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 
(Starck ef al, 1998) for more information. Thus, the algorithm 
outputs J +1 subband arrays of size nxn [The indexing is 
such that, here, j = corresponds to the finest scale(high 
frequencies).] 
  
IMAGE / WT20 
  
  
  
  
  
  
  
  
  
  
Radar Teasatonn 
  
  
  
  
  
  
  
  
  
  
  
  
  
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. 
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