Tomographic inversion using $\ell_1$-norm regularization of wavelet   coefficients
  We propose the use of $\ell_1$ regularization in a wavelet basis for the solution of linearized seismic tomography problems $Am=d$, allowing for the possibility of sharp discontinuities superimposed on a smoothly varying background. An iterative method is used to find a sparse solution $m$ that contains no more fine-scale structure than is necessary to fit the data $d$ to within its assigned errors. 
Ignace Loris, Guust Nolet, Ingrid Daubechies, F. A. Dahlen