Digital Signal Processing Reference
In-Depth Information
Fig. 9.19
LSMI STAP algorithm (
blue
optimum,
red
LSMI with arithmetic mean,
green
LSMI
with Riemannian mean)
evident in some area close to the clutter notch. If the improvement in performance is
due to the Riemannian mean algorithm more closely approximating the clairvoyant
covariance matrix. This confirms the conjecture that the improved performance is due
to better approximation of the true covariance matrix by the proposed Riemannian
mean algorithm.
Apart from inversion, an eigenvector projection algorithm using the Riemannian
mean can also be investigated. Naïvely, one expects that the most important eigen-
vectors are those corresponding to the strongest eigenvalues. Furthermore, these
eigenvectors are better estimated using fewer samples. The EVP performance for
a single run using the Riemannian mean are shown in Fig.
9.20
. Once again, clear
improvement in performance is evident. In fact, improvement over the inversion is
also observed near the important clutternotch region.
9.18 Miscellaneous: Shape Manifold
I would like to conclude with some remarks on “shape manifold”. In image process-
ing, it is very useful to make statistics on shape. We can use previous approach if we
can define “shape manifold” or “shape space”, and in case of Metric space, we can
extend definition of Fréchet Mean for shapes.
I will give a very simple example. If we consider a set of right triangles
N
i
{
1
, where one right triangle could be defined by one point on the sur-
face/manifold
S
a
i
,
b
i
,
h
i
}
=
(
b
2
, then the Fréchet p-mean is defined
=
h
2
a
2
a
,
b
,
h
) /
=
+
by the minimum of:
