What are the advantages and disadvantages of the Ridgelet transform?

Question

I used the continuous wavelet transform (CWT) to analyze the behavior of a 1-D smooth signal (extracting its modal property, damping and frequency components) in my research. Now someone asked:

Why the presented CWT algorithm is better than Ridgelet transform?

I thought it is better to ask in this forum about disadvantages of Ridgelets. For example:

• it is said that Wavelet can handle point discontinuities whereas Ridgelet is better to deal with line discontinuities. So, do we have any line discontinuities in a 1-D signal?
• what is computational complexity of Ridgelet in term of big $O$ i.e. (like $O(N)$), for both continuous (in scale) and discrete versions?

Answer 1

Because of their ability to better represent signals, Ridgelet transforms are preferred over Wavelet transforms in general. However, to be used in practice it has to be discretized. Now, the discretization of Ridglet transform is a challenging task on its own as it involves interpolation in polar coordinates which makes perfect reconstructions (inversions) difficult. For example, Minh N. Do and Martin Vetterli propose an orthonormal version of Ridgelet Transform to achieve better invertability [1].

Ridgelet transform enhances the idea of point-to-point mapping of singularities to point-to-line mappings, which is more effective in handling directions. Yet, even Ridgelets might be ineffective for certain applications such as image processing, where edges are curved rather than being straight lines. In such cases, better transforms such as Curvelets are already proposed and proven to be more powerful.

In this regard, the method of choice really depends on your application. As wiki quotes:

Continuous Wavelet Transform (CWT) is very efficient in determining the damping ratio of oscillating signals (e.g. identification of damping in dynamical systems). CWT is also very resistant to the noise in the signal.

If your problem is suitable to CWT, then you will be perfectly fine with using it. The Academia involves a lot of works in comparing and demonstrating the strengths of all of these transforms. Some of these would be [2,3,4].

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Under the shade of the comments below, I now elaborate more on the theory.

First of all, singularity is simply the concept of point where the function or its derivative is not defined, the point of cusp, double point or any other simlar undefinedness, which I won't go into details here. You can check the wiki "Mathematical Singularity". A good measure of singularity strength would be Lipschitz exponent, which can well be characterized by Wavelet transforms. For more information on that take a look at [5].

But, as you ask, why are you interested in the singularities anyway? Well, because such irregularities basically play an important role in carrying the information that characterizes the signal [6].

Having explained the singularities a bit, I certainly agree that singular lines are a concern of multidimensional structures, more specifically $R^2$ ($L^2$ etc.). This is why transforms like Ridgelets and Curvelets are good at image denoising than Wavelets. Yet, this doesn't violate the fact that 2D Wavelets would also be useful in image analysis.

Regarding the computational complexity, even before analyzing the equations, you could rely on your knowledge that computation of the Ridgelet transform involves a Wavelet transform in the Radon domain. That is why it should definitely be heavier in terms of computational complexity, compared to a Wavelet transform.

. REFERENCES

1. http://www.ifp.illinois.edu/~minhdo/publications/FRIT_trans.pdf
2. http://jstarck.free.fr/curvencyclop09.pdf
3. http://authors.library.caltech.edu/1381/1/STAieeetip02.pdf
4. http://www.iro.umontreal.ca/~mignotte/IFT6150/ComplementCours/CurveletTransform.pdf
5. http://www.iis.sinica.edu.tw/papers/whwang/5114-F.pdf
6. http://www.cmap.polytechnique.fr/~mallat/papiers/MallatSingDet92.pdf