Abstract: Traditional compressive sensing theories mainly consider sampling and reconstructing a 1 Dimensional (1D) signal. For a 2 dimensional (2D) signal, if reshaped into a vector, the required size of the sensing matrix becomes dramatically large, which increases the storage and computational complexity of reconstruction significantly. To efficiently sample and reconstruct 2D signals exhibiting sparsity in 2D separable dictionaries, we first construct an analog-to-information conversion (AIC) frame to jointly sample 2D signals in column and row instead of vectoring, which requires much lower storage. Then a novel 2D fast iterative shrinkage-thresholding algorithm (2D-FISTA) is proposed. The basic iterate format, convergence, parameter choices of the 2D-FISTA are thoroughly analyzed. It is shown that the proposed method can handle 2D signals directly with much lower storage and computational complexity.