Compressed sensing has captured considerable attention of researchers in the past decades. In this paper, with the aid of the powerful null space property, some deterministic recovery conditions are established for the previous \(\ell _{1}\)–\(\ell _{1}\) method and the \(\ell _{1}\)–\(\ell _{2}\) method to guarantee the exact sparse recovery when the side information of the desired signal is available. These obtained results provide a useful and necessary complement to the previous investigation of the \(\ell _{1}\)–\(\ell _{1}\) and \(\ell _{1}\)–\(\ell _{2}\) methods that are based on the statistical analysis. Moreover, one of our theoretical findings also shows that the sharp conditions previously established for the classical \(\ell _{1}\) method remain suitable for the \(\ell _{1}\)–\(\ell _{1}\) method to guarantee the exact sparse recovery. Numerical experiments on both the synthetic signals and the real-world images are also carried out to further test the recovery performance of the above two methods.

在过去的几十年中，压缩感知引起了研究人员的极大关注。本文借助强大的零空间属性，为之前的\(\ell_{1}\) - \(\ell_{1}\)方法和\(\ ell _{1}\) – \(\ell _{2}\)方法，用于在所需信号的边信息可用时保证精确的稀疏恢复。这些获得的结果为先前对\(\ell _{1}\) – \(\ell _{1}\)和\(\ell _{1}\) – \( \ell_{2}\)基于统计分析的方法。此外，我们的一项理论发现还表明，先前为经典\(\ell _{1}\)方法建立的尖锐条件仍然适用于\(\ell _{1}\) – \(\ell _{ 1}\)方法来保证精确的稀疏恢复。还对合成信号和真实世界图像进行了数值实验，以进一步测试上述两种方法的恢复性能。