Digital Signal Processing Reference
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and other complex directional wavelet transforms (Fernandes et al. 2003, 2004; van
Spaendonck et al. 2003).
The ridgelet and the curvelet (Candes and Donoho 1999, 2002) transforms were
developed as an answer to the weakness of the separable wavelet transform in
sparsely representing what appears to be simple building-block atoms in an image,
that is, lines, curves, and edges. Curvelets and ridgelets take the form of basis ele-
ments, which exhibit high directional sensitivity and are highly anisotropic (Donoho
and Duncan 2000; Candes and Donoho 2002; Starck et al. 2002).
These very recent geometric image representations are built on ideas of multi-
scale analysis and geometry. They have had considerable success in a wide range of
image processing applications, including denoising (Starck et al. 2002; Saevarsson
et al. 2003; Hennenfent and Herrmann 2006), deconvolution (Starck et al. 2003c;
Fadili and Starck 2006), contrast enhancement (Starck et al. 2003b), texture analy-
sis (Starck et al. 2005a; Arivazhagan et al. 2006), detection (Jin et al. 2005), water-
marking (Zhang et al. 2006), component separation (Starck et al. 2004b), inpainting
(Elad et al. 2005; Fadili et al. 2009c), and blind source separation (Bobin et al. 2006,
2007a). Curvelets have also proven useful in diverse fields beyond the traditional im-
age processing application. Let us cite, for example, seismic imaging (Hennenfent
and Herrmann 2006; Herrmann et al. 2008; Douma and de Hoop 2007), astronomi-
cal imaging (Starck et al. 2003a, 2006; Lambert et al. 2006), scientific computing, and
analysis of partial differential equations (Candes and Demanet 2003, 2005). Another
reason for the success of ridgelets and curvelets is the availability of fast transform
algorithms, which are available in noncommercial software packages.
Continuing at this informal level of discussion, we will rely on an example to il-
lustrate the fundamental difference between the wavelet and ridgelet approaches.
Consider an image that contains a vertical band embedded in an additive white
Gaussian noise with large standard deviation. Figure 5.1 (top left) represents such
an image. The parameters are as follows: the pixel width of the band is 20 and the
signal-to-noise ratio (SNR) is set to be 0
20 dB). Note that it is not possible to
distinguish the band by eye. The image denoised by thresholding the undecimated
wavelet coefficients does not reveal the presence of the vertical band, as shown in
Fig. 5.1 (bottom left). Roughly speaking, wavelet coefficients correspond to aver-
ages over approximately isotropic neighborhoods (at different scales), and those
wavelets clearly do not correlate very well with the very elongated structure (pat-
tern) of the object to be detected.
We now turn our attention toward procedures of a very different nature, which
are based on line measurements. To be more specific, consider an
ideal
procedure
that consists of integrating the image intensity over columns, that is, along the ori-
entation of our object. We use the adjective
ideal
to emphasize the important fact
that this method of integration requires a priori knowledge about the geometry of
our object. This method of analysis gives, of course, an improved SNR for our linear
functional, which is better correlated with the object in question: see Fig. 5.1 (top
right).
This example will make our point. Unlike wavelet transforms, the ridgelet trans-
form processes data by first computing integrals over lines at all orientations and
locations. We will explain in the next section how the ridgelet transform further
processes those line integrals. For now, we apply naive thresholding of the ridgelet
.
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