Sparse Dictionary-Based Solution of Dynamic Inverse Problems

TL;DR

To promote a vector of coefficients with mostly vanishing entries, this work considers a stochastic extension of the dictionary coding problem model with a random hierarchical sparsity promoting prior, and compute the Maximum A Posteriori (MAP) estimate of the coefficient vector using the Iterative Alternating Sequential Algorithm (IAS).

Abstract

In ill-posed dynamic inverse problems expected spatial features and temporal correlation between frames can be leveraged to improve the quality of the computed solution, in particular when the available data are limited and the dimensionality of the unknown is large. One way to take advantage of the spatial and temporal traits believed to characterize the solution is to encode them into the entries of a dictionary, and to seek the solution as a sparse linear combination of the dictionary atoms. To promote a vector of coefficients with mostly vanishing entries, we consider a stochastic extension of the dictionary coding problem model with a random hierarchical sparsity promoting prior. We compute the Maximum A Posteriori (MAP) estimate of the coefficient vector using the Iterative Alternating Sequential Algorithm (IAS), which has been demonstrated to efficiently solve inverse problems with minimal need for parameter tuning. The proposed methodology is tested on real-world dynamic Computed Tomography and MRI datasets, where it is compared to the popular Alternating Direction Method of Minimizers (ADMM). The computed examples show the that proposed methodology is competitive with the ADMM for compressed sensing, with a significantly lower sensitivity to hyper-parameter selection.

Figures (6)

• Figure 1: CT reconstructions with optimised hyper-parameter selection (top), compared to the ground truth (middle). The variable, $\theta$, has been transformed to image space via application of the inverse wavelet transform at each timestep.
• Figure 2: Heatmaps depicting the SSIM scores for ADMM (left) and IAS (right) reconstructions in terms of the selected hyper-parameter pairs. The black dots indicate the location of measured values over which linear interpolation was performed.
• Figure 3: Optimal (top) and worst-case (bottom) reconstructions for ADMM and IAS, respectively. Worst-case reconstructions are chosen as those with the lowest SSIM among hyper-parameter settings within two orders of magnitude of the optimal values. The ground truth and least-squares reconstructions are shown on the left for reference, and SSIM and normalised root-mean-squared error (nRMSE) scores for each reconstruction are displayed in yellow.
• Figure 4: SSIM scores for the IAS (top) and ADMM (bottom) reconstructions at each iteration of the respective algorithms. Plots are shown using a stopping criteria of 10, 50, 150, and 300 LSMR iterations respectively. Note that the black, red and green lines overlap in the IAS plot; and the red and green lines overlap in the ADMM plot.
• Figure 5: Reconstructed DCE-MRI images at timepoints 30 (top) and 100 (bottom). IAS (left) and ADMM (middle) reconstructions are shown for two different hyper-parameter pairs, which are indicated in white on the images. White arrows on the IAS images indicate the locations of pixels in the vessel and tumour regions. In the absence of a ground truth reference, the unregularized least squares reconstruction is shown on the far right.