Contourlet transform for image compression
I do not know of any book solely devoted to contourlets. There is a chapter in Beyond Wavelets, 2003 (G. Welland, Ed.) by Do and Vetterli, named Contourlets. You also have a 100-page paper (almost a book) where contourlets are put in the perspective of Multidimensional Filter Banks and Multiscale Geometric Representations (2012).

At the risk of mildening your teacher's assertions, good compression is the result of a transformation with nice qualities and an efficient coding scheme. The "best" transform can fail to properly compress an image if you do not have a neat coding scheme associated with it. Contourlets, for instance, are slightly redundant. And its compaction capabilities are thus limited by the redundancy. To my knowledge, redundant transforms have not proved yet they are much better than critically-sampled ones for compression (except perhaps at large compression ratios).
Even if you are not talking about research papers, let me mention a couple of different contourlet versions. The CRISP-contourlet, which is non-redundant, and The Nonsubsampled Contourlet Transform, fully redundant. 

When trying to understand a transformation, especially with wavelet-like decompositions, it is often useful to look at the interplay between the math and the sampling schemes.

In addition to Minh N. Do thesis, I am trying to find a copy of A. L. da Cunha PhD.