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We propose a sparse recovery approach to detect moving targets in clutter. In presence of clutter, the target space is not sparse. We propose a simple way to estimate the clutter region. We then enforce sparsity by modeling the clutter as a single extended cluster of nonzero components. This done by solving a sparse signal recovery problem with partially known support within a maximum a posteriori estimation framework. The resulting algorithm is applied in angle-Doppler imaging for moving target indication in an airborne radar. Our approach has a number of advantages including improved robustness to noise and increased resolution with limited data.