# Full text: Proceedings, XXth congress (Part 8)

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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B-YF. Istanbul 2004
[n wavelet theory, one uses a decomposition into dyadic
s s+i, .
subbands [2°,2"7]. In contrast, the subbands used in the
discrete curvelet transform of continuum functions have the
2542
2y oie . -
non-standard form [27,27 — ]. This is non-standard feature of
the discrete curvelet transform well worth remembering.
With the notations of section above, the curvelet decomposition
is the sequence of the following steps.
e Subband Decomposition. The object f is decomposed
into subbands
fa (MS, AS, A SL M.
e Smooth Partitioning. Each subband is smoothly windowed
into “squares” of an appropriate scale (of sidelength : 7°)
Af à (o A feo.
e Renormalization. Each resulting square is renormalized
to unit scale
705) (^. f). QeQs.
e Ridgelet Analysis. Each square is analysed via the
discrete ridgelet transform.
2541
In the definition, the two dyadic subbands [25/2 ] and
25+ 2542 . : ;
[2^ poii ] are merged before applying the ridgelet trans-
form.
3.1 Digital Realization
In developing a transform for digital » by » data which is
analogous to the discrete curvelet transform of a continuous
function /(xj,x2), we have to replace each of the continuum
concepts with the appropriate digital concept mentioned in
section above. Recently, Starck et al.(2002) showed that "*'a
trous" subband filtering algorithm is especially well-adapted to
the needs of the digital curvelet transform. The algorithm
decomposes an » by n image / as a superposition of the form
J
Ix, y) 2 e, G5 y)* S ojGy) (11)
j=l
where c, is a coarse or smooth version of the original image
I and represents “the details of / ” at scale 2" , see
J
outputs J +1 subband arrays of size nxn [The indexing is
such that, here, j = corresponds to the finest scale(high
frequencies).]

IMAGE / WT20

Figure 2. Curvelet transform flowgraph. The figure illustrates
the decomposition of the original image into subbands followed
by the spatial partitioning of each subband(i.e., each subband is
decomposed into blocks). The ridgelet transform is then applied
to each block .
3.2 Algorithm
Starck et al.(2002) presented a sketch of the discrete curvelet
transform algorithm:
1) apply the ‘a trous algorithm with J scales;
2) set Bı=Bain 5
3)for j LK , J do
a) partition the subband @ with a block size B.
J j
and apply the digital ridgelet transform to each
block;
b) if j modulo2 -1 then B = 2B
-1 J
c) else B ER.
Note that the coarse description of the image ¢, is not
processed. Finally, Figure 2. gives an overview of the
organization of the algorithm.
This implementation of the curvelet transform is also redundant.
The redundancy factor is equal to 16J -- 1 whenever J scales
are employed. Finally, the method enjoys exact reconstruction
and stability, because this invertibility holds for each element of
the processing chain.
Inte
pri
op
im
m
ha:
sin
cui
fus
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