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Impulsive noise in a compressed measurement signal, y, has a significant effect on the reconstruction in Compressed Sensing (CS). In this paper, Greedy gap's Boundary Finder (GBF), a fast preprocessing for impulsive noise rejection for the CS reconstruction of image signal, is proposed. An image is sparsified by the octave-tree wavelet transform. GBF adopts an idea that the leakage energy out of the third-level (L3) subband in the reconstructed sparse signal is the result of either the impulsive noise in y or the insufficient information (of y) for reconstruction. The leakage energy is measured as the ratio to the total energy. In a graph of relationship between the energy ratio and the number of the removed elements, there is a gap of low energy ratio in the middle. A binary search is adapted in GBF to find the lower gap's boundary which is the minimum number of removed elements with low energy ratio. As the image signal is highly redundant, the search in GBF can be non-exact and will be stopped after the number is within +g of the lower boundary, where g is the gap's resolution and defined as the percent of the size of y. GBF was evaluated on 100 16×16 image blocks and 10 256×256 standard test images. The evaluation shows that at the optimal g of 5%, GBF provided the comparable performance to the exact boundary's search, but consumed less computational time.