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Journal Article
Tangent Estimation from Point Samples
2016
Overview
Let
M
be an
m
-dimensional smooth compact manifold embedded in
R
d
, where
m
is a constant known to us. Suppose that a dense set of points are sampled from
M
according to a Poisson process with an unknown parameter. Let
p
be any sample point, let
ϱ
be the local feature size at
p
, and let
ϱ
ε
be the distance from
p
to the
(
n
+
1
)
th nearest sample point for some
n
between
m
+
1
2
+
1
and
d
+
1
2
. Using the
n
sample points nearest to
p
, we can estimate the tangent space at
p
and it holds with probability
1
-
O
(
n
-
1
/
3
)
that the angular error is
O
(
ε
2
)
. The running time is bounded by the time to compute the thin SVD of an
n
×
d
+
1
2
matrix and the full SVD of an
n
×
d
matrix, which is usually
O
(
d
2
n
2
)
in practice. We implemented the algorithm and experimentally verified its effectiveness on both noiseless and noisy data.
Publisher
Springer US,Springer Nature B.V
