May 2015
Intermediate to advanced
246 pages
6h 33m
English
book
A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing
by Daniel Cohen-Or, Chen Greif, Tao Ju, Niloy J. Mitra, Ariel Shamir, Olga Sorkine-Hornung, Hao Zhang
Content preview from A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing
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68 Spectral Transform
Laplacian smoothing can then be expressed as applying a smooth-
ing operator S to the signal x, resulting in a new contour repre-
sented by x
0
= Sx. The smoothing operator
S =
1
2
1
4
0 . . . . . . 0
1
4
1
4
1
2
1
4
0 . . . . . . 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 . . . . . . 0
1
4
1
2
1
4
1
4
0 . . . . . . 0
1
4
1
2
(5.4)
is related to the Laplacian operator by S = I −
1
2
L.
To analyze the behavior of Laplacian smoothing, in particu-
lar what happens in the limit, we utilize the set of basis vectors
formed by the eigenvectors of L. This leads to a framework for
spectral analysis of geometry. From linear algebra, we know that
since L is symmetric, it has
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