  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 a 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 is typically
violated and none of the existing guarantees applies.
  Motivated by this discrepancy, we propose two related changes in the existing
framework: (i) a generalized RIP called the restricted biorthogonality property
(RBOP); and (ii) 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.
