We propose an approach for solving the linear inverse problems in the wavelet domain. The solution minimizes the residual sum of squares subject to the $L_1$ norm of the Discret Wavelet Transform of the underlying signal less than a constant. Such solution in wavelet domain is sparse and is more appropriate for nonsmooth signals. We use the ``Least Absolute Shrinkage and Selection Operator'' (LASSO) modification of the ``Least Angle Regression'' (LARS) algorithm proposed by Efron et. al. (2004) to carry out the numerical computation. A simulation result confirmed the good performance of the approach for the optimally selected tuning parameter.