as changed/unchanged according to a thresholding tech-
nique applied to the difference image D. A first energy
term is computed from the joint density function of the
pixel values in D given the label. This term takes into
account a kind of self-information for each pixel location.
A second energy term is computed through the interactions
between a pixel and its neighbors taking into account their
labels. This term in the above approaches is termed the
consistency term. Assuming a random field defined by the
PDFs, we can see in the above approaches that the energy
for each pixel is computed based only on the own pixel and
its neighbors, out of the neighborhood the contribution to
the energy of the pixels is null. This implies that the ran-
dom fields fulfil the so called Markov condition for spatial
descriptions as the images, i.e., the field is termed a MRF.
The MRF is an approach which has been broadly used in
image analysis [10]. Liu et al. [25] compute the consis-
tency by applying the mean field theory (MFT) which
assumes that the impacts from the neighbors can be
approximated by an average field. In Pajares [27] the
change detection problem is focused minimizing an energy
function through the analog hopfield neural network
(HNN) paradigm. Under this paradigm, the energy function
assumes a trade-off between the self-information and the
consistency. Also, under the HNN approach the consis-
tency is extended so that the interactions in a neighborhood
around a pixel location are based not only in the labels but
also in the joint density function values of the neighbors,
i.e., this implicitly assumes the Markovian condition. This
extension and the analog properties of the HNN paradigm
make of this method a valid approach for the set of images
tested as compared with other existing image change
detection strategies.
Unfortunately, through additional experiments we have
verified that for images where the amount of changes
surpasses the 20% the performance of the HNN approach
decreases (see Fig. 5 and related comments in Sect. 3.3).
This is because there is an important number of these
difference images in which the energy falls in local minima
that are not global optimum. This behavior of the Hopfield
neural network is reported in Haykin [15]. The change
mask consistency approaches, involving both contextual
and self information, perform favorably. The deterministic
simmulated annealing (DSA) is also an energy optimiza-
tion based approach which can embed contextual and self
information with the advantage that it can avoid local
minima. Indeed, according to Geman and Geman [13] and
reproduced in Haykin [15] when the temperature involved
in the simulated annealing process satisfies some con-
straints (explained in the Sect. 2.5) the system converges to
the minimum global energy which is controlled by the
annealing scheduling instead of the nonlinear first-order
differential equation used in HNN. This is the main
difference of the proposed DSA technique with respect to
the HNN approach.
In Kasetkasen and Varshney [19] the stochastic simu-
lated annealing (SSA) is used to minimize the energy under
the MRF framework. The results are binary labels indi-
cating only changed/unchanged pixels. In Duda et al. [9]it
is reported that SSA is slow due to its discrete nature as
compared to the analog nature of the DSA.
In summary, we focus on the DSA approach, making the
main contribution of this paper, because of the following
set of advantages: (a) the contextual and self information
can be mapped under an energy function; (b) the annealing
scheduling allows avoiding local minima; (c) the optimi-
zation process corrects a posteriori the initial errors derived
from a thresholding approach and (d) the analog nature
allows to obtain the strength of the change for each pixel
location.
The paper is organized as follows. In Sect. 2, the DSA
process is described including the mapping of the self-
information and consistencies. The performance of the
method is illustrated in Sect. 3, where a comparative
analysis against other existing image change detection
strategies is carried out. Finally, in Sect. 4, there is a dis-
cussion of some related topics.
2 Deterministic simulated annealing process
We build a network of nodes, so that each pixel location
(x, y) in the reference image or equivalently in the differ-
ence image D
is associated to a node i. The node i is
interconnected to the node j through a symmetric synaptic
weight w
ij
which is to be defined later. Moreover, each
node i has associated a state value s
i
which will be set
through an activation function. A commonly activation
function is the hyperbolic tangent one, which is bounded by
-1 and +1 values [15]. Because of the analog nature of the
DSA approach and taking into account the above limits, the
s
i
values range in the continuous interval [-1, + 1], where
-1 and +1 indicate, from our point of view, a secure
unchanged and changed pixels respectively. Other values
in [-1, + 1] measure the degree of the change. The states
are updated after each step t. Based on the change mask
consistency strategies described in the point 6 of the
introduction, we assume that our method falls under the
MRF framework [10]; where instead of maximizing a PDF
we minimize an energy function, which is equivalent. The
updating process for each node i, is carried out through the
DSA optimization approach assuming that the energy must
be minimum. This is carried out through a regularization
coefficient which computes the consistency between the
states of the nodes in a given neighborhood and a data
coefficient which computes the consistency between the
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