Digital Signal Processing Reference
In-Depth Information
Compared with two-state HMT, four-state HMT-2 also improves the denois-
ing performance for most cases.
In this section, we have briefly introduced and improved the wavelet-
domain HMMs originally developed in Reference 1. This section serves the
background materials of this chapter. Meanwhile, these preliminary stud-
ies will lead to more powerful wavelet-domain HMMs as well as image-
processing algorithms afterward. In particular, we study image denoising,
image segmentation, and texture analysis and synthesis, where wavelet-
domain HMM are discussed and tailored for different applications.
10.3
Image Denoising
Wavelet-domain statistical image modeling can be roughly categorized into
three groups: the interscale models,
1
,
6
,
20
the intrascale models,
21
,
22
and the
hybrid inter- and intrascale models.
13
,
23-25
These models allow more accurate
statistical modeling and more effective image processing, e.g., denoising and
estimation, than other methods that assume wavelet coefficients to be in-
dependent. In particular,
1
,
6
wavelet-domain HMT imposes a tree-structured
Markov chain to capture interscale dependencies of wavelet coefficients across
scales. In Reference 13, both interscale and intrascale dependencies can be ef-
ficiently captured by a so-called contextual hidden Markov model (CHMM).
However, the local statistics of wavelet coefficients cannot be well charac-
terized by HMT and CHMM. In other words, neither HMT nor CHMM has
the sufficient spatial adaptability that is found useful in the wavelet-domain
statistical image modeling.
21
Second, although the tree structure involved in
HMT captures the key characteristics of DWT along the hierarchical wavelet
subtree, it also introduces undesirable denoising artifacts in denoised images
due to the discontinuity of the tree structure. Third, the tree-structured EM
training algorithm of HMT is computationally expensive. In this work, we
propose a new wavelet-domain HMM by considering the above three issues,
i.e.,
spatial adaptability
,
reduced denoising artifacts
, and
fast model training
.
26
10.3.1
Gaussian Mixture Field
×
w
(
)
Given the
J
-scale DWT of an
N
N
image,
j,k,i
denotes the
k, i
th coefficient
=
...
=
in scale
j
, where we omit the subband notation,
j
1
,
,J
and
k, i
2
j
.
W
j,k,i
and
S
j,k,i
are the continuous random
variable and the discrete state variable of
....
−
=
/
0
,
1
,
,N
j
1 with
N
j
N
w
j,k,i
,respectively. In References 1,
6, and 13, the GMM,
,isassumed for the wavelet
coefficients in scale
j
, and
S
j
is the state variable associated with scale
j
.
Sufficient data in scale
j
allows the robust estimation of
j
={
p
S
j
(
m
)
,
σ
j,m
|
m
=
0
,
1
}
j
at the loss of
spatial adaptability of statistical image modeling in the wavelet domain. In
this work, we propose a
Gaussian mixture field
(GMF), which can be thought
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