Efficient learning methods for large-scale optimal inversion design

Figure 1.  Four prototype true images used for generating the training set (top row) and validation set (bottom row) in the OID experiment with $ \mathit{\boldsymbol{\theta }} = [\lambda, p, q]^{ \top }$

Figure 2.  For the image deblurring problem with model error and impulse noise, we provide scatter plots of RRE norms for OID$ _{\lambda, p, q} $, OID$ _{\lambda, 2, 2} $, OID$ _{\lambda, 1, 2} $, and OID$ _{\lambda, 2, 1} $ (where the inner problem is solving using MM-GKS). Results for OID$ _{\lambda, 2, 1}^{\rm vpal} $ correspond to using a variable projection augmented Lagrangian method to solve the inner problem. Each column of dots corresponds to one sample from the validation set, where the indices have been sorted based on the RRE norms for OID$ _{\lambda, p, q} $. As a further comparison, $ \lambda_{\rm opt}^j $ corresponds to RRE norms for (21), where the optimal regularization parameter is selected for each image using the learned $ \widehat p $ and $ \widehat q $

Figure 3.  For one sample of the validation data set, we provide in the top row the true image and the observed image. In the second row, we provide the OID$ _{\lambda, p, q} $ reconstruction, the OID$ _{\lambda, 1, 2} $ reconstruction, and the reconstruction computed using the optimal regularization parameter for this image, which is provided for comparison purposes only. In the bottom row are reconstructions for OID$ _{\lambda, 2, 2} $ and OID$ _{\lambda, 2, 1} $, and OID$ _{\lambda, 2, 1}^{\rm vpal} $. RRE values are provided in the titles

Figure 4.  Investigating the impact of model error on the overall noise statistics. From left to right in the top row, we provide pixel-wise values of the additive Gaussian noise (of level 1%), the model error, and the sum of these two errors. The density plot for the combined error is provided, along with the density function for $ p = 2 $ (corresponding to Gaussian noise) and the best density fit to the true errors $ {\bf A}(5.6, 5.6){\bf x}_{\rm true}- {\bf b} $, i.e., $ p = 1.3835 $

Figure 5.  Seismic image reconstruction example. The true image (left) contains $ 256 \times 256 $ pixels and represents a smooth medium. The noisy sinogram image (right) represents projection data from a setup with $ 256 $ sources and $ 512 $ receivers

Figure 6.  Seismic example - random samples from the training set

Figure 7.  For the seismic example, we provide scatter plots of RRE norms for OID, OID-wgcv, SC-wgcv, and HyBR opt. Each column of dots corresponds to one sample from the validation set, where the indices have been sorted based on the RRE norms for OID

Figure 8.  For the validation set of the seismic example, we provide histograms of the RRE norms for OID, OID-wgcv, and SC-wgcv

Figure 9.  For the validation image in Figure 5, we provide reconstructions obtained with HyBR opt and OID reconstructions for the Mat$ \acute{\rm{e}} $rn and squared exponential kernels

Figure 10.  Design objective for OID with the squared exponential kernel for the seismic example. The filled contour corresponds to OID, and the white point denotes the OID computed values