I present a new compressive reconstruction algorithm, which aims to simultaneously achieve low measurement requirement and fast reconstruction. This algorithm alternates between detecting partial support information of the true signal and solving a resulting truncated ℓ 1 minimization problem. I generalize Null Space Property to Truncated Null Space Property and exploit it for theoretical analysis of this truncated ℓ 1 minimization algorithm with Iterative Support Detection (abbreviated as ISD). Numerical results indicate the advantages of ISD over many other state of the art algorithms such as the basis pursuit (BP) model, the iterative reweighted ℓ 1 minimization algorithm (IRL1) and the iterative reweighted least squares algorithm (IRLS). Meanwhile, its limitation is demonstrated by both theoretical and experimental results.