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Fast 3D Point Cloud Denoising via Bipartite Graph
Approximation & Total Variation
Chinthaka Dinesh #1 , Gene Cheung ?2, Ivan V. Baji´
c#3, Cheng Yang ?4
#Simon Fraser University, Burnaby, BC, Canada; ?National Institute of Informatics, Tokyo, Japan
1hchintha@sfu.ca; 2cheung@nii.ac.jp; 3ibajic@ensc.sfu.ca; 4cheng@nii.ac.jp
Abstract—Acquired 3D point cloud data, whether from active
sensors directly or from stereo-matching algorithms indirectly,
typically contain non-negligible noise. To address the point
cloud denoising problem, we propose a fast graph-based local
algorithm. Specifically, given a k-nearest-neighbor graph of
the 3D points, we first approximate it with a bipartite graph
(independent sets of red and blue nodes) using a KL divergence
criterion. For each partite of nodes (say red), we first define
surface normal of each red node using 3D coordinates of
neighboring blue nodes, so that red node normals ncan be
written as a linear function of red node coordinates p. We
then formulate a convex optimization problem, with a quadratic
fidelity term kp−qk2
2given noisy observed red coordinates qand
a graph total variation (GTV) regularization term for surface
normals of neighboring red nodes. We minimize the resulting
l2-l1-norm using alternating direction method of multipliers
(ADMM) and proximal gradient descent. The two partites of
nodes are alternately optimized until convergence. Experimental
results show that compared to state-of-the-art schemes with
similar complexity, our proposed algorithm achieves the best
overall denoising performance objectively and subjectively.
Index Terms—graph signal processing, point cloud denoising,
total variation, bipartite graph approximation
I. INTRODUCTION
Point cloud is now a popular representation of 3D visual
data in signal processing, and there are ongoing efforts in
standardization bodies such as MPEG1and JPEG-PLENO2
to efficiently compress the data. Point cloud can be acquired
directly using active sensors like Microsoft Kinect cameras, or
computed indirectly from multiple viewpoint images using ex-
isting stereo-matching algorithms [1]. In either case, acquired
point cloud data tend to be noisy, and thus denoising should be
performed prior to compression. We address the point cloud
denoising problem in this paper.
Typically, an inverse problem like image super-resolution or
deblurring [2] is ill-posed and requires a suitable regularization
term to formulate a mathematically rigorous optimization
problem. Because point cloud data are irregularly sampled in
3D space, typical signal priors like total variation (TV) [3]
cannot be directly used. While graph variant of TV (called
GTV) has been proposed [4], [5], applying GTV directly on
the 3D point coordinates, as done in [6], is not appropriate,
because GTV of coordinates promotes variational proximity
of 3D points, and only a singular 3D point cloud has zero
GTV value.
1https://mpeg.chiariglione.org/standards/mpeg-i/point-cloud-compression.
2https://jpeg.org/jpegpleno/pointcloud.html.
In this paper, we argue that GTV is more appropriately
applied to the surface normals of the 3D point cloud, so
that GTV promotes smooth object surfaces like a tabletop—a
generalization of functional smoothness in conventional TV to
3D geometry. While there exist numerous methods to compute
surface normals of 3D point clouds [7]–[11], the computations
are typically nonlinear, resulting in difficult optimizations.
To alleviate this problem, we propose to first partition
the 3D point cloud data into two sets via bipartite graph
approximation [12] of a k-nearest-neighbor graph, resulting
in two independent sets of nodes (say red and blue). For each
partite of nodes (say red), we then define surface normal of
each red node using 3D coordinates of neighboring blue nodes;
this results in a linear relationship between surface normals n
and 3D coordinates pof the red nodes. We then formulate
an objective composing of a quadratic fidelity term kp−qk2
2
given noisy observed red coordinates qand a GTV for surface
normals of neighboring red nodes. We can now minimize
efficiently the resulting convex l2-l1-norm using alternating
direction method of multipliers (ADMM) and proximal gradi-
ent descent [13], [14]. The two partites of nodes are alternately
optimized until convergence. Experimental results show that
compared to state-of-the-art schemes with similar complexity,
our proposed algorithm achieves the best overall denoising
performance objectively and subjectively.
The outline of the paper is as follows. We first overview
related works in Section II. We define necessary fundamental
concepts in Section III. We formulate our optimization and
present our algorithm in Section IV. Finally, experiments and
conclusion are presented in Section V and VI, respectively.
