yu and cobbold: eigen-processing methods for color flow imaging—part i 561
Fig. 1. Flow diagram of the proposed Hankel-SVD filter. During operation, the filter is applied to the slow-time ensemble of each sample
volume.
Fig. 2. Illustration of how the Hankel matrix A is constructed for
the case where the slow-time ensemble size is eight (N
D
=8)and
the dimension parameters is four (P =4).
where P must satisfy the relation P ≤ ceil(N
D
/2), i.e., the
smallest integer greater than or equal to N
D
/2. As such,
the relation P ≤ N
D
− P + 1 always holds, and the matrix
A has a rank (i.e., the number of independent rows) that is
equal to P with a largest possible rank of ceil(N
D
/2). An
example of the entire matrix formation process is shown in
Fig. 2. Note that, for this matrix, statistical stationarity
of the samples in the signal vector is inherently assumed.
To compute the P orthogonal components in the Hankel
component approximation, we can factor the Hankel data
matrix into the following sum of P orthogonal, rank-one
matrix components via the SVD:
A =
P
k=1
A
k
=
P
k=1
σ
k
u
k
v
H
k
. (3)
In this expression, A
k
is the kth rank-one Hankel compo-
nent in the decomposition, while σ
k
, u
k
,andv
k
are cor-
respondingly the singular value, left singular vector (with
dimension P ), and right singular vector (with dimension
N
D
− P +1) of A
k
. Note that the P singular values in
(3) are ordered from largest to smallest by definition, and
hence A
k
can be considered as the kth-order principal
Hankel component. From these principal Hankel compo-
nents, it is possible to reconstruct the orthogonal basis
vectors γ
k
ϕ
k
as seen in (1a). One particular way to per-
form this reconstruction process, as originally described
by Poon et al. [12], is to sum and average the matrix
elements along the reverse diagonals of A
k
because the
matrix more or less maintains a constant reverse-diagonal
structure. Such an approach is used by the proposed fil-
tering method.
B. Estimation of Clutter Eigen-Space Dimension
Because the aim of the Hankel-SVD filter is to suppress
clutter in the slow-time signal, it is necessary to determine
whether a principal Hankel component A
k
is part of the
clutter eigen-space. In general, there are two types of ap-
proaches to carry out this analysis. First, given that clut-
ter often has higher energy than blood echoes and white
noise, a principal Hankel component can be considered as
being part of the clutter eigen-space if its singular value
magnitude σ
k
is larger than a given value. Such way of
estimating the clutter eigen-space dimension is similar to
the eigenvalue analysis algorithms reported previously in a
few eigen-filter designs [7], [13]. Alternatively, since clutter
generally consists of low-frequency contents, it is possible
to identify a clutter eigen-space component based on the
frequency contents of each orthonormal basis vector ϕ
k
.
This latter approach is adopted in the Hankel-SVD filter
to determine the clutter eigen-space dimension because the
approach can be implemented in a way that gives filtering
characteristics similar to the stopband of a conventional
bandpass filter.
The clutter eigen-space analysis algorithm used by the
Hankel-SVD filter is illustrated in Fig. 3. This algorithm
is fundamentally based on two assumptions: 1) the slow-
time clutter is contained in the more dominant Hankel
components (i.e., ones with larger singular values); 2) the
blood flow component of the slow-time signal is contained
in the Hankel components with high Doppler frequen-