We consider the problem of denoising a one-dimensional signal modeled as the realization of a Markov random field (MRF) by maximizing its posterior distribution. We study the maximum a posteriori (MAP) estimate corresponding to the Laplacian (ℓ1) MRF prior to avoid oversmoothing regions with large intensity gradient. Although the MAP estimate is unique, the non-differentiability of the posterior distribution makes it difficult to find, so we first derive and prove convergence of a relaxation algorithm to find the exact MAP estimate. We then investigate the finite sample property of the MAP MRF-ℓ1 and -ℓ2 estimates on a Gaussian simulation. Finally we apply the estimator to detect the trend on the extreme value distribution of a financial time series.