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Compressed sensing is a new data acquisition paradigm enabling universal, simple, and reduced-cost acquisition, by exploiting a sparse signal model. Most notably, recovery of the signal by computationally efficient algorithms is guaranteed for certain randomized acquisition systems. However, there is a discrepancy between the theoretical guarantees and practical applications. In applications, including Fourier imaging in various modalities, the measurements are acquired by inner products with vectors selected randomly (sampled) from a frame. Currently available guarantees are derived using the so-called restricted isometry property (RIP), which has only been shown to hold under ideal assumptions. For example, the sampling from the frame needs to be independent and identically distributed with the uniform distribution, and the frame must be tight. In practice though, one or more of the ideal assumptions are typically violated and none of the RIP-based guarantees applies. Motivated by this discrepancy, we propose two related changes in the existing framework: 1) a generalized RIP called the restricted biorthogonality property (RBOP); and 2) correspondingly modified versions of existing greedy pursuit algorithms, which we call oblique pursuits. Oblique pursuits are guaranteed using the RBOP without requiring ideal assumptions; hence, the guarantees apply to practical acquisition schemes. Numerical results show that oblique pursuits also perform competitively with, or sometimes better than their conventional counterparts.