  A reliable support detection is essential for a greedy algorithm to
reconstruct a sparse signal accurately from compressed and noisy measurements.
This paper proposes a novel support detection method for greedy algorithms,
which is referred to as "\textit{maximum a posteriori (MAP) support
detection}". Unlike existing support detection methods that identify support
indices with the largest correlation value in magnitude per iteration, the
proposed method selects them with the largest likelihood ratios computed under
the true and null support hypotheses by simultaneously exploiting the
distributions of sensing matrix, sparse signal, and noise. Leveraging this
technique, MAP-Matching Pursuit (MAP-MP) is first presented to show the
advantages of exploiting the proposed support detection method, and a
sufficient condition for perfect signal recovery is derived for the case when
the sparse signal is binary. Subsequently, a set of iterative greedy
algorithms, called MAP-generalized Orthogonal Matching Pursuit (MAP-gOMP),
MAP-Compressive Sampling Matching Pursuit (MAP-CoSaMP), and MAP-Subspace
Pursuit (MAP-SP) are presented to demonstrate the applicability of the proposed
support detection method to existing greedy algorithms. From empirical results,
it is shown that the proposed greedy algorithms with highly reliable support
detection can be better, faster, and easier to implement than basis pursuit via
linear programming.
