Digital Signal Processing Reference
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(cartoon) parts. The importance of such separation is for applications in image
compression, image analysis, synthesis, and more; see, for example, the work of
Bertalmıo et al. (2003). A functional-space characterization of oscillating textures
was proposed by Meyer (2002) and was then used within a variational framework
by Vese and Osher (2003) and Lieu and Vese (2008) and, later, by others (Aujol
et al. 2005, 2006; Aujol and Chambolle 2005) for the design of such an image sep-
aration algorithm. Since these pioneering contributions, we have witnessed a flurry
of research activity in this application field. A different methodology toward the
same separation task is proposed by Meyer et al. (2002) and Guo et al. (2003). The
work of Meyer et al. (2002) describes a novel image compression algorithm based
on image decomposition into cartoon and texture layers using the wavelet packet
transform. The work presented by Guo et al. (2003) shows a separation based on
matching pursuit and Markov random field (MRF) modeling.
In this section, we focus on the same decomposition problem, that is, texture and
natural (piecewise smooth) additive layers. In the rest of this section, we will essen-
tially deal with locally periodic/oscillating textures. The core idea here is to choose
two appropriate dictionaries, one known to sparsify the textured part and the second
the cartoon part. Each dictionary will play the role of a discriminant, preferring the
part of the image it is serving, while yielding nonsparse representations on the other
content type. Then MCA is expected to lead to the proper separation as it seeks the
overall sparsest solution, and this should align with the sparse representation for
each part.
8.6.1 Choosing a Dictionary
As already discussed in Section 8.5, identifying the appropriate dictionary is a key
step toward a good separation. We discuss here the dictionary for the texture and
the piecewise smooth content.
8.6.1.1 Dictionaries for Textures: The (Local) Discrete Cosine Transform
The DCT is a variant of the discrete Fourier transform, replacing the complex anal-
ysis with real numbers by a symmetric signal extension. The DCT is an orthonor-
mal transform, known to be well suited for first-order Markov stationary signals.
Its coefficients essentially represent frequency content, similar to the ones obtained
by Fourier analysis. When dealing with locally stationary signals, DCT is typically
applied in blocks. Such is the case in the JPEG image compression algorithm. Over-
lapping blocks are used to prevent blocking artifacts, in this case an overcomplete
transform corresponding to a tight frame with redundancy 4 for an overlap of 50
percent. A fast algorithm with complexity
O
(
N
2
log
N
) exists for its computation.
The DCT is appropriate for sparse representation of either smooth or locally pe-
riodic behaviors. Other dictionaries that could be considered are Gabor, brushlets,
wavelet packets, or wave atoms (Demanet and Ying 2007).
8.6.1.2 Dictionaries for Piecewise Smooth Content: The Curvelet Transform
As we have seen in Chapter 5, the curvelet transform enables the directional anal-
ysis of an image in different scales. This transform provides a near-optimal sparse
representation of piecewise smooth (
C
2
) images away from
C
2
contours. It is well