II. RELATED WORK
Existing work on point cloud denoising can be roughly
divided into four categories: moving least squares (MLS)-
based methods, locally optimal projection (LOP)-based meth-
ods, sparsity-based methods, and non-local similarity-based
method.
MLS-based method. In MLS-based methods, a smooth sur-
face is approximated from the input point cloud, and these
points are projected to the resulting surface. To construct the
surface, [15] first finds a local reference domain for each
point that best fits its neighboring points in terms of MLS.
Then a function is defined above the reference domain by
fitting a polynomial function to neighboring data. However, if a
point cloud has high curvatures on its underlying surface, then
arXiv:1804.10831v1 [eess.SP] 28 Apr 2018
this method becomes unstable. In response, several solutions
have been proposed, e.g., algebraic point set surfaces (APSS)
[16] and its variant [17] and robust implicit MLS (RIMLS)
[18]. Although, these methods can robustly generate a smooth
surface from extremely noisy input, they can over-smooth as
a result [19].
LOP-based methods: Compared to MLS-based method,
LOP-based methods do not compute explicit parameters for
the point cloud surface. For example, [20] generates a set of
points that represent the underlying surface while enforcing
a uniform distribution over the point cloud. There are two
main modifications to [20]. The first is weighted LOP (WLOP)
[21] that provides a more uniformly distributed output by
adapting a new term to prevent a given point from being too
close to other neighboring points; the second is anisotropic
WLOP (AWLOP) [22] that preserves sharp features using an
anisotropic weighting function. However, LOP-based methods
also suffer from over-smoothing due to the use of local
operators [19].
Sparsity-based methods: There are two main steps in
sparsity-based methods. First, a sparse reconstruction of the
surface normals is obtained by solving a global minimization
problem with sparsity regularization. Then the point positions
are updated by solving another global minimization problem
based on a local planar assumption. Some of the examples
include [23] that utilizes l1regularization, and [24] that uses
l0regularization. However, when the noise level is high, these
methods also lead to over-smoothing or over-sharpening [19].
Non-local methods: Non-local methods generalize the con-
cepts in non-local means (NLM) [25] and BM3D [26] image
denoising algorithms to point cloud denoising. These methods
rely on the self-similarity characteristic among surface patches
in the point cloud. A method proposed in [27] utilizes a NLM
algorithm, while a method in [28] is inspired by the BM3D
algorithm. In addition, recently, [29] defines self-similarity
among patches formally as a low-dimensional manifold prior
[30]. Although, non-local methods achieve state-of-the-art
performance, their computational complexity is often too high.
III. PRELIMINARIES
A. 3D Point Cloud
We define a point cloud as a set of discrete sampling
(roughly uniform) of 3D coordinates of an object’s 2D surface
in 3D space. Let q=qT
1. . . qT
NT∈R3Nbe the
position vector for the point cloud, where qi∈R3is the
3D coordinate of a point iand Nis the number of points
in the point cloud. Noise-corrupted qcan be simply modeled
as q=p+e, where pare the true 3D coordinates, eis a
zero-mean signal-independent noise, where p,e∈R3N. To
recover the true positions p, we define surface-normal-based
GTV as the regularization term for the ill-posed problem.
B. Graph Definition
We define graph-related concepts needed in our work.
Consider an undirected graph G= (V,E)composed a node
set Vand an edge set Especified by (i, j, wi,j ), where i, j ∈ V
and wi,j ∈R+is the edge weight that reflects the similarity
between nodes iand j. Graph Gcan be characterized by its ad-
jacency matrix Wwith W(i, j) = wi,j . Moreover, Ddenotes
the diagonal degree matrix where entry D(i, i) = Pjwi,j .
Given Wand D, the combinatorial graph Laplacian matrix
is defined as L=D−W. A graph-signal assigns a scalar
value to each node, denoted by f= [f1. . . fN]T.
C. Graph Construction from a 3D Point Cloud
A common graph construction from a given point cloud is
to construct a k-nearest-neighbor (k-NN) graph as it makes
geometric structure explicit [31]. The set of points in a given
point cloud is considered as nodes, and each node is connected
through edges to its knearest neighbors with weights which
reflect inter-node similarities. In this paper, we choose the Eu-
clidean distance between two nodes to measure the similarity.
