Árbók VFÍ/TFÍ - 01.06.2002, Side 242

(3)
<p(t) = X/?o [k]<p{2t-k), and !//(/) = [*]0(2/-£),
Ae Z (eZ
where //0[^] and hi[k] are the low-pass and high-pass impulse responses for the DWT. The
DWT can be calculated with a fast algorithm of the order O(NlogN) where N is the length
of the input signal [4]. Figure 1 shows a schematic figure of the algorithm. In each level
of the analysis part, the input goes through a low pass filter H0 (impulse response h0[k]
and a high pass filter Hj (impulse response [/c]) each followed by a downsampler by 2.
(The downsampler keeps every other input sample). For a digital image (i.e., two dimen-
sional signal) a separable DWT is used. That is first a one dimensional DWT is performed
along the rows of the image and then along the columns of the image. The separable 2-
Dimensional (2-D) DWT is, of course, still a fast algorithm of the order (/(N^logN) but
using it leads to problems representing edges in digital images [5]. And this is the main
reason for developing the curvelet transform.
Ridgelet transform
The main building block along with wavelet transform as explained below is the ridgelet
transform. It was proposed in [6] to represent straight edges with as few coefficients as
possible. For an image/(.r,i/) the ridgelet coefficients are defined by
Rf(a,b,9)= ] ] f(x,y)</)*Ag(x,y)dxdy, (4)
where
<Þaj>,o(x,y) = a~'/2<f> ((x cosð +ysmO-b )/a), (5)
where <p is a function that is constant along the line rcosö + i/sinö - b = 0 and wavelet
along the orthogonal directions. An important observation is that the ridgelet transform
can be written in terms of the Radon transform and the wavelet transform. First the
Radon transform is defined as follows: For an image / the Radon transform is defined by
Raf(9,t) = j j/(x,_y)ð (rcosð + y s\n 0-t)dxdy. (6)
That is the Radon transform is calculated by taking line integrals at an angle 6 and a nor-
mal distance to the origin t. Figure 2 shows the Radon parameters defined.
Rf (a,b,9)= (iRaf,<i/ab) = ] Raf(0,t)a~'l2y/ ((/ -b)/a)dt, (7)
where VC.jO) = V(U~b)/a)/ fa is an 1-D wavelet. Equation (7) gives an intuitive view into
the ridgelet transform. First straight lines are mapped into points with the Radon trans-
form. Then the wavelet transform is applied to represent the
points. Because the wavelet transform is known to represent point
singularities efficiently [1] the representation of lines with the
ridgelet transform is quite compact.
Curvelet transform
As noted above, usually digital images do not have straight edges
but have curved edges. Curvelet transform uses the idea that on a
small scale curved edges look like straight edges. That is curvelet
transform is a kind of multiscale ridgelet transform. A procedure
for finding the curvelet transformation for image f that is defined
on [0,1] x [0,1] can be described as follows [2].
• Subband Decomposition: Define a bank of subband filters
and As,s £ 0. The image is filtered into subbands:
2 3 8
Árbók VFl/TFl 2002