Image Processing Reference
In-Depth Information
Similar to the ML estimation procedure, the log-likelihood function is often used
in the MAP estimation for its computational ease without affecting the results. The
connection between the two techniques of estimation is obvious. The MAP estimate
equals the ML estimate when the prior
is uniform, i.e., a constant function.
The MAP estimate can also be considered as a regularized ML estimate. Addition-
ally, when the prior is non-informative, then the MAP estimate approaches the ML
estimate, i.e,
θ
MAP
→
θ
ML
.
The ML and MAP estimators provide simple, but powerful frameworks of
Bayesian statistics to deal with a variety of problems through the right choice of
the model and the prior. Some highly popular techniques such as Viterbi algorithm,
Kalman filter, hidden Markov model, and Expectation-Maximization algorithm have
their roots in the Bayesian framework. While these techniques have been used to
solve problems from various fields, here we discuss in brief a few techniques and
applications relevant to image processing and vision.
Geman and Geman introduced a Bayesian paradigm for the processing of images
using the Markov random field (MRF) [63]. They demonstrated the results in con-
nection with the image restoration problem which searches for the MAP estimate
of an image modeled as a Markov random field using simulated annealing. Their
method has been further explored by several researchers. The assessment of MAP-
based restoration for binary (two-color) images has been carried out by Greig et al.
where the performance of the simulated annealing has also been tested [70]. Murray
et al. have experimented with the MAP restoration of images on the parallel SIMD
processor array for speeding up the operation [120]. The problemof reconstruction of
medical images has often been addressed through the Bayesian framework. An iter-
ative algorithm for the reconstruction of PET images using a normal prior has been
proposed in [73]. This solution has a guaranteed convergence which is optimal in the
MAP sense. A 3-D reconstruction of images obtained by the microPET scanner using
the MAP formulation has been discussed in [143], where the likelihood function and
image have been modeled using the Poisson and the Gibbs priors, respectively. Lee
et al. have incorporated aweak plate-based prior for the reconstruction of tomography
images, which is claimed to preserve edges during the reconstruction [100].
Several edge preserving operations have the Bayesian concept at their basics.
The anisotropic diffusion proposed by Perona and Malik [130] can be formulated
into a Bayesian framework [11]. The bilateral filter discussed in Chap.
3
also has
a Bayesian connection as shown by Barash [11, 12]. A classical problem of image
segmentation has also been formulated into a Bayesian framework. Sifakis et al. have
proposed a Bayesian level-set formulation for image segmentation [166]. The use
of the double Markov random field [112], and the binary Markov random field [48]
has also been investigated for the image segmentation problem. Moghaddam et al.
have developed a MAP-based technique for visual matching of images which can
be used for face recognition or image retrieval [119]. In [18], a MAP-based object
recognition technique has been proposed which employs an MRF-based prior to
capture phenomena such as occlusion of object features.
We have already discussed the use of a Bayesian framework for image fusion in
P(θ)