For a given point cloud p=pT
1. . . pT
NT, edge weight
wi,j between nodes iand jis computed using a Gaussian
kernel [32] as follows:
wi,j = exp −||pi−pj||2
2
σ2
p,(1)
where σpis a parameter.
D. Surface Normals
A surface normal at a point iin a given 3D point cloud is a
vector that is perpendicular to the tangent plane to that surface
at i. Coordinates of the knearest neighbors of iare used to
obtain the surface at i. There are numerous methods [7]–[11]
in the literature to define the normal to that surface at i. The
most popular method is to fit a local plane to points in iand
knearest neighbors of i, and take the perpendicular vector to
that plane (see e.g. [7]–[9]). An attractive alternative to this
approach is to calculate the normal vector as the weighted
average of the normal vectors of the triangles formed by iand
pairs of its neighbors (see e.g. [10], [11]).
IV. PROPOS ED ALGORITHM
Given two neighboring nodes i, j ∈ E for a constructed
point cloud graph G= (V,E), when the underlying 2D
surface is smooth, the corresponding (consistently oriented)
surface normals at nodes iand jshould be similar. Hence the
piecewise smoothness (PWS) of the point cloud surface can
be measured using GTV of surface normals over Gas follows:
||n||GTV =X
i,j∈E
wi,j ||ni−nj||1,(2)
where ni∈R3is the surface normal at node i. Now we can
formulate our point cloud denoising problem as a minimization
of the defined GTV while keeping the points close to their
original locations. Denote the 3D coordinate of a point/node i
by a column vector piand p=hpT
1. . . pT
|V| iT
, where
|V| is the number of points in the point cloud. Here, pis the
optimization variable, and ni’s are functions of p. Unfortu-
nately, using state-of-art surface normal estimation methods,
each niis a nonlinear function of piand its neighbors. Hence,
it is difficult to formulate a clean convex optimization using
GTV in (2).
To overcome this issue, we first partition 3D points of
the point cloud into two classes (say red and blue). When
computing the surface normal for a red point, we consider only
neighboring blue points, and vice versa. Towards this goal, we
compute a bipartite graph approximation of the original graph
Gas follows.
A. Bipartite Graph Approximation
A bipartite graph B= (V1,V2,E0)is a graph whose nodes
are divided into two disjoint sets V1and V2(i.e., red nodes
and blue nodes respectively), such that each edge connects a
node in V1to one in V2. Given an original graph G= (V,E),
our goal is to find a bipartite graph Bthat is “closest” to Gin
some sense.
First, we assume that the generative model for graph-signal
fis a Gaussian Markov random field (GMRF) [33] with
respect to G. Specifically, f∼ N (µ, Σ), where µis the mean
vector, and Σis the covariance matrix specified by the graph
Laplacian matrix Lof G,i.e.,Σ−1=L+δI, where 1/δ
is interpreted as the variance of the DC component for f. For
simplicity, we assume µ=0.
We now find a graph Bwhose distribution NB(0,ΣB)is
closest to N(0,Σ)in terms of Kullback-Leibler Divergence
(KLD) [34]:
DKL (N ||NB) = 1
2(tr(Σ−1
BΣ) + ln|ΣBΣ−1| − |V|),(3)
where Σ−1
B=LB+δIis the precision matrix of the GMRF
specified by B, and LBis the graph Laplacian matrix of B.
To minimize (3), we use an iterative greedy algorithm similar
to the one proposed in [12] as follows.
For a given non-bipartite graph G, a bipartite graph Bis built
by adding nodes one-by-one into two disjoint sets (V1and V2)
and removing the edges within each set. First, we initialize one
node set V1with one randomly chosen node, and V2is empty.
Then, we use breadth-first search (BFS) [35] to explore nodes
within one hop. To determine to which set among V1and V2
the next node should be allocated, we calculate KLD Di
KL
where i= 1,2, assuming the node is allocated to V1or V2
respectively. The node is allocated to V1if D2
KL > D1
KL , to
V2otherwise. If D2
KL =D1
KL , then the node is alternately
allocated to V1and V2, so that the resulting sizes of the two
node sets will be balanced.
An example for the bipartite graph approximation is shown
in Fig. 1. Here, we construct the original k-NN graph (with
k= 6) using a small portion of Bunny point cloud model
in [36].
