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number of landmarks is large and/or the objects are three-dimension-
al. It is not uncommon in medical research to need at least 10, 15, or
more landmarks to represent an object reasonably. In such a situation,
the exact shape distribution is extremely complicated as can be seen
below. Notice that it involves telescoping sums whose number of com-
ponents increase geometrically with the number of landmarks.
In the following, we briefly discuss the exact distribution of the
shape coordinates, sometimes referred to as the Mardia-Dryden distri-
bution. For the details of the derivation, the reader should refer to
Dryden and Mardia (1991,1992). Their paper deals only with two-
dimensional objects and we restrict our attention to that case. For
generalizations and details, the reader may refer to Dryden and
Mardia (1998).
a) Definition of Kendall shape variables
Let X be a K 2 landmark coordinate matrix and let X c = C 1 X be the
centered landmark coordinate matrix where C 1 = diag ( H 1 , H 1 ) and
H 1 T is the usual Helmert matrix without the first column. Notice that
this centering matrix is different than the centering matrix L intro-
duced earlier in this chapter. Define new variables as follows:
for i
2) variables. These are
called Kendall Shape variables. Let us denote the shape variables by a
1,2,…, K
2. Notice that there are 2( K
vector S , where
b) The probability density function for the Kendall Shape vari-
ables
Assume that vec ( X )~ N ( vec ( M ),
) .
First define new vectors U + and V + as follows:
U +
V + = (1,
(1, U 1 , U 2 ,…, U K -2 ,0, V 1 , V 2 ,…, V K -2 )
and
V 1 ,
V 2 ,…,
V K -2 ,0, U 1 , U 2 ,…, U K -2 ) . Let M *
*
C T .
C { vec ( M )},
C
Define new matrices:
é
ù
é
ù
C = U + T
* - 1
U + U + T
* - 1
˙ , b= M * T
* - 1
W
W
V +
W
U +
ê
ê
ú ˙
V + T
* - 1
V + T
* - 1
M * T
* - 1
W
U +
W
V +
W
V +
ë
û
ë
û
M *T
* -1 U +
T C -1
and g
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