Phase transition of the ℓ
-minimization problem for different sparsity bases and random selection of Fourier measurements without (left panels) and with (right panels) random modulation. The sparsity bases considered are the Dirac basis (top), the Haar wavelet basis (center), and the Fourier basis (bottom). The dashed green line indicates the phase transition of Donoho-Tanner . The color bar goes from white to black indicating a probability of recovery from 0 to 1.