B. Normal Vector Estimation
We examine how niis defined for a red node i. For this
purpose, two nearest neighbor blue nodes (named kand l)
that are not on a line with the red node iare used to compute
a perpendicular vector to the plane, where nodes i,kand l
are placed (see Fig. 2). The corresponding 3D coordinates
(a) original graph (b) bipartite graph
Fig. 1. An example for bipartite graph approximation.
Fig. 2. Illustration of the normal vector estimation at a red node.
of nodes i,k, and lare denoted by pi=xiyiziT,
pk=xkykzkT, and pl=xlylzlTrespectively.
We define the normalized perpendicular vector nito that plane
as the surface normal at node i, computed as:
ni=[pi−pk]×[pk−pl]
||[pi−pk]×[pk−pl]||2
,(4)
where ‘×’ represents the symbol for vector cross product. We
rewrite the cross product in (4) as follows:
[pi−pk]×[pk−pl] =
0zk−zlyl−yk
zl−zk0xk−xl
yk−ylxl−xk0
xi
yi
zi
+
−yk(zk−zl)−zk(yl−yk)
−xk(zk−zl)−zk(xk−xl)
−xk(yk−yl)−yk(xl−xk)
(5)
Using (5), normal vector nican be re-written as:
ni=Cipi+di
||Cipi+di||2
,(6)
where Ci=
0zk−zlyl−yk
zl−zk0xk−xl
yk−ylxl−xk0
and
di=
−yk(zk−zl)−zk(yl−yk)
−xk(zk−zl)−zk(xk−xl)
−xk(yk−yl)−yk(xl−xk)
. Fig. 3(a) shows the
surface normals (black arrows) calculated from (6) for red
nodes in the graph in Fig. 1(b).
As shown in Fig. 3(a), the orientations of the normal vectors
obtained by (6) are not necessarily consistent across the 2D
surface. To find consistent orientations of the surface normals,
we fix the orientation of one normal vector and propagate this
information to neighboring red points. Specifically, we use
minimum spanning tree (MST) based approach proposed in
[37] to do this propagation. First, a k-NN graph of red nodes
is constructed with wi,j = 1 − |nT
inj|, where niand njare
computed using (6). Then, an arbitrary node of the graph is
assumed to be the tree root, and the normal is propagated to
(a) (b)
Fig. 3. surface normals a) before b) after consistent orienting alignment.
its children nodes recursively. When the normal direction is
propagated from node ito node j, if nT
injis negative, then
the direction of njis reversed; otherwise it is left unchanged.
Hence, consistently oriented normal vector can be written as:
ni=Cipi+di
||Cipi+di||2α, (7)
where αis 1 or −1according to the consistent normal
orientation. Fig. 3(b) shows surface normals for red nodes in
Fig. 3(a) after consistent orienting alignment.
In this paper, for simplicity we assume that the scalar
values αand ||Cipi+di||2remain constant when piis being
optimized. Hence, surface normal nican be written as a linear
function of 3D coordinates piof the point i. Specifically,
ni=Aipi+bi,(8)
where Ai=Ciαin/||Cipin
i+di||2and
bi=diαin/||Cipin
i+di||2. Here pin
iis the initial
vector of piand αin is the value of αcomputed at pi=pin
i.
C. Optimization Framework
After computing surface normals for each red node, we
construct a new k-NN graph G= (V1,E1)for red nodes,
where V1is the set of red nodes and E1is the set of edges
of the graph. Now for a given red node graph G, the resulting
denoising objective is a l2-l1-norm minimization problem:
min
p||q−p||2
2+γX
i,j∈E1
wi,j kni−njk1(9)
subject to linear constraint (8), where γis a weight parameter
that trades off the fidelity term and GTV prior, and wi,j is the
edge weight between node iand node jas defined in (1).
To solve (9), we first rewrite it using the definition of
mi,j =ni−nj:
min
p,m||q−p||2
2+γX
i,j∈E1
wi,j kmi,j k1
s.t. mi,j =ni−nj.
(10)
again subject to constraint (8).
