Dear Prabhakar T.,

You can find the answer in the following link:

http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=1014998&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D1014998

The ridgelet transform has limited directional features ,it only works for line singularities.The idea of ridgelets is to transform line singularities in to point singularities using projection slice theorem on radon transform. The finite ridgelet transform FRIT) works in two steps: Calculating the discrete radon transform and then applying a wavelet transform. The finite radon transform (FRAT) is computed in two steps: calculate 2D Fast Fourier transform (FFT) of the image and(applying 1D inverse Fourier Transform (iFFT) on radon projection.then applying  1D wavelet on it we get the finite ridgelet transform (FRIT) .

while the curvelet has superior directional representations as compared to other multiresolution representations such as wavelets or curvelets. It uses the scaling law in its construction .That is why you can decompose an image at different scales and angles to represent more curvilinear objects and have better edge representations.The details about these you can find in Candes and Donho paper on curvelet scaling law. 

There is also a good book spars

Sparse Image and Signal Processing Wavelets, Curvelets, Morphological Diversity..

you can download various implementations mentioned in this book ,in http://www.multiresolutions.com/sparsesignalrecipes/

https://link.springer.com/referenceworkentry/10.1007%2F978-1-4614-1800-9_48

http://jstarck.free.fr/curvencyclop09.pdf follow this pdf to get a better idea of ridgelet and curvelet.