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is oriented in the direction of the third eigenvector
corresponding to the smallest eigenvalue. Then, a
displacement of the neighbourhood mean along the
normal vector is achieved in order to preserve sharp
features.
The main contributions of this work are: i) a
simple method for point clouds denoising that does
not require a either previous mesh representation,
nor local polynomial fitting, ii) a simple approach to
prevent the shrinkage problem, and iii) a mechanism
for bias reduction. Our method is robust to outliers,
fast and easy to implement.
The remainder of this paper is organized as
follows. In section 2, related work dealing with
mesh and point clouds denoising algorithms are
presented. In section 3, a short review of principal
component analysis is presented. In section 4, the
stages of our method are explained. In section 5, the
results of our method are shown. In section 6,
conclusions and future work are discussed.
2 RELATED WORK
Point clouds have become a primitive for surface
representation and geometric modelling; however
such point clouds are noisy due to the inherent noise
of the acquisition devices. Point clouds should be
noise free for using in 3D reconstruction. Recently, a
great research effort has been done in mesh and
point clouds denoising and smoothing, producing
numerous algorithms.
Taubin (Taubin, 1995) applies in mesh
smooting a discrete version of the Laplacian
Operator, which is taken from signal processing. The
method is linear in both time and memory.
Desbrum et al. (Desbrum, 1999, Desbrum,
2000) and Bajaj (Bajaj, 2003) successfully use
anisotropic diffusion over meshes, in order to
improve the smoothing in reasonable time.
Peng et al. (Peng, 2001) use locally adaptive
Wiener filtering for denoising geometric data
represented as semiregular mesh. The algorithm
allows interactive local denoising.
Pauly and Gross (Pauly, 2001) apply Wiener
filtering to restore surfaces from point clouds in
presence of blur and noise.
Fleishman et al. (Fleishman, 2003) and Jones et
al. (Jones, 2003) have independently proposed the
use of Bilateral filtering based on robust statistics for
features preserving and mesh smoothing.
Mederos et al. (Mederos, 2003) follows the
same approach that Fleishman and Jones, by
modifying a high order fitting method, called
Moving Least Squares (MLS), to preserve sharp
features. Their approach also considers optimization
techniques for reducing the execution time of the
algorithm.
Choudhury and Tumblin, (Choudhury, 2003)
present a single-pass nonlinear filter for edge
preserving and smoothing. The method is called
Trilateral filtering and it is an evolution of the
Bilateral filtering. The filter produces better outlier
rejection and strong noise reduction than Bilateral
filtering.
Schall et al. (Schall, 2005) have proposed a
probabilistic method which consists of using a
kernel density estimation technique. It associates to
each point a local measure of probability to locate
the point into the surface. The method achieves
effectiveness result in filtering and robustness in
outliers detection.
3 PRINCIPAL COMPONENT
ANALYSIS
Principal Component Analysis (PCA) is a statistical
method that tries to explain the covariance structure
of data by means of various components expressed
as linear combinations of the original variables
(Hubert, 2005).
The first component of PCA corresponds to the
direction in which the projected data have the largest
variance. The second component is orthogonal to the
first component, and maximizes the variance of the
data points projected on it.
PCA is applied widely in bias identification into
data sets. It is used for data dimensionality reduction
and visualization (Jolliffe, 1986), data clustering
(Pauly, 2002) and pattern recognition (De la Torre,
2001). Despite the versatility of PCA, it is sensitive
to outliers present in data. Figure 1a shows a set of
points mainly concentrated at the low part of the
figure. Three of them, which are considered outliers,
are enclosed in red circles. The first component of
PCA, blue line, should indicate the main direction of
data dispersion, but it is observed a bias produced by
the outliers. Figure 1b shows the correction by
robust PCA.
The PCA, first take a set of neighbors
)(
i
pN
around a point
i
p
, next the neigborhood mean
i
p
is
estimated using (1). Finally, using (2), the
covariance matrix
CM
is obtained from the points
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