To solve (10), we design a new algorithm based on Al-
ternating Direct Method of Multipliers (ADMM) [13] with
a nested proximal gradient descent [14]. We first write the
linear constraint for each mi,j in matrix using our definition
of ni=Aipi+biin (8):
m=Bp +v,(11)
where m∈R3|E1|,B∈ {Ai,0,−Aj}3|E1|×3|V1|,v∈ {bi−
bj}3|E1|. Specifically, for each mi,j, the corresponding block
row in Bhas all zero matrices except block entries iand j
have Aiand −Ajrespectively. Moreover, for each mi,j, the
corresponding block entry in vis bi−bj. We can now rewrite
(10) in ADMM scaled form as follows:
min
p,m||q−p||2
2+γX
i,j∈E1
wi,j kmi,j k1+
ρ
2||Bp +v−m+u||2
2+const,
(12)
where ρ > 0is a Lagrange multiplier. As typically done in
ADMM, we solve (12) by alternately minimizing pand m
and updating uone at a time in turn until convergence.
1) pminimization: To minimize phaving mkand ukfixed,
we take derivatives of (12) with respect to p, set it to 0and
solve for the closed form solution pk+1:
(2I+ρBTB)pk+1 = 2q+ρBT(mk−uk−v),(13)
where Iis an identity matrix. We see that matrix 2I+ρBTBis
positive definite (PD) for ρ > 0and hence invertible. Further,
the matrix is sparse and symmetric, hence (13) can be solved
efficiently via conjugate gradient (CG) without full matrix
inversion [38].
2) mminimization: Keeping pk+1 and ukfixed, the min-
imization of mbecomes:
min
m
ρ
2||Bpk+1 +v−m+uk||2
2+γX
i,j∈E1
wi,j kmi,j k1,(14)
where the first term is convex and differentiable, and the
second term is convex but non-differentiable. We can thus
use proximal gradient [14] to solve (14). The first term has
gradient ∆m:
∆m(pk+1,m,uk) = −ρ(Bpk+1 +v−m+uk).(15)
We can now define a proximal mapping proxg,t(m)for a
convex, non-differentiable function g() with step size tas:
proxg,t(m) = arg min
θg(θ) + 1
t||θ−m||2
2(16)
We know that for our weighted l1-norm in (14), the proximal
mapping is just a soft thresholding function:
proxg,t(mi,j,r ) =
mi,j,r −tγwi,j if mi,j,r > tγwi,j
0if |mi,j,r| ≤ tγ wi,j
mi,j,r +tγwi,j if mi,j,r < tγwi,j ,
(17)
where mi,j,r is the r-th entry of mi,j . We can now update
mk+1 as:
mk+1 =proxg,t(mk−t∆m(pk+1 ,mk,uk)).(18)
We compute (18) iteratively until convergence.
3) u-update: Finally, we can update uk+1 simply:
uk+1 =uk+ (Bpk+1 +v−mk+1).(19)
p,mand uare iteratively optimized in turn using (13), (18)
and (19) until convergence.
Following the procedure in Section IV-B and IV-C, two
classes of nodes (i.e., red and blue) are alternately optimized
until convergence.
V. EX PE RIM EN TAL RES ULTS
The proposed point cloud denoising method is compared
with four existing methods: APSS [16], RIMLS [18], AWLOP
[22], and the state-of-the art moving robust principle compo-
nent analysis (MRPCA) algorithm [24]. APSS and RIMLS
are implemented with MeshLab software [39], AWLOP is
implemented with EAR software [22], and MRPCA source
code is provided by the author. Point cloud models we use
are Bunny provided in [36], Gargoyle, DC, Daratech, Anchor,
Lordquas, Fandisk, and Laurana provided in [24], [28]. Both
numerical and visual comparisons are presented.
For the numerical comparisons, we measure the point-to-
point (C2C) error and point to plane (C2P) error between
ground truth and denoising results. In C2C error, we first mea-
sure the average of the squared Euclidean distances between
ground truth points and their closest denoised points, and also
that between the denoised points and their closest ground
truth points. Then the average between these two measures
is computed as C2C error. Although C2C error is a popular
metric for point cloud evaluation, it fails to account for the
fact that points in point clouds often represent surfaces in the
structure. As an alternative, C2P error is introduced in [40] for
evaluation of the geometric errors between two point clouds.
Therefore, we use C2P error also for our evaluation in addition
to C2C error. In C2P error, we first measure the average of
the squared Euclidean distances between ground truth points
and tangent planes at their closest denoised points, and also
that between the denoised points and tangent planes at their
closest ground truth points. Then the average between these
two measures is computed as the C2P error.
