Fig. 1.

Illustration of (a) analysis SToRM and (b) generative SToRM. Analysis SToRM considers a non-linear (e.g. exponential) lifting of the data. If the original points lie on a smooth manifold, the lifted points lie on a low-dimensional subspace. The analysis SToRM cost function in (5) is the sum of the fit of the recovered images to the undersampled measurements and the nuclear norm of the lifted points. A challenge with analysis SToRM is its high memory demand and the difficulty in adding spatial regularization. The proposed method models the images as the non-linear mapping ${\mathcal{G}}_{\theta }$ of some latent vectors z_i, which lie in a very low-dimensional space. Note that the same generator is used to model all the images in the dataset. The number of parameters of the generator and the latent variables is around the size of a single image, which implies a highly compressed representation. In addition, the structure of the CNN offers spatial regularization as shown in DIP. The proposed algorithm in (13) estimates the parameters of the generator and the latent variables from the measured data. A distance regularization prior is added to the generator to ensure that nearby points in the latent subspace are mapped to nearby points on the manifold. Similarly, a temporal regularization prior is added to the latent variables. The optimization is performed using ADAM with batches of few images.