We propose a sparse reconstruction method based on compressed sensing theory for aperture synthesis imaging. Our algorithm directly works on observational data without grid-ding. We achieve fast convergence by introducing an adaptive tolerance parameter based on the noise level and a thresholding value based on the cumulative sum of the power of the estimated source components. We demonstrate the accuracy in estimating the source positions and intensities in extremely low signal-to-noise (SNR) scenarios in Monte Carlo simulation. We could recover both point sources and extended sources with our algorithm using a Dirac basis from real data.