Gaussian noise with zero mean and standard deviation σ
of 0.1 and 0.3 is added to the 3D position of the point cloud.
Numerical results are shown in Table I, and II (with C2C error)
and in Table III, and IV (with C2P error), where the proposed
method is shown to have the lowest C2C and C2P errors. In
each experiment, the selected parameters are σp= 1.5,ρ= 5,
t= 0.1,γ= 0.05 when Gaussian noise with σ= 0.1, and
γ= 0.1when Gaussian noise with σ= 0.3. Moreover, kis
set to 8 when constructing k−NN graphs.
Apart from the numerical comparison, visual results for
Anchor model and Daratech model are shown in Fig. 4 and 5
respectively. For Anchor model, existing schemes are under-
smoothed and for Daratech model, and existing methods result
in distorted surface with some details lost in addition to under-
smoothing. However, for the proposed method, the details are
well preserved without under-smoothing for both models.
TABLE I
C2C OF D IFFE RENT M ODE LS,WITH GA USSI AN NO ISE (σ= 0.1)
Model Noise APSS RIMLS AWLOP MRPCA Prop.
Bunny 0.157 0.135 0.143 0.153 0.141 0.128
Gargoyle 0.154 0.133 0.143 0.151 0.144 0.131
DC 0.154 0.130 0.140 0.148 0.136 0.128
Daratech 0.156 0.134 0.137 0.156 0.134 0.132
Anchor 0.156 0.134 0.139 0.152 0.130 0.127
Lordquas 0.155 0.130 0.143 0.153 0.132 0.126
Fandisk 0.159 0.148 0.148 0.157 0.138 0.136
Laurana 0.150 0.136 0.139 0.147 0.130 0.130
TABLE II
C2C OF D IFFE RENT M ODE LS,WITH GA USSI AN NO ISE (σ= 0.3)
Model Noise APSS RIMLS AWLOP MRPCA Prop.
Bunny 0.329 0.235 0.251 0.315 0.243 0.231
Gargoyle 0.304 0.220 0.232 0.288 0.218 0.214
DC 0.305 0.213 0.230 0.302 0.212 0.207
Daratech 0.313 0.264 0.268 0.293 0.262 0.246
Anchor 0.317 0.225 0.231 0.281 0.216 0.210
Lordquas 0.307 0.212 0.228 0.284 0.208 0.203
Fandisk 0.406 0.352 0.343 0.390 0.331 0.319
Laurana 0.318 0.239 0.249 0.266 0.242 0.231
VI. CONCLUSION
Denoising of 3D point cloud remains a fundamental and
challenging problem. In this paper, we propose to apply graph
total variation (GTV) to the surface normals of neighboring
3D points as regularization. By first partitioning points into
two disjoint sets, one can define surface normals of one
set as linear functions of the set’s 3D coordinates. This
leads naturally to a l2-l1-norm objective function, which can
be optimized elegantly using ADMM and nested gradient
descent. Experimental results show our proposal outperforms
competing schemes with comparable complexity objectively
and subjectively. REFERENCES
[1] M. Ji, J. Gall, H. Zheng, Y. Liu, and L. Fang, “Surfacenet: An end-to-
end 3d neural network for multiview stereopsis,” arXiv preprint arXiv:
1708.01749, 2017.
[2] Y. Bai, G. Cheung, X. Liu, and W. Gao, “Graph-based blind image
deblurring from a single photograph,” arXiv preprint arXiv:1802.07929,
2018.
[3] L. Condat, “A direct algorithm for 1-d total variation denoising,” IEEE
Signal Process. Lett., vol. 20, no. 11, pp. 1054–1057, 2013.
[4] A. Elmoataz, O. Lezoray, and S. Bougleux, “Nonlocal discrete regu-
larization on weighted graphs: a framework for image and manifold
processing,” IEEE Trans. Image Process., vol. 17, no. 7, pp. 1047–1060,
2008.
[5] P. Berger, G. Hannak, and G. Matz, “Graph signal recovery via primal-
dual algorithms for total variation minimization,” IEEE J. Sel. Topics
Signal Process., vol. 11, no. 6, pp. 842–855, 2017.
[6] Y. Schoenenberger, J. Paratte, and P. Vandergheynst, “Graph-based
denoising for time-varying point clouds,” in IEEE 3DTV-Conference,
2015, pp. 1–4.
