN* [31] 4.76 6.38 29.87 14.96 6.24 11.18
PF* [37] 4.01 6.63 33.38 30.60 3.68 14.93
Ours 3.11 5.48 12.26 8.14 3.18 5.80
Table 3. Quantitative evaluation results of the compared methods and our method on the synthetic point clouds in Figures 4,5,6,7and 8.
Note that * represents deep learning methods. Mean Square Error (×10−3) is used here. The best method for each model is highlighted.
Methods Figure 4Figure 5Figure 6Figure 7Figure 8Avg.
CLOP [29] 10.32 13.91 21.86 23.31 9.88 15.86
GPF [25] 11.64 17.07 22.43 23.88 11.97 17.40
RIMLS [27] 10.05 14.02 21.68 23.30 10.02 15.81
TD* [9] 13.22 14.78 22.44 23.36 11.45 17.05
PCN* [31] 10.30 14.28 23.68 23.79 10.59 16.53
PF* [37] 10.02 14.17 23.71 25.28 9.82 16.60
Ours 9.92 14.01 21.46 23.15 9.92 15.69
ation parameter is set too large, the boundary of the model
will become unclear again.
(a) Noisy input (b) k= 1 (c) k= 5
(d) k= 15 (e) k= 30 (f) k= 45
Figure 13. Filtered results with different k. Noise level: 1.0%.t=
30, µ= 0.3.
With/without the repulsion term. Local-based fil-
tering approaches tend to converge in certain places when
updating the positions. Obviously, this will make follow-
up applications such as surface reconstruction very difficult.
Our method adopts the repulsion term mentioned in Section
3to allow the points to be evenly distributed while filtering,
thus improving the quality of the filtered point cloud. As
shown in Figure 16(a), without the repulsion term, it is clear
that some points are concentrated at the edges, whereas the
(a) Noisy input (b) µ= 0.1 (c) µ= 0.2
(d) µ= 0.3 (e) µ= 0.4 (f) µ= 0.5
Figure 14. Filtered results with different µ. Noise level: 1.0%,t=
30, k= 30.
(a) Noisy input (b) 5 iterations (c) 15 iterations (d) 30 iterations
Figure 15. Filtered results of different iterations. Noise level:
1.0%, other parameters: k= 30, µ= 0.3.
distribution in Figure 16(b) is more even.
Point density. The performance under different point
10
(a) Without repulsion
term
(b) With repulsion term
Figure 16. Filtered results with or without the repulsion term.
densities is tested. Figure 17 shows that our method yields
promising results on both sparse and dense point clouds. It
is worth noting that since our method requires only local
information, for point clouds with greater point density, a
desired filtered result can be obtained by setting a larger k.
Noisy
input
Ours
7682 points 30722 points 67938 points
Figure 17. Filtered results of models with different sampling num-
bers of points.
Noise level. Different noise levels are applied to the
same model to verify the robustness of our approach. Fig-
ure 18 gives the filtered results by our method under noise
levels of 0.5%, 1.0%, 1.5%, 2.0%, 2.5% and 3.0%. It can
be seen that our method is capable of handling models with
different levels of noise but may produce less desired re-
sults on excessively high noise. As our method relies on
the quality of normals, it is difficult to accurately keep the
geometric features if the model has less accurate normals
caused by higher noise levels.
Irregular sampling. We conduct experiments on mod-
els with irregular sampling. As shown in Figure 19, we pro-
vide the visual comparisons on an unevenly sampled model
for PCN [31], PF [37], and our method. It can be seen that
the filtered point cloud of PCN still contains obvious noise
and PF makes the detail features blurred, while our method
smooths the model better with feature preserving effect.
Holes filling. Taking the cube as an example, we exper-
iment on a model with holes. Figure 20 shows the filtered
results of different holes. Our method is capable of filling
relatively small holes because we consider the distribution
of the updated points. However, it will be challenging to fill
big holes that severely disrupt the surfaces of the model.
Lowrank [23] versus ours. Figure 21 gives a compari-
son of our approach with Lowrank [23]. It demonstrates that
our method gets a more uniform point distribution while re-
moving noise, which has better quality than Lowrank.
Indoor scene data. We perform an experiment on the
more challenging indoor scene data, as shown in Figure 22.
The result manifests that our method has the capacity of
dealing with point cloud indoor scenes as well.
