  In this article we propose a locally adaptive strategy for estimating a
function from its Exponential Radon Transform (ERT) data, without prior
knowledge of the smoothness of functions that are to be estimated. We build a
non-parametric kernel type estimator and show that for a class of functions
comprising a wide Sobolev regularity scale, our proposed strategy follows the
minimax optimal rate up to a $\log{n}$ factor. We also show that there does not
exist an optimal adaptive estimator on the Sobolev scale when the pointwise
risk is used and in fact the rate achieved by the proposed estimator is the
adaptive rate of convergence.
