Figure 1: The proposed DCCAE and training strategy. X denotes the collection of multi-TE FID training data with length T and M TEs. Complex units are used where different TEs are treated as different channels in the input. For each complex convolution block, data dimensions were reduced by half while the channel dimension (K) increased by a small amount. The fully connected part followed an encoder-decoder structure and a middle feature layer with dimension $$$L$$$ (referred to as the model order). Errors between $$$\textbf{X}$$$ and $$$\hat{\textbf{X}}$$$ is minimized.

Figure 2: Representation efficiency of the learned model: a) Relative $$$\ell_2$$$ errors of the proposed model approximation with different model orders $$$L$$$’s for 3-TE data (orange curve), compared to linear subspace models (TE-combined subspace in the blue curve and TE-dependent subspace in a yellow curve). For the TE-dependent subspaces, $$$L$$$ is the total dimension of the three subspaces. b) Approximations of a test spectra at different TEs (30, 80, and 130 ms) using the three models with $$$L = 42$$$. A more accurate representation is achieved by our learned model.