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Question 1:

Does anybody know any paper proper for getting started understanding the concept of Wavelet, Ridgelet, Curvelet and Contourlet transforms and their differences in detail?

Question 2:

does anybody know a MATLAB toolbox for implementation of these transforms?

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    $\begingroup$ I would add Bandlets as well. They are amazing for Image Denoising. $\endgroup$ – Royi May 28 '17 at 14:51
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On A panorama on multiscale geometric representations, you find a tutorial paper on those directional 2D wavelets, starting with an historical perspective (eg Gaussian & Laplacian pyramids):

Laplacian pyramids

and highlighting details between the main fixed and adaptive decompositions (below: a contourlet atom).

contourlet vector basis

It was published in 2011 in Signal processing, special issue on multirate filter banks and multiscale representations.

It also links to the two-dimensional wavelet and multiscale toolboxes used to generate the figures.

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