Runtime. The running time of the proposed method is
calculated under different kand iterations parameters. It is
clearly shown in Table 4that as kand the number of itera-
tions increase, the runtime increases accordingly.
For each iteration, our method gathers k-local neighbors
for each point to obtain the updated points. Thus the larger
the parameter kis, the slower its computation becomes. The
number of iterations has a similar effect on the running time.
The higher the number of iteration is, the longer the com-
putation becomes. However, since our method is locally
based, multiple iterations usually run fast.
In addition, we also give the runtime of other methods for
comparison. Table 5shows that our method is significantly
faster than the other methods.
Table 4. Runtime (in seconds) on Dodecahedron for different k
and t.
Iterations t= 5 t= 15 t= 30 t= 60
k= 30 1.41 3.77 7.37 14.69
k= 60 2.33 6.37 12.36 24.36
Table 5. Runtime (in seconds) comparison on different models.
Methods CLOP TD PCN PF Ours
Fig. 459.66 16.65 294.00 67.38 6.27
Fig. 510.82 6.17 78.04 13.86 1.80
Fig. 63.89 4.91 83.69 38.98 1.89
Fig. 72.60 5.53 28.24 70.81 0.91
Fig. 842.85 16.24 186.30 49.99 4.66
Fig. 960.92 155.76 317.67 81.10 7.93
Fig. 10 70.58 102.41 642.01 247.77 6.37
Fig. 11 103.91 40.52 241.05 68.23 28.16
Fig. 12 50.58 30.01 352.99 74.68 6.98
Limitation. Though our method achieves good results,
it still has room for improvement. Similar to [25], since it
is a normal-based approach, it is inevitably dependent on
the normal quality. In each iteration of the position update,
each point is estimated with reference to the direction of
the normal. Therefore, less accurate input normals may
affect the filtered results. Figure 23 shows an example of
this issue. Also, similar to previous methods, our method
may produce less desirable results when handling a very
11
Noisy input
Ours
0.5% noise 1.0% noise 1.5% noise 2.0% noise 2.5% noise 3.0% noise
Figure 18. Filtered results of models with different levels of noise.
PCN [31] PF [37] Ours
Figure 19. Filtered results on the irregularly sampled point cloud.
(a) Cube with big
holes
(b) Cube with
small holes
(c) Updated
points of (a)
(d) Updated
points of (b)
Figure 20. Filtered results of the model with holes.
(a) Noisy input (b) Lowrank[23] (c) Ours
Figure 21. Filtered points of ours and Lowrank [23].
high level of noise. For instance, Figure 18 indicates 1.5%
noise is more challenging than the 0.5% and 1.0% noise.
In future, we would like to develop effective techniques to
handle the above limitations, e.g., fusing evolutionary opti-
mization within the filtering framework [26].
(a) Noisy input (b) Filtered result
Figure 22. Filtered result on noisy point cloud of an indoor scene.
(a) Noisy input (b) Filtered result
Figure 23. A failure example.
5. Conclusion
In this paper, we presented a method to improve point
cloud filtering by enabling a more even point distribution
for filtered point clouds. Built on top of [23], our method
introduces a repulsion term into the objective function. It
not only removes noise while preserving sharp features but
also ensures a more uniform distribution of cleaned points.
Experiments show that our method obtains very promising
filtered results under different levels of noise and densities.
Both visual and quantitative comparisons also show that it
generally outperforms the existing techniques in terms of
visual quality and quantity. Our method also runs fast, ex-
ceeding the other compared methods.
12
References
[1] M. Alexa, J. Behr, D. Cohen-Or, S. Fleishman, D. Levin,
and C. T. Silva. Computing and rendering point set surfaces.