[7] J. Huang and C. H. Menq, “Automatic data segmentation for geometric
feature extraction from unorganized 3-d coordinate points,” IEEE Trans.
Robot. Autom., vol. 17, no. 3, pp. 268–279, Jun 2001.
[8] K. Kanatani, Statistical optimization for geometric computation: theory
and practice. Courier Corporation, 2005.
[9] D. OuYang and H. Y. Feng, “On the normal vector estimation for point
cloud data from smooth surfaces,” Computer-Aided Design, vol. 37,
no. 10, pp. 1071–1079, 2005.
[10] H. Gouraud, “Continuous shading of curved surfaces,” IEEE Trans.
Comput., vol. C-20, no. 6, pp. 623–629, June 1971.
[11] S. Jin, R. R. Lewis, and D. West, “A comparison of algorithms for vertex
normal computation,” The visual computer, vol. 21, no. 1-2, pp. 71–82,
2005.
(a) ground truth (b) noisy input (c) APSS (d) RIMLS (e) MRPCA (f) proposed
Fig. 4. Denoising results illustration for Anchor model (σ= 0.3); a surface is fitted over the point cloud for better visualization.
(a) ground truth (b) noisy input (c) APSS (d) RIMLS (e) MRPCA (f) proposed
Fig. 5. Denoising results illustration for Daratech model (σ= 0.3); a surface is fitted over the point cloud for better visualization.
TABLE III
C2P (×10−3)OF DI FFER ENT M ODE LS,WITH GAU SSI AN NO ISE (σ= 0.1)
Model Noise APSS RIMLS AWLOP MRPCA Prop.
Bunny 9.91 4.62 5.14 7.95 4.66 4.61
Gargoyle 9.67 4.59 5.97 8.46 4.56 4.48
DC 9.66 4.37 4.82 7.63 3.98 3.71
Daratech 9.93 2.93 4.05 9.54 3.01 2.85
Anchor 9.87 3.32 4.10 8.43 2.18 2.05
Lordquas 9.72 3.33 5.81 9.10 3.79 3.11
Fandisk 9.88 6.70 7.05 8.93 4.86 4.39
Laurana 9.23 5.16 5.86 7.70 5.13 5.01
TABLE IV
C2P (×10−2)OF DI FFER ENT M ODE LS,WITH GAU SSI AN NO ISE (σ= 0.3)
Model Noise APSS RIMLS AWLOP MRPCA Prop.
Bunny 6.442 1.256 1.704 5.634 1.373 1.128
Gargoyle 6.096 1.512 1.954 5.004 1.540 1.499
DC 6.130 1.349 1.738 6.097 1.391 1.201
Daratech 6.116 3.422 3.483 4.881 3.212 2.215
Anchor 6.354 1.930 2.160 3.991 1.714 1.597
Lordquas 6.234 1.846 2.558 4.928 1.768 1.644
Fandisk 7.297 3.180 2.640 6.093 1.720 1.702
Laurana 5.890 1.392 1.800 2.307 1.464 1.211
[12] J. Zeng, G. Cheung, and A. Ortega, “Bipartite approximation for graph
wavelet signal decomposition,” IEEE Trans. Signal Process., vol. 65,
no. 20, pp. 5466–5480, Oct 2017.
[13] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed
optimization and statistical learning via the alternating direction method
of multipliers,” Foundation and Trends in Machine Learning, vol. 3,
no. 1, pp. 1–122, Jan. 2011.
[14] N. Parikh and S. Boyd, “Proximal algorithms,” Foundations and Trends®
in Optimization, vol. 1, no. 3, pp. 127–239, 2013.
[15] M. Alexa, J. Behr, D. Cohen-Or, S. Fleishman, D. Levin, and C. T.
Silva, “Computing and rendering point set surfaces,” IEEE Trans. Vis.
Comput. Graphics, vol. 9, no. 1, pp. 3–15, Jan 2003.
[16] G. Guennebaud and M. Gross, “Algebraic point set surfaces,” ACM
Transactions on Graphics (TOG), vol. 26, no. 3, p. 23, 2007.
[17] G. Guennebaud, M. Germann, and M. Gross, “Dynamic sampling and
rendering of algebraic point set surfaces,” in Computer Graphics Forum,
vol. 27, no. 2, 2008, pp. 653–662.
