Graphics Reference
In-Depth Information
Rigid Registration
Registration of
M
and
P
involves estimating an optimal rigid transformation between them,
denoted T . Here,
(the
source data) is spatially transformed to match it. The Iterative Closet Point algorithm (ICP)
is the best-known technique for pairwise surface registration. Since the first paper of Besl
and McKay (1992) ICP has been widely used for geometric alignment of 3D models and
many variants of ICP have been proposed (Rusinkiewicz and Levoy, 2001). ICP is an iterative
procedure minimizing the error (deviation) between points in
P
is assumed to remain stationary (the reference data), whereas
M
.It
is based one of the following two metrics: (i) the point-to-point , metric which is the earlier and
the classical one, by minimizing in the k -th iteration, the error E reg ( T k )
P
and the closest points in
M
= ( T k
.
p i
q j );
min q M ( E reg ( T k )); (ii) the point-to-plane introduced later and minimizes E reg ( T k )
q j =
q
|
=
n ( q j )( T k
q j ). For each used metrics, this ICP procedure is alternated and iterated
until convergence (i.e., stability of the error). Indeed, total transformation T is updated in an
incremental way as follows: for each iteration k of the algorithm: T
.
p i
T . One note that
ICP performs fine geometric registration assuming that a coarse registration transformation
T 0 is known. The final result depends on the initial registration. The initial registration could
be obtained when corresponding detected landmarks in
=
T k
.
M
and
P
.
Template Warping/Fitting
A warping of
M
to
P
is defined as the function F such that F (
M
)
= P
. The function F
is called the warping function , which takes
. Given a pair of landmarks (detected as
described in Section 1.4.5) with known correspondences, U L =
M
to
P
( u i ) 1 < i < L and V L =
( v i ) 1 < i < L ,
in
, respectively. One needs to establish dense correspondence between other meshes
vertices; u k and v k denote the locations of the k -th corresponding pair and L is the total
number of corresponding landmarks. Thus, a warping function, F , that warps U L to V L
subject to perfect alignment is given by the conditions F ( u i )
M
and
P
=
v i for i
=
1
,
2
,...,
L .
Thin Plate Spline (TPS) . TPS Bookstein (1989) are a class of widely used non-rigid interpo-
lating (warping) functions. The thin plate spline algorithm specifies the mapping of points
for a reference,
P
, set to corresponding points on a source set,
M
. The TPS fits a mapping
function F ( u ) between corresponding point-sets
{
v i }∈ M
and
{
u i }∈ P
by minimizing the
following energy function:
L
2
E tps =
1
v i
F ( u i )
+
L
λ
J
(1.38)
i
=
For a fixed
λ
which provides trade-off of warp smoothness and interpolation.
v 2 2 d u d v
u 2 2
2
2
2 F
2 F
2 F
=
+
+
J
(1.39)
u
v
 
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