IEEE Transactions on Visualization and Computer Graphics,
9(1):3–15, 2003. 2
[2] H. Chen, M. Wei, Y. Sun, X. Xie, and J. Wang. Multi-patch
collaborative point cloud denoising via low-rank recovery
with graph constraint. IEEE Transactions on Visualization
and Computer Graphics, 26(11):3255–3270, 2019. 2
[3] J.-E. Deschaud and F. Goulette. Point cloud non local de-
noising using local surface descriptor similarity. IAPRS,
38(3A):109–114, 2010. 2
[4] J. Digne. Similarity based filtering of point clouds. In 2012
IEEE Computer Society Conference on Computer Vision and
Pattern Recognition Workshops, pages 73–79. IEEE, 2012. 2
[5] C. Duan, S. Chen, and J. Kovacevic. 3d point cloud de-
noising via deep neural network based local surface estima-
tion. In ICASSP 2019-2019 IEEE International Conference
on Acoustics, Speech and Signal Processing (ICASSP), pages
8553–8557. IEEE, 2019. 2
[6] P. Erler, P. Guerrero, S. Ohrhallinger, N. J. Mitra, and
M. Wimmer. Points2surf learning implicit surfaces from
point clouds. In European Conference on Computer Vision,
pages 108–124. Springer, 2020. 2
[7] G. Guennebaud and M. Gross. Algebraic point set surfaces.
In ACM SIGGRAPH 2007 papers, page 23. 2007. 2
[8] P. Guerrero, Y. Kleiman, M. Ovsjanikov, and N. J. Mitra.
Pcpnet learning local shape properties from raw point clouds.
In Computer Graphics Forum, volume 37, pages 75–85. Wi-
ley Online Library, 2018. 2
[9] P. Hermosilla, T. Ritschel, and T. Ropinski. Total denois-
ing: Unsupervised learning of 3d point cloud cleaning. In
Proceedings of the IEEE/CVF International Conference on
Computer Vision (ICCV), October 2019. 5,10
[10] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and
W. Stuetzle. Surface reconstruction from unorganized points.
In Proceedings of the 19th annual conference on computer
graphics and interactive techniques, pages 71–78, 1992. 1
[11] H. Huang, D. Li, H. Zhang, U. Ascher, and D. Cohen-Or.
Consolidation of unorganized point clouds for surface recon-
struction. ACM Transactions on Graphics (TOG), 28(5):1–7,
2009. 1,2
[12] H. Huang, S. Wu, M. Gong, D. Cohen-Or, U. Ascher, and
H. Zhang. Edge-aware point set resampling. ACM Transac-
tions on Graphics (TOG), 32(1):1–12, 2013. 1,2,3,5
[13] M. Kazhdan and H. Hoppe. Screened poisson surface recon-
struction. ACM Transactions on Graphics (TOG), 32(3):1–
13, 2013. 1,8
[14] D. Levin. The approximation power of moving least-squares.
Mathematics of Computation, 67(224):1517–1531, 1998. 2
[15] D. Levin. Mesh-independent surface interpolation. In Ge-
ometric Modeling for Scientific Visualization, pages 37–49.
Springer, 2004. 2
[16] B. Liao, C. Xiao, L. Jin, and H. Fu. Efficient feature-
preserving local projection operator for geometry recon-
struction. Computer-Aided Design, 45(5):861–874, 2013. 2
[17] Y. Lipman, D. Cohen-Or, D. Levin, and H. Tal-Ezer.
Parameterization-free projection for geometry reconstruc-
tion. ACM Transactions on Graphics (TOG), 26(3):22, 2007.
1,2
[18] Y. Liu, J. Guo, B. Benes, O. Deussen, X. Zhang, and
H. Huang. Treepartnet: neural decomposition of point clouds
for 3d tree reconstruction. ACM Transactions on Graphics
(TOG), 40(6):1–16, 2021. 2
[19] Z. Liu, X. Xiao, S. Zhong, W. Wang, Y. Li, L. Zhang, and
Z. Xie. A feature-preserving framework for point cloud de-
noising. Computer Aided Design, 127:102857, 2020. 2
[20] D. Lu, X. Lu, Y. Sun, and J. Wang. Deep feature-preserving
normal estimation for point cloud filtering. Computer-Aided
Design, 125:102860, 2020. 2
[21] X. Lu, H. Chen, S.-K. Yeung, Z. Deng, and W. Chen. Un-
supervised articulated skeleton extraction from point set se-
quences captured by a single depth camera. Proceedings of
the AAAI Conference on Artificial Intelligence, 32(1), Apr.
