1. A computer-implemented method of performing image reconstruction with sequential cycle-spinning, the method comprising: acquiring, by a computer system, an input signal comprising k-space data using a magnetic resonance imaging (MRI) device; initializing, by the computer system, an estimate of a sparse signal associated with the input signal; selecting, by the computer system, one or more orthogonal wavelet transforms corresponding to a wavelet family; performing, by the computer system, an iterative reconstruction process to update the estimate of the sparse signal over a plurality of iterations, wherein each iteration comprises: applying the one or more orthogonal wavelet transforms to the estimate of the sparse signal to yield one or more orthogonal domain signals, updating the estimate of the sparse signal by applying a non-convex shrinkage function to the one or more orthogonal domain signals, and applying a shift to the one or more orthogonal wavelet transforms; and following the iterative reconstruction process, generating an image based on the estimate of the sparse signal.