Information Technology Reference
In-Depth Information
computational load by designing an appropriate vector quantization scheme with
almost any desired number of steering matrices in the look-up table, and (iii) we
can pre-calculate kernel weights to lower the computational load further (since the
steering matrices are fixed).
From (18), the elements of the spatial covariance matrix
C
i
are given by the
steering parameters with the following equations:
θ
j
)=
c
11
c
12
c
12
c
22
,
C
j
(
γ
j
,
ρ
j
,
(33)
with
γ
j
θ
j
ρ
j
cos
2
ρ
−
1
j
sin
2
c
11
=
θ
j
+
(34)
−
γ
j
θ
j
ρ
−
1
j
c
12
=
ρ
j
cos
θ
j
sin
θ
j
+
cos
θ
j
sin
(35)
γ
j
θ
j
ρ
j
sin
2
ρ
−
1
j
cos
2
c
22
=
θ
j
+
(36)
where
θ
j
is the orien-
tation angle parameter. Fig. 8 visualizes the relationship between the steering param-
eters and the values of the covariance matrix. Based on the above formulae, using a
pre-defined set of the scaling, elongation, and angle parameters, we can generate a
lookup table for covariance matrices, during an off-line stage.
During the on-line processing stage, we compute a naive covariance matrix
C
naive
i
γ
j
is the scaling parameter,
ρ
j
is the elongation parameter, and
(15) and then normalize
C
naive
i
so that the determinant of the normalized naive co-
variance matrix det(
C
naive
i
) equals 1
.
0:
C
naive
i
1
C
naive
i
γ
i
C
naive
=
det (
C
naive
i
=
,
(37)
i
)
where again
γ
i
is the scaling parameter. This normalization eliminates the scaling pa-
rameter from the look-up table and simplifies the relationship between the elements
of covariance matrices and the steering parameters, and allows us to reduce the size
of the table. Table 1 shows an example of a compact lookup table. When the elon-
gation parameter
ρ
i
of
C
i
is smaller than 2.5,
C
i
is quantized as an identity matrix
(i.e. the kernel spreads equally every direction). On the other hand, when
ρ
i
≥
2
.
5,
we quantize
C
i
with 8 angles. Using
C
naive
, we obtain the closest covariance matrix
i
C
i
from the table. I.e.,
F
,
C
i
= arg min
ρ
j
,
−
C
naive
C
(
ρ
j
,
θ
j
)
(38)
i
θ
j
·
F
is the Frobenius norm. The final matrix
C
i
is given by:
where
C
i
=
γ
i
C
i
.
(39)