2018. 1
[22] X. Lu, Z. Deng, J. Luo, W. Chen, S.-K. Yeung, and Y. He. 3d
articulated skeleton extraction using a single consumer-grade
depth camera. Computer Vision and Image Understanding,
188:102792, 2019. 1
[23] X. Lu, S. Schaefer, J. Luo, L. Ma, and Y. He. Low rank ma-
trix approximation for 3d geometry filtering. IEEE Transac-
tions on Visualization and Computer Graphics, 2020. 1,2,
3,11,12
[24] X. Lu, Z. Wang, M. Xu, W. Chen, and Z. Deng. A personality
model for animating heterogeneous traffic behaviors. Com-
puter Animation and Virtual Worlds, 25(3-4):361–371, 2014.
1
[25] X. Lu, S. Wu, H. Chen, S.-K. Yeung, W. Chen, and
M. Zwicker. Gpf: Gmm-inspired feature-preserving point
set filtering. IEEE Transactions on Visualization and Com-
puter Graphics, 24(8):2315–2326, 2017. 1,2,3,5,10,11
[26] T. Nakane, N. Bold, H. Sun, X. Lu, T. Akashi, and C. Zhang.
Application of evolutionary and swarm optimization in com-
puter vision: a literature survey. IPSJ Transactions on Com-
puter Vision and Applications, 12(1):1–34, 2020. 12
[27] A. C. ¨
Oztireli, G. Guennebaud, and M. Gross. Feature pre-
serving point set surfaces based on non-linear kernel regres-
sion. In Computer Graphics Forum, volume 28, pages 493–
501. Wiley Online Library, 2009. 1,2,5,10
[28] Y. Pei, Z. Huang, W. Yu, M. Wang, and X. Lu. A cas-
caded approach for keyframes extraction from videos. In
F. Tian, X. Yang, D. Thalmann, W. Xu, J. J. Zhang, N. M.
Thalmann, and J. Chang, editors, Computer Animation and
Social Agents, pages 73–81, Cham, 2020. Springer Interna-
tional Publishing. 1
[29] R. Preiner, O. Mattausch, M. Arikan, R. Pajarola, and
M. Wimmer. Continuous projection for fast l1 reconstruc-
tion. ACM Transactions on Graphics (TOG), 33(4):1–13,
2014. 1,2,5,10
[30] C. R. Qi, H. Su, K. Mo, and L. J. Guibas. Pointnet: Deep
learning on point sets for 3d classification and segmentation.
In Proceedings of the IEEE Conference on Computer Vision
and Pattern Recognition, pages 652–660, 2017. 2
13
[31] M.-J. Rakotosaona, V. La Barbera, P. Guerrero, N. J. Mi-
tra, and M. Ovsjanikov. Pointcleannet: Learning to denoise
and remove outliers from dense point clouds. In Computer
Graphics Forum, volume 39, pages 185–203. Wiley Online
Library, 2020. 1,2,5,10,11,12
[32] R. Roveri, A. C. ¨
Oztireli, I. Pandele, and M. Gross. Point-
pronets: Consolidation of point clouds with convolutional
neural networks. In Computer Graphics Forum, volume 37,
pages 87–99. Wiley Online Library, 2018. 1,2
[33] R. B. Rusu, N. Blodow, Z. Marton, A. Soos, and M. Beetz.
Towards 3d object maps for autonomous household robots.
In 2007 IEEE/RSJ International Conference on Intelligent
Robots and Systems, pages 3191–3198. IEEE, 2007. 2
[34] L. Yu, X. Li, C.-W. Fu, D. Cohen-Or, and P.-A. Heng. Ec-
net: an edge-aware point set consolidation network. In Pro-
ceedings of the European Conference on Computer Vision
(ECCV), pages 386–402, 2018. 2
[35] L. Yu, X. Li, C.-W. Fu, D. Cohen-Or, and P.-A. Heng. Pu-
net: Point cloud upsampling network. In Proceedings of the
IEEE Conference on Computer Vision and Pattern Recogni-
tion, pages 2790–2799, 2018. 2
[36] J. Zeng, G. Cheung, M. Ng, J. Pang, and C. Yang. 3d point
cloud denoising using graph laplacian regularization of a low
dimensional manifold model. IEEE Transactions on Image
Processing, 29:3474–3489, 2019. 2
[37] D. Zhang, X. Lu, H. Qin, and Y. He. Pointfilter: Point cloud
filtering via encoder-decoder modeling. IEEE Transactions
on Visualization and Computer Graphics, 27(3):2015–2027,
2020. 1,2,5,7,10,11,12
14
