I'm starting work today on a paper related to reconstructing a two-dimensional image of a fish from many observations of broadband scattering. Thus far, we have employed conventional dense reconstruction methods such as the inverse Radon transform and assumed a known geometry. This has lead to quite interesting results, but in some cases suffers from errors in geometry estimates, or biases due to low resolution. In the next phase, I plan to apply some of the sparse reconstruction methods that fall under the category of Compressive Sensing. To learn more about compressive sensing, there is a very comprehensive resource available from Rice University:
Compressive Sensing Resources.For this application, the idea is to exploit the fact that the true image of the fish can be accurately expressed by only a small number of coefficients in a suitable basis. Since we don't know the true image of the fish, we don't know the correct coefficients
a priori and that is where sparse learning comes in. In the learning phase, we search an over-complete set of solutions under constraints that enforce sparsity. For appropriate choices of sparsity constraints, these methods have been shown to find the correct set of sparse coefficients. As an additional task, I also hope to use to sparse learning framework to deal with the fact that we don't really know the geometry in this case.Here is an example of the results that we have obtained thus far:
Inverse Radon Transform of a Damselfish: Computed using the echo envelope recorded on more than 360 views of the fish. Orientations were estimated using video data and sorted before running the IRT. The final image was thresholded manually. 
