  Exploiting intrinsic structures in sparse signals underpins the recent
progress in compressive sensing (CS). The key for exploiting such structures is
to achieve two desirable properties: generality (\ie, the ability to fit a wide
range of signals with diverse structures) and adaptability (\ie, being adaptive
to a specific signal). Most existing approaches, however, often only achieve
one of these two properties. In this study, we propose a novel adaptive Markov
random field sparsity prior for CS, which not only is able to capture a broad
range of sparsity structures, but also can adapt to each sparse signal through
refining the parameters of the sparsity prior with respect to the compressed
measurements. To maximize the adaptability, we also propose a new sparse signal
estimation where the sparse signals, support, noise and signal parameter
estimation are unified into a variational optimization problem, which can be
effectively solved with an alternative minimization scheme. Extensive
experiments on three real-world datasets demonstrate the effectiveness of the
proposed method in recovery accuracy, noise tolerance, and runtime.
