sparse image representation via combined transforms - Convex ...

3.4. OTHER TRANSFORMS 73
We can apply the idea of BOB, which is described in the previous subsubsection, to select
an orthonormal basis.
Computational complexity. Computation of the coefficients at every step involves merely
filtering. The complexity of filtering a length-N signal with a finite-length filter is no more
than O(N), so in each step, the complexity is O(N). Since the wavelet packets can go
from scale 1 down to scale log 2 N. The total complexity of calculating all possible wavelet
packets coefficients is O(N log 2 N).
3.4.2 Transforms for 2-D Images
To capture some key features in an image, different transforms have been developed. The
main idea of these efforts is to construct a basis, or frame, such that the basis functions, or
the frame elements, have the interesting features. Some examples of the interesting features
are anisotropy, directional sensitivity, etc.
In the remainder of this subsection, we introduce brushlets and ridgelets.
Brushlets
The brushlet is described in [105]. A key idea is to construct a windowized smooth orthonormal
basis in the frequency domain; its correspondent in the time domain is called brushlets.
By constructing a “perfectly” localized basis in the frequency domain, if an original signal
has a peak in the frequency domain, then the constructed basis, brushlets, tends to capture
this feature. A 2-D brushlet is a tensor product of two 1-D brushlets. A nice property of a
2-D brushlet is that it is an anisotropic, directionally sensitive, and spatially localized basis.
More details are in Meyers’ original paper [105].
Ridgelets
A ridgelet system is a system designed to process a high-dimensional signal that is a superposition
of some linear singularities. In image analysis, linear singularities can be considered
as edges. It is known that edge features are important features of an image.
The term ridgelet was first coined by Candès [21] [22]. A ridge function is a function
that is constant over a hyper-plane. For example, function
f(u · x − b)