Statistical Image Modeling and Processing Using Wavelet-Domain Hidden Markov Models - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 367

It has been found to be useful for a variety of texture analysis and synthesis
applications, including texture classification, texture segmentation, and tex-
ture synthesis. These approaches have been found to be more efficient than
the traditional methods, considering the characteristics of the human visual
system in perceiving textures.
Wavelet-domain HMMs, e.g, HMT, are powerful in statistical signal and
image modeling and processing. When HMT was applied to image process-
ing, it was usually assumed that the three DWT subbands, i.e., HL , LH , and
HH , were independent. This assumption is valid in modeling most real im-
ages, as real images usually carry a large amount of randomly distributed
edges or structures, which weaken the cross-correlation between DWT sub-
bands. However, we observed that, for natural textures, in particular struc-
tural textures, the regular spatial structures or patterns may result in certain
dependencies across the three DWT subbands. It was also shown 46 that the
dependencies across subbands are useful for wavelet-based texture charac-
terization. Specifically, a vector wavelet-domain HMT was proposed, 9 which
incorporates multivariate Gaussian densities to capture statistical dependen-
cies across DWT subbands. The vector HMT was applied to the redundant
wavelet transform to obtain rotation-invariant texture retrieval. It was demon-
strated that the two-state vector HMT in Reference 9 has a moderate feature
size and provides more accurate texture characterization than the two-state
scalar HMT in References 1 and 6.
In this section, we propose a new wavelet-domain HMM, HMT-3S, by inte-
grating the three DWT subbands into one tree structure. In addition to the joint
DWT statistics captured by HMT, the proposed HMT-3S can also exploit sta-
tistical dependencies across DWT subbands. Differing from the vector HMM
proposed in Reference 9, we still impose the single-variable Gaussian mixture
densities in the wavelet domain. In HMT-3S, the state combination of three
wavelet coefficients from the three DWT subbands results in an increase in
the state number from 2 to 8, and the dependencies across DWT subbands
can be characterized by the enlarged state transition matrices, i.e., 2 × 2in

HMT and 8 8inHMT-3S. It is demonstrated that the more accurate texture

characterization from HMT-3S improves the performance of texture analysis
and synthesis. 47

×

10.5.1 Wavelet-Domain HMT-3S

As discussed before, the two-state GMM is used to characterize the marginal
statistics of wavelet coefficients. If we consider all wavelet coefficients to be
independent, we obtain the so-called independence mixture model (IMM). 1
Wavelet-domain HMT was proposed mainly to capture interscale dependen-
cies of wavelet coefficients across scales. When HMT is extended to the 2D
case for image processing, the three wavelet subbands are usually considered
independent. 6 , 7 To improve the accuracy of texture characterization by captur-
ing dependencies across DWT subbands, we propose a new wavelet-domain