The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Voi. XXXVII. Part Bl. Beijing 2008 
2.2 The basic theory of curvelet transformation 
Curvelet transformation develops from wavelet transformation, 
but overcome the inner limitation of wavelet in expressing 
direction of edge in image. Candes (1998) put forward Ridgelet 
transformation and implemented Radon transformation on 
image, which is to map one dimensional singularity of image, 
such as line in the image to a point in Radon domain, and then 
detect the singularity of point by one dimensional wavelet and 
thus solve the problem in 2D image when using wavelet 
transformation. However, most edges of natural image are 
curve lines, and it is not very effective to analyze the whole 
image only by single scale Ridgelet, so images need to be 
divided into blocks to make lines in each block are similar with 
beelines, and then implement Ridgelet transformation on each 
block. Because Ridgelet has great redundancy, Donoho put 
forward curvelet: firstly, decompose the image into subbands, 
and then adopt different blocks to subband image with different 
scales, at last analyze the each block by using Ridgelet. The 
frequency bandwidth width and length satisfy the relation width 
= length. This kind of dividing mode makes the curvelet 
transformation has fierce anisotropism, and this anisotropism 
increases exponentially as scale decreases. Researches show 
that when using finite coefficients to approach a c 2 continual 
curve line, the speed of curvelet is far larger than that of Fourier 
transformation and Wavelet transformation. In another way, 
curvelet is the sparsest way of presenting these curve lines. 
Anyway, curvelet combines the anisotropism of Ridgelet 
transformation and multi-scale of wavelet transformation, so its 
appearance is a milestone in 2 dimensional signal analyzing. 
The main steps of curvelet transformation are: 
(1) subband decompotion: 
/ -» (/>„/, A,/,A 2 /,-)- 
(2) smooth partitioning: 
A 2 /^(w 0 A 5 /) QeQ t > 
Where, Wq presents smoothing function sets in binary block 
Q = [k, /2\(£, +l)/2 s ]x[i 2 /2 s ,(k 2 +l)/2'] 
(3) Renormalization: 
S Q =^ S {T Q y\w^ s f) , QeQs , 
wru (Tof)U,x 2 ) = f (2 s x, -k,,2 s x 2 -k 2 ) . 
Where, QJ 2 ' y v 1 15 2 2 ' .This step reverts 
each block to unit scale. 
(4) Ridgelet analysis: 
a »={gQ’Px)’ M = (.Q,X)’ 
Pi ■ 
Where, is the function that composes the orthonormal basis. 
Ridgelet function with double variables is defined as: 
Wa,b,B - a 17 V((*i cos 6 + x 2 sin 6 - b) / a) 
Where, ^ is the wavelet function, a is scale factor of Ridgelet 
variable, b is position parameter of Ridgelet-, u is the direction 
of Ridgelet transformation. We can see that Ridgelet function is 
invariable along ridgeline jc, cos 6 + x 2 sin 0 = c (c is 
constant), but in vertical direction of ridgeline, it is changing 
curve line of wavelet function. 
For an integrable single-variable function f (x) , the form of 
Ridgelet transformation is : 
R f (a,b,6)= \y/ abg (x)f(x)dx 
Rewrite Ridgelet transformation to be: 
R f {a,b, 6) = \R f (0, t)a~ V2 y/((t - b) / a)dt 
The equation above indicates that Ridgelet transformation the 
one dimensional wavelet analysis on slice of Radon 
transformation, where azimuth 0 is fixed and t is the parameter 
that is analyzed. 
2.3 Digital implementation of Curvelet transformation 
According to theory above, Starck put forward a digital 
implementation of Curvelet transformation, of which main steps 
are: 
(D Subband decomposing. Decompose image into different 
subbands by using àtrous wavelet. 
©Blocking. Each subband is processed with window, and the 
size of window doubles every two subbands. 
© Digital Ridgelet analysis. Implement Ridgelet 
transformation on each block, among which 2 dimensional 
Fourier transformation, transformation from orthogonal 
coordinate to pole coordinate, and one dimensional inverse 
Fourier transformation and one dimensional wavelet 
transformation on corresponding lines. 
Digital inverse Curvelet transformation is just the inversion of 
these steps. 
2.4 Experiment on denoising SAR image 
The result proves that the processing effect of Curvelet 
transformation is better than traditional method. The setting of 
threshold often gets rid of good coefficients of wavelet, so the 
effects on edge and texture details are not satisfying. In this 
paper, logarithmic transformation is firstly implemented on 
original SAR image, then the speckle noise become similar with 
Gauss additive noise. After preprocessing, Curvelet 
transformation is implemented on image, and hard threshold 
processing in implemented, and at last denoised image is