Graphics Reference
In-Depth Information
; Soatto et al.,
; Wu et al.,
).Methods for extracting features from static
and dynamic images in
-Dcan beapplied to ultrasound and MRI images, and these
are discussed in Sects.
.
and
.
(Wu and Lu,
). hese methods can also be
extended to
-D or higher dimensions.
Space Domain: Local Blocks
Let us denote a rectangular lattice containing a
-D digital image of size N
M by
S=(
I, J
)
I
N,
J
M, I, J
Z
.
he spatial characteristics of a pixel in the image are described by its neighboring
pixels.herefore,alocalblockinonetimeframeofsize b
b canbecreatedasafeature
(
i
)
vector
x
(
t
j
)
of the central pixel i
S
, i
=
,
, n and n
=(
M
−
b
+
)(
N
−
b
+
)
for each time point t
j
, j
, m. he dimension of the feature vector in the space
domain is b
. Pixels at different time frames are not included in the feature vector
because they may vary according to the motion. For the same pixel i,thecollection
of feature vectors along the sequence of images
=
,
(
i
)
x
(
t
j
)
, j
=
,
, m
represents the
temporal variation of the features due to motion over time.
Givenasequenceof m trainingimages,theclasslabels
y
(
i
)
(
t
j
)
, j
=
,
, m
,and
(
i
)
thefeaturevectors
,theprojectiondirectionsoffeaturevectors
for classification and prediction can be found. If the number of training images, m,
is bigger than the dimension of the feature vectors, b
, then it is feasible to estimate
the projection direction in the dimension of b
. However, for a short sequence of
training images, m is not necessarily bigger than b
. herefore, information from
neighborhood pixels must be borrowed.
Let
x
(
t
j
)
, j
=
,
, m
(
i
)
q
be the set of neighboring sites for pixel i in a q-order neighborhood
system.Forexample, afirst-orderneighborhoodsystemisa
-neighborhoodsystem,
sinceeachinteriorsitehasfourneighborsandthesizeof
N
(
i
)
is
. here are eight
neighbors for every interior site in a second-order neighborhood system, which is
also called an
-neighborhood system, and the size of
N
(
i
)
is
.hen, for pixel i,the
neighboring feature vectors can be collected as the training set:
N
(
i
)
l
(
i
)
(
i
)
q
X
=
x
(
t
j
)
, j
=
,
, m, l
N
(
.
)
where i
=
,
, n and n
=(
M
−
b
+
)(
N
−
b
+
)
.Forinstance,ifq
=
, then the size
of the training set becomes
m,whichisbiggerthanb
when m
b
.
Frequency Domain: Fourier Transform of Local Blocks
If the features of an image are periodic over space, then the features in a local block
in the space domain can betransformed tothe frequency domain bya Fourier trans-
form. his transform will highlight the periodic pattern (Weaver,
). his can be
performed using afast Fourier transform (FFT) ifthe blocksize is of the power
.For
