19-10-2012, 10:54 AM
THE MONOGENIC CURVELET TRANSFORM
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ABSTRACT
In this article, we reconsider the continuous curvelet transform
from a signal processing point of view. We show that the
analyzing elements of the curvelet transform, the curvelets,
can be understood as analytic signals in the sense of the partial
Hilbert transform. We then replace the usual curvelets
by the monogenic curvelets, which are analytic signals in the
sense of the Riesz transform. They yield a new transform,
called the monogenic curvelet transform, which has the interesting
property that it behaves at the fine scales like the usual
curvelet transform and at the coarse scales like the monogenic
wavelet transform. In particular, the new transform is highly
anisotropic at the fine scales and yields a well-interpretable
amplitude/phase decomposition of the transform coefficients
over all scales.
INTRODUCTION
The continuous curvelet transform is a multiscale transform
which allows to resolve the singularities of an image together
with their orientations, e.g., the positions and the directions
of edges [1]. To this end, the curvelet transform increases the
anisotropy of its analyzing elements – the curvelets – as the
scale decreases, thus the curvelets have higher directional selectivity
at the fine scales. However, from a signal processing
point of view, the curvelets have another nice property, which
has not been exposed so far. That is, the curvelets are analytic
signal filters in the sense of the partial Hilbert transform.
Analytic signal filters are important in signal processing,
because they yield a meaningful decomposition of the filtered
signal into amplitude and phase, where the amplitude has the
interpretation of the envelope of the signal. The analytic signal
of one-dimensional functions is based on the Hilbert transform.
The aforementioned two-dimensional generalization of
the analytic signal by the partial Hilbert transform has several
drawbacks, which we point out in section 2.2. We resolve
these problems by switching to another generalization of the
analytic signal, the monogenic signal. Thus we replace the
usual curvelets by monogenic curvelets, which yield the new
monogenic curvelet transform.
CONCLUSION
We introduced a new transform which unifies the main advantages
of the monogenic wavelet transform and of the curvelet
transform. In particular, the monogenic curvelet coefficients
split into meaningful amplitude and phase components over
all scales. Furthermore, the anisotropy of the analyzing elements
increases at the fine scales, which results in excellent
directional selectivity.