Fig. 3.

Model coefficients and dictionary bases. We show few of the estimated spatial coefficients u_i(x) and its corresponding temporal bases υ_i(t) from 7.5 fold undersampled myocardial perfusion MRI data (data in Fig. 2). (a) corresponds to the estimates using the proposed BCS scheme, while (b) is estimated using the greedy BCS scheme. For consistent visualization, we sort the product entries u_i(x)υ_i(t) according to their ℓ₂ norm, and show the first 30 sorted terms. Note that the BCS basis functions are drastically different from exponential basis functions in the Fourier dictionary; they represent temporal characteristics specific to the dataset. It can also be seen that the energy of the basis functions in (a) varies considerably, depending on their relative importance. Since we rely on the ℓ₁ sparsity norm and Frobenius norm dictionary constraint, the representation will adjust the scaling of the dictionary basis functions υ_i(t) such that the ${\Vert \mathrm{U}\Vert }_{{\ell }_1}$ is minimized. Specifically, the ℓ₁ minimization optimization will ensure that basis functions used more frequently are assigned higher energies, while the less significant basis functions are assigned lower energy (see υ₂₅(t) to υ₃₀(t)), hence providing an implicit model order selection. By contrast, the formulation of the greedy BCS scheme involves the setting of ℓ₀ sparsity norm and column norm dictionary constraint; the penalty is only dependent on the sparsity of U. Unlike the proposed scheme, this does not provide an implicit model order selection, resulting in the preservation of noisy basis functions, whose coefficients capture the alias artifacts in the data. This explains the higher errors in the greedy BCS reconstructions in Fig. 2.