L1Magic部分译文

 Given these 80 observed samples, the set of length-256 signals that have samples that match our observations is an affine subspace of dimension 256-80=176.  From the candidate signals in this set, we choose the one whose DFT has minimum L1 norm; that is, the sum of the magnitudes of the Fourier transform is the smallest. 

信号（经过DFT后）256个成分，且大部分sparse的。对它进行80次采样，这样长度为256的信号集 中符合我们采样观测的那些信号可以看作是176维的仿射空间，从余下的信号集中选择一个其DFT（系数）符合L1最小  min(|y|_l1)，这样可重构原始信号。

 

In general, if there are B sinusoids in the signal, we will be able to recover using L1 minimization from on the order of B log N samples 。

通常，如果信号(大小)B，可以用min L1恢复  Blog(N) 的信号，换句话说，最小要这么多采样才能恢复。

 

The framework is easily extended to more general types of measurements (in place of time-domain samples), and more general types of sparsity (rather than sparisty in the frequency domain).   Suppose that instead of taking K samples in the time domain, we project the signal onto a randomly chosen K dimensional subspace.  Then if f is B sparse is a known orthobasis,  and K is on the order of B log N, f can be recovered without error by solving an l1 minimization problem.
这种思想可扩展到别的地方。比如时域以外的测量，频域以外的sparisty。如果先把信号投影到一个随机选择的K维子空间，f是 在一个已知的正交基上B-sparse信号，K~Blog(N)，那么通过解决min L1可无错恢复f。