[18] A. C. ¨
Oztireli, G. Guennebaud, and M. Gross, “Feature preserving
point set surfaces based on non-linear kernel regression,” in Computer
Graphics Forum, vol. 28, no. 2, 2009, pp. 493–501.
[19] Y. Sun, S. Schaefer, and W. Wang, “Denoising point sets via l0
minimization,” Computer Aided Geometric Design, vol. 35, pp. 2–15,
2015.
[20] Y. Lipman, D. Cohen-Or, D. Levin, and H. Tal-Ezer, “Parameterization-
free projection for geometry reconstruction,” ACM Transactions on
Graphics (TOG), vol. 26, no. 3, p. 22, 2007.
[21] H. Huang, D. Li, H. Zhang, U. Ascher, and D. Cohen-Or, “Consol-
idation of unorganized point clouds for surface reconstruction,” ACM
transactions on graphics, vol. 28, no. 5, p. 176, 2009.
[22] H. Huang, S. Wu, M. Gong, D. Cohen-Or, U. Ascher, and H. R. Zhang,
“Edge-aware point set resampling,” ACM Transactions on Graphics,
vol. 32, no. 1, p. 9, 2013.
[23] H. Avron, A. Sharf, C. Greif, and D. Cohen-Or, “l1-sparse reconstruction
of sharp point set surfaces,” ACM Transactions on Graphics, vol. 29,
no. 5, p. 135, 2010.
[24] E. Mattei and A. Castrodad, “Point cloud denoising via moving rpca,”
in Computer Graphics Forum, vol. 36, no. 8, 2017, pp. 123–137.
[25] A. Buades, B. Coll, and J. M. Morel, “A non-local algorithm for image
denoising,” in CVPR, vol. 2, June 2005, pp. 60–65.
[26] K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by
sparse 3-d transform-domain collaborative filtering,” IEEE Trans. Image
Process., vol. 16, no. 8, pp. 2080–2095, Aug 2007.
[27] J. Digne, “Similarity based filtering of point clouds,” in 2012 IEEE Com-
puter Society Conference on Computer Vision and Pattern Recognition
Workshops, June 2012, pp. 73–79.
[28] G. Rosman, A. Dubrovina, and R. Kimmel, “Patch-collaborative spectral
point-cloud denoising,” in Computer Graphics Forum, vol. 32, no. 8,
2013, pp. 1–12.
[29] 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,” arXiv, 2018.
[30] S. Osher, Z. Shi, and W. Zhu, “Low dimensional manifold model for
image processing,” SIAM Journal on Imaging Sciences, vol. 10, no. 4,
pp. 1669–1690, 2017.
[31] J. Wang, Geometric structure of high-dimensional data and dimension-
ality reduction. Springer, 2011.
[32] D. I. Shuman, S. K. Narang, P. Frossard, A. Ortega, and P. Van-
dergheynst, “The emerging field of signal processing on graphs: Ex-
tending high-dimensional data analysis to networks and other irregular
domains,” IEEE Signal Process. Mag., vol. 30, no. 3, pp. 83–98, 2013.
[33] H. Rue and L. Held, Gaussian Markov random fields: theory and
applications. CRC press, 2005.
[34] J. Duchi, “Derivations for linear algebra and optimization,” [Online]
https://web.stanford.edu/∼jduchi/projects/general notes.pdf, 2007.
[35] M. Bona, A walk through combinatorics: an introduction to enumeration
and graph theory.
[36] M. Levoy, J. Gerth, B. Curless, and K. Pull, “The Stanford 3D scanning
repository,” [Online] https://graphics.stanford.edu/data/3Dscanrep/.
[37] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle,
“Surface reconstruction from unorganized points,” SIGGRAPH Comput.
Graph., vol. 26, no. 2, pp. 71–78, Jul 1992.
[38] J. R. Shewchuk, “An introduction to the conjugate gradient method
without the agonizing pain,” 1994.
[39] P. Cignoni, M. Callieri, M. Corsini, M. Dellepiane, F. Ganovelli, and
G. Ranzuglia, “Meshlab: an open-source mesh processing tool.” in
Eurographics Italian Chapter Conference, vol. 2008, 2008, pp. 129–
136.
[40] D. Tian, H. Ochimizu, C. Feng, R. Cohen, and A. Vetro, “Geometric
distortion metrics for point cloud compression,” in ICIP, Sept 2017, pp.
3460–3464.
