Digital Signal Processing Reference
In-Depth Information
Insignificant coeff
Significant coeff
Figure 5.10. Local ridgelet transform on bandpass-filtered image. At fine scales, curved
edges are almost straight lines.
5.4.1.1 First-Generation Curvelet Construction
The CurveletG1 transform (Cand es and Donoho 2002; Donoho and Duncan 2000;
Starck et al. 2002) opens the possibility to analyze an image with different block
sizes but with a single transform. The idea is to first decompose the image into a
set of wavelet bands and to analyze each band by a local ridgelet transform, as il-
lustrated in Fig. 5.10. The block size can be changed at each scale level. Roughly
speaking, different levels of the multiscale ridgelet pyramid are used to represent
different subbands of a filter bank output. At the same time, this subband decompo-
sition imposes a relationship between the width and length of the
important
frame
elements so that they are anisotropic and obey approximately the
parabolic scaling
law
width
length
2
.
The first-generation discrete curvelet transform (DCTG1) of a continuum func-
tion (i.e., a function of a continuous variable)
f
(
t
) makes use of a dyadic sequence
of scales and a bank of filters with the property that the bandpass filter
≈
j
is con-
centrated near the frequencies [2
2
j
,
2
2
j
+
2
], that is,
,
=
(2
−
2
j
j
(
f
)
=
2
j
∗
f
2
j
(
ν
)
ν
)
.
In wavelet theory, one uses a decomposition into dyadic subbands [2
j
2
j
+
1
]. In con-
trast, the subbands used in the discrete curvelet transform of continuum functions
have the nonstandard form [2
2
j
,
2
2
j
+
2
]. This nonstandard feature of the DCTG1 is
worth remembering (this is where the approximate parabolic scaling law comes into
play).
The DCTG1 decomposition is the sequence of the following steps:
,
Subband decomposition:
The object
f
is decomposed into subbands.
Smooth partitioning:
Each subband is smoothly windowed into “squares” of an
appropriate scale (of side length
2
−
j
).
Ridgelet analysis:
Each square is analyzed via the DRT.
∼
2
2
j
+
1
] and [2
2
j
+
1
2
2
j
+
2
] are merged
In this definition, the two dyadic subbands [2
2
j
,
,
before applying the ridgelet transform.
