The performance of the class of sparse reconstruction algorithms which is based on the iterative thresholding is highly dependent on a selection of the appropriate threshold value, controlling a trade-off between the algorithm execution time and the solution accuracy. This is why most of the state-of-the-art reconstruction algorithms employ some method of decreasing the threshold value as the solution converges toward the optimal one. To address this problem we propose a data-driven adaptive threshold selection method based on the fast intersection of confidence intervals (FICI) method, with which we have augmented the two-step iterative shrinkage thresholding (TwIST) algorithm. The performance of the proposed algorithm, denoted as the FICI-TwIST algorithm, has been evaluated on a problem of image reconstruction with the missing pixels, exploiting image sparsity in the discrete cosine transformation domain. The obtained results have shown competitive performance in comparison with a number of state-of-the-art sparse reconstruction algorithms, even outperforming them in some scenarios.