![](bg2.png)
tially improved from the observational image acqui-
sitions. We proposed and described in (Lepoittevin
et al., 2015) the construction of the initial ensem-
ble. A set of state-of-the-art optical flow algorithms
(around 50 methods, described in the surveys (Sun
et al., 2010) and (Baker et al., 2011)) are used to de-
sign the initial ensemble of motion fields. The HS-
brightness method (Sun et al., 2010) is one of these al-
gorithms and it provides the best performance, when
used alone, on our synthetic data. HS-brightness is
a specific implementation of the Horn-Schunk algo-
rithm (Horn and Schunk, 1981), which includes a
multi-resolution scheme and a median filtering (Sun
et al., 2010). Each member of the ensemble is then
integrated in time with the model and an estimate of
motion is computed, at each time, as the average of
the motion members. The uncertainty is described
by the spread of that ensemble. As in (Lepoittevin
et al., 2015) the paper concerns the application of
EnKF for estimating a dense motion field, based on
the structures displayed by images. As the applica-
tion domain concern fluid flows images, we do not
rely on object characteristics, such as the SIFT fea-
tures (Lowe, 1999) and variants, since they are not
significant on these data. The innovation of the paper,
compared to (Lepoittevin et al., 2015), concerns the
use of a segmentation of images by EnKF. Moreover,
we will stress, from the experiments, that merging a
number of optical flow algorithms gives better results
than the best one alone (the HS-brightness method):
all methods take part to the result of EnKF.
Two different alternatives for characterizing the
structures, which are displayed on fluid flows images,
such as fronts and vortices on ocean satellite data, are
compared in the context of EnKF. Comparison con-
cerns two criteria: quality of results and computa-
tional performances.
The first approach, introduced in (Lepoittevin
et al., 2015), concerns the design of a localiza-
tion function that includes information on the dis-
played structures, as it has been discussed for in-
stance in (Anderson, J. L., 2001), (Houtekamer and
Mitchell, 1998), (Hamill et al., 2001), (Oke et al.,
2007) and (Anderson, 2007).
The second approach applies a domain decompo-
sition technique, as in (Nerger et al., 2006) and (Hunt
et al., 2007), which depends on the image brightness
values. This domain decomposition is equivalent to a
segmentation of the image acquisitions.
Both techniques make the estimation depending
on the structures and on their evolution in time.
As the Kalman filter itself is well-known in the
image processing community, Section 2 will shortly
summarize that point and only discuss the mathemat-
ical equations of the ensemble Kalman filter, as ini-
tially given in (Evensen, 2003). Section 3 reminds
about the design of the localization function from im-
age brightness values and its use in the EnKF formal-
ism. The resulting method is named Explicit Struc-
tures Localization in the remaining of the paper. Sec-
tion 4 explains the use of domain decomposition in
the context of EnKF and describes the domain decom-
position associated to a segmentation process. Results
are given in Section 5, which compares the two ap-
proaches. The paper ends with some conclusions and
hints on future research work.
2 EnKF AND EXPLICIT
LOCALIZATION
Let us first provide the notations that are further re-
quired in the paper.
Images are acquired on the spatial domain Ω. A
pixel is denoted p.
A sequence of (N
O
+ 1) acquisitions A
t
,
{
t ∈ [0, N
O
]
}
is processed. An observation vec-
tor Y
t
is computed on each acquisition A
t
. This
vector is either a representation, line by line, of
the image or a description of quantities that are
computing on this image.
A state vector X of size N
X
is defined, whose
value at time t is X
t
= (w
t
, I
t
)
T
. w
t
is the vector de-
scribing the motion field and I
t
is the vector associ-
ated with a synthetic image. The assumption is that
the synthetic image sequence corresponding to I
t
sat-
isfies the optical flow equation: the image brightness
is advected by the motion field.
The objective is to get an estimate X
(a)
t
of the
true state vector (and consequently of the true mo-
tion field), from its background value X
(b)
t
and from
the observation vector Y
t
, so that I
t
is as close as pos-
sible to Y
t
. The background value X
(b)
t
is obtained
from time integration of the estimation X
(a)
t−1
at previ-
ous time.
Ensemble methods rely on a number N
m
of mem-
bers. X
i
t
denotes the state vector at time t of the i
th
member of the ensemble. The average X
t
of the state
vectors X
i
t
is defined by:
X
t
=
1
N
m
N
m
∑
i=1
X
i
t
= X
i
t
(1)
Error terms are discussed in the paper according to the
following notation: E
R
is a centered Gaussian noise
associated to the covariance matrix R and denoted:
E
R
∼ N (0, R) (2)
Ensemble Kalman Filter based on the